Recent Progress In Orbital-free Density Functional Theory 9789814436731, 9789814436724

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RECENT PROGRESS IN ORBITAL-FREE DENSITY FUNCTIONAL THEORY

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Recent Advances in Computational Chemistry Editor-in-Charge Delano P. Chong, Department of Chemistry, University of British Columbia, Canada

Published Recent Advances in Density Functional Methods, Part I (Volume 1) ed. D. P. Chong Recent Advances in Density Functional Methods, Part II (Volume 1) ed. D. P. Chong Recent Advances in Density Functional Methods, Part III (Volume 1) eds. V. Barone, A. Bencini and P. Fantucci Recent Advances in Quantum Monte Carlo Methods, Part I (Volume 2) ed. W. A. Lester Recent Advances in Coupled-Cluster Methods (Volume 3) ed. Rodney J. Bartlett Recent Advances in Multireference Methods (Volume 4) ed. K. Hirao Recent Advances in Relativistic Molecular Theory (Volume 5) eds. K. Hirao and Y. Ishikawa

Forthcoming Recent Advances in Computational Chemistry Software (Volume 7) eds. R. Amos and R. Kobayashi

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Recent Advances in Computational Chemistry – Vol. 6

RECENT PROGRESS IN ORBITAL-FREE DENSITY FUNCTIONAL THEORY

edited by

Tomasz A Wesolowski University of Geneva, Switzerland

Yan Alexander Wang University of British Columbia, Canada

World Scientific NEW JERSEY

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LONDON

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SINGAPORE

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BEIJING

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SHANGHAI

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HONG KONG

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TA I P E I

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Published by World Scientific Publishing Co. Pte. Ltd. 5 Toh Tuck Link, Singapore 596224 USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE

Library of Congress Cataloging-in-Publication Data Recent progress in orbital-free density functional theory / edited by Tomasz A Wesolowski, University of Geneva, Switzerland, Yan Alexander Wang, University of British Columbia, Canada. pages cm. -- (Recent advances in computational chemistry ; volume 6) Includes bibliographical references and index. ISBN 978-9814436724 (hardcover : alk. paper) 1. Density functionals. 2. Chemistry--Mathematics. 3. Atomic orbitals. I. Wesolowski, Tomasz A. II. Wang, Yan Alexander. QD462.6.D45R45 2013 541'.28--dc23 2012046733

British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library.

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Preface

The interest in orbital-free density-functional-theory based methods (OF-DFT ) originates from the fact that they are built upon a sound theory, which, in principle, makes it possible to describe the ground state of a multi-electron system exactly without constructing the wave function explicitly. This feature opens the door for practical modeling very large and complex polyatomic systems of millions of atoms accurately. The computational cost of OF-DFT calculations grows quasi-linearly with the size of the system, allowing one to break the barrier of the highly nonlinear growth in the computational costs of conventional wave-function based methods. Converting this virtually exact theory into an effective computational method proved, however, to be a formidable task in practice. The first attempts were made in the late 1920’s independently by Thomas and Fermi, shortly after the birth of quantum theory. Unfortunately, such early Thomas-Fermi models, hampered by poor accuracy, offered no potential usefulness for studies of chemical problems. The chemistry community seemed to have lost interest in OF-DFT following the demonstration by Teller that molecules described using the original Thomas-Fermi model always dissociate. The use of orbitals appeared indispensable. Approximating the exact quantities by density functionals could be accepted only for the smaller components of the total energy as it was made in the celebrated Xα method of Slater, which showcased the appeal of electron-density based methods and which led ultimately to the modern density functional theory (DFT) formulated by Hohenberg and Kohn in the middle of 1960’s. To bypass the poor density-only approximation to the kinetic energy, Kohn and Sham devised an orbital-based formal framework for numerical simulations, which is commonly known as the Kohn-Sham method nowadays. The Kohn-Sham DFT is based on density-only OF-DFT of Hohenberg and Kohn but the orbitals are used nevertheless as auxiliary quantities in order to avoid explicit density-only approximations for the kinetic energy. Although the efforts to bring life back to OF-DFT never ceased ever since, the progress was rather slow. The advance in the OF-DFT methods was hindered by the difficulty in constructing sufficiently accurate approximations to the electrondensity functional for the kinetic energy of a many-electron system. Not until the 1990’s, many research groups, mainly driven by the ever-increasing appetite for modeling bigger complex systems in many fields, began to work intensively to revitalize OF-DFT methods. Over the last two decades, OF-DFT methods have

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achieved a certain level of maturity and can be considered as the optimal tool for simulating certain static and dynamic phenomena in soft- and hard condensed matter. To facilitate the research progress, a series of symposia exclusively dedicated to exhibit the latest developments of OF-DFT methods have been in operation since the early 2000’s. The earliest OF-DFT symposium was the CECAM workshop on “Approximating and Applying of the Kinetic Energy Functional ” organized by T.A. Wesolowski and H. Chermette at Lyon, France, on 31 July–2 August 2002. The second OF-DFT symposium, “Explicit Density Functional of the Kinetic Energy in Computer Simulations at Atomistic Level,” was organized by T.A. Wesolowski at the International Conference of Computational Methods in Sciences and Engineering, in Loutraki, Korinthos, Greece, on 23–24 October 2005. The third OF-DFT symposium, “Orbital-Free Density Functional Theory,” was organized by T.A. Wesolowski and Y.A. Wang at the Sixth Congress of the International Society for Theoretical Chemical Physics, in Vancouver, Canada, on 23–24 July 2008. At this critical junction, during a conversation over a dinner at the second OFDFT symposium, we came up an idea to collect in a single volume the accounts of representative developments in various interconnected sub-domains related to the kinetic energy density functional and its applications. Thus, the common denominator of all chapters in the present volume is the kinetic energy functional - the quest for a better approximation to this functional unifies the contributors from many different fields. The first part of the present volume starts with contributions focusing on exact properties of the kinetic energy density functional (Chapters 1-3). Chapters 4-8, focus rather on practical applications of OF-DFT in numerical simulations. They provide a representative overview of those key developments leading to the current stage of the OF-DFT methods. The reported numerical simulations supply convincing evidence that the constantly expanding domain of applicability of the OF-DFT methods now includes solids, large clusters, and liquids. The second part of this volume deals with formalisms and methods, which are closely related to OF-DFT, but hinge on approximations to the bi-functional for the non-additive kinetic energy: subsystem DFT and Frozen-Density Embedding Theory (FDET ). In subsystem DFT, which was proposed originally in papers by Senatore and Subbaswamy and by Cortona for simulating solids, introduced latter for molecular interactions by Wesolowski and collaborators, sets of Kohn-Sham-like orbitals for each subsystem are used as auxiliary quantities. The non-interacting reference wave function for the whole system as in the Kohn-Sham case is not constructed. As a consequence, evaluation of the total kinetic energy requires approximating the non-additive kinetic energy. FDET on the other hand was introduced originally by Wesolowski and Warshel as the formal basis for multi-level simulations. In FDET, the orbital level of description applies only for selected part of the total system. Although FDET is also based on Hohenberg-Kohn theorems, it does not target the

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ground-state energy of a given system as do the OF-DFT, Kohn-Sham DFT, and subsystem DFT methods, but aims rather at the upper bound of this energy. In Chapter 9, an analytically solvable model system is used to introduce the basic elements of FDET and to overview the challenges in approximating the non-additive kinetic potential. Chapters 10-11 review the recent developments and applications of computational methods which apply approximations for the non-additive kinetic energy. Chapter 12, on the other hand focuses on qualitative differences between quantities defined in subsystem DFT and their counterparts obtained if approximations for the non-additive kinetic energy are used. The third part of this volume covers mainly the use of the kinetic energy and related quantities in interpreting key chemical concepts. Chapters 13-15 deliver a comprehensive overview of the relations between the kinetic energy density functional and the Fisher information. In the Appendix, the analytical expressions used in the literature to approximate the kinetic energy and related quantities are collected together with a full bibliography. Finally, we have to stress that the contributors to that volume come from different fields and despite the common interest in the kinetic energy functional, the used terminology and conventions are not uniform. We only partially addressed this issue of by pointing out equivalent terms or by identifying similar terms used for various quantities in Subject Index or in the Editors’ Note. We hope that this very first volume devoted to OF-DFT will enable scientific community to gain a deeper appreciation of the state of the art of OF-DFT methods. Consequently, more people will be attracted to work in this promising research field to propel OF-DFT methods into the world of main-stream ab initio simulations. Tomasz Adam Wesolowski & Yan Alexander Wang June 2012

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Contents

Preface

v

Part 1: Density Functional for the Kinetic Energy and Its Applications in Orbital-Free DFT Simulations

1

1.

3

From the Hohenberg-Kohn Theory to the Kohn-Sham Equations Y. A. Wang & P. Xiang

2.

Accurate Computation of the Non-Interacting Kinetic Energy from Electron Densities

13

F. A. Bulat & W. Yang 3.

The Single-Particle Kinetic Energy of Many-Fermion Systems: Transcending the Thomas-Fermi plus Von Weizs¨acker Method

31

G. G. N. Angilella & N. H. March 4.

An Orbital Free ab initio Method: Applications to Liquid Metals and Clusters

55

A. Aguado, D. J. Gonz´ alez, L. E. Gonz´ alez, J. M. L´ opez, S. N´ un ˜ez & M. J. Stott 5.

Electronic Structure Calculations at Macroscopic Scales using Orbital-Free DFT

147

B. G. Radhakrishnan & V. Gavini 6.

Properties of Hot and Dense Matter by Orbital-Free Molecular Dynamics F. Lambert, J. Cl´erouin, J.-F. Danel, L. Kazandjian & S. Mazevet ix

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Shell-Correction and Orbital-Free Density-Functional Methods for Finite Systems

203

C. Yannouleas & U. Landman 8.

Finite Element Approximations in Orbital-Free Density Functional Theory

251

H. Chen & A. Zhou Part 2: The Functional for the Non-Additive Kinetic Energy and Its Applications in Numerical Simulations 9.

Non-Additive Kinetic Energy and Potential in Analytically Solvable Systems and Their Approximated Counterparts

273

275

T. A. Wesolowski & A. Savin 10. Towards the Description of Covalent Bonds in Subsystem DensityFunctional Theory

297

Ch. R. Jacob & L. Visscher 11. Orbital-Free Embedding Calculations of Electronic Spectra

323

J. Neugebauer 12. On the Principal Difference Between the Exact and Approximate Frozen-Density Embedding Theory

355

O. V. Gritsenko Part 3: Kinetic Energy Functional and Information Theory

367

13. Analytic Approach and Monte Carlo Sampling for Electron Correlations

369

L. M. Ghiringhelli & L. Delle Site 14. Kinetic Energy and Fisher Information

387

´ Nagy A. 15. Quantum Fluctuations, Dequantization, Information Theory and Kinetic-Energy Functionals I. P. Hamilton, R. A. Mosna & L. Delle Site

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Part 4: Appendix

427

16. Semilocal Approximations for the Kinetic Energy

429

F. Tran & T. A. Wesolowski Author Index

443

Subject Index

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PART 1

Density Functional for the Kinetic Energy and Its Applications in Orbital-Free DFT Simulations

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Chapter 1 From the Hohenberg-Kohn Theory to the Kohn-Sham Equations

Yan Alexander Wang* and Ping Xiang Department of Chemistry, University of British Columbia Vancouver, British Columbia, Canada V6T 1Z1 [email protected] Two existing ways of deriving the Kohn-Sham equations were analyzed in great details. It turns out that both are incomplete in principle because they will lead to a paradox. It was further shown that the paradox can be resolved by introducing some arbitrary constants in the total Kohn-Sham effective potential. As a result, the functional derivative of the kinetic-energy density functional of the auxiliary non-interacting system within the Kohn-Sham method can be exactly calculated only from the highest occupied Kohn-Sham orbital 2 KS KS KS φKS H (r): δTs [ρ]/δρ0 (r) = −0.5{∇ φH (r)}/φH (r) = −I − {veff (r)}can , where I KS and {veff (r)}can are the first (lowest) ionization energy and the canonical representative of the total Kohn-Sham effective potential, respectively.

Contents 1.1 Introduction . . . . . . . . . . . . . . . . 1.2 Routes to the Kohn-Sham equations . . . 1.3 A paradox and its resolution . . . . . . . 1.3.1 The Wang paradox . . . . . . . . . 1.3.2 The Wang-Parr resolution . . . . . 1.4 Direct inclusion of the constraints . . . . 1.5 Functional derivative of the kinetic-energy 1.6 Conclusions . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . .

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1.1. Introduction Two papers published over 45 years ago by Hohenberg, Kohn, and Sham (HKS)1,2 laid the foundation for modern density-functional theory.3–14 The two HohenbergKohn (HK) theorems1 first legitimatized the ground-state (GS) electron density ρ0 (r) as the basic variable of the GS quantum chemistry and the Kohn-Sham (KS) method2 then offered a specific numerical implementation for this theory. Later, the HKS theory was recasted in the constrained-search formulation.3–5,15–20 The first HK theorem1 guarantees that, for an N -electron quantum chemical 3

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system, its total GS electronic energy is a functional of ρ0 (r): E[Ψ0 ] ≡ E[ρ0 ] ,

(1.1)

where Ψ0 is the GS wave function. The second HK theorem1 insures that, for a fixed external potential v(r), any trial density ρ(r) coming from N -electron wave functions in Hilbert space (HS) corresponds to a higher energy than the GS energy: Ev [ρ] > Ev [ρ0 ] .

(1.2)

If Ev [ρ] is differentiable in the vicinity of ρ0 (r),3–9,12–14 this variational principle can be reexpressed as9,14   δEv dEv [ρ0 ] = δρ(r) = 0 , (1.3) δρ0 (r) with the normalization condition, hδρ(r)i ≡ 0 .

(1.4)

According to a theorem by Gelfand and Fomin,21,22 one immediately has δEv = µHS , δρ0 (r)

(1.5)

where µHS is a constant,9,12–14,23–25 and the subscript “HS” denotes the N -electron HS as the variational domain. If the functional form of Ev [ρ] is explicitly given, µHS will be a fixed number for a specific ρ0 (r); otherwise, it is normally believed that µHS can only be determined up to an arbitrary constant in HS.9 However, theoretical considerations in Fock space13,14,23,24 have definitely concluded that µHS = −I,

(1.6)

where I is the first (lowest) ionization energy of the N -electron system. Because the exact form of Ev [ρ] is unknown for general many-electron systems, one has to rely on the KS scheme2 to implement the HK theory.1 In the KS method,2 the basic working equation (in canonical form) is  KS KS KS (1.7) tˆ + veff (r) φKS i (r) = εi φi (r) , where the kinetic-energy operator is

1 tˆ ≡ − ∇2 , 2

(1.8)

KS and the total KS effective potential veff (r) is defined as the functional derivative at KS the KS density ρ (r):

KS veff (r) =

δ(Ev [ρKS ] − Ts [ρKS ]) δVs [ρKS ] = . δρKS (r) δρKS (r)

(1.9)

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5

Here, Ts [ρKS ] is the non-interacting kinetic-energy density functional, and Vs [ρKS ] is the difference between Ev [ρKS ] and Ts [ρKS ]. At the solution point, ρKS (r) and Ts [ρKS ] are recovered from the first N (normalized) KS orbitals {φKS i (r)}: ρKS (r) =

N X i=1

2

|φKS i (r)| ,

hρKS (r)i ≡ N,

(1.10)

(1.11)

and Ts [ρKS ] =

N X i=1

ˆ KS hφKS i (r)| t |φi (r)i .

(1.12)

It was claimed that ρKS (r) would be identical to ρ0 (r) at the final solution point.2,9 In the constrained-search formulation,3–5,15–20 Levy and Lieb (LL) propose to minimize the model kinetic energy,3–5,15–18 Tm [ρ0 ] = min ρ0

N X

LL LL  φi (r) tˆ φi (r) ,

(1.13)

i=1

with the exact orbital realization of ρ0 (r), ρ0 (r) =

N X i=1

2

|φLL i (r)| .

(1.14)

LL Then, the effective potential veff (r) is the local Lagrange-Euler multiplier for Eq. (1.14), and the first N LL orbitals {φLL i (r)} satisfy similar working equations to Eq. (1.7),9

 LL LL LL tˆ + veff (r) φLL i (r) = εi φi (r) .

(1.15)

A question naturally arises here: Is the KS scheme fully equivalent to the LL proposal? To answer this question, it is necessary to review the details of the derivations of the KS equations. The existing literature2,9 implies two plausible routes to the KS equations from the HK theorems.1 The first one was illustrated in the original KS paper,2 whereas the second one was implied in a book by Parr and Yang (PY).9 However, it should be noted that neither of the two sources provided complete details of the derivations. For simplicity of the notational system, all orbitals will be chosen to be real and the superscripts on ρKS (r) and φKS i (r) will be dropped unless otherwise noted hereafter.

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1.2. Routes to the Kohn-Sham equations After introducing Eqs. (1.5), (1.10), and (1.12), Kohn and Sham partitioned Ev [ρ] into two main pieces:2 Ev [ρ] ≡ Ts [ρ] + Vs [ρ] ,

(1.16)

δEv [ρ] δTs [ρ] δVs [ρ] ≡ + = µHS , δρ(r) δρ(r) δρ(r)

(1.17)

dEv [ρ] ≡ dTs [ρ] + dVs [ρ] ,

(1.18)

      δTs [ρ] δVs [ρ] δEv [ρ] δρ(r) ≡ δρ(r) + δρ(r) . dEv [ρ] ≡ δρ(r) δρ(r) δρ(r)

(1.19)

and hence, one has

and

From Eqs. (1.10) and (1.12), one observes δρ(r) = 2

N X

φi (r)δφi (r) ,

(1.20)

i=1

and dTs [{φi }] = 2 On setting

N X

 δφi (r) tˆ φi (r) .

(1.21)

i=1

dTs [ρ] ≡ dTs [{φi }] ,

(1.22)

one obtains from Eqs. (1.9) and (1.18)−(1.21) that dEv [ρ] = dTs [{φi }] + dVs [ρ]   δVs [ρ] = dTs [{φi }] + δρ δρ =2

N X

i=1

=2

N X

i=1

=2

N X

i=1

 KS δφi (r) tˆ φi (r) + hveff (r) δρ(r)i

N X  KS δφi (r) tˆ φi (r) + 2 hδφi (r)|veff (r)|φi (r)i i=1

 KS δφi (r) tˆ + veff (r) φi (r) .

(1.23)

For arbitrary infinitesimal variations of {δφi }, a minimization of the above equation with the orthonormality constraint between {φi } arrives at the KS equations.2

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Somewhat differently, Parr and Yang9 wrote the total energy purely in an orbital-functional form: Ev [{φi }] ≡ Ts [{φi }] + Vs [{φi }] ,

(1.24)

dEv [{φi }] ≡ dTs [{φi }] + dVs [{φi }] .

(1.25)

and then, one gets As it is usually done in the literature,25–30 one would believe that  N  X δVs [{φi }] dVs [{φi }] = δφi (r) . δφi (r) i=1

(1.26)

From Eq. (1.10) and the chain rule, one has

and δVs [{φi }] = δφi (r)

δρ(r0 ) ∂ρ(r) = δ(r0 − r) = 2φi (r)δ(r0 − r) , δφi (r) ∂φi (r)

(1.27)

Z

(1.28)

δVs δρ(r0 ) 0 δVs KS dr = 2φi (r) = 2veff (r)φi (r) . δρ(r0 ) δφi (r) δρ(r)

Then, with the aid from Eqs. (1.12), (1.25), (1.26), and (1.28), one recovers the KS equations once again. 1.3. A paradox and its resolution The pivotal step of the KS derivation is Eq. (1.22), while the crucial procedure of the PY derivation is the orbital-functional derivatives in Eqs. (1.26)−(1.28). However, the following arguments seem to refute them both. 1.3.1. The Wang paradox First, given Eqs. (1.26)−(1.28) were correct, one could do the same trick to Ts [ρ], such that δTs δTs [ρ] ∂ρ(r) δTs [ρ] = = 2φi (r) . (1.29) δφi (r) δρ(r) ∂φi (r) δρ(r) From Eq. (1.21), one could obtain δTs = 2 tˆφi (r) . δφi (r)

(1.30)

From Eqs. (1.7), (1.29), and (1.30), one would readily derive δTs [ρ] tˆφi (r) KS = = εKS i − veff (r) . δρ(r) φi (r) There is another way to derive Eq. (1.31) through   δTs dTs [ρ] = δρ(r) . δρ(r)

(1.31)

(1.32)

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By substituting Eq. (1.20) into the above equation, one would get  N  X δTs dTs [ρ] = 2 δφi (r) φi (r) . δρ(r) i=1

(1.33)

Applying Eq. (1.22), one would obtain Eq. (1.31) again after comparing Eq. (1.21) with Eq. (1.33). However, Eq. (1.31) is obviously incorrect, because the orbital dependence of the right-hand side is directly contradictory to the property of the left-hand side. Because the first N KS orbitals are linearly dependent31 via Eqs. (1.10) and (1.11), Eqs. (1.26)−(1.28) are in error. This point is elucidated more clearly and generally in the following theorem.32 Theorem 1.1 (Wang-Davidson, 1996). If orthonormal orbitals {ξi (r)} are solutions of a set of coupled variational equations,   ˆ0 + h ˆ 1 [θ] ξi (r) = ωi ξi (r) , h (1.34) with the inter-orbital relations, θ(r) =

∞ X

ni ξi2 (r) , and η =

i=1

∞ X

ni ,

(1.35)

i=1

then, the functional derivative relations, δni δξi = δik , and = δik , δξk δnk

(1.36)

are not true in general. Here, the operators on the left-hand side of Eq. (1.34) ˆ 1 [θ] (a θ functional), {ωi } are the consist of a θ-independent ˆ h0 and a θ-dependent h eigenvalues of {ξi }, {ni } are the (non-negative) occupation numbers of {ξi }, and η is a fixed positive number. Proof. Since Eq. (1.34) is variational, it must hold true after any arbitrary infinitesimal derivation in {ξi } and {ni },   ˆ0 + h ˆ 1 [θ] ˜ ξ˜i = ω h ˜ i ξ˜i , (1.37) where {ξ˜i } and θ˜ are the new orbitals and density such that δξi = ξ˜i − ξi , δni = n ˜ i − ni , and δθ = θ˜ − θ =

∞ X i=1


2ni ξi δξi + ξi2 δni .

(1.38)

Expanding Eq. (1.37) around the original quantities to first order and deleting redundant terms, we have     ˆ0 + h ˆ 1 − ωi δξi = δωi − δ h ˆ 1 ξi , h (1.39)

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From the Hohenberg-Kohn Theory to the Kohn-Sham Equations

where ˆ1 = h ˆ 1 [θ] ˜ −h ˆ 1 [θ] = h ˆ 1 [θ + δθ] − h ˆ 1 [θ] . δh A simple utilization of first-order perturbation theory ∞ X ˆ 1 |ξi i hξk |δ h δξi = ξk . ωi − ωk

33

(1.40)

yields (1.41)

k6=i

Because of Eqs. (1.38) and (1.40), one concludes that δξi is related to all other ˆ 1 in Eq. (1.41). Moreover, because of the fixed orbital derivations {δξi } through δ h value of η, one knows that ∞ X 0 ≡ δη = δni . (1.42) i=1

Therefore, Eq. (1.36) is untrue in general.



Because Eq. (1.36) is implicitly employed in Eqs. (1.26)−(1.28), the PY derivation is invalid. 1.3.2. The Wang-Parr resolution The error in the Paradox lies in Eq. (1.30) after the false identification of δTs /δφi from Eq. (1.21). The orthonormality of {φi } simply implies that hφi |φi i = hφi + δφi |φi + δφi i = 1 ,

(1.43)

and hence, to first order, hδφi |φi i = 0 .

(1.44)

Then, Eq. (1.21) will be invariant to any amount of addition of Eq. (1.44), δTs [{φi }] = 2

N X 

δφi tˆ + ci φi ,

(1.45)

i=1

where {ci } are some arbitrary constants.34 Going through the arguments presented in the Paradox again, one instead has
δTs = 2 tˆ + ci φi (r) , (1.46) δφi (r)

and

δTs [ρ] tˆφi (r) KS = + ci = εKS i + ci − veff (r) . δρ(r) φi (r)

(1.47)

With the aid from Eqs. (1.9) and (1.17), one can unambiguously define {ci } as ci = µHS − εKS i ,

(1.48)

such that both sides of Eq. (1.47) are orbital independent. As a result, the Paradox is avoided. However, it does point out that {ci } have been implicitly absorbed into the orbital energies {εKS i } in the KS and PY derivations.

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1.4. Direct inclusion of the constraints Explicitly enforcing the constraints of Eqs. (1.10) and (1.11) in the KS and PY derivations only introduces a new local Lagrange-Euler multiplier γ(r) into the KS effective potential, KS KS v˜eff (r) = veff (r) + γ(r) .

(1.49)

Then, Eqs. (1.31) and (1.47) become δTs [ρ] KS = (˜ εKS ˜i ) − v˜eff (r) , ∀ i . i +c δρ(r)

(1.50)

Multiplying both sides of Eq. (1.50) by δρ(r) and integrating over the entire space, one obtains   δTs [ρ] KS dTs = δρ(r) = (˜ εKS ˜i ) hδρ(r)i − hveff (r) δρ(r)i − hγ(r) δρ(r)i . (1.51) i +c δρ(r) Because of Eqs. (1.4), (1.9), and (1.19), one has dEv = − hγ(r) δρ(r)i .

(1.52)

At the solution point, dEv will be zero for any arbitrary infinitesimal derivation δρ(r), and hence, γ(r) has to be a constant.21,22 This means that Eq. (1.10) will be automatically satisfied at the solution point within the KS scheme. This conclusion together with the discussion presented in the previous section finalize the full equivalence between all the existing derivations of the KS equations, besides a possible arbitrary real constant in the effective potentials: KS LL (r) − veff (r) = constant . veff

(1.53)

In other words, these two effective potentials belong to the same class of local potentials, [veff (r)], with difference up to a real constant:13 [veff (r)] = {u(r)|u(r) = veff (r) + c, c ∈ R}.

(1.54)

Eqs. (1.53) and (1.54) simply restate that the canonical representatives these two effective potentials are identical: LL KS {veff (r)}can ≡ {veff (r)}can = veff (r) − veff (∞) ∈ [veff (r)] ,

(1.55)

where any additive constants of the effective potentials at infinity have been removed.13 1.5. Functional derivative of the kinetic-energy density functional To this end, we are ready to discuss the exact definition of the functional derivative of the non-interacting kinetic-energy density functional, δTs [ρ]/δρ0 (r).

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11

After adopting the canonical representative for the total KS effective potential, we can employ Eqs. (1.6) and (1.48) to update Eqs. (1.47) and (1.50) to δTs [ρ] tˆφKS i (r) KS = KS − (I + ε¯KS i ) = −I − {veff (r)}can , δρ0 (r) φi (r)

(1.56)

which simply states that tˆφKS i (r) KS = ε¯KS i − {veff (r)}can . KS φi (r)

(1.57)

KS Here, {¯ εKS i } are the renormalized KS orbital energies corresponding to {veff (r)}can . KS Because {veff (r)}can is zero at infinity, the renormalized energy of the highest occupied KS orbital φKS (r) is identical to −I,9,13,14,23,24 H

ε¯KS H = −I ,

(1.58)

and consequently δTs [ρ]/δρ0 (r) can be calculated solely from φKS (r): H δTs [ρ] tˆφKS H (r) KS = KS = −I − {veff (r)}can , δρ0 (r) φH (r)

(1.59)

which should be used to benchmark all existing approximations and to help design better approximations to the non-interacting kinetic-energy density functional.11,13,14 1.6. Conclusions None of the two derivations2,9 of the KS equations is completely correct. However, both derivations are mutually equivalent up to an arbitrary constant in the KS effective potential. The functional derivative of the non-interacting kinetic-energy density functional can be exactly calculated from the highest occupied KS orbital alone. Acknowledgement Financial support for this project was provided by a grant from the Natural Sciences and Engineering Research Council (NSERC) of Canada. Y.A.W. is grateful for discussions with Profs. Ernest R. Davidson and Robert G. Parr. Y.A.W. wrote most of the chapter whereas P.X. mainly contributed to the first half of Section 1.4. References 1. P. Hohenberg and W. Kohn, Phys. Rev. 136, B864 (1964). 2. W. Kohn and L. J. Sham, Phys. Rev. 140, A1133 (1965). 3. E. H. Lieb, in Physics as Nature Philosophy: Essays in Honor of Laszlo Tisza on His 75th Birthday, edited by H. Feshbach and A. Shimony (MIT Press, Cambridge, Massachusetts, 1982), p. 111.

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4. E. H. Lieb, Int. J. Quantum Chem. 24, 243 (1983). 5. E. H. Lieb, in Density Functional Methods in Physics, edited by R. M. Dreizler and J. da Providˆencia (Plenum, New York, 1985), p 31. 6. H. Englisch and R. Englisch, Phys. Status Solidi B 123, 711 (1983). 7. H. Englisch and R. Englisch, Phys. Status Solidi B 124, 373 (1984). 8. H. Englisch and R. Englisch, Physica A 121, 253 (1983). 9. R. G. Parr and W. Yang, Density-Functional Theory of Atoms and Molecules (Oxford University Press, New York, 1989). 10. R. M. Dreizler and E. K. U. Gross, Density Functional Theory: An Approach to the Quantum Many-Body Problem (Springer, Berlin, 1990). 11. Y. A. Wang and E. A. Carter, in Theoretical Methods in Condensed Phase Chemistry, edited by S. D. Schwartz (Kluwer, Dordrecht, 2000), Chap. 5, p. 117. 12. R. van Leeuwen, Adv. Quantum Chem. 43, 25 (2003). 13. F. E. Zahariev and Y. A. Wang, Phys. Rev. A 70, 042503 (2004). 14. Y. A. Zhang and Y. A. Wang, Int. J. Quantum Chem. 109, 3199 (2009). 15. M. Levy, Proc. Natl. Acad. Sci. U.S.A. 76, 6062 (1979). 16. M. Levy, Phys. Rev. A 26, 1200 (1982). 17. M. Levy and J. P. Perdew, in Density Functional Methods in Physics, edited by R. M. Dreizler and J. da Providˆencia (Plenum, New York, 1985), p. 11. 18. J. K. Percus, Int. J. Quantum Chem. 13, 89 (1978). 19. Y. A. Wang, Phys. Rev. A 55, 4589 (1997). 20. Y. A. Wang, Phys. Rev. A 56, 1646 (1997). 21. I. M. Gelfand and S. V. Fomin, Calculus of Variations (Prentice-Hall, Englewood Cliffs, NJ, 1963). 22. R. G. Parr and L. J. Bartolotti, J. Phys. Chem. 87, 2810 (1983). 23. Z.-Z. Yang, Y. A. Wang, and S.-B. Liu, Sci. China, Ser. B Chem. 41, 1741 (1998). 24. Z.-Z. Yang, S. Liu, and Y. A. Wang, Chem. Phys. Lett. 258, 30 (1996). 25. R. G. Parr, R. A. Donnelly, M. Levy, and W. E. Palke, J. Chem. Phys. 68, 3801 (1978). 26. R. A. Donnelly and R. G. Parr, J. Chem. Phys. 69, 4431 (1978). 27. R. A. Donnelly, J. Chem. Phys. 71, 2874 (1979). 28. J. D. Talman and W. F. Shadwick, Phys. Rev. A 14, 36 (1976). 29. M. R. Norman and D. D. Koelling, Phys. Rev. B 30, 5530 (1984). 30. J. B. Krieger, Y. Li, and G. J. Iafrata, in Density Functional Theory, edited by E. K. U. Gross and R. M. Dreizler (Plenum, New York, 1995), p. 191. 31. For a similar discussion about the linear dependence problem, see E. R. Davidson, Chem. Phys. Lett. 246, 209 (1995). 32. Y. A. Wang and E. R. Davidson, personal communications (1995-1996). 33. For example, A. Szabo and N. S. Ostlund, Modern Quantum Chemistry (Dover, Mineola, New York, 1996). 34. Y. A. Wang and R. G. Parr, private discussions (1995-1996).

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Chapter 2 Accurate Computation of the Non-Interacting Kinetic Energy from Electron Densities Felipe A. Bulat1,∗ and Weitao Yang2 1

Acoustics Division, Naval Research Laboratory Washington DC, WA 20375, USA [email protected] 2

Department of Chemistry, Duke University Durham, NC 27708, USA [email protected]

We describe a robust and practical method for the computation of implicit density functionals from electron densities based on a regularized version of the Wu-Yang method. For any given electron density that is non-interacting vrepresentable, our method provides a stable variational solution for calculating the non-interacting one-electron potential and the non-interacting kinetic energy. It is thus a computational approach to the challenge of constructing kinetic energy functionals. We also emphasize its application in the computation of the separate exchange and correlation components of the exchange-correlation potential.

Contents 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 Direct optimization method for the Kohn-Sham kinetic energy functional Ts and the exact exchange-correlation potential vxc . . . . . . . . . . . . . . . . . . . . 2.2.2 Exchange vx and correlation vc components of the exchange-correlation potential vxc . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Regularization of the WY functional . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Results and discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.1 Exchange-correlation vxc (r) potentials . . . . . . . . . . . . . . . . . . . . . . . 2.4.2 Kohn-Sham kinetic energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.3 Exchange vx (r) and correlation vc (r) potentials . . . . . . . . . . . . . . . . . . 2.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

13 15 15 17 18 20 21 22 24 26 27

2.1. Introduction Density Functional Theory (DFT)1–5 is, perhaps, the most widely used electronic ∗ Under

contract from Global Strategies Group (North America) Inc., Crofton, MD 21114, USA. 13

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structure method. To a large extent the success of DFT is due to the cost-effective consideration of electron correlation effects within the Kohn-Sham framework.2 Althought increasingly accurate exchange-correlation density functional approximations (DFA) have been developed over the years,6–19 there remains a multitude of challenges20 and limitations that future DFAs should aim to overcome. Most of these limitations can be related to the fact that approximate exchange-correlation functionals are not explicitly discontinuous,21 and thus markedly deviate from the exact behaviour for fractional charges and fractional spins.21,22 While the energy of a system with N + δ electrons (0 < δ < 1) should be a linear combination of the energy of the N -and N + 1-electron systems ((1 − δ)E[N ] + δE[N + 1]), all current DFAs violate this condition to various extents (delocalization error). Analogously, a system with fractional spins – ensembles of degenerate states with integer spins – should have the same energy as the normal spin states that comprise the ensemble;22 current DFA dramatically violate this condition related the strong correlation limit (large static correlation). The self-interaction error (SIE)4,23 is also related to the delocalization error for one-electron systems (SIE had only been defined for one electron systems). The delocalization error concept thus supersedes the SIE concept and is the source of many known deciencies of current DFAs: notably, the overestimation of polarizabilities24–26 and molecular conductance,27 the incorrect description of systems having fractional number of electrons,28 and its implications in main group thermochemistry.29 Recently, and based on the early ideas of and Savin,30 the so-called coulomb attenuated functionals have spawned renewed attention,19,31–33 showing improved description of systems with fractional charge.19,32 Approximate and exact Density functionals can be classified according to the knowledge of their functional forms in terms of the electron density. Explicit density functionals are those whose functional form in terms of the electron density is known, and their functional derivatives (with respect to the electron density) are readily available through functional differentiation. LDA (Local density approximation) and GGA (generalized gradient approximation) functionals are all examples of explicit functionals. Implicit density functionals, on the other hand, are functionals whose form in terms of the electron density is not known. Orbital dependent functionals are examples of implicit density functionals, but we emphasize that, in principle, any density functional can be defined as an implicit density functional through some appropriately defined wave function. The non-interacting kinetic energy, another example of a functional not know explicitly in terms of the electron density, can be defined in terms of the Kohn-Sham determinant (vide infra). We have described34–38 computational approaches for the calculation of a variety of implicit density functionals and their functional derivatives, among which we distinguish two cases. i) If a set of orbitals and an energy functional in terms of them is known, the regularized optimized effective potential (rOEP) method is used to evaluate the functionals derivatives. The evaluation of the exact exchange potential from the Hartree-Fock energy functional in terms of Kohn-Sham orbitals is one ex-

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ample of the applicability of the rOEP method, but the method is general enough that can be applied to almost any energy functional. ii) If a functional is known in terms of Kohn-Sham orbitals but the orbitals themselves are not know, knowledge of the electron density can be exploited to evaluate said functional through the regularized Wu-Yang method (rWY). In the process of doing so, the rWY method also affords the functional derivative and the corresponding set of Kohn-Sham orbitals (vide infra) that reconstruct the electron density. The need for regularization in the OEP and WY methods originates in the fact that they are inverse ill-possed problems.35,38–42 The regularization techniques developed for the OEP35 and WY38 methods ensure that physical solutions are obtained, and emphasize that unphysical solutions41 are nothing but the controllable manifestation of their inverse nature. A notable example of an implicit density functional is, as mentioned above, the Kohn-Sham kinetic energy functional Ts [ρ], which is not known as an explicit functional of the electron density, and its computation shall be the main subject of the present Chapter. Ts [ρ] is defined through the use of an auxiliary set of one-electron orbitals {φi }, from which the electron density can be reconstructed Pocc ρ[{φi }] = |φi |2 . This set of orbitals are the eigenstates of the Kohn-Sham i equation {Tˆ + veff (r)}φi = εi φi , so the problem of determining Ts [ρ] hence becomes that of determining the set {φi }. Finally, Ts [ρ[{φi }]] is readily computed as a functional of {φi }, Ts [{φi }]. The Kohn-Sham kinetic energy functional Ts [ρ] has received considerable attention, and methods to evaluate it along with the exchange-correlation potential have been developed over the years.37,43–49 In this work we are concerned with the direct optimization method for calculating Ts [ρ] and the Kohn-Sham potential vs (r) from electron densities,37,38 which is quite general and applicable to arbitrary density functionals. The rWY method utilizes a similar construction for the potential to that used in the direct optimization method for the OEP,34 taking advantage of the variational principle in terms of the one-electron potential v(r) alone. The Levy constrained search approach3 is replaced by an unconstrained optimization of the Kohn-Sham potential within a finite basis. We introduce here for the first time an extension of the rWY method that enables the computation of the separate exchange and correlation components of the exchange-correlation potential by solving the OEP equation for exact exchange after the computation of Ts [ρ] and vs (r) for a given electron density ρ(r). 2.2. Theory 2.2.1. Direct optimization method for the Kohn-Sham kinetic energy functional Ts and the exact exchange-correlation potential vxc The Kohn-Sham kinetic energy Ts [ρ(r)] is known as an explicit functional of the KS orbitals φi (r) and, through φi (r) ≡ φi (r)[ρ(r)], as an implicit functional of the

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Pocc. electron density ρ(r) = |φi (r)|2 . Ts [ρ(r)] can be defined, through Levy’s3 i constrained search formulation, by resorting to an auxiliary determinantal wavefunction Ψdet. Ts [ρ] is then found by searching over all wavefunctions Ψdet for the one that minimizes T [Ψdet] Ts [ρ] = min T [Ψdet] , Ψdet →ρ

(2.1)

and simulateously yields ρ(r) = hΨdet |ˆ ρ|Ψdet i. Wu and Yang37 introduced a direct optimization method for computing implicit density functionals that can be applied to the computation of the Kohn-Sham kinetic energy functional. As anticipated above, in the Wu-Y ang method Levy’s constrained search formulation3 is replaced by an unconstrained maximization that provides a new variational principle for Ts [ρ] in terms of the one-electron potential v(r). Consider wave functions Ψdet whose orbitals {φi } are eigenstates of the single particle Kohn-Sham equation [Tˆ + v(r)]φi = εi φi , and hence implicit functionals of v(r) (Ψdet [v(r)]). One can then define the following functional: Ws [Ψdet , v(r)] = 2

Z N/2 X hφi |Tˆ|φi i + drvs (r){ρ(r) − ρin (r)} ,

(2.2)

i

PN/2 where ρ(r) = 2 i |φi (r)|2 . Note that it’s first derivative with respect to the potential v(r), evaluated at the input density ρin , vanishes identically:   δWs [Ψdet , v(r)] = 0, (2.3) δv(r) ρ=ρin and the second variation can be shown to be always non-positive: Z δ 2 Ws [Ψdet , v(r)] δ 2 Ws [Ψdet , v(r)] = dr0 drδv(r0 )δv(r) ≤ 0, δv(r0 )δv(r)

(2.4)

where: unocc. occ. ∗ X X δ 2 Ws [Ψdet , v(r)] δρ(r) φi (r)φa (r)φ∗a (r0 )φi (r0 ) = = . δv(r0 )δv(r) δv(r0 ) εi − εa a i

(2.5)

Equations (2.3) and (2.4) combined imply that Ts [ρin ] is a concave functional of v(r) and the stationary point is then a maximum. The Kohn-Sham non-interacting kinetic energy for a given input density ρin (r) can then be determined through the following unconstrained maximization: Ts [ρin ] = max Ws [Ψdet [v(r)], v(r)] . v(r)

(2.6)

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The variational principle Eq. (2.6) can be turned into a direct optimization method by choosing an appropriate expansion for the pontential. In particular, we choose the following expansion: X v(r) = vext (r) + v0 (r) + bt gt (r) , (2.7) t

where vext (r) is the external potential due to the nuclei, v0 (r) is a reference potential, and gt (r) is a set of gaussian functions with expansions coeeficients {bt }, which are to be determined. The problem now turns into the unconstrained maximization of Ws [Ψdet [v(r)], v(r)] with respect to the expansion coefficients {bt }. The first (∇b Ws ) and second derivatives (H) are readily available: Z ∂Ws [∇b Ws ]t = = drgt (r){ρ(r) − ρin (r)} , (2.8) ∂bt Hu,t

occ unocc X X hφi |gu (r)|φa ihφa |gt (r)|φi i ∂ 2 Ws =2 + c.c. , = ∂bu ∂bt εi − εa a i

(2.9)

and the maximization can be carried out efficiently with iterative optimization methods such as the quadratically convergent Newton method.50 The direct optimization approach for Ts [ρin ] can in general be extended to any functional of the electron density (for details, the reader is referred to Ref.37 ). It must be emphasized that the WY method relies on different variational principles than the OEP method, every time it deals with general implicit density functionals of the electron density. In the present case, the focus is on the KS kinetic energy of a given density and on its exchange-correlation potential, and the value of Ws thus represents a variational estimation of Ts . 2.2.2. Exchange vx and correlation vc components of the exchange-correlation potential vxc The WY procedure can be applied, as outlined above, in the computation of the Kohn-Sham kinetic energy Ts and the exchange-correlation potential vxc (r) from a given electron density ρin (r). We now present an extension that enables the computation of the exchange and correlation components of the exchange-correlation potential, which is equivalent to that used by Filippi et al.51–53 It is based in the fact that given a set of orbitals φi (r) and eigenvalues εi , corresponding to a given exchange-correlation potential vxc (r) and electron density ρ(r), one can solve the so-called OEP equation for exact exchange to obtain vx (r), and obtain the pure correlation component by difference vc (r) = vxc (r) − vx (r). For a given electron density ρ(r) the non-interacting kinetic energy Ts [ρ] and the exchange-correlation potential vxc (r) are obtained with the WY method as outlined

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above. The method also yields the corresponding orbitals φi (r) and orbital energies εi . Upon convergence, the following OEP equation holds: Z

dr0 χ0 (r, r0 )vXC (r0 ) =

occ unocc X X φi (r)φi (r)hj|v n` |ii XC , ε − ε j i i j

(2.10)

where χ0 (r, r0 ) is the Kohn-Sham linear response function: χ0 (r, r0 ) =

δρ(r) , δvs (r0 )

(2.11)

n` and vXC is the (non-local) potential that generated the input density ρin (r). The n` (non-local) potential vXC can be split in two contributions vxn` + vcn` , just as the Kohn-Sham potential vxc (r) can be split into its exchange and correlation compon` n` nents vx (r) + vc (r). Indeed, for any vxc , we simply define vcn` (r) = vxc (r) − vxn` (r), n` where vx (r) corresponds to the derivative of the Hartree-Fock energy functional with respect to the density matrix (the Hartree-Fock energy functional in terms of the Kohn-Sham orbitals). We then solve the following OEP equation for the exchange potential:

Z

dr0 χ0 (r, r0 )vx (r0 ) =

occ unocc X X φi (r)φi (r)hj|v n` |ii x

i

j

εj − εi

,

(2.12)

and compute the correlation component as: vc (r) = vxc (r) − vx (r) ,

(2.13)

where vxc (r) is that of Eq. (2.7) and vx (r) comes from the solution of Eq. (2.12). The main difficulty in the application of the above procedure in a finite basis set is that both Eqs. (2.12) and (2.2) are inverse, ill-possed problems, and both require regularization. Care should be taken to employ consistent regularizations schemes in both of them. 2.3. Regularization of the WY functional The original work of Yang and Wu36,37 did consider a first level of regularization for the OEP and WY methods. Indeed, a truncated singular value decomposition (TSVD) was used for the numerically stable inversion of the Hessian matrix when solving for the update vector p in the Newton method.50 The TSVD alone has been shown to produce good quality potentials, with oscillations restricted to the regions near the nuclei even when unbalanced basis sets are used.35,38 We will herein focus on the so-called λ-regularization of the WY method, that yields potentials of the highest quality regardless of the basis sets used, and that we refer to as the rWY

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method. For a thorough discussion of the TSVD regularization, the reader is refered to ealier work on the subject.35–38 To effectively solve the inverse problem possed by Eq. (2.6) for the potential and the KS kinetic energy in a stable and physically meaningful way that ensures ¯ s based on well behaved potentials, we have defined a regularized WY functional W ¯ s is defined by introducing a measure of the the original WY functional Ws .37 W smoothness of the expected solution: ¯ s [Ψdet , v(r)] = Ws [Ψdet , v(r)] − λ||∇vb (r)||2 , W

(2.14)

where: ||∇vb (r)||2 =

XX t

u

bt bu hgt (r)| − ∇2 |gu (r)i ,

(2.15)

is the squared norm of the gradient of the basis set expansion part of the potential ¯ s are vb (r). The first and second derivatives of the regularized WY functional W simply related to those of Ws by X ¯s ∂W ∂Ws = − 2λ bu hgt (r)| − ∇2 |gu (r)i , ∂bt ∂bt u

¯s ∂2W ∂ 2 Ws = − 2λhgt (r)| − ∇2 |gu (r)i , ∂bu ∂bt ∂bu ∂bt

(2.16) (2.17)

or in the more compact matrix notation, equations (2.15), (2.16) and (2.17) read: ||∇vb (r)||2 = 2bT Tb ,

(2.18)

¯ s = ∇b Ws − 4λTb , ∇b W

(2.19)

¯ s = ∇2b Ws − 4λT , ∇2b W

(2.20)

where: Tt,u = hgt (r)|Tˆ|gu (r)i .

(2.21)

¯ s ({bt }) functional, as when dealing with Ws ({bt }) itself, is maximized The W with respect to the expansion coefficients ({bt }) entering in the expression for the potential. For every value of λ there corresponds an unconstrained maximum value ¯ s , which we denote by W ¯λ=W ¯ s ({bλ }), whose maximizing set of coefficients are of W s t λ denoted {bt }. For such set of coefficients there corresponds a value of the Ws functional, which we denote by Wsλ = Ws ({bλt }), that corresponds to the constrained maximum for a given λ. To complete the analysis one must choose the most appropriate value for the λ parameter, denoted hereafter λ∗ , which is to be determined by the L-Curve analysis,54 that we now brifly describe.

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For the L-Curve analysis one simply plots the logarithm of the solution norm, ||∇vb (r)||, against a measure of the decrease in the functional’s value due to the additional constraint. Such a measure is given here by Ws0 − Wsλ (since Wsλ is bounded by Ws0 , this quantity is always positive). The L-Curve plot thus represents the trade off between the desire to fully maximize the (unconstrained) functional and that of extracting a physically meaningful, well behaved solution. In the classical literature of inverse problems,54 one chooses as an optimal solution a value of λ∗ near the corner of the L-Curve, where the solution is already stable with respect to variations of λ yet the trade off in the functional value is minimum. However, the method for determining the most appropriate value for the λ∗ chosen here is that used in our earlier work:35,38 the point along the L-Curve with a minimum slope. We emphasize that the slope of the L-curve is given by:38 ∂ log(n(λ)) ∆W (λ) ∂n(λ) (W 0 − Wsλ ) = =− s . ∂ log(∆W (λ)) n(λ) ∂∆W (λ) n(λ)λ

(2.22)

Note that this analytical expression is generally applicable to L-Curve analysis. To the best of our knowledge, was first derived in Reference.38

Fig. 2.1. (Left panel) Potentials obtained from an LDA input density of N2 using various basis set for the potential expansion. No regularization is used. (Right panel) L-Curves for an LDA input density of N2 using various basis sets for the potential expansion.

2.4. Results and discussion We present the result of applying the regularized WY method to LDA and HF densities of N2 to obtain the KS kinetic energy and exchange-correlation potentials. The LDA density – using the SVWN5 functional2,55,56 – was considered because the direct availability of the exact potential through functional differentiation make comparison with the exact result very straight forward. The Hartree-Fock (HF) density was considered to illustrate how the reliability of the regularized WY method, as applied to the LDA case, transfers to densities generated from non-local potentials. We also examine the case of a CCSD density of N2 , and briefly discuss other atomic

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Fig. 2.2. (Left panel) Potentials obtained from an LDA input density of N2 using various basis set for the potential expansion. (Right panel) Potentials obtained from an LDA input density of N2 using the UC-3 basis set for the potential expansion with different choices of λ∗ as indicated.

(Ar) and molecular systems (H2 O). Finally, we compute the exchange-correlation potential from a CCSD density of F2 , and examine the identification of the exchange and correlation components of the exchange-correlation potential. We have chosen to work with a simple example for which a CCSD density can be obtained on a very large MO basis. This make the regularization uncessesary, and removes the issue of compatibility between the regularization schemes otherwise necessary in the two steps of our method (Eqs. (2.6) and (2.12)). 2.4.1. Exchange-correlation vxc (r) potentials We start by noting that the application of the WY method to a density comming from a fully local potential displays some subtle but important differences with respect to the case when the density comes from a non-local potential. For a given arbitrary density ρin (r) it is in general impossible to know a priori whether it comes from a local or non-local potential, which brings up the issue of v-representability of the density. We note that if a density is not v-representable, the rWY method should provide (through Ws ) the best variational estimate for the non-interacting Kohn-Sham kinetic energy. Note also that since a non-v-representable density by definition cannot be obtained with a local potential, one should expect a non-zero contribution of the second term in the right-hand-side of Eq. (2.2). Several basis sets are used for the potential expansion, while the cc-pVDZ basis set has been chosen for the atomic orbitals in all cases. For HF densities, the reference potential, v0 (r) in Eq. (2.7), corresponds to the Fermi-Amaldi potential of a sum of atomic densities. For LDA densities, the bare coulomb potential of a sum of atomic densities is used instead to reflect the corresponding long-range behavior. The regularized WY method has been implemented in the NWChem program.57 We considered four different basis sets for the potential expansion in N2 . Firstly, as an example of a balanced basis set, we have chosen the cc-pVDZ basis. Second, as an example of unbalanced basis sets, we have considered three basis sets constructed

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from even tempered sets of s-type and p-type gaussian functions. The UC-1 basis (for uncontracted gaussian basis 1) contains 13 s-functions (2n , −4 ≥ n ≥ 8) and 3 p-functions (2n , −2 ≥ n ≥ 0) per nitrogen atom, the UC-2 basis contains 14 sfunctions (2n , −4 ≥ n ≥ 9) and 5 p-functions (2n , −3 ≥ n ≥ 1), and the UC-3 basis contains 18 s-functions (2n , −4 ≥ n ≥ 13) and 8 p-functions (2n , −2 ≥ n ≥ 5). Figure 2.1 displays the unregularized potentials obtained from an LDA density of N2 (left panel). The reader should keep in mind that these potentials contain no regularization; when the WY method37 (and direct OEP method34 ) was introduced, it contained TSVD regularization that enabled physically meaningful potentials in most cases by eliminating all of the oscillations in the potential except for those very close to the nuclei. We choose here to apply the λ-regularization to unregularized-potentials (no TSVD used) to illustrate the reliability of our method. Going back to Fig. 2.1, it is noteworthy that the larger the potential basis set, the more oscillatory the potential becomes – an observation that carries over to all the cases we have studies of unbalanced basis sets –. The right panel in Fig. 2.1 displays the L-Curves obtained with the four different potential basis sets. Since Wsλ is the variational estimate of the kinetic energy, this is the energy that should be compared to the unregularized Ws0 , which is the target functional. Note that when the atomic basis set used for the molecular orbitals is used for the potential there is no raise in the solution norm. This indicates that no unstable, unphysical, oscillating potential is formed for that basis set, and the basis set is hence regarded as balanced following our earlier work.35,38 Note in this connection that the potential obtained with this balanced basis set is not shown in Fig. 2.1, as it would show up as an horizontal straight line due to the very large vertical scale of Fig. 2.1. Instead, said potential can be seen in Fig. 2.2, where we note that it correlates nicely with the actual LDA potential for N2 . The L-Curves for the UC-1, UC-2, and UC-3 basis set do display raises in the norms and yield unphysical potentials (Fig. 2.1). The raise in the norms in the L-Curve analysis is the defining feature of an unbalanced basis, and we recall that we would like to choose an optimum point along the L-Curve for which the compromise in the residual norm is minimal, yet a well behaved potential is obtained. By a minimal compromise in the residual norm (loss of numerical accuracy in the evaluation of the target functional) we mean the smallest value of Ws0 − Wsλ for which the potential is well behaved. We observe that this compromise is less than 3 × 10−4 a.u., which we expect to be beyond the accuracy of the orbital basis set. Figure 2.2 also displays the potentials obtained for the UC-3 basis set when different values are chosen for the regularization parameter λ. This illustrates the adecuacy of the choice of λ∗ , as an smaller λ results in some oscillations, while a large λ results in over-smoothing. 2.4.2. Kohn-Sham kinetic energy We have so far discussed the usage of the rWY method in obtaining accurate exchange-correlation potentials from electron densities originating in both local and

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Fig. 2.3. (Left panel) L-Curves for a HF input density of N2 using various basis sets for the potential expansion. (Right panel) Potentials obtained from a HF input density of N2 using various basis set for the potential expansion.

non-local potentials, exemplified by the LDA and HF cases discussed above. We now turn to the evaluation of the Kohn-Sham kinetic energy Ts [ρ], and would like to emphasize that the regularization process results in some loss of accuracy in the values of the target functional. Indeed, it is a measure of that loss of accuracy that is used in the L-Curve analysis as discussed previously. This means that even for a density originated in a local potential, the method may not recover the kinetic energy exactly. Consider the case of a complete basis set for the molecular orbitals. In such case, the quality of the variational estimate for the KS kinetic energy will increase with the size of the basis set used in the potential expansion, until the point on which the potential basis reaches (or approximates) completeness, at which point the variational estimate for Ts [ρ] will approach the exact value. Now consider the case of a finite basis for the molecular orbitals, which entails certain subtleties. The variational estimate for Ts [ρ] will improve with the size of the potential basis, up to the point where the “MO basis/potential basis” pair becomes unbalanced. At this point the numerical value of Ts [ρ] becomes unreliable, even though it may appear to converge, and may lead to unphysical behaviour like, for example, reproducing exactly the HF kinetic energy with a local potential. The KS potential itself is the quantity one should monitor, and the onset of oscillations in it is the hallmark of an unstable “MO basis/potential basis” pair, and regularization is needed, although it may result in some loss of accuracy. This loss of accuracy is the price one has to pay to ensure the potential and KS kinetic energy remain physically meaningful. If more accuracy is needed, the MO basis should be enlarged. It is in this way that, as we shall see below, no potential basis can reproduce the exact KS kinetic energy obtained originally for an LDA calculation of N2 using the cc-pVDZ basis for the molecular orbitals. The variational estimates, however, bear comparable accuracy to that of the MO basis itself. Table 2.1 displays the variational estimates for the Kohn-Sham kinetic energy obtained from and LDA density of F2 . Note that when the MO basis is used for the

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potential expansion the estimate of the KS kinetic energy is remarkably accurate, and emphasizes the convenience of choosing a basis set for the potential that is balanced with respect to the molecular orbital basis set. Indeed, the error in the kinetic energy estimate is about 1·10−4 Hartree for the LDA density of N2 when the MO basis is used for the orbital expansion, and it only increases slightly to ∼ 4·10−4 for UC-1, UC-2, and UC-3 at λ∗ . The fact that the accuracy does not increase with the basis set size used in the potential expansion highlights the fact that the rWY method can only give an estimate for Ts [ρ] with an intrinsic accuracy limited by the MO basis set. Note, however, that if no regularization is used, the value of Ws0 approaches the exact value obtained from the LDA calculation and, in fact, does get deceivingly “better” with increasing size of the potential basis set. This comes at the expense of highly oscillatory, unphysical potentials which is not acceptable because a physically meaningful well behaved potential is the only evidence for the v-representability of the final density. The final density may differ slightly from the input density, but the former is guaranteed to be v-representable, while the latter is not. Finally, note that the estimate for Ts [ρ] depends on the regularization parameter, and we also show in Table 2.1 how it behaves around λ∗ . As expected, a value for λ that is too large (small) results in an estimate for Ts [ρ] that is too small (large), and corresponds to an oversmoothed (oscillatory) potential. (Note that the relationship between λ and the error in the kinetic energy is approximately linear.) Table 2.2 displays the variational estimates for the Kohn-Sham kinetic energy obtained from a HF density of N2 . Note that the unregularized results for the UC1, UC-2, and UC-3 potential basis get increasingly close to the HF kinetic energy, This is unphysical because the HF density is not v-representable. This situation is manifested in the highly oscillatory behaviour observed for the potentials in the previous section. The estimates for Ts [ρ] when regularization is used are pretty consistent at the optimum point in the L-Curve (λ∗ ), emphasizing the fact that, up to the accuracy provided by the MO basis set, the regularization scheme is pretty robust and consistent. Table 2.3 displays the results obtained for a CCSD density of N2 using the UC-2 basis set for the potential, and we observe that the rWY best estimate for the KS kinetic energy is below the unregularized value. Finally, we display in Table 2.4 the variational estimated for the KS kinetic energy for LDA and HF densities of Ar and H2 O both at the regularized and unregularized levels (for details of these calculations, including the basis sets used and an extensive analysis of the potentials obtained, the reader is referred to our previous work38,58 ). In all cases the rWY method is successful in providing consistent results not significantly dependent on the basis sets, with the added benefit of accompanying potentials of very high-quality that ensure the v-representability of the resulting density. 2.4.3. Exchange vx (r) and correlation vc (r) potentials Having discussed the application of the rWY method in the computation of the exchange-correlation potential and KS kinetic energy from electron densities, we

Variational estimates for the KS kinetic energy from an LDA (SVWN) density of N2 . Values in a.u. UC-1

UC-2

UC-3

LDA

108.287860859506 -

108.288001556831 108.287905774360 108.287573303071 108.282059937202

108.288006848736 108.287897445097 108.287638341841 108.282301853745

108.288007771150 108.287935287901 108.287684003632 108.282440788516

108.288007771336 -

Table 2.2.

MO Basis

UC-1

UC-2

UC-3

HF

108.652675479578 -

108.654321870934 108.653436162330 108.652151103152 108.642037661189

108.654602308454 108.653533237448 108.652169778237 108.642098571850

108.654872594532 108.653630096598 108.652306154364 108.642217629141

108.6548741907 -

Table 2.3. Variational estimates for the KS kinetic energy from a CCSD density of N2 . Values in a.u.

Unregularized λ = 10−5 λ∗ = 10−4 λ = 10−3

MO Basis

UC-1

UC-2

UC-3

-

-

108.933404787096 108.931139617331 108.930052082238 108.920843011620

-

Table 2.4. Variational estimates for the KS kinetic energy from LDA and HF densities of Ar and H2 O. The last column corresponds to the kinetic energy of LDA or HF the input density. Values in a.u.

Ar LDA Ar HF H2 O LDA

UC-1

UC-2

Input

525.750661759946 525.750661759945 526.709052927157 526.709049843973 75.608444806071 75.608395662595 75.941025308113 75.939724614503

525.750661759945 525.750661759945 526.709052927157 526.709049842989 75.608446346554 75.608398121345 75.941024215260 75.939732271592

525.750661759945 526.709052927157 75.608446346738 75.941034642694 -

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MO Basis 525.750652099832 526.708342154777 75.608425385416 75.940200841677 -

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H2 O HF

Unregularized λ∗ = 4 · 10−5 Unregularized λ∗ = 1 · 10−6 Unregularized λ∗ = 4 · 10−5 Unregularized λ∗ = 2 · 10−4

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Unregularized λ = 4 · 10−5 λ∗ = 2 · 10−4 λ = 1 · 10−3

Variational estimates for the KS kinetic energy from a HF density of N2 . Values in a.u.

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MO Basis

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Unregularized λ = 8 · 10−6 λ∗ = 8 · 10−5 λ = 8 · 10−4

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Table 2.1.

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now turn to the identification of the exchange and correlation components of the exchange-correlation potential. The application of our method is quite straightforward, and the only subtlety is that the regularization used in the iterative maximization of Eq. (2.2) should be consistent with the regularization used in the solution of Eq. (2.12). In order to explore the reliability of the method an attain the highest possible accuracy in the exchange and correlation components we focus on a highly correlated CCSD density for an atom (Be) and a small molecule (F2 ) with large basis sets for the molecular orbitals. This effectively removes the need for regularization and we observe no raise for the solution norm in the L-Curve analysis. For Be we use a Partridge-3 basis set for the molecular orbitals (12s6p3d2f) and an uncontracted basis set containing s-functions only for the potential expansion (8 s-functions, exponents 47.0-0.5). For F2 we used an uncontracted version of the aug-cc-pvTz basis set for the molecular orbitals (16s10p3d2f after removing duplicate primitives) and an uncontracted basis set (6s3p1d) for the potential expansion (exponents 2.0-64.0, 2.0-8.0, and 0.5). For the reference potential in Eq. (2.7), we chose the Fermi-Amaldi potential. Figure 2.4 displays the exchange-correlation potential obtained with the rWY method, along with the exchange potential obtained by solving Eq. (2.12), and the correlation potential obtained by difference. Note that the correlation potential for Be is almost always negative, and compares very well with that of Filippi et al.51 except for an extra oscillation close to the nuclei in out result. For F2 , we note that the exchange potential is above the exchange-correlation potential in most of the range considered, from the bonding region (z ∼ 0) to the nuclear region (z = 1.33a.u.) along the C∞ semiaxis. However, the exchange potential is below the exchange-correlation potential from about z ∼ 2. This means that the correlation potential is negative in the bonding and nuclear region, but positive for z > 2a.u.. It remains positive throughout the numerical range considered (up to z = 10a.u.) as seen in the right panel of Fig. 2.4, although it goes to zero fairly quickly beyond z = 3a.u., probably as a consequence of the long-range dependence built into the exchange and exchange-correlation potentials throught the reference potential (v0 (r) in Eq. (2.7)).

2.5. Conclusions We have described the regularized Wu-Yang37,38 method (rWY) for the computation of exchange-correlation potentials and the KS kinetic energy functional, and showed through numerical example its robustness and reliability. A modified energy functional that incorporates a measure of the smoothness of the solution is the key ingredient in obtaining physically meaningnful potentials that ensure vrepresentability of the electron densities. Any arbitrary density, possibly even nonv-representable, can be approximated in a well defined variational sense by a vrepresentable density. This also guarantees that the kinetic energy obtained indeed

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Fig. 2.4. Exchange, correlation, and exchange-correlation potentials for Be (upper panel) and F2 (lower panel).

corresponds to the KS kinetic energy, avoiding numerical artifacts that can occur in finite basis setting. Different molecular cases have been studied with densities obtained throught several quantum mechanical methods, illustrating the general validity of the method. Importantly, we have here introduced for the first time an extension to our method that allows for the computation of the separate exchange and correlation components of the exchange-correlation potential. We anticipate that these methods will prove helpful in the development of accurate exchange, correlation, and kinetic energy functionals. Acknowledgements Financial support from the National Science Foundation (WY) and the Office of Naval Research (FAB) is gratefully acknowledged. References 1. P. Hohenberg and W. Kohn, Phys. Rev. 136, B864 (1964). 2. W. Kohn and L. Sham, Phys. Rev. 140, A1133 (1965). 3. M. Levy, Proc. Natl. Acad. Sci. U.S.A. 76, 6062 (1979).

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4. R. G. Parr and W. Yang, Density Functional Theory of Atoms and Molecules. (Oxford University Press, Oxford, 1989). 5. R. M. Dreizler and E. K. U. Gross, Density Functional Theory. (Springer, Berlin, 1990). 6. J. P. Perdew and Y. Wang, Phys. Rev. B. 33, 8800 (1986). 7. A. D. Becke, Phys. Rev. A. 38, 3098 (1988). 8. J. P. Perdew, Electronic Structure of Solids ’91, p. 11. Akademic, Berlin, (1991). 9. C. Lee, W.Yang, and R. G. Parr, Phys. Rev. B. 37, 785 (1988). 10. J. P. Perdew, K. Burke, and M. Ernzerhof, Phys. Rev. Lett. 77, 3865 (1996). 11. A. D. Becke, J. Chem. Phys. 98, 1372 (1993). 12. A. D. Becke, J. Chem. Phys. 98, 5648 (1993). 13. T. Van Voorhis and G. E. Scuseria, J. Chem. Phys. 109, 400 (1998). 14. C. Adamo and V. Barone, J. Chem. Phys. 110, 6158 (1999). 15. M. Ernzerhof and G. E. Scuseria, J.Chem. Phys. 111, 911 (1999). 16. J. Tao, J. P. Perdew, V. N. Staroverov, and G. E. Scuseria, Phys. Rev. Lett. 91, 146401 (2003). 17. A. D. Becke, J. Chem. Phys. 122, 064101 (2005). 18. P. Mori-S´ anchez, A. J. Cohen, and W. Yang, J. Chem. Phys. 124, 091102 (2006). 19. A. J. Cohen, P. Mori-S´ anchez, and W. Yang, J. Chem. Phys. 126, 191109 (2007). 20. A. J. Cohen, P. Mori-S´ anchez, and W. Yang, Science. 312, 792 (2008). 21. P. Mori-S´ anchez, A. J. Cohen, and W. Yang, Phys. Rev. Lett. 102, 066403 (2009). 22. A. J. Cohen, P. Mori-S´ anchez, and W. Yang, J. Chem. Phys. 129, 121104 (2008). 23. J. P. Perdew and A. Zunger, Phys. Rev. B. 23(10), 5048 (1981). 24. B. Champagne, E. A. Perp`ete, S. J. A. van Gisbergen, E.-J. Baerends, J. G. Snijders, C. Soubra-Ghaoui, K. A. Robins, and B. Kirtman, J. Chem. Phys. 109, 10489 (1998). 25. P. Mori-S´ anchez, Q. Wu, and W. Yang, J. Chem. Phys. 119, 11001 (2003). 26. F. A. Bulat, A. Toro-Labb´e, B. Champagne, B. Kirtman, and W. Yang, J. Chem. Phys. 123, 014319 (2005). 27. S.-H. Ke, H. U. Baranger, and W. Yang, J. Chem. Phys. 126, 201102 (2006). 28. Y. Zhang and W. Yang, J. Chem. Phys. 109, 2604 (1998). 29. E. R. Johnson, P. Mori-S´ anchez, A. J. Cohen, and W. Yang, J. Chem. Phys. 129, 204112 (2008). 30. A. Savin, Recent Developments and Applications of Modern Density Functional Theory, p. 327. Elsevier, Amsterdam, (1996). 31. H. Iikura, T. Tsuneda, T. Yanai, and K. Hirao, J. Chem. Phys. 115, 3540 (2001). 32. P. Mori-S´ anchez, A. J. Cohen, and W. Yang, J. Chem. Phys. 125, 201102 (2006). 33. O. A. Vydrov and G. E. Scuseria, J. Chem. Phys. 125, 234109 (2006). 34. W. Yang and Q. Wu, Phys. Rev. Lett. 89, 143002 (2002). 35. T. Heaton-Burgess, F. A. Bulat, and W. Yang, Phys. Rev. Lett. 98, 256401 (2007). 36. Q. Wu and W. Yang, J. Theo. Comp. Chem. 2, 627 (2003). 37. Q. Wu and W. Yang, J. Chem. Phys. 118, 2498 (2003). 38. F. A. Bulat, T. Heaton-Burgess, A. J. Cohen, and W. Yang, J. Chem. Phys.. 127, 174101 (2007). 39. S. Hirata, S. Ivanov, I. Grabowski, R. J. Bartlett, K. Burke, and J. D. Talman, J. Chem. Phys. 115, 1635 (2001). 40. A. Ben-Haj-Yedder, E. Cances, and C. Le Bris, Diff. Int. Eq. 17, 331 (2004). 41. V. N. Staroverov, G. E. Scuseria, and E. R. Davidson, J. Chem. Phys. 124, 141103 (2006). 42. E. Cances, G. Stoltz, G. E. Scuseria, V. N. Staroverov, and E. R. Davidson, MathS In Action. 2, 1 (2009).

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43. 44. 45. 46. 47. 48. 49. 50.

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Q. Zhao and R. G. Parr, Phys. Rev. A. 46, 2337 (1992). Q. Zhao and R. G. Parr, J. Chem. Phys. 98, 543 (1993). Q. Zhao, R. C. Morrison, and R. G. Parr, Phys. Rev. A. 50, 2138 (1994). V. E. Ingamells and N. C. Handy, Chem. Phys. Lett. 248, 373 (1996). D. J. Tozer, V. E. Ingamells, and N. C. Handy, J. Chem. Phys. 105, 9200 (1996). D. J. Tozer, K. Somasundram, and N. C. Handy, Chem. Phys. Lett. 265, 614 (1997). D. J. Tozer, N. C. Handy, and P. Palmieri, Mol. Phys. 91, 567 (1997). W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in FORTRAN: The Art of Scientific Computing. (Cambridge University Press, Cambridge, 1992). C. Filippi, C. J. Umrigar, and X. Gonze, Phys. Rev. A. 54, 4810 (1996). A. G¨ orling and M. Levy, Phys. Rev. A. 50, 196 (1994). A. G¨ orling and M. Levy, Int. J. Quantum Chem., 56(S29), 93 (1995). P. C. Hansen, Rank-Deficient and Discrete Ill-Posed Problems: Numerical Aspects of Linear Inversion. (Society for Industrial and Applied Mathematics, Philadelphia, PA, 1998). J. C. Slater, Quantum Theory of Molecules and Solids. (McGraw-Hill, New York, 1974). S. H. Vosko, L. Wilk, and M. Nusair, Can. J. Phys. 58, 1200 (1980). E. Apra, E. J. Bylaska, W. de Jong, M. T. Hackler, S. Hirata, L. Pollack, D. Smith, T. P. Straatsma, T. L. Windus, R. J. Harrison, J. Nieplocha, V. Tipparaju, M. Kumar, E. Brown, G. Cisneros, M. Dupuis, G. I. Fann, H. Fruchtl, J. Garza, K. Hirao, R. Kendall, J. A. Nichols, K. Tsemekhman, M. Valiev, K. Wolinski, J. Anchell, D. Bernholdt, P. Borowski, T. Clark, D. Clerc, H. Dachsel, M. Deegan, K. Dyall, D. Elwood, E. Glendening, M. Gutowski, A. Hess, J. Jaffe, B. Johnson, J. Ju, R. Kobayashi, R. Kutteh, Z. Lin, R. Littlefield, X. Long, B. Meng, T. Nakajima, S. Niu, M. Rosing, G. Sandrone, M. Stave, H. Taylor, G. Thomas, J. van Lenthe, A. Wong, , and Z. Zhang. NWChem, A Computational Chemistry Package for Parallel Computers, Version 4.5., (2003). Pacific Northwest National Laboratory, Richland, Washington 99352-0999, USA. A modified version. F. A. Bulat, T. Heaton-Burgess, A. J. Cohen, and W. Yang, EPAPS Document No. E-JCPSA6-127-308742. (2007).

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Chapter 3 The Single-Particle Kinetic Energy of Many-Fermion Systems: Transcending the Thomas-Fermi plus Von Weizs¨ acker Method G. G. N. Angilella1 and N. H. March2 1

Dipartimento di Fisica e Astronomia, Universit` a di Catania and INFN Sez. Catania, and CNISM, Sez. Catania, 64 Via S. Sofia, I-95123 Catania, Italy and Scuola Superiore di Catania, Universit` a di Catania 5/i, Via S. Nullo, I-95123 Catania, Italy 2

Donostia International Physics Center (DIPC) E-20018 San Sebastian/Donostia, Spain and Oxford University, Oxford, UK and Department of Physics, University of Antwerp B-2020 Antwerp, Belgium

After a short introduction to the semiclassical Thomas-Fermi (TF) method, and the gradient correction to its kinetic energy term proposed by von Weizs¨ acker (vW), some attention is given to the differential form of the virial theorem. In one dimension, this allows the kinetic energy to be determined from the groundstate Fermion density ρ(r) generated by a given potential V (r). This is followed by a summary of the perturbative expansion of the Dirac density matrix γ(r, r0 ) in powers of the given three-dimensional potential V (r). This, following March and Murray, is shown, when r0 = r, to sum to the TF density-potential relation when V (r) varies only slowly in configuration space. A corresponding treatment of the kinetic energy density t(r) follows. Due to the interest in experiments in ultracold Fermion vapours, the case of harmonic confinement of independent Fermions is next summarized. In one dimension, the kinetic energy functional T [ρ] is known, and has as building blocks the TF form ρ3 (r) and the vW gradient correction term ρ02 (r)/ρ(r), even though T [ρ] is itself fully non-local in this example. For D dimensions and an arbitrary number of closed shells, the differential equation for the density is summarized, together with an explicit expression for the kinetic energy density as a functional of the Fermion ground-state density. The concept of the Pauli potential is then referred to, in relation to the functional density δT [ρ]/δρ(r) for a general potential V (r). As an example, the Pauli potential is given explicitly for the case of harmonically confined independent Fermions. A brief section notes some significant generalizations when the potential V (r) of density functional theory is replaced by the non-local generalization V (r, r0 ), the most important case of this latter form being Hartree-Fock theory.

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Contents 3.1 Background and outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Fermions in surface regimes: nuclei and simple liquid metals . . . . . . . . . . . . . . . 3.2.1 The nucleon surface density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.2 Brief background on surface energies . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Variational principle for the TF plus von Weizs¨ acker (TFvW) method . . . . . . . . . 3.4 Differential virial theorem and the Dirac density matrix . . . . . . . . . . . . . . . . . 3.4.1 Relation of the exact DVT to the semiclassical Thomas-Fermi method . . . . . 3.5 Perturbative expansion of Dirac density matrix γ(r, r0 ) in powers of the given one-body potential V (r) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5.1 Stoddart-March series for the kinetic energy density t(r) in three dimensions . . 3.6 Complete DFT for harmonically confined Fermions in D dimensions, for an arbitrary number of closed shells . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6.1 Current experimental focus on many Fermions that are harmonically confined . 3.6.2 Differential equation for Fermion density . . . . . . . . . . . . . . . . . . . . . 3.6.3 Kinetic energy density functional t[ρ] for arbitrary number of Fermions moving independently in one-dimensional harmonic oscillator potential . . . . . . . . . 3.6.4 Summary of complete DFT for many closed shells of Fermions which are (isotropically) harmonically confined in D dimensions . . . . . . . . . . . . . . . . . . . 3.7 The Pauli potential in relation to the functional derivative of the single-particle kinetic energy density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.7.1 Relation to the differential virial theorem . . . . . . . . . . . . . . . . . . . . . 3.7.2 Example of harmonic confinement . . . . . . . . . . . . . . . . . . . . . . . . . . 3.8 Non-local potential theory: V (r) 7→ V (r, r0 ) . . . . . . . . . . . . . . . . . . . . . . . . 3.8.1 Fine-tuning of Hartree-Fock (HF) density for spherical atoms like neon . . . . . 3.8.2 Scaling approach to obtain a correlated density ρ(r) from HF densities for Ne and Ar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.9 Summary and directions for future studies . . . . . . . . . . . . . . . . . . . . . . . . . A.1 Exact correlated kinetic energy related to Fermion density in model two-electron atom with harmonic confinement and arbitrary interparticle interaction . . . . . . . . . . . A.1.1 Correlated relative motion kinetic energy density, in terms of relative motion wave function ΨR (r) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.1.2 Some specific results for correlated kinetic energy of model two-electron ground states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.2 Transcending the von Weizs¨ acker single-particle kinetic energy in an artificial twoelectron atom . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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3.1. Background and outline Modern density functional theory (DFT) has its origins in the so-called ThomasFermi-Dirac (TFD) statistical theory, reviewed by Gomb´as1 and by March.2,3 Nowadays, the TFD method is viewed, in modern parlance, as a completely local density approximation. This means that not only are exchange (x) and correlation (c) approximated via the homogeneous electron liquid (jellium model — see e.g. Ref. 4), with its constant electron density, ρ0 say, replaced by the inhomogeneous density ρ(r) in the molecules, clusters or condensed phases under consideration, but also, and now involving a much more serious approximation, the single-particle (s) kinetic energy density functional, Ts [ρ] say, is written as3  2/3 Z 3h2 3 TFD 5/3 Ts [ρ] = ck ρ (r) dr, ck = . (3.1) 10m 8π

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Already, it was known to von Weizs¨acker5 that Eq. (3.1) needed to be corrected in systems with electron density ρ(r) varying strongly in space (e.g. the Fermions in the nuclear surface, or the electrons at the surface of a simple liquid metal: to be considered briefly below), by a term having the form Z ~2 (∇ρ(r))2 TsvW [ρ] = dr. (3.2) 8m ρ(r)

One major focus of this review is to assess the continuing usefulness of the forms (3.1) and (3.2) as building blocks in ongoing research to construct the correct singleparticle kinetic energy functional Ts [ρ]. As discussed at length, e.g. in Parr and Yang,6 in the usual DFT programmes currently employed in practical studies on molecules and condensed phases, Ts [ρ] is bypassed by going back to symmetrized Hartree-like orbitals (the so-called Slater-Kohn-Sham (SKS) orbitals 7,8 ), and calculating the single-particle kinetic energy by standard quantum-mechanics using wave functions determined from an appropriate one-body Schr¨odinger equation. Authors who use this wave function approach often refer to work on constructing Ts [ρ] as ‘orbital-free’ DFT. With this brief historical introduction, the outline of the present review is as follows. In a brief section 3.2, we shall expand on the relevance of Eqs. (3.1) and (3.2) for the treatment of (a) the Fermions in the nuclear surface, and (b) the electrons in a liquid metal surface. We shall also there take the opportunity of noting the limited accuracy of the Fermion density in both applications (a) and (b), when we use the Thomas-Fermi (TF) plus the von Weizs¨acker approximation to the singleparticle kinetic energy in a variational framework. Then in section 3.4 we shall introduce the so-called differential virial theorem derived, via the so-called Dirac density matrix (equivalent to an off-diagonal Fermion density), in one dimension by March and Young.9 This will be related to the TF result in one dimension. The restriction to one dimension will then be relaxed in section 3.5. There, results are recorded from an infinite order perturbation treatment in three dimensions, based on the free-particle Dirac density matrix10 as the unperturbed problem will be presented, the resulting perturbation series having been given to all orders in a one-body potential energy V (r) by March and Murray.11 As these authors then showed, this series sums to the TF density-potential relation when V (r) varies by but a small fraction of itself over a characteristic Fermion wavelength. Correspondingly, the single-particle kinetic energy, derived from the Dirac matrix by Stoddart and March12 to all orders in V (r), sums to the TF limiting local density approximation (LDA) when the TF ρ-V relation is used. Because of experimental interest on ultracold atomic gases of Fermions, a complete DFT of Ts [ρ] for harmonically confined Fermions is then set out in section 3.6, beginning for simplicity of presentation with the one-dimensional case. For the subsequent generalization to three dimensions, it is pointed out that the March-Murray series for ρ[V ] can be summed exactly for a harmonic confinement one-body potential V (r), as well as t[V ] for the kinetic energy density. Section 3.8 describes how the differential virial theorem

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1

ρ / ρ0

0.8 0.6 0.4 0.2 0 -6

-4

-2

0 kx

2

4

6

Fig. 3.1. Nucleon density close to the nuclear surface, within a semiempirical statistical model.13 Redrawn from Ref. 14.

(DVT), and the corresponding force-balance equation which follows directly from the DVT, can be derived for a three-dimensional Hamiltonian with given external potential and general inter-Fermion two-body interactions. Section 3.9 constitutes a summary plus some proposed directions for future research on Ts [ρ] which should prove fruitful. Some Appendices report on the correction of Ts [ρ] to embrace correlation kinetic energy, in solvable models. 3.2. Fermions in surface regimes: nuclei and simple liquid metals As anticipated, we report below the relevance of TF and von Weizs¨acker ‘building blocks’ to Ts [ρ] in discussing (a) Fermions in the nuclear surface region, and (b) simple liquid metal surfaces. 3.2.1. The nucleon surface density We appeal here to the early work of Berg and Wilets15,16 and to the review article by Wilets.14 In this latter study, Wilets records a semiempirical statistical model based on equal numbers of neutrons and protons and with neglect of Coulomb forces. Figure 3.1 has been redrawn from Ref. 12 and shows an independent particle model using a semi-infinite well with a sloping wall, following Swiatecki.13 The Fermi energy is taken to be 32.5 MeV and the wave number k appearing in the plot is 1.25 fm−1 . The solid and dashed curves of Fig. 3.1 reveal oscillations (the so-called Friedel oscillations) while the dotted curve shows the TF approximation. This TF method, being semiclassical, is, roughly speaking, akin to geometrical optics

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compared with wave optics, and so it fails to show any of the oscillatory (diffraction) effects revealed in the solid curve of Fig. 3.1. 3.2.2. Brief background on surface energies 3.2.2.1. Nucleon surface energies Berg and Wilets15,16 used the TF method plus a modified form of the von Weizs¨acker correction displayed in Eq. (3.2) to estimate surface energies of nuclei. Essentially, these authors wrote the ground-state energy E of the nucleus in the approximate form  Z  λ~2 (∇ρ)2 E= (ρ) + dr, (3.3) 8m ρ where (r) is a local, TF-like contribution, while the second term of the right-hand side is of the form of von Weizs¨acker, m being now the nucleon mass, reduced by the factor λ. Then in a semi-infinite nuclear model, the surface energy is shown by Berg and Wilets14–16 to have the form Z λ~2 ∞ 0 2 (u ) dx, (3.4) m −∞ with u = ρ1/2 , often referred to below as the density amplitude. 3.2.2.2. Application to a liquid metal planar surface Independently, Brown and March17 used a closely related method to discuss the surface energy σ of simple liquid metals, as well as to estimate the ‘thickness’, ` say, of the liquid-vapour interface. Their main result is that σ = κ−1 T `,

(3.5)

where κT is the isothermal compressibility of the bulk liquid metal. Returning to Eq. (3.4), the thickness ` involves the parameter λ, reducing the von Weizs¨acker ‘inhomogeneity’ kinetic energy Eq. (3.2) as well as the Fermi energy of the liquid metal. For further details reference may be made to the review article by Brown and March.18 3.3. Variational principle for the TF plus von Weizs¨ acker (TFvW) method The basic variational principle on which the TFvW method is founded takes the form δ(E − N µ) = 0

(3.6)

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where the variation δ is with respect to the density ρ(r). In turn the number of Fermions N is related to ρ(r) by Z N = ρ(r)dr. (3.7) For a given one-body potential energy V (r), this yields, on insertion of the kinetic energy given as the sum of Eqs. (3.1) and (3.2), plus the potential energy R ρ(r)V (r)dr, for the energy E in Eq. (3.6) µ=

5 δT vW ck ρ2/3 (r) + s + V (r). 3 δρ(r)

(3.8)

Although µ was introduced into the variational principle Eq. (3.6) as a Lagrange multiplier to take care of the density normalization Eq. (3.7), Eq. (3.6) is readily recognized as the statement that the chemical potential µ of the inhomogeneous Fermi assembly described by the TFvW method for a given one-body potential V (r) is the same at every point in the Fermion density distribution ρ(r).3 For a full generalization of Eq. (3.8) which removes the limitation to the TFvW method, the reader is referred to the book by Parr and Yang.6 The functional derivative δTsvW /δρ(r) is readily derived from Eq. (3.2) to insert in Eq. (3.8) but we need not give the details at this point. 3.4. Differential virial theorem and the Dirac density matrix At this point, we shall introduce the differential virial theorem as derived in one dimension by March and Young19 using the so-called equation of motion of the Dirac density matrix. Although the spirit of this review concerns what is now termed ‘orbital-free’ DFT, let us note that a specified potential V (r) generates the ground-state density ρ(r) from orbitals, say ψi (r), given by the Schr¨odinger equation ∇2 ψi (r) +

2m [i − V (r)]ψi (r) = 0. ~2

(3.9)

The ground-state density ρ(r) is then given by the sum of the squares of the lowest N normalized orbitals ψi (r) via ρ(r) =

N X

ψi (r)ψi∗ (r).

(3.10)

i=1

Dirac10 introduced the ‘off-diagonal’ density γ(r, r0 ) by writing γ(r, r0 ) =

N X

ψi (r)ψi∗ (r0 ),

(3.11)

i=1

which is evidently such that γ(r, r0 )|r0 =r = ρ(r) by comparison with Eq. (3.10). Employing the Schr¨ odinger equation (3.9), it is a quite straightforward matter to form

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the difference ∇2r γ(r, r0 ) − ∇2r0 γ(r, r0 ), which enables the one-electron eigenvalues i to be removed and is readily shown to be given by ∇2r γ(r, r0 ) − ∇2r0 γ(r, r0 ) =

2m [V (r) − V (r0 )]γ(r, r0 ). ~2

(3.12)

This is the equation of motion for the Dirac density matrix γ(r, r0 ) for electrons of mass m moving in a given one-body potential energy V (r). In early work, March and Young9 utilized Eq. (3.12) in one dimension to derive the so-called differential virial theorem (DVT), which will be a theme running through this present review. Writing the density matrix as γ(ξ, η), where ξ=

x + x0 , 2

η=

x − x0 , 2

(3.13)

reduces the one-dimensional form of Eq. (3.12) to read ∂2 2m γ(ξ, η) = 2 [V (ξ + η) − V (ξ − η)]γ(ξ, η). ∂ξ∂η ~

(3.14)

If we define the kinetic energy per unit length from the Schr¨odinger equation as −(~2 /2m)ψi d2 ψi /dx2 for the wave function ψi , then summing over i from 1 to N we obtain the result of March and Young9 that ∂t(x) 1 ∂V 1 ∂3ρ =− ρ − ∂x 2 ∂x 8 ∂x3

(3.15)

by expansion of Eq. (3.14) around the diagonal x0 = x corresponding to η = 0. In Eq. (3.15) t(x) is evidently given by N

t(x) = −

~2 X ∂2 ψi (x) 2 ψi∗ (x), 2m i=1 ∂x

(3.16)

which is readily rewritten in terms of the one-dimensional Dirac density matrix as ~2 ∂ 2 0 t(x) = − γ(x, x ) . (3.17) 2 2m ∂x x0 =x

While the focus of the present review is to obtain t(x) in terms of the ground-state density ρ(x) — an aim so far achieved completely only for specific forms of the onebody potential V (x) — it is to be emphasized from Eq. (3.17) that the ‘off-diagonal’ density γ(x, x0 ) is the natural tool to use in obtaining t(x). Next, let us very briefly explain why Eq. (3.15) is termed a ‘differential’ virial theorem (DVT). First, to form the so-called virial of Clausius, essentially r · F in three dimensions, where F is the force −∂V (r)/∂r, let us multiply Eq. (3.15) throughout by x and integrate from −∞ to ∞. Then integration by parts readily yields Z ∞ Z 1 ∞ T = t(x)dx = xF (x)ρ(x)dx, (3.18) 2 −∞ −∞

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or 2T = hxF (x)i.

(3.19)

With T evidently being the total kinetic energy and the brackets representing a quantum-mechanical average, we have regained the quantum version of the virial theorem of Clausius. Already, Eq. (3.15) is seen to allow, by quadrature with suitable physical boundary conditions, the kinetic energy t(x) to be determined directly, for N Fermions, from the given one-body potential V (x) plus the ground-state density ρ(x) which it generates. 3.4.1. Relation of the exact DVT to the semiclassical Thomas-Fermi method The semiclassical TF method is founded in Fermi statistics and is most appropriate when the number of particles N is large. Then it is easy to rewrite the chemical potential Eq. (3.8) in one dimension, and to obtain ρ2TF (x) + V (x) µTF = 3cD=1 k

(3.20)

if we neglect the von Weizs¨ acker term, as is increasingly appropriate as N becomes very large. Here, cD=1 is known from free Fermi gas theory, and corresponding to k Eq. (3.20) it is easy to show that tTF (x) = cD=1 ρ3TF (x), k

(3.21)

the general D-dimensional TF result being proportional to ρ1+2/D (r). To make contact with the exact DVT in Eq. (3.15), let us next differentiate Eq. (3.21) to find ∂tTF (x) ∂ρTF = 3cD=1 ρ2TF (x) . k ∂x ∂x

(3.22)

But from Eq. (3.20) we have ρ2TF = const × [µTF − V (x)]

(3.23)

and hence, since µTF is independent of x, 2ρTF

∂ρTF ∂V (x) = const × . ∂x ∂x

(3.24)

Substituting Eq. (3.24) into Eq. (3.22) we find almost immediately that ∂tTF (x) ∂V (x) = const × ρTF (x) ∂x ∂x

(3.25)

and the constant, from free Fermi gas theory, turns out to be − 21 . Thus, Eq. (3.25) relates to the exact DVT result Eq. (3.15) by neglect of ∂ 3 ρ/∂x3 . This neglect is due to the fact that the TF approximation uses free Fermi gas relations locally,

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and is therefore valid for slowly varying potentials V (x) and hence slowly varying densities ρ(x). We are therefore encouraged to summarize in the following section a perturbative treatment of the Dirac density matrix γ(r, r0 ) in three dimensions, following the early study of March and Murray,11 which removes the TF constraint of slowly varying potentials V (r). 3.5. Perturbative expansion of Dirac density matrix γ(r, r0 ) in powers of the given one-body potential V (r) March and Murray11 switched the potential V (r) on to a three-dimensional uniform Fermi gas with density ρ0 and Fermi wave number kF given by ρ0 =

kF3 . 3π 2

(3.26)

With plane waves exp(ik · r) as the orbitals, the free gas density matrix γ0 (r, r0 ) is readily calculated as19 γ0 (r, r0 ) =

kF3 j1 (kF |r − r0 |) , π 2 kF |r − r0 |

(3.27)

where j1 (x) = (sin x − x cos x)/x2 is the first-order spherical Bessel function. Evidently, when r0 = r, Eq. (3.27) reduces to Eq. (3.26). While γ(r, r0 ) − γ0 (r, r0 ) was given to all orders in V (r), we quote here only the O(V ) result, which reads11 Z V (r1 ) j1 (kF |r − r1 | + kF |r1 − r0 |) k2 dr1 +O(V 2 ). (3.28) γ(r, r0 )−γ0 (r, r0 ) = − F2 2π 2π |r − r1 | |r1 − r0 | By analogy with the TF statistical method, we shall focus below on the diagonal element ρ(r) − ρ0 = [γ(r, r0 ) − γ0 (r, r0 )]r0 =r , and the corresponding kinetic energy density calculated subsequently by Stoddart and March.12 Then we can write, following March and Murray,11 ρ(r) − ρ0 =

∞ X

ρj (r),

(3.29)

j=1

where ρj (r) is O(V j ) and is known explicitly in terms of kF , V (r) and the spherical Bessel function j1 (x) given above. What March and Murray11 proved was that in the limit when V (r) was slowly varying in space, Eq. (3.29) summed to precisely the TF density-potential relation. To date, however, Eq. (3.29) has only been summed exactly for one other case, namely when V (r) = 21 kr2 . This is however the important physical case of harmonic confinement, to be treated at some length in the following section. To summarize at this point, we can say that the perturbative series of March and Murray in Eq. (3.29) represents an exact quantum-mechanical generalization of the TF semiclassical method.

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3.5.1. Stoddart-March series for the kinetic energy density t(r) in three dimensions Since the Dirac density matrix γ(r, r0 ), as outlined above, is known exactly to all orders in a given one-body potential V (r), we can use the exact formula ~2 2 t(r) = − ∇r γ(r, r0 ) r0 =r (3.30) 2m to calculate the kinetic energy density difference t(r)−t0 corresponding to Eq. (3.29) for the particle density. The result obtained by Stoddart and March12 takes the remarkably simple form  ∞  X j t(r) − t0 = −V (r) ρj (r), (3.31) j+1 j=1 where ρj (r) enters the density Eq. (3.29). It seems natural enough then to rewrite Eq. (3.31) in the equivalent form ∞ ∞ X X 1 ρj (r) t(r) − t0 = −V (r) ρj (r) − V (r) j + 1 j=1 j=1 = −V (r)[ρ(r) − ρ0 ] − V (r)

∞ X j=1

1 ρj (r). j+1

(3.32)

Thus, as in the one-dimensional DVT in Eq. (3.15), both ρ(r) and V (r) enter. For the special case of harmonic confinement, the summation in Eq. (3.32) can be completed, and hence an exact generalization of the TF method for kinetic energy density can be achieved in this physically important example, to which we now turn. 3.6. Complete DFT for harmonically confined Fermions in D dimensions, for an arbitrary number of closed shells In this section, we will briefly review a complete DFT for independent harmonically confined Fermions in D dimensions. Let us begin by emphasizing the current experimental interest in this area. 3.6.1. Current experimental focus on many Fermions that are harmonically confined It is worth stressing at this point that many harmonically confined Fermions are currently of major interest because of experiments on ultracold vapours of 40 K and 6 Li isotopes populating hyperfine states inside magnetic traps.20 For these experiments based on axially symmetric magnetic traps, it proves possible to range from a fully spherical three-dimensional trap to a quasi-two-dimensional system. This motivated the treatment of harmonic confinement in D dimensions summarized below. It will be useful first of all to discuss, somewhat intuituively, the onedimensional case, going back to the work of Lawes and March.21

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3.6.2. Differential equation for Fermion density We shall consider more generally the functional derivative δT /δρ(x) of the kinetic energy T but intuitively (cf. the Thomas-Fermi method)  −1 δT ∂t ∂ρ ∼ . (3.33) δρ(x) ∂x ∂x Now, from Eq. (3.6) it follows that δT /δρ(x) = µ − V (x), and hence, employing Eq. (3.33), we find ∂t ∂ρ ∂ρ + V (x) =µ . ∂x ∂x ∂x

(3.34)

Substituting for ∂t/∂x from the differential virial theorem (DVT), Eq. (3.15), we then find 1 ∂V 1 ∂ρ ∂ρ − ρ − ρ000 (x) + V (x) =µ . (3.35) 2 ∂x 8 ∂x ∂x As Lawes and March proved,21 making use of solvable harmonic oscillator (HO) wave functions, Eq. (3.35) is exact if µ = N , the lowest nodeless wave function corresponding to N = 1. Thus, for arbitrary N , we no longer need the Schr¨odinger equation for harmonically confined Fermions in one dimension in order to calculate: (a) the particle density in the ground state, and (b) hence the kinetic energy per unit length t(x). However, it is instructive to get T [ρ] explicitly for harmonic confinement. 3.6.3. Kinetic energy density functional t[ρ] for arbitrary number of Fermions moving independently in one-dimensional harmonic oscillator potential By combining the DVT (arbitrary V ) and the Lawes-March differential equation (3.35), which is however only true for one-dimensional harmonic confinement, one can show22 that t(x) = tvW (x) + ξ(x)tTF (x),

(3.36)

where tvW (x) = and 4 ξ(x) = ξ(0) + 3

Z

0

x

"

1 ρ02 8 ρ

t0 (x) dx vW − tTF (x)

(3.37) 

tvW (x) tTF (x)

0 #

,

(3.38)

with tTF (x) = const × ρ3 (x) in 1D and tTF (r) = const × ρ1+2/D (r) in arbitrary dimensions D (LDA). Thus it is possible to construct the x-dependence of the exact t(x) solely from the Thomas-Fermi plus von Weizs¨acker kinetic energy ‘densities’

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(viz. energy per unit length, in one dimension) as building blocks. One can also get for the isotropic harmonic oscillator and M closed shells a complete DFT for arbitrary M . 3.6.4. Summary of complete DFT for many closed shells of Fermions which are (isotropically) harmonically confined in D dimensions First of all, the differential equation for M + 1 closed shells is23    ~2 ∂ 2 1 D ∂V ∇ ρ(r) + M + ~ω − V (r) ρ0 (r) + ρ(r) = 0, 8m ∂r 2 2 ∂r

(3.39)

where V (r) = 21 mω 2 r2 , and ∇ denotes the D-dimensional gradient. Secondly, the kinetic energy ‘density’ t(r) in D dimensions is t(r) =

D ρ1+2/D (r) ρ1+2/D (r) µ 2/D + D+2 ρ 4(D + 2) (0)

Z

0

r

1 ρ1+2/D

∂ 2 ~2 ∇2 ρ(r) ∇ ρ ds − , (3.40) ∂s 4m D + 2

where the chemical potential µ for M + 1 closed shells is given by   1 µ = M + (D + 1) ~ω. 2

(3.41)

Of course, Eq. (3.39) leads back to the Lawes-March21 one-dimensional equation, and Eq. (3.40) becomes the March et al.22 result for t(x) when D = 1. 3.7. The Pauli potential in relation to the functional derivative of the single-particle kinetic energy density Having ‘seeds’ in the early study of March and Murray,11 the background to the introduction of the Pauli potential, denoted below by VP (r), into DFT has been well reviewed by Levy and G¨ orling24 (see also Ref. 25). As noted by one of us,26,27 the problem of developing an adequate approximation to the single-particle kinetic energy functional Ts [ρ] can be viewed as equivalent to finding a useful approximation to the Pauli potential VP (r). In essence, one can regard the introduction of VP (r) as a ‘bosonization’ of a many-body potential V (r). Therefore, one writes a Schr¨odinger-like equation for the so-called ground-state density amplitude ρ1/2 (r), with V (r) modified by the addition of the Pauli potential VP (r). Thus one has ∇2r ρ1/2 (r) +

2m [I − V (r) − VP (r)]ρ1/2 (r) = 0, ~2

where I is the ionization potential.

(3.42)

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3.7.1. Relation to the differential virial theorem Below, we follow the recent study of March and Nagy28 in combining the Pauli potential theory with the differential virial theorem (DVT), already introduced in section 3.4 [see especially Eq. (3.15) for the general one-dimensional form]. We next invoke the three-dimensional form of the differential virial theorem derived by Holas and March.29 This now involves the kinetic energy density tensor tαβ (r) rather than just t(x) appearing in Eq. (3.15) of the one-dimensional case. Then the vector field zs (r) defined by Holas and March29 has components X ∂ (s) zs(α) (r) = 2 (3.43) t (r), ∂rβ αβ β

and the single-particle (s) DVT then corresponds in three dimensions to the force balance equation, which we write in spherical symmetry only, and which then reads −

ˆr · zs (r) ~2 ∂ 2 ∂V (r) =− ∇ ρ(r) + , ∂r 4mρ(r) ∂r ρ(r)

(3.44)

where ˆr denotes the unit radial vector, ˆr = r/r. March and Nagy28 have more recently shown that in the above case of spherical symmetry considered in Eq. (3.44), the final term could be replaced by a form involving the first derivative of the Pauli potential VP (r), namely   ˆr · zs (r) 4 tvW 1 ∂tvW = + + VP0 (r). (3.45) ρ(r) ρ(r) r 2 ∂r With reference to Eq. (3.45), Akbari et al.30,31 gave the first term on the right-hand side for single-level occupancy, the Pauli potential derivative VP0 (r) being derived by March and Nagy,28 for spherically symmetric ground-state densities n(r) and for arbitrary level occupancy. 3.7.2. Example of harmonic confinement As an explicit example of Eq. (3.45), let us utilize the differential equation for the ground-state Fermion density ρ(r) given by Howard et al.23 in D dimensions, as    ~2 ∂ 2 D+1 D ∂V ∇ ρ+ m+ ~ω − V (r) ρ0 (r) + ρ(r) = 0, (3.46) 8m ∂r 2 2 ∂r

for m + 1 closed shells. Here, the confining potential energy has the harmonic form V (r) = 12 mωr2 . Then it follows for D = 3 that VP0 (r) for that example is given by March32 as   ~2 ∂ 2 2 ρ0 (r) 4 tvW 1 ∂tvW 0 VP (r) = ∇ ρ(r) + [(m + 2)~ω − V (r)] − + 3mρ(r) ∂r 3 ρ(r) ρ(r) r 2 ∂r (3.47) It is a straightforward matter to confirm this Eq. (3.47) for m = 0 corresponding to single-shell occupancy (for which VP0 (r) = 0) by direct insertion of V (r) = 12 mω 2 r2 , and the corresponding density n(r) = n(0) exp(−βr2 ), where β = mω/~.

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As noted in March,32 it is relevant here to mention that Elliott et al.,33 in the course of work on the semiclassical origins of density functionals, conclude, and we quote them, ‘how much simpler the kinetic energy is as a functional of the potential than of the density.’ While this is true in the ‘formally exact’ TF theory reported in section 3.5.1, and especially Eq. (3.31), Eq. (3.32) makes the case for involving both ρ and V in the expression for the kinetic energy density t(r). And this viewpoint is supported by the one-dimensional result Eq. (3.15) of March and Young,9 which is quite general. Returning briefly to the Pauli potential VP (r), it was stressed especially by one of us26,27 that VP (r) can be written formally exactly as VP (r) =

δTs [ρ] δTvW [ρ] − . δρ(r) δρ(r)

(3.48)

Since TvW [ρ] is given quite generally by Eq. (3.2), the last term in Eq. (3.48) is exactly known as an explicit functional of ρ(r). Thus VP (r) is equivalent to knowledge of the functional derivative of the single-particle kinetic energy, which is the focus of this review. 3.8. Non-local potential theory: V (r) 7→ V (r, r0 ) Very briefly, we will stress here first two points: (i) The Hohenberg-Kohn theorem (applying to a local potential V (r)) can be generalized to non-local potentials V (r, r0 ) considered by Bethe,34 motivated by Brueckner’s work; (ii) For the twolevel Fermion problem, one can construct the Dirac density matrix explicitly for both V (r) and V (r, r0 ), using the density amplitude ρ1/2 (r) and phase θ(r)35 γ(r, r0 ) = ρ1/2 (r)ρ1/2 (r0 ) cos[θ(r) − θ(r0 )].

(3.49)

For local V (r), θ(r) is related to ρ(r) by the non-linear pendulum equation ∇2 θ(r) +

∇ρ(r) · ∇θ(r) + λ sin 2θ(r) = 0. ρ(r)

(3.50)

From Eq. (3.49), which holds for both local and non-local potentials, the singleparticle kinetic energy density t(r) is given by 1 2 t(r) = tvW (r) + ρ (∇θ(r)) 2 1 ~2 (∇ρ)2 2 = + ρ (∇θ(r)) . 8m ρ 2

(3.51)

But for non-local V , Eq. (3.50) has to be corrected by a Fock-operator–like piece. So even for the same ρ(r), t(r) is different, as expected.

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3.8.1. Fine-tuning of Hartree-Fock (HF) density for spherical atoms like neon It is, of course, implicit that the exact ground-state density ρ(r) is needed to obtain the single-particle kinetic energy Ts . But currently, this is not possible by DFT, as we do not know the exchange-correlation contribution Vxc (r) to the required one-body potential V (r). Therefore, Cordero, March, and Alonso (CMA,36 ) have formulated a procedure whereby, for light spherical atoms, fine-tuning of the HF-ground-state electron density can be achieved, which leads to ground-state densities for Be, Ne, Mg, and Ar matching quantum Monte Carlo (QMC) densities in quality. The motivation for the CMA study was the celebrated theoretical study of Møller and Plesset37 who added Coulomb correlation effects to the HF approximation perturbatively. They emphasized the accuracy of the HF ground-state density ρ(r) in atomic physics by demonstrating that this quantity was correct to second order in the difference between the correct nonrelativistic Hamiltonian and the Fock operator. The idea underlying the CMA study was to start out from an unconventional use of the HF method with nonintegral nuclear charge — say Z 0 — in order to (a) insert semiempirically the correct nonrelativistic ionization potential, and (b) scale the resulting density to satisfy Kato’s cusp condition38 for the electron density at the atomic nucleus. CMA then note, in the above context, that the numerically calculated HF density (i.e. avoiding basis sets entirely) for species with nuclear charge Z 0 and N electrons, ρHF (Z 0 , N, r), has the exponential decay at large distances r from the nucleus of the form p ρHF (Z 0 , N, r) ∼ exp(−2 2IK (Z 0 , N )r),

(3.52)

in atomic units, IK (Z 0 , N ) being the appropriate Koopmans ionization potential. This, in turn, is the one-electron HF eigenvalue associated with the highest occupied atomic orbital (HOAO). In contrast to Eq. (3.52), the exact exponential decay of the non-relativistic (NR) electron density ρNR (Z, N, r) is known to be39 p ρNR (Z, N, r) ∼ exp(−2 2INR (Z, N )r),

(3.53)

where INR (Z, N ) is now the NR ionization potential of the neutral atom with atomic number Z under consideration (now with N = Z). CMA describe an available method which allows INR to be found semiempirically. But since, for the light atoms considered by CMA, the corrections to the measured I are negligible, we refer the reader to the original work for further details.

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Ionization potentials in a.u. and fitted parameters for the atoms studied. After

Atom

Z

IK (Z, N )

Iexpt (Z, N )

INR (Z, N )

Z0

λ = Z/Z 0

IK (Z 0 , N )

Ne Ar Be Mg

10 18 4 12

0.850410 0.591017 0.309270 0.253053

0.792482 0.579155 0.342603 0.280994

0.794464 0.582173 0.344332 0.280740

9.9128 17.9802 4.1270 12.0960

1.0088 1.0011 0.96922 0.99207

0.780612 0.580883 0.366550 0.285245

3.8.2. Scaling approach to obtain a correlated density ρ(r) from HF densities for Ne and Ar If we scale the HF density ρHF (Z 0 , N, r) obtained numerically for neutral atoms with N equal to the true integer ataomic number Z (Z = 10, for the neon atom) by solving the HF equations with non-integral nuclear charge Z 0 , then CMA next scale this density using the norm-conserving Ansatz ρλ (Z 0 , N, r) = λ3 ρHF (Z 0 , N, λr),

(3.54)

where λ is a positive parameter near to unity. Then the asymptotic behaviour of the scaled density is evidently p ρλ (Z 0 , N, r) ∼ exp(−2 2IK (Z 0 , N )λr). (3.55) If Eq. (3.55) is required to decay in its long-range form just as the exact nonrelativistic density, that is ρλ (Z 0 , N, r) ∼ ρNR (Z, N, r), one must impose the condition p p IK (Z 0 , N )λ = INR (Z, N ). (3.56)

Furthermore, to satisfy Kato’s nuclear cusp condition, CMA show that Z 0 λ = Z.

(3.57)

This result (3.57) shows that the two parameters of the CMA model are not independent. We show in Table 3.1 for Ne and Ar, and also for comparison two other light spherical atoms, Be and Mg, the values of Z 0 and λ, plus relevant information on the ionization potentials introduced above. For Ne and Ar, Z 0 is slightly less than Z in each case, whereas for Be and Mg, Z 0 > Z, by about 0.1. CMA make an extensive comparison with r space moments of QMC ground-state densities, their results testifying to the high quality of the fine-tuned HF densities thereby obtained. In subsequent work, Amovilli et al.40 have extended the diagonal density ρ(r) proposed by CMA to an off-diagonal idempotent one-particle density matrix. Amovilli et al.40 show that the orthonormal used in this construction are then

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to be found from the solution of generalized HF equations. These authors report calculations on the Be isoelectronic series of atomic ions with Z from Li− to Ca16+ . 3.9. Summary and directions for future studies Emphasis is placed here on the way the forerunner of the currently used form of DFT, namely the Thomas-Fermi statistical method, can be generalized without recourse to Slater-Kohn-Sham orbitals. Such a generalization is shown to involve use, say in a one-dimensional problem of independent Fermions moving in a potential V (r), of the March-Young result for t(r) obtained by quadrature from Eq. (3.15), plus a differential equation allowing the ground-state density n(r) to be calculated from a given V (r) without recourse to one-body Schr¨odinger equations. To date, such a differential equation is only known for special forms of V (r): as for example in the case of harmonic confinement given in Eq. (3.35). As for future directions in one dimension, it was noted by March and Murray41 that the so-called Slater sum S(r, β), related to the ground-state electron density n(r, E) by Laplace transform41 Z ∞ S(r, β) = β n(r, E)e−βE dE, (3.58) 0

satisfies a partial differential equation which is known exactly for an arbitrary onebody potential V (r), as summarized by Howard and March.42 S(r, β) is quoted analytically in this article for harmonic confinement, and allows the differential Eq. (3.35) to be recovered (see also Ref. 43). In three dimensions, we have, to date, the formally exact generalizations of the Thomas-Fermi semiclassical method set out here in Eqs. (3.29) and (3.32). At the time of writing, however, the summation in Eq. (3.32) has only been completed for harmonic confinement, namely when V (r) = 21 kr2 . But for this case, with Fermions occupying an arbitrary number of closed shells, the kinetic energy density functional is known in D dimensions, and the forms is quoted in Eq. (3.40). For the future, further studies of the total kinetic energy directly from the density may be prompted by the ongoing numerical work of Howard and March44 prompted by Eq. (3.40). The reduction in weight of the von Weizs¨acker term except for D = 1 is stressed there. We also gave attention in Sec. 3.7 to the concept of the Pauli potential VP (r), which in turn is related to the functional derivative of the single-particle kinetic energy by Eq. (3.48), where the von Weizs¨acker term, of course, is explicitly known in terms of the Fermion density ρ(r). Stress is also placed on the relation of the Pauli potential to the exact differential virial theorem via Eq. (3.45), derived in Ref. 28. It will be of considerable interest for the future if another route can be found which allows the constraint of spherical symmetric ground-state densities in Eq. (3.45) to be relaxed, thereby allowing possible analytic progress on molecules and clusters by means of ‘orbital-free’ DFT.

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A.1. Exact correlated kinetic energy related to Fermion density in model two-electron atom with harmonic confinement and arbitrary interparticle interaction There has been considerable activity in a model two-electron atom with harmonic confinement and different interparticle interactions u(r12 ). This goes back to Kestner and Sinano˘ glu45 for the so-called Hookean atom with u(r12 ) = e2 /r12 as in helium itself, and to Moshinsky 46 for also harmonic u(r12 ). Subsequently, Crandall et al.47 solved for the spatial ground-state wave function Ψ(r1 , r2 ) for the inverse 2 square interaction λ/r12 . This model atom problem was subsequently solved by Holas, Howard and March (HHM)48 for arbitrary interparticle interaction u(r12 ) in that both the ground-state Fermion density ρ(r), and the first-order correlated density matrix γ(r, r0 ), the offdiagonal generalization of ρ(r), were obtained in terms of known functions plus the one-variable relative (R) motion wave function ΨR (r), which only required solution of a one-body radial Schr¨ odinger equation with an effective potential Veff (r) = Vext (r) + u(r),

(A.1)

where Vext (r) = 12 kr2 is the harmonic confinement contribution to Veff (r). Just this same input information was shown by HHM48 to determine γ(r, r0 ) by purely quadrature. A.1.1. Correlated relative motion kinetic energy density, in terms of relative motion wave function ΨR (r) HHM48 also derived an exact expression for the total kinetic energy T for arbitrary u(r12 ), in terms of the wave function ΨR (r). Their result had the physical interpretation that it was the sum of two ‘von Weizs¨acker-like’ contributions, one of these coming from the centre-of-mass motion, calculable once-for-all as it depended only on Vext (r) in Eq. (A.1), and the other involving the first derivative dΨR (r)/dr of the above relative motion wave function. The HHM result had the final form 2 Z  ~2 dΨR (r) 3 T = ~ω0 + dr, (A.2) 4 2mR dr where ω02 = k/m, with m the electron mass, k being the harmonic confinement introduced following Eq. (A.1). The relative motion mass mR entering Eq. (A.2) is obtained in Ref. 48 and is simply m/2. Subsequent work by March, Akbari and Rubio30,31 has pointed out that it is helpful next to extract from the HHM Eq. (A.2) a relative-motion kinetic energy density tR (r) defined by the positive-definite quantity  2 dΨR (r) ~2 tR (r) = . (A.3) 2mR dr

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Taking the square root of this equation, one can evidently write  1/2 dΨR (r) 2mR 1/2 =± tR (r), dr ~2

49

(A.4)

with physical solution ΨR (r) =



2mR ~2

1/2 Z

r

∞

1/2

tR (s) ds.

(A.5)

To achieve the desired result for the exact correlated kinetic energy T in Eq. (A.2), we next insert the wave function ΨR (r) into the HHM relation for the ground-state density ρ(r) for this model with arbitrary interaction u(r12 ), namely   2Z ∞  sinh(ry/ac ) 8 r 1 2 2 ρ(r) = √ exp − 2 , (A.6) dy y exp − y Ψ2R (ac y) π ac 4 (ry/ac ) 0 ac being the length associated with the centre-of-mass Gaussian wave function exp(−R2 /2a2c ), where 1/2  ~ . (A.7) ac = 2mω0 A.1.2. Some specific results for correlated kinetic energy of model two-electron ground states Evidently, by inserting Eq. (A.5) into Eq. (A.6), we find a functional relation between density ρ(r) and relative-motion kinetic energy density tR (r) of the form ρ(r) = ρ [r; [tR ]] .

(A.8)

Though, at the time of writing, no general proof has been given that a unique inversion of Eq. (A.8) formally exists, we anticipate on physical grounds that this will indeed be the case. Therefore we turn immediately to give results bearing on Eq. (A.8), as discussed by March et al.30,31 A.1.2.1. The Hookean atom with Vext = 18 r2 and u(r12 ) = e2 /r12 Of course, the long term aim is to generalize the model of this Appendix to apply to the important problem of the ground state of the helium atom itself. Then it is natural to take as the prime example of Eqs. (A.8) and (A.5) the Coulomb interparticle repulsion u(r12 ) as u(r12 ) = e2 /r12 .

(A.9)

From the study of Kais et al.,49 ΨR (r) is known to take the form, for the specific force constant k = 14 ,  2 r ΨR (r) = AR e−r /8 1 + , (A.10) 2

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0.006 0.005

tR (r)

0.004 0.003 0.002 0.001 0 0

1

2

3 r

4

5

6

Fig. A.1. Relative kinetic energy density tR (r) for the Hookean atom model, Eq. (A.12), in units where 2mR /~2 = 1. Redrawn from Ref. 30.

where the normalization factor AR is given by Z ∞ Ψ2R (r)4πr2 dr = 1.

(A.11)

0

From Eqs. (A.10) and (A.3) it follows readily that, for this Hookean atom example  2 h  2mR 1 r i −r 2 /4 AR e tR (r) = 1−r 1+ (A.12) ~2 2 2

and a plot of tR (r) is shown in Fig. A.1 which is redrawn from the study of March et al.30,31 The Fourier transform of the density ρ(r) entering the functional relation Eq. (A.8) is known in particular cases, as discussed in Ref. 30. For this case, ω02 = 14 with m = 1 in atomic units, and Eq. (A.2) for the exact correlated kinetic energy T becomes (with ~ = 1, ω0 = 12 ), T =

3 + TR . 8

(A.13)

The known value T = 0.664 au (readily calculated from the known wave function) yields from Eq. (A.13) that TR = 0.289 au.R As Kais et al. calculate,49 the von Weizs¨ acker inhomogeneity kinetic energy 18 ρ02 /ρ dr using ρ(r) in Ref. 30 gives TvW = 0.632 au and hence the ‘correlation’ kinetic energy defined here by T − TvW = 0.032 au. This is more than 10% of TR entering the exact total kinetic energy Eq. (A.13), the first term on the right-hand side being independent of the interaction u(r12 ). To date, we have not found a way of writing TR in terms of ρ(r) in as given in Ref. 30 for the Hookean atom.

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A.2. Transcending the von Weizs¨ acker single-particle kinetic energy in an artificial two-electron atom Recent work by G´ al and March50 has been concerned with the energy density functional of a specific artifical two-electron atom. This has again, as in section 3.6, harmonic confinement with external potentia l 12 kr2 , but the electron-electron interaction e2 /r12 , with r12 the separation of the two electrons, is replaced by the inverse square law repulsion u(r12 ) given by u(r12 ) =

λ 2 . r12

G´al and March50 construct the ground-state energy functional Z E[ρ] = F [ρ] + ρ(r)Vext (r)dr,

(A.14)

(A.15)

by displaying F [ρ] as a quite explicit functional of ρ and the interaction strength λ, but independent of Vext (r) = 12 kr2 . This situation draws on the study of Capuzzi et al.,51 in which a differential equation was derived for the groud-state density ρ(r) in the above model. This equation reads     ~ ~ 3 3 mω 2 rρ00 (r) + + r2 ρ0 (r) + r −α+2 r ρ(r) = 0, (A.16) 4mω 2mω 2 2 ~ where α = 21 [(1 + 4λm/~2 )1/2 − 1] is a measure of the repulsive coupling. Eq. (A.16) has a solution in terms of hypergeometric functions. The second important point is that the ground-state wave function ΨCWB (r1 , r2 ) and the corresponding energy FCWB are known in exact analytical form from the work of Crandall, Whitnell, and Bettega (CWB).47 Utilizing these explicit results, G´al and March50 find the form FCWB [ρ] for this model, as in Eq. (A.15), to be  Z  02 ~2 ρ (r) ρ0 (r)ρ00 (r) +r dr 16m ρ(r) ρ(r)   FCWB [ρ] = . (A.17) Z 1 ρ02 (r) 1− r2 − ρ(r) dr 3α + 8 ρ(r) Taking the limit λ = 0, also corresponding to α = 0, Eq. (A.17) must reduce to the single-particle von Weizs¨acker form TvW [ρ], Eq. (3.2). This can be verified explicitly using the well-known Gaussian density ρ(r) = ρ(0) exp(−βr2 ),

(A.18)

where β = mω/~, inserted into Eq. (A.17) in the limit α → 0. The exact analytic result Eq. (A.17) means that for this, admittedly simplistic, two-electron spin-compensated model, one of a very few forms of the functional F [ρ] in the general DFT Eq. (A.15) is now available through Eq. (A.17) given in Ref. 50.

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References 1. P. Gomb´ as, Die statistische Theorie des Atoms und ein anwendungen. (SpringerVerlag, Vienna, 1949). 2. N. H. March, Adv. Phys. 6, 1 (1957). 3. N. H. March, Self-Consistent Fields in Atoms. (Pergamon Press, Oxford, 1975). 4. M. P. Tosi. Many-body effects in jellium., In ed. N. H. March, Electron correlation in the solid state, chapter 1. (Imperial College Press, London, 1999). 5. C. F. von Weizs¨ acker, Z. Phys. 96, 431 (1935). 6. R. G. Parr and W. Yang, Density Functional Theory of Atoms and Molecules. (Oxford University Press, Oxford, 1989). 7. J. C. Slater, Phys. Rev. 81, 385 (1951). 8. W. Kohn and L. J. Sham, Phys. Rev. 140, A1133 (1965). 9. N. H. March and W. H. Young, Nucl. Phys. 12, 237, (1959). Reprinted in Ref. 52. 10. P. A. M. Dirac, Proc. Camb. Phil. Soc. 26, 376 (1930). 11. N. H. March and A. M. Murray, Proc. Roy. Soc. A. 261, 119 (1961). Reprinted in Ref. 52. See also Ref. 53. 12. J. C. Stoddart and N. H. March, Proc. Roy. Soc. A 299, 279 (1967). Reprinted in Ref. 52. 13. W. J. Swiatecki, Proc. Phys. Soc. (London) A 63, 1208 (1950). 14. L. Wilets, Rev. Mod. Phys. 30, 542 (1958). 15. R. A. Berg and L. Wilets, Phys. Rev. 101, 201 (1956). 16. R. A. Berg and L. Wilets, Proc. Phys. Soc. (London) A 68, 229 (1958). 17. R. C. Brown and N. H. March, J. Phys. C 6, L363 (1973). 18. R. C. Brown and N. H. March, Phys. Rep. 24, 77 (1976). 19. N. H. March, W. H. Young, and S. Sampanthar, The Many-Body Problem in Quantum Mechanics. (Dover, New York, 1995). 20. B. DeMarco and D. S. Jin, Science 285, 1703 (1999). 21. G. P. Lawes and N. H. March, J. Chem. Phys. 71, 1007 (1979). Reprinted in Ref. 52. 22. N. H. March, P. Senet, and V. E. Van Doren, Phys. Lett. A 270, 88 (2000). 23. I. A. Howard, N. H. March, and L. M. Nieto, Phys. Rev. A 66, 054501 (2002). 24. M. Levy and A. G¨ orling, Phil. Mag. B 69, 763 (1994). 25. C. Herring and M. Chopra, Phys. Rev. A 37, 31 (1988). 26. N. H. March, Phys. Lett. A 113, 66 (1985). Reprinted in Ref. 52. 27. N. H. March, Phys. Lett. A 113, 476 (1986). Reprinted in Ref. 52. ´ Nagy, Phys. Rev. A 78, 044501 (2008). 28. N. H. March and A. 29. A. Holas and N. H. March, Phys. Rev. A 51, 2040 (1995). Reprinted in Ref. 52. 30. A. Akbari, N. H. March, and A. Rubio, Phys. Rev. A 76, 032510 (2007). 31. A. Akbari, N. H. March, and A. Rubio. Exact density matrix theory for the ground state of spin-compensated harmonically confined two-electron model atoms with general interparticle repulsion, to be published (2009). 32. N. H. March, Phys. Lett. A 372, 6667 (2008). 33. P. Elliott, D. Lee, A. Cangi, and K. Burke, Phys. Rev. Lett. 100, 256406 (2008). 34. H. A. Bethe, Phys. Rev. 103, 1353 (1956). 35. K. A. Dawson and N. H. March, J. Chem. Phys. 81, 5850 (1984). Reprinted in Ref. 52. 36. N. A. Cordero, N. H. March, and J. A. Alonso, Phys. Rev. A 75, 052502 (2007). 37. C. Møller and M. S. Plesset, Phys. Rev. 46, 618 (1934). 38. T. Kato, Commun. Pure Appl. Math. 10, 151 (1957). 39. T. Hoffmann-Ostenhof, M. Hoffmann-Ostenhof, and R. Ahlrichs, Phys. Rev. A 18, 328 (1978).

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40. C. Amovilli, N. H. March, and J. D. Talman, Phys. Rev. A 77, 032503 (2008). 41. N. H. March and A. M. Murray, Phys. Rev. 120, 830 (1960). Reprinted in Ref. 52. 42. I. A. Howard and N. H. March, phys. stat. sol. (b). 237, 265, (2003). Reprinted in Ref. 52. 43. N. H. March and L. M. Nieto, Phys. Lett. A 373, 1691 (2009). 44. I. A. Howard and N. H. March, in the course of publication (2009). 45. N. R. Kestner and O. Sinano˘ glu, Phys. Rev. 128, 2687 (1962). 46. M. Moshinsky, Am. J. Phys. 36, 52 (1952). 47. R. Crandall, R. Whitnell, and R. Bettega, Am. J. Phys. 52, 438 (1984). 48. A. Holas, I. A. Howard, and N. H. March, Phys. Lett. A 310, 451 (2003). 49. S. Kais, D. R. Herschbach, and R. D. Levine, J. Chem. Phys. 91, 7791 (1989). 50. T. G´ al and N. H. March, J. Phys. B: At. Mol. Opt. Phys. 42, 025001 (2009). 51. P. Capuzzi, N. H. March, and M. P. Tosi, J. Phys. A 38, L439 (2005). 52. N. H. March and G. G. N. Angilella, Eds., Many-body Theory of Molecules, Clusters, and Condensed Phases. (World Scientific, Singapore, 2009). 53. N. H. March and A. M. Murray, Proc. Roy. Soc. A 256, 400 (1960). Reprinted in Ref. 52.

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Chapter 4 An Orbital Free ab initio Method: Applications to Liquid Metals and Clusters Andr´es Aguado1 , David J. Gonz´alez1 , Luis E. Gonz´alez1 , Jose M. L´opez1 , Sara N´ un ˜ez1 and Malcolm J. Stott2 1

Departamento de F´ısica Te´ orica, Universidad de Valladolid 47011 Valladolid, SPAIN 2 Department of Physics, Queen’s University Kingston, ON K7L 3N6, CANADA

An orbital free ab initio scheme for the simulation of materials is described and applications of the scheme to liquid metal systems, the solid-liquid metal interface and to clusters are presented. The scheme is based on density functional theory and first principles pseudopotentials. The electron system is described entirely in terms of the electron density - it is free of orbitals - and the main approximation is the use of an explicit but approximate density functional for the electron kinetic energy. In addition, the absence of electron orbitals means that the pseudopotential must be local. The scheme is compared and contrasted with the popular Kohn-Sham approach in which the kinetic energy is treated exactly and accurate first principles non-local pseudopotentials are used but, at great computational cost. The error in the calculated total energy is found to be acceptable for a class of systems which includes so-called simple metals and their alloys but is not restricted to these e.g. good results have been obtained for Si and Ga. For these systems the orbital free scheme allows the simulations of much larger systems than can be treated with the full Kohn-Sham approach and long molecular dynamics runs can be achieved. The scheme is useful for investigating the static and dynamic properties of liquid systems and clusters.

Contents 4.1 Introduction . . . . . . . . . . . . . . . . . . 4.2 Theory . . . . . . . . . . . . . . . . . . . . . 4.2.1 Kohn-Sham approach . . . . . . . . . . 4.2.2 Orbital free approach . . . . . . . . . . 4.2.3 Pseudopotentials . . . . . . . . . . . . 4.2.4 Future prospects . . . . . . . . . . . . 4.3 OF-AIMD simulation of liquid metal systems 4.3.1 Bulk liquid simple metals and alloys . 4.3.2 Liquid Mg . . . . . . . . . . . . . . . . 4.3.3 Liquid Ga . . . . . . . . . . . . . . . . 4.3.4 Liquid Si . . . . . . . . . . . . . . . . . 4.3.5 Liquid Ga-In . . . . . . . . . . . . . . 55

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4.4 OF-AIMD studies of the liquid–vapor interface in simple liquid metallic systems 4.4.1 Liquid Ga . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.2 Liquid In . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.3 Liquid Ga-In . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5 OF-AIMD studies of solid-liquid interfaces . . . . . . . . . . . . . . . . . . . . . . 4.5.1 Surface relaxation and its temperature variation. Al(110) and Mg(10¯ 10) . 4.5.2 Liquid Al on pinned solid Al . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5.3 Liquid Li on solid Ca . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6 OF-AIMD study of the melting-like transition in alkali clusters . . . . . . . . . . 4.6.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6.2 Analysis of the molecular dynamics . . . . . . . . . . . . . . . . . . . . . . 4.6.3 OF-AIMD simulations of melting in Na clusters . . . . . . . . . . . . . . . 4.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix A. The kinetic energy functional . . . . . . . . . . . . . . . . . . . . . . . . Appendix B. Position-dependent chemical potential . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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4.1. Introduction The study of the structural and dynamical properties of disordered systems such as liquids or clusters of atoms is greatly facilitated by the results of molecular dynamics (MD) simulations in which the arrangement and velocities of the atoms consistent with a given thermodynamical state and an assumed interatomic force law are followed in time. Unlike crystalline systems in which periodicity and the harmonic approximation form a good basis for studying the motion of the ions, liquids and clusters are generally disordered at the atomic level and the motion of the ions is complex. Inevitably, measured quantities for these systems involve an average over configurations and these can be constructed from the results of MD simulations, but in addition the simulations yield several dynamical magnitudes which are not accessible to experiment and which provide knowledge of the microscopic atomic arrangement and the way this evolves in time. The study of the static and dynamic properties of liquid metals has already produced much experimental and theoretical work.1–6 The experimental investigation of the structure and dynamical properties of liquid metals has mostly resorted to radiation diffraction and inelastic scattering techniques, respectively. Both X-ray diffraction (XD) and neutron diffraction (ND) have routinely been used in the determination of the average atomic arrangement, as described by the static structure factor, S(q). Information on the atomic dynamics is contained in the dynamic structure factor, S(q, ω), which can also be measured for which the main experimental technique has been inelastic neutron scattering (INS); however, the recent development of synchroton radiation facilities has enabled the possibility of using inelastic X-ray scattering (IXS) as well. Other bulk properties such as the compressibility, and the viscosity and diffusion coefficients can also be measured. On the theoretical side, the development of microscopic theories such as the memory function formalism or the kinetic theory, along with the realization that the decay of several time-dependent properties can be explained by the interplay of two differ-

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ent dynamical processes, has created a theoretical framework whose application to simple liquids has led to good qualitative results for several dynamical quantities.1–8 Small clusters of N atoms in the range N=10 or less up to a few hundreds are of technological interest because they hold the possibility of being components in the design of composite materials with special properties. But, they are of intrinsic interest because of what we can learn from them of the factors influencing the stability of atomic structures. Clusters of different sizes give systems with a range of surface to volume ratios and the role played by the larger proportion of surface atoms in the stability is of interest. The finite size of a cluster will lead to new quantum mechanical effects in the electronic structure as energy gaps appear in the electron energy spectrum. These factors influence the relative stability of clusters of different size.9,10 the abundance population in cluster beams of alkali metal atoms obtained by gas aggregation shows a spectrum with sharp peaks at some particular sizes N=2,8,20,40,.... followed by sharp drops in the population. The appearance of these so-called magic clusters was recognised as a consequence of an electronic shell structure in which the magic sizes have filled electronic shells. In addition to the influence of electronic shell closing on the cluster stability there are also geometrical effects with clusters with complete icosahedral shells of atoms being especially stable. The dynamics of the atoms in clusters manifests itself in the melting of the clusters. Calorimetry and photofragmentation experiments have yielded information on the way melting takes place and the size dependence of the melting temperature, Tm (N) which exhibits a pattern of maxima and minima that are not fully understood in terms of the electronic shells and atomic geometry.11–14 MD simulations have provided valuable insights into the dynamical effects which lead to the melting of clusters. Except for the lightest atoms the motion of the ions in MD is governed by Newtonian mechanics.15 A sample set of ions representing a liquid are placed in a cell and periodic boundary conditions are applied. The system is allowed to evolve in time at chosen thermodynamic conditions under the action of the forces acting on the ions. After sufficient time has elapsed for equilibrium to be established the positions and velocities of the ions are recorded at each time and used to compute average quantities of interest. Individual atomic configurations can be studied to trace the origin of particular features. The representation of the liquid by the simulation is limited by the number of atoms in the sample, which should be as large as possible to reduce the effects of the periodic boundary conditions, and by the time over which configurations are available for the calculation of dynamical quantities, which should be as long as possible. The size of the sample cell and the simulation time place lower limits respectively on the wave vectors and frequencies of phenomena relevant to the liquid. The simulation of a cluster often also applies periodic boundary conditions because of technical reasons related to the treatment of the electrons. If this is the case the cell containing the cluster must be sufficiently large to accommodate the cluster and enough vacuum to minimize the interaction

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between a cluster and its neighbouring images in the superlattice. The study of cluster melting also requires long simulation times so that the system samples the many isomeric possibilities. Dealing with these factors of the size of the sample and simulation time depend on the computing resources available and the patience of the practitioners. These issues aside, the accuracy with which the simulation represents the particular liquid system or cluster depends on the assumed interatomic forces. Early MD simulations were entirely classical using model potentials such as hard sphere or Lennard-Jones potentials. These classical MD studies gave useful insights into generic liquid and cluster behaviour. However, a simulation with the objective of investigating a particular system requires forces appropriate to the system. The valence electrons play a crucial role in the behaviour of metallic systems, forming a glue that holds together the ions against the Coulomb repulsion of the ionic charges. The embedded atom model (EAM) of Daw and Baskes16 incorporates the electron density in the interaction energy following the work of Stott and Zaremba17 and has been used successfully for metallic systems. However, the interaction is empirical and results can depend substantially on the way the parameters are chosen. In principle, the solution of the Schrodinger equation for the electrons provides the energy of the system and from this the forces on the ions, but there are many electrons to be considered and these are coupled one to another by their Coulomb repulsion as well as to the ions. The complexity posed by the electron-electron interactions can be handled by Density Functional Theory (DFT)18,19 which introduces the electron density distribution as a fundamental variable for the system, and shows how to replace the interacting many-body system of electrons by an auxiliary system of noninteracting particles moving in an external field, a much simpler system amenable to solution. DFT is the basis of present-day simulations. Pseudopotentials are the second important ingredient for the calculation of the forces on the ions including a treatment of the electrons.20 It is the valence electrons that respond to the arrangement of the ions and contribute to the forces on them. Many of the electrons, the so-called core electrons, are tightly bound in the ions and move with them, are insensitive to the surroundings and contribute negligibly to the forces. The core electrons may be removed and their effect incorporated in a pseudopotential which gives an accurate description of the valence electrons. The use of a pseudopotential has an additional practical benefit. The pseudowavefunctions that result are much smoother than their all-electron Kohn-Sham orbital counterparts and may be accurately represented by plane waves with great computational advantages. The approach to simulations based on the two pillars of DFT and pseudopotentials has become known as the ab initio method. If the electron-ion interaction is sufficiently weak the electrons respond linearly to the pseudopotential and the ions interact with one another through a pair potential21 which can be used very efficiently in MD simulations.15 Such a scheme was often used to investigate bulk liquid alkali and other simple metal systems. Although some effect of the electrons is incorporated in this effective ion-ion inter-

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action it is assumed that the electron distribution is weakly perturbed by the ions and the pair potential is rigid. This cannot be the case when there are substantial variations in electron density such as occur at a liquid-vapor interface or at the surface of a cluster where the electron density decays from its bulk value inside to zero outside. Similar problems with this perturbative approach must also occur at the interface between dissimilar metals and in many alloys. For example, in the homovalent LiNa liquid alloy the valence electron density will almost double in going from a Na rich to a Li rich region merely because of the atom size difference. A realistic simulation of these situations requires forces on the ions that take account of changes in the underlying electron density near them. The most common application of DFT in simulations uses the approach of Kohn and Sham19 (KS) who introduced the electron kinetic energy functional, Ts [n], for an auxiliary system of independent particles with the same density n(~r) as the physical electron system of interest. This step isolates all effects of the electronelectron interaction in a single term, the exchange-correlation energy functional, Exc [n], for which approximations have been developed that approach chemically useful accuracy. KS then showed how to evaluate Ts [n] exactly in terms of the single particle orbitals, φi (~r) for the auxiliary system which are self-consistent solutions of the Schrodinger equation with an effective external potential for which the prescription is given. This KS ab initio molecular dynamics (KS-AIMD) scheme by-passes the functional Ts [n] but at the cost of moving away from the electron density as the fundamental variable and working instead with a set of orbitals, one per electron. The most successful pseudopotentials have been developed for use in the KS-AIMD approach.22–24 They are fitted to the free atom, are transferable to the atom in other environments of interest, for example a liquid metal, but are nonlocal in the sense that different potentials act on different angular momentum components of the KS orbitals. The orbital free (OF) approach25–27 follows the KS development of DFT but retains the electron density as the fundamental variable. Because the KS orbitals are no longer needed, memory requirements scale linearly with system size as opposed to quadratically for the KS approach. There are also enormous savings in computational time because there is only the density distribution to manipulate rather than the whole set of orbitals, provided the Ts [n] is simple to evaluate. Costly orthogonalization of the orbitals and Brillouin zone sampling are no longer required. The savings in computational time and memory of the OF scheme allow for the simulation of much larger systems for longer simulation times than would otherwise be possible. However, an explicit form for Ts [n] as well as Exc [n] is required, and since only approximate forms for Ts [n] are known errors are made in the kinetic energy which can be serious because the kinetic energy is the same order of magnitude as the total energy whereas Exc is typically only 10% of the total. Another consequence of the freedom from orbitals is that the highly transferable nonlocal pseudopotentials cannot be used and a local potential is needed for the electron-ion interaction.

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An approximate form for Ts [n] and a prescription for constructing a local pseudopotential have been developed and applied in OF simulation studies of a variety of liquid metal systems and of clusters. Because of the approximations involved in Ts [n] the method is best suited to simple metal systems such as the alkalis, alkali-earths, Al, In, Tl and their mixtures which do not exhibit directional bonding, but applications to liquid Si which is weakly metallic and has some remnants of the bonding present in the solid have been rather successful. The theoretical foundation of the orbital free ab initio molecular dynamics (OF-AIMD) approach is presented in Sec. 4.2 with particular emphasis on the method we have developed and applied. Section 4.3 reviews applications of the method in simulations of bulk liquid metals and alloys with emphasis on the dynamical properties for which the method is well suited. The next section is devoted to applications of the OF-AIMD method to investigate the interface between a solid and a liquid metal. Applications to the dynamical properties of metal clusters follows in Sec. 4.6 where studies of the melting-like transition in alkali clusters is described. 4.2. Theory ~ l} The total energy of a system of N classical ions carrying charge Z at positions {R in a volume V , and interacting with Ne = N Z valence electrons, may be written within the Born-Oppenheimer approximation as ~ l }) = Tion + Vion−ion + Ee ({R ~ l }) E({R

(4.1)

where the first two terms are the ion kinetic energy and the direct Coulomb interaction energy, and Ee is the ground state energy of the electrons in the presence of the ions. The force on an ion isa X ∂Ee 3 F~i = Z 2 R~ij /Rij − (4.2) ~i ∂R j ~ i − R~j and the Hellman-Feynman theorem36 has been used. The where R~ij = R simulation then proceeds by solving Newton’s equations for the ions numerically using, for example, the Verlet algorithm and a suitable time step ∆tion .15 Density functional theory18,19,28,29 shows that the electronic energy can be written as a functional of the electron density, n(~r), as follows19 E[n(~r)] = Ts [n] + Eext [n] + EH [n] + Exc [n]

(4.3)

where Ts [n] is the kinetic energy of a non-interacting system of density n(~r), Eext is due to the interaction of the electrons with the ions, EH [n] is the classical electrostatic Hartree energy given by Z Z n(~r) n(~s) 1 EH [n] = d~r d~s , (4.4) 2 |~r − ~s| a Atomic

units are used throughout, unless stated otherwise, with ~ = e = me = 1

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and Exc [n] is the exchange–correlation energy, for which well tested approximations are available. The exchange-correlation energy has been the subject of a great deal of research30 because the only obstacles to an exact treatment of the electronic energy using the KS-AIMD method described in the next section lie in this term. Approximate forms of Exc [n] are of two types. The first are local density approximations (LDA) in which the electron gas is treated locally at ~r as though it were a uniform electron gas of mean density n(~r), and uses the results of many-body calculations of the uniform interacting gas. Two popular LDA approximations31,32 are parametrizations of the results of Monte Carlo treatments of the interacting electron gas of Ceperley33 and Ceperley and Alder.34 More sophisticated approximations involve density gradients as well as the density at ~r and are known as generalized gradient approximations (GGA); one in common use is due to Perdew et al.35 However, the simple LDA is usually sufficient in OF calculations because the errors associated with the approximate kinetic energy functional are expected to be much greater than those associated with Exc [n]. Superlattice geometry is usually adopted in which the N atoms of interest are located in a cell of volume V and replicated so that the whole system consists of a periodic superlattice.36 If the target of the simulation is a bulk crystalline system the supercell would consist of one or more unit cells of the material. In the case of a liquid, superlattice geometry would again be adopted with periodic boundary conditions applied as in the motion of the ions, but to minimize the effects of the artificial periodicity the sample N of atoms in the cell must be as large as possible. The simulation of an isolated cluster of N atoms would position the atoms in a cell large enough so that the cluster is surrounded by empty space and the interaction between a cluster and its periodic images is negligible. Similarly, for a liquid–vapor interface a slab of liquid would be positioned in a cell with empty space above and below so that interaction between a slab and its neighboring images can be neglected. The superlattice geometry is so often employed, even for systems which are patently not periodic, because Bloch wave methods for solving the electronic structure can be used and, in particular, the electronic structure can be represented by a discrete set of plane waves and manipulated with the highly efficient fast Fourier transform algorithm. 4.2.1. Kohn-Sham approach Kohn and Sham19 introduced the orbitals, φi (~r), of the system of independent particles having the same density as the interacting system of interest and moving in the effective field vef f . The orbitals satisfy the single particle equations 1 (− ∇2 + vef f [n, ~r])φi = i φi , 2

i = 1, 2, . . .

(4.5)

which must be solved self-consistently because the effective potential vef f depends explicitly on the density, which is obtained from the occupied orbitals,

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n(~r) =

X occ

|φi (~r)|2

(4.6)

The ground state kinetic energy, Ts , can be obtained exactly in terms of these orbitals   Z X 1 Ts = d~r φi (~r)∗ − ∇2 φi (~r) (4.7) 2 occ which leaves Exc [n] as the only small source of error in Eq. (4.3) for the electronic energy of the ground state. In addition, highly accurate and transferable nonlocal pseudopotentials may be used for the electron-ion interaction. In practice the KS equations are not solved directly, a time consuming process, instead the ground state electronic energy is found by minimizing iteratively the energy functional with respect to the orbitals while ensuring that the orbitals are mutually orthogonal.36 Orthogonalization is itself time consuming requiring order N 3 operations. Storing the order N orbitals and manipulating them place computational limits on the number of atoms that can be simulated and the length of the MD run. The most that can be done at present are about a few hundreds of atoms, which is a small sample of a bulk liquid, and for a few tens of picoseconds, which is a time too small to investigate dynamical effects. Similar limits also apply to the simulation of clusters. These limits will no doubt be surpassed in the future as computer technology advances, and strides are being made to develop DFT based procedures that scale as order N , but at present the OF-AIMD approach is the only choice for realistic simulations of liquid metals and large clusters. 4.2.2. Orbital free approach If an explicit density functional is adopted for Ts [n], the ground state energy can be obtained byR minimizing E[n] in Eq. (4.3) with respect to the density subject to the constraint d~r n(~r) = Ne which maintains the number of electrons as Ne . The choice of Ts [n] will be discussed later. Minimization could proceed by expanding n(~r) in a set of plane waves having the periodicity of the superlattice, as follows X

~

−iG·~ r nG ~ e

(4.8)

~ = 2π (n1 , n2 , n3 ) . G L

(4.9)

n(~r) =

~ G

where L is the side of the supercell assumed for simplicity to be a cube, and minimizing E[n] with respect to the nG ~ . The number of electrons can easily be held constant by keeping nG=0 = Ne /V . However, it is necessary physically to maintain ~ n(~r) ≥ 0 – if this is breached the iterative minimization becomes unstable – and this constraint is difficult to impose with this approach.

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Instead of using n(~r) as the system variable, we have had success using a single effective orbital ψ(~r) where n(~r) = ψ(~r)2 . If ψ(~r) is initially real it stays real throughout the iterations and n(~r) is guaranteed non-negative. Representing ψ(~r) in the plane waves ψ(~r) =

X

~

−iG·~ r cG ~ e

(4.10)

~ G

minimization can be performed with respect to the Fourier coefficients, cG ~ . The plane wave expansion is truncated at a maximum wave vector Gc corresponding to an energy cutoff Ec = G2c /2. Plane waves are a good basis set for the systems we have treated; they span the whole space, they are independent of the ion positions, and the completeness is controlled by the single parameter Ec which can be varied to test convergence. In contrast, constraining the number of electrons during minimization is harder to accomplish with ψ(~r) than when working directly with n(~r). The ground state energy for Ne electrons can be obtained by treating the coefficients cG ~ as dynamical variables and conducting a simulated anneal as first P 2 introduced by Car and Parrinello.37 A fictitious kinetic energy, |c˙G ~ | /2Mc is introduced and the equation of motion for the cG ~ is Mc c¨G ~ = −2

Z

~

d~r µ(~r)ψ(~r)eiG·~r + 2¯ µV cG ~

(4.11)

where µ(~r) = δE/δn(~r) and µ ¯ the chemical potential for which the system contains the required number of particles Ne . These equations are solved numerically using the Verlet leapfrog algorithm36,38 with an electronic timestep ∆tc . The velocities c˙G ~ are reduced at every timestep or so and the system cools until static equilibrium – the electronic ground state – is reached to within preset tolerances. However, the chemical potential ¯ is not known in advance but replacing µ ¯ in Eq. (4.11) by R µ R its stationary value d~ r µ(~ r )n(~ r )/ d~ r n(~ r ) at each timestep, and renormalizing the R density so that d~r n(~r) = Ne , gives good convergence to the ground state. 4.2.2.1. The kinetic energy functional The use of an explicit electron kinetic energy functional, Ts [n], is the main feature distinguishing the OF from the full KS scheme. The theorems of Hohenberg and Kohn show that such a functional exists but little is known about the form of the functional, and compared to the exchange–correlation energy functional, Exc [n], little work on Ts [n] has been done although the prospect of an accurate OF-AIMD scheme has stimulated interest and several approximate functionals have been proposed recently. Approximate forms have been based on the known forms of Ts [n] in a few limiting cases.

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4.2.2.2. Known limiting cases In the limit of a uniform noninteracting electron gas the kinetic energy is given by the Thomas–Fermi (TF) functional39–43 Z TTF = a0 d~r n(~r)5/3 (4.12) where a0 = 3(3π)2/3 /10, and this functional is appropriate for systems in which the electron density, n(~r) is slowly varying. Indeed, the TF functional is the leading term in an expansion of Ts [n] in density gradients. The gradient expansion was first developed by Kirzhnits44 who reported the second and fourth order corrections to the TF result. An error in the fourth order term was noted and corrected by Hodges45 whose method is easy to follow. Subsequently, the sixth order term has been obtained by Murphy.46 The calculation of these gradient corrections becomes increasingly complicated with the order and although corrections higher than the sixth can undoubtedly be calculated the effort may not be worthwhile. The functional may be written Z Ts [n] = d~r ts (~r) (4.13)

in terms of a kinetic energy density, and gradient corrections up to fourth order are 5/3 t(0) s = a0 n

(4.14)

|∇n|2 72n

(4.15)

t(2) s =

t(4) s

n1/3 = 540(3π 2 )2/3

"

∇2 n n

2

9 − 8



∇2 n n



∇n n

2

1 + 3



∇n n

4 #

(4.16)

A few points regarding the gradient expansion are worth noting. The fourth order (4) correction, ts , involves |∇n|2 and ∆n which are readily calculated, but not the fourth derivative. It is also positive everywhere. For an exponentially decaying density such as one would find in the outer region of an atom or molecule, or outside (6) the surface of a solid or a liquid, ts and higher order terms diverge exponentially. These results are for a three dimensional system, but curiously, for two dimensions all density gradient corrections of the Kirzhnits type to the TF functional vanish.47,48 This is not to say that TF is exact in two dimensions but rather that there are technical problems with the expansion. The von Weizs¨ acker functional49 Z 1 |∇n|2 d~r (4.17) TvW = 8 n p is exact for a one electron system for which ψ(~r) = n(~r) (or for two if the state is doubly occupied for spin) and is therefore correct in the outer regions of a localized

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system. It has the same form as the leading, second order, correction in the density gradient expansion only a factor of 9 larger. In addition, TvW is widely held to be correct in the limit of a rapidly varying density and this is supported by a result of linear response (LR) theory to which we now turn. A weakly perturbed uniform electron gas provides another limiting case for the kinetic energy functional.50 If the gas has density n ¯ and the LR of the system to the perturbing potential leads to a change in density δn(~r), then perturbation theory to second order in the potential gives for the kinetic energy 1X TLR = V n ¯ 5/3 + δ˜ n~q [χ(q, n ¯ )]−1 δ˜ n−~q (4.18) 2 q6=0 ~

where δ˜ n~q =

Z

d~rδn(~r) exp(i~q.~r)

and χ(~ q ) is the so-called Lindhard function given by   k¯F 1 − x2 1+x χ(~ q) = − 2 1 + ln | | , 2π 2x 1−x

(4.19)

x = q/2k¯F

(4.20)

where k¯F is the Fermi wavevector to the electron density n ¯ . The R corresponding (2) second order gradient correction d~r ts (~r) (Eq. 4.15), and TvW (Eq. 4.17) have the same form but differ by a factor 9. Jones and Young51 analysed the LR result for the kinetic energy and showed that the factor 1/72 is correct when the dominant Fourier components of the perturbing potential are at small wavevector ~q, that is when the potential is slowly varying. However, when the dominant Fourier components are at large ~ q and the potential is rapidly varying, the von Weizs¨acker term with its factor of 1/8 is correct. 4.2.2.3. Approximate functionals Several approximate kinetic energy functionals have been proposed which give the TF, von Weizs¨ acker and the LR results in the appropriate limiting conditions. The first of these was due to Wang and Teter52 who also investigated the possibility of including quadratic response. With Perrot’s work as the basis,53 Madden and coworkers54,55 have developed functionals which also correctly recover the TF, von Weizs¨ acker and LR limits54 and they too included the quadratic response.55 OF-AIMD simulations of simple metal systems using such functionals, have yielded promising results. Later, Carter and coworkers56 investigated these functionals, and proposed a linear combination of functionals as a suitable form for Ts . The evaluation of these functionals requires only order N operations and OF-AIMD simulations based on them are very efficient. Unfortunately, they have two undesirable features. They are not positive definite, so that minimization of the energy functional can lead to an unphysical negative kinetic energy. Secondly, the functionals that incorporate the weakly perturbed electron gas through the linear or higher order response

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functions must specify a reference uniform electron gas density, n ¯ . In the case of systems such as bulk liquid simple metals in which there are small variations in the electron density, the mean electron density of the system is an appropriate choice for n ¯ . However, the choice of a uniform reference system is problematic when there are more substantial variations in electron density such as may appear in some simple metals and alloys, for example when the components of an alloy have a large size and/or valence difference, and especially for the surface of a solid or a liquid where the electron density drops through the interface from the bulk value to zero in the vacuum. Approximate functionals of the GGA form have been proposed which include some features of electron gas LR but avoid the need for a uniform reference density. The first of these due the DePristo and Kress57,58 incorporated the correct small ~q and large ~ q forms of the LR function and fitted parameters in a Pade approximant to the total kinetic energy of all-electron Hartree-Fock rare gas atoms. Wang et al.59 investigated GGA’s more appropriate for smoother pseudodensities. Parameters were fit to give δTs /δn(~r) = −V (~r) + µ for a model pseudoatom. The gradient expansion through fourth order was found to give good kinetic energies and it was suggested that an interpolation between the gradient expansion including fourth order for slowly varying density and TvW when densities are rapidly varying might be a useful kinetic energy functional. Perdew and Constantin60 developed a functional that met these requirements and tested it for a variety of systems by evaluating it for an exact density and comparing the results with the corresponding kinetic energy. Their functional was reported to be typically a strong improvement over the gradient expansion. However, to be useful in OF-AIMD simulations not only Ts [n] but also δTs /δn(~r) must be accurate so that the variational principle can be used to obtain the ground state density n0 (~r) and hence Ts = Ts [n0 ], and the functional is untested in this regard. A functional of this type is attractive because is does not require a uniform reference density and might be more widely applicable than those that incorporate uniform electron gas response functions52–55 and are limited to systems in which the variations in electron density about the mean value are small. However, it cannot give any Fermi surface effects that would be expected for metallic system such as Friedel oscillations. Even so, tests should be performed for impurities in a gas and for the jellium surface to determine if some perhaps truncated or damped oscillatory behaviour in the density is present. Chac´ on, Alvarellos and Tarazona61 have developed a different type of kinetic energy functional, which employs an “averaged density” and recovers the TF, von Weizs¨ acker and LR limits. The functional has been investigated and generalized by Garc´ıa-Gonz´alez et al.62–64 These functionals have the merit of incorporating Fermi surface effects through the uniform gas response but do not require a fixed reference density, and they can be chosen to be positive definite, an essential requirement if they are to be used variationally. However, they are somewhat complicated to apply and more

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time consuming to evaluate because they require an additional level of computation in order to determine the averaged density. Consequently, they scale as order N2 which diminishes the advantage of the OF-AIMD approach over the full KS-AIMD method. In most of the applications of the OF-AIMD scheme to be described later, the simplified version of Gonz´ alez et al.27,65 of the averaged density approach of Chac´on, 61 Alvarellos and Tarazona and Garc´ıa-Gonz´alez et al.62–64 has been used, with the kinetic energy given by

Tβ [n] = ˜ r ) = (2k¯F )3 k(~

Ts = TvW [n] + Tβ [n]

(4.21)

3 10

Z

˜ r )2 d~r n(~r)5/3−2β k(~

(4.22)

Z

d~s k(~s)wβ (2k¯F |~r − ~s|)

(4.23)

k(~r) = (3π 2 )1/3 n(~r)β ,

(4.24)

where k¯F is the Fermi wavevector corresponding to a reference mean electron density n ¯ , and wβ (x) is a rigid weighting function, determined by requiring the correct recovery of the LR limit at the reference mean density and the T-F limit. The ˜ r ) appears as a convolution which for a advantage of the simplification is that k(~ plane wave basis can be performed rapidly by the usual fast Fourier transform techniques. This renders the functional order N in scaling. The functional is a generalization of one with β = 1/3, used earlier in a study of expanded liquid Cs.66 The details of the functional are given in Gonz´alez et al.27,65 and are reproduced in appendix A. Its main characteristics are as follows: (i) β is a real positive number whose maximum value still leading to a mathematically well behaved weight function is ≈ 0.6, (ii) the functional recovers the uniform and LR limits, and is positive definite (iii) when k¯F → 0 because the mean electron density vanishes, e.g. for a finite system, the von Weizs¨acker term is recovered if β = 4/9, whereas for other values of β, the limit is TvW + CTTF , (iv) for values of β > 0.5 it is expected that µ(~r)ψ(~r), which is the driving force for the dynamic minimization of the energy, remains finite even for very small electron densities n(~r). For systems in which there will be regions where the density will decay exponentially such as clusters or a liquid/vapor interface the von Weizs¨acker term should appear limit of the functional and a value of β as close as possible to 4/9 is called for. Unless otherwise stated the simulations described later used β = 0.51, which in the limit n ¯ → 0 gives C = 0.046 and guarantees, at least for the systems considered, that µ(~r)ψ(~r) remains finite and not too large everywhere so that the energy minimization can be achieved.

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4.2.3. Pseudopotentials Ab-initio simulations using the KS-AIMD approach invariably employ nonlocal pseudopotentials obtained by fitting to properties of the free atom.22,23 The nonlocality is a consequence of the pseudopotential acting on the pseudoatom stripped of the core electrons - the pseudoion - having different potentials acting on different angular momentum components of the valence electron orbitals. Vion = vlocal (~r) +

lmax X l,m

|l, m > δvl (r) < l, m|

(4.25)

where lmax is usually 1 or 2. The nonlocal part of the potential is time consuming to evaluate with a plane wave basis and in ab initio simulations a procedure due to Kleinman and Bylander67 is invariably used which leads to a factorization of the matrix elements and considerable time saving. The fitting procedure of the so-called “norm conserving” pseudopotentials guarantees that the pseudoion scatters electrons in the same way as the true ion not only at the valence energy levels but also over an energy range about them. These pseudopotentials work well for many elements, but for some e.g. oxygen and the 3d transition elements, the ion core is small, and extensive and therefore costly plane wave basis sets are necessary. For these and other cases Vanderbilt’s ultrasoft pseudopotential24 is often used. Vanderbilt enlarged the ion core leading to a softer pseudopotential, but at the cost of sacrificing the norm conservation which required some adjustment to the charge density. Since the OF scheme is free of orbitals the ion-electron interaction energy has the form Z Ee−ion = d~r V (~r)n(~r) (4.26) where n(~r) is the electron pseudodensity, X ~ i) V (~r) = vi (~r − R

(4.27)

i

and vi (~r) is the electron-ion interaction for ion i. This form would seem to exclude the possibility of using nonlocal pseudopotentials like eq. (4.25) in OF-AIMD simulations and a local pseudopotential must be selected. But, this question is worth further study. 4.2.3.1. Local pseudopotentials There are two types of local pseudopotentials. Most are empirical and often have some simple analytic form; a few are from first principles. Early OF calculations25,54,55,66,68 used empirical local pseudopotentials. However, great care must be taken in using empirical potentials outside the set of data used in their fitting and the degree of confidence to be placed in the results is unknown. A successful

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application of an empirical pseudopotential in an OF scheme is the work of Aguado and L´ opez in their studies of the melting of small metal clusters.69–72 The two parameter analytic local ion pseudopotential of Fiolhais et al.73,74 was assumed and the parameters were chosen so that the OF-AIMD results reproduced KS-AIMD results for the interatomic forces and relative energies of a few selected cluster configurations. For a wide size range of Na clusters average forces and energy differences were reproduced very well. The potentials thus fitted and used within the OF-AIMD scheme led to an explanation of the anomalous size dependence of the melting temperatures of Na clusters. This and related applications of the OF-AIMD scheme to metal clusters is the subject of Sec. 4.6. Ideally, a local pseudopotential would be used for the electron-ion interaction which is fitted to give the suitably pseudized free atom electron density and designed to be transferable from the atom in free space to the atom in other electronic environments of interest. Wang and Stott obtained first principles local pseudopotentials for group IV elements based on the free atoms.75 The electron density of the free pseudoatom used to construct the nonlocal pseudopotential in, for instance, the Bachelet-Hamman-Schluter22 or the Troullier-Martins23 schemes can of course be constructed from the nonlocal potential, but DFT tells us that there is a unique local pseudopotential that yields the same density. The potentials constructed in this way may be regarded as the local equivalents of the nonlocal pseudopotentials. Wang and Stott found such local potentials for C, Si and Ge and used them in full KS calculations of dimers and crystalline solids. Fairly good results were obtained for bond lengths and lattice constants while the results for energies compared less well with the results obtained using the original nonlocal potentials. In the absence of a transferable local pseudopotential based on the free atom, an alternative is to base the pseudopotential on a reference system in which the environment of the atom is similar to that for the physical systems of interest. Watson et al.76 used a bulk crystal as a reference system. KS calculations were performed for a popular nonlocal pseudopotential and the Fourier components of the electron density were obtained. The Fourier components of the local KS effective potential which produced this density using the OF functional were found and “unscreened” to yield the local ion pseudopotential. Some interpolation was necessary in order to obtain the potential at all q-values, not just those at the reciprocal lattice vectors of the reference crystal. Application of these pseudopotentials using the OF scheme gave good agreement for a variety of crystalline properties with the results of full KS calculations with the original nonlocal pseudopotential for Na, Li and Al. Applications of the OF-AIMD method to liquid metal systems presented in Sec. 4.3 use a first principles local pseudopotential similarly constructed from the electron density obtained from a full KS calculation treatment of a liquid-like reference system.27 The reference system has the atom at the centre of a spherical cavity in the positive background of a uniform electron gas having density equal to the mean valence electron density of the system of interest. The radius of the

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cavity is such that the positive charge removed to make the hole is equal to the valence charge of the atom, and a so-called neutral pseudoatom77,78 is formed. This reference system represents the situation of an atom in a liquid metal when its surroundings are smeared out to form a uniform gas. The first step in the construction is an all-electron KS density functional calculation to obtain the displaced valence electron density, nat (r), i.e. the change in the electron density induced by the atom and the cavity. Next, nat (r) is pseudized by removing the contribution from core electrons and eliminating the core-orthogonality oscillations to give nps (r). Finally, an effective local ion pseudopotential is found using the OF density functional which, when inserted into the uniform electron gas along with the cavity, reproduces the pseudized displaced electron density. The development proceeds as follows. The minimization of the energy functional gives the Euler-Lagrange equation for the pseudopotential in the jellium-vacancy system: µs (r) + Vext (r) + VH (r) + Vxc (r) − µ = 0 ,

(4.28)

where each of the terms is the derivative of the corresponding term in Eq. (4.3), namely, µs (r) = µvW (r) + µβ (r) ,

(4.29)

with the expressions for the von-Weizs¨acker term and the β-term given in appendix B, Vext (r) = vps (r) + vcav (r) + vjell (r) , VH (r) =

Z

d~sn(s)/|~r − ~s| ,

(4.30) (4.31)

with n(r) = n ¯ + nps (r). Given n(r), vps (r) can be obtained from Eq. (4.28), and the constant µ is an energy origin set to give a pseudopotential that decays to zero at large distances. 4.2.4. Future prospects The OF-AIMD scheme based on this neutral pseudo-atom pseudopotential and the order-N averaged density electron kinetic energy functional, Eq. (4.22), gives a good account of static and dynamic properties of several liquid simple metals and alloys including the alkalis, alkali-earths, Al, In, Tl and some systems that would not normally be regarded as simple, such as Si and Ga. However, a significant unsatisfactory feature of this scheme is the dependence of both the kinetic energy functional and the pseudopotential that is adopted on some uniform electron gas reference system. For example, in their OF-AIMD study of the pressure induced structural and dynamical changes in liquid Si, Delisle et al.79 were required to use different ion pseudopotentials for the different thermodynamic states, each based on the reference gas density appropriate for that state. The effects of the reference system on

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the kinetic functional and the pseudopotential are mingled because the unscreening of the effective potential involves the Ts [n]. A kinetic energy functional which takes some account of the mainstay LR limit but which avoids a uniform reference system is clearly desirable. The GGA form proposed by Perdew and Constantin60 might meet this objective, but it awaits thorough testing and application. A modification of the order-N averaged density electron kinetic energy functional in Eq. (4.22) to render it insensitive to the choice of reference density is also a possibility. Some progress can be made in the choice of neutral pseudo-atom pseudopotential which makes it insensitive to the reference density and improves its transferability. The norm conserving nonlocal pseudopotentials22,23 are transferable because the conservation of the norm for each partial wave guarantees that the pseudopotential scatters the same as the all-electron potential in an energy range about the valence energy eigenvalue as well as at the energy. A corresponding requirement on the local pseudopotential for use in the OF scheme and the all-electron potential can be imposed. The energy dependence of each angular momentum component of the scattering cross-section is replaced in the OF scheme by the dependence of the pseudopotential on the chemical potential or mean electron density of the reference system. If ∆Ez (¯ n) is the cost in energy to insert the neutral atom with valence Z into the cavity of radius R in the reference system with mean density n ¯ then Z d∆Ez (¯ n) 0 (4.32) = − d~rφ (~r) d¯ n n ¯0 where φ0 (r) is related to the Hartree potential φ(r) of the all-electron atom in the gas. Specifically  φ(R) , r < R φ0 (r) = φ(r) , r > R

For cavities that fully enclose the atom core, the all-electron atom and the pseudized atom will have the same φ0 (r), and so there is the possibility of shaping n) z (¯ the cavity to modify φ0 (r) and reduce the magnitude of d∆E |n¯ 0 or ideally have it d¯ n vanish. Presumably, a pseudopotential generated in this way would be transferable and give energies that are insensitive to the reference density. 4.3. OF-AIMD simulation of liquid metal systems This section surveys our application of the OF-AIMD method to simple (sp bonded) liquid metallic systems. The focus has been on the simple metals because the limitations posed by the approximate kinetic functional and by the local pseudopotential are least severe for these systems. The kinetic energy functional used in these applications is the simplified version discussed earlier of the averaged density approach details of which are given in appendix A, and the local pseudopotential is our adaptation of the neutral pseudoatom.77,78 However, the efficiency the OF-AIMD

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scheme allows the treatment of large samples for long simulation runs and realistic treatments of the dynamics of bulk liquid metals and alloys can be performed. This is the first subject of this section. The description of OF-AIMD simulations of the liquid-vapor interface of several liquid simple metals and alloys completes the section. 4.3.1. Bulk liquid simple metals and alloys Research into the dynamical properties of liquid metals has produced a considerable amount of both experimental and theoretical work.1 Comparatively, less attention has been devoted to the dynamics of liquid binary alloys, although the last twenty years have witnessed an increasing effort in this direction. Following a classical MD study of liquid Na-K,80 the field was stimulated by the results of a MD simulation of liquid Li4 Pb by Jacucci et al.,81 where a new, high-frequency mode, supported by the Li atoms only (the so-called “fast sound”) was found. Subsequently, several theoretical,82–86 computer simulations87–89 and experimental88,90–92 studies have investigated the existence and properties of the collective excitations in liquid binary systems. During the last ten years, the OF-AIMD method has been extensively applied to describe a wide range of static, dynamic and electronic properties in simple liquid metals and binary alloys.27,65,79,93–97 Applications of the competing ab initio method, the KS-AIMD approach, to these systems has been restricted almost entirely to the study of the static properties. Even the calculation of the static properties in liquid binary alloys with moderate ordering tendencies poses enormous computational challenges to the KS-AIMD approach because a proper description of the local ordering requires simulations with a few thousand particles. Therefore, most of the ab initio studies performed on the static properties in liquid binary alloys have been done using the OF-AIMD method. The study of dynamical, time-dependent properties usually requires simulations lasting long enough so that the property of interest can be obtained for that time interval in which it takes non-negligible values. This condition implies that several thousand, even tens of thousand, configurations must be generated during the simulation run. Within the ab initio methods, this is a task which can be routinely carried out at present by the OF-AIMD method. Even though some KS-AIMD calculations of dynamic properties have been performed, they ultimately rely on the use of theoretical models for tackling the problems of noise and limited correlation time, both related to the short simulation times affordable. In contrast, OF-AIMD calculations of several dynamic properties, e.g. intermediate scattering functions, dynamic structure factors, current correlation functions etc, have already been performed for several simple liquid metals and binary alloys. This section showcases the capability of the OF-AIMD simulation method for treating the structural and dynamical properties of simple liquid metals and alloys. A survey of the correlation functions and other quantities which are helpful in the

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interpretation of the dynamics is given next. These are quantities that can be obtained from the MD simulations, some of which are accessible to experiment. Results are then presented for liquid Mg, Ga and Si and the Ga-In liquid binary alloy. These systems have been chosen because experimental data for static and/or dynamic structure and some transport coefficients are available. Moreover, l-Ga and l-Si are a challenging test of the approach because remnants of covalent bonding are believed to persist into the liquid phase. The OF-AIMD method yields valence electronic charge densities closely resembling covalent bonding. 4.3.1.1. Dynamic properties of liquids The basic quantities for describing the dynamic properties of a fluid system are the time correlation functions of several dynamic variables of the system. Relevant dynamical variables are defined first and then their correlation functions will be considered. The self number density of particle j is defined as ~ j (t)) , ρs;j (~r, t) = δ(~r − R

(4.33)

~ j (t) denotes the position of particle j at time t. Its associated space Fourier where R transform (FT) will be denoted as ρs;j (~q, t). The number density of particles of a given type, is defined as the sum of all the self number densities of particles of that type. We will assume throughout that we are dealing with one-component systems; the generalization to multicomponent mixtures is straightforward. The number density for a one-component system is X ~ j (t)) , ρ(~r, t) = δ(~r − R (4.34) j

and its space FT will be denoted ρ(~q, t). The current density of particles of a given type is also an important dynamic variable, defined as X ~j (t) δ(~r − R ~ j (t)) , ~j(~r, t) = V (4.35) j

~j (t) is the velocity of particle j at time t. The particle’s velocity itself is also where V an important dynamic variable which will be considered later. The corresponding space FT, ~j(~ q , t), is usually decomposed into a longitudinal (parallel to ~q) component ~jl (~ q , t) and a transverse (perpendicular to ~q) component ~jt (~q, t), with ~j(~q, t) = ~jl (q, t) + ~jt (q, t). Other dynamic variables related to the dynamical properties of a system e.g. the stress tensor, the energy current density, etc., will not be discussed here, instead, the reader is referred to the many books and reviews on the subject, for instance Refs. 1,15.

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The time correlation functions between dynamic variables are the basic quantities describing the dynamic properties. Particularly important are the time autocorrelation functions of the dynamic variables introduced earlier. The time autocorrelation function associated with ρs;j (~q, t) is the self intermediate scattering function Fs (q, t) = h ρs;j (~q, t + t0 ) ρs;j (−~q, t0 ) i ,

(4.36)

where, for homogeneous and isotropic bulk systems, the average h· · · i is carried out over time origins t0 , particles of the same type j, and directions of wave vector ~q. The FT to real space of Fs (q, t) is the van Hove self function Gs (r, t), which is related to the probability that a given particle has moved a distance r in a time interval t. The FT of Fs (q, t) into the frequency domain is the self dynamic structure factor Ss (q, ω), which is directly related to the incoherent intensity obtained in an INS experiment and so is accessible experimentally. The intermediate scattering function is defined to be F (q, t) =

1 h ρ(~q, t + t0 ) ρ(−~q, t0 ) i , N

(4.37)

where N is the number of particles, and the average is now over time origins t0 and directions of ~ q . This quantity contains information on the collective dynamics of density fluctuations over the length and time scales. For multicomponent mixtures p the factor 1/N is replaced by 1/ Nα Nβ in the definition of Fαβ (q, t), where α and β run from 1 to the number of different species and Nα is the number of particles of species α. The FT to real space of F (q, t) is the van Hove function G(r, t), which is related to the probability that if one particle is at the origin at a certain time, then any particle (the same or a different one) will be at a distance r from the origin after a time interval t. The FT into the frequency domain of F (q, t) is the dynamic structure factor S(q, ω), which is also experimentally accessible, since it is directly related to the coherent intensity obtained in an INS or IXS experiment. Finally we note that F (q, t = 0) = S(q) is the static structure factor. The longitudinal and transverse current correlation functions are defined to be E 1 D~ Cl (q, t) = jl (~q, t + t0 ) · ~jl (−~q, t0 ) (4.38) N Ct (q, t) =

E 1 D~ jt (~q, t + t0 ) · ~jt (−~q, t0 ) . 2N

(4.39)

The former is directly related to the intermediate scattering function, or, in the frequency domain, to the dynamic structure factor Cl (q, t) = −

1 d2 F (q, t) , q2 dt2

Cl (q, ω) =

ω2 S(q, ω) , q2

(4.40)

and can therefore be measured experimentally. However, the transverse current correlation function, is not accessible by experiment and can only be determined by

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simulation. Its shape evolves from a Gaussian, in both q and t, in the free particle limit (q → ∞), to a Gaussian in q for the hydrodynamic limit where q → 0 with Ct (q → 0, t) = (1/βm) exp(−q 2 η | t | /M ρ) where η is the shear viscosity coefficient, M is the ionic mass and ρ is the mean ionic number density. For intermediate qvalues Ct (q, t) has more complicated behavior. Another correlation function, not directly accessible to experiment is the velocity autocorrelation function, Z(t), and the closely related mean square displacement, δr2 (t), of a given particle in the system D E ~j (t + t0 ) · V ~j (t0 ) Z(t) = V (4.41)  2  ~ ~ j (t0 ) Rj (t + t0 ) − R Z t =2 dτ (t − τ )Z(τ ) ,

δr2 (t) =

(4.42)

0

where, for a homogeneous bulk system the average is over time origins and particles of the same type. Both functions are related to the diffusion coefficient, D, which is a measurable quantity, either as the time integral of Z(t) or as 1/6 the slope of δr2 (t) at long times. These definitions are easily generalized to multi-component mixtures. For example, for a binary mixture there appear the partial intermediate scattering functions, Fαβ (q, t) whose FT into the frequency domain give the partial dynamic structure factors, Sαβ (q, ω). Fαβ (q, t = 0) = Sαβ (q) are the partial static structure factors. The total static structure factor, ST (q), the quantity usually measured by neutron diffraction, is a weighted combination of the Sαβ (q). 4.3.2. Liquid Mg Starting with the pioneering work of Madden and coworkers,54 the OF-AIMD method has been used to calculate static, dynamic and electronic properties in simple liquid metals. Here we report the results for l-Mg because the sort of agreement with experiment is typical of that obtained for other liquid simple metals. Mg is the simplest alkali-earth metal, but the group has not attracted much experimental work because of technical difficulties related to the high chemical reactivity and to gas adsortion.98 Only a few properties have been measured, e.g. the static structure,99,100 density,101 sound velocity,102 electrical resistivity,103 and thermopower.104 Consequently, in comparison with the alkalis or some polyvalent metals theoretical work has been scarce. However, recent experiments on bulk liquid Mg (l-Mg) have measured the static100 and dynamic105 structure factors. Liquid Mg near its triple point has been studied by the OF-AIMD method.93,97,106 Here we report the recent simulations of Sengul et al.,97 with T =953 K and ρ = 0.03829 ˚ A−3 , and using 2000 particles, a timestep of 2.5 fs and

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a total averaging time of 50 ps after equilibration. The cutoff energy was 13 Rydbergs and the number of planewaves was around 145000. In comparison, the only KS-AIMD simulation performed thus far for this system107,108 used 90 particles, a non-local pseudopotential, a total averaging time of 1.8 ps, a cutoff energy of 12 Rydbergs resulting in 500000 planewaves. Moreover, only some static properties were computed – dynamic ones would have required substantially longer simulation times. This highlights the important computational advantages of the OF-AIMD scheme, specially for the evaluation of time dependent quantities. 3

2

Sie(q)

1

S(q)

0.5 3

4

5

0

1 -0.2

0

2

-1

q (Å )

4

6

Fig. 4.1. S(q) of l-Mg at 953 K. Open circles: experimental XD data from Waseda.99 Triangles and squares: experimental XD and ND data at 973 K from Tahara et al.100 Continuous line: OF-AIMD simulations. The inset shows the second peak region, where Waseda’s XD (Tahara’s ND) data have been displaced downwards (upwards) by 0.2 units.

0

2

4

-1

q (Å )

6

Fig. 4.2. Sie (q) of l-Mg at 953 K. Continuous line: present OF-AIMD simulations. Triangles: KS-AIMD simulation data from de Wijs et al.107,108 Dash-dotted line: experimental data of Tahara et al.100

Figure 4.1 shows the calculated static structure factor S(q) along with the XD data of Waseda99 and the more recent (XD and ND) data of Tahara et al.100 The OF-AIMD results closely follow experiment, especially the position (qp ) and height of the main peak as well as the asymmetric form of the second peak which is shown in the inset of Fig. 4.1. Similar good agreement was also found for the S(q) obtained from the KS-AIMD simulations.107,108 The simulation also gives the pair distribution function, g(r), whose main peak’s position, rp , is usually taken to be the average nearest neighbor distance. The OFAIMD gives rp ≈ 3.10 ˚ A, close to the experimental value99 of ≈ 3.09 ˚ A. A related quantity is the radial distribution function (RDF), G(r) = 4πr2 ρg(r), which gives the distribution of particles around a given one taken as the origin. The number of nearest neighbors, the coordination number (CN), is obtained by integrating G(r) up to a distance rm which is usually taken to be the position of its first minimum,109,110 giving CN ≈ 12.3 atoms/ions.

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The ion-valence electron structure factor, Sie (q) providing information on the distribution of valence electrons around the nuclei is simple to calculate from an ab initio simulation, but its experimental measurement is difficult. The Sie (q) is depicted in Fig. 4.2 which shows good agreement between the KS-AIMD and OFAIMD results, and both reproducing rather well the trends in the experimental data.100

S(q,ω)/S(q)

0.1 -1

-1

q=0.35 Å

q=0.59 Å

0.05

-1

-1

q = 1.89 Å

0.03

q = 3.11 Å

0 -1

-1

q=0.82 Å

q=1.35 Å

0 -1

-1

0

q = 2.66 Å

q = 2.35 Å

0.02

0.05

0

20

40

0

20 40 -1 ω (ps )

60

Fig. 4.3. S(q, ω) for l-Mg at T = 953 K. Full circles: experimental IXS data at 973 K.105 Full and dashed lines: OF-AIMD results for q = 0.34, 0.61, 0.82 and 1.36 ˚ A−1 , after and before convolution with the experimental resolution function respectively.

0

0

20

40

0

20 40 -1 ω (ps )

Fig. 4.4. Same as in the previous graph but for q = 1.89, 2.36, 2.66 and 3.12 ˚ A−1 .

The calculated dynamic structure factors, S(q, ω), are shown in Figs. 4.3–4.4 for a representative range of wavevectors. Up to q ≈ (3/5) qp , the S(q, ω) show well defined sidepeaks indicative of collective density excitations. A proper comparison with the measured IXS data105 requires the prior convolution of the calculated S(q, ω) with the experimental resolution function105 and the inclusion of the detailed balance factor.1 Once both are included, there is good overall agreement with experimental IXS data although for small q’s, there is some underestimation of the quasielastic contribution. From the positions of the sidepeaks, ωm (q), the dispersion relation of the density fluctuations is obtained and the result is shown in Fig. 4.5 along with ωl (q), which is the dispersion relation derived from the maxima of the longitudinal current correlation function, Cl (q, ω) = ω 2 S(q, ω). Also shown is the experimental ωl (q) data105 and the line representing the dispersion of the hydrodynamic sound whose slope gives the experimental111 bulk adiabatic sound

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20

20

0

-1

Γ(q) (ps )

-1

ωm(q), ωl(q) (ps )

78

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0

1

2

10

3

-1

q (Å ) Fig. 4.5. Dispersion relation for l-Mg at T = 953 K. Open circles: peak positions, ωm (q), from the calculated S(q, ω). Open squares: peak positions, ωl (q), from the maxima of the calculated longitudinal current, Cl (q, ω). Full circles: experimental ωm (q) at T = 973 K from Kawakita et al.105 Full line: Linear dispersion with the hydrodynamic sound velocity, v = 4050 m/s.

0

1

2

3

-1

q (Å ) Fig. 4.6. Full (open) diamonds with error lines: experimental (calculated) values for the HWHM, Γ(q), of the inelastic peaks in l-Mg at T = 953 K.

velocity cs = 4050 m/s at T=953 K. In the hydrodynamic region, the slope of the dispersion relation gives a q-dependent adiabatic sound velocity, cs (q), which in the limit q → 0 reduces to the bulk adiabatic sound velocity. The calculated ωm (q), gives a value for the adiabatic sound velocity cs = 4200 (± 150) m/s in reasonable agreement with the experiment. Another quantity characterizing the collective density excitations is the half width at half height (HWHM) of the inelastic peak, Γ(q), which provides information on the lifetimes of the excitations. Figure 4.6 shows the calculated and experimental data105 for Γ(q). Figure 4.7 shows the calculated Ct (q, t) and Ct (q, ω) for a range of q-values. At the smallest q-value allowed by the simulation (q = 0.168 ˚ A−1 ≈ 0.07qp ) Ct (q, t) already has a weak oscillatory behaviour. The associated spectrum, Ct (q, ω), for some intermediate q-range shows an inelastic peak existing at the smallest value reached by the simulation which is a signature of the propagation of shear waves in the liquid. The shear viscosity can be calculated from Ct (q, t). A value of η=1.35 ± 0.15 GPa ps is obtained which is in fair agreement with the experimental value112 ηexp ≈ 1.16 GPa ps. The velocity autocorrelation function (VACF) obtained by OF-AIMD is plotted in Fig. 4.8. The shape typical of high density systems1,27,65 such as simple liquid metals near melting is seen, which can be explained in terms of the so-called “cage

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2

1

4

1 1

0

0.3 t (ps)

0.6

0

0 60

30

0.01

3 2

0

Z(ω)

3

5

-1

ω (ps )

Fig. 4.7. OF-AIMD calculated transverse current correlation function, Ct (q, t), and its spectra, Ct (q, ω), at several q-values (in ˚ A−1 ) for l-Mg at T = 953 K and q = 0.23 (full curve), q = 0.60 (dotted curve), q = 1.36 (dashed curve), q = 2.35 (dash-dotted curve), q = 3.50 (dashdouble dot curve), q = 5.0 (full circles)

0.5 Z(t)

0.23 0.60 1.36 2.35 3.50 5.0

Ct(q, ω)

Ct(q, t)/Ct(q, t=0)

4

0

0

30

-1

60

ω (ps )

0

-0.4

0

0.2 t (ps)

0.4

Fig. 4.8. Full line: Normalized velocity autocorrelation function, Z(t), of l-Mg at 953 K. The inset represents its power spectrum Z(ω).

effect”, in which a tagged particle is enclosed in the cage formed by its neighbors. The Z(t) exhibits oscillatory behaviour with a distinct negative minimum at ≈ 0.1 ps followed by rapidly decaying oscillations. An estimate of the frequency at which a given atom is vibrating within the cage1,7 can be obtained from the short time 2 2 expansion Z(t) = 1 − ωE t /2 + · · · , where ωE is the so-called “Einstein frequency” of the system. The OF-AIMD result is ωE ∼ 30.0 ps−1 which is comparable to ωE ∼ 30.5 ps−1 estimated by Kawakita et al. from their experimental data.105 The self-diffusion coefficient, D, is deduced from the time integral of Z(t) giving a value DOF−AIMD = 0.51 ˚ A2 /ps. Although no experimental data are available, we note that the KS-AIMD simulation of Wijs et al.107 gave DKS−AIMD = 0.50 ˚ A2 /ps. 4.3.3. Liquid Ga Gallium is an interesting material with unusual structural and electronic properties. It has a low melting point (303 K) and exhibits a variety of morphological crystalline structures including both covalent and metallic bondings. At ambient pressure,113,114 the stable crystalline structure is the orthorhombic α-Ga with seven near neighbors; the nearest one at 2.48 ˚ A is covalently bonded, two second neigh˚ ˚ bors are at 2.69 A, two third at 2.73 A, and two fourth neighbors at 2.79 ˚ A. Upon application of pressure it transforms to the metallic face-centered tetragonally distorted Ga II. At melting, l-Ga has a higher density than the solid α-Ga and its local structure is similar to that of Ga II. Moreover, its static structure factor, S(q) has a main peak with a shoulder characteristic of non-closed packed structures. Two early KS-AIMD simulations of bulk l-Ga used 64 particles and the LDA for the exchange and correlation. The first performed by Gong et al.115 for T =1000K

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used a separable pseudopotential and obtained electronic and static properties. The second for l-Ga at T = 702 K and 982 K of Holender et al.116 used a norm conserving nonlocal pseudopotential and yielded some results for the dynamical structure factor. These ab initio studies provided insights into the valence electronic charge densities. It was suggested115 that metallic and covalent characters coexist in l-Ga, with the latter one represented by the appearance of some very short lived Ga2 covalent molecules with bonds similar to those of α-Ga. However, these KS-AIMD studies were performed at high temperatures, far above the region near the melting point where most experimental data is available. Recently, OF-AIMD simulations have been performed for bulk l-Ga at several thermodynamic states. Here we report results for two temperatures and ion number densities: T = 373 and 959 K and ρ = 0.0512 and 0.0490 ˚ A−3 respectively. The simulations used 2000 ions in a cubic cell with periodic boundary conditions, an equilibration time of 15 ps followed by 80 ps over which properties were averaged. For comparison we comment that the two KS-AIMD simulations cited above lasted 2.5 ps. Figure 4.9 shows the calculated static structure factors, S(q), along with the ND data of Bellissent-Funel et al.117,118 The calculated S(q) at T = 373 K, has a main peak at qp ≈ 2.51 ˚ A−1 and the characteristic shoulder at ≈ 3.10 ˚ A−1 . Comparison 117,118 with the ND data reveals some overestimation of the height of the main peak, but the amplitude and phase of the subsequent oscillations are well reproduced. The calculated S(q) at T = 959 K shows very good agreement with experiment, similar agreement to that of the previous KS-AIMD results.115,116 The calculated OF-AIMD g(r) has a main peak at rp ≈ 2.81 ˚ A for T = 373 K, which reduces to ≈ 2.73 ˚ A for T = 959 K yielding CN values of ≈ 11.8 and 8.5 atoms/ions respectively. This reduction in rp is in good agreement with the experimental results: 2.77 and 2.71 ˚ A leading to CN ≈ 10.5 and 8.7 atoms for T = 326 and 959 K respectively. The KS-AIMD results at T = 1000 and 982 K gave CN ≈ 8.9 and 9.1 respectively.115,116 Figure 4.9 shows the bonded-atoms structure factor, S(q, rb ), calculated by considering only those atoms with one or more neighbors within a distance rb . The S(q, rb ) are shown for several rb values. For T = 959 K the main peak of S(q, rb ) for rb ≤ 2.38 ˚ A is at the position of the shoulder of the full S(q), but around rb = 2.38 ˚ A a second peak emerges at a smaller q value, which grows as rb increases, to become the main peak of S(q). This feature was noted in the KS-AIMD simulations115 of l-Ga at 1000 K and suggests that the shoulder in S(q) is related to the existence of ions separated by distances less than ≈ 2.38 ˚ A. Moreover, the KS-AIMD simulations showed that for these separations there is an accumulation of electron density along the line joining the atoms akin to the bonding charge in the Ga2 dimers found in solid α-Ga, and it suggested that the shoulder in S(q) is related to the survival of these “molecules” after melting, which although short-lived are appreciable in

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81

6

373 K

5

959 K full

full

S(q,rb)

4

2.78

2.78 3

2.68

2.68

2.58 2

2.58

2.48

2.48 2.38

2.38 1 0

1

2.28

2

3

4

5 -1

q (Å )

6

1

2

3

4

5

6

7

-1

q (Å )

Fig. 4.9. Bonded-atoms structure factors of l-Ga for different cutoff radii, rb . The total S(q) corresponds to upper curves at both temperatures. Full circles: experimental ND data.117,118 at T = 363 K and 959 K.

number.115 Similar trends are obtained for the variation of S(q, rb ) with rb at the lower temperature, T = 373 K. The shoulder of S(q) is at the position of the main peak of S(q, rb ) for small rb , but a second peak at lower q emerges around rb = 2.38 ˚ A and develops into the main peak of S(q). P The RDF can be decomposed asR110 G(r) = i Gi (r) where the Gi (r) is the r partial RDF whose integral Pi (r) = 0 Gi (s)ds, gives the probability of finding the i-th neighbor at a distance ≤ r. The limits Pi (0) = 0 and Pi (∞) = 1 are attained at finite distances rimin and rimax which define the spherical shell where the i-th neighbor sits. In particular, r1min is the distance of closest approach between two atoms. The OF-AIMD results show a noticeable displacement of G1 (r) to smaller distances at the higher temperature, with r1min moving from 2.25 at 373 K to 2.13 ˚ A at 959 K, while r1max ≈ 2.90 forR both temperatures. From the G1 (r) the average R concentration, x(R) = P1 (R) = 0 G1 (r) dr, of atoms whose first neighbor is closer than a distance R can be computed. The KS-AIMD simulations115 of l-Ga at 1000 K yielded for a typical covalent bonding distances, R = 2.35 ˚ A, a value x(R) ≈ 0.05 − 0.10, whereas the OF-AIMD simulations at 959 K give x(2.28) = 0.04 and x(2.38) = 0.18. These “covalently bonded” atoms have been interpreted as short-lived “molecules” mainly responsible for the shoulder in the S(q). It was also predicted that the concentration of “molecules” should increase at lower temperatures leading to a sharper shoulder. However, the OF-AIMD results for T = 373 K give a larger distance of closest approach and smaller values x(2.28) ≈ 0 and x(2.38) = 0.03. This result brings into question the interpretation that these atoms form “molecules”, even though

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ρOF-ρsup

3

electron density (e/a.u. )

0.025 0.015

0.015

0.015

0.02 0.015

0.005 0 -0.005

Fig. 4.10. Electron density for a selected triplet of Ga atoms at 373 K. Left panel: OF density. Right panel: KS density. The contour lines are plotted for values of the density equal to 0.010 electrons/(a.u.)3 (the minimum inside the peanut-shaped line in the right part of the KS data) and increments of 0.005 electrons/(a.u.)3 .

959 K 373 K

0.01

2.2

2.4

2.6 2.8 r (Å)

3

3.2

Fig. 4.11. Difference between the OF density and the superposition obtained from the LR screened pseudopotential, at its maximum value between two atoms separated by a distance r. The atoms considered in the plot are those with a neighbor within 2.38 ˚ A, and their first, second and third neighbors.

the shoulder in S(q) is sharper at the lower T . It may be that the relevant signature defining these “molecules” is not the ion separation but the accumulation of electronic density between them. There is a triplet in α-Ga with distances between the central and outer atoms of 2.48 and 2.69 ˚ A and a bond angle of 106◦ , and we have found a very similar triplet in l-Ga with distances of 2.35 and 2.55 ˚ A and a bond angle of 103◦. For the triplet in l-Ga we have calculated the electron density using both OF-AIMD and KS-AIMD simulation methods and obtained the density contours plotted in Fig. 4.10. The overall agreement is remarkable although there are some differences, for instance in the lower-right corner or to the right where the KS-AIMD density shows a somewhat smaller minimum than the OF-AIMD one. However, the OF-AIMD method tends to overestimate the electron density between the atoms, suggesting even stronger “bonds”, if bonds they are, than those obtained with the KS-AIMD. A good approximation to the electron density for a simple metal is given by the superposition of pseudoatom densities obtained from LR screening of the pseudopotential. This superposed density increases between the atoms as they come closer. An analysis of the OF-AIMD density between pairs of atoms including first, second and third neighbors shows that the charge accumulation depends strongly on the distance between the atoms and the smaller the separation, the stronger the effect. But this accumulation clearly exceeds that of the superposed densities for small separations and approaches the superposed density as the atoms separate for both temperatures, as shown in Fig. 4.11. This correlation between distance and bonding charge suggests using the distance between two atoms to identify them as bonded. Turning now to the dynamic properties, for both states the calculated S(q, ω) show side peaks persisting up to q ≈ (3/5)qp . The dispersion relation of the den-

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sity fluctuations obtained from the positions of the sidepeaks, ωm (q), provides an estimate of the adiabatic sound velocity, cs . For T=373 K we have obtained cs = 2920 ± 150 m/s, which is close to the experimental result119 of cs = 2890 ± 50 m/s. Hosokawa et al.120 have measured the q-dependent adiabatic sound velocity, cs (q) = ωl (q)/q, where ωl (q) is the maximum frequency of the longitudinal current correlation function, Cl (q, ω). Also, Scopigno et al.121 have measured cs (q) for a slightly smaller temperature of T= 315 K. Both sets of data are plotted in Fig. 4.12. The OF-AIMD results for the higher temperatures, T=523 and 959 K, are cs = 2875 and 2750 ± 150 m/s, which are close to the respective experimental values,119 cs = 2840 and 2700 ± 50 m/s.

< < 1

3

cs(q) ( 10 m/s)

3 2

Z(t)

373 K 523 K 959 K

1

0.5

0

0

0

1 -1 q (Å )

2

Fig. 4.12. q-dependent sound velocity cs (q) for lGa. Open circles: OF-AIMD results. Full circles: experimental cs (q) at T=373 K from Hosokawa et al.120 Crosses: experimental cs (q) at T=315 K from Scopigno et al.121 Dashed line: OF-AIMD results for T=959 K. The arrows show the experimental adiabatic sound velocities of 2890 m/s. and 2700 m/s. for T = 373 K and 959 K respectively.

-0.2

0

0.4 t (ps)

0.8

Fig. 4.13. Normalized OF-AIMD Z(t) for l-Ga at three temperatures along the coexistence line.

The calculated Ct (q, t) show, at low and intermediate q-values, an oscillatory behaviour which becomes weaker with increasing temperature. The associated spectra display inelastic peaks within a range (0.3 ˚ A−1 ≤ q ≤ 2.30 ˚ A−1 for T = 373 K) which narrows as the temperature is increased and at T = 959 K the inelastic peaks have disappeared. The shear viscosity coefficient has been evaluated from the Ct (q, t) and the results are (in GPa ps units) η = 1.60±0.20 (for T = 373 K), η = 1.12±0.15 (T = 523 K) and η = 0.74 ± 0.08 (T = 959 K) which are in reasonable agreement with the experimental data,122 namely ηExp = 1.58 ± 0.05, ηExp = 1.10 ± 0.05 and ηExp = 0.63 ± 0.03 GPa ps.

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The calculated VACF’s are depicted in Fig. 4.13 for the different temperatures. They exhibit the usual pattern for the simple liquid metals: (i) oscillatory behaviour with a distinct negative minimum followed by weaker oscillations and (ii) these features dampened with increasing temperature i.e. decreasing density. The time integral of the VACF gives the self-diffusion coefficient and the calculated values are: DOF−AIMD = 0.20 ± 0.02, 0.38 ± 0.02 and 0.95 ± 0.04 ˚ A2 /ps for T = 373, 523 and 111 959 K respectively. The experimental results for l-Ga at melting (T = 303 K) lie within the range DExp ≈ 0.16 − 0.17 ˚ A2 /ps and extrapolation to T = 373 K gives an estimate DExp ≈ 0.24 − 0.27 ˚ A2 /ps. No experimental data are available for the higher temperatures, but we may compare with other calculations. For example, the KS-AIMD calculation at T = 1000 K of Gong et al.115 gave DKS−AIMD = 1.0 ˚ A2 /ps, which is very close to the present OF-AIMD result. However, the KS-AIMD study of Holender et al.116 at T=982 K gave the much smaller value DKS−AIMD = 0.65 ˚ A2 /ps. All in all, the OF-AIMD results are within the range of values predicted by the KS-AIMD method. 4.3.4. Liquid Si Si also has interesting properties which, together with its technological importance, has stimulated intensive theoretical123–133 and experimental99,134–139 work. Its high-density forms include crystalline, amorphous and liquid phases with the former two being covalently bonded and semiconducting and the latter one metallic. Upon melting Si undergoes a semiconductor-metal transition along with a density increase of ≈ 10% and important changes in the local atomic structure. This evolves from an open one, with a fourfold tetrahedral coordination, to a more compact liquid structure with an approximate sixfold coordination.140,141 In c-Si the semiconducting diamond structure contracts with pressure and transforms at 12 GPa to the metallic white-tin structure142 and then to the metallic simple hexagonal structure at 16 GPa.143 This has led to the suggestion that l-Si might consist of a mixture of diamond-type and white-tin-type structures with the fraction of the latter increasing with pressure. The S(q) of l-Si near the triple point has been measured by both XD and ND.99,134–136 More recently, Funamori and Tsuji141 have performed XD experiments to measure the S(q) of l-Si at pressures of 4, 8, 14 and 23 GPa and temperatures about 50 K above the melting point for each pressure. From their analysis of the data, Funamori and Tsuji141 have concluded that l-Si up to 8 GPa has a local structure intermediate between the diamond-type and the white-tin-type but, between 8 and 14 GPa drastic structural changes occur and l-Si transforms to a denser structure. Stich et al.130,131 performed the first KS-AIMD calculation for l-Si near the triple point using 64 particles, a non-local pseudopotential22 and the LDA for exchange and correlation. Subsequently, a more comprehensive study was reported132 with 350 particles and a GGA for exchange and correlation. Another KS-AIMD

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calculation near the triple point has been performed by Godlevsky et al.133 using 64 ions and a non-local, pseudopotential.23 These ab initio studies provided accurate insights into the static structure and the valence electron charge densities, however the inherent strong computational demands prevented the study of dynamical properties. The extensive OF-AIMD simulations of Delisle et al 79,96 for bulk l-Si at several thermodynamic states as listed in Table 4.1 allow study of dynamical properties. The simulations used 2000 ions, an equilibration time of 10 ps and the calculation of properties was made averaging over 50-65 ps. The three KS-AIMD simulations cited above lasted for a time 0.9-1.2 ps.

4 GPa

2

8 GPa

S(q)

S(q)

2

1

1

0 23 GPa

14 GPa

2 0

0

5 -1 q (Å )

10

Fig. 4.14. Static structure factor of l-Si near the tripe point. Open circles: experimental ND data.134 Full circles: experimental XD data.135,136 Continuous line: OF-AIMD simulations

1 0

0

4

8

0

4 -1 q (Å )

8

Fig. 4.15. Static structure factor of l-Si at different high pressures. Full circles: experimental XD data.141 Continuous line: OF-AIMD simulations.

Figure 4.14 shows the calculated S(q) along with the experimental data.134–136 The OF-AIMD S(q) accounts for the position and height of the main peak, but the amplitudes of the following peaks and troughs are underestimated. The shoulder on the high-q side of the main peak is reproduced although with a smaller height. It will be recalled that very similar shortcomings are seen in the S(q) obtained from the KS-AIMD simulations of Stich et al.,130–132 whereas that of Godlevsky et al. gives a shoulder whose height is comparable to the main peak.133 The calculated S(q) for l-Si under pressure are shown in Fig. 4.15 along with the experimental XD data.141 The experimental S(q) show that as pressure increases, the main peak grows in intensity and shifts to larger q monotonically whereas the second peak’s position decreases between 8 and 14 GPa. Meanwhile, the distinctive shoulder at the high-q side of the main peak shrinks and practically vanishes by 23 GPa. These

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features are qualitatively reproduced by the OF-AIMD S(q)’s although there are some quantitative discrepancies i.e. the OF-AIMD S(q) overestimate the intensity of the main peak while slightly underestimating that of the shoulder. Funamori and Tsuji141 have argued that l-Si undergoes a high pressure structural transformation between 8 and 14 GPa. Whereas, l-Si contracts with pressure up to at least 8 GPa by reducing the bond length (quantified by rp ), the increase in rp between 8 and 14 GPa does suggest a structural change with a qualitative increase in the CN. Table 4.1 lists CN values obtained from OF-AIMD simulations along with corresponding experimental data.135,137,141 Notice that CN grows with compression with an abrupt increase from 8 to 14 GPa. A similar sharp increase was also deduced by Funamori and Tsuji141 from their experimental RDF. Table 4.1. Thermodynamic states of l-Si studied by the OF-AIMD simulations along with the results obtained for several magnitudes. ρ is the total ionic number density, T is the temperature, CN is the coordination number, cs is the adiabatic sound velocity (± 150 m/s), D is the self-diffusion coefficient and η is the shear viscosity coefficient. P (GPa)

ρ (˚ A−3 )

T (K)

CN

cs (m/s)

0 4 8 14 23

0.0555 0.0580 0.0600 0.0670 0.0710

1740 1503 1253 1093 1270

6.0 6.6 7.2 9.6 11

4250 5100 5400 6300 6750

D (˚ A2 /ps)

η (GPa ps)

2.28 1.82 1.33 0.70 0.70

0.75 0.77 0.84 1.47 1.55

± ± ± ± ±

0.05 0.05 0.05 0.03 0.03

± ± ± ± ±

0.10 0.10 0.10 0.15 0.15

The calculated partial RDF, G1 (r), gives a distance of closest approach, r1min ≈ 1.9 ˚ A at ambient pressure, and a slight increase to a common value of 2.0 for l-Si at the higher pressures. The width of the first neighbor shell narrows with pressure, leading to narrower and higher G1 (r) functions, with a marked change between 8 and 14 GPa further supporting the idea of a structural transition between these pressures. Analysis of the bonded-atoms structure factor, S(q, rb ), for all the thermodynamic states shows the same pattern as l-Ga. For small values of rb there is a peak at the position of the shoulder of S(q), while at larger rb values a feature emerges that develops into the main peak of S(q) as rb is increased. Those rb -values for which the feature at the position of the main peak of S(q) starts developing are 2.20, 2.25, 2.22, 2.19 and 2.16 ˚ A corresponding to 0, 4, 8, 14 and 23 GPa states respectively. The electron density in a plane containing a triplet of atoms again helps to visualize the degree of “covalency”, and the OF-AIMD results are compared with the KS-AIMD calculations in Fig. 4.16. The distances (in ˚ A) between the central and outer atoms are 2.14 and 2.25 (for 0 GPa), 2.08 and 2.21 (for 4 GPa), 2.21 and 2.23 (for 8 GPa), 2.14 and 2.25 (for 14 GPa) and 2.15 and 2.24 (for 23 GPa) and the respective bond angles are 111, 106, 107, 99 and 101 degrees. The overall agreement between the OF-AIMD and KS-AIMD electron densities is again impressive and in

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Fig. 4.16. Electron density around selected triplets for liquid Si. Upper panels correspond to KS-AIMD results and lower panels are OF-AIMD densities. From left to right the pressures are 0, 4, 8, 14 and 23 GPa. Contour lines start at 0.02 electrons/(a.u.)3 with increments of 0.01 electrons/(a.u.)3 , and the lines within the core radius ≈ 0.675 ˚ A have been omitted.

the interatom regions the OF-AIMD density is greater than for KS-AIMD calculation. Also, the charge accumulation is greater for the shorter of the two “bonds”, suggesting a correlation between interatomic distance and charge accumulation. Shown in Figs. 4.17–4.18 are S(q, ω) for l-Si at 0 and 8 GPa and for several q values. For all states, the calculated S(q, ω) show sidepeaks up to q ≈ (3/5)qp , which are indicative of collective density excitations. The general shapes of the S(q, ω) are qualitatively similar at equivalent q/qp values for all the states and we have not identified any specific feature in the S(q, ω) whose variation would mark the structural transformation occurring somewhere between 8 and 14 GPa. The

0.04 S(q, ω)/S(q)

S(q, ω)/S(q)

0.02

0.01

0

0.02

0

0

30

-1

ω (ps )

60

Fig. 4.17. Dynamic structure factor at several q-values (in ˚ A−1 units), for l-Si near the triple point. q = 0.46 (full line), 0.72 (long dashed line), 1.07 (short-dashed line), 1.46 (dotted line) and 2.57 (dash-dotted line).

0

40 -1 ω (ps )

80

Fig. 4.18. Dynamic structure factor at several q-values (in ˚ A−1 units), for l-Si at 8 GPa. q = 0.43 (full line), 0.73 (long dashed line), 1.01 (short-dashed line), 1.50 (dotted line) and 2.30 (dash-dotted line).

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dispersion relations, ωm (q), of the density fluctuations obtained from the sidepeak positions are shown in Fig. 4.19. The curves are qualitatively similar, but whereas the ωm (q)’s for 8 GPa and l-Si near the triple point are close, the ωm (q) for 14 GPa differs markedly. The adiabatic sound velocities cs obtained from these curves are listed in Table 4.1.

1

60 Z(t)

-1

ωm(q) (ps )

80

40 0.5

20 0

0

1 -1 q (Å )

2

Fig. 4.19. Dispersion relation for the peak positions, ωm (q), from the calculated S(q, ω), for l-Si at the triple point (open circles), 8 GPa (full triangles), 14 GPa (grey squares) and 23 GPa (open squares). The figure also includes the experimental (asterisks) data138 for l-Si near the triple point. Dashed line: linear dispersion with the hydrodynamic sound velocity, v=3977 m/s at the triple point.

0 0

0.1 t (ps)

0.2

Fig. 4.20. Normalized velocity autocorrelation function Z(t) for l-Si at the triple point (full line), 4 GPa (dashed line), 8 GPa (circles), 14 GPa (dotted line) and 23 GPa (dot-dashed line).

The calculated Ct (q, t) at low and intermediate q-values, show oscillations which at the triple point are very weak but strengthen as the pressure increases and persist longer. This is reflected in frequency spectra, Ct (q, ω). For both 14 and 23 GPa the Ct (q, ω) exhibits an inelastic peak which appears at low q-values (≈ 0.45 ˚ A−1 ) −1 and persists up to about q = 2.50 ˚ A . However, at 8 GPa the inelastic peaks are present over an appreciably smaller range (0.85 ˚ A−1 ≤ q ≤ 1.50 ˚ A−1 ) and for 4 and 0 GPa there are no inelastic peaks. The inelastic peak in Ct (q, ω) is associated with propagating shear waves which seem to be absent in l-Si up to somewhere between 4 and 8 GPa. The calculated shear viscosity coefficients are listed in Table 4.1. The only available experimental data is for the triple point,144 namely ηExp = 0.58−0.78 GPa ps to be compared with the OF-AIMD result ηOF −AIMD = 0.75 ± 0.15 GPa ps. Figure 4.20 shows the calculated VACF for the different states. At the triple point, Z(t) lacks the usual backscattering behaviour observed in liquid simple metals

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near melting (compare with the previous results for l-Mg and l-Ga) and this is related to the relatively open structure of l-Si with ≈ 6 nearest neighbors which precludes much cage effect with Z(t) decreasing monotonically to zero. Indeed, its shape is remarkably similar to that from the KS-AIMD calculations of Stich et al.130–132,145 The Z(t) for 4 and 8 GPa are qualitatively similar and the small differences are related to a slightly larger cage effect. But as the pressure is increased to 14 and 23 GPa the changes are more substantial with the associated Z(t) closely resembling that of the simple liquid metals near melting. The time integral of Z(t) gives the self-diffusion coefficient, D, and the results are given in Table 4.1. There are no experimental data for the self-diffusion coefficients of l-Si for any thermodynamic state. However, the calculated value for the triple point, D = 2.28 ˚ A2 /ps compares well with the KS-AIMD simulations of Stich 130–132 et al., which gave DKS−AIMD = 2.02 ˚ A2 /ps, a value that increased to 2.4 ˚ A2 /ps when the number of particles was increased to 350. The KS-AIMD study of Godlevsky et al.133 gave DKS−AIMD = 1.90 ˚ A2 /ps. 4.3.5. Liquid Ga-In We have performed OF-AIMD simulations for several liquid alloys and choose to report the results for the eutectic Ga0.835 In0.165 alloy because of the availability of experimental data for both the bulk liquid146 and the liquid–vapor interface (see below). Simulations were performed for T =360 K and ρ = 0.05258 ˚ A−3 , and used 2000 particles (1670 Ga and 330 In) and a total averaging time of 90 ps after equilibration. The partial pair distribution functions, gij (r) as well as the partial AshcroftLangreth (AL) structure factors Sij (q) were evaluated directly, from which was obtained the Faber-Ziman bulk total static structure factor, SF Z (q). It is plotted in Fig. 4.21 and displays fair agreement with the corresponding experimental data146 for both the position and amplitude of the oscillations. The number of j-type particles around an i-type particle, nij , is easily determined from the calculated gij (r). The results are: nGaIn = 1.6, nGaGa = 11.0, nInGa = 8.5 and nInIn = 2.8, implying an average number of nearest neighbors of 12.3 which is very close to the experimental estimate146 of ≈ 12.4. The MD run was sufficiently long to allow calculation of the self-diffusion coefficients, DGa and DIn , of the Ga and In ions. These were obtained by a time integration of the velocity autocorrelation functions,94,95 and the results are: DGa = 0.179 ˚ A2 /ps and DIn = 0.151 ˚ A2 /ps. Finally, we report the alloy shear viscosity coefficient, obtained from the calculated total transverse current correlation function.94,95 The OF-AIMD result for the Ga0.835 In0.165 alloy at T=360 K is η = 2.3 ± 0.2 GPa ps, and we are unaware of any experimental data. However, we recall that the viscosity of the simple liquid alloys shows either a linear or slightly concave variation with concentration, whereas large positive deviations from linearity are exhibited by liquid alloys with heteroco-

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-1

(Å )

4

q ( SFZ(q) - 1 )

2

0

-2

0

5 -1 q (Å )

10

Fig. 4.21. Total XD Faber-Ziman static structure factor, SF Z (q), for the Ga0.865 In0.135 liquid alloy at T = 360 K. Continuous lines are the results from the present OF-AIMD simulations whereas the full circles are the experimental XD data of Gebhardt et al.146

ordinating tendencies.112,147 We remark that the calculated value for Ga0.835 In0.165 is close to an estimate obtained by a simple linear relationship. 4.4. OF-AIMD studies of the liquid–vapor interface in simple liquid metallic systems It had long been believed that the change in the ion density in passing from the liquid through the liquid–vapor (LV) interface into the vapor is monotonic148 and during the last two decades much work has been done on the nature of the change and the atomic reorganization at the interface. Experimental investigation of the LV interface has been mainly carried out by radiation scattering techniques149 such as X-ray reflectivity, which provides information on the longitudinal density profile (DP), that in the direction normal to the interface, and grazing-incidence X-ray diffraction accessing the transverse atomic structure, that in planes parallel to the interface. X-ray measurements on different non-metallic liquids150–152 have confirmed a DP with a smooth monotonic decay from the high-density bulk liquid to the low-density vapor. However, the application of this technique to several liquid metals153–157 (Hg, Ga, In, K, Sn) and alloys155,158–166 (Na-K, In-Ga, Bi-Ga, Bi-In, Sn-Ga and Bi-Sn) has found an oscillatory DP extending several atomic diameters into the bulk liquid, suggesting layering parallel to the interface. The possibility of surface layering in liquid metals was first suggested by Rice and coworkers on the basis of their Monte Carlo167–171 simulations for pure liquid metals, and for binary alloys.165,172–175 They argued that the abrupt decay of the electron DP into the vacuum and the electron–ion coupling led to an effective wall against which the ions, behaving like hard-spheres, stack. Other workers176 have suggested that the oscillatory DP is a consequence of electron–ion effects tending

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to increase the ionic surface density so that its coordination approaches that of the bulk. However, it is not clear whether layering is a feature of all liquid metals. More recently, Chac´ on et al.177 have proposed that surface layering may be a generic property of fluids at low temperature, so that the only requirement for an oscillatory DP is a low melting temperature relative to the critical temperature so as to avoid crystallization. There are additional factors affecting the structure of the LV interface of binary alloys such as atomic size difference and surface tension. Moreover, it is important to study the effect on the interface of the various tendencies of binary alloys to order. The Gibbs absorption rule predicts the surface segregation of the component with the lower surface energy, and some experimental results155,158–165 have shown the coexistence of surface segregation with surface layering. In a metallic system, the nature of the interactions between the ions changes drastically across the LV interface as the electron density falls. The OF-AIMD method has proved capable of dealing with this change for simple liquid metals and binary alloys. During the last five years, its application to several liquid metals and binary alloys178–182 has produced results in good agreement with the available experimental data, whereas the heavy computational demands of the full KSAIMD method have restricted its application to two systems only, namely l-Si and l-Na.183,184 Reports on the application of the OF-AIMD to the LV interface of l-Ga, l-In and the binary alloy Ga-In follow. The choice of systems is dictated by the existence of reflectivity data allowing comparison with experiment. 4.4.1. Liquid Ga X-ray reflectivity measurements have been performed on the LV interface of l-Ga over a range of temperatures from 295 to 443 K.154,158,185 In this technique, Xrays of wavelength λ incident on the liquid surface at an angle α are specularly reflected and the reflected intensity is R(qz ), where qz =(4π/λ) sin α, is the momentum transfer perpendicular to the interface. The measurements on l-Ga have determined R(qz ) up to wavevector transfers of qz = 3.0 ˚ A−1 and show a marked −1 ˚ peak at around qz = 2.4 A which is a signature of an oscillatory, longitudinal ion density profile. The amplitude of the peak decreased drastically upon heating which has been linked to the temperature dependence of capillary-wave induced surface roughness. More recently, X-ray diffuse scattering measurements by Rice and coworkers186 have been used to study the wavelength dependent surface tension of l-Ga at 308 K. Monte Carlo simulations, based on density dependent pair potentials, have been performed for the LV interface of l-Ga in a thermodynamic state near melting.170,187 The results showed oscillations in the longitudinal ionic DP lasting for around four atomic diameters into the bulk liquid, in agreement with experiment. Furthermore, the calculated longitudinal electronic valence DP exhibited weaker oscillations which were clearly in opposite phase to the ionic ones.

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Prompted by this experimental and theoretical work, OF-AIMD simulations have been performed for the LV interfaces of l-Ga at T =373, 523 and 959 K. For each state, the sample was a slab of 3000 ions in a supercell with two free surfaces normal to the z-axis. The dimensions of the slabs were L0 × L0 × Lz (Lz = α L0 ), with α = 1.75 and L0 chosen so that the average ionic number density of the slab coincides with the experimental bulk ionic number density of the system at the same temperature; a further 20 ˚ A of vacuum were added both above and below the slab. The ionic time step was δt = 0.005 ps, and after an equilibration run of 15 ps, the evaluation of the slab’s physical properties was made by averaging over the following 90 ps.

2 ρ(z)/ρο

T = 373 K

T = 959 K

T = 523 K

1

0

0

10

20

0

10

20

0

10 20 z (Å)

30

Fig. 4.22. Electronic (continuous thin line) and ionic (continuous thick line) density profiles normal to the liquid-vapor interface in liquid Ga at T = 373, 523 and 959 K. The densities are plotted relative to their values at the slab’s center. The dashed and dot-dashed lines are the x-transverse (displaced by -0.25) and y-transverse (displaced by -0.5) ionic density profiles.

The longitudinal ionic DP’s, ρGa (z), shown in Fig. 4.22 were computed from a histogram of particle positions relative to the slab’s center of mass, with the profiles from both halves of the slab being averaged. The results are shown in Fig. 4.22. There is a marked stratification lasting for several layers ranging from ≈ 6-7 for T = 373 K to ≈ 3-4 for T = 959 K, with the outer oscillation displaying the higher amplitude. For each thermodynamic state, the oscillations have similar wavelength for the different thermodynamic states: λ = 2.5, 2.53 and 2.6 ˚ A for T = 373, 523 and 959 K respectively. These features agree qualitatively with the DP derived from the reflectivity measurements;154,158,185,186 the experimental DP near melting exhibits oscillations with a λ ≈ 2.55 ˚ A and the outer oscillation has a greater amplitude than the second. Similar results were obtained from Monte Carlo simulations187 for l-Ga at T = 373 K. An oscillating ρGa (z) was obtained with λ ≈ 2.5 ˚ A and the outer oscillation higher than the previous one. Also shown in Fig. 4.22 are the calculated transverse ionic DPs (x and y) which are rather structureless. The calculated valence electron DP also shown in Fig. 4.22 exhibits

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clear oscillations which are nearly in phase with the ionic ones; and ref. 188 provides an explanation for this behavior in terms of the ratio between the pseudoatom size and the interlayer spacing. The stratification has some effects on the in-plane structure of the liquid which are seen in the variation in the transverse pair correlation function, gT (r), across the interface. To display this the outer region of the slab has been partitioned into layers located between consecutive minima of the oscillations, with the narrower outer layer stretching from the outermost minimum to the point in the decaying tail where it takes half its bulk value. For T=373 K (953 K), the outer layer has an ionic number density ≈ 18 % (5%) greater than the bulk value, whereas at the first inner layer the increase is ≈ 1.5 % (-2.7%) with all the inner layers having roughly the bulk value. Figure 4.23 shows for T=373 K the gT (r) corresponding to the outer layers along with the bulk one. The variations of density with depth lead to a gT (r) with a main peak that increases in height while preserving its position. The gT (r) have been used to calculate the z-dependent CN, n(z) taken to be the average number of neighbors within thr radial position of the first minimum of the bulk RDF. For most of the slab n(z) remains practically a constant (n(z) ≈ 12.0) and it is very near the LV interface, around the second outer maximum, when n(z) begins to decrease. Indeed, for all states at this second maximum, n(z) still has the bulk value, whereas at the outer maximum the n(z) values have already decreased by ≈ 30 %. The longitudinal total electronic DP, ρTe (z), which is the physical quantity probed in the X-ray reflectivity measurements, has been constructed from the ρGa (z) by adding to the valence density a superposition of core electron densities.27,65 As the core densities are rather narrow, their superposition gives a profile in phase with the ionic DP, and because of the preponderance of core electrons in Ga likewise for the total ρTe (z). The reflected intensity, R(qz ), obtained in an X-ray reflectivity experiment can be written R(qz ) 2 = |Φint (qz )| exp(−σc2 qz2 ) RF (qz )

(4.43)

where qz =(4π/λ) sin α, is the momentum transfer perpendicular to the interface, RF (qz ) is the Fresnel reflectivity of a perfectly sharp step-function interface. Φint (qz ) is the intrinsic surface structure factor defined as 1 Φint (qz ) = T ρe0

Z

∞ −∞

∂ρTe,int (z) ∂z

!

exp(iqz z) dz

(4.44)

where ρTe0 is the bulk total electron density and ρTe,int (z) is the intrinsic longitudinal total core plus valence electronic DP in the absence of capillary wave smearing. The

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3

R(qz)/RF(qz)

gT(r)

3 2

2

1

1 0

0

2

4

6

8

0

1

r (Å) Fig. 4.23. Transverse pair correlation functions for the outer layers in l-Ga at T = 373 K. Full line: outermost layer. Dashed line: first inner layer. Dotted line: second inner layer. Full circles: bulk g(r).

-1

qz (Å )

2

3

Fig. 4.24. Reflectivity curves of l-Ga. Full circles and triangles: Experimental results154,185 at T = 360 and 397 K respectively. Continuous, dashed, dotted and dot-dashed lines: OF-AIMD results for T = 373 K and σcw = 0.78 ˚ A but with σ0 = 0.60, 0.55, 0.50 and 0.48 ˚ A respectively.

term exp(−σc2 qz2 ) in Eq. (4.43) accounts for the thermally excited capillary waves, with σc representing an effective capillary wave roughness. The longitudinal total electronic DP, ρTe (z), obtained from an OF-AIMD simulation includes some thermal fluctuations and predicts a reflected intensity 2  ^   2 Z ∞  T T (z) 1 ∂ρ (z) ∂ρ 1 R(qz ) e e = exp(iqz z) dz ≡ T ≡| Φ(qz ) |2 ρe0 RF (qz ) ρTe0 −∞ ∂z ∂z (4.45)   T  ^ ∂ρ (z) ∂ρT (z) e e is the Fourier transform of . Comparison with Eq. (4.43) where ∂z ∂z shows that the OF-AIMD surface structure factor Φ(qz ) is the result of a convolution of the intrinsic one, Φint (qz ), with a Gaussian distribution describing the thermal 2 fluctuations in the simulation. The σc2 has two contributions,:154,158 σc2 = σ02 + σcw , where σ0 is an intrinsic surface roughness and σcw accounts for the thermally excited capillary waves149,154,158,189 2 σcw

kB T = ln 2πγ



qmax qmin



(4.46)

where γ is the surface tension and qmax and qmin are determined by the ionic diameter and the instrumental resolution, respectively. A common choice is qmax = π/d with d the ionic diameter. Capillary waves require careful consideration when Exp comparing simulated and experimental reflectivity curves because value of qmin in Eq. (4.46) is dictated by instrumental resolution in the experimental case, but

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OF in the simulations it is the transverse area (L2 ) of the simulation box, qmin = π/L, which dominates and this value is far greater than the experimental one. Consequently, the capillary wave contribution to surface roughness, σcw , is smaller in OF Exp the simulation than in the experiment (σcw < σcw ). The reflectivity has been computed as

R(qz ) 2 = |Φ(qz )|2 exp{−[(σ0OF )2 + ∆σcw ] qz2 } RF (qz )

(4.47)

2 Exp 2 OF 2 where ∆σcw = (σcw ) − (σcw ) , and the contribution already present in the simulation has been subtracted from the total capillary damping. To evaluate Eq. (4.47) Exp OF we have used γ = 0.710 N/m (ref. 111), σcw = 0.78 ˚ A and σcw = 0.52 ˚ A. There OF is no consensus on the choice of σ0 although a relationship to the width of the outer layer of nTe (z) has been suggested. Therefore, we have used several values: σ0OF = 0.48, 0.50, 0.55 and 0.60 ˚ A. Figure 4.24 shows the OF-AIMD results for R(qz )/RF (qz ) at T = 373 K, along with the experimental data at T =360 and 397 K. There is qualitative agreement with experiment. The peak position is given correctly and although very good agreement with experiment is obtained for the peak height in the range 0.55 − 0.60 ˚ A, the height is very sensitive to σ0OF .

4.4.2. Liquid In The simulation of the LV interface of l-In at T = 450 K used 3000 ions, the same geometry and time step as for l-Ga, equilibration lasted 15 ps and physical properties were averaged over the next 72 ps. Figure 4.25 shows the longitudinal ionic DP, ρIn (z), which displays stratification for at least five layers with the outer oscillation having larger amplitude. The oscillations have a wavelength of 2.75 ˚ A. These features are in qualitatively agreement with the experimental and other theoretical results. The X-ray measurements154,158,185 give λ ≈ 2.69 ˚ A−1 . Similar characteristics were found in the Monte Carlo simulations of Rice and coworkers170,187 for l-In near melting. Figure 4.25 also shows the calculated self-consistent valence electron DP which oscillates in phase with the ion DP. The ρIn (z) has again been sliced. The ion density in the outer 2.35 ˚ A wide slice is ≈ 9 % greater than the bulk value; the change is very small and negative in the next layer, and deeper layers take the bulk values. Despite the marked stratification, the bulk density is attained at ≈ 6-7 ˚ A from the interface. The zdependent coordination number is fairly constant at ≈ 10.3 over nearly all the slab but drops by ≈ 30%, at the outer maximum. The X-ray reflectivity measured at T= 443 K185 and shown in Fig. 4.26 decreases monotonically with a small peak at qz ≈ 2.1 ˚ A−1 , and is qualitatively different from the data for l-Ga. The reflectivity has been constructed from the OF-AIMD results following the procedure outlined for l-Ga using the following values: γ = 0.556 N/m

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In

1

0

1

0 0

20

z (Å)

40

Fig. 4.25. Electron (thin continuous line) and ion (thick continuous line) density profiles normal to the liquid-vapor interface in liquid In at T=450 K. The densities are plotted relative to their total bulk values.

0

1

2

3

-1

qz (Å ) Fig. 4.26. Fresnel normalized reflectivity curves of liquid In at 450 K. Full circles: Experimental results185 at T = 443 K. Continuous lines: Calculated OF-AIMD results for T = 450 K.

Exp OF (ref. 111), σcw = 0.92 ˚ A, σcw = 0.65 ˚ A and σ0OF = 0.65 ˚ A, and is also shown in Fig. 4.26. There is excellent agreement with experiment.

4.4.3. Liquid Ga-In OF-AIMD simulations have been performed for the LV interface of Ga1−x Inx alloy at T = 360 K and the eutectic concentration, xIn = 0.165. In order to account for the In surface segregation and the finite sample size a simulation sample with a greater percentage of In atoms must be used, otherwise the strong surface segregation depletes In in the inner region of the slab. The “correct” 16.5% In bulk alloy is achieved by a 20% In sample. The time step was δt = 0.0075 ps, and after an equilibration run of 20 ps the physical properties were averaged over the next 130 ps. Figure 4.27 shows the calculated longitudinal partials (ρGa (z), ρIn (z)) and total (ρGaIn (z)) ionic DPs. The ρGaIn (z) is distinctly stratified for at least five layers into the bulk liquid, with the outer oscillation having the larger amplitude. Apart from the outer one, the oscillations have the same wavelength, λ ≈ 2.50 ˚ A. These features qualitatively agree with the DP derived by Regan et al.158 from their X-ray reflectivity measurements on the Ga0.835 In0.165 liquid alloy which had a λ ≈ 2.60 ˚ A. However, comparing Figs. 4.22, 4.25 and 4.27 we see that the amplitude of the oscillations for ρGaIn (z) is smaller than those of the component bulk liquid metals, although these are calculated at higher temperatures. To analyze the slab’s outer region, the ρGaIn (z) has been partitioned as before into slices. In Fig. 4.27 it will be noticed that the outer oscillation in ρGaIn (z) is dominated by the partial ρIn (z). Despite being the minority component In has the

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T

xIn = 0.165

ρe (z)/ρe0

ρ(z)/ρ0

smaller surface tension and consequently is segregated to the surface creating a high In concentration monolayer. The outer monolayer has a concentration x0In ≈ 0.84, whereas the bulk concentration is xIn = 0.165 and is only attained at around the fourth slice which is ≈ 10 ˚ A from the interface. Figure 4.27 shows the calculated valence electronic DP. There are clear oscillations which last for a smaller range than the ionic ones, and a monotonic decreases at the LV interface. Also shown in Fig. 4.28 is the longitudinal total electronic DP, ρTe (z), constructed using for each component the respective core electronic density obtained in the process of construction of the respective pseudopotentials. We see that the ρTe (z) closely follows the ρGaIn (z), except for the outer oscillation which has a substantially enhanced amplitude because of the increased relative contribution of In, with 46 versus 28 core electrons for Ga. Even so, the oscillations in ρTe (z) have the same wavelengths as those of ρGaIn (z). The figure also shows the total core+valence electron density profiles for each component of the alloy, the sum of which gives ρTe (z).

T

1

0 z (Å) Fig. 4.27. Valence electron (thin continuous line), total ion (thick continuous line), partial Ga ion (dashed line) and partial In ion (dash-dotted line) density profiles normal to the Ga-In liquid-vapor interface at xIn = 0.165. The densities are plotted relative to their total bulk values.

2

1

0

0

10

z (Å)

20

30

Fig. 4.28. Total electronic density profile (core+valence) normalized to the slab’s bulk value for the Ga-In liquid-vapor interface at xIn = 0.165. The dashed and dotted lines correspond to the Ga and In total electronic density profiles respectively. The thin continuous line is the total ionic density profile.

We have also examined how the stratification of the LV interface affects the in-plane structure of the alloy by evaluating the transverse partial pair correlation functions, gij (r) slice by slice. The gij (r) in the outer two slices are shown in Fig. 4.29 along with gij (r) for the wide central region of the slab which will regard as bulk. The first inner slice appears bulk-like and the important differences are in the outer slice where gGaGa (r) and gInIn (r) are displaced towards larger and smaller r values respectively as a result of the drastic increase and decrease in the ion number densities.

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15

R(qz) / RF(qz)

2 In-In

gij(r)

10

5

0 xIn = 0.165

Ga-In

-2 Ga-Ga 0

2

0

1

2

3

-1

qz (Å )

4

6

r (Å) Fig. 4.29. Transverse partial pair correlation functions, gij (r), for selected layers of the Ga0.835 In0.165 slab, namely from the bulk (open circles), a slice from the outermost minimum outwards (continuous line) and from the region between the outermost minimum and the precedent one (dashed line). The Ga-In and In-In partials have been shifted by 3 and 6 units respectively.

Fig. 4.30. Fresnel normalized reflectivity (continuous line) for the Ga0.835 In0.165 liquid-vapor interface at T= 360 K. The dashed, dot-dashed and dotted lines represent the contributions from the Ga and In total electronic density profiles and the cross term respectively. The full circles are the experimental data.158

The alloy reflectivity has been constructed according to the previous procedure Exp OF and using the following values: γ = 0.616 N/m, σcw = 0.80 ˚ A, σcw = 0.55 ˚ A and σ0OF = 0.43 ˚ A, and is shown in Fig. 4.30 along with the experimental data.158 The results qualitatively follow the experimental data although somewhat overestimate them. The position of the main peak and the small, wide bump near 1.2 ˚ A−1 are reproduced. Further insight can be achieved by decomposing the calculated reflectivity into different contributions. For an A-B alloy ρTe (z) = ρTe,A (z) + ρTe,B (z) and substituting into Eq. (4.45), we obtain  ! 2 ! 2  2 ^ ^ T T ∂ρe,B (z) R(qz ) 1  ∂ρe,A (z) + = +  T RF (qz ) ∂z ∂z ρe0 2Re

! T (z) ∂ρ^ e,A ∂z

∗

 ! T (z) ∂ρ^  e,B  ∂z

(4.48)

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where Re denotes the real part and the asterisk the complex conjugate. This decomposition gives the contributions from ρTe,A (z), ρTe,B (z) and a cross-term, which are shown in Fig. 4.30. As expected, because of the relative concentrations in the outermost layer of ρTe (z), the main contributions arise from the In and cross term with Ga playing a negligible role.

4.5. OF-AIMD studies of solid-liquid interfaces Interfaces between solids and liquids play an important role in a number of technological and physical processes, such as soldering, lubrication, wetting or crystal growth just to mention a few. Early studies of this type of system modeled the interaction of the solid wall with the liquid in terms of a potential that depended only on the distance, z, from the wall. Such studies only provided information on the longitudinal order established in the liquid in the form of a z-dependent ionic DP, ρ(z). This profile showed stratification in the liquid near the wall, with an interlayer spacing related to the liquid structural characteristics, for example, the position of the main peak of the structure factor. Real solids, however, are not continuous but show a periodicity dictated by the crystallographic structure and the orientation of the exposed surface. This periodicity appears both in the direction normal to the surface, as an interlayer distance, and in the 2-dimensional plane of the surface. In order to analyse the effect this periodicity may induce in the liquid, an atomistic description of the solid surface is needed. The ionic DP of the liquid perpendicular to the wall can be influenced by the interlayer spacing of the solid, whereas in the case of an integrated wall-liquid interaction only liquid properties determine the liquid layering profile. There can also be a strong influence of the lateral periodic surface potential on the transverse ordering in the liquid, at least in the regions near the interface. Atomistic-surface studies have been performed for model systems, with LennardJones and hard spheres interactions, and also for some metallic liquids, particularly Al, in contact with a solid metal. The solid was routinely assumed to be fcc, and several crystal orientations have been considered, mainly the (100) and (111) cases, although some studies also considered the (110) orientation. For a recent review, see Ref. 190. As for bulk liquids, the simulation of realistic liquids on realistic solids requires a proper description of the forces between atoms. Even if there are good prescriptions for the interaction energies separately for the bulk liquid and the bulk solid, the presence of an interface introduces the question of the cross interactions and the modifications of the interactions due the proximity of the different phases (see Ref. 191 for the case of liquid Al on solid Cu). These problems do not arise when the two components are made up of the same type of atom (e.g. liquid Al in contact with solid Al), but this type of simulations is restricted to a temperature for which there is coexistence between solid and liquid, i.e., at the melting temperature, and

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several studies have been performed this way.192,193 Another possibility is to pin the atoms of the solid to their ideal crystallographic positions, so that any temperature of the liquid may be set, and the simulation would mimic a solid with a high melting temperature. Hashibon et al. performed such a simulation194 for liquid Al on ideal fcc Al at 1000 K, and analysed the properties of the liquid in terms of the solid surface orientation. In most of the simulations of liquid metals on solids, the embedded atom method (EAM)16 was used for the interactions. This model has been used successfully in several instances but it is empirical, parameters are fitted and different parametrisations can lead to substantially different behaviours away from the fitting state. For instance in liquid Al the EAM parametrisation used by Hashibon et al. leads to a solid-like non-crystalline state at low temperatures195 which is absent in other EAM parametrisations for Al, and not found in real Al. The OF approach of DFT incorporates the behaviour of the electrons in the interaction from first principles, and we have seen in the previous section that OF-AIMD simulations describe reasonably well the properties of bulk liquid metals and free liquid surfaces of simple metals and their alloys. In the following section, OF-AIMD simulations of solid metal-liquid metal interfaces are described. Little work has been done on the performance of the OF-AIMD approach for solid metal surfaces, and so before proceeding to the treatment of the solid-liquid interface, simulations of the surfaces of Al(110) and Mg(10¯10) are discussed. Simulations of liquid Al on pinned solid Al, and liquid Li on solid Ca are then described. 4.5.1. Surface relaxation and its temperature variation. Al(110) and Mg(10¯ 10) If a bulk metallic solid at zero temperature is cleaved, the electrons redistribute and ions near the surface experience different forces from ions in the bulk. This usually leads to a relaxation of the layers in the direction perpendicular to the surface. Open surfaces frequently relax inwards in order to increase the coordination, while closely packed surface can relax inwards or outwards depending on the material.196 Inner layer relaxations can also occur and in several cases lead to an oscillatory pattern of expansion and contraction. Such a pattern has been found in all-electron DFT calculations for Al in several orientations, indicating that a large number of layers is required to obtain fully converged relaxations.197,198 When the temperature is raised additional dynamic effects take place and experiments show a wide variety of behaviors. While thermal expansion is expected on general grounds199,200 some surfaces, e.g., Pb(110), Ni(100), Ag(111), Cu(110) and Be(0001), exhibit an anomalously large effect for its first interlayer separation201–205 while others show a negative thermal coefficient followed by positive ones for inner interlayer separations206 ( Al(110) ) or even an alternating behavior207,208 ( Mg(10¯ 10), Be(10¯ 10) ).

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¯ We will show below results for Al(110) and Mg(1010), two of these anomalous systems. Both are simple metals and good candidates for OF-AIMD simulation. Moreover, both have been studied using the full KS approach providing benchmarks for checking the OF-AIMD. For Al(110) KS-AIMD simulations were performed,209 which reproduced both the oscillatory relaxations and the thermal behaviour observed experimentally. Calculations for Mg(10¯10) using the quasi harmonic approximation (QHA) and static KS computations207 also reproduced the oscillatory pattern in both relaxations and thermal expansion coefficients, but no ab initio simulations have yet been performed for this system. In order to compare our results with these KS calculations we have used in our simulations exactly the same atomic sample, including the rather small number of atomic layers. Consequently, the relaxations are not fully converged and the comparison of the OF results with experiment must be taken with more caution than the comparison with the KS data.

Interlayer relaxation (%)

d23 5 d45 0 d34 -5

-10 0

d12

200

400

600

800

1000

T (K) Fig. 4.31. Results for the percent interlayer relaxations for Al(110). Full circles: OF-AIMD results. Line: best linear fit to them. Hatched squares: uncorrected KS-AIMD results. Lozenges with lines: experimental data.

4.5.1.1. Al(110) The simulation box contains 8 layers, each with 9 Al atoms, with an in-plane spacing equal to the experimental one at each temperature considered, plus a vacuum of 8.5 ˚ A. We have considered three temperatures: 70, 310 and 707 K. For the first two states experimental measurements of the interlayer relaxations based on low energy electron diffraction have been reported, while the higher temperature is close to one of the states considered in the KS-AIMD simulations. After an initial equilibration of 2-4 ps, constant energy simulations were run for 8-16 ps, and the positions of atoms in a given layer were used to find the average layer position and consequently

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the interlayer separations dij (T ). The percent interlayer relaxations are then defined bulk as [dij (T ) − dbulk ij (T )]/dij (T ) × 100. These are plotted in Fig. 4.31 together with the experimental data and the results from the KS-AIMD simulations. Note that the KS-AIMD results published in Ref. 209 were corrected rigidly by the difference between a KS calculation at T = 0 K using the same sample parameters and another using more layers, one atom per layer, and improved Brillouin zone sampling; the uncorrected KS-AIMD results are shown in Fig. 4.31. The OF-AIMD simulations give the oscillatory pattern of interlayer relaxations found experimentally, and also the temperature dependence with a negative thermal expansion coefficient for the first interlayer separation and a positive coefficent for the rest, but, emphasis is only on the trends because of the possibility that more layers would be needed to obtain fully converged relaxations. The main purpose of these simulations was to compare the results the OF simulation with KS results. The comparison with the KS data is satisfying since the OF results reproduce well all the aspects of the KS data. 4.5.1.2. Mg(10¯ 10) The simulation setup for this system is the same as that used in the KS calculations of Ismail et al.207 that accompanied the experimental low energy electron diffraction study. There were 16 layers of 4 Mg atoms each, plus a vacuum region of 8.5 ˚ A. The in-plane spacing is taken as the experimental one at the temperatures considered: 106, 308 and 399 K. Equilibration lasted 4 ps, followed by a production run of 8 ps. The hexagonal close packed structure for this orientation has alternating short and long interlayer separations. It is energetically favorable for the two outermost layers to be separated by the short distance. Figure 4.32 shows the percent relaxations found for this system. Experiment has the short interlayer separations contracting and showing a negative thermal expansion coefficient, while the long interlayer separations both expand and show positive thermal expansion. The OFAIMD results reproduce the trends with respect to both the sign and thermal variation of the relaxations, although the magnitude of the contractions is reproduced with better accuracy than that of the expansions. There are three points of comparison with the KS data: i) the magnitude of the KS relaxations is closer to the experimental results than those from the OF-AIMD simulation. (ii) the thermal variation of the KS-QHA and OF-AIMD results is very similar, and (iii) both approaches seriously underestimate the thermal variation found in the experimental measurements, which show a much larger slope for the first three interlayer separations. In summary, the OF-AIMD results for the structure of the open surfaces considered and their temperature dependence reproduce qualitatively the experimental trends. In many aspects the results are also very similar to those obtained through more computationally expensive methods such as KS. The only exception is the magnitude of the expansion of the long interlayer separation in the hcp structure.

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Long interlayer distances

Short interlayer distances

Relaxation (%)

-5

d45

-10

5

-15

d34

-20

0 15

Relaxation (%)

-25

d23

-15 10 -20

5

d12 -25

0

100

200

300

400

Temperature (K)

500

0

0

100

200

300

400

Temperature (K)

500

Fig. 4.32. Results for the percent interlayer relaxations for Mg(10¯ 10). Full circles: OF-AIMD results. Thick continuous line: best linear fit to them. Hatched squares: KS-QHA results. Dashed lines: best linear fit to them. Lozenges with error bars: experimental data. Thin continuous lines: best linear fit to them.

4.5.2. Liquid Al on pinned solid Al Satisfied that solid metallic surfaces are well described by the OF-AIMD, we go on to the study of the properties of a l-Al at 1000 K in contact with solid fcc Al for which we have fixed the atoms at their unrelaxed ideal positions, corresponding to a lattice parameter a = 4.1 ˚ A. Simulations lasted 30-40 ps in an orthorhombic simulation cell. In the z-direction there are consecutively: vacuum, several solid layers exposing a given orientation, 2000 particles comprising the liquid phase, and further vacuum. Three crystal orientations have been considered: (100), (110) and (111). The (111) ˚2 surface is close √ packed with a planar density of 0.1374 atoms/A , and an interlayer A. The (110) surface is very open with spacing of a 3/3 = 2.37 ˚ √ a planar density 2 ˚ of 0.0841 atoms/A but a much shorter interlayer spacing of a 2/4 = 1.45 ˚ A; it is ˚ also highly anisotropic, exhibiting surface channels of width a = 4.1 A , along which √ the atoms are separated by a 2/2 = 2.90 ˚ A. The (100) surface is intermediate both in packing (0.1190 atoms/˚ A2 ) and in interlayer distance, which is a/2 = 2.05 ˚ A. For comparison we note that the most probable distance between atoms in the liquid, the distance at which g(r) has its main peak, is 2.68 ˚ A,99 and the interlayer distance in 210 the stratified free liquid surface is 2.35 ˚ A. Table 4.2 lists details for the different orientations. An OF-AIMD simulation for solid Al(100) in equilibrium with its melt was reported some time ago193 using a different kinetic energy functional and pseudopotential from those used here. That study led to a melting temperature ≈ 300 K too low, suggesting deficiencies in the OF functional.

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N

geometry

solid layers

atoms per solid layer

(100) (110) (111)

2000 + 294 2000 + 432 2000 + 324

28.70 × 28.70 × 92.05 28.41 × 30.13 × 75.05 30.13 × 26.09 × 75.05

3 6 3

98 72 108

The results for the (100) orientation are shown in Fig. 4.33. The ionic DP of the liquid is strongly layered close to the wall, showing five to six layers before decaying to the homogenous bulk liquid density. The interlayer spacings within the liquid are larger than the 2.05 ˚ A substrate spacing, and increase gradually from 2.18 ˚ A, which is the distance between the solid surface and the first liquid layer, to a value around 2.38 ˚ A in the inner layers, a value very close to that for the free liquid Al surface. In total the stratification extends for about 12-15 ˚ A. The transverse pair correlation functions in the different layers evolve from substrate-like in layer 1, to almost bulk liquid-like by the third layer. The mean squared displacements in the liquid layer nearest to the interface show that the atoms in the liquid diffuse more slowly as the interface is approached due to the influence of the substrate. This is further confirmed the projections of the trajectories of the atoms on the xy-plane for the different liquid layers, shown in Fig. 4.34. The liquid atoms close to the surface vibrate around solid-like positions and diffuse from one site to an adjacent one. In the second layer some influence of the substrate persists, while at the third layer the liquid appears isotropic. The valence electron DP, also shown in Fig. 4.33 has opposite phase to the ionic DP within the solid but shifts more in-phase in the liquid region due to the larger interlayer spacing.188 The bump at the free solid surface is very similar to that found in KS calculations for an Al(100) slab.211 The atomic profile for the (110) orientation shown in Fig. 4.35 is similar, but different in detail from that for the (100) surface. The interlayer spacings in the liquid are the same as those of the substrate, and while the number of layers is similar to those for the (100) case, the extension of the stratified region is smaller, about 9-11 ˚ A. The transverse pair correlation functions are also quite different, showing influence of the surface structure deeper in the liquid; the fourth layer still shows peaks at distances related to the surface structure, and at the sixth layer, almost 9 ˚ A from the substrate, differences persist from the bulk liquid g(r). Indeed the mean square displacements (Fig. 4.35) and the projected trajectories (Fig. 4.36) suggest that layers 1 to 3 are solid, since there is no diffusion, extending the substrate into the liquid. Diffusion starts in the fourth layer but preferentially along the surface channels. This anisotropy is still present in the sixth layer. The valence electron DP has opposite phase to the ionic one, and displays no

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change from the solid to the liquid because the interlayer spacings are identical.188 The bump at the free solid surface is much smaller than for the (100) orientation, but still discernable. We are not aware of full KS calculations for this orientation with which to compare. 3 (1)

profile

(a)

(2)

2

(3) 1 0 -10

0

-5

5

20

15

1

(b)

(3)

(3)

(c) 0.8

(2) 2

2

3

0.6

2

δu (t) (Å )

4

gT(r)

10

5 z (Å)

0.4

(2)

(1)

(1) 0.2

1 0

2

4

6

8 r (Å)

10

12

14

0

0

0.2

0.4 t (ps)

0.6

0.8

Fig. 4.33. Results for liquid Al on pinned fcc Al for the (100) surface. (a) Ion (full line) and valence electron (dashed line) density profiles. The vertical lines correspond to the solid (100) interlayer spacing, and z = 0 corresponds to the position of the solid layer closest to the liquid. (b) Transverse pair correlation functions in layers 1 to 3. Vertical lines correspond to the (100) solid surface and dashed lines to the bulk liquid. (c) Mean squared displacements.

Fig. 4.34. Projection of the trajectories of atoms in the liquid Al layers 1 to 3 (left to right) for the (100) orientation.

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3 (1) (2)

profile

(a)

(3)

2

(4)

(5) (6)

1 0 -10

0

-5

10

5 z (Å)

20

15

6 (b)

2

2

(4)

3

2

gT(r)

(4)

2

(6)

4

(6)

0.3 δu (t) , δv (t) (Å )

5

(3)

(3)

0.2

(1)

0.1

1 (c)

(1) 0

2

4

6 8 r (Å)

10

12

0

0

0.2

0.4 t (ps)

0.6

0.8

Fig. 4.35. Results for liquid Al on pinned fcc Al for the (110) surface. (a) Ion (full line) and valence electron (dashed line) density profiles. The vertical lines correspond to the solid (110) interlayer spacing, and z = 0 corresponds to the position of the solid layer closest to the liquid. (b) Transverse pair correlation functions in the labeled layers. Vertical lines correspond to the (110) solid surface and dashed lines to the bulk liquid. (c) Mean squared displacements at the labeled layers, along (full lines) and across (dashed lines) the surface channels direction.

Fig. 4.36. Projection of the trajectories of Al atoms in layers 1, 3, 4 and 6 (left to right) for the (110) orientation.

The (111) surface orientation leads to quite different results. As the simulation proceeds the liquid crystallizes layer by layer. In 32.25 ps about 10 layers solidify as illustrated in Fig. 4.37 suggesting that the theoretical liquid is somewhat undercooled and crystalizes when faced with an appropriate seed. Solidification of the (110) surface is not as effective leading to 3 solid layers of decreasing ordering. The

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Fig. 4.37. Snapshots of the “liquid” Al system on the pinned (111) surface of fcc Al at the initial and final stages of the simulation. The first three layers are the fixed solid ones.

(100) surface is especially ineffective, as only the first layer shows solid-like structure with some diffusion. In any case further checks of the thermodynamic melting temperature of our ab initio model are called for. Our results, for the structural properties of the (100) and (110) orientations, validate the EAM results of Hashibon et al.194 However, those studies did not consider dynamic quantities like those studied here, that have helped the interpretation of the structure of the first layers of the liquid, which we found are solid for the (110) orientation. Of course, the EAM gives no information on the electronic structure. 4.5.3. Liquid Li on solid Ca We finally consider a solid-liquid interface of dissimilar metals and allow the atoms in the solid to move as freely as those in the liquid. Of interest will be the effects of a dissimilar substrate on the liquid structure near the interface and the effects on the solid surface of the proximity of the liquid. We expect surface relaxation, but different from that of the free solid surface because the variation of the electron density across the interface will be less when there is liquid present. We consider the l-Li on solid Ca system because although the metals are dissimilar, each with its own interactions, the change in mean electron density across the interface is small. Solid Ca is fcc with a lattice parameter at zero temperature of a = 5.58 ˚ A, a 3 ˚ mean valence electron density of 0.0458 electrons/A and a melting temperature of 1115 K. Li melts at 470 K, where Ca is firmly solid, and its mean valence electron density is 0.0445 electrons/˚ A3 . The (110) orientation of Ca is chosen for the interface because of the anisotropy introduced by the surface channels. OF-AIMD results are presented below for l-Li on a (110) solid Ca substrate at 470 K. In order to allow for converged relaxations we need a thicker solid slab than used in the pinned Al case. The simulation setup is as follows: in an orthorhombic simulation cell of dimensions 28.99 × 27.34 × 108.25 ˚ A3 , there are consecutively: vacuum, 16 layers of solid Ca each containing 36 atoms and exposing the (110) orientation, amounting to 576 Ca atoms, 2000 Li atoms constituting the liquid, and further vacuum. The cell contains a free l-Li surface, the solid Ca-liquid Li

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interface, and a free solid Ca (110) surface. For the former we recover results obtained previously for a l-Li slab,210 while we expect relaxation of the solid surfaces. The ion DP is shown in Fig. 4.38 where we see the 16 solid Ca layers and a stratified liquid Li profile with around seven to eight layers from the solid-liquid interface. The free liquid surface is also layered as found in the earlier study.210

Fig. 4.38.

Ion DP for the solid Ca-liquid Li system simulated.

0.08

0.06 1.78 1.86 1.86 1.86

0.05

0.06

0.04 0.03

2.74

2.52

2.56

0.04

0.02

0.02

0.01 0

1.39 1.65 2.76 1.86 1.86

10

12

14

16

18

0

35

40

45

50

Fig. 4.39. Ion density profiles in the region near the free solid surface (left panel) and for the solid-liquid interface (right panel) for the solid Ca-liquid Li system.

At the free Ca(110) surface, also shown in Fig. 4.39, we see that the outermost layer relaxes inwards, as for the Al(110) surface, but there are no inner layer relaxations in Ca. There are two noteworthy points concerning the solid-liquid interface shown in Fig. 4.39: (i) there is no relaxation of the outermost solid layers; the spacing is the same as in the bulk. This underpins the importance for the relaxation of the decay in the electronic density at a metal interface, even for dissimilar metals. (ii) the liquid ion DP has a strange shape near the solid. The first liquid layer sits very close to the surface. The following interlayer spacings first increase and then decrease towards a value ≈ 2.50 ˚ A, which coincides with the interlayer spacing for the free liquid surface. We next investigate liquid structure in the layers, particularly the first, and compare it with that of the closest solid layer. In Fig. 4.40 we show several transverse correlation functions. Those for an ideal

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fcc (110) face are shown as dotted vertical lines. The first, fourth and seventh lines correspond to interatomic distances along the surface channels, the second and sixth lines to distances across the channels, and the others to oblique directions. The upper graphs (that have been inverted and shifted upwards) are the transverse pair correlation functions for the Ca atoms in the first layers of the solid-vacuum and the solid-liquid interface. These are very similar suggesting that the lateral structure of the layers is not altered at either interface. Figure 4.40 also shows the transverse gT (r) for Li atoms in several liquid layers, the first one very close to the solid surface, the second layer and an inner central layer. The transverse structures of the second liquid layer and an inner one are very similar pointing to the surface effects on the transverse structure not propagating deeply into the liquid but being restricted to the first liquid layer, in contrast to the results for Al(110). We also note that the position of the second peak in the inner Li gT (r)’s and the second peak in the solid Ca(110) gT (r) coincide. This is consistent with the fact that the main peak position of the bulk g(r) in l-Li99 is at 2.8 ˚ A, very near a/2 for Ca, and is precisely the distance between atoms across the channels of the (110) surface. In contrast, the transverse gT (r) for the first liquid layer has a rather odd shape which appears to be unrelated to those for the solid substrate or that of the liquid, except the position of the first peak. In order to understand the structure of this first liquid layer it is very useful to consider the projections of the trajectories onto the xy-plane for the atoms in the 5

4

gT(r)

3

2

1

0

2

4

6

8

10

12

r (Å)

Fig. 4.40. Transverse pair correlation functions in different layers of the solid Ca-liquid Li system. Vertical lines: ideal fcc (110) surface. Upper lines: first solid layers at the solid vacuum interface (dotted line) and at the solid liquid interface (continuous line). The functions have been inverted and displaced upwards. Lower lines: first liquid layer (continuous line), second liquid layer (dashed line) and an inner layer (circles).

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two adjacent solid and liquid layers. These are shown in Fig. 4.41. As expected from the pair correlation function, the Ca atoms in the solid layer are vibrating around their equilibrium positions for an ideal (110) surface, whereas the Li atoms in the liquid layer are seen to move easily along the channels direction but very little across them. Note that the periodic boundary conditions mean that in fact we are seeing infinitely long lines of a one-dimensional Li liquid. The odd shape of gT (r) is merely the consequence of studying the layer as a two-dimensional liquid, instead of a set of one-dimensional liquids. The picture is much clearer if we compare the structure of the 1-dimensional liquid with the corresponding 1-dimensional solid along the (110) channels. Figure 4.42 shows that the structure of the one-dimensional Li liquid is very simple with g1d (r) peaking at multiples of the atomic diameter (2.8 ˚ A) and a peak width gradually increasing with distance. No relationship is found with the underlying 1-dimensional Ca solid.

Fig. 4.41. Projections onto the xy-plane of the trajectories of the particles in the first solid layer of the solid Ca-liquid Li interface, the first liquid layer and the second liquid layer (from left to right). The hatched circles denote the ideal positions of the fcc (110) surface.

The mean square displacements seen in Fig. 4.43 show that the solid atoms do not diffuse, and that the Li atoms in the first layer diffuse very little in the direction across the surface channels but diffuse readily along the channels at a rate practically equal to that of the Li atoms in the second layer, where they diffuse isotropically. The trajectories of the Li atoms in the second layer shown in the rightmost panel of Fig. 4.41 confirm this isotropic behavior. There are few other results to be compared with our calculations. The only realistic simulations of a liquid on a dissimilar solid that we know of is for l-Al on solid Cu studied using classical MD.191 The Cu structure is fcc but only the (100) and (111) orientations were considered. Both gave stratified DPs with around 6 layers, somewhat less than in our case. The interlayer spacing in l-Al was the same for both orientations and larger than that of the Cu substrate, and tended to ≈ 2.4 ˚ A, similar to the value in the free l-Al interface.210 These results are in agreement with our simulations of the (100) orientation of the interface between l-Al and pinned solid Al, and also with the tendency of liquid Li to layer with the same spacing as for the free liquid surface away from the Ca substrate. The transverse structure

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0.5

4

0.4 2

δ u(t), δ v(t) (Å )

5

0.3

2

g1d(r)

3

0.2

2

2

0.1

1

0

0

2

4

6

8 r (Å)

10

12

Fig. 4.42. One dimensional pair correlation functions for the first solid (upper line, inverted and displaced upwards) and liquid (lower line) layers of the solid Ca-liquid Li interface along the direction of the surface channels.

0

0.1

0.2 t (ps)

0.3

0.4

Fig. 4.43. Mean squared displacements in the directions along and across the surface channels for the solid Ca-liquid Li system. Two lower curves (dotted lines): first solid layer. Two upper curves (dashed lines): second liquid layer. The continuous line near the upper curves corresponds to the first liquid layer along the channels. The other continuous line corresponds to the first liquid layer across the channels.

and the atomic mobility results for liquid Al on solid Cu show both similarities and differences from ours. The first l-Al layer on Cu was strongly influenced by the structure of the solid and showed a spatial distribution that tended to extend the solid structure into the liquid. It also showed diffusion through intersite jumps that proceeded easily because of the presence of vacancies related to the size mismatch between Cu and Al. Our OF-AIMD simulations for Al on pinned Al show a similar picture although the lack of a size mismatch inhibits diffusion. The results for l-Li on Ca substrate differ greatly; the solid structure is not extended into the liquid (see Fig. 4.42) and the Li properties are liquid-like, not only the correlation function but also the diffusion coefficient. Nevertheless, the influence of the underlying solid is strong, forcing the liquid into one-dimensional channels, with infrequent interchannel jumps. Although no results were reported in Ref. 191 for the pair correlation function of inner Al layers, the authors indicated a progressive dissapearance of solid-like features. This is in qualitative agreement with our results for the (100) orientation of the interface between pinned solid Al and l-Al, but in disagreement with those for liquid Li on solid Ca in the (110) orientation, where the transition to isotropic-liquid behavior is abrupt and complete by the second layer. In summary, our study of a realistic interface of l-Li on solid Ca (110) finds negligible relaxation of the solid surface layer, presumably because the mean electron density varies little across the interface. The liquid is stratified with seven to eight pronounced layers and an interlayer spacing which tends towards that of the free l-Li

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surface away from the interface. However, the channels on the solid surface impose a strong topological constraint on the first liquid layer, giving it one-dimensional character. Beyond the first layer the liquid is isotropic and no effects of the lateral structure of the interface remain. 4.6. OF-AIMD study of the melting-like transition in alkali clusters 4.6.1. Background The melting point Tm of a finite atomic system is expected to decrease from its corresponding bulk limit as the number of atoms N is reduced, because of the increased surface to volume ratio. This is indeed the observed behavior at the mesoscale level. However, at the nanoscale clusters formed by a few hundred atoms or less show important deviations from this classical law. Jarrold and coworkers212 have demonstrated that small Ga and Sn clusters melt at temperatures higher than Tbulk m . Calorimetry experiments on alkali clusters, which are the main subject of this section, by Haberland and coworkers11–14,213 show that the size dependence of Tm is not monotonous for Na+ N clusters in the size range N ≈ 50 − 350. In the first set of calorimetric experiments,11 where only a reduced size range was covered, maxima in the Tm (N ) curve where associated with joint electronically and structurally enhanced stability. For example, a local maximum in Tm was observed for Na+ 142 , and the conjecture was made that the close proximity of both geometrical + (Na+ 147 ) and electronic (Na139 ) shell closings produced that maximum. However, later experiments in a wider size range213 showed that Tm -maxima are not correlated in general either with known electronic or geometrical shell closings. A very recent joint analysis of calorimetry experiments and photoelectron spectra214 has demonstrated that maxima in the latent heat and entropy of melting are indeed correlated with geometrical shell closings, which suggests that electronic effects do not play an important role in the melting of sodium clusters with more than 50 atoms. For clusters with less than 50 atoms there is some uncertainty in defining a melting temperature as no calorimetric measurements have been reported. However, measurements of the temperature dependence of the photoabsorption cross 10,215 sections for Na+ Although the spectra show N (N = 4 − 16) have been reported. no evidence of a sharp melting transition, they do show a characteristic temperature evolution, where the different peaks present at low temperature gradually disappear by merging into a single broad peak as the temperature is increased over a critical value. Detailed analysis of calorimetric experiments has also shown that the microcanonical specific heat may be negative at melting for some cluster sizes.14 Also, the relative importance of energetic and entropic effects on the melting transition of Na clusters has been elucidated.216 There were several early attempts to investigate the melting-like transition of alkali clusters with computer simulation.217 Notable among these are the KS-AIMD simulations of R¨ othlisberger and Andreoni218 for small NaN clusters (N=8-20), but

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the clusters were still rigid at 240 K, probably because of the short simulation time of 3-6 ps, which was all that was possible then and is nowadays recognized as insufficient for meaningful statistical averages. Later, KS-AIMD simulations were reported by Bonacic-Koutecky et al.219 on small Li clusters who concluded that even at 500 K these small clusters are not in a fully developed liquid state. Rytk¨ onen et al.220 reported a KS-AIMD simulation of Na clusters with up to 55 atoms and concluded that such a cluster can show a well defined melting transition at temperatures within the scope of experiment. The computational demands of the full KS approach limited these simulations to small clusters and short simulation times. Approximate schemes allowing more extensive simulations have been performed. These include the use of many-body parameterized interatomic potentials,221 a tight-binding Hamiltonian description of the electronic system,222 and a Thomas-Fermi-like treatment of the electrons;223 but these all contain uncontrolled approximations. Aguado et al.68,224 reported the first OF-AIMD simulations of the melting of NaN clusters with N =8, 20, 55, 92 and 142, and achieved moderate success with a primitive kinetic energy functional and pseudopotential. Similar studies of clusters of K, Rb, and Cs obtained qualitatively similar melting features as for Na clusters.225 A recurring feature of the simulations was melting taking place over a significant temperature range with premelting associated with melting of a surface layer. This feature has been the focus of several other simulations.70,226–231 Simulations of NaN clusters over a wide size range have been reported by Manninen et al.232 with N =40-355. Qualitative agreement with experiment was obtained for smaller clusters, and discrepancies may have been due to the use of a phenomenological potential because of computational limitations. Finally, Lee et al.233 and Chacko et al.234 have reported intensive computational studies of the melting transition in Na clusters with as many as 142 atoms with full KS-AIMD calculations. Where direct comparison with experimental melting points is possible, agreement is very good, which leads the authors to conclude that a quantum mechanical treatment of metallic bonding is a prerequisite for an accurate description. A summary of the various quantities used in the analysis of data from molecular dynamics simulations of clusters follows. Results of OF-AIMD investigations of the melting of Na clusters using an improved methodology is then presented. The electron kinetic energy is treated with the averaged density scheme using a rigid weighting function as described in section 4.2.2.3 and the appendix.27,65 The pseudopotential is cluster-adapted from a force matching procedure to the results of Kohn-Sham calculations on selected clusters.69,72 4.6.2. Analysis of the molecular dynamics In order to characterize the thermal behavior of the clusters as a function of increasing internal energy, and the solid-like to liquid-like transitions, we monitor: (a) global quantities that are calculated from time averages over a whole trajectory

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at a given energy; (b) time dependent quantities that are calculated from averages over well separated time origins along a single trajectory. We define the internal temperature T of the cluster as235,236 T=

2 < Ekin >t , (3N − 6)kB

(4.49)

where kB is Boltzmann constant, Ekin is the ion kinetic energy, and t represents the time average over a whole trajectory. All global quantities will be plotted as functions of this internal temperature. We evaluate the following melting indicators: (1) The caloric curve, giving the dependence of average temperature on total energy. At the melting temperature there is a change of slope in the the caloric curve connecting the solid and liquid branches. The height of the rounded discontinuity provides an estimate of the latent heat of fusion. (2) The specific heat per particle (in units of the Boltzmann constant). This quantity is related to fluctuations in the ionic kinetic energy, and has peaks, corresponding to slope changes in the caloric curve, associated with phase transitions. Multiple histogram techniques237 have been employed in order to extract the curves corresponding to this magnitude and the previous one. (3) The root-mean-square bond length fluctuation, q 2 > − < R >2 < Rij X t ij t 2 , (4.50) δ= N (N − 1) it where Rij is the distance between atoms i and j. By restricting the sum to specific pairs of atoms, fluctuations of A–A, B–B, and A–B bond lengths can be evaluated separately. This quantity is a “rigidity index” of the cluster characterizing the mobility of atoms, and increases abruptly at isomerization or melting transitions. For the bulk, a sharp increase in δ gives the Lindemann melting criterion. (4) The diffusion coefficient of atoms of species A, 1 d 2 < rA (t) >, (4.51) 6 dt which is obtained from the long time behavior of the corresponding mean square displacement DA =

2 < rA (t) >=

nt X NA 1 X ~ i (t0j + t) − R ~ i (t0j )]2 , [R NA nt j=1 i=1

(4.52)

where nt is the number of time origins, t0j , considered along a trajectory, and 2 NA the number of atoms of species A. < rA (t) > is a time-dependent quantity that also serves as a measure of the rigidity of the cluster.235 Its slope for large 2 t is proportional to the diffusion coefficient. A flat < rA (t) > indicates that A

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are vibrating about their equilibrium positions. When diffusive motion begins, 2 the slope of < rA (t) > becomes positive. (5) Short-time averages of the “atomic equivalence indexes”,219 defined for each atom by X ~ i (t) − R ~ j (t) |, σi (t) = |R (4.53) j

which contain detailed structural information. The degeneracies in σi (t) are due to the symmetries of the isomer so their temporal evolution allows for a detailed examination of the melting mechanism. 4.6.3. OF-AIMD simulations of melting in Na clusters Some69,72,238 of the main results of our recent simulations69–72,238–244 of the melting phenomenon in alkali clusters are presented here. The results reproduce many of the experimental observations with surprising accuracy and identify some systematics in the evolution of thermal properties with size and structure. Along with all previous experimental and other theoretical contributions, the results suggest that the melting process in sodium clusters begins to be understood in terms of a sound physical basis. 4.6.3.1. Irregular variation of the melting point in a broad size range We consider here the melting-like transition in unsupported NaN clusters with N =55, 92, 147, 181, 189, 215, 249, 271, 281 and 299. These specific sizes are close to local minima or maxima in the experimental Tm (N ) curve of Na+ N cluster ions.12,213 The OF-AIMD simulations reproduce at a quantitative level the irregular size dependence of the melting temperatures, Tm , observed in the calorimetry experiments. We will also find that structural effects alone can explain all broad experimental features. Specifically, maxima in Tm (N ) correlate with a high surface stability and with structural features such as a high compactness degree. Candidate ground state isomers for the clusters were selected by simulated annealing.69 For all sizes, lowest-energy structures were found by annealing icosahedral structures. The resulting geometries were distorted, rounded, incomplete Mackay icosahedra, except for N = 55, 147 and 299, that form perfect icosahedra. Figure 4.44 shows two examples of these structures. For each cluster size, isokinetic Born Oppenheimer MD runs were performed, in which the average kinetic energy is kept constant by velocity rescaling. A time step of 3 fs was used and the total simulation length for each size was 5 ns. Multiple histogram techniques237 were used to extract smooth caloric and specific heat curves. A representative example of these curves is shown in Fig. 4.45. The specific heat curves are similar for all sizes, and display a single peak which is slightly wider on the low-temperature side. The peak asymmetry is due to the

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Fig. 4.44. OFDFT ground state isomers of Na181 and Na215 found by simulated annealing. Atoms at the surface are represented by light spheres and interior atoms by dark spheres.

16

-6

14

-6,02

12

LATENT HEAT

10

-6,04

Cv/NkB

Total energy (eV/atom)

Tm

8

6 -6,06 4 160 180 200 220 240 260 280 160

Temperature (K)

180

200

220

240

260

Temperature (K)

280

Fig. 4.45. Caloric curve (left) and specific heat per particle in units of kB (right) of Na299 . The latent heat is obtained from the caloric curve as shown. Tm signals the beginning of melting.

melting process: the melting initiates with surface atoms, but the surface and homogeneous melting are so close they merge into a single asymmetric specific heat peak. The melting temperature and the latent heats, obtained respectively from the maximum in the specific heat peak and from the step height between liquid and solid branches of the caloric curve (see Fig. 4.45) are in quantitative agreement with

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experiment.213 Figure 4.46 shows the experimental and theoretical values of T m as a function of the number of atoms in the cluster. This level of agreement has not been achieved over such a broad size range previously.

Fig. 4.46. clusters.

Experimental and theoretical size variation of the melting temperature in sodium

Melting begins at the surface of the cluster, so the stability of the surface is an important factor in the melting temperature. Indeed, Aguado and L´opez have shown that the relative surface stability is directly related to Tm . In addition, surface metal atoms tend to undergo a bond length contraction to compensate for the reduced electronic density at the surface, and also, very low coordination dangling atoms are avoided. The result of these two rules are ground state geometries that optimize the ion packing without surface steps or dangling atoms leading to a smooth geometry with rounded surfaces as seen in Fig. 4.44 The presence of all surface atoms in a single, rounded, surface shell leads to oscillations as a function of N in the average distance between surface atoms (dss ) and the average distance between surface and bulk atoms (dsb ). When dss is relatively large, dsb may be reduced, resulting in additional stabilization of the cluster surface which inhibits melting. For example, the anomalous very high melting temperature of Na55 is due to its higher compactness relative to other sizes. 4.6.3.2. Variation of the melting point in a narrow size range: N = 135 − 147 This same procedure was used by Aguado72 to study the melting transition in unsupported NaN clusters, with N = 135 − 147 and results show the main trends observed in the calorimetric experiments214 in that range. The size dependence of the latent heat and entropy of melting are maxima at N = 147 suggesting a correlation with geometric shell closings. The electronic shell closing effects are of secondary importance to the melting process, at least within this size range. The

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maximum in Tm is for Na141 which can be accounted for in terms of two different thermally-activate structural instabilities which trigger the melting transition in the ranges N = 135 − 141 and N = 141 − 147. The two mechanism which have been observed are: a) isomerizations where an atomic vacancy at a surface vertex site moves to a different surface vertex. This mechanism dominates the N = 135 − 141 range. For example, three surface atoms move together along an icosahedral edge so that the vacancy can jump between vertex sites directly, see Fig. 4.47; b) isomerizations, where a surface atom moves from an edge to a hollow vertex

Fig. 4.47. Snapshots taken from an molecular dynamical run on Na145 , showing the edge running premelting mechanism. The surface vacancy can jump directly between vertex sites due to the cooperative displacement of the whole edge of atoms marked in grey. Two other surface atoms are marked in white in order to better appreciate the mechanism

position, so that the vacancy can explore also the edge sites at the surface. This mechanism dominates the range N = 135 − 141. The surface shell is formed from three different radial sub-shells because face, edge and vertex surface atoms have different radial distances. The excitation of an edge atom to a hollow vertex site can be viewed as a thermal generation of a floater atom which is later neutralized by a neighboring surface vacancy. These effects produce an increase in the entropy per atom of the solid phase before the cluster melts, and provide an explanation for the size evolution of the melting entropy. The size dependence of the activation energy for these mechanisms explains the size dependence of the melting temperature. 4.6.3.3. Small sodium clusters that melt gradually: melting mechanisms The clusters that do not show a well-defined melting transition fall into two categories: (1) Amorphous clusters242,245–247 that have been shown to melt gradually over a very wide temperature range. (2) Experiments suggest that a smooth melting transition should also be expected for clusters below a small critical size, which for Na clusters seems to be ≈ 50 atoms.

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In this section the results of the melting transition for Na30 will be summarized.238 The results reproduce the experimental trends, and an explanation for them is provided. Different dynamical mechanisms that result in the approximately continuous increase of the available phase space with temperature are identified. Then, unpublished results on the melting transition for clusters with N = 6 − 25 will be presented. Na30 cluster Several isokinetic MD runs were performed in order to obtain the caloric curve for each cluster. The simulated time was between 1 and 2 ns for each isokinetic run, adding up to a total simulation time of about 45 ns, much greater than presently affordable by other ab initio techniques. Some KS-AIMD calculations using the SIESTA code248 were also performed to check the accuracy of the OF forces and to compare the potential energy surfaces (PES) generated by the KS and OF techniques. In order to characterize the thermal properties and melting transition in Na30 , we have evaluated a number of well known indicators,68,71,239,249,250 such as the caloric curve, specific heat, root-mean-square bond length fluctuation and mean square displacements, and studied their dependence on the kinetic cluster temperature controlled in the isokinetic ensemble. In order to identify the dynamical processes leading to melting we have visually inspected the atom trajectories and analyzed the very powerful atomic equivalence σi indices (AEI).219 These, which are short-timeaveraged to eliminate vibrational noise, contain very detailed structural information, and are very sensitive to slight atomic rearrangements that might be overlooked in the visual inspection of MD movies and/or which are not manifest in the global indicators mentioned above. Structure In order to locate the ground state isomer of Na30 , two independent simulated annealing runs were performed. The cluster was equilibrated at 200 K and cooled at a rate of 0.05 K per picosecond. In order to have a good sample of local minima on the PES, regular quenchings were performed starting from atomic configurations sampled during the high temperature part of the simulated annealing runs. Both runs led to the same GS isomer shown in Fig. 4.48, and none of the quenchings located a lower energy isomer. The ground state isomer shows a polyicosahedral packing to be energetically favored for this size. It contains four interior atoms with a slightly distorted tetrahedral structure. The rest of atoms are at the cluster surface, and are positioned so that the first coordination shell of each interior atom is a slightly distorted icosahedron. Figure 4.48 shows that the isomer results from the combination of two 19-atom double icosahedra (see Fig. 4.55), rotated by approximately 90 degrees with respect to each other. Preferential polyicosahedral packing is in qualitative agreement with a basin hopping global optimization performed by Calvo et al.,251 who found evidence for this particular atomic packing employing three different energy models for Na clus-

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Fig. 4.48. Orbital-free GS isomer of Na30 . Light and dark spheres differentiate atoms in core and surface regions, respectively. The four core atoms form a tetrahedron at the cluster center. Two different views, obtained by rotation of 90o , are shown to better appreciate the distorted double-icosahedron Na19 building blocks.

ters, but there are discrepancies. Preference for a polyicosahedral packing has also been observed in Lennard-Jones clusters in a similar size range.252 Although a direct comparison with experiment is not possible for a neutral cluster, photoabsorption experiments253 imply that Na+ 30 is triaxially deformed at 105 K. Our GS isomer is triaxial with a very small oblate character. Also, the photoelectron spectra of Na− N cluster anions254 show the emergence of a new low-energy peak for Na− 30 , indicating electronic shell closure for Ne = 30 electrons, and importance of the ionic structure at low temperatures. Even though the GS structures of the neutral and cluster anion probably differ, we note that a KS relaxation of our Na30 geometry gives a large HOMO-LUMO gap of 0.34 eV, to be compared with e.g. 0.46 eV between Ne = 8 and Ne = 9. Our results are thus in reasonable agreement with the experiment in this size-range, and do not support the alternative significantly prolated structures found by K¨ ummel et al.255 At higher temperatures, where detailed ionic structure is less important, our OF-AIMD simulations predict a prolate deformation increasing with temperature, in agreement with the jellium model for a cluster with 30 electrons. However, it would be a mistake to compare directly the properties of this GS isomer and the results of photoabsorption or photoelectron experiments performed at ≈ 105 K. At that temperature, we find several isomers and structural rearrangements contributing to the spectra. Sampling of the PES reveals many structural isomers in a tiny energy range of 2 meV/atom, in agreement with other work256–258 using sophisticated energy models. KS relaxation of a group of these isomers shows that the energy ordering is modified from that of the OF calculations, but there persists the important feature that a large number of isomers have very similar energies. Although the potential energy landscape is typical of a glassy system (a desirable property if we wish to reproduce a featureless caloric curve) individual isomers have substantial structural order and are not amorphous, and dynamical processes must

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differ from those acting in an amorphous cluster. Our working hypothesis is that the thermal and melting properties of Na30 are most sensitive to the global features of the potential energy landscape showing a large number of isomers with similar energies, rather than to the specific structure of the GS isomer. With such small energy differences, the GS structure and the order of isomer energies will be highly dependent on the energy model, as we have verified in a series of calculations. In summary, even though the GS of Na30 is uncertain we are confident that the cluster adopts one of many polyicosahedral packings which lie very close in energy, and it is this feature that determines the character of the melting transition. Melting Figure 4.49 shows the calculated caloric curve, specific heat, rms bond length fluctuation parameter δ, and the mean square displacements < r2 (t) > for Na30 as a function of temperature. All indicators suggest rather gradual melting, but none give much detail of the process. Detail is provided by the atomic equivalence indices.

150

200

Temperature (K) 250

50

100

150

200

250

0.4 0.3

-6.94

0.2

-6.96

0.1 50

4

4

3.8

3

3.6 2

3.4

1

3.2 3

0

50

100

150

200

Temperature (K)

250

300 0

0.5

1

1.5

2

time (ps)

2.5

3

2

4.2

(a.u.)

-6.92

Specific heat (k B)

Total energy (a.u.)

-6.9

100

δ

Temperature (K) 50

0

Fig. 4.49. Caloric curve (upper left), specific heat (lower left), rms bond-length-fluctuation parameter δ (upper right), and mean-square displacements (lower right) of Na30 as a function of temperature.

Figures 4.50(a), 4.50(c) and 4.50(e) show the atomic equivalence indices, σ, of Na30 for three representative runs in the low temperature range below 100 K. At 30 K, Fig. 4.50(a) shows four small σ values associated with the tetrahedral Na4 core seen in Fig. 4.48. The grouping of the larger σ values shows that the surface shell

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can be partitioned into two subshells. Even at 30 K, the cluster has enough energy for permutational isomerization in which isomers topologically identical to the GS isomers are visited through the concerted movement of a subset of surface atoms. Surface atoms switch between inner and outer surface subshells with no appreciable diffusion. These structural rearrangements have a very small activation barrier.238 Figure 4.50(c) shows more dramatic isomerization taking place at 40 K, that is illustrated in Fig. 4.51. The cage of surface atoms allows some rotational freedom of the quite rigid inner tetrahedron. This rotation is mostly performed about one of the tetrahedron edges chosen to be perpendicular to the page in Fig. 4.51. Starting from the GS isomer on the left, the tetrahedron rotates to a new orientation in the cage, with some small modification of the surface structure (middle plot). After which, surface atoms rearrange to give a permuted version of the GS (right plot). The intermediate isomer in the middle plot is responsible for σ lines shown in the middle of Fig. 4.50(c). The floppy character of Na30 allows substantial isomerization even at these low temperatures. Figures 4.50(b), 4.50(d) and 4.50(f) shows the atomic equivalence indexes of Na30 above 100 K, for three representative runs. The 90 K simulation suggested that the cluster was about to melt, but at ≈ 100K a new type of transition occurred in which the structure of the cluster core changes, although no core-surface atomic interchanges are involved. The new structures stabilized at this temperature are found by quenching and are shown in Figs. 4.52 and 4.53. The four inner atoms now form a planar rhombus (Fig. 4.52) or a closely related open planar structure (Fig. 4.53), and the surface atoms provide local icosahedral environments for the core atoms. The structure in Fig. 4.52 can also be visualized as a lateral coalescence of two 19-atom double icosahedra. The isomer shown in Fig. 4.53 can be interpreted similarly but with the main axes of the two double icosahedra not parallel. At this high temperature the two structures cannot be distinguished because of the similarity of their σ lines. This transition transfers energy from the surface to the core shell preempting surface melting. The new structure will have a higher thermal stability and melting point, consistent with a solid-solid transition. In the temperature interval 100–130 K, we find dynamical coexistence between the new isomers with the planar core and the surface-melted isomer with the tetrahedral core. The cluster fluctuates between these two possibilities, spending enough time in each for well defined properties of each isomer to be assigned. At ≈ 130 K, a new isomer with five interior atoms is accessed, as shown in the central part of Fig. 4.50(d). As before, several views of the new structure identified by quenching are shown in Fig. 4.54. The five core atoms form a trigonal bi-pyramid, with each of the five continuing to have a distorted icosahedron for its local coordination shell. Again there is dynamic coexistence between isomers with 5 and 4 interior atoms through which interchange of core and surface atoms occurs for the first time. The new isomer has a thermally more stable surface shell, and during the time spent in its catchment basin, surface melting is not so clearly established as

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(a) T = 30K

(b) T = 105K

(c) T = 40K

(d) T = 140K

(e) T = 90K

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(f) T = 170K

Fig. 4.50. Short-time average atomic equivalence indices, σi (t), as functions of time for Na30 cluster at different temperatures

for the isomers with a 4-atom core. In this temperature range, generation of floater atoms is also possible, although they exist briefly and cannot diffuse very far from the place where they were created. This is shown in the σ lines which temporarily adopt abnormally high values in Fig. 4.50(d).

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Fig. 4.51. Pictorial representation of the isomerization process identified in the run at 40 K. Light and dark spheres represent core and surface atoms, respectively.

Fig. 4.52. Excited-state isomer of Na30 visited in the temperature region about 100 K (see Fig. 4.50(b)). Light and dark spheres differentiate atoms in core and surface regions, respectively. The four core atoms form a planar rhombus at the cluster center. Both front and side views are shown.

Figures 4.50(e) and 4.50(f) show that upon further temperature increase the rate at which the rearrangement processes discussed so far occur steadily increases, but at ≈ 180 K, the rate of core-surface inter-shell diffusion increases markedly (Fig. 4.50(e)) and surface melting is complete irrespective of the number of core atoms. This explains the main peak in the heat capacity. But even at a T = 250 K (not shown), the core-surface diffusion remains substantially slower than surface diffusion and the picture is different from that of a homogeneous bulk liquid. In summary, the simulations reproduce the experimental observation216 that small sodium clusters melt gradually over a very wide temperature interval, with no abrupt features in the caloric curve and thus with a very small latent heat. Two categories of dynamical mechanism operating in Na30 have been identified: (i) those allowing the exploration of isomers similar to the ground state and which are already active at 30 K. These involve concerted surface rearrangements, changes in the relative orientation of core and surface shells, and surface melting, when

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Fig. 4.53. Excited-state isomer of Na30 visited in the temperature region about 120 K; it has a similar set of σ lines as that shown in Fig. 4.52. Light and dark spheres differentiate atoms in core and surface regions, respectively. The four core atoms adopt an open planar structure at the cluster center. The structure can also be seen as a decorated pentagonal bi-pyramid Na7 (right part). The local environment of each inner atom is icosahedral.

Fig. 4.54. Three different views of the excited-state isomer of Na30 visited in the temperature region starting at about 130 K. Light and dark spheres differentiate atoms in core and surface regions, respectively. There are five core atoms which adopt a trigonal bi-pyramidal structure. The local environment of each inner atom is icosahedral.

they happen sufficiently rapidly; (ii) mechanisms associated with more substantial structural change involving all atoms including the core, becoming important at ≈ 100 K. While mechanism (i) can only lead to surface melting of the corresponding isomer, (ii) is more likely to act once the cluster has attained a surface melting stage. NaN , N = 6 − 25 clusters The observed caloric curve of small clusters does not show a well defined melting transition but rather suggests a gradual evolution of the solid clusters towards the liquid-like state.216 Here we present an OF-AIMD study

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of the melting-like transition in NaN , N = 6 − 25 clusters. Although details of the melting are only presented for two of the clusters, the GS isomers for all found using the same annealing procedure as for Na30 are shown for completeness in Fig. 4.55. The GS geometry of N = 6 is an octahedron, and of N = 7 a pentagonal bipyramid, which is the starting point for the icosahedral growth observed later. An octahedron with two added atoms was found for N = 8, and for N = 9 the GS is a pentagonal bi-pyramid with two added atoms covering two adjacent triangular

Fig. 4.55. cluster

NaN (6 ≤ N ≤ 25) ground state geometries obtained by quenching from the liquid

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faces. For N = 10 and N = 11 the growth continues by covering the faces of the bi-pyramid, for N = 12 an icosahedra with a missing vertex is obtained and for N = 13 a perfect icosahedron is formed. For N = 14 an atom is added to one face of the icosahedron. The external surface of the N = 15 is modified to accommodate the last two added atoms and the two five-atom rings are opened forming two rings of six atoms, producing a smoother external surface. Adding an atom to form N = 16 increases the surface distortion, and one of the six atom rings is deformed by the addition of the last atom near one of the hexagonal vertices of the previous cluster. This process of deformation of the external surface of the cluster in order to accommodate more atoms while avoiding the formation of the next shell was also identified earlier259 for N = 56 − 58. For N = 17 the extra atom cannot be accommodated in the surface and the next concentric shell is begun. For N = 18 the five atoms added to the N = 13 cluster form a five atom ring around a vertex of the perfect icosahedron. This process is completed in N = 19 forming a double icosahedra which for the first time has two central atoms. It is the simplest example of poly-icosahedral growth, where the local atomic environment for each internal atom is the same as in the 13 atom icosahedron. This kind of growth optimizes the atomic packing but has only a limited range of stability because it creates a huge strain in the bonds.9 For N = 20 the GS has the added atom covering one edge of the central ring of the N = 19 cluster. For N = 21 and N = 22 the trend to avoid the creation of a new layer with low coordination atoms is seen again. The accumulated strain in the bonds leads to the start of a new layer in N = 23, N = 24 and N = 25. The cluster with N = 23 atoms is the smallest with three internal atoms. Melting. The melting-like transition in several of the N = 6 - 25 clusters has been studied. A 30 - 270 K temperature range was covered by the simulations and the total simulation time, for each temperature, varied from 600 to 1800 ps depending on the number of atoms in the cluster and on the temperature region. The runs were longer for temperatures near the melting temperature. Overall, the results are similar to those for Na30 presented earlier. At low temperatures the atoms oscillate around their equilibrium positions. When the temperature increases atom interchanges occur and the cluster starts to visit permutational isomers. Then, the cluster begins to visit new configurations, the isomerization rate increases with the temperature and finally the cluster melts. For the clusters with one or more central atoms, the melting occurs first in the surface and then extends to the rest of the cluster, but even for high temperatures, the rate of core-surface diffusion remains substantially lower than surface diffusion. The melting of the Na8 and Na20 clusters will be discussed in more detail in order to investigate discrepancies in the specific heat reported by Aguado et al.68 and Vichare et al.227 The problem is revisited with the more accurate energy functional and longer simulation times.

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Na20 cluster. Figure 4.56 shows short-time average atomic equivalence indices for the Na20 cluster for increasing values of the temperature. In Fig. 4.56(a) the equivalence indices represent the ground state geometry, with the two smaller σ values for the two central atoms and the others for the atoms in the external shell. For T = 55 K (Fig. 4.56(b)) the cluster visits permutational isomers of the ground state with no participation of the internal atoms. At T = 95 K (Fig. 4.56(c)) the interchange between the external shell atoms is more frequent and the cluster is now able to visit atomic configurations different from the ground state. In Fig. 4.56(d) (T = 150 K) we see that the external shell is totally melted and the internal atoms start to participate in diffusion. Finally, if the temperature is increased further the frequency of the interchanges between the core and external shell increases and the cluster is considered to have melted. Figure 4.57 shows four different isomers visited by Na20 in the heating process. All the isomers have two internal atoms. In the first row, the ground state geometry and three different views of the first isomer are shown. In the second row are three more isomers. The number below each isomer indicates the energy difference between it and the ground state in 10−3 a. u. 250 240

σi (a. u.)

σi (a. u.)

225 220

200

180

200

175

160

150 140 0

100

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300

400

500

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250

500

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Time (ps)

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1750

Time (ps)

(a) T = 40K

(b) T = 55K 275

250

250

σi (a. u.)

σi (a. u.)

225

200

175

225

200

175 150

125

150

0

100

200

300

400

Time (ps)

(c) T = 95K

500

600

125

0

100

200

300

400

500

600

Time (ps)

(d) T = 150K

Fig. 4.56. Short-time average atomic equivalence indices (σi (t)) as functions of time for the Na20 cluster for different temperatures

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GS

+1.715

idem

+1.42

+2.338

129

idem

+5.223

Fig. 4.57. Isomers visited in the heating process of the Na20 cluster. First row: ground state and three different views of the first isomer. Second: three other isomers. The number below each isomer indicates, in units of 10−3 a. u., the difference in energy from the GS

The first isomer, 1.142 10−3 a. u. above the GS state, breaks the vertical axis of the two central atoms in order to increase the coordination of the external atom. This isomer is an example of rotation of the internal core with respect to the external shell and the slight rearrangement of the external shell in order to maximize the coordination of the atoms. The three different views shown in Fig. 4.57 demonstrate this. The second isomer 1.715 10−3 a. u. above the GS recovers the symmetry of the vertical axis but incorporates the external atom in the central six atom ring increasing its coordination. The next isomer, 2.338 10−3 above the GS, has the same basic structure as the GS but with an atom added to the Na19 double icosahedra in a different position. The last geometry shown, 5.223 10−3 a. u. above the GS, does not have icosahedral symmetry and is only visited at high temperatures. The geometry and energy of these isomers were found by quenching from one of the different atomic configurations located by inspecting the σ ’s and using a steepest descent method to cool the system to the nearest minimum. Figure 4.58 shows the specific heat and the rms bond-length-fluctuation parameter δ as functions of the temperature of the cluster. Both indicators show melting of the cluster spreading over a very wide temperature range. The small peak in Cv

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at T ≈ 75 K is related to the temperature region in which the cluster has enough thermal energy to visit atomic configurations different from the ground state. The local minimum around T = 100 K is the point where the cluster has enough energy to visit all the isomers with two central atoms, and the surface is melted. This is also the starting point of the final process of melting in which the central atoms interchange with atoms of the external shell. At T ≈ 160 K the cluster melts. Similar information can be extracted from the δ graph: δ increases with the temperature as the access to the phase space of the system is increasing; near T = 75 K, the slope increases indicating that the access to more isomers is accelerating; a similar change occurs after T ≈ 120 K when interchanges between internal and external atoms begins; finally just after the melting of the cluster at T ≈ 160 K the value of the δ tends to plateau. 4.75

0.35

4.25

0.3

4

0.25

3.75

δ

Cv (kB )

4.5

0.2

3.5

0.15 3.25

0.1

3 2.75 2.5

0.05 50

75

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175

Temperature (K)

(a) Specific heat Na20

200

225

0

50

75

100

125

150

175

200

225

250

Temperature (K)

(b) Na20 rms bond-length-fluctuation parameter δ

Fig. 4.58. Specific heat Cv and bond-length-fluctuation parameter δ for Na20 as functions of temperature.

These OF-AIMD results for N a20 are in accordance with those obtained by other authors227,231,233 and show that the existence of two sharp maxima in the Cv obtained earlier by Aguado68 were mainly a consequence of the short simulation times, 8-60 ps rather than the 600-1800 ps of calculations reported here. Na8 cluster. Unlike the Na20 and Na30 clusters, Na8 does not have central atoms. Figure 4.59 shows that at very low temperature (40 K), the cluster has enough energy to visit a new isomer for a short 570 ps time interval. This new isomer is the pentagonal bi-pyramid with an atom capping one of the triangular faces seen in Fig. 4.60. The isomer energy is 1.052 10−3 a.u. above the GS. Also shown are the second and third isomers visited which are octahedrons with two opposite faces and two contiguous faces capped respectively. The energies are 2.127 10−3 and 2.152 10−3 a. u. above the GS. The specific heat curve as a function of temperature is presented in Fig. 4.61. At

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isomer 65

σi (a. u.)

GS 60

55

50

45

0

200

400

600

800

1000

1200

Time (ps) Fig. 4.59. Short-time average atomic equivalence indices (σi (t)) as a function of time for Na8 (T = 40K)

GS

+1.052

+2.127

+2.152

Fig. 4.60. Isomers visited in the heating process of the Na8 cluster. The number below each isomer indicates, in units of 10−3 a. u., the difference in energy from the GS

low temperature Cv starts to increase when the first isomer is visited. Between T ' 75 K and T ' 115 K the curve flattens and then drops. From then on Cv increases reaching a maximum around T = 170 K, where we can consider the cluster to have melted. Figure 4.62 shows the short-time average atomic equivalence indices for Na20 (left panel) and Na8 (right panel) for T = 170 K at which temperature both clusters are melted. Only in the Na8 clusters are all the atoms participating at the same rate in the position interchange; the central atoms in the Na20 cluster interchange less frequently with the external atoms than the external shell atoms do among themselves. The rms bond-length-fluctuation parameter, δ, which is not shown, behaves similarly. Both quantities indicate that melting of the clusters spreads over a very wide temperature range, as found by other authors.226,227,257

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3

Cv (kB )

2.8

2.6

2.4

2.2

40

60

80

100

120

140

160

180

200

Temperature (K) Fig. 4.61.

(a) Na20

Specific heat for Na8 as a function of temperature

(b) Na8

Fig. 4.62. Short-time average atomic equivalence indices (σi (t)) as a function of the time for Na20 and Na8 , at T = 170 K − Na+ 8 and Na8 . In order to check if there might be important differences between the behaviors of neutral and the charged clusters, which are the only ones for which experimental data is available, OF-AIMD simulations of the melting of − Na+ 8 and Na8 clusters have been performed. The ground state for charged Na8 clusters was obtained in the same way as for the neutrals: the cluster was heated up to the liquid state and then cooled slowly the process was repeated several times. The ground state geometries for the cation and the anion, found in the way described earlier, was the same as that obtained previously for the neutral but with small differences in the interatomic bond lengths (this result is not universal, for other systems, i.e. Aluminium, the charged clusters have a different geometry from the neutral ones.260 ) The interatomic distances are larger in the charged

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clusters than in the neutral one due mainly to the Coulomb repulsion. The energy differences between the ground states of the charged and neutral clusters correspond to the adiabatic ionization potential (0.161 a.u.) and the electron affinity (0.0778 a.u.). Figure 4.64 shows the specific heats of the neutral, positively and negatively charged Na8 clusters as functions of temperature. Apart from the presence of the peaks for T ≈ 50 K for the charged systems, the general trend is for Cv to be larger than for the neutral one. The low temperature peak in Cv for each of the charged clusters is related to their larger bond-length. The bonds are softer and less thermal energy is needed for visits to the nearest excited isomer. Confirmation is found in Fig. 4.63 showing the short-time average of the atomic equivalent indices for Na+ 8 and Na− clusters (left panel) and the corresponding rms bond-length-fluctuation 8 parameters δ (right panel). We show in Fig. 4.63(a), for a simulation at T = 40 K, the charged systems oscillating between the geometries of the ground state and the first excited isomer. The Na+ 8 cluster goes back and forth several times between the geometries; the Na− 8 make only one oscillation but remains in the excited isomer for a long period of time. In contrast, the Na8 cluster (see Fig. 4.59), also for T = 40 K, is able to access the first isomer only for few picoseconds. Also, in Fig. 4.63(b) we see that the rms bond-length-fluctuation parameter for the charged clusters increases more quickly than for the neutral one as a function of temperature, also implying that the charged clusters need less thermal energy to visit the excited isomers than the neutral one.

0.25

Na+ 8 0.2 70

σi (a. u.)

δ

Na− 8

0.15

60 50

0.1 0 70

500

1000

Na8

0.05

60 50

0 0

500

1000

Time (ps) − (a) Na+ 8 upper panel. Na8 bottom panel

25

50

75

100

125

150

175

200

Temperature (K)

(b) δ parameter

Fig. 4.63. Short-time average atomic equivalence indices (σi (t)) as functions of time for Na+ 8, − + Na− 8 and Na8 clusters, and δ parameters for Na8 , Na8 and Na8

The specific heat curves for the charged and neutral Na8 clusters show melting spread over a wide temperature range in all cases with the neutral having the highest − melting temperature at ' 170 K, and the Na+ 8 and Na8 melting at ' 160 K and ' 150 K respectively. Figure 4.64 indicates evaporation events contributing to Cv for

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3.2

Evaporation events 3

Cv (kB )

Na+ 8

Na− 8

2.8

2.6

Na8

2.4

2.2

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60

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140

160

180

Temperature (K)

Fig. 4.64.

− Specific heat as a function of temperature for Na8 , Na+ 8 and Na8 clusters

the charged clusters just a few degrees above the melting point, another indication of weaker bonds in the charged than in the neutral. This section demonstrates that OF-AIMD allows the reliable, large scale simulation of alkali clusters. The specific problem given as an example was the thermal properties and the melting-like transition of Na clusters of varying sizes. The method reproduces the irregular size dependence of the melting temperature observed in the calorimetric experiments in a broad size range. The maxima in the melting temperature correlate with high surface stability of the cluster and with structural features such as a high degree of compactness. The method also reproduces the main trends in the evolution of the melting temperature as a function cluster size in a range around one of the maxima, and the existence of the maximum can be explained in terms of two different thermally induced structural instabilities. The size dependence of activation energies for these mechanisms accounts for the size dependence of the melting temperature. The simulations show that small clusters melt gradually over a very wide temperature interval, with no abrupt features in the caloric curve and consequently a negligible latent heat, again in agreement with experiment. The exploration of the potential energy surface reveals many structural isomers within an energy range of about 1 meV/atom based on preferential poly-icosahedral packing, and the landscape is that of a typical glassy system. Nevertheless, these structures are not amorphous, all of them have high symmetry and radial order. Finally, the mechanism allowing the smooth opening of the phase space available to the system as its temperature increases has been identified. We found that for clusters with two shells, the external shell melted first, and the rate of core-surface atom exchange increases greatly upon total melting of the cluster but the rate is always lower than that of surface diffusion.

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4.7. Conclusion In this chapter we have described the orbital-free molecular dynamics method which allows large-scale simulations of metallic systems while taking account of the electronic structure. The main computational advantage of eliminating the set of KohnSham orbitals and working directly with the electron density are that the computation scales linearly with the number of atoms in the sample so that total computational effort increases much more slowly with the size of the system than with the full Kohn-Sham method. Significant other advantages are the much reduced memory requirements and the elimination of the sum over occupied states which is particularly tricky for partially filled bands as in the case of metals. However, there are major disadvantages over the Kohn-Sham method. An approximate electron kinetic energy functional is needed and a local electron-ion pseudopotentials must be used. The various possibilities for these are highlighted in section 4.2. The scope of the OF-AIMD method has been illustrated by applications to specific systems. Because of the efficiency of the method it is particularly useful for simulating systems for which a large sample size is necessary, and/or long simulation times are needed. Bulk liquids, the liquid-vapor interface, the liquid-solid interface and large clusters of atoms are such examples. The spatial disorder means that with periodic boundary conditions a sample containing many atoms is required to mimic the physical system. The dynamical properties of these systems, and even ordered systems, requires monitoring the configuration of the atoms over long time periods so that long simulation runs are necessary. Sections describe examples of the simulations of liquid metal and alloys, the liquid-vapor interface, the interface between liquid and solid metal, where the focus was on the static and dynamical properties. The final section describes the simulation of metal clusters in which the dependence of the melting of the cluster on the cluster size was studied. Although full Kohn-Sham simulations have been performed for many of the systems considered here the heavy computational demands limit the sample size to questionably small numbers of atoms, and the short simulation runs limit the study of dynamical effects to high frequencies only. Let us now put the OF-AIMD method in a more general context. Schemes use an approximate kinetic energy functional and a pseudopotential. Obviously, the need to generate a pseudopotential with its associated uncertainties is eliminated if all the electrons, including the core electrons, are treated on the same basis. In this case the electron-ion interaction is known to exquisite accuracy; it is simply the Coulomb interaction −Z/r between the nuclear charge, Z, and the electron. However, the prospect of developing a kinetic energy functional that can handle the very large and rapidly varying electron density in the core region is remote. Furthermore, total kinetic energies including the core are very much larger than the structural energies of interest, and these would be swamped by even a tiny percentage error in the kinetic energy. An additional technical difficulty would be the problem of representing the now rapidly varying all-electron density; we could

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no longer use the very convenient plane wave basis set which is so easy to manipulate with the fast Fourier Transform. For the moment we are stuck with the need for both an explicit kinetic energy functional and a local pseudopotential. With the presently developed approximations for the electronic kinetic energy functional and local pseudopotentials the OF-AIMD technique provides reliable results that compete in accuracy with those of the full Kohn-Sham method for simple metals such as the alkalis, alkaline-earths, and some trivalent metals like aluminum, and their alloys. Further progress in our understanding of the Ts [n] functional and in the methods for generating local pseudopotentials that work as well as the best nonlocal potentials is needed in order to extend the applicability of the OF-AIMD method to other metals. Although the qualitative success in the treatment of l-Ga and l-Si is most promising. It can be concluded that present Ts [n] functionals work better the closer is the simulated system to a nearly-free electron metal. Indeed, most approximate functionals are constructed to meet the limiting case of an electron gas with small departures from a homogeneous system. Systems with strongly inhomogeneous electron densities such as alkali halides and those for which nonlocality in the pseudopotential is a necessity, such as involving the first row or 3d elements, are presently beyond the scope of the OF method. However, even with the limitations of the OF method as it is today there are many opportunities for using the method to gain understanding of systems that cannot yet be addressed by orbital-based ab initio methods. In addition to the sort of systems discussed earlier, other examples include the solid-liquid interface of bulk metals where OF-AIMD simulations could be used to extract melting curves, and the thermal properties of nanoalloys. But, there is also much work to be done exploring the functional relationship between the independent particle kinetic energy and the electron density and developing an explicit functional with the goal of a truly density based ab initio scheme in the spirit of density functional theory. Acknowledgements This research was supported by Junta de Castilla y Le´on, the Spanish MEC, and the the European Regional Development Fund (Projects No. FIS2008-02490 and No GR120). MJS acknowledges the support of the NSERC of Canada. Appendix A. The kinetic energy functional We consider the kinetic energy functional Ts [n] = TvW [n] + Tβ [n] where Tβ [n] =

3 10

Z

˜ r )2 d~r n(~r)5/3−2β k(~

(A.1)

(A.2)

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˜ r ) = (2k¯F )3 k(~

Z

d~s k(~s) ωβ (2k¯F |~r − ~s|) ≡ k(~r) ∗ ωβ (2k¯F r) k(~r) = (3π 2 )1/3 n(~r)β

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(A.3)

(A.4)

In the limit of small deviations from a uniform system, we wish to recover the LR result. Equating the Fourier transform of the second functional derivative of Ts [n] with respect to n(~r) for n(~r) = n ¯ , to the inverse of the Lindhard response function, gives for the weight function     20 10 5 2 6β − β + + 4β − 2β ω β (η) + 2β 2 ωβ (η)2 = 3 9 3
10 1/πL (η) − 3η 2 9

where η = q/2k¯F , ω β is the Fourier transform of ωβ and   1 1 − η 2 1 + η πL (η) = 1+ ln . 2 2η 1 − η

(A.5)

(A.6)

is the noninteracting homogeneous electron gas response function. Taking in Eq. (A.5) the solution which satisfies the normalization condition ωβ (η = 0) = 1, and with β within the range 0 ≤ β ≤ 5/6 so that the power of n(~r) in Eq. (A.2) is positive, the weight function is given by q 5 1 −1 ω β (η) = 2 − + (5 − 3β)2 + 5(πL (η) − 1 − 3η 2 ) . (A.7) 3β 3β Requiring ω β to be real places a stricter limit on β: β ≤ 0.5991. With this choice of weight function, the functional recovers the LR limit, and in the limit of uniform density it reduces to the TF functional. In the limit η → ∞ we have ωβ (η) → C1 + A/η 2 + · · ·

(A.8)

where C1 = 2 −

1 p 5 + 17 − 30β + 9β 2 3β 3β

(A.9)

The constant C1 gives rise to a Dirac delta function in the real space; therefore it is convenient to define a “modified” weight function ω ˜ β (η) = ωβ (η) − C1

(A.10)

so that a convolution involving ωβ , such as in Eq. (A.3), becomes f (~r) ∗ ωβ (2k¯F r) = C1 f (~r) + f (~r) ∗ ω ˜ β (2k¯F r) .

(A.11)

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An important limit is when the mean electron density, and therefore k¯F vanishes, for instance in a finite system. Now, the convolutions involving the “modified” weight function vanish because η = q/2k¯F → ∞ and ω ˜ (η) vanishes. Con˜ r ) = C1 k(~r), and the kinetic energy functional becomes T [n] = sequently, k(~ TvW [n] + C12 TT F [n], and when β = 4/9, C1 = 0. Appendix B. Position-dependent chemical potential The functional derivative of Eq. (4.3) gives µ(~r) = µvW (~r) + µβ (~r) + Vext (~r) + VH (~r) + Vxc [n(~r)]

(B.1)

where µvW (~r) =

2 ~ r ) ∇n(~ 1 8 n(~r)2

VH (~r) =

Z

d~s

−

1 ∇2 n(~r) , 4 n(~r)

n(~s) , |~r − ~s|

(B.2)

(B.3)

and in terms of the modified weight function µβ (~r) =

3 ˜ r )2 + 2β(3π 2 )1/3 n(~r)β−1 h(~r) ] [ (5/3 − 2β) n(~r)2/3−2β k(~ 10

(B.4)

with h(~r) = f (~r) ∗ ω ˜ β (2kF0 r)

(B.5)

˜ r ) n(~r)5/3−2β . The product µ(~r)ψ(~r), is the “driving force” where f (~r) = k(~ for the dynamical minimization of the energy functional, see Eq. (4.11). If the various powers of the density appearing in µβ (~r)ψ(~r) are to remain positive so that this driving force is not to diverge in regions where the density vanishes, then 1/2 ≤ β ≤ 7/12. In practice we have found that for β = 0.51 the minimization has always proved possible. References 1. 2. 3. 4. 5. 6.

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Chapter 5 Electronic Structure Calculations at Macroscopic Scales using Orbital-Free DFT Balachandran G. Radhakrishnan and Vikram Gavini Department of Mechanical Engineering, University of Michigan Ann Arbor, MI 48105, USA [email protected] In this chapter we provide an overview of the recently developed coarse-graining technique for orbital-free density functional theory that enables electronic structure calculations on multi-million atoms. The key ideas involved are: (i) a local real-space formulation of orbital-free density functional theory; (ii) a finite element discretization of the formulation; (iii) a systematic means of adaptive coarse-graining of the finite-element basis set using quasi-continuum reduction. The accuracy and effectiveness of the computational technique is demonstrated by studying the energetics of mono- and di-vacancies in multi-million atom aluminum crystals.

Contents 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Overview of quasi-continuum orbital-free density functional theory 5.2.1 Real-space formulation of OFDFT . . . . . . . . . . . . . . 5.2.2 Finite-element discretization . . . . . . . . . . . . . . . . . . 5.2.3 Quasi-continuum reduction . . . . . . . . . . . . . . . . . . 5.3 Vacancies in aluminum . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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5.1. Introduction Electronic structure calculations, especially those using density-functional theory of Hohenberg, Kohn and Sham1,2 (KSDFT), have provided great insights into various aspects of materials properties in the last decade. Derived from quantum mechanics, electronic structure theories incorporate significant fundamental physics with little empiricism. Therefore, these theories are transferable, and capable of predicting a wide range of properties across various materials and external conditions. Successes of electronic structure calculations include the accurate prediction of phase transformations in a wide range of materials, and insights into the me147

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chanical, electronic, magnetic, and optical properties of materials and compounds. More recently, for materials systems whose electronic structure is close to a free electron gas, the orbital-free approach to density-functional theory (OFDFT) has received considerable interest where the kinetic energy of non-interacting electrons is modeled.3 Recent efforts in this field have resulted in the development of accurate kinetic energy functionals for simple metals, Aluminium, as well as some covalently bonded systems like Silicon.4–8 The ground state energy in orbital-free approaches is explicitly determined by the ground-state electron-density without recourse to the wavefunctions, which reduces the computational complexity of determining the ground state properties to scale linearly with system size. This allows the consideration of much larger systems using an orbital-free approach in comparison to the Kohn-Sham approach to density functional theory (KSDFT). Despite the many algorithmic developments in electronic structure calculations, the computational effort associated with these calculations is still enormous. Typical systems sizes accessible using the Kohn-Sham approach are of the order of a few hundred atoms, whereas orbital-free approaches can handle larger systems of about a thousand atoms. Moreover, the use of a plane-wave basis in most calculations restricts these investigations to periodic geometries. These restrictive periodic geometries in conjunction with the cell-size limitations limits the applicability of electronic structure theories to bulk properties of perfect materials. However, defects play a critical role in determining the properties of materials. These include dopants in semi-conductors to dislocations in mechanics to surfaces in nanostructures. These defects occur at very small concentrations and have long-ranged interactions. Therefore a complete and accurate description of such defects must include the electronic structure of the core of the defect at the fine (sub-nanometer) scale and the elastic, electrostatic, and other effects on the coarse (micrometer and beyond) scale. This in turn requires electronic structure calculations on systems containing millions of atoms, or in other words electronic structure calculations at macroscopic scales. Various multi-scale schemes have been proposed to address this significant challenge, among which hierarchical methods 9,10 and concurrent schemes 11–15 are the most popular. Multi-scale schemes where information is transferred from smaller to larger length scales are referred to as upscaling methods or hierarchical schemes. In such methods, electronic structure calculations are used to fit interatomic potentials/force-fields and these potentials are then used to compute materials properties on the macroscopic scale. On the other hand, the philosophy behind concurrent schemes, also referred to as embedding schemes, is to embed a refined electronic structure calculation in a coarser theory like tight-binding or atomistic calculations using empirical potentials, which in turn is embedded in a continuum theory. Valuable as these schemes are, they suffer from notable shortcomings. In

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some cases, uncontrolled approximations are made such as the assumption of linear response theory or the Cauchy-Born hypothesis. Others assume separation of scales, the validity of which can not be asserted. Moreover, these schemes are not seamless and are not solely based on a single electronic structure theory. In particular, they introduce undesirable overlaps between regions of the model governed by heterogeneous and mathematically unrelated theories. Finally, no clear notion of convergence to the full electronic structure solution is afforded by the existing methods. For all the above reasons, there is need for a seamless multi-scale scheme to perform electronic structure calculations at macroscopic scales with no ad hoc assumptions. In this article we present an overview of the recently developed systematic coarse-graining of orbital-free density functional theory that effectively overcomes the present limitations and enables electronic structure calculations on multi-million atom systems at no significant loss of accuracy.16,17 This multi-scale scheme is referred to as the quasi-continuum orbital-free DFT (QC-OFDFT) and is composed of the following building blocks: (i) a local real-space formulation of OFDFT; (ii) a finite-element discretization of the formulation; (iii) and a quasi-continuum reduction of the resulting equations that resolves detailed information in regions where it is necessary (such as in the immediate vicinity of the defect), but adaptively samples over details where it is not (such as in regions far away from the defect), without significant loss of accuracy. This proposed multi-scale scheme has the following defining properties: It adapts the level of spatial resolution to the local structure of the solution, e. g., supplying higher resolution near lattice defects and rapidly coarsening the resolution away from the defects; in particular, the coarse-graining is completely unstructured and does not rely on periodicity. Fully-resolved electronic structure and finite latticeelasticity are obtained as special limits. The coarse-graining is entirely seamless and based solely on approximation theory; in particular, a single electronic structure theory is the sole physics input to the calculations, and no spurious physics or ansatz regarding the behavior of the system is introduced as a basis for—or as a result of—the coarse-graining. The nature of the systems of interest is such that vast reductions in the size of the problem can be achieved without appreciable loss of accuracy, thus effectively permitting consideration of systems much larger than heretofore possible. In section 5.2 we present an overview of quasi-continuum orbital-free densityfunctional theory and salient features of the method that enable electronic structure calculations at macroscopic scales. We refer to Gavini et al.16,17 for a more comprehensive discussion on the method. Section 5.3 provides a discussion on electronic structure studies of vacancies using QC-OFDFT. Section 5.4 provides an outlook for the proposed method and opportunities for future work.

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5.2. Overview of quasi-continuum orbital-free density functional theory 5.2.1. Real-space formulation of OFDFT The ground state energy in density functional theory is given by3,10 E(ρ, R) = Ts (ρ) + Exc (ρ) + EH (ρ) + Eext (ρ, R) + Ezz (R) ,

(5.1)

where ρ is the ground-state electron-density; R = {R1 , . . . , RM } is the collection of nuclear positions in the system; Ts is the kinetic energy of non-interacting electrons; Exc denotes the exchange correlation energy; EH is the classical electrostatic interaction energy between electrons, also referred to as Hartree energy; Eext is the electrostatic interaction energy of electrons with external field induced by nuclear charges and Ezz denotes the repulsive energy between nuclei. The various terms and their functional forms are briefly described below. In OFDFT the kinetic energy of non-interacting electrons, Ts , is modeled, as opposed to the more widely used KSDFT where this is evaluated exactly within the mean-field approximation by solving an effective single electron Schr¨ odinger equation. In material systems whose electronic structure is close to a free-electron gas, e. g. simple metals, aluminum etc., very good orbital-free models for the kinetic energy term are available which have been shown to accurately predict a wide range of properties in these materials. A simple choice for this is the ThomasFermi-Weizsacker (TFW) family of functionals,3 which have the form Z Z λ 2 Ts (u) = CF u10/3 (r)dr + |∇u(r)| dr, (5.2) 2 p 2/3 3 (3π 2 ) , λ is a parameter, and u(r) = ρ(r) denotes the squarewhere CF = 10 root electron-density. More recently, there have been efforts4–7 to improve these orbital-free kinetic energy functionals by introducing an additional non-local term called the kernel energy. These kinetic energy functionals have a functional form given by Z Z 1 Ts (u) = CF u10/3 (r)dr + |∇u(r)|2 (5.3) 2 Z Z 0 0 0 + f (u(r))K(|r − r |)g(u(r ))drdr , where f , g and K are chosen to satisfy known limits of exact Ts (u) for uniform electron gas. The exchange correlation energy, Exc , describes the quantum-mechanical interactions for which accurate models for most systems are available. The Local Density Approximation (LDA)18,19 given by Z Exc (u) = xc (u2 (r))u2 (r)dr, (5.4)

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where xc has a parameterized form, has been shown to capture the exchange and correlation effects for most systems accurately. Finally, the electrostatic interactions in equation (5.1) are given by, EH (u) =

Eext (u, R) =

1 2

Z Z

M Z X

u2 (r)u2 (r0 ) drdr0 , |r − r0 | u2 (r)

I=1

M

Ezz (R) =

ZI dr, |r − RI |

(5.5)

(5.6)

M

1 X X ZI ZJ , 2 |RI − RJ |

(5.7)

I=1 J=1 J6=I

where ZI denotes the charge of the nucleus located at RI , I = 1, 2, . . . , M . Traditionally, OFDFT (as well as KSDFT) calculations have mostly been performed in Fourier-space using plane-wave basis functions. The choice of a planewave basis for electronic structure calculations has been the most popular one, as it lends itself to a computation of the electrostatic interactions naturally using Fourier transforms. However, such a Fourier-space formulation has very serious limitations in describing defects in materials. Firstly, it requires periodic boundary conditions, thus limiting an investigation to a periodic array of defects. This periodicity restriction in conjunction with the cell-size limitations (∼ 200 atoms) arising from the enormous computational cost associated with electronic structure calculations, limits the scope of these studies to very high concentrations of defects that are not realized in nature. Importantly, plane-wave basis functions used in Fourierspace formulations provide a uniform spatial resolution, which is not desired in the description of defects in materials. Often, higher resolution is required in the description of the core of a defect and a coarser resolution suffices away from the defect-core. This in turn makes Fourier-space formulations computationally inefficient in the study of defects in materials. Further, from a numerical viewpoint, plane-wave basis functions are non-local in the real-space, thus resulting in a dense matrix which limits the effectiveness of iterative solutions. Also, a plane-wave basis requires the evaluation of Fourier transforms which affect the scalability of parallel computation. For all the above reasons, and since it forms a key feature of the quasi-continuum reduction, a local real-space formulation of OFDFT is desired. The energy functional given by equation (5.1) is local in real-space except for the electrostatic interactions and the kernel energies which are extended in real-space. For this reason, evaluation of these energy terms is the most computationally intensive part 1 of the calculation of the energy functional. However, noticing that |r−r kernel 0 | in the electrostatic interactions is the Green’s function of the Poisson’s equation, the electrostatic interactions can be expressed locally as the following variational

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problem,

Z Z

(u2 (r) + b(r; R))(u2 (r0 ) + b(r0 ; R)) drdr0 |r − r0 |   Z Z 1 2 2 = − inf |∇φ(r)| dr − (u (r) + b(r; R))φ(r)dr , φ∈H01 (R3 ) 8π

1 2

(5.8)

where φ denotes a trial function for the electrostatic potential of the system of charges, and b(r; R) denotes the regularized nuclear charges corresponding to the pseudopotentials that provide an external potential for valence electrons. We note that equation (5.8) is a variational reformulation of solving the Poisson’s equation. We remark that the left hand side of equation (5.8) differs from the sum of electrostatic terms by the self energy of the nuclei, which is an inconsequential constant and does not influence the computation of ground-state properties. Turning to the non-local kinetic energy terms (kernel energies) given by equation (5.3), the approach suggested by Choly & Kaxiras20 is used to approximate the kernel in the reciprocal space by a rational function. This results in a system of Helmhotz equations that are local in real-space. Under this approximation, whose error can be systematically controlled, the kernel energies can be expressed in a local form given by Z Z m Z X 1 2 |∇u(r)| + wj (r)g(u(r)) Ts (u) = CF u10/3 (r)dr + 2 j=1 (5.9) Z m X +( Pj ) f (u(r))g(u(r))dr j=1

where wj ’s denote the kernel potentials which are minimizers of the following sequence of variational problems  Z  Z Z C Qj 2 2 inf |∇wj (r)| dr + wj (r)dr + Cj wj (r)f (u(r))dr wj ∈Xw 2 2 (5.10) j = 1 . . . m.

C, Cj , Qj and Pj , j = 1, 2, . . . m, are constants determined from a fitted rational function with degree 2m, and Xw is a suitable functional space. Finally, the problem of determining the ground-state energy, electron-density and equilibrium positions of the nuclei is determined by the following variational problem, inf

inf

E(u, R)

(5.11a)

u2 (r)dr = N,

(5.11b)

R∈R3M u∈H01 (R3 )

subject to:

Z

where N is the total number of electrons in the system. We note that, using the reformulations in equations (5.8)-(5.10), all components of energy are local in realspace.

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5.2.2. Finite-element discretization A finite-element basis set which respects the local variational nature of the formulation and allows for arbitrary boundary conditions and complex geometries is a natural choice to discretize and compute. Importantly, a finite-element discretization is amenable to adaptive coarse-graining, which is a key departure from previous numerical methods relying on plane-wave basis sets that have a uniform resolution in real-space. This is the key-idea which will be exploited in the construction of the quasi-continuum reduction, presented in the following section, that will enable electronic structure calculations of multi-million atom systems. Furthermore, the compact support of a finite-element basis is a desirable property in an implementation of the formulation on parallel computing architecture, which derives considerable importance owing to the computational complexity associated with electronic structure calculations. Finite-element bases are piecewise polynomial bases and are constructed from a discretization of the domain of analysis as a cell complex, or triangulation, Th .21,22 Often, the triangulation is chosen to be simplicial as a matter of convenience, but other types of cells, or elements, can be considered as well. A basis–or shape– function is associated to every vertex–or node–of the triangulation. The shape functions are normalized to take the value 1 at the corresponding node and 0 at all remaining nodes. The support of each shape function extends to the simplices incident on the corresponding node, which confers the basis a local character. The interpolated fields uh (r), φh (r), wjh (r) spanned by a finite-element basis are of the form uh (r) =

X

ui Nih (r)

(5.12a)

φi Nih (r)

(5.12b)

i

φh (r) =

X i

wjh (r) =

X

wj i Nih (r)

j = 1 . . . N,

(5.12c)

i

where i indexes the nodes of the triangulation, Nih (r) denotes the shape function corresponding to node i, and ui ,φi ,wj i represent the values of uh (r), φh (r), wjh (r) at node i respectively. Let Xh denote the subspace spanned by the finite-element basis functions that become increasingly dense in H01 (R3 ). The variational problem given by equation (5.11) reduces to a constrained minimization problem given by, inf E(uh , R) ,

inf

R∈R3M uh ∈Xh

subject to:

Z

2

uh (r)dr = N,

(5.13a) (5.13b)

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where the discrete local reformulations of the extended electrostatic interactions (equation (5.8)) and kernel energies (equation (5.10)) in the finite-element basis are given by the constrained minimization problems,   Z Z 1 2 inf |∇φh (r)|2 dr − (uh (r) + b(r; R))φh (r)dr (5.14) 8π φh ∈Xh inf

wjh ∈Xh



C 2

Z

|∇wjh (r)|2 dr +

Qj 2

Z

2

wjh (r)dr + Cj

Z

 wjh (r)f (uh (r))dr

(5.15)

j = 1 . . . m. 5.2.3. Quasi-continuum reduction The real-space formulation of density-functional theory and the finite-element discretization of the formulation described in subsections 5.2.1 and 5.2.2 is attractive, as it gives freedom from periodicity, which is important in modeling defects in materials. But, the computational complexity of these calculations limit investigations to systems consisting of a few thousand atoms. On the other hand, materials properties are influenced by defects—vacancies, dopants, dislocations, cracks, free surfaces—in small concentrations (parts per million). An accurate understanding of such defects must not only include the electronic structure of the core of the defect, but also the elastic and electrostatic effects on the macro-scale. This in turn requires calculations involving millions of atoms well beyond the current capability. The recently proposed quasi-continuum reduction for OFDFT17 effectively overcomes the present limitations and enables electronic structure calculations on multimillion atom systems. This is a multi-scale method which enables systematic coarsegraining of OFDFT in a seamless manner that resolves detailed information in regions where it is necessary (such as in the immediate vicinity of the defect) but adaptively samples over details where it is not (such as in regions far away from the defect) without significant loss of accuracy. The real-space formulation, and a finite-element discretization of the formulation which is amenable to coarse-graining are crucial steps in its development. The approach is similar in spirit to the quasicontinuum (QC) approach developed in the context of interatomic potentials23,24 as a scheme to seamlessly bridge the atomistic and continuum length scales. This bridging is achieved by adaptively selecting representative atoms and interpolating the positions of other atoms using finite-element shape functions. The representative atoms are chosen such that near the defect core all atoms are represented, whereas away from the defect core the interpolation becomes coarser and a small fraction of the atoms determines the displacements of the rest. In the context of OFDFT, the conventional QC reduction scheme can be applied mutatis mutandis to describe the positions of the nuclei. However, the electronic fields comprising of the electron-density, electrostatic potential and kernel potentials exhibit subatomic structure as well as lattice scale modulation, and therefore require

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an altogether different type of representation. To realize the coarse-graining of OFDFT, three unstructured triangulations of the domain are introduced as shown in Figure 5.1 to provide a complete description of the discrete fields: (i) A triangulation Th1 of selected representative atoms in the usual manner of QC, which in this discussion is called the atomic-mesh. (ii) A triangulation Th3 subatomic close to lattice defects and increasingly coarser away from the defects, which is called the electronic-mesh. (iii) An auxiliary subatomic triangulation Th2 that resolves a lattice unit-cell to capture the subatomic oscillations in the electronic fields, which is labeled as the fine-mesh. The triangulations are restricted in such a way that Th3 is a sub-grid of Th1 for convenience. We additionally denote by Xh1 , Xh2 and Xh3 the corresponding finite-element approximation spaces.

e De

nucleus

nuclei nuclei atomic-mesh (Th1) (a)

electronic-mesh (Th3) (b)

fine-mesh (Th2) (c)

Fig. 5.1. Hierarchy of triangulations used in the QC reduction. (a) Atomic-mesh (Th1 ) used to interpolate nuclei positions away from the fully-resolved defect-core; (b) Electronic-mesh (Th3 ) used to represent the corrector fields. It has subatomic resolution in the defect-core, and coarsens away from the defect-core. (c) Fine auxiliary mesh (Th2 ) is used to sample the Cauchy-Born predictor fields within an integration lattice unit cell, De , in each element e.

The representation of the electronic fields is decomposed as predictor and corrector fields given by

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uh = uh0 + uhc h

φ =

φh0

+

(5.16a)

φhc

(5.16b)

wjh = wjh 0 + wjh c

j = 1...m,

(5.16c)

where uh0 , φh0 , wjh 0 ∈ Xh2 denote the predictor for electronic fields and is obtained by performing a local periodic calculation in every element of Th1 using the CauchyBorn hypothesis. This predictor for the electronic fields is expected to be accurate away from defect cores, in regions where the deformation field is slowly varying. There are formal mathematical results which support this hypothesis.25 Hence, the corrector fields comprising of uhc , φhc , wjh c may be accurately represented by means of a finite-element triangulation such as Th3 , namely, a triangulation that has subatomic resolution close to the defect and coarsens away from the defect to become superatomic. The minimization problem given by equation (5.13) now reduces to a minimization problem for the corrector fields and takes the form inf Z

E(uh0 + uhc , R),

(5.17a)

(uh0 (r) + uhc (r))2 dr = N,

(5.17b)

inf

R∈Xh1 uh c ∈Xh3

subject to:

with the reformulations of electrostatic interactions and kernel energies in equations (5.14)-(5.15) reducing to   Z Z 1 h h 2 h2 |∇φ (r)| dr − (u (r) + b(r; R))φ dr inf (5.18) 8π φh c ∈Xh3 inf

wjh ∈Xh3 c



C 2

Z

|∇wjh (r)|2 dr

Qj + 2

Z

2 wjh (r)dr

+ Cj

Z



wjh (r)f (uh (r))dr

(5.19)

j = 1 . . . m. Since the predictor for electronic fields is defined on the uniformly subatomic mesh Th2 , it would appear that the computation of the system corresponding to the reduced problem (5.17) has complexity commensurate with the size of Th2 , which would render the scheme infeasible. In the spirit of quadrature rules for highly oscillating functions, integration rules that reduce all computations to the complexity of Th3 are introduced. The precise form of the integration rule for an element e in the triangulation Th3 is Z f (r)dr ≈ |e|hf iDe , (5.20) e

where |e| is the volume of element e, De is the unit cell of an atom if such cell is contained in e or e otherwise, and hf iDe is the average of f over De . Using (5.20),

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integration over the entire domain can be written as, Z X Z X f (r)dr = f (r)dr ≈ |e|hf iDe , Ω

e∈Th3

e

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(5.21)

e∈Th3

reducing all computations to a complexity commensurate with the size of Th3 , and thus reducing the complexity of the computations. This integration rule (5.20) is designed to be exact in regions close to the defect that correspond to sub-atomic elements of Th3 , where the non-local corrections are significant. However, in regions far away from the defect, which correspond to superatomic elements of Th3 , the nonlocal corrections are very small compared to the predictor. Thus, the integrand of equation (5.20) is a rapidly oscillating function with a very gradual modulation on the scale of the element. Hence, equation (5.20), for regions away from the core of a defect, denotes the zero order quadrature rule for rapidly oscillating functions. Equations (5.17)-(5.20) describe the quasi-continuum reduction of OFDFT, referred to as QC-OFDFT. To end this section, we note the following defining properties of the quasi-continuum reduction: (1) OFDFT is the sole input physics, the rest is approximation theory. There are no spurious physics, patching conditions or a priori ansatz. The method is seamless and free of any structure. (2) Fully-resolved OFDFT and finite lattice-elasticity are obtained as the two limiting cases. Therefore, a converged solution obtained by this method may be regarded as a solution of OFDFT. (3) Importantly, it enables electronic structure calculations of multi-million atom systems at no significant loss of accuracy as demonstrated in the next section. (4) A further property of the method is that it is possible to consider any arbitrary geometry and boundary conditions in light of the real-space finite-element formulation. 5.3. Vacancies in aluminum In this section we present results that demonstrate the accuracy and effectiveness of QC-OFDFT scheme through previously conducted studies on energetics of vacancies in aluminum16,17,26,27 as well as some recent investigations. For the purpose of demonstration, initial studies have been conducted using the TFW family of kinetic energy functionals with λ = 16 , and these constitute the majority of results presented here. Later in this section, we also present some results from recent investigations that use the more accurate kernel energies. All simulations were performed using a modified form of the Heine-Abarenkov pseudopotential 28 for aluminum and LDA treatment of the exchange and correlation functionals.18,19 Numerical parameters were chosen such that errors due to mesh discretization do not exceed 0.01 eV in the computed energies.

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To demonstrate the effectiveness of the coarse-graining scheme a sample containing 16,384 aluminum atoms was considered to compute the formation energy of a mono-vacancy. The kinetic energy of non-interacting electrons is modeled using the TFW family of functionals with λ = 16 . Dirichlet boundary conditions are applied on the corrector fields which implies all fields approach the bulk values at the boundary. Figure 5.2 shows the dramatic savings offered by coarse-graining through quasi-continuum reduction. As is evident from the figure, the formation energy attains a converged value at just 256 representative atoms which provides an 80 fold computational savings. These savings improve with the size of the sample and enable electronic structure studies on much larger samples than possible heretofore.

16384 nominal atoms

Vacancy formation energy (eV)

0.76

0.75

0.74

0.73

0.72

0.71 0

100

200 300 400 Number of representative atoms

500

600

Fig. 5.2. Convergence of the QC reduction: Formation energy of a mono-vacancy as a function of number of representative atoms. Adapted from Gavini et al.17

The energetics of vacancies are determined both by the electronic structure of the core as well as the long-ranged elastic and electrostatic interactions. Many efforts aimed towards determining the properties of vacancies using electronic structure calculations used periodic geometries with cell-sizes of at most a few hundred atoms.6,7,29–32 Such small cell-size may not capture the long-ranged effects in the presence of defects. To this end, QC-OFDFT scheme, which is free of cell-size limitations, was used to determine the cell-size effects of the computed energetics. Figure 5.3 shows the cell-size dependence of the vacancy formation energy in terms of the nominal number of atoms N . This study is conducted for two cases, one where the atomic relaxations are suppressed and another where the atoms are relaxed. Results indicate that large cell-sizes, consisting of more than thousand atoms, are required to obtain a converged mono-vacancy formation energy. We

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note that the converged vacancy formation energy, computed to be 0.72 eV, is in good agreement with experimental estimates.33

Vacancy formation energy (eV)

0.95 unrelaxed atoms relaxed atoms 0.9

0.85

0.8

0.75

0.7 0 10

Fig. 5.3.

2

4

10

6

10 Cell−size (N)

8

10

10

Cell-size effects in the mono-vacancy formation energy. Adapted from Gavini et al.17

0.25

Di−vacancy binding energy (eV)

0.2 0.15



0.1 0.05 0 −0.05 −0.1 0 10

2

10

4

10

6

10

Cell−size (N)

Fig. 5.4. Cell-size effects in the di-vacancy binding energy. Adapted from Gavini et al.17 Positive binding energy represents attraction between vacancies and negative binding energy denotes repulsion.

The cell-size dependence was more evident in a study of the energetics of divacancies, which are vacancy complexes formed from two adjacent vacancies. Of interest are the binding energies of the di-vacancies along h110i (nearest neighbor) and h100i (next nearest neighbor) crystallographic directions. Experimental studies interpret the binding energies of di-vacancies in aluminum to range between 0.2-0.3 eV,34,35 suggesting that vacancies attract. However, most DFT calculations per-

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formed to date on cell-sizes of the order of a hundred atoms predict either a repulsive or a negligible attractive interaction between vacancies.31,32 It is widely argued that this discrepancy is a result of the entropic effects as the experiments are performed at very high temperatures where as DFT can provide only ground-state properties. One other possible reason for discrepancy is that the computations are performed on very small cell-sizes representing unrealistically high concentration of vacancies. The typical concentration of vacancies is about a few parts per million.36 Access to such large cell-sizes that simulate realistic vacancy concentrations is not possible using conventional methods in electronic structure calculations. However using QC-OFDFT such a study is possible, and a cell-size study of the di-vacancy binding energies was recently performed. Figure 5.4 shows that there is a very strong cell-size dependence of the binding energies of both h110i and h100i di-vacancies. More strikingly the h110i di-vacancy is repulsive for small cell-sizes, representative of previous computations, but becomes attractive for large cell-sizes representative of realistic vacancy concentrations with binding energies in agreement with experiments.34,35 These results indicate that the physics not only changes quantitatively but also qualitatively with cell-size and underscores the need to account for the long-ranged interactions, especially in the presence of defects. Similar cell-size effects have also been observed in quad-vacancy clusters where much larger cell-sizes were required to obtain a converged solution.26

Vacancy formation energy (eV)

1.44 1.42 1.4 1.38 1.36 1.34 1.32 0 10

2

10

4

10 Cell−size (N)

6

10

8

10

Fig. 5.5. Cell-size effects in the unrelaxed mono-vacancy formation energy using density√ independent kernel energies with parameters {α, β} = { 56 ± 65 }.7

We note that although the computed di-vacancy binding energies are in good agreement with experiments, the calculations were performed by using the TFW family of functionals for the kinetic energy. It is well known that these functionals do not satisfy the Linhards response function for small perturbations of uniform electron gas which is explicitly known. Moreover, a recent study which employed

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more accurate kinetic energy functionals using kernel energies showed that even up to cell-sizes of about 500 atoms the binding energy of h110i di-vacancy is negligible.37 To this end, we reformulated the density independent kernel energies6,7 in a local form via equations (5.9)-(5.10) and incorporated into the QC-OFDFT scheme. We have very recently performed a cell-size study of an unrelaxed mono-vacancy using these kernel energies. As is evident from figure 5.5 there is a significant cellsize dependence. The vacancy formation energy for a 32 atom cell-size is computed to be 1.33 eV which is in excellent agreement with previous computations.7 However, upon increasing the cell-size to account for a million atoms, the formation energy increases by almost 0.1 eV which is significant especially in the computation of di-vacancy binding energies. The cell-size effects on the di-vacancy binding energies after incorporating the kernel energies are currently being investigated, alongside the pursuit of some algorithmic developments to improve the efficiency of computations. 5.4. Outlook Quasi-continuum orbital-free density-functional theory is a seamless multi-scale scheme to coarse-grain OFDFT using quasi-continuum reduction. It uses a local real-space formulation of OFDFT and a systematic coarse-graining of the finiteelement basis set to enable electronic structure calculations on multi-million atom systems at no significant loss of accuracy, and without the introduction of spurious physics or assumptions. An important feature of the multi-scale scheme is that OFDFT is the sole input physics and the rest is approximation theory which enables a systematic convergence study of the method. As demonstrated through the studies on vacancies in aluminum, the proposed scheme can be useful in the study of defects in materials where much larger cell-sizes than those accessible through conventional techniques are required to capture the long-ranged effects present in these systems. Furthermore, as general geometries and non-periodic boundary conditions can be analyzed using this method, it enables the study of a wide range of defects, e. g. a single dislocation core-structure which is otherwise not possible in a restrictive periodic geometry. Although the proposed quasi-continuum reduction scheme has been developed for OFDFT, the mathematical structure of the method is general and can be extended to KSDFT as well as other electronic structure theories. However to obtain similar computational savings as afforded for OFDFT, two non-trivial aspects which need to be addresses are: (i) the treatment of the delocalized nature of the wavefunctions and enforcing the orthogonality constraint; (ii) development of efficient eigenvalue solvers for large systems. Herein lies the opportunity for future work, alongside the accurate electronic structure study of defects made possible by the method.

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References 1. P. Hohenberg and W. Kohn, Phys. Rev. 136, B864 (1964). 2. W. Kohn and L. J. Sham, Phys. Rev. 140, A1133 (1965). 3. R. Parr and W. Yang, Density-Functional Theory of Atoms and Molecules (Oxford University Press, New York, 1989). 4. L.-W. Wang and M. P. Teter, Phys. Rev. B 45, 13196 (1992). 5. E. Smargiassi and P. A. Madden, Phys. Rev. B 49, 5220 (1994). 6. Y. A. Wang, N. Govind, and E. A. Carter, Phys. Rev. B 58, 13465 (1998). 7. Y. A. Wang, N. Govind, and E. A. Carter, Phys. Rev. B 60, 16350 (1999). 8. B. Zhou, V. L. Ligneres, and E. A. Carter, J. Chem. Phys. 122, 044103 (2005). 9. A. K. Rappe, C. J. Casewit, K. S. Colwell, W. A. Goddard, and W. M. Skiff, J. Am. Chem. Soc. 114, 10024 (1992). 10. M. Finnis, Interatomic Forces in Condensed Matter (Oxford University Press, New York, 2003). 11. N. Govind, Y. A. Wang, and E. A. Carter, J. Chem. Phys. 110, 7677 (1999). 12. M. Fago, R. L. Hayes, E. A. Carter, and M. Ortiz, Phys. Rev. B 70, 100102 (2004). 13. N. Choly, G. Lu, W. E, and E. Kaxiras, Phys. Rev. B 71, 094101 (2005). 14. G. Lu, E. B. Tadmor, and E. Kaxiras, Phys. Rev. B 73, 024108 (2006). 15. Q. Peng, X. Zhang, L. Hung, E. A. Carter, and G. Lu, Phys. Rev. B 78, 054118 (2008). 16. V. Gavini, J. Knap, K. Bhattacharya, and M. Ortiz, J. Mech. Phys. Solids 55, 669 (2007). 17. V. Gavini, K. Bhattacharya, and M. Ortiz, J. Mech. Phys. Solids 55, 697 (2007). 18. D. M. Ceperley and B. J. Alder, Phys. Rev. Lett. 45, 566 (1980). 19. J. P. Perdew and A. Zunger, Phys. Rev. B 23, 5048 (1981). 20. N. Choly and E. Kaxiras, Sol. State Comm. 121, 281 (2002). 21. P. Ciarlet, The Finite-Element Method for Elliptic Problems (SIAM, Philadelphia, 2002). 22. S. C. Brenner and L. R. Scott, The Mathematical Theory of Finite Element Methods (Springer-Verlag, New York, 2002). 23. E. B. Tadmor, M. Ortiz, and R. Phillips, Philos. Mag. A 73, 1529 (1996). 24. J. Knap and M. Ortiz, J. Mech. Phys. Solids 49, 1899 (2001). 25. X. Blanc, C. L. Bris, and P. L. Lions, F Arch. Rational Mech. Anal. 164, 341 (2002). 26. V. Gavini, K. Bhattacharya, and M. Ortiz, Phys. Rev. B 76, 180101 (2007). 27. V. Gavini, Phys. Rev. Lett. 101, 205503 (2008). 28. L. Goodwin, R. J. Needs, and V. Heine, J. Phys. Condens. Matter 2, 351 (1990). 29. N. Chetty, M. Weinert, T. S. Rahman, and J. W. Davenport, Phys. Rev. B 52, 6313 (1995). 30. D. E. Turner, Z. Z. Zhu, C. T. Chan, and K. M. Ho, Phys. Rev. B 55, 13842 (1997). 31. K. Carling, G. Wahnstr¨ om, T. R. Mattsson, A. E. Mattsson, N. Sandberg, and G. Grimvall, Phys. Rev. Lett. 85, 3862 (2000). 32. T. Uesugi, M. Kohyama, and K. Higashi, Phys. Rev. B 68, 184103 (2003). 33. W. Triftsh¨ auser, Phys. Rev. B 12, 4634 (1975). 34. P. Ehrhart, P. Jung, H. Schultz, and H. Ullmaier, Atomic Defects in Metal, LandoltB¨ ornstein, New Series, Group 3, Vol. 25 (Springer-Verlag, Berlin, 1991). 35. T. Hehenkamp, J. Phys. Chem. Solids 55, 907 (1994).

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36. M. J. Fluss, S. Berko, B. Chakraborty, K. R. Hoffmann, P. Lippel, and R. W. Siegel, J. Phys. F 14, 2831 (1984). 37. G. Ho, M. T. Ong, K. J. Caspersen, and E. A. Carter, Phys. Chem. Chem. Phys. 9, 4951 (2007).

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Chapter 6 Properties of Hot and Dense Matter by Orbital-Free Molecular Dynamics Flavien Lambert, Jean Cl´erouin, Jean-Fran¸cois Danel, Luc Kazandjian and St´ephane Mazevet CEA, DAM, DIF, F-91297 Arpajon, France This paper presents the exploration of hot and dense matter by using orbital-free density-functional theory for the electrons coupled with molecular dynamics for the nuclei. Equations of state, as well as structure and transport coefficients, are computed from this formalism. By treating on an equal footing both pure elements and mixtures, the microscopic properties of mixtures are directly studied which allows the verification of standard mixing rules.

Contents 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 What kind of matter are we dealing with? . . . . . . . . . . . . . 6.3 From quantum to orbital-free molecular dynamics . . . . . . . . 6.3.1 Quantum molecular dynamics . . . . . . . . . . . . . . . . 6.3.2 Orbital-free molecular dynamics . . . . . . . . . . . . . . . 6.4 Numerical features of the orbital-free treatment . . . . . . . . . . 6.4.1 Description of the energy contributions . . . . . . . . . . . 6.4.2 Free energy minimization . . . . . . . . . . . . . . . . . . 6.4.3 Parallelization . . . . . . . . . . . . . . . . . . . . . . . . . 6.4.4 Regularization of the electron-nucleus interaction . . . . . 6.5 Thermodynamics: towards high-density plasmas . . . . . . . . . 6.6 Structural and dynamic properties: the quest for ionization . . . 6.6.1 Ionization choice . . . . . . . . . . . . . . . . . . . . . . . 6.6.2 Comparison of results . . . . . . . . . . . . . . . . . . . . 6.7 Inside the mixture: the plasma as a soup of electrons and nuclei 6.7.1 Eos mixing rule . . . . . . . . . . . . . . . . . . . . . . . . 6.7.2 Transport coefficients and partial ionization . . . . . . . . 6.8 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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165 167 168 169 171 172 172 173 174 176 181 183 184 185 188 189 195 198 199

6.1. Introduction With the advent of laser facilities dedicated to inertial confinement fusion (icf),1,2 physics of dense and hot matter constitutes a field of research of growing interest. In icf schemes, matter can be encountered in various and extreme states: from low-density-high-temperature, i.e. kinetic, plasmas of the outer ablator to 165

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high-density-high-temperature matter for the ignition, through high-density-lowtemperature statesa during the implosion process. The design of such icf targets is entirely based on hydrodynamics simulations which rely on the knowledge of microscopic coefficients such as the equation of state and other transport coefficients for example. These properties were, until recently, essentially determined by classical and approximate models like the onecomponent plasma (ocp) model3–5 or its Yukawa counterpart.6,7 These models are parametrized through quantities such as the ionization or effective charge state which act as ad hoc parameters. Contrary to the aforementioned models, first-principles or ab initio simulations have proved to be extremely efficient in determining the properties of solids, liquids8,9 or warm dense matter states10,11 by describing the system through classical nuclei and a quantum electronic component without external parameters. Most ab initio simulations are carried out in the density-functional framework12 through the independent-particle scheme proposed by Kohn and Sham.13 Despite the development of massively parallel computers which has allowed to tackle more and more difficult problems, the application of first-principles simulations remains limited to a few tenths of the Fermi temperature. Indeed, as temperature rises, the number of quantum states to be taken into account becomes numerically intractable since these states are populated through a Fermi-Dirac thermal distribution. In order to bridge the gap between high density-low temperature plasmas, described by quantum simulations, and low density-high temperature plasmas, evaluated by classical simulations, we propose to use an orbital-free treatment to represent the electronic component in the true spirit of the Hohenberg and Kohn theorem. To a certain extent, this choice is an extension from one to many bodies of the pioneering work of Feynman, Metropolis and Teller.14 By taking the electronic density as the true variable, the orbital-free scheme for the electrons coupled with classical molecular dynamics for the nuclei provides a computationally efficient tool to deal with a variety of plasmas. The first two sections are dedicated to a brief description of both quantum and orbital-free molecular dynamics and their numerical implementations. The following paragraphs expose several applications of orbital-free molecular dynamics ranging from equations of state to transport coefficients of both pure elements and mixtures. In the whole chapter, each atom, or equivalently its nucleus, is represented by its atomic number Z and mass A, the mass of the nucleus being therefore M = A/N with N the Avogadro’s number. Thermodynamic conditions are expressed in terms of temperature kT = β −1 and material density ρ. Atomic units are used except where otherwise stated.

a Still

a few eV. . .

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6.2. What kind of matter are we dealing with? From the point of view of quantum statistical physics, the state of matter that is called “plasma” is not different from solids or liquids. The many-body Hamiltonian describing the interaction of nuclei and electrons is indeed H=

Na Na X X P`2 Z` Z`0 + 2M` |R ` − R`0 | 0 `=1

+

`,` =1 `6=`0

Ne X p2 `

`=1

2

+

Ne X

`,`0 =1 `6=`0

N

N

e a X X 1 Z` − , |r` − r`0 | |R − r` 0 | ` 0

(6.1)

`=1 ` =1

where R, P, r, p are the position and linear momentum operators associated with nuclei and electrons, Na is the number of nuclei and Ne is the number of electrons, the system being overall neutral in our case. The equilibrium state of the system for −1 the temperature
kT = β is entirely determined by the partition function, namely Tr exp −βH . As a definition of the plasma state, one can propose a state of matter in which the thermal excitations of electrons are large. In an independent-particle picture, this means that the ratio θ between the temperature and the Fermi temperature Tf is not a small parameter, Tf being defined as kTf =


2 1 3π 2 ne 3 , 2

(6.2)

where ne is the average electronic density. To a certain extent, θ gauges the competition between temperature and density effects. When θ → 0, the thermal effects can be treated in a perturbative way with the 0 K isotherm as a reference. This kind of technique is used for all-temperature solids for example. Nevertheless, one must keep in mind that, when reaching extremely high densities where θ can be small, the state of matter can be very different from a solid. For example, at very high densities as in the core of supernovae, the matter is so compressed that the electronic bath can be completely degenerated, acting as a uniform rigid background as described in every elementary textbook on quantum statistical physics, whereas the nuclear component is still in the liquid state. In such a case, the Coulomb interactions between nuclei – which can be described classically – are not screened anymore. This model is known in the plasma community as the one-component plasma (ocp) model.3,15 The Hamiltonian of the system is Na Na   1X P`2 1 X Z` Z`0 Hocp R, P = + . 2 M` 2 0 |R` − R`0 | `=1

`,` =1 `6=`0

(6.3)

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The statistical physics of this model depends solely on a non-dimensional parameter, called the coupling constant Γ, given by Γ=

Z2 , akT

(6.4)

where a is the average distance between two nuclei, 43 πa3 ni = 1 with ni the nuclear number density. Γ is nothing but the ratio between the average coulomb potential energy Z 2 /a and the kinetic energy kT . When Γ tends to 0, the potential effects are negligible and the plasma is said weakly coupled. This kind of plasma is encountered in magnetically-confined-fusion programs for example. As in the case of θ, when Γ is small, perturbative techniques can be applied leading to corrections to the perfect gas in terms of Coulomb interactions. On the contrary, when Γ is high, the plasma is strongly coupled and its behavior is close to a liquid or even a solid when Γ > 172. Such plasmas, with 1 < Γ < 10, are now produced by interaction of matter with a laser where strong shocks are generated. Perturbative approaches fail in such regimes which are then studied by means of Monte-Carlo or molecular dynamics methods. It is important to note that, in most cases, the matter is not sufficiently hot or dense to lead to a complete peeling of the atom. A part of the electronic cloud, usually called “bound electrons”, is localized around the nucleus whereas the other part, the “free electrons”, tends to delocalize on a fraction of or on the whole system. As a consequence, the ocp model is often used not with the atomic number Z but with a partial ionization Z ? that has to be determined by an ad hoc prescription. In the following paragraphs, we address the thermodynamic domain where no parameter is small, i.e. both the thermal electronic excitations and the potential interactions between nuclei are large. In other terms, no perturbative treatment in θ or Γ can be employed. We mostly present one technique based on the coupling of molecular dynamics and orbital-free density-functional theory which is particularly well suited to deal with such high-density-high-temperature plasmas.

6.3. From quantum to orbital-free molecular dynamics Both quantum and orbital-free molecular dynamics rely on the adiabatic approximation,16 i.e. electronic and nuclear degrees of freedom are separated, the electrons “moving” much faster than the nuclei. The electrons are then acting on the nuclei as a many-body screening potential. Moreover, in the thermodynamic regimes in which we are interested, the nuclear De Broglie wavelength is much smaller than the inter-nuclei distance so that nuclei can be considered as classical particles, labeled by their position R and their linear momentum P. These first approximations are summarized through

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(6.5)

`,` =1 `6=`0

with L the Lagrangian of the nuclear dynamics and F e the electronic free energy. 6.3.1. Quantum molecular dynamics 6.3.1.1. Kohn-Sham scheme In the framework of density-functional theory (dft),12,17 the electronic free energy – which is a functional of the many-body quantum states – was proved to be a unique functional of the local density in the vicinity of the nuclear Coulomb potential. At given temperature and material density, the local electronic density n(r) verifies the variational principle, constrained to charge conservation, " Z # Na X δF e [n] e =µ (6.6) min F [n] − µ dr n(r) − Zi ⇔ n δn(r) i=1 with µ the chemical potential. Since the form of the functional is unknown, it was proposed by Kohn & Sham,13 at zero temperature, and Mermin,17 at non-zero temperature, to map the system into an independent-particle model that accounts for the major contributions, namely kinetic and coulombic, leaving an unknown but minor correction to the socalled exchange-correlation functional. In this framework, the electronic free energy reads as i X Z 2 2 h e f` ln f` + (1 − f` ) ln(1 − f` ) F [n] = f` dr ∇ψ` (r) + β ` ZZ Z Na 0 X 1 n(r 0 ) 0 n(r) n(r ) 0 (6.7) + drdr − Z dr ` 2 |r − r 0 | |R` − r 0 | `=1 Z + dr εxc [n], where f` is the Fermi-Dirac distribution associated with the normalized quantum state ψ` and εxc is the exchange-correlation contribution per unit volumeb . The local electronic density is related to the one-body states through X n(r) = 2 f` |ψ` (r)|2 . (6.8) `

b For

completeness, the exchange-correlation term should contain an explicit dependence on the temperature.

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The minimization of the electronic free energy leads to a set of coupled one-body Schr¨ odinger equations, named Kohn-Sham equations, ! 1 2 − ∇ + vee [n] + vie (r) + vxc [n] ψ` (r) = ε` ψ` (r), (6.9) 2 where vee , vie and vxc are respectively the electron-electron Hartree, the ion-electron and the exchange-correlation potentials. The system of equations is closed by the charge conservation imposing the chemical potential µ inside the Fermi-Dirac dis
−1 tribution f` = 1 + exp β(ε` − µ) . Quantum molecular dynamics (qmd) consists in applying molecular dynamics to the ions or nuclei with F e [n], Eq. (6.7), as the electronic dynamic screening in the Born-Oppenheimer approximation. 6.3.1.2. Pseudo-potential and the problem of delocalization Solving the Kohn-Sham equations is particularly costly in terms of computational time. The latter scales roughly like the cube of the number of quantum states introduced. At standard temperatures and densities encountered in “standard” chemistry or condensed matter, only a part of the electrons, named here valence electrons, are involved in forming molecules or delocalized in the whole structure to allow the cohesion of a solidc . The rest of them, labeled as core electrons, are little affected by the presence of other atoms in the sense that the one-body quantum states are close to the atomic ones. Therefore, in order to limit the number of orbitals to be calculated, it was proposed to treat the nucleus and the core electrons as an effective ion whose interaction with the valence electrons is represented by an effective potential, the pseudo-potential.18 The purpose of this paragraph is not to describe the different techniques developed concerning the problem of pseudo-potentials but, since our primary goal is to study matter at high temperatures and densities, it is important to note that pseudo-potentials induce strong limitations to perform qmd calculations in these regimes. Indeed, at sufficiently high temperatures and densities, core electrons tend to delocalize, phenomena also referred as, respectively, ionization and pressure delocalizationd . As a consequence, the arbitrary distinction between core and valence electrons vanishes preventing from using such an approximation. Nevertheless, the mathematical foundation of pseudo-potentials allows to build such a potential without core electrons, all electrons being then in the valence contribution. The pseudo-potential is, in that case, a regularization of the divergent bare nucleus-electrons Coulomb potential. While technically possible, at high temperatures, since the orbitals are populated through a thermal Fermi-Dirac distribution, the number of quantum states to c We

employ here an independent-particle vocabulary. One must of course keep in mind that the all-electron quantum state is non-separable preventing us from talking about individual entities. d Both of them occur at the same time at both high pressures and high temperatures.

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take into account becomes completely prohibitive using current massively parallel computers. 6.3.2. Orbital-free molecular dynamics One way to reach high-temperature states of matter is to get rid of the orbitals, which are the bottleneck – but of course the source of precision – in qmd, by determining a direct approximation of the electronic free energy in terms of the electronic density in the true spirit of the Hohenberg and Kohn theorem.12 Several attempts have been made to develop the orbital-free (of) kinetic energy functional.19,20 These elaborated functionals, successful in reproducing the quantal electronic density, are yet limited to the zero isotherm and therefore not straightforwardly applicable to the kind of regimes we are interested in. In order to get a “finite-temperature” of free-energy functional, we move towards the semi-classical expansion of the partition function, i.e. in terms of the Planck constant, where the Hamiltonian is taken to be the Kohn-Sham one, Eq. (6.7). This procedure is nicely exposed in Ref. 21. The first term of the expansion leads to the “finite-temperature” version of the famous Thomas-Fermi functional, as derived in the seminal work of Feynman, Metropolis and Teller,14 whereas the second term, taking into account the gradient of the electronic density, is the extension of the von Weizs¨acker correction22 at non-zero temperatures, first derived by Perrot.23 The of free energy, where local density and kinetic energy are valid up to second order in ~, reads as √   Z h  

i 1 2 2 e F n(r) = dr n(r) Φ n(r) − 3 I 3 Φ n(r) β 3π 2 β 2 2 Z ∇n(r) 2 + dr h(n) n(r) (6.10) ZZ Z Na 0 0 X 1 n(r) n(r ) n(r ) + drdr 0 − Z` dr 0 2 |r − r 0 | |R` − r 0 | `=1 Z + dr εxc [n],

where the function h is analytical23 whereas Φ is √ h
i 2 n(r) = . 3 I 1 Φ n(r) π2 β 2 2

(6.11)

As in the Kohn-Sham scheme, charge conservation imposes the chemical potential.   The Thomas-Fermi kinetic-entropic part of Eq. (6.10) will be denoted F0 n(r) in the following paragraphs. Let us also recall here that the simple Thomas-Fermi model plus exchangecorrelation, i.e without gradient correction, does not lead to any bonding between

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atoms24 and any equilibrium volume at room temperature. Therefore, it is obvious that this functional produces catastrophic results when molecules are involvede . The introduction of a gradient correction allows the formation of bonding and the existence of a volume of equilibrium which remains, nevertheless, poorly evaluated.25 6.4. Numerical features of the orbital-free treatment This paragraph is dedicated to the numerical implementation of the previous of treatment of plasmas. Section 6.4.1 describes general aspects of the computation. Section 6.4.2 exposes the techniques involved in determining the minimum of the free energy whereas Sec. 6.4.3 is dedicated to the reasons why of methods are particularly well-suited for massively parallel computations. In absence of pseudo-potentials, theory developed in the Kohn-Sham framework, the divergence of the Coulomb potential between electrons and nuclei has to be regularized. This problem is tackled with the help of the average atom model in paragraph 6.4.4. The results obtained in the present work have been produced with two different codes: ofmd26,27 and abinit.28–30 ofmd is dedicated to orbital-free implementations whereas abinit can be used either with the quantum or with the orbital-free description. Both codes lend themselves to parallelization. 6.4.1. Description of the energy contributions A large part of the of molecular dynamics implementation is similar to that of qmd18,28 with periodic boundary conditions. Nuclear coordinates R span a continuous space. At each time step, the nucleusnucleus Coulomb interactions are handled with classical Ewald sums25,28,31 and screened nucleus-electrons interactions are evaluated from the Hellmann-Feynman theorem32 after minimization of the electronic free energy, i.e in the BornOppenheimer approximation. Molecular dynamics can be performed either in the micro-canonical – with Verlet or Leap-frog algorithms – or in the isokinetic ensemble.33,34 The latter allows to maintain a constant temperature in the simulation and to produce a nuclear velocity distribution as well as static and dynamic quantities that are close to those of the canonical ensemble.25 The electronic quantities are developed on a finite three-dimensional grid in both real and reciprocal spaces, related to each other through fast Fourier transforms.25,28 The non-zero temperature Thomas-Fermi and von Weizs¨acker free energies are evaluated thanks to the fits computed by Perrot,23 the gradient of the electronic density being evaluated by direct and inverse Fourier transforms. The exchange-correlation terms can be calculated in the local density approximation at both zero35 and non-zero36 temperatures. Both electrons-electrons and electronse At

relatively high densities, molecules can “survive” even at temperatures of the order of a few electronvolts.

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nuclei Coulomb interactions are computed in reciprocal space since both are based on convolution products. The in- and off-diagonal components of the stress tensor are explicitly calculated after the electronic free energy minimization taking into account both the potential37–40 – of kinetic, electrons-electrons, electrons-nuclei, nuclei-nuclei and exchange-correlation – and the nuclear kinetic contributions.41 6.4.2. Free energy minimization Historically, the first attempt to couple finite-temperature of dft and molecular dynamics was based on a Car-Parinello-like description of the system.42,43 By separating the dynamics of electronic and nuclear degrees of freedom,44,45 one can avoid the time-consuming energy minimization at each time step at the cost of smaller time steps to stay close to the Born-Oppenheimer energy surface. This method, although powerful, has a serious drawback, namely no constraint on the positivity of the density can be easily added. Because of this limitation and of the development of powerful minimization techniques, it was chosen to move towards a direct minimization of the electronic free energy.27,30 Whereas Kohn-Sham formulations require both self-consistent cycles – to update the potential – and conjugate gradient method – to solve the Kohn-Sham equations at fixed potential – to achieve minimization, of framework allows to use one of the two techniques to solve the variational principle, Eq. (6.6). The first, self-consistent calculation, is introduced in the of average atom model, see Sec. 6.4.4.1. The second method is performed by noticing, first, that Eq. (6.6) in the of framework reduces to a Schr¨ odinger-like equation by introducing a “pseudo-orbital” defined as n(r) = ψ(r)2 .

(6.12)

With the notations of Sec. 6.3.2, Eq. (6.6) and Eq. (6.10) give −4h(n)∆ψ(r) + V(r)ψ(r) = µψ(r)

(6.13)

with µ the chemical potential and V(r) =

δF0 [n] ∂h(n) + vxc [n] − 4 |∇ψ(r)|2 δn(r) ∂n Z Na X Z` n(r 0 ) − + dr 0 . |R` − r| |r − r 0 |

(6.14)

`=1

The similarity between Eq. (6.14) and Eq. (6.9) allows to have access to the powerful algorithms used for solving the Kohn-Sham equations. This opportunity is used in the abinit28,30 and of molecular dynamics27 implementations. The second code makes use of constrained conjugate gradient algorithm especially designed for orbital-free treatment of dft.46

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To have recourse to a gradient correction in the of functional calls for special care in the algorithm. Indeed, depending on the elements and the thermodynamic conditions, the minimization is sometimes difficult to achieve when beginning with a homogeneous electronic density. Therefore, it is numerically interesting to run a first minimization without gradient correction (with or without exchangecorrelation) and start the full functional minimization with the previous density, i.e. the Thomas-Fermi one, as an initial guess.25,30

6.4.3. Parallelization One of the key features of of dft comes from the fact that it fits well into a massively parallel scheme allowing computations which require very large grid sizes like mixtures with small concentrations of elementsf . A good parallelism is obtained when each entity – supposed to be of equal efficiency – has the same amount of work, limiting the transfer of information with the others. To a certain extent, the smaller the social life, the better ! A way to parallelize the evaluation of the free energy, or its gradient for the conjugate gradient method, consists in dividing the whole volume of simulation Ω into Nv independent sub-volumes each dedicated to a processor,

Ω=

Nv ]

Ω` .

(6.15)

`=1

Indeed, since the sub-volumes are independent, the integral on the whole volume of a local functional can be evaluated through Z

Ω

Nv   X dr F n(r) = `=1

Z

Ω`

  dr F n(r) ,

(6.16)

i.e. each processor can compute its own partial integral by knowing only the electronic density on its own sub-volume. Expressed in computational terms, each processor executes elementary operations (sum or products) on a table whose size is Nv times smaller. The transfer of information required to obtain the total integral is then limited to Nv reals, which is completely negligible in comparison to the operations performed on the tables. The quality of the parallelization is crucially dependent on the locality of the theory which limits the transfer of information between the processors. Turning into the of functional, the electronic free energy, Eq. (6.10), can be written in terms of the spaces – real (integrals over r) or reciprocal (integrals over g) f And

therefore a lot of particles!

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– in which the different functionals are computed Z Z Z ∇n(r) 2 e F [n] = dr F0 [n] + dr εxc [n] + dr h[n] n(r) (6.17) Z Z 2 4π + dg n(g) 2 + dg v(g)n(−g) , g where v is the total nuclear potential. Both the potential energies are completely local in reciprocal spaceg and can thus be evaluated with a linear scalingh . Moreover, F0 [n], h[n] and Fxc [n] are computed from numerical fits23,35 which are “simple” local functions of n. Their calculation scales also linearly. The real bottleneck of the method comes from the path from real to reciprocal space which is not a linear method and can be very costly for grid sizes of millions of points. By studying the form of Eq. (6.17), it can be seen that the potential energies need one FFT and the gradient of the density (or equivalently of the pseudo-orbital) four or five of them at each computation of the energyi . The gradient-corrected functional is much more costly in terms of computational time. Table 6.1 illustrates the parallelization of the of molecular dynamics code – which is operated in the message passing interface (mpi)47 framework – in terms of numerical scaling. The computational times are averaged over several minimizations of the electronic free energy. The electronic quantities are described on a 256×256× 256 grid and the free energy does not include a gradient correction. It exposes the ratio between computational times for performing the operations with respectively 4 and N processorsj. As expected, the FFT algorithm is not linear because of Table 6.1. Scaling of a typical 256 × 256 × 256 of molecular dynamics calculation Number of processors

FFT

Local functionals

Structure factor

4 8 16 32 64 128

1 1.85 4.23 7.71 14.91 21.48

1 1.98 3.98 8.54 20.69 51.14

1 1.98 3.98 8.10 18.23 54.10

the important transfer of information between the processors, the scaling being fairly reduced at the highest number of processors, namely 128. On the contrary, the computation of both the local functionals and the structure factor – which is necessary for the electrons-nucleus potential – is perfectly linear at the beginning g They

are obviously non-local in real space since they are convolution products. scaling corresponds to the ratio of computational times required to perform an operation by one and N processors. i There can be numerous evaluations of the free energy or its gradient along the path of minimization.31,46 j The processors are nodes composed of 8 Itanium2 1.6 GHz with 24 GB of memory per node. h The

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and becomes even better than linear at high number of processors because of smaller memory required for each operation (avoiding thus swap problems). It should be noted that a time step with the 256 × 256 × 256 grid and 128 processors takes only a few seconds to be calculated allowing molecular dynamics of thousands of atoms in a very reasonable computational time. 6.4.4. Regularization of the electron-nucleus interaction Since the Coulomb electron-nucleus interaction is divergent, this potential has to be regularized in numerical simulations. The procedure described below makes use of the average atom model. It consists in representing the complex plasma structure by a single neutral atom of atomic number Z and mass A which represents a kind of average component of the plasma at ρ and T . The electrons can be described by a quantum,48–50 an hybrid quantum-semi-classical 51–53 or a completely of14,23,53 functional. It is worth noting that this kind of approach can only be applied to pure element plasmas. The following paragraphs are focused on the orbital-free representationk of such a model. 6.4.4.1. Orbital-free average atom model The system – imposed to be spherically symmetric – is represented by a set of two coupled equations, the Poisson’s one,   ∆v(r) = 4π Zδ0 − n(r) , (6.18)

that computes the total potential v(r) in terms of the nucleus and electronic densityl , and the variational principle, Eq. (6.6), that is expressed as √ h
i 2 n(r) = , (6.19) 3 I 1 β µ − v(r) − vxc (r) π2 β 2 2

where vxc is, as previously, the exchange-correlation potential, i.e. the functional derivative of the exchange-correlation energy with respect to n(r). The density effects are taken into account through the charge conservation (or electro-neutrality) equation Z a Z = 4π dr r2 n(r), (6.20) 0

where a is the Wigner-Seitz radius defined by 4πa3 ρN = 3A. The divergent Coulomb interaction between nucleus and electrons can be easily regularized at the origin by a quadratic polynomial on a scale of the size of the k No

gradient correction is introduced to simplify the equations. solution is exactly the potential in the third part of the right-hand side of Eq. (6.10) in three dimensions, i.e. the convolution product of the charge densities with the elementary solution of the Laplacian. l The

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nucleus – on the order of a fm – thanks to a logarithmic radial grid and without impact on the thermodynamic results.54 Figure 6.1 illustrates the average atom model (aam) calculation of an iron plasma at 5 eV and 7.891 g cm−3 with both a quantum and an orbital-free functional. 35

Kohn-Sham lda Orbital-free lda

30

4πr 2 n(r) (a.u.)

25

20

15

10

5

0 0

0.5

1

1.5

2

2.5

r (a.u.)

Fig. 6.1. Local electronic densities from the average atom model with a quantum (straight line) and an orbital-free (dashed line) functional. The material is iron at 5 eV and 7.891 g cm−3 .

As expected, the approximate of treatment does not lead to any oscillations of the electronic density and smoothes the shell effects. For future use, we define the ionization, following More,55 as 4 Z ? = πa3 n(a). (6.21) 3 Because of the neutrality of the system inside the Wigner-Seitz sphere, this value corresponds to electrons that are “propagated” in a null electric field. 6.4.4.2. Regularization of the Coulomb potential As in the case of the average atom model, the interaction between nuclei and electrons has to be regularized to perform an of molecular dynamics. If one follows the same path as previously, one would have to regularize the potential on a typical length scale of the order of the fermi in a simulation whose length scale is greater than the atomic one. This procedure would introduce a grid size of the order of 1015 points which is completely out of reach. To be reasonable in terms of computational time, the potential has to be regularized on an atomic scale, typically a fraction of the Wigner-Seitz radius which is half the average distance between two nuclei. The easiest – and at first thought obvious – way to perform such a regularization is to replace the divergent 1/r behavior by a smooth function, like a quadratic polynomial, below a chosen cut-off radius ensuring continuity both of the function

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and its derivative. This choice was made in the first version of the of molecular dynamics.42,56 Unfortunately, this method has a lot of drawbacks. Indeed, by softening arbitrarily the potential, it reduces the interaction between the nucleus and the electrons, pushing away the electrons from the nucleus, see Fig. 6.2. Since the electronic density is completely modified, the interactions between the nuclei – screened by the electronic part – are poorly evaluated leading to a dubious molecular dynamics.26 40 0

Coulomb potential Norm conserving regularized potential Quadratic regularized potential

35

-50

25

V (r) (a.u.)

4πr 2 n(r) (a.u.)

30

20

-100

Coulomb potential Norm conserving regularized potential Quadratic regularized potential

15 -150

10

5 -200

0 0

0.5

1

1.5

r (a.u.)

(a)

2

2.5

0

0.5

1

1.5

2

2.5

r (a.u.)

(b)

Fig. 6.2. Iron at 5 eV and 7.891 g cm−3 . (a) aam local electronic densities: bare (solid line), regularized with a quadratic polynomial (dot-dashed line) and with the norm-conserving procedure (dashed line). The cut-off radius for the regularization is 0.2 Wigner-Seitz radius. (b) Corresponding nucleus-electrons potentials.

The key feature to develop appropriate regularization is charge conservation. Indeed, from Gauss’ theorem, if charge is conserved inside the cut-off volume, the electric field generated by the regularized electronic density outside the cut-off volume is the same as the one obtained from the exact electronic density. Therefore, the forces generated by “ion” ` on the other “ions” are also identical leading to the same molecular dynamics, as long as the two ions are more than two cut-off radii away from each other. With this idea in mind, a methodm – based on the generation of pseudo-potentials for Kohn-Sham equations 57 – was developed.26,30 First, a regularized density n e(r) is built from the exact electronic density n(r), solution of Eq. (6.19) and Eq. (6.18), above the cut-off radius rc and a smooth function below, (
exp A + Br2 + Cr4 if r < rc n e(r) = (6.22) n(r) if r > rc

m The

method is presented for a functional without gradient correction. The necessary extension to take into account a gradient term is exposed in Ref. 30.

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with the three parameters A, B and C being obtained from continuity and charge conservation, lim n e(r) = n(rc ), ∂e n ∂n lim = , r→rc ∂r ∂r rc rc Z a Z a 2 dr 4πr n e(r) = dr 4πr2 n(r). r→rc

0

(6.23)

0

An example of “norm-conserving” regularized density for iron is represented on Fig. 6.2 (b). The regularized screened potential ves (r) is computed by inverting Eq. (6.19), " # 3 1 −1 π 2 β 2 √ n vs (r) = µ − I 1 e e(r) . (6.24) β 2 2

The regularized nuclear potential is finally calculated by subtracting both the Hartree and the exchange-correlation potentials, # " 3 1 −1 π 2 β 2 √ n ve(r) = µ − I 1 e(r) β 2 2 (6.25) Z Z a 1 r 2 − ds 4πs n e(s) − ds 4πse n(s) − vexc (r). r 0 r

Figure 6.2 shows ve(r) corresponding to the regularized density for iron. In comparison to the quadratic regularized potential, the “norm-conserving” potential is deeper even if both potentials reach their Coulomb behavior above 0.2 a. By constructing ve(r) from n e(r), it is ensured that the solution of Eq. (6.18) and Eq. (6.19) with ve(r) as the nucleus-electrons interaction is n e(r), thus conserving charge above rc . ve(r) is the Coulomb potential above rc . As a concluding remark of this paragraph, it is important to note the differences between the of regularization procedure and the Kohn-Sham pseudo-potentials. On the one hand, in the of scheme, there is absolutely no separation between core and valence electrons. All the electrons are treated on an equal footing avoiding problems of thermal and pressure delocalization. On the other hand, a new regularized potential is generated for each set of thermodynamic conditions – density and temperature – whereas pseudo-potentials are produced once and for alln . 6.4.4.3. Convergence with the cut-off radius Since the bare electrons-nucleus interaction is replaced by a regularized potential, it has to be checked that the quantities obtained with the two potentials converge as rc goes to zero. The comparison is operated on the equation of state, namely pressure and internal energy as functions of density and temperature. n For

a given number of valence electrons, cut-off energy and cut-off radius.

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Concerning pressure, by construction, both “bare” and “regularized” pressures are identical in the average atom model, Sec. 6.4.4.1, since pressure in this model is determined by the electronic density at the edge of the sphere14 which is the same for the two potentials because of charge conservation. In the case of 3D molecular dynamics, the regularized potential acts on the 3D local electronic density which, in return, has an impact on the electronic thermodynamic quantities and on the molecular dynamics itself through the screening of nuclei interactions. The result of such a convergence study is performed on a helium plasma at kT = 8.639 eV and ρ = 0.669 g cm−3 , Fig. 6.3. It shows that, once rc is below 0.4 a, pressure is independent of rc within one standard deviation and is therefore the pressure at rc = 0, i.e. the pressure that would be obtained with a bare electrons-nucleus potential. The status of internal energy is completely different with respect to regularization. The procedure of regularization does not impose a clear link between “regularized” and “bare” energies and, in fact, it is not possible to reach convergence of energy with rc 6= 0.25 Nevertheless, it is possible to correct the “regularized” energies through the help of the average atom model. Computations show that, for a pure element, the energy per atom of a face-centered-cubic (fcc) structure εnc,f cc (rc ) obtained with a regularized potential is very close to the average atom model energy per atom εnc,aam (rc ) evaluated with the same regularized potential. We define the internal energy per atom ε(rc ) through29   ε(rc ) = εnc (rc ) − εnc,aam (rc ) − εaam ,

(6.26)

with εnc (rc ) the internal energy computed from the molecular dynamics with the regularized potential and εaam the internal energy per atom from the average atom model with the bare coulomb potential. This internal energy per atom is represented on Fig. 6.3 for the helium plasma and does converge as rc is reduced. Consequently, if extrapolated to rc = 0, ε(rc ) represents the internal energy per atom that would be computed from a molecular dynamics with the bare potential. To a certain extent, this result can be understood as follows. The quantity nc ε (rc )−εnc,aam (rc ), equal to εnc (rc )−εnc,f cc(rc ), represents the effect of disorder on the electronic density, essentially the delocalized component which is not affected by the regularization. The part which is affected by the regularization does not depend on the disordered structure and is very close, around each atom, to the average atom one. The internal energy can therefore be corrected by this latter model. The idea is in fact quite close to the notion of core electrons in the pseudo-potentials in the paw formalism58 except that no electrons are frozen in the orbital-free case and that the average atom model is used at each new thermodynamic condition. Since both pressure and internal energy are computed correctly once convergence is reached, other quantities can be derived from them like sound velocity29 which is of interest for hydrodynamics simulations.

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-100.4

270

-100.6 265

ε (eV)

p (GPa)

-100.8

260

-101

-101.2 255 -101.4

250

-101.6 0

0.2

0.4

0.6

0.8

1

0

0.2

0.4

rc /a

(a)

0.6

0.8

1

rc /a

(b)

Fig. 6.3. Equation of state of helium at kT = 8.639 eV and ρ = 0.669 g cm−3 in terms of the cut-off radius rc . (a) Pressure (b) Internal energy per atom. Results are extracted from Ref. 29.

6.4.4.4. Other convergence issues Besides the cut-off radius rc characterizing the regularized potential, the main numerical parameters intervening in the implementation of of molecular dynamics are the convergence parameters for the minimization of F e , the cut-off energy used to limit the size of Fourier expansions, the size and the number of time steps of the molecular dynamics and the number of nuclei Na in the basic reference cell. All parameters are in principle determines by a systematic search for numerical convergence, within statistical uncertainty, of the quantity calculated. Some parameters can be evaluated through rules of thumb.29,30 It can be noted that, in the thermodynamic domain addressed in the present work, convergence of pressure is reached with about 30 atoms in the basic reference cell, as illustrated in Fig. 6.4. 6.5. Thermodynamics: towards high-density plasmas As was explained in Sec. 6.3.2, the chosen of free energy, Eq. (6.10), is a semiclassical expansion of the Kohn-Sham functional and thus an approximation of it. qmd is the reference at moderate densities and temperatures since it was successfully confronted to experiments.11 At the same time, of molecular dynamics is the reference at very high temperatures and densities and qmd should recover the of limit in this regime. Therefore, it is necessary to perform a comparison of the two approaches to gauge the range of validity of of molecular dynamics towards moderate densities and evaluate the critical temperatures and densities above which both qmd and of molecular dynamics lead to the same results. Our primary focus is concerned here with the equation of state. We have performed qmd and of molecular dynamics along the 1 and 4 eV isotherm from solid

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1.935

1.93

p (TPa)

1.925

1.92

1.915

1.91

1.905

1.9 0

20

40

60

80

100

120

140

Number of atoms

Fig. 6.4. Pressure of helium, calculated with of molecular dynamics at kT = 0.864 eV and ρ = 5.354 g cm−3 versus the number of atoms. The functional contains an exchange-correlation part but no gradient correction. Results are extracted from Ref. 30.

density up to 100 g cm−3 .59 Two pseudo-potentials were built for qmd: the “classical” one with three valence electrons and a large cut-off radius, and a “hard” one without core electrons, the five electrons being considered as valence including the |1si atomic state, and a small cut-off radius of half the Bohr radius. of molecular dynamics were performed with and without gradient correction in order to quantify the effect of this functional on the equation of state. In all cases, the exchangecorrelation functional is expressed in the lda35 so that the comparison of the two approaches gives an insight on the impact of the semi-classical expansion of the kinetic-entropic functionalo . Figure 6.5 shows the pressures obtained with qmd and of molecular dynamics. The two qmd calculations give the same pressure at 4 g cm−3 as expected from the transferability of the three-valence-electron pseudo-potential but diverge as density increases, the three-valence-electron qmd producing much lower pressures. The full-electron qmd converges smoothly to the of molecular dynamics results which are the reference at high density. As was previously explained, the three-valenceelectron qmd is based on a pseudo-potential that considers the |1si state as frozen with the nucleus, frozen-core approximation which fails at high density due to the partial delocalization of the atomic state. At moderate density, the of approach shows its limit, overestimating the pressure by a factor of three. The gradient correction improves the range of agreement, lowering the matching density to less than 10 g cm−3 .

o The

impact is both explicit through the form of the functional (at fixed density) and implicit since local electronic densities are different because of their self-consistent nature. The molecular dynamics is also modified by different screenings.

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p (GPa)

105

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183

qmd 3e− qmd 5e− tfdmd tfdwmd

104

103

10

ρ g cm−3




Fig. 6.5. Pressure along the 1 eV isotherm of boron. Circles and squares are, respectively, full and three valence electrons quantum results. Dashed (TFDW) and solid (TFD) lines are orbitalfree with and without (h = 0 in Eq. (6.10)) gradient correction, with a LDA exchange-correlation functional. Results are extracted from Ref. 59 and 30.

It is worth noting that, besides agreement on the equation of state, both qmd and of molecular dynamics predictp the crystallization of the system between 15 and 20 g cm−3 . With the help of the of average atom model, the effective ionic charge state can be estimated from Eq. (6.21) to be Z ? = 2.6 at 15 g cm−3 and Z ? = 2.8 at 20 g cm−3 . These ionization fractions correspond to effective coupling constants of, respectively, Γ = 152 and Γ = 190 crossing the critical constant of crystallization of the one-component plasma (ocp) Γ = 172.3,15,60 This phase transition is thus interpreted as the solid-liquid transition of the ocp.

6.6. Structural and dynamic properties: the quest for ionization Although of molecular dynamics is particularly efficient in terms of computational time, it is still not possible to couple it in-line with a hydrodynamics simulation in a straightforward way even if some recent progress has been made in this respect.61–63 Hydrodynamics simulations rely on three conservation laws, namely mass, linear momentum and energy64 completed with an equation of state. Depending on the physical phenomena involved, the conservation equations require the knowledge of microscopic coefficients like opacities for the transfer of radiation, thermal conductivity for electronic or ionic heat conduction, resistivity for magneto-hydrodynamics or ionic diffusion for multi-species simulations. To be performed in a reasonable computational time, most of the coefficients have to be tabulated for a wide range of thermodynamic conditions as well as chemical elements, and are consequently dep This

conclusion is extracted from a visual inspection of the nuclear trajectories.

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rived from simple approximate models like the Spitzer-Hubbard65,66 one for thermal conductivity. Most of the models of the microscopic coefficients are based on some ad hoc parameter. In the case of atomicq transport coefficients, like thermal conductivity, the ionization or effective ionic charge plays a particularly important role although no quantum observable can be associated with this notion. Indeed, it is the source that drives the electron-ion scattering potential which is the main ingredient in scattering cross-sections involved in thermal or electrical conductivity67,68 for example. As a consequence, it is of prime interest to support the choice of ionization model by a more fundamental theory like qmd and of molecular dynamics which treat all electrons on an equal footingr . In particular, it is possible to study transport coefficients or ionization with of molecular dynamics in thermodynamic regimes where no other 3D-parameter-free model is until now availables . This paragraph focuses on the comparison of of molecular dynamics with several parametrized effective models, namely one-component plasma and its Yukawa extension. These classical models are particularly interesting from a hydrodynamic viewpoint since analytical fits have been computed for several transport coefficients like ionic self-diffusion4 and bulk viscosity.5 Indeed, if one has a rule of thumb to compute the ionization from density and temperature data – which are the basic input in hydrodynamics simulations –, one has immediately access to transport coefficients from the previous fits. One of the main concerns of of molecular dynamics is therefore to provide this “rule of thumb”, i.e. a kind of mapping between of molecular dynamics and ocp results.

6.6.1. Ionization choice A definition of ionization can be brought up from the average atom model. In the case of quantum treatment of the electronic structure, there exists a widespread definition of ionization which counts the number of states whose energy is positive, definition inspired by the hydrogen atom. Nevertheless, this definition can lead to difficulties when resonant states appear in the continuum of free states.69 The case of the of average atom model is also complicated since there is no notion of quantum states. One possible choice for ionization is to take into account only the electrons that are free in the sense that they are subjected to no electric field. From charge conservation, Eq. (6.20), and with the help of Gauss’ theorem, one can see that free electrons correspond to electrons bound to the edge of the box, i.e. 43 πa3 n(a) where a is the Wigner-Seitz radius. This choice was highlighted by More55 in deriving the qeos model, but only in the pure Thomas-Fermi case, and was proved by Blenski q Where r For

electrons or ions are involved. the sake of simplicity, we suppose that qmd is all-electron. average atom model can deal with such regimes but are limited to pure-element systems.

s Quantum

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and Cichocki49 to be the only definition that makes the average atom Thomas-Fermi functional truly variational. In the following paragraph, the ionization computed from the average atom model is used as parameter of the inter-ionic forces. In the case of the ocp , this ionization Z ? determines the effective Coulomb interaction between the ions, the Lagrangian being Na Na   1X P`2 1 X Z`? Z`?0 Locp R, P = − . 2 M` 2 0 |R` − R`0 | `=1

(6.27)

`,` =1 `6=`0

It is worth noting that both the choice of ionization and the ocp model are consistent in the sense that they are based on a non-responding, zero electric field, neutralizing electron backgroundt.3 Comparing the Born-Oppenheimer, Eq. (6.5), and the ocp, Eq. (6.27), Lagrangians, the electronic density is seen in the ocp case as composed by a rigid screening around each nucleus, forming an effective ion of charge Z ? , and a free non-interacting background. 6.6.2. Comparison of results Comparison is made on iron for several thermodynamic conditions,27,70,71 summarized in Table 6.2. Table 6.2. Ionization and coupling constant of iron. kT (eV)

ρ
g cm−3

Z?

Γ

10 100 1000

22.5 34.5 39.65

5.9 9.3 20.5

49.8 14.4 7.3

Simulations were performed, in both of molecular dynamics and ocp, with 432 particles propagated in the isokinetic ensemble during 2000 time steps after relaxation of the system. The time steps are chosen as in Ref. 41, 25 and 29. The functional does not include any gradient correction. First, the comparison is made on the “ionic” structure through the pair distribution function (pdf) g(r) defined as72 na g(r) =

Na

 1 X δ(r + R` − R`0 ) , Na 0

(6.28)

`,` =1 `6=`0

t Historically,

the electronic background was fully degenerated to ensure that there was no response of the background to the Coulomb ionic potential. Nevertheless, the mathematical foundation of the ocp model requires only a non-responding background whatever the underlying physical phenomena.

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where na is the particle density and hi represents the average on the desired statistical ensemble. Results of the simulations with some of the conditions of Table 6.2 are exposed on Fig. 6.6. 1.2

ofmd ocp Z ? = 9.3

1.2

ofmd ocp Z ? = 20.5 1

1

0.8

g(r)

g(r)

0.8

0.6

0.6

0.4 0.4

0.2

0.2

0

0 0

0.5

1

1.5

2

2.5

3

0

r/a

0.5

1

1.5

2

2.5

3

r/a

(a)

(b)

Fig. 6.6. Pair distribution functions of iron by both of molecular dynamics and effective ocp. (a) kT = 100 eV and ρ = 34.5 g cm−3 (b) kT = 1000 eV and ρ = 39.65 g cm−3 . Results are extracted from Ref. 70.

In each case, pdf’s of of molecular dynamics and effective ocp match well, the agreement being excellent at 1000 eV. Despite the high temperatures, both the density and the peeling of atomic electrons lead to strong ionic interactions and a liquid-like structure of the plasma. This is the reason why perturbative theories – based on the smallness of interactions – fail and the recourse to molecular dynamics is necessary. Nevertheless, one must keep in mind that pdf’s give only a partial view of the plasma structure all the more so since, due to the limited number of particles, statistical accuracy is only ensured for the first bump of the pdf. In order to support the choice of ionization, the comparison between of molecular dynamics and ocp can be performed not only on static properties but also on dynamic ones. Two features are particularly important in hydrodynamics simulations, especially for inertial confinement fusion, namely ionic diffusion and viscosity, since these two phenomena can lead to a reduction of hydrodynamic instabilities of icf targets.73 Both ionic diffusion and viscosity are obtained through auto-correlation functions, either from the nuclear velocities for the self-diffusion coefficient D` , D` =

1 βM`

Z

R+

Z` (t) dt =

1 βM`

Z

R+

hP` (t) · P` (0)i dt, hP` (0) · P` (0)i

(6.29)

or from the off-diagonal elements of the microscopic stress tensor ς µν for the shear

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viscosity η of a stationary and uniform fluid, Z Z

µν  β β η= η(t) dt = ς (t)ς µν (0) dt, Na Ω R+ Na Ω R+

187

(6.30)

with Ω the volume per atom of the system. It is important to note that, in the case of of molecular dynamics, the stress tensor contains nuclear41 and electronic37,39 contributions. Figure 6.7 reports the velocity auto-correlation functions for the of molecular dynamics and ocp simulations at kT = 10 eV and ρ = 22.5 g cm−3 . Statistical convergence is excellent for this signal since it is obtained by averaging over the particles72 and consequently constitutes a reliable check for comparison. As shown 1

ofmd ocp Z ? = 5.9

0.8

Z(t)

0.6

0.4

0.2

0

0

500

1000

1500

2000

t (a.u.)

Fig. 6.7. Velocity auto-correlation function of iron at kT = 10 eV and ρ = 22.5 g cm−3 from ofmd and ocp simulations. Results are extracted from Ref. 27.

by the two functions, ocp auto-correlation leads to more pronounced oscillations, i.e. stronger interactions among the “ions”. The results indicate that it is impossible to map the of molecular dynamics system into an equivalent effective purely coulombic system, the of molecular dynamics interactions being more damped at long distance. This conclusion is not contradictory with the pdf results since longrange interactions are not reproduced by the poor statistics of the previous pdf’s. Even if the temporal signals are different, the temporal integrations of these signals, computing the diffusion coefficient, are in good agreement as shown in Table 6.3. Although similar in their mathematical form, diffusion and viscosity are very different from a numerical point of view since η(t) is calculated with only one value per time step, leading to very noisy signals.5,74 Viscosities are reported in Table 6.4 in conjunction with the fit proposed by Bastea.5 As in the case of diffusion coefficients, transport coefficients of of molecular dynamics and ocp are in excellent agreement although the temporal signals, not shown here,25 are quite different.

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Diffusion coefficients of iron by ofmd and of molecular dynamics   10−3 cm2 s−1

ocp

1.1 7.0

1.0 6.8

10 eV, 22.5 g cm−3 100 eV, 34.5 g cm−3

Table 6.4. and fit.

Viscosities of iron from ofmd and ocp, both from simulations of molecular dynamics ocp (10−2 Pa s)

100 eV, 34.5 g cm−3 1000 eV, 39.65 g cm−3

2.3 9.0

2.1 8.4

Bastea’s fit 2.3 8.4

Comparisons between of molecular dynamics and ocp results show that there cannot be a unique effective ocp model that reproduces both structural and dynamic properties of the of molecular dynamics model. Nevertheless, by choosing the ionization from electrons bound to the edge of the box in the average atom, i.e. the More’s definition, transport coefficients are in reasonable agreement and offer the possibility to use the existing fits of the ocp in hydrodynamics simulations. 6.7. Inside the mixture: the plasma as a soup of electrons and nuclei Mixtures of elements are often encountered in both astrophysical and icf plasmas. In the interior of stars, there are mixtures of heavy (C, N, O, Ne) and light (H, He) elements either at the limits between the shells or during the gravitational segregation process. In icf capsule designs, the ablator – which transforms the radiative energy into mechanical energy for compression – can be composed of beryllium doped with copper or plastic (C,H) doped with germanium. Consequently, knowing the microscopic properties of such mixtures, in the plasma regime, is of prime importance for “scientific” as well as “technological” purposes. Until recently, these properties were studied by means of approximate parametrized models, like the binary ionic model75–77 (the multi-species version of the ocp) or its Yukawa counterpart.7 As in the case of ocp, these models rely on the knowledge of the ionization degree of each species, a notion even more difficult to define and determine than for pure elements. Contrary to the aforementioned models, orbital-free DFT, by treating all electrons on an equal footing, allows to deal with mixtures in the same way as with simple elements, i.e. without specific approximation related to the multi-species nature. This fact can be readily checked from the Lagrangian, Eq. (6.5), and the form of the functional, Eq. (6.10), which are determined entirely by the atomic numbers of the atoms. In this sense, of

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molecular dynamics treats a plasma as a soup of electrons and nuclei and not as a mixture of effective ions. As was highlighted in the previous paragraph, hydrodynamics simulations are based on coefficients that depend on the underlying microscopic physics. Besides simple and fast models for pure elements, the code must have access to rules allowing to deal with mixtures, which constitutes a particularly tough challenge. The purpose of the following paragraphs is to show how of molecular dynamics can bring support to some of these rules, especially in terms of eos, by tackling the problem of mixtures without approximation. 6.7.1. Eos mixing rule A mixing rule for an equation of state allows one to calculate the equation of state of a mixture from the equation of state of its components. Besides the academic perfect-gas mixing rule, which has a very limited range of validity, an isobaricisothermal mixing rule has been defined in the framework of the average atom model,55 but has not been verified so far since it requires 3D-non-parametrized simulations. This paragraph focuses on a specific application of the isobaric-isothermal mixing rule on the excess quantities, defined below, in order to show the possibilities offered by of molecular dynamics for the verification of the mixing rules. A thorough analysis can be found in Ref. 78 and a comparison of this mixing rule with qmd is given in Ref. 79,80. Isobaric-isothermal mixing rule case of the ocp, as

The excess quantities are defined, as in the

pex = p − na kT,

(6.31)

3 εex = ε − kT, 2

(6.32)

and

p and ε being the pressure and the internal energy per atom of the system. Excess quantities evaluate the deviation from the nuclear perfect gas. It is important to note that the excess quantities cannot be solely attributed to the electronic component since it contains the Coulomb nuclei-nuclei interactions; pex is not the electronic pressure.40 The isobaric-isothermal mixing rule states that the partial mass densities ρ` related to component ` are determined by the following coupled equations: • equality of the excess pressures (isobaric) of the pure components ` at their respective partial densities and at temperature T (isothermal), pex,` (ρ` , T ) = pex,m (ρm , T ), ∀`, m;

(6.33)

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• additivity of partial volumes, Ω=

X `

Ω` ≡

X A` 1X x` A` = x` , ρ ρ` `

(6.34)

`

where Ω` , A` and x` are the partial volume per atom, the atomic mass and the mole fraction of component `. It is implicitly supposed that there is a one-to-one relationship between density and pressure in Eq. (6.33) so that Eq. (6.33) and Eq. (6.34) have a unique solution. Once the ρ` ’s are known, the excess pressure is taken equal to the pex,` ’s and the excess internal energy per atom to
X εex ρ, T, {x` } = x` εex,` (ρ` , T ), (6.35) `

where εex,` (ρ` , T ) is the excess internal energy per atom of the pure component ` at density ρ` and temperature T . It can be shown that the isobaric-isothermal mixing rule can be equivalently expressed as a linear mixing rule applied to the free energy (see paragraph below). Test procedure of molecular dynamics can be used as a test for the previous mixing rule. For given composition, temperature and density, a first computation is performed with the full mixture, therefore without approximation on the equation of state. Depending on the composition and the thermodynamic conditions, it may be necessary to perform a convergence study in terms of the number of particles and regularization cut-off radii. At the same time, of molecular dynamics calculations are carried out on each element individually. The temperature is imposed to be the same in both mixture and pure element calculations. For pure elements, the range of densities is chosen to be around the density provided by the application of the mixing rule to the average atom model. Let us take an example. The plasma is an equimolar mixture of iron and helium at temperature T and density ρ: • first, 200 particlesu , 100 of each species, are propagated providing pressure pFe,He (ρ, T ) and energy of the mixture at T and ρ ; • second, the equations of state of iron and helium have to be determined independently: – to this purpose, the average atom model, Sec. 6.4.4.1, is combined with the mixing rule, as is done in the qeos model,55 just to provide an initial guess for the partial densities ρaam . Let us emphasize that the average ` atom model is not used to compute the equation of state ; u We

suppose here that convergence is ensured with such a particle number.

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191

– once ρaam and ρaam are known, of molecular calculations He Fe  aam dynamicsaam  are done on helium for densities included in ρ − δρ ; ρ + δρ He He He and He  aam  aam on iron in ρFe − δρFe ; ρFe + δρFe ; – these calculations lead to equations of state of the form pex,He (ρHe , T ) and pex,Fe (ρFe , T ), that are combined with the mixing rule, Eq. (6.33) and Eq. (6.34), to obtain pmr Fe,He (ρ, T ) ; • the test of the mixing rule is eventually the comparison between the directsimulation pressure pFe,He (ρ, T ) and the mixing-rule pressure pmr Fe,He (ρ, T ), comparison being made on the energies per atom as well. Results The previous procedure was applied to an equimolar mixture of iron and helium for various thermodynamic conditions given in Table 6.5. In each direct simulation of the mixture, 30 particles – 15 of each component – are propagated during 2000 time steps. For pure He or pure Fe involved in applying the mixing rule, 32 particles are propagated during 2000 time steps. Convergence of the results with the number of particles was checked for both mixture and pure-element simulations. Table 6.5. Excess pressures obtained with a lda and without gradient correction for an equimolar mixture of He and Fe. pFe,He designates the results given by direct simulations and pmr Fe,He those given by the mixing rule used with of molecular dynamics. The numbers in parentheses are standard deviations. Results are extracted from Ref. 78. ρ
g cm−3

kT (eV)

pFe,He

1 1 10 10 10 10 10 10 10 10

5 500 2 5 10 20 50 100 200 500

18.54 (0.06) 18397 (2) 688 (2) 888 (2) 1311 (3) 2421 (4) 7048 (9) 17851 (6) 46656 (9) 161160 (30)

pmr Fe,He (GPa) 18.54 (0.05) 18385 (2) 682 (2) 883 (2) 1307 (3) 2400 (4) 7073 (6) 17821 (8) 46593 (20) 161110 (50)

Thanks to the high computational efficiency of the orbital-free procedure, it is also possible to perform precise simulations for highly asymmetric systems – in terms of chemical composition – and gauge the validity of the mixing rule for such a mixture. Results for excess pressures and internal energies, obtained with a LDA and without gradient corrections, are summarized in Table 6.6. Convergence with the number of particles was checked with simulations up to 200 atoms. As in the equimolar case, the mixing rule leads to results in excellent agreement with the simulations for both pressures and internal energies.

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F. Lambert et al. Table 6.6. Excess pressures and internal energies per atom for a mixture of He and Fe at 10 g cm−3 and 50 eV for various chemical compositions from direct simulation and mixing rule. The results are obtained with the abinit code.81 xHe

xFe

pmr Fe,He

pFe,He

0.9 0.7 0.5 0.3 0.1

0.1 0.3 0.5 0.7 0.9

11843 8196.5 7069.9 6523.0 6200.7

(11) (9.1) (6.9) (4.0) (2.7)

εmr Fe,He

εFe,He

(GPa)

(eV)

11791 8150.7 7049.6 6514.9 6199.2

(13) (5.5) (4.0) (4.4) (5.2)

-4206.69 (0.10) -12594.80 (0.11) -20981.15 (0.11) -29365.19 (0.07) -37750.30 (0.06)

-4206.72 (0.15) -12594.91 (0.09) -20981.13 (0.07) -29365.00 (0.07) -37750.18 (0.08)

The mixing rule seems to be robust with respect to the variation of either the composition or the thermodynamic conditions, providing an efficient way to compute the eos of the mixture from the eos of each pure element. It is interesting to note that the isobaric-isothermal mixing rule is also robust with respect to a change of the functional used. Each choice of xc [n] and h(n) in the construction of the functional F e [n] yields an eos. The previous eos, obtained with h(n) = 0 and xc [n] described in Sec. 6.4.1, is denoted TFD (standing for Thomas-Fermi-Dirac); the one obtained with, in addition, Perrot’s fit23 for h(n) is denoted TFDW (for Thomas-Fermi-Dirac-Weizs¨ acker), and the case xc [n] = 0 and h(n) = 0 is denoted TF (for Thomas-Fermi). We consider an equimolar mixture of helium and iron at ρ = 10 g cm−3 and temperatures from 2 to 500 eV. All calculations are carried out with the numbers of time steps and particles previously given. As shown in Fig. 6.8, this range of temperatures allows to explore gradual differences, that may be large, among the three eos (TF, TFD and TFDW). For each eos, we examined the validity 0

α=TFDMD α=TFDWMD -0.1

pα ex pTFMD ex

−1

-0.2

-0.3

-0.4

-0.5

-0.6 1

10

100

1000

kT (eV)

Fig. 6.8. Comparison of the excess pressures pα ex calculated with the functional α as TF, TFD, or TFDW for an equimolar mixture of helium and iron at density ρ = 10 g cm−3 and various temperatures. Results are extracted from Ref. 81.

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1.5

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pTFDMD ex

− 1 (%)

2

pTFDMD ex,mr

pTFMD ex

pTFMD ex,mr

− 1 (%)

Properties of Hot and Dense Matter by Orbital-Free Molecular Dynamics

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-1.5

-2

193

-2 1

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100

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1

10

kT (eV)

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kT (eV)

(a)

(b) 2

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pTFDWMD ex

pTFDWMD ex,mr

− 1 (%)

1

0.5

0

-0.5

-1

-1.5

-2 1

10

100

1000

kT (eV)

(c) Fig. 6.9. Equimolar mixture of helium and iron at density ρ = 10 g cm−3 and various temperatures. The models α are (a) TF, (b) TFD and (c) TFDW. Comparison of the excess pressures α pα ex,mr , calculated by the isobaric-isothermal mixing rule, with the excess pressures pex calculated by a direct simulation of the full mixture. Results are extracted from Ref. 81.

of the isobaric-isothermal mixing rule in the thermodynamic conditions indicated above. Whether the three eos yield significantly different excess pressures or not, it can be seen on Fig. 6.9 that, for each case, all the excess pressures given by the isobaric-isothermal mixing rule agree within statistical error with those given by a direct simulation of the full mixture. The robustness of the mixing rule with respect to the eos used leads us to suggest its usefullness with qmd. It is also interesting to note that the procedure described in the mixing rule can be readily used in hydrodynamics simulations based on tabulated equations of state.

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The isobaric-isothermal rule as a linear mixing rule The isobaricisothermal rule can be expressed as a particular form of the widespread linear mixing rule. First, we suppose that we have N non-interacting sub-systems defined by their respective free energy per atom f` depending on their respective volume per atom and temperature. Those sub-systems are brought in contact in such a way that the P total volume per atom of the system is Ω = ` x` Ω` and that the whole system is maintained at temperature T . Since the sub-systems are non interacting, the free energy of the whole system is the sum of the individual free energies. There is no entropy of mixing, the two sub-systems being only constrained through the conservation of the total volume Ω per atom. The total free energy per atom reads then X f (T, Ω, {x` }, {Ω` }) = x` f` (T, Ω` ). (6.36) `

The total free energy per atom is a parametric state function of the partial volumes per atom. At given Ω and T , the equilibrium of the system is reached when the free energy is stationary with respect to a change in the varying state variables, namely Ω` , with the constraint of volume conservation. With the notation d for the differential and the use of a Lagrange multiplier λ, the stationarity of the free energy per atom at equilibrium is thus expressed by X !
d f T, Ω, {x` }, {Ω` } − λ x` Ω` − Ω = 0, (6.37) `

which gives ∂f` (T, Ω` ) = −p` (T, Ω` ) = λ. ∂Ω`

(6.38)

Equation (6.38) expresses the fact that, when sub-systems exchange volume, the thermodynamic potential that is equalized at equilibrium is pressure. This just renders the mechanical equilibrium of the interfaces between the sub-systems. From the free energy per atom, Eq. (6.36), one computes the pressure p through X ∂Ω` ∂f` (T, Ω` )
p T, Ω, {x` } = − x` , ∂Ω ∂Ω`

(6.39)

`

which, with Eq. (6.34), reduces to
p ρ, T, {x` } = −λ.

(6.40)

The pressure of the whole system is, as expected, the pressure of each sub-system. The entropy per atom s is equal to

∂f Ω, T, {x` } s Ω, T, {x` } = − . (6.41) ∂T

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With the linear mixing rule, Eq. (6.36), s reads as X ∂f` (T, Ω` ) X ∂Ω` ∂f` (T, Ω` )
s T, Ω, {x` } = − x` − x` . ∂T ∂T ∂Ω` `

(6.42)

`

The second part of the right-hand side comes from the fact that the partial volumes per atom Ω` obtained from the pressure equality vary with T . With Eq. (6.38) and volume conservation, Eq. (6.42) yields
X s ρ, T, {x` } = x` s` (ρ` , T ), (6.43) `

where s` (ρ` , T ) is the entropy per atom of element ` at density ρ` and temperature T . Eventually, the internal energy per atom is obtained from the free energy and the entropy per atom. As shown by the previous derivation, the conjunction of the linear mixing rule with the volume conservation allows to compute all the thermodynamic properties of the system from those of each pure element. Moreover, the previous derivation is strictly identical for excess pressures since the perfect-gas free energy per atom of the nuclei does not contain any partial volume dependence. The linear mixing rule can be readily applied – with excess quantities – when considering the plasma as a collection of independent average atoms, Sec. 6.4.4.1, with no restriction on the electronic functional, the derivation being entirely based on thermodynamic arguments. The case of the true mixture, managed by of molecular dynamics simulations, seems therefore to be interpreted as average atom subsystems with no interactions. With a static – Born-Oppenheimer-like – picture, the charge of each nucleus is compensated by the surrounding electronic cloud, producing a neutral sub-system, whose volume is determined by the equalized pressure of each cloud. 6.7.2. Transport coefficients and partial ionization The equation of state is one of the properties necessary for hydrodynamic simulations. In addition, several coefficients are of interest in icf like ionic diffusion for the mixing shells between different materials as well as thermal conductivity or viscosity as phenomena limiting the growth of hydrodynamic instabilities.73 As in the case of pure plasmas, most models dealing with the underlying microscopic physics in hydrodynamics simulations require the knowledge of average ionization of the plasma or of each species inside the plasma. First, it is important to note that ionization has to be provided by means of the average atom model where ionization can be defined and efficiently computed. Since the average atom model cannot deal intrinsically with mixtures, being a onecenter approximation, one has to use a mixing rule in evaluating mixing properties from this model. In Sec. 6.7.1, we have proved numerically that the eos mixing rule was particularly robust. Therefore, we use the average atom model in conjunction

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with the eos mixing rule in order to have an insight on the ionization of each element inside the plasma and compare some of the properties with our reference, of molecular dynamics, which treats a soup of electrons and nuclei. In the following, ionization is defined as in Sec. 6.4.4.1. The procedure was checked on a highly asymmetric mixture of 90% of deuterium and 10% of copper at 50 g cm−3 and 100 eV, mixture inspired by some special design of icf targets. This mixture is particularly difficult to tackle since the deuterium is in a kinetic regime, involving a short time step, whereas statistical convergence of the copper properties requires long runs. The of molecular dynamics simulation was performed with 450 atoms of deuterium and 50 atoms of copper propagated during 47000 time steps. As in the helium-iron plasma, the eos of of molecular dynamics and of the mixing rule are in good agreement, the excess pressures being respectively of 134 000(150) GPa and 133 860 GPa. The mixing rule is also used in conjunction with the average atom model, yielding both the partial densities and ionizations of deuterium and copper, Table 6.7. These ionizations lead to coupling constants of 0.5 for deuterium and 22.7 for copper, expressing the kinetic and coupled nature of each species. Table 6.7. Effective charge state inside the plasma from the average atom model with the mixing rule. Results are extracted from Ref. 78.

Z∗

deuterium

copper

0.89

11.2

By knowing thermodynamic conditions and effective charge states, a simulation based on the binary ionic mixture model (bim)5,75 was performed with the same numerical parameters (number and length of time steps, number of particles) as with of molecular dynamics. The structures were extracted from the of molecular dynamics and the bim simulations in order to gauge the validity of the ionization choice. The self pair distribution functions are represented on Fig. 6.10, expressing the different behavior of the two species, namely kinetic for deuterium and strongly coupled for copper. The pdf’s are in good agreement even if the rises of the copper pdf’s are slightly shifted apart. Nevertheless, one must keep in mind that copper is described through 50 atoms, limiting the statistical accuracy of the results in comparison with deuterium. The self-diffusion coefficients of both deuterium and copper were also extracted from the two simulations giving an insight on the dynamic properties of the system. The results are presented in Table 6.8. The behavior of deuterium seems to be correctly caught by the BIM simulations.

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ofmd D-D ofmd Cu-Cu bim D-D bim Cu-Cu

1.2

gαβ (r)(u.a.)

1

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0.6

0.4

0.2

0 0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

r(u.a.)

Fig. 6.10. Self pair distribution functions obtained with of molecular dynamics and bim of the 90% deuterium-10% copper plasma at ρ = 50 g cm−3 and kT = 100 eV. Results are extracted from Ref. 78.

Table 6.8. Self-diffusion coefficients of 90% deuterium and 10% copper at ρ = 50 g cm−3 and kT = 100 eV from ofmd and bim simulations.

deuterium copper

of molecular dynamics
cm2 s−1

bim

1.32 × 10−1 5.36 × 10−4

1.24 × 10−1 4.65 × 10−4

Being kinetic, the properties of this element, partial pdf and self-diffusion, are little sensitive to a small modification of the 0.9 ionization value. Copper, being strongly coupled, is slightly more affected by a wrong estimation of the effective charge state. Contrary to pdf’s, the self-diffusion coefficients are computed by averaging over the particles and are therefore much more converged than the pdf’s. Consequently, the difference between of molecular dynamics and bim self-diffusion coefficients is not due to statistical inaccuracy. The agreement on copper selfdiffusion between the two methods is a bit worse than for deuterium, the difference being of 13% and of the same order as in the strongly coupled case for iron, Table 6.3. Inspection of the velocity auto-correlation functions for copper shows that the of molecular dynamics signal remains correlated longer than the bim one. In the of molecular dynamics case, the effective ion-ion interactions are more damped than a pure Coulomb potential, reducing the “deviation” of each rectilinear ion trajectoryv and increasing the scalar product P(t) · P(0), i.e. correlations. v As

a comparison, for the same thermodynamic conditions, the diffusion coefficient of a Yukawa ocp is generally higher than an ocp one.

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Nevertheless the BIM seems to draw the essential dynamic picture of the of molecular dynamics simulation and allows a computationally efficient calculation of the transport coefficients of mixtures.

6.8. Conclusion In this chapter, we have shown that of molecular dynamics is particularly wellsuited to study the properties of matter in extreme conditions, i.e. high temperatures and pressures. By coupling of dft for the electrons and classical molecular dynamics for the nuclei, it has been possible to evaluate both static properties, like equation of state or pair distribution function, and dynamic ones, like diffusion coefficient or viscosity. Besides, by treating single-element plasma and mixtures on an equal footing, of molecular dynamics allowed us to gauge the validity of commonly used mixing rules for the equation of state. Moreover, of molecular dynamics has given support to approximate classical models whose properties can be efficiently used as inputs for hydrodynamics simulations. Although of molecular dynamics is extremely powerful in computing “ionic” transport coefficients, special care must be taken when dealing with electronic transport coefficients like electrical and thermal conductivities. These properties are computed by qmd simulations in the Kubo-Greenwood82,83 formalism which is the implementation of linear response theory84 to an independent-particle picture. This method has been successfully compared to experiments in the warm dense regime.85–89 The Kubo-Greenwood formalism is based on the transitions between one-body electronic states and can not be straightforwardly applied to the of framework. In order to compute the electronic transport properties in of dft, some attempts have been made by coupling the Ziman theory of conduction67 with the Born approximation which requires only the knowledge of the total electronic scattering potential.25 Unfortunately, comparisons on boron between qmd and ofmd highlighted specific quantum effects around the Fermi energy that cannot be caught by the conjunction of the simple functionals exposed in this work and the Ziman’s formalism.90 Since ofmd is efficient to deal with the ionic structure at high temperature and density, and, qmd is reliable to produce the precise electronic structure on a given ionic structure, we coupled the two methods by, first, performing ofmd simulations and, second, computing with the Kohn-Sham formalism the detailed electronic structure on a few statistically independent ionic configurations extracted from the previous of molecular dynamics. This method was successfully used in calculating the thermal conductivity of hydrogen along the 80 g cm−3 isochor up to 800 eV.91 In this way, the commonly used models of thermal conductivity65,66 can be checked by first-principle simulations. By coupling orbital-free and quantum molecular dynamics, we have now a complete chain of calculation to determine static and dynamic properties of both the electronic and “ionic” component of a plasma in a wide range of thermodynamic

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conditions. By improving the kinetic-entropic part of the of free-energy functional, of molecular dynamics could be performed at lower temperatures and densities extending the range of validity of the chain of calculation, and, then opening the opportunity of a reliable and systematic derivation of plasma properties. Acknowledgments The authors would like to thank D. Gilles and G. Salin for a fruitful collaboration on Yukawa systems, G. Z´erah for his precious advice about the abinit code, V. Recoules for her collaboration on the thermal conductivity project, and, D.A. Horner, J.D. Kress and L.A. Collins for common works on comparisons between ofmd and qmd. A part of the present results has been obtained through the abinit software package, a common project of the Universit´e Catholique de Louvain, Corning Incorporated and other contributors. This work was also made possible by the use of the Centre de Calcul, Recherche et Technologie (CCRT) in Bruy`eres Le Chˆatel. References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24.

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62. X. Zhang and G. Lu, Phys. Rev. B 76, 245111 (2007). 63. Q. Peng, X. Zhang, L. Hung, E. A. Carter, and G. Lu, Phys. Rev. B 78, 054118 (2008). 64. Y. B. Zel’dovich and Y. P. Raizer, Physics of Shock Waves and High-Temperature Hydrodynamic Phenomena (Dover Publications, 2002). 65. L. Spitzer and R. Harm, Phys. Rev. 89, 977 (1953). 66. W. B. Hubbard and M. Lampe, Astrophys. J. Suppl. Ser. 18, 297 (1969). 67. J. M. Ziman, Principles of the Theory of Solids (Cambridge University Press, 1979). 68. R. Evans, B. L. Gyorffy, N. Szabo, and J. M. Ziman, in The properties of Liquid Metals, edited by S. Takeuchi (Taylor & Francis, 1972), p. 319–331. 69. C. Cohen-Tannoudji, B. Diu, and F. Laloe, Quantum Mechanics, Vols. 1 and 2 (John Wiley & Sons, 1977). 70. F. Lambert, J. Cl´erouin, S. Mazevet, and D. Gilles, Contrib. Plasma Phys. 47, 272 (2007). 71. D. Gilles, F. Lambert, J. Cl´erouin, and G. Salin, High Energy Density Phys. 3, 95 (2007). 72. M. P. Allen and D. J. Tildesley, Computer Simulation of Liquids (Oxford University Press, USA, 1989). 73. R. P. Drake, High-Energy-Density Physics: Fundamentals, Inertial Fusion, and Experimental Astrophysics (Shock Wave and High Pressure Phenomena) (Springer, 2006). 74. D. Alfe and M. J. Gillan, Phys. Rev. Lett. 81, 5161 (1998). 75. J. P. Hansen, I. R. Mcdonald, and P. Vieillefosse, Phys. Rev. A 20, 2590 (1979). 76. J. P. Hansen, F. Joly, and I. R. Mcdonald, Physica A 132, 472 (1985). 77. D. B. Boercker and E. L. Pollock, Phys. Rev. A 36, 1779 (1987). 78. F. Lambert, J. Cl´erouin, J. F. Danel, L. Kazandjian, and G. Z´erah, Phys. Rev. E 77, 026402 (2008). 79. D. A. Horner, J. D. Kress, and L. A. Collins, Phys. Rev. B 77, 064102 (2008). The pressures involved in the present paper are excess pressures and not total pressures (private communication with the authors). 80. D. A. Horner and L. A. Collins, private communication. 81. J. F. Danel, L. Kazandjian, and G. Z´erah, Orbital-free molecular dynamics simulation of a warm dense dense mixture: examination of the excess-pressure matching rule, Phys. Rev. E, in press (2009). 82. R. Kubo, J. Phys. Soc. Japan 12, 570 (1957). 83. D. A. Greenwood, Proc. Phys. Soc. London 71, 585 (1958). 84. R. Kubo, M. Toda, and N. Hashitsume, Statistical Physics II. Nonequilibrium Statistical Mechanics (Springer, Berlin, 1998). 85. V. Recoules, P. Renaudin, J. Cl´erouin, P. Noiret, and G. Z´erah, Phys. Rev. E 66, 056412 (2002). 86. S. Mazevet, M. P. Desjarlais, L. A. Collins, J. D. Kress, and N. H. Magee, Phys. Rev. E 71, 016409 (2005). 87. V. Recoules and J. P. Crocombette, Phys. Rev. B 72, 104202 (2005). 88. P. Renaudin, V. Recoules, P. Noiret, and J. Clerouin, Phys. Rev. E 73, 056403 (2006). 89. J. Cl´erouin, P. Renaudin, and P. Noiret, Phys. Rev. E 77, 026409 (2008). 90. F. Lambert, S. Mazevet, and J. Cl´erouin, High Energy Density Phys. 5, 31 (2009). 91. V. Recoules, F. Lambert, A. Decoster, B. Canaud, and J. Clerouin, Phys. Rev. Lett. 102, 075002 (2009).

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Chapter 7 Shell-Correction and Orbital-Free Density-Functional Methods for Finite Systems Constantine Yannouleas and Uzi Landman School of Physics, Georgia Institute of Technology Atlanta, GA 30332-0430, USA [email protected] [email protected] Orbital-free (OF) methods promise significant speed-up of computations based on density functional theory (DFT). In this field, the development of accurate kinetic-energy density functionals remains an open question. In this chapter we review the shell-correction method (SCM, commonly known as Strutinsky’s averaging method) applied originally in nuclear physics and its more recent formulation in the context of DFT [Yannouleas and Landman, Phys. Rev. B 48, 8376 (1993)]. We demonstrate the DFT-SCM method through its earlier applications to condensed-matter finite systems, including metal clusters, fullerenes, and metal nanowires. The DFT-SCM incorporates quantum mechanical interference effects and thus offers an improvement compared to the use of Thomas-Fermi-type kinetic energy density functionals in OF-DFT.

Contents 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1.1 Preamble . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1.2 Motivation for finite systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1.3 Plan of the chapter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Methodology and derivation of microscopic DFT-SCM . . . . . . . . . . . . . . . . . 7.2.1 Historical review of SCM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.2 DFT-SCM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3 Applications of DFT-SCM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.1 Metal clusters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.2 Neutral and multiply charged fullerenes . . . . . . . . . . . . . . . . . . . . . 7.3.3 On mesoscopic forces and quantized conductance in model metallic nanowires 7.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix A. Semiempirical shell-correction method (SE-SCM) . . . . . . . . . . . . . . . A.1 Semiempirical shell-correction method for triaxially deformed clusters . . . . . . . . A.1.1 Liquid-drop model for neutral and charged deformed clusters . . . . . . . . . A.1.2 The modified Nilsson potential . . . . . . . . . . . . . . . . . . . . . . . . . . A.1.3 Averaging of single-particle spectra and semi-empirical shell correction . . . . A.1.4 Overall procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.2 Applications of SE-SCM to metal clusters . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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7.1. Introduction 7.1.1. Preamble Often theoretical methods (in particular computational techniques) are developed in response to emerging scientific challenges in specific fields. The development of the shell correction method (SCM) by Strutinsky1 in the late 1960’s was motivated by the observation of large nonuniformities (oscillatory behavior) exhibited by a number of nuclear properties as a function of the nuclear size. These properties included: total nuclear masses, nuclear deformation energies, and large distortions and fission barriers. While it was understood already that the total energy of nuclei can be decomposed into an oscillatory part and one that shows a slow “smooth” variation as a function of size, Strutinsky’s seminal contribution was to calculate the two parts from different nuclear models: the former from the nuclear shell model and the latter from the liquid drop model. In particular, the calculation of the oscillatory part was enabled by employing an averaging method that smeared the single particle spectrum associated with a nuclear model potential. It is recognized that the Strutinsky procedure provides a method which “reproduces microscopic results in an optimal way using phenomenological models”;2 in the Appendix we describe an adaptation of the Strutinsky phenomenological procedure to metal clusters; we term this procedure as the semiempirical (SE)-SCM. In this chapter, we focus on our development in the early 1990’s of the microscopic density-functional-theory (DFT)-SCM approach,3 where we have shown that the total energy of a condensed-matter finite system can be identified with the Harris functional [see Eq. (7.16)], with the shell correction [Eq. (7.23)] being expressed through both the kinetic energy, Tsh , of this functional [Eq. (7.19)] and the kinetic energy of an extended-Thomas-Fermi (ETF) functional expanded to fourth-order density gradients [see TET F in Eq. (7.22)]. It is important to note that in our procedure an optimized input density is used in the Harris functional. This optimization can be achieved through a variational procedure [using an orbital-free (OF) energy functional, e.g., the ETF functional with 4th-order gradients] with a parametrized trial density profile [see Eq. (7.25)], or through the use [see Eq. (7.24)] of the variational principle applied to an orbital-free energy functional. (For literature regarding orbital-free kinetic-energy functionals, see, e.g., Refs. 4–9.) A similar optimization of an OF/4th-order-ETF density has been shown to be consistent with the Strutinsky averaging approach.10 Such 4th-order optimization of the input density renders rather ambiguous any direct (term-by-term) comparison between the method proposed by us and subsequent treatments, which extend the DFTSCM to include higher-order shell-correction terms without input-density optimization 11,12 (see also Ref. 13). Indeed, the input-density optimization (in particular with the use of 4th-order gradients) minimizes contributions from higher-order shell corrections. In light of certain existing similarities between the physics of nuclei and clusters

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(despite the large disparity in spatial and energy scales and the different origins of inter-particle interactions in these systems), in particular the finding of electronic shell effects in clusters,14–18 it was natural to use the jellium model in the early applications of the DFT-SCM to clusters. However, as noted3 already early on, “the very good agreement between our results and those obtained via Kohn-Sham self-consistent jellium calculations suggested that it would be worthwhile to extend the application of our method to more general electronic structure calculations extending beyond the jellium model, where the trial density used for minimization of the ETF functional could be taken as a superposition of site densities, as in the Harris method.” Additionally, generalization of the DFT-SCM method to calculations of extended (bulk and surface) systems appeared rather natural. Indeed, recent promising applications of DFT-SCM in this spirit have appeared.19,20 In this case, the term “shell correction effects” is also maintained, although “quantum interference effects” could be more appropriate for extended systems. 7.1.2. Motivation for finite systems One of the principle themes in research on finite systems (e.g., nuclei, atomic and molecular clusters, and nano-structured materials) is the search for size-evolutionary patterns (SEPs) of properties of such systems and elucidation of the physical principles underlying such patterns.21 Various physical and chemical properties of finite systems exhibit SEPs, including: 1. Structural characteristics pertaining to atomic arrangements and particle morphologies and shapes; 2. Excitation spectra involving bound-bound transitions, ionization potentials (IPs), and electron affinities (EAs); 3. Collective excitations (electronic and vibrational); 4. Magnetic properties; 5. Abundance spectra and stability patterns, and their relation to binding and cohesion energetics, and to the pathways and rates of dissociation, fragmentation, and fission of charged clusters; 6. Thermodynamic stability and phase changes; 7. Chemical reactivity. The variations with size of certain properties of materials aggregates are commonly found to scale on the average with the surface to volume ratio of the cluster, i.e., S/Ω ∼ R−1 ∼ N 1/3 , where S, Ω, R, and N are the surface area, volume, average radius, and number of particles, respectively (the physical origins of such scaling may vary for different properties). In general, the behavior of SEPs in finite systems in terms of such scaling is non-universal, in the sense that it is non-monotonic exhibiting characteristic discontinuities. Nevertheless, in many occasions, it is convenient to analyze the energetics of finite systems in terms of two contributions,

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Fig. 7.1. Ionization potentials of NaN clusters. Open squares: Experimental measurements of Refs. 15,17. The solid line at the top panel (a) represents the smooth contribution to the theoretical total IPs. The solid circles in the middle (b) and bottom (c) panels are the total SCM IPs. The shapes of sodium clusters have been assumed spherical in the middle panel, while triaxial deformations have been considered at the bottom one.

namely, (i) a term which describes the energetics as a function of the system size in an average sense (not including shell-closure effects), referrred to usually as describing the “smooth” part of the size dependence, and (ii) an electronic shell-correction term. The first term is the one which is expected to vary smoothly and be expressible as an expansion in S/Ω, while the second one contains the characteristic oscillatory patterns as the size of the finite system is varied. Such a strategy has been introduced1 and often used in studies of nuclei,2,22 and has been adopted recently for investigations of metal clusters,3,23–38 fullerenes,39 and metal nanowires.40–42 As a motivating example, we show in Fig. 7.1 the SEP of the IPs of NaN clusters, which illustrates odd-even oscillations in the observed spectrum, a smooth description of the pattern [Fig. 7.1(a)], and two levels of shell-corrected descriptions — one assuming spherical symmetry [Fig. 7.1(b)], and the other allowing for triaxial shape deformations [Fig. 7.1(c)]. The progressive improvement of the level of agreement between the experimental and theoretical patterns is evident.

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7.1.3. Plan of the chapter The chapter is organized as follows: In Sec. 7.2, the general methodology of shell-correction methods is reviewed, and the microscopic DFT-SCM is introduced and presented in detail in Sec. 7.2.2. Applications of the DFT-SCM to condensed-matter finite systems are presented in Sec. 7.3 for three different characteristic nanosystems, namely, metal clusters (Sec. 7.3.1), charged fullerenes (Sec. 7.3.2), and metallic nanowires (Sec. 7.3.3). In the Appendix, we describe the semiempirical SCM for clusters, which is closer to the spirit of Strutinsky’s original phenomenological approach for nuclei. There we also briefly present applications of the SE-SCM to triaxial shape deformations and fission of metal clusters. A summary is given in Sec. 7.4. 7.2. Methodology and derivation of microscopic DFT-SCM 7.2.1. Historical review of SCM It has long been recognized in nuclear physics that the dependence of groundstate properties of nuclei on the number of particles can be viewed as the sum of two contributions: the first contribution varies smoothly with the particle number (number of protons Np and neutrons Nn ) and is referred to as the smooth part; the second contribution gives a superimposed structure on the smooth curve and exhibits an oscillatory behavior, with extrema at the nuclear magic numbers.22,43 Nuclear masses have provided a prototype for this behavior.43 Indeed, the main contributions to the experimental nuclear binding energies are smooth functions of the number of protons and neutrons, and are described by the semi-empirical mass formula.44,45 The presence of these smooth terms led to the introduction of the liquid-drop model (LDM), according to which the nucleus is viewed as a drop of a nonviscous fluid whose total energy is specified by volume, surface, and curvature contributions.22,43,46 The deviations of the binding energies from the smooth variation implied by the LDM have been shown1,46 to arise from the shell structure associated with the bunching of the discrete single-particle spectra of the nucleons, and are commonly referred to as the shell correction. Substantial progress in our understanding of the stability of strongly deformed open-shell nuclei and of the dynamics of nuclear fission was achieved when Strutinsky proposed1 a physically motivated and efficient way of calculating the shell corrections. The method consists of averaging [see the Appendix, Eq. (A.1) and Eq. (A.2)] the single-particle spectra of phenomenological deformed potentials and of subtracting the ensuing average from the total sum of single-particle energies. While certain analogies, portrayed in experimental data, between properties of nuclei and elemental clusters have been recognized, the nuclear-physics approach of

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separating the various quantities as a function of size into a smooth part and a shell correction part has only partially been explored in the case of metal clusters. In particular, several investigations47–50 had used the ETF method in conjunction with the jellium approximation to determine the average (smooth, in the sense defined above) behavior of metal clusters, but had not pursued a method for calculating the shell corrections. In the absence of a method for appropriately calculating shellcorrections for metal clusters in the context of the semiclassical ETF method, it had been presumed that the ETF method was most useful for larger clusters, since the shell effects diminish with increasing size. Indeed, several studies had been carried out with this method addressing the asymptotic behavior of ground-state properties towards the behavior of a jellium sphere of infinite size.51,52 It has been observed,48,53–56 however, that the single-particle potentials resulting from the semiclassical method are very close, even for small cluster sizes, to those obtained via self-consistent solution of the local density functional approximation (LDA) using the Kohn-Sham (KS) equations.57 These semiclassical potentials were used extensively to describe the optical (linear) response of spherical metal clusters, for small,53–55 as well as larger sizes56 (for an experimental review on optical properties, cf. Refs. 58,59). The results of this approach are consistent with timedependent local density functional approximation (TDLDA) calculations which use the KS solutions.60,61 It is natural to explore the use of these semiclassical potentials, in the spirit of Strutinsky’s approach, for evaluation of shell corrections in metal clusters of arbitrary size. Below we describe a microscopic derivation of an SCM in conjunction with the density functional theory,3,23,24 and its applications in investigations of the properties of metal clusters and fullerenes. Particularly interesting and promising is the manner by which the shell corrections are introduced by us at the microscopic level through the kinetic energy term,3,23,24 instead of the traditional semiempirical Strutinsky averaging procedure of the single-particle spectrum.1,31 In particular, our approach leads to an energy functional that corrects many shortcomings of the orbital-free DFT, and one that is competitive in numerical accuracy and largely advantageous in computational speed compared to the KS method. 7.2.2. DFT-SCM Underlying the development of the shell-correction method is the idea of approximating the total energy Etotal (N ) of a finite interacting fermion system as e ) + ∆Esh (N ), Etotal (N ) = E(N

(7.1)

e is the part that varies smoothly as a function of system size, and ∆Esh is an where E oscillatory term. Various implementations of such a separation consist of different choices and methods for evaluating the two terms in Eq. (7.1). Before discussing such methods, we outline a microscopic derivation of Eq. (7.1).

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Motivated by the behavior of the empirical nuclear binding energies, Strutinsky conjectured that the self-consistent Hartree-Fock density ρHF can be decomposed into a smooth density ρe and a fluctuating contribution δρ, namely ρHF = ρe + δρ. Then, he proceeded to show that, to second-order in δρ, the Hartree-Fock energy is equal to the result that the same Hartree-Fock expression yields when ρHF is replaced by the smooth density ρe and the Hartree-Fock single-particle energies εHF i are replaced by the single-particle energies corresponding to the smooth potential constructed with the smooth density ρe. Namely, he showed that EHF = EStr + O(δρ2 ),

where the Hartree-Fock electronic energy is given by the expression Z occ X 1 EHF = εHF − drdr0 V(r − r0 )[ρHF (r, r)ρHF (r0 , r0 ) − ρHF (r, r0 )2 ], i 2 i=1

(7.2)

(7.3)

with εHF being the eigenvalues obtained through a self-consistent solution of the i HF equation,   ~2 2 − ∇ + UHF φi = εHF φi , (7.4) i 2m where Z

dr0 V(r − r0 )[ρHF (r0 , r0 )φi (r) − ρHF (r0 , r)φi (r0 )].

(7.5)

The Strutinsky approximate energy is written as follows, Z occ X 1 EStr = εei − drdr0 V(r − r0 )[e ρ(r, r)e ρ(r0 , r0 ) − ρe(r, r0 )2 ], 2 i=1

(7.6)

UHF (r)φi (r) =

where the index i in Eq. (7.3) and Eq. (7.6) runs only over the occupied states (spin degeneracy is naturally implied). The single-particle energies εei correspond to a e . Namely, they are eigenvalues of a Schr¨odinger equation, smooth potential U   ~2 2 e − ∇ + U ϕi = εei ϕi , (7.7) 2m e depends on the smooth density ρe, i.e., where the smooth potential U Z e (r)ϕi (r) = dr0 V(r − r0 )[e U ρ(r0 , r0 )ϕi (r) − ρe(r0 , r)ϕi (r0 )],

(7.8)

and V is the nuclear two-body interaction potential. It should be noted that while Eq. (7.6)−Eq. (7.8) look formally similar to the Hartree-Fock equations (7.3-7.5), their content is different. Specifically, while in the HF equations, the density ρHF is self-consistent with the wavefunction solutions of

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Eq. (7.4), the density ρe in Eq. (7.6)−Eq. (7.8) is not self-consistent with the wavePocc function solutions of the corresponding single-particle Eq. (7.7), i.e., ρe 6= i=1 |ϕi |2 . We return to this issue below. Since the second term in Eq. (7.6) is a smooth quantity, Eq. (7.2) states that all shell corrections are, to first order in δρ, contained in the sum of the single-particle Pocc energies i=1 εei . Consequently, Eq. (7.6) can be used as a basis for a separation of the total energy into smooth and shell-correction terms as in Eq. (7.1). Indeed Strutinsky suggested a semiempirical method of such separation through an averaging procedure of the single-particle energies εei in conjunction with a phenomenological (or semi-empirical) model [the liquid drop model (LDM)] for the smooth part (see the appendix). Motivated by the above considerations, we have extended them3,23,24 in the context of density functional theory for electronic structure calculations. First we review pertinent aspects of the DFT theory. In DFT, the total energy is given by   Z  Z 1 E[ρ] = T [ρ] + VH [ρ(r)] + VI (r) ρ(r) dr + Exc [ρ(r)]dr + EI , (7.9) 2

where VH is the Hartree repulsive potential among the electrons, VI is the interaction potential between the electrons and ions, Exc is the exchange-correlation functional [the corresponding xc potential is given as Vxc (r) ≡ δExc ρ(r)/δρ(r)] and T [ρ] is R given in terms of a yet unknown functional t[ρ(r)] as T [ρ] = t[ρ(r)]dr. EI is the interaction energy of the ions. In the Kohn-Sham (KS)-DFT theory, the electron density is evaluated from the single-particle wave functions φKS,i (r) as ρKS (r) =

occ X i=1

2

|φKS,i (r)| ,

(7.10)

where φKS,i (r) are obtained from a self-consistent solution of the KS equations,   ~2 2 − ∇ + VKS φKS,i (r) = εKS,i φKS,i (r) (7.11) 2m where

VKS [ρKS (r)] = VH [ρKS (r)] + Vxc [ρKS (r)] + VI (r).

(7.12)

The kinetic energy term in Eq. (7.9) is given by T [ρKS ] =

occ X i=1

< φKS,i | −

~2 2 ∇ |φKS,i >, 2m

which can also be written as Z occ X T [ρKS ] = εKS,i − ρKS (r)VKS [ρKS (r)]dr.

(7.13)

(7.14)

i=1

According to the Hohenberg-Kohn theorem, the energy functional (7.9) is a minimum at the true ground density ρgs , which in the context of the KS-DFT

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theory corresponds the the density, ρKS , obtained from an iterative self-consistent solution of Eq. (7.11). In other words, combining Eq. (7.9) and Eq. (7.14), and denoting by “in” and “out” the trial and output densities of an iteration cycle in the solution of the KS equation [Eq. (7.11)], one obtains, EKS [ρout KS ]

= EI + Z  Z

occ X

εout KS,i +

i=1

 1 out VH [ρout (r)] + E [ρ (r)] + V (r) ρout xc KS I KS KS (r)dr − 2

in ρout KS (r)VKS [ρKS (r)]dr.

(7.15)

in Note that the expression on the right involves both ρout KS and ρKS . Self-consistency out,in in is achieved when δρKS (r) = ρout KS (r)−ρKS (r) becomes arbitrarily small (i.e., when out ρKS converges to ρKS ). On the other hand, it is desirable to introduce approximate energy functionals for the calculations of ground-state electronic properties, providing simplified, yet accurate, computational schemes. It is indeed possible to construct such functionals,62–66 an example of which was introduced by J. Harris,62 where self-consistency is circumvented and the result is accurate to second order in the difference between the trial and the self-consistent KS density (see in particular Eq. (24a) of Ref. 66; the same also holds true for the difference between the trial and the output densities of the Harris functional). The expression of the Harris functional is obtained from Eq. (7.15) by dropping the label KS and by replacing everywhere ρout by ρin , yielding [note cancellations between the third and fourth terms on the right-hand-side of Eq. (7.15)].  Z  occ X 1 in in in out VH [ρ (r)] + Vxc [ρ (r)] ρin (r)dr + EHarris [ρ ] = EI + εi − 2 i=1 Z Exc [ρin (r)]dr. (7.16)

εout are the single-particle solutions (non-self-consistent) of Eq. (7.11), with i VKS [ρin (r) [see Eq. (7.12)]. As stated above this result is accurate to second order in ρin −ρKS (alternatively in ρin − ρout ), thus approximating the self-consistent total energy EKS [ρKS ]. Obviously the accuracy of the results obtained via Eq. (7.16) depend on the choice of the input density ρin . In electronic structure calculations where the corpuscular nature of the ions is included (i.e., all-electron or pseudo-potential calculations), a natural choice for ρin consists of a superposition of atomic site densities, as suggested originally by Harris. In the case of jellium calculations, we have shown3 that an accurate approximation to the KS-DFT total energy is obtained by using the Harris functional with the input density, ρin , in Eq. (7.16) evaluated from an Extended-Thomas-Fermi (ETF)-DFT calculation.

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The ETF-DFT energy functional, EET F [ρ], is obtained by replacing the kinetic energy term in Eq. (7.9) by a kinetic energy density-functional in the spirit of the Thomas-Fermi approach,67 but comprising terms up to fourth-order in the density gradients.48,68 The optima l ETF-DFT total energy is then obtained by minimization of EET F [ρ] with respect to the density. In our calculations, we use for the trial densities parametrized profiles ρ(r; {γi }) with {γi } as variational parameters (the ETF-DFT optimal density is denoted as ρET F ). The single-particle eigenvalues, {εout i }, in Eq. (7.16) are obtained then as the solutions to a single-particle Hamiltonian, 2 b ET F = − ~ ∇2 + VET F , H 2m

(7.17)

where VET F is given by Eq. (7.12) with ρKS (r) replaced by ρET F (r). These singleparticle eigenvalues will be denoted by {e εi } As is well known, the ETF-DFT does not contain shell effects .48–50 Consequently, the corresponding density ρET F can be taken `a la Strutinsky as the smooth part, ρe, of the KS density, ρKS . Accordingly, EET F is identified with the smooth e in Eq. (7.1) (in the following, the “ETF” subscript and “ e ” can be used part E interchangeably). Since, as aforementioned, EHarris [ρET F ] approximates well [i.e., to second order in (ρET F − ρKS )] the self-consistent total energy EKS [ρKS ], it follows from Eq. (7.1), with EHarris [ρET F ] taken as the expression for Etotal , that the shell-correction, ∆Esh , is given by

Defining,

e ρ]. ∆Esh = EHarris [ρET F ] − EET F [ρET F ] ≡ Esh [e ρ] − E[e Tsh =

occ X i=1

εei −

Z

ρET F (r)VET F (r)dr,

(7.18)

(7.19)

and denoting the total energy EHarris by Esh , i.e., by identifying Esh ≡ EHarris ,

(7.20)

e ρ], Esh [e ρ] = {Tsh − Te[e ρ]} + E[e

(7.21)

we obtain

where Te[e ρ] is the ETF kinetic energy, given to fourth-order gradients by the expres68 sion, Z  ~2 3 1 (∇ρ)2 1 TET F [ρ] = (3π 2 )2/3 ρ5/3 + + (3π 2 )−2/3 ρ1/3 2m 5 36 ρ 270 "  4  2  2 #) 9 ∇ρ ∆ρ ∆ρ 1 ∇ρ × − + dr, (7.22) 3 ρ 8 ρ ρ ρ

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Fig. 7.2. Total energy per atom of neutral sodium clusters (in units of the absolute value of the energy per atom in the bulk, |ε∞ | = 2.252 eV ). Solid circles: DFT-SCM results (see text for details). The solid line is the ETF result (smooth contribution). In both cases, a spherical jellium background was used. Open squares: Kohn-Sham DFT results from Ref. 69. The excellent agreement (a discrepancy of only 1%) between the DFT-SCM and the Kohn-Sham DFT approach is to be stressed.

which as noted before does not contain shell effects. Therefore, the shell correction term in Eq. (7.1) [or Eq. (7.18)] is given by a difference between kinetic energy terms, ∆Esh = Tsh − Te[e ρ].

(7.23)

One should note that the above derivation of the shell correction does not involve a Strutinsky averaging procedure of the kinetic energy operator. Rather it is based e Other on using ETF quantities as the smooth part for the density, ρe, and energy, E. descriptions of the smooth part may result in different shell-correction terms. To check the accuracy of this procedure, we have compared results of calculations using the functional Esh [Eq. (7.21)] with available Kohn-Sham calculations. In general, the optimized density from the minimization of the ETF-DFT functional can be obtained numerically as a solution of the differential equation δTET F [ρ] + VET F [ρ(r)] = µ, δρ(r)

(7.24)

where µ is the chemical potential. As mentioned already, for the jellium DFT-SCM calculations, we often use a trial density profile in the ETF-DFT variation which is chosen as, ρ0
γ , ρ(r) =  (7.25) 0 1 + exp r−r α

with r0 , α, and γ as variational parameters that minimize the ETF-DFT functional (for other closely related parametrizations, cf. Refs. 49,50). Figure 7.2 displays results of the present shell correction approach for the total energies of neutral sodium clusters. The results of the shell correction method for

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Fig. 7.3. Ionization potentials for sodium clusters. Solid circles: IPs calculated with the DFTSCM (see text for details). The solid line corresponds to the ETF results (smooth contribution). In both cases, a spherical jellium background was used. Open squares: Kohn-Sham DFT results from Ref. 69. The excellent agreement (a discrepancy of only 1-2%) between the DFT shell correction method and the full Kohn-Sham approach should be noted.

ionization potentials of sodium clusters are displayed in Fig. 7.3. The excellent agreement between the oscillating results obtained via our DFT-SCM theory and the Kohn-Sham results (cf., e.g., Ref. 69) is evident. To further illustrate the two components (smooth contribution and shell correction) entering into our approach, we also display the smooth parts resulting from the ETF method. (In all calculations, the Gunnarsson-Lundqvist exchange and correlation energy functionals were used; see Refs. 3,23.) 7.3. Applications of DFT-SCM 7.3.1. Metal clusters 7.3.1.1. Charging of metal clusters Investigations of metal clusters based on DFT methods and self-consistent solutions of the Kohn-Sham equations (employing either a positive jellium background or maintaining the discrete ionic cores) have contributed significantly to our understanding of these systems.69–72 However, even for singly negatively charged − metal clusters (MN ), difficulties may arise due to the failure of the solutions of the KS equations to converge, since the eigenvalue of the excess electron may it− erate to a positive energy.73 While such difficulties are alleviated for MN clus74,75 ters via self-interaction corrections (SIC), the treatment of multiply charged Z− clusters (MN , Z > 1) would face similar difficulties in the metastability region against electronic autodetachment through a Coulombic barrier. In the following we are applying our DFT-SCM approach, described in the previous section, to these systems3,23,24 (for the jellium background, we assume spherical symmetry, unless

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Fig. 7.4. Calculated first (A1 ) and second (A2 ) electron affinities of sodium clusters as a function of the number of atoms N . Both their smooth part (dashed lines) and the shell-corrected affinities (solid circles) are shown. A spherical jellium background was used.

otherwise stated; for a discussion of cluster deformations, see Sec. A.1). 7.3.1.2. Electron affinities and borders of stability eZ prior to shell corrections are defined as The smooth multiple electron affinities A the difference in the total energies of the clusters eZ = E(vN, e e A vN + Z − 1) − E(vN, vN + Z),

(7.26)

e Ash Z − AZ = ∆Esh (vN, vN + Z − 1) − ∆Esh (vN, vN + Z).

(7.27)

where N is the number of atoms, v is the valency and Z is the number of excess electrons in the cluster (e.g., first and second affinities correspond to Z = 1 and Z = 2, respectively). vN is the total charge of the positive background. Applying the shell correction in Eq. (7.23), we calculate the full electron affinity as

A positive value of the electron affinity indicates stability upon attachment of an extra electron. Figure 7.4 displays the smooth, as well as the shell corrected, first and second electron affinities for sodium clusters with N < 100. Note that e2 becomes positive above a certain critical size, implying that the second electron A (2) in doubly negatively charged sodium clusters with N < Ncr = 43 might not be

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eZ , Z = 1-4, for sodium clusters as a function of Fig. 7.5. Calculated smooth electron affinities A the number of atoms N (Z is the number of excess electrons). A spherical jellium background was used. Inset: The electron drip line for sodium clusters. Clusters stable against spontaneous electron emission are located above this line. While for spherical symmetry, as seen from Fig. 7.4, shell effects influence the border of stability, shell-corrected calculations25 including deformations (see the appendix) yield values close to the drip line (shown in the inset) which was obtained from the smooth contributions.

stably attached. The shell effects, however, create two islands of stability about the 2− sh magic clusters Na2− 32 and Na38 (see A2 in Fig. 7.4). To predict the critical cluster (Z) size Ncr , which allows stable attachment of Z excess electrons, we calculated the smooth electron affinities of sodium clusters up to N = 255 for 1 ≤ Z ≤ 4, and (3) (4) display the results in Fig. 7.5. We observe that Ncr = 205, while Ncr > 255. The similarity of the shapes of the curves in Fig. 7.5, and the regularity of distances between them, suggest that the smooth electron affinities can be fitted by a general expression of the form: 2 2 (Z − 1)e2 eZ = A e1 − (Z − 1)e = W − β e − , A R+δ R+δ R+δ

(7.28)

where the radius of the positive background is R = rs N 1/3 . From our fit, we find

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that the constant W corresponds to the bulk work function. In all cases, we find β = 5/8, which suggests a close analogy with the classical model of the image charge.76,77 For the spill-out parameter, we find a weak size dependence as δ = δ0 + δ2 /R2 . The contribution of δ2 /R2 , which depends on Z, is of importance only for smaller sizes and does not affect substantially the critical sizes (where the curve crosses the zero line), and consequently δ2 can be neglected in such estimations. Using the values e1 of sodium clusters (namely, W = 2.9 eV which is also the value obtained by us for A obtained by KS-DFT calculations for an infinite planar surface,78 δ0 = 1.16 a.u.; with R = rs N 1/3 , and rs = 4.00 a.u.), we find for the critical sizes when the l.h.s. of (2) (3 (4) (5) Eq. (7.28) is set equal to zero, Ncr = 44, Ncr ) = 202, Ncr = 554, and Ncr = 1177, in very good agreement with the values obtained directly from Fig. 7.5. (Z) The curve that specifies Ncr in the (Z, N ) plane defines the border of stability for spontaneous electron decay. In nuclei, such borders of stability against spontaneous proton or neutron emission are known as nucleon drip lines.79 For the case of sodium clusters, the electron drip line is displayed in the inset of Fig. 7.5. 7.3.1.3. Critical sizes for potassium and aluminum While in this investigation we have used sodium clusters as a test system, the methodology and conclusions extend to other materials as well. Thus given a calculated or measured bulk work function W , and a spill-out parameter (δ0 typically eZ = 0, of the order of 1-2 a.u., and neglecting δ2 ), one can use Eq. (7.28), with A to predict critical sizes for other materials. For example, our calculations for potassium (rs = 4.86 a.u.) give fitted values W = 2.6 eV (compared to a KS-DFT value of 2.54 eV for a semi-infinite planar surface with rs = 5.0 a.u.78 ) and δ0 = 1.51 a.u. (2) (3) (4) for δ2 = 0, yielding Ncr = 33, Ncr = 152, and Ncr = 421. As a further example, we give our results for a trivalent metal, i.e. aluminum (rs = 2.07 a.u.), for which our fitted values are W = 3.65 eV (compared to a KS-DFT value of 3.78 eV for a semi-infinite plane surface, with rs = 2.0 a.u.78 ) (2) (3) and δ0 = 1.86 a.u. for δ2 = 0, yielding Ncr = 40 (121 electrons), Ncr = 208 (626 (4) electrons), and Ncr = 599 (1796 electrons). 7.3.1.4. Metastability against electron autodetachment The multiply charged anions with negative affinities do not necessarily exhibit a positive total energy. To illustrate this point, we display in Fig. 7.6 the calculated e total energies per atom (E(N, Z)/N ) as a function of excess charge (Z) for clusters containing 30, 80, and 240 sodium atoms. These sizes allow for exothermic attachment of maximum one, two, or three excess electrons, respectively. As was the case with the electron affinities, the total-energy curves in Fig. 7.6 show a remarkable regularity, suggesting a parabolic dependence on the excess charge. To test this conjecture, we have extracted from the calculated total enere e gies the quantity g(N, Z) = G(N, Z)/N where G(N, Z) = [E(N, Z) − E(N, 0)]/Z +

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Fig. 7.6. Calculated smooth total energy per atom as a function of the excess negative charge Z for the three families of sodium clusters with N = 30, N = 80, and N = 240 atoms. A spherical jellium background was used. As the straight lines in the inset demonstrate, the curves are parabolic. We find that they can be fitted by Eq. (7.29). See text for an explanation of how the function g(N, Z) was extracted from the calculations.

e1 (N ), and have plotted it in the inset of Fig. 7.6 as a function of the excess negA ative charge Z. The dependence is linear to a remarkable extent; for Z = 1 all three lines cross the energy axis at zero. Combined with the results on the electron affinities, this indicates that the total energies have the following dependence on the excess number of electrons (Z): 2 e e e1 Z + Z(Z − 1)e , E(Z) = E(0) −A 2(R + δ)

(7.29)

where the dependence on the number of atoms in the cluster is not explicitly indicated. This result is remarkable in its analogy with the classical image-charge result of van Staveren et al.77 Indeed, the only difference amounts to the spill-out parameter δ0 and to the weak dependence on Z through δ2 . This additional Z-dependence becomes negligible already for the case of 30 sodium atoms. For metastable multiply-charged cluster anions, electron emission (autodetachment) will occur via tunneling through a barrier (shown in Fig. 7.7). However, to reliably estimate the electron emission, it is necessary to correct the LDA effective potential for self-interaction effects. We performed a self-interaction correction of the Amaldi type73 for the Hartree term and extended it to the exchange-correlation SIC LDA LDA contribution to the total energy as follows: Exc [ρ] = Exc [ρ] − Ne Exc [ρ/Ne ],

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Fig. 7.7. The DFT (LDA) and the corresponding self-interaction corrected potential for the metastable Na2− 18 cluster. A spherical jellium background was used. The single-particle levels of the SIC potential are also shown. Unlike the LDA, this latter potential exhibits the correct asymptotic behavior. The 2s and 1d electrons can be emitted spontaneously by tunneling through the Coulombic barrier of the SIC potential. Distances in units of the Bohr radius, a0 . The specified single-particle levels are associated with the SIC potential.

where Ne = vN +Z is the total number of electrons. This self-interaction correction is akin to the orbitally-averaged-potential method.73 Minimizing the SIC energy functional for the parameters r0 , α, and γ, we obtained the effective SIC potential for Na2− 18 shown in Fig. 7.7, which exhibits the physically correct asymptotic 80 behavior. The spontaneous electron emission through the Coulombic barrier is analogous to that occurring in proton radioactivity from neutron-deficient nuclei,81 as well as in alpha-particle decay. The transition rate is λ = ln 2/T1/2 = νP , where ν is the attempt frequency and P is the transmission coefficient calculated in the WKB method (for details, cf. Ref. 81). For the 2s electron in Na2− 18 (cf. Fig. 7.7), we find ν = 0.73 1015 Hz and P = 4.36 10−6 , yielding T1/2 = 2.18 10−10 s. For a cluster size closer to the drip line (see Fig. 7.5), e.g. Na2− 35 , we find T1/2 = 1.13 s. Finally, the exression in Eq. (7.29) for the total energy can be naturally extended to the case of multiply positively charged metal clusters by setting Z = −z, with z > 0. The ensuing equation retains the same dependence on the excess positive

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e1 , replaced by the charge z, but with the negative value of the first affinity, −A positive value of the first ionization potential, Ie1 = W + (3/8)e2 /(R + δ), a result that has been suggested from earlier measurements on multiply charged potassium cations.82 Naturally, the spill-out parameter δ assumes different values than in the case of the anionic clusters. 7.3.2. Neutral and multiply charged fullerenes 7.3.2.1. Stabilized jellium approximation — The generalized DFT-SCM Fullerenes and related carbon structures have been extensively investigated using ab initio density-functional-theory methods and self-consistent solutions of the KohnSham (KS) equations.83,84 For metal clusters, replacing the ionic cores with a uniform jellium background was found to describe well their properties within the KS-DFT method.58 Motivated by these results, several attempts to apply the jellium model in conjunction with DFT to investigations of fullerenes have appeared recently.39,85–87 Our approach39 differs from the earlier ones in several aspects and, in particular, in the adaptation to the case of finite systems of the stabilized-jellium (or structureless pseudopotential) energy density functional (see Eq. (7.30) below and Ref. 73). An important shortcoming of the standard jellium approximation for fullerenes (and other systems with high density, i.e., small rs ) results from a well-known property of the jellium at high electronic densities, namely that the jellium is unstable and yields negative surface-energy contribution to the total energy,73 as well as unreliable values for the total energy. These inadequacies of the standard jellium model can be rectified by pseudopotential corrections. A modified-jellium approach which incorporates such pseudopotential corrections and is particularly suited for our purposes here, is the structureless pseudopotential model or stabilized jellium approximation developed in Ref. 73. In the stabilized jellium, the total energy Epseudo , as a functional of the electron density ρ(r), is given by the expression Z Z Epseudo [ρ, ρ+ ] = Ejell [ρ, ρ+ ] + hδυiW S ρ(r)U(r)dr − εe ρ+ (r)dr, (7.30)

where by definition the function U(r) equals unity inside, but vanishes, outside the jellium volume. ρ+ is the density of the positive jellium background (which for the case of C60 is taken as a spherical shell, of a certain width 2d, centered at 6.7 a.u. ). Epseudo in Eq. (7.30) is the standard jellium-model total energy, Ejell , modified by two corrections. The first correction adds the effect of an average (i.e., averaged over the volume of a Wigner-Seitz cell) difference potential, hδυiW S U(r), which acts on the electrons in addition to the standard jellium attraction and is due to the atomic pseudopotentials (in this work, we use the Ashcroft empty-core pseudopotential, specified by a core radius rc , as in Ref. 73). The second correction subtracts from the jellium energy functional the spurious electrostatic self-repulsion

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of the positive background within each cell; this term makes no contribution to the effective electronic potential. Following Ref. 73, the bulk stability condition (Eq. (25) in Ref. 73) determines the value of the pseudopotential core radius rc , as a function of the bulk WignerSeitz radius rs . Consequently, the difference potential can be expressed solely as a function of rs as follows (energies in Ry, distances in a.u.): hδυiW S

2 =− 5



9π 4

2/3

rs−2

1 + 2π



9π 4

1/3

1 dεc rs−1 + rs , 3 drs

(7.31)

where εc is the per particle electron-gas correlation energy (in our calculation, we use the Gunnarsson-Lundqvist exchange and correlation energy functionals; see Refs. 3,23). The electrostatic self-energy, εe, per unit charge of the uniform positive jellium is given by εe = 6υ 2/3 /5rs ,

(7.32)

where υ is the valence of the atoms (υ = 4 for carbon). 7.3.2.2. ETF electron-density profile To apply the ETF-DFT method to carbon fullerenes, we generalize it by employing potential terms according to the stabilized-jellium functional in Eq. (7.30). Another required generalization consists in employing a parametrized electrondensity profile that accounts for the hollow cage-structure of the fullerenes. Such a density profile is provided by the following adaptation of a generalization of an inverse Thomas-Fermi distribution, used earlier in the context of nuclear physics,88 i.e., ρ(r) = ρ0



Fi,o sinh[wi,o /αi,o ] cosh[wi,o /αi,o ] + cosh[(r − R)/αi,o ]

γi,o

,

(7.33)

where R = 6.7 a.u. is the radius of the fullerene cage. w, α, and γ are variables to be determined by the ETF-DFT minimization. For R = 0 and large values of w/α, expression (7.33) approaches the more familiar inverse Thomas-Fermi distribution, with w the width, α the diffuseness and γ the asymmetry of the profile around r = w. There are a total of six parameters to be determined, since the indices (i, o) stand for the regions inside (r < R) and outside (r > R) the fullerene cage. Fi,o = (cosh[wi,o /αi,o ] + 1)/ sinh[wi,o /αi,o ] is a constant guaranteeing that the two parts of the curve join smoothly at r = R. The density profile in Eq. (7.33) peaks at r = R and then falls towards smaller values both inside and outside the cage (see top panel of Fig. 7.8).

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Fig. 7.8. Bottom panel: The stabilized-jellium LDA potential obtained by the ETF method for the neutral C60 molecule. The Wigner-Seitz radius for the jellium bacground is 1.23 a.u. Note the asymmetry of the potential about the minimum. The associated difference potential hδυiW S = −9.61 eV . Top panel: Solid line: Radial density of the positive jellium background. Dashed line: ETF electronic density. Note its asymmetry about the maximum. Thick solid line: The difference (multiplied by 10) of electronic ETF densities between C5− 60 and C60 . It illustrates that the excess charge accumulates in the outer perimeter of the total electronic density. All densities are normalized to the density of the positive jellium background.

7.3.2.3. Shell correction and icosahedral splitting To apply the SCM to the present case, the potential VET F in Eq. (7.19) is replaced by the stabilized-jellium LDA potential shown in Fig. 7.8. After some rearrangements, the shell-corrected total energy Esh [e ρ] in the stabilized-jellium case can be written in functional form as follows [compare to Eq. (7.21), see also Eq. (7.16)].  Z  X 1e Esh [e ρ] = εei − VH (r) + Vexc (r) ρe(r)dr 2 i Z Z + Eexc [e ρ(r)]dr + EI − εe ρ+ (r)dr, (7.34)

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Heretofore, the point-group icosahedral symmetry of C60 was not considered, since the molecule was treated as a spherically symmetric cage. This is a reasonable zeroth-order approximation as noticed by several authors.83,87,89,90 However, considerable improvement is achieved when the effects of the point-group icosahedral symmetry are considered as a next-order correction (mainly the lifting of the angular momentum degeneracies). The method of introducing the icosahedral splittings is that of the crystal field theory.91 Thus, we will use the fact that the bare electrostatic potential from the ionic cores, considered as point charges, acting upon an electron, obeys the wellknown expansion theorem91 U (r) = −υe2

X i

∞ X l X 1 =− κl (r)Clm Ylm (θ, φ), |r − ri |

(7.35)

l=0 m=−l

where the angular coefficients Clm are given through the angular coordinates θi , φi of the carbon atomic cores, namely, X Ylm∗ (θi , φi ), (7.36) Clm = i

where ∗ denotes complex conjugation. We take the radial parameters κl (r) as constants, and determine their value by adjusting the icosahedral single-particle spectra εico to reproduce the pseudopoteni tial calculation of Ref. 83, which are in good agreement with experimental data. Our spectra without and with icosahedral splitting are shown in Fig. 7.9(a) and Fig. 7.9(b), respectively. We find that a close reproduction of the results of ab initio DFT calculations83,92,93 is achieved when the Wigner-Seitz radius for the jellium background is ico 1.23 a.u. The shell corrections, ∆Esh , including the icosahedral splittings are calculated using the icosahedral single-particle energies εico in Eq. (7.19). The average i e quantities (e ρ and V ) are maintained as those specified through the ETF variation with the spherically symmetric profile of Eq. (7.33). This is because the first-order correction to the total energy (resulting from the icosahedral perturbation) vanishes, since the integral over the sphere of a spherical harmonic Ylm (l > 0) vanishes. 7.3.2.4. Ionization potentials and electron affinities Having specified the appropriate Wigner-Seitz radius rs and the parameters κl of the icosahedral crystal field through a comparison with the pseudopotential DFT calculations for the neutral C60 , we calculate the total energies of the cationic and anionic species by allowing for a change in the total electronic charge, namely by imposing the constraint Z 4π ρ(r)r2 dr = 240 ± x, (7.37)

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Fig. 7.9. (a) The single-particle levels of the ETF-LDA potential for C60 shown in Fig. 7.8. Because of the spherical symmetry, they are characterized by the two principle quantum numbers nr and l, where nr is the number of radial nodes and l the angular momentum. They are grouped in three bands labeled σ (nr = 0), π (nr = 1), and δ (nr = 2). Each band starts with an l = 0 level. (b) The single-particle levels for C60 after the icosahedral splittings are added to the spectra of (a). The tenfold degenerate HOMO (hu ) l and the sixfold degenerate LUMO1 (t1u ) and LUMO2 (t1g ) are denoted; they originate from the spherical l = 5 and l = 6 (t1g ) π levels displayed in panel (a). For the σ electrons, the icosahedral perturbation strongly splits the l = 9 level of panel (a). There result five sublevels which straddle the σ-electron gap as follows: two of them (the eightfold degenerate gu , and the tenfold degenerate hu ) move down and are fully occupied resulting in a shell closure (180 σ electrons in total). The remaining unoccupied levels, originating from the l = 9 σ level, are sharply shifted upwards and acquire positive values.

where ρ(r) is given by Eq. (7.33). The shell-corrected and icosahedrally perturbed first and higher ionization potentials Ixico are defined as the difference of the groundico state shell-corrected total energies Esh as follows: ico ico Ixico = Esh (Ne = 240 − x; Z = 240) − Esh (Ne = 240 − x + 1; Z = 240),

(7.38)

where Ne is the number of electrons in the system and x is the number of excess charges on the fullerenes (for the excess charge, we will find convenient to use two different notations x and z related as x = |z|. A negative value of z corresponds to positive excess charges). Z = 240 denotes the total positive charge of the jellium background. The shell-corrected and icosahedrally perturbed first and higher electron affinities Aico x are similarly defined as ico ico Aico x = Esh (Ne = 240 + x − 1; Z = 240) − Esh (Ne = 240 + x; Z = 240).

(7.39)

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Table 7.1. ETF (spherically averaged, denoted by a tilde) and shell-corrected (denoted by a superscript ico to indicate that the icosahedral splittings of energy levels have been included) IPs and EAs of fullerenes Cx± 60 . Energies in eV . rs = 1.23 a.u. x 1 2 3 4 5 6 7 8 9 10 11 12

Iex

5.00 7.98 10.99 14.03 17.09 20.18 23.29 26.42 29.57 32.73 35.92 39.12

Ixico 7.40 10.31 13.28 16.25 19.22 22.20 25.24 28.31 31.30 34.39 39.36 42.51

ex A

2.05 −0.86 −3.75 −6.60 −9.41 −12.19 −14.94 −17.64 −20.31 −22.94 −25.53 −28.07

Aico x 2.75 −0.09 −2.92 −5.70 −8.41 −11.06 −14.85 −17.24 −19.49 −21.39 −22.93 −23.85

ex , which We have also calculated the corresponding average quantities Iex and A result from the ETF variation with spherical symmetry (that is without shell and icosahedral symmetry corrections). Their definition is the same as in Eq. (7.38) and Eq. (7.39), but with the index sh replaced by a tilde and the removal of the index ico. In our calculations of the charged fullerene molecule, the rs value and the icosahedral splitting parameters (κl , see Eq. (7.35), and discussion below it) were taken as those which were determined by our calculations of the neutral molecule, discussed in the previous section. The parameters which specify the ETF electronic density (Eq. (7.33)) are optimized for the charged molecule, thus allowing for relaxation effects due to the excess charge. This procedure is motivated by results of − 92,93 previous electronic structure calculations for C+ which showed that 60 and C60 , the icosahedral spectrum of the neutral C60 shifts almost rigidly upon charging of the molecule. Shell-corrected and ETF calculated values of ionization potentials and electron affinities, for values of the excess charge up to 12 units, are summarized in Table 7.1 (for rs = 1.23 a.u.) 7.3.2.5. Charging energies and capacitance of fullerenes Figure 7.10(a) shows that the variation of the total ETF-DFT energy difference e e − E(0), e (appearance energies) ∆E(z) = E(z) as a function of excess charge z (|z| = x), exhibits a parabolic behavior. The inset in Fig. 7.10(a) exhibiting the quantity g(z) = e

e e E(z) − E(0) e1 , +A z

(7.40)

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e e e Fig. 7.10. (a) ETF-DFT total energy differences (appearance energies) ∆E(z) = E(z) − E(0) as a function of the excess charge z (z < 0 corresponds to positive excess charge). Inset: The ETF ico (z) (filled circles). For z ≥ 1 the function ge(z) (open squares), and the shell-corrected function gsh two functions are almost identical. (b) magnification of the appearance-energy curves for the region −2 ≤ z ≤ 4. Filled circles: shellico (z) = E ico (z) − E ico (0)]. Open squares: ETF-DFT values corrected icosahedral values [∆Esh sh sh e e e [∆E(z) = E(z) − E(0)].

plotted versus z (open squares), shows a straight line which crosses the zero energy line at z = 1. As a result the total ETF-DFT energy has the form, z(z − 1)e2 e1 z. e e E(z) = E(0) + −A (7.41) 2C Equation (7.41) indicates that fullerenes behave on the average like a capacitor having a capacitance C (the second term on the rhs of Eq. (7.41) corresponds to the charging energy of a classical capacitor, corrected for the self-interaction of the excess charge3,23 ). We remark that regarding the system as a classical conductor, where the excess charge accumulates on the outer surface, yields a value of C = 8.32 a.u. (that is the outer radius of the jellium shell). Naturally, the ETF calculated value for C is somewhat larger because of the quantal spill-out of the electronic charge density. Indeed, from the slope of e g(z) we determine94 C = 8.84 a.u.

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A similar plot of the shell-corrected and icosahedrally modified energy differences ico ico ico ∆Esh (z) = Esh (z)−Esh (0) is shown in Fig. 7.10(b) (in the range −2 ≤ z ≤ 4, filled ico circles). The function gsh (z), defined as in Eq. (7.40) but with the shell-corrected ico quantities (∆Esh (z) and Aico 1 ), is included in the inset to Fig. 7.10(a) (filled circles). ico ico The shift discernible between gsh (−1) and gsh (1) is approximately 1.7 eV , and originates from the difference of shell effects on the IPs and EAs (see Table 7.1). The ico segments of the curve gsh (z) in the inset of Fig. 7.10(a), corresponding to positively (z < 0) and negatively (z > 0) charged states, are again well approximated by straight lines, whose slope is close to that found for ge(z). Consequently, we may approximate the charging energy, including shell-effects, as follows, ico ico Esh (x) = Esh (0) +

x(x − 1)e2 − Aico 1 x, 2C

(7.42)

for negatively charged states, and x(x − 1)e2 + I1ico x, (7.43) 2C for positively charged states. Note that without shell-corrections (i.e., ETF only) e1 = e2 /C = 27.2/8.84 eV ≈ 3.1 eV , because of the symmetry of Eq. (7.41) Ie1 − A with respect to z, while the shell-corrected quantities are related as I1ico − Aico 1 ≈ e2 /C + ∆sh , where the shell correction is ∆sh ≈ 1.55 eV (from Table 7.1, I1ico − Aico 1 ≈ 4.65 eV ). Expression (7.42) for the negatively charged states can be rearranged as follows (energies in units of eV ), ico ico Esh (x) = Esh (0) +

ico ico Esh (x) − Esh (0) = −2.99 + 1.54(x − 1.39)2 ,

(7.44)

in close agreement with the all-electron LDA result of Ref. 95. Equations (7.42) and (7.43) can be used to provide simple analytical approximaico ico tions for the higher IPs and EAs. Explicitly written, Aico x ≡ Esh (x − 1) − Esh (x) = ico 2 ico ico 2 A1 − (x − 1)e /C and Ix = I1 + (x − 1)e /C. Such expressions have been used previously96 with an assumed value for C ≈ 6.7 a.u. (i.e., the radius of the C60 molecule, as determined by the distance of carbon nuclei from the center of the molecule), which is appreciably smaller than the value obtained by us (C = 8.84 a.u., see above) via a microscopic calculation. Consequently, using the above expression with our calculated value for Aico 1 = 2.75 eV (see Table 7.1), we obtain an approximate value of Aico = −0.35 eV (compared to the microscopically 2 calculated value of −0.09 eV given in Table 7.1, and −0.11 eV obtained by Ref. 95) — indicating metastability of C2− 60 — while employing an experimental value for ico Aico = 2.74 eV , a value of A = 0.68 eV was calculated in Ref. 96. 1 2 Concerning the cations, our expression (7.43) with a calculated I1ico = 7.40 eV (see Table 7.1) and C = 8.84 a.u. yields approximate values 18.5 eV and 31.5 3+ eV for the appearance energies of C2+ 60 and C60 (compared to the microscopic calculated values of 17.71 eV and 30.99 eV , respectively, extracted from Table 7.1, and 18.6 eV for the former obtained in Ref. 92). Employing an experimental value

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for I1ico = 7.54 eV , corresponding values of 19.20 eV and 34.96 eV were calculated in Ref. 96. As discussed in Ref. 97, these last values are rather high, and the origin of the discrepancy may be traced to the small value of the capacitance which was used in obtaining these estimates in Ref. 96. A negative value of the second affinity indicates that C2− 60 is unstable against electron autodetachment. In this context, we note that the doubly negatively charged molecule C2− 60 has been observed in the gas phase and is believed to be a long-lived metastable species.98,99 Indeed, as we discuss in the next section, the small DFT values of Aico 2 found by us and by Ref. 95 yield lifetimes which are much longer than those estimated by a pseudopotential-like Hartree-Fock model calculation,98 where a value of ∼ 1 µs was estimated. 7.3.2.6. Lifetimes of metastable anions, C60x− The second and higher electron affinities of C60 were found to be negative, which implies that the anions Cx− 60 with x ≥ 2 are not stable species, and can lower their energy by emitting an electron. However, unless the number of excess electrons is large enough, the emission of an excess electron involves tunneling through a barrier. Consequently, the moderately charged anionic fullerenes can be described as metastable species possessing a decay lifetime. To calculate the lifetime for electron autodetachmant, it is necessary to determine the proper potential that the emitted electron sees as it leaves the molecule. The process is analogous to alpha-particle radioactivity of atomic nuclei. The emitted electron will have a final kinetic energy equal to the negative of the corresponding higher EA. We estimate the lifetime of the decay process by using the WKB method, in the spirit of the theory of alpha-particle radioactivity, which has established that the main factor in estimating lifetimes is the relation of the kinetic energy of the emitted particle to the Coulombic tail, and not the details of the many-body problem in the immediate vicinity of the parent nucleus. Essential in this approach is the determination of an appropriate single-particle potential that describes the transmission barrier. It is well known that the (DFT) LDA potential posseses the wrong tail, since it allows for the electron to spuriously interact with itself. A more appropriate potential would be one produced by the Self-Interaction Correction method of Ref. 78. This potential has the correct Coulombic tail, but in the case of the fullerenes presents another drawback, namely Koopman’s theorem is not satisfied to an extent adequate for calculating lifetimes.100 In this context, we note that Koopman’s theorem is known to be poorly satisfied for the case of fullerenes even in Hartree-Fock calculations.101 Therefore, the HOMO corresponding to the emitted electron, calculated as described above, cannot be used in the WKB tunneling calculation. Since the final energy of the ejected electron equals the negative of the value of the electron affinity, we seek a potential that, together with the icosahedral ico perturbation, yields a HOMO level in Cx− 60 with energy −Ax . We construct this

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potential through a self-interaction correction to the LDA potential as follows, VW KB = VLDA [e ρ] − VH [

ρe ρe ] − Vxc [ξ ], Ne Ne

(7.45)

ico where the parameter ξ is adjusted so that the HOMO level of Cx− 60 equals −Ax . In the above expression, the second term on the rhs is an average self-interaction Hartree correction which ensures a proper long-range behavior of the potential (i.e., correct Coulomb tail), and the third term is a correction to the short-range exchange-correlation. 3− For the cases of C2− 60 and C60 such potentials are plotted in Fig. 7.11. We observe that they have the correct Coulombic tail, namely a tail corresponding to one 3− electron for C2− 60 and to two electrons for C60 . The actual barrier, however, through which the electron tunnels is the sum of the Coulombic barrier plus the contribution of the centrifugal barrier. As seen from Fig. 7.11, the latter is significant, since the HOMO in the fullerenes possesses a rather high angular momentum (l = 5), while being confined in a small volume. Using the WKB approximation,102 we estimate for C2− 60 a macroscopic half-life of ∼ 4 × 107 years, while for C3− we estimate a very short half-life of 2.4 × 10−12 s. 60 Both these estimates are in correspondence with observations. Indeed, C3− 60 has 2− not been observed as a free molecule, while the free C60 has been observed to be long lived98,99 and was detected even 5 min after its production through laser vaporization.99 We note that the WKB lifetimes calculated for tunneling through Coulombic barriers are very sensitive to the final energy of the emitted particle and can vary by many orders of magnitude as a result of small changes in this energy, a feature well known from the alpha radioctivity of nuclei.102 Since the second electron affinity of C60 is small, effects due to geometrical relaxation and spin polarization can influence its value and, consequently, the estimated lifetime. Nevertheless, as shown in Ref. 95, inclusion of such corrections yields again a negative second affinity, but of somewhat smaller magnitude, resulting in an even longer lifetime (the sign conventions in Ref. 95 are the opposite of ours). Furthermore, as discussed in Ref. 103, the stabilization effect of the Jahn-Teller relaxation for the singly-charged ion is only of the order of 0.03 – 0.05 eV . Since this effect is expected to be largest for singly-charged species, C2− 60 is not expected to be influenced by it.95 On the other hand, generalized exchange-correlation functionals with gradient corrections yield slightly larger values for the second electron affinity. For example, using exchange-correlation gradient corrections, Ref. 95 found Aico 2 = −0.3 eV , which is higher (in absolute magnitude) than the value obtained without such corrections. This value of −0.3 eV leads to a much smaller lifetime than the several million of years that correspond to the value of −0.09 eV calculated by us. Indeed, using the barrier displayed in Fig. 7.11(a), we estimate a lifetime for C2− 60 of approx. 0.37 s, when Aico = −0.3 eV . We stress, however, that even this lower-limit value 2

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3− Fig. 7.11. WKB effective barriers used to estimate lifetimes for C2− 60 (a) and C60 (b). Dashed lines correspond to barriers due solely to Coulombic repulsion and solid lines to total barriers after adding the centrifugal components. The thick horizontal solid lines correspond to the negative of the associated electron affinities Aico (a) and Aico (b). In the case of C2− 2 3 60 [panel (a)], the horizontal solid line at −Aico = 0.09 eV crosses the total barrier at an inside point R1 = 9.3 a.u. 2 and again at a distance very far from the center of the fullerene molecule, namely at an outer point R2 = −e2 /Aico = 27.2/0.09 a.u. = 302.2 a.u. This large value of R2 , combined with the large 2 centrifugal barrier, yields a macroscopic lifetime for the metastable C2− 60 (see text for details).

still corresponds to macroscopic times and is 5 orders of magnitude larger than the estimate of Ref. 98, which found a lifetime of 1 µs for Aico = −0.3 eV , since it 2 omitted the large centrifugal barrier. Indeed, when we omit the centrifugal barrier, we find a lifetime estimate of 1.4 µs, when Aico 2 = −0.3 eV . 7.3.3. On mesoscopic forces and quantized conductance in model metallic nanowires 7.3.3.1. Background and motivation In this section, we show that certain aspects of the mechanical response (i.e., elongation force) and electronic transport (e.g., quantized conductance) in metallic nanowires can be analyzed using the DFT shell correction method, developed and

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applied previously in studies of metal clusters (see Sec. 7.2 and Sec. 7.3.1). Specifically, we show that in a jellium-modelled, volume-conserving nanowire, variations of the total energy (particularly terms associated with electronic subband corrections) upon elongation of the wire lead to self-selection of a sequence of stable “magic” wire configurations (MWC’s, specified in our model by a sequence of the wire’s radii), with the force required to elongate the wire from one configuration to the next exhibiting an oscillatory behavior. Moreover, we show that due to the quantized nature of electronic states in such wires, the electronic conductance varies in a quantized step-wise manner (in units of the conductance quantum g0 = 2e2 /h), correlated with the transitions between MWC’s and the above-mentioned force oscillations. Prior to introducing the model, it is appropriate to briefly review certain previous theoretical and experimental investigations, which form the background and motivation for this study of nanowires. Atomistic descriptions, based on realistic interatomic interactions, and/or first-principles modelling and simulations played an essential role in discovering the formation of nanowires, and in predicting and elucidating the microscopic mechanisms underlying their mechanical, spectral, electronic and transport properties. Formation and mechanical properties of interfacial junctions (in the form of crystalline nanowires) have been predicted through early molecular-dynamics simulations, 104 where the materials (gold) were modelled using semiempirical embeddedatom potentials. In these studies it has been shown that separation of the contact between materials leads to generation of a connective junction which elongates and narrows through a sequence of structural instabilities; at the early stages, elongation of the junction involves multiple slip events, while at the later stages, when the lateral dimension of the wire necks down to a diameter of about 15 ˚ A, further elongation involves a succession of stress accumulation and fast relief stages associated with a sequence of order-disorder structural transformations localized to the neck region.104–106 These structural evolution patterns have been shown through the simulations to be portrayed in oscillations of the force required to elongate the wire, with a period approximately equal to the interlayer spacing. In addition, the “sawtoothed” character of the predicted force oscillations (see Fig. 3(b) in Ref. 104 and Fig. 3 in Ref. 105) reflects the stress accumulation and relief stages of the elongation mechanism. Moreover, the critical resolved yield stress of gold nanowires has been predicted104,105 to be ∼ 4GPa, which is over an order of magnitude larger than that of the bulk, and is comparable to the theoretical value for Au (1.5 GPa) in the absence of dislocations. These predictions, as well as anticipated electronic conductance properties,104,107 have been corroborated in a number of experiments using scanning tunneling and force microscopy,104,108–113 break junctions,114 and pin-plate techniques105,115 at ambient environments, as well as under ultrahigh vacuum and/or cryogenic conditions. Particularly, pertinent to this section are experimental ob-

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servations of the oscillatory behavior of the elongation forces and the correlations between the changes in the conductance and the force oscillations; ¡indexConductancesee especially the simultaneous measurements of force and conductance in gold nanowires in Ref. 112, where in addition the predicted “ideal” value of the critical yield stress has also been measured (see also Ref. 113). The jellium-based model introduced in this paper, which by construction is devoid of atomic crystallographic structure, does not address issues pertaining to nanowire formation methods, atomistic configurations, and mechnanical response modes (e.g., plastic deformation mechanisms, interplanar slip, ordering and disordering mechanisms (see detailed descriptions in Refs. 104,105 and 106, and a discussion of conductance dips in Ref. 110), defects, mechanichal reversibility,105,112 and roughening of the wires’ morphology during elongation106 ), nor does it consider the effects of the above on the electron spectrum, transport properties, and dynamics.116 Nevertheless, as shown below, the model offers a useful framework for linking investigations of solid-state structures of reduced dimensions (e.g., nanowires) with methodologies developed in cluster physics, as well as highlighting certain nanowire phenomena of mesoscopic origins and their analogies to clusters. 7.3.3.2. The jellium model for metallic nanowires: Theoretical method and results Consider a cylindrical jellium wire of length L, having a positive background with a circular cross section of constant radius R  L.117 For simplicity, we restrict ourselves here to this symmetry of the wire cross section. Variations in the shape of the nanowire cross section serve to affect the degeneracies of the electronic spectrum118,119 without affecting our general conclusions. We also do not include here variations of the wire’s shape along its axis. Adiabatic variation of the wire’s axial shape introduces a certain amount of smearing of the conductance steps through tunnelling, depending on the axial radius of curvature of the wire.118–120 Both the cross-sectional and axial shape of the wire can be included in our model in a rather straightforward manner. As elaborated in Sec. 7.2, the principal idea of the SCM is the separation of the total DFT energy ET (R) of the nanowire as e ET (R) = E(R) + ∆Esh (R),

(7.46)

e where E(R) varies smoothly as a function of the radius R of the wire (instead of the number of electrons N used in Sec. 7.2), and ∆Esh (R) is the shell-correction term arising from the discrete quantized nature of the electronic levels. Again, as elaborated in Sec. 7.2, the smooth contribution in Eq. (7.46) is identified with EET F [e ρ]. The trial radial lateral density ρe(r) is given by Eq. (7.25), and the constant ρ0 at a given radius R is obtained under the normalization condition (charge R (+) (+) neutrality) 2π ρe(r)rdr = ρL (R), where ρL (R) = 3R2 /(4rs3 ) is the linear posi-

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Fig. 7.12. Lower panel: The VET F (r) potential for a sodium wire with a uniform jellium background of radius R = 12.7 a.u., plotted versus the transverse radial distance from the center of the wire, along with the locations of the bottoms of the subbands (namely the transverse eigenvalues e nm ; n is the number of nodes in the radial direction plus one, and m is the azimuthal quantum number of the angular momentum). The Fermi level is denoted by a dashed line. Top panel: The jellium background volume density (dashed line) and the electronic volume density ρe(r) (solid line, exhibiting a characteristic spillout) normalized to bulk values are shown.

tive background density. Using the optimized ρe, one solves for the eigenvalues e i of the Hamiltonian H = −(~2 /2m)∇2 + VET F [e ρ], and the shell correction is given by ∆Esh ≡ EHarris [e ρ] − EET F [e ρ] Z occ X = e i − ρe(r)VET F [e ρ(r)]dr − TET F [e ρ],

(7.47)

i=1

where the summation extends over occupied levels. Here the dependence of all quantities on the pertinent size variable (i.e., the radius of the wire R) is not shown explicitly. Additionally, the index i can be both discrete and continuous, and in the latter case the summation is replaced by an integral (see below). Following the above procedure with a uniform background density of sodium (rs = 4 a.u.), a typical potential VET F (r) for R = 12.7 a.u., where r is the radial coordinate in the transverse plane, is shown in Fig. 7.12, along with the transverse eigenvalues e nm and the Fermi level; to simplify the calculations of the electronic spectrum, we have assumed (as noted above) R  L, which allows us to express

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Fig. 7.13. Variation of the Fermi energy F [shown in (a)] and of the conductance G (shown in (b) in units of g0 = 2e2 /h), plotted versus the radius R, for a sodium nanowire. Note the coincidence of the cusps in F with the step-rises of the conductance. The heights of the steps in G reflect the subband degeneracies due to the circular shape of the wire’s cross section.

the subband electronic spectrum as e nm (kz ; R) = e nm (R) +

~2 kz2 , 2m

(7.48)

where kz is the electron wave number along the axis of the wire (z). As indicated earlier, taking the wire to be charge neutral, the electronic linear (−−) (+) density, ρL (R), must equal the linear positive background density, ρL (R). The chemical potential (at T = 0 the Fermi energy F ) for a wire of radius R is determined by setting the expression for the electronic linear density derived from the (+) subband spectra equal to ρL (R), i.e., occ 2X π n,m

r

2m (+) [F (R) − e nm (R)] = ρL (R), ~2

(7.49)

where the factor of 2 on the left is due to the spin degeneracy. The summand defines the Fermi wave vector for each subband, kF,nm . The resulting variation of F (R) versus R is displayed in Fig. 7.13(a), showing cusps for values of the radius where a new subband drops below the Fermi level as R increases (or conversely as a subband moves above the Fermi level as R decreases upon elongation of the wire). Using the Landauer expression for the conductance G in the limit of no mode mixing P and assuming unit transmission coefficients, G(R) = g0 n,m Θ[F (R) − e nm (R)], where Θ is the Heaviside step function. The conductance of the nanowire, shown in Fig. 7.13(b), exhibits quantized step-wise behavior, with the step-rises coinciding with the locations of the cusps in F (R), and the height sequence of the steps is

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1g0 , 2g0 , 2g0 , 1g0 , ..., reflecting the circular symmetry of the cylindrical wires’ cross sections,107 as observed for sodium nanowires.114 Solving for F (R) [see Eq. (7.49)], the expression for the sum on the right-hand-side of Eq. (7.47) can be written as occ X i

occ Z 2 X kF,nm dkz e nm (kz ; R) π n,m 0 r occ 2 X 2m = [F (R) + 2e nm (R)] [F (R) − e nm (R)], 3π n,m ~2

e i =

(7.50)

which allows one to evaluate ∆Esh [Eq. (7.47)] for each wire radius R. Since the expression in Eq. (7.50) gives the energy per unit length, we also calculate EET F , TET F , and the volume integral in the second line of Eq. (7.47) for cylindrical volumes of unit height. To convert to energies per unit volume [denoted as εT (R), εe(R), and ∆εsh (R)] all energies are further divided by the wire’s cross-sectional area, πR2 . The smooth contribution and the shell correction to the wire’s energy are shown respectively in Fig. 7.14(a) and Fig. 7.14(b). The smooth contribution decreases slowly towards the bulk value (−2.25 eV per atom3 ). On the other hand, the shell corrections are much smaller in magnitude and exhibit an oscillatory behavior. This oscillatory behavior remains visible in the total energy [Fig. 7.14(c)] with the local energy minima occurring for values Rmin corresponding to conductance plateaus. The sequence of Rmin values defines the MWC’s, that is a sequence of wire configurations of enhanced stability. From the expressions for the total energy of the wire [i.e., ΩεT (R), where Ω = 2 πR L is the volume of the wire] and the smooth and shell (subband) contributions to it, we can calculate the “elongation force” (EF), d[ΩεT (R)] = −Ω FT (R) = − dL ≡ Fe(R) + ∆Fsh (R).



de ε(R) d[∆εsh (R)] + dL dL

 (7.51)

Using the volume conservation, i.e., d(πR2 L) = 0, these forces can be written as FT (R) = (πR3 /2)dεT (R)/dR, Fe (R) = (πR3 /2)de ε(R)/dR, and ∆Fsh (R) = (πR3 /2)d[∆εsh (R)]/dR. Fe(R) and FT (R) are shown in Fig. 7.14(d,e). The oscillations in the force resulting from the shell-correction contributions dominate. In all cases, the radii corresponding to zeroes of the force situated on the left of the force maxima coincide with the minima in the potential energy curve of the wire, corresponding to the MWC’s. Consequently, these forces may be interpreted as guiding the self-evolution of the wire toward the MWC’s. Also, all the local maxima in the force occur at the locations of step-rises in the conductance [reproduced in Fig. 7.14(f)], signifying the sequential decrease in the number of subbands below the Fermi level (conducting channels) as the wire narrows (i.e., as it is being

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Fig. 7.14. (a-c): The smooth (a) and shell-correction (b) contributions to the total energy (c) per unit volume of the jellium-modelled sodium nanowire (in units of u ≡ 10−4 eV/a.u.3 ), plotted versus the radius of the wire (in a.u.). Note the smaller magnitude of the shell corrections relative to the smooth contribution. (d-e): The smooth contribution (d) to the total force and the total force (e), plotted in units of nN versus the wire’s radius. In (e), the zeroes of the force to the left of the force maxima occur at radii corresponding to the local minima of the energy of the wire (c). In (f), we reproduce the conductance of the wire (in units of g0 = 2e2 /h), plotted versus R. Interestingly, calculations of the conductance for the MWC’s (i.e., the wire radii corresponding to the locations of the step-rises) through the Sharvin-Weyl formula,119,121 corrected for the finite height of the confining potential121 (see lower panel of Fig. 7.12), namely G = g0 (πS/λ2F −αP/λF ) where S and P are the area and perimeter of the wire’s cross section and λF is the Fermi wavelength (λF = 12.91 a.u. for Na) with α = 0.1 (see Ref. 121), yield results which approximate well the conductance values (i.e., the values at the bottom of the step-rises) shown in (f).

elongated). Finally the magnitude of the total forces is comparable to the measured ones (i.e., in the nN range). 7.4. Summary While it was understood rather early that the total energy of nuclei can be decomposed into an oscillatory part and one that shows a slow “smooth” variation as a function of size, Strutinsky’s seminal contribution1 was to calculate the two parts from different nuclear models: the former from the nuclear shell model and the latter from the liquid drop model. In particular, the calculation of the oscillatory part (shell correction term) was enabled by employing an averaging method that smeared the single particle spectrum associated with a nuclear model potential.

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A semiempirical shell-correction method (referred to as SE-SCM) for metal clusters, that was developed in close analogy to the original phenomenological Strutinsky approach, was presented in the Appendix, along with some applications to triaxial deformations and fission barriers of metal clusters. This chapter reviewed primarily the motivation and theory of a microscopic shell correction method based on density functional theory (often referred to as DFT-SCM and originally introduced in Ref. 3). In developing the DFT-SCM, we have used for the shell correction term (arising from quantum interference effects) a derivation that differs from the Strutinsky methodology.1 Instead, we have shown3 that the shell correction term can be introduced through a kinetic-energy-type density functional [see Eq. (7.19) and Eq. (7.23)]. The DFT-SCM is computationally advantageous, since it bypasses the selfconsistent iteration cycle of the more familiar KS-DFT. Indeed, the DFT-SCM energy functional depends only on the single-particle density, and thus it belongs to the class of orbital-free DFT methods. Compared to previous OF-DFT approaches, the DFT-SCM represents an improvement in accuracy. Applications of the DFT-SCM to condensed-matter nanostructures, and in particular metal clusters, fullerenes, and nanowires, were presented in Sec. 7.3. Acknowledgements This research was supported by a grant from the U.S. Department of Energy (Grant No. FG05-86ER45234). Appendix A. Semiempirical shell-correction method (SE-SCM) As mentioned above already [see, e.g., Sec. 7.1.1], rather than proceed with the microscopic route, Strutinsky proposed a method for separation of the total energy into smooth and shell-correction terms [see Eq. (7.1)] based on an averaging esp , is extracted out of the sum of the procedure. Accordingly, a smooth part, E P single-particle energies i εei [see Eq. (7.6), or equivalently Eq. (7.16) with ρin replaced by ρe and εout by εei ] by averaging them through an appropriate procedure. i Usually, but not necessarily, one replaces the delta functions in the single-particle density of states by gaussians or other appropriate weighting functions. As a result, each single-particle level is assigned an averaging occupation number fei , and the esp is formally written as smooth part E X esp = E εei fei . (A.1) i

Consequently, the Strutinsky shell correction is given by Str ∆Esh =

occ X i=1

esp . εei − E

(A.2)

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The Strutinsky prescription (A.2) has the practical advantage of using only the single-particle energies εei , and not the smooth density ρe. Taking advantage of this, the single-particle energies can be taken as those of an external potential that empirically approximates the self-consistent potential of a finite system. In the nuclear case, an anisotropic three-dimensional harmonic oscillator has been used successfully to describe the shell-corrections in deformed nuclei. esp , however, is only one component of the The single-particle smooth part, E e e total smooth contribution, E[e ρ] (EHF in the Hartree-Fock energy considered by Strutinsky). Indeed as can be seen from Eq. (7.6) [or equivalently Eq. (7.16)], Str e ρ]. Etotal ≈ ∆Esh + E[e

(A.3)

Str Etotal ≈ ∆Esh + ELDM .

(A.4)

Strutinsky did not address the question of how to calculate microscopically the e (which necessarily entails specifying the smooth density ρe). Instead smooth part E e the empirical energies, ELDM , he circumvented this question by substituting for E of the nuclear liquid drop model, namely he suggested that

In applications of Eq. (A.4), the single-particle energies involved in the averaging [see Eqs. (A.1) and (A.2)] are commonly obtained as solutions of a Schr¨odinger equation with phenomenological one-body potentials. This last approximation has been very successful in describing fission barriers and properties of strongly deformed nuclei using harmonic oscillator or Wood-Saxon empirical potentials. In the following (Sec. A.1), we describe the adaptation of the SE-SCM approach to condensed-matter finite systems, and in particular to triaxially deformed metal clusters. Moreover (Sec. A.2), we will present several figures illustrating applications of the SE-SCM to investigations of the effects of triaxial shape-deformations on the properties of metal clusters25,33,35–37 and to studies of large-scale deformations and barriers in fission of charged metal clusters.26,27,38 We note that the SE-SCM has been extended to incorporate electronic entropy effects at finite temperatures. This latter extension, referred to as finite-temperature (FT)-SE-SCM is not described here, but its theory can be found in Ref. 33. We mention that, in addition, Strutinsky-type calculations using phenomenological potentials have been reported for the case of neutral sodium clusters assuming axial symmetry in Refs. 29–31,122, and for the case of fission in Ref. 28. A.1. Semiempirical shell-correction method for triaxially deformed clusters A.1.1. Liquid-drop model for neutral and charged deformed clusters For neutral clusters, the liquid-drop model28,48,123 (LDM) expresses the smooth e of the total energy as the sum of three contributions, namely a volume, a part, E,

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surface, and a curvature term, i.e., e = Evol + Esurf + Ecurv E Z Z Z = Av dτ + σ dS + Ac dSκ,

(A.5)

where dτ is the volume element and dS is the surface differential element. The local −1 −1 curvature κ is defined by the expression κ = 0.5(Rmax + Rmin ), where Rmax and Rmin are the two principal radii of curvature at a local point on the surface of the jellium droplet which models the cluster. The corresponding coefficients can be determined by fitting the extended Thomas-Fermi (ETF)-DFT total energy EET F [ρ] (see Sec. 7.2.2) for spherical shapes to the following parametrized expression as a function of the number, N , of atoms in the cluster,124 sph 2/3 EET + αc N 1/3 . F = αv N + αs N

(A.6)

The following expressions relate the coefficients Av , σ, and Ac to the corresponding coefficients, (α’s), in Eq. (A.6), Av =

3 1 1 αv ; σ = αs ; Ac = αc . 4πrs3 4πrs2 4πrs

(A.7)

In the case of ellipsoidal shapes the areal integral and the integrated curvature can be expressed in closed analytical form with the help of the incomplete elliptic integrals F (ψ, k) and E(ψ, k) of the first and second kind,125 respectively. Before writing the formulas, we need to introduce some notations. Volume conservation must be employed, namely a0 b0 c0 /R03 = abc = 1,

(A.8)

where R0 is the radius of a sphere with the same volume (R0 = rs N 1/3 is taken to be the radius of the positive jellium assuming spherical symmetry), and a = a0 /R0 , etc..., are the dimensionless semi-axes. The eccentricities are defined through the dimensionless semi-axes as follows e21 = 1 − (c/a)2 e22 = 1 − (b/a)2

e23 = 1 − (c/b)2 .

(A.9)

a ≥ b ≥ c.

(A.10)

The semi-axes are chosen so that

With the notation sin ψ = e1 , k2 = e2 /e1 , and k3 = e3 /e1 , the relative (with respect to the spherical shape) surface and curvature energies are given126 by   ell Esurf ab 1 − e21 3 = F (ψ, k3 ) + e1 E(ψ, k3 ) + c (A.11) sph 2 e1 Esurf

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and ell Ecurv

 
bc a3 2 2 = 1+ (1 − e1 )F (ψ, k2 ) + e1 E(ψ, k2 ) . 2a e1

(A.12) sph Ecurv The change in the smooth part of the cluster total energy due to the excess charge ±Z was already discussed by us for spherical clusters in the previous section. The result may be summarized as 2 e sph (Z) = E e sph (Z) − E e sph (0) = ∓W Z + Z(Z ± 0.25)e , ∆E 2(R0 + δ)

(A.13)

where the upper and lower signs correspond to negatively and positively charged states, respectively, W is the work function of the metal, R0 is the radius of the positive jellium assuming spherical symmetry, and δ is a spillout-type parameter. To generalize the above results to an ellipsoidal shape, φ(R0 + δ) = e2 /(R0 + δ), which is the value of the potential on the surface of a spherical conductor, needs to be replaced by the corresponding expression for the potential on the surface of a conducting ellipsoid. The final result, normalized to the spherical shape, is given by the expression e ell (Z) ± W Z bc ∆E = F (ψ, k2 ), sph e e1 ∆E (Z) ± W Z

(A.14)

where the ± sign in front of W Z corresponds to negatively and positively charged clusters, respectively. A.1.2. The modified Nilsson potential A natural choice for an external potential to be used for calculating shell corrections with the Strutinsky method is an anisotropic, three-dimensional oscillator with an l2 term for lifting the harmonic oscillator degeneracies.127 Such an oscillator model for approximating the total energies of metal clusters, but without separating them into a smooth and a shell-correction part in the spirit of Strutinsky’ s approach, has been used58 with some success for calculating relative energy surfaces and deformation shapes of metal clusters. However, this simple harmonic oscillator model has serious P limitations, since i) the total energies are calculated by the expression 43 i εi , and thus do not compare with the total energies obtained from the KS-DFT approach, ii) the model cannot be extended to the case of charged (cationic or anionic) clusters. Thus absolute ionization potentials, electron affinities, and fission energetics cannot be calculated in this model. Alternatively, in our approach, we are making only a limited use of the external oscillator potential in calculating a modified Strutinsky shell correction. Total energies are evaluated by adding this shell correction to the smooth LDM energies. In particular, a modified Nilsson Hamiltonian appropriate for metal clusters128,129 is given by HN = H0 + U0 ~ω0 (l2 − < l2 >n ),

(A.15)

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REL. ABUND.

0.4

(a)

241

39

0.2

33 49

29

54

Energy (eV)

0

(b)

1

33

39

49

0

53

29

-1 20

30

40

50

60

N Fig. A.1. (a) Experimental yields (denoted as “REL. ABUND.”) of dianionic silver clusters Ag2− N , plotted versus cluster size. The error bars indicate the statistical uncertainty. (b) Theoretical FTSE-SCM33 second electron affinities A2 for Ag2− N clusters at T = 300 K. LDM results are depicted by the dashed line. The figure was reproduced from Ref. 36.

where H0 is the hamiltonian for a three-dimensional anisotropic oscillator, namely ~2 me 2 2 4+ (ω1 x + ω22 y 2 + ω32 z 2 ) 2me 2 3 X 1 (a†k ak + )~ωk . = 2

H0 = −

(A.16)

k=1

U0 in Eq. (A.15) is a dimensionless parameter, which for occupied states may depend on the principal quantum number n = n1 +n2 +n3 of the spherical-oscillator major shell associated with a given level (n1 , n2 , n3 ) of the hamiltonian H0 (for clusters comprising up to 100 valence electrons, only a weak dependence on n is found, see Table I in Ref. 25). U0 vanishes for values of n higher than the corresponding value of the last partially (or fully) filled major shell in the spherical limit. P3 l2 = k=1 lk2 is a “stretched” angular momentum which scales to the ellipsoidal shape and is defined as follows, l32 ≡ (q1 p2 − q2 p1 )2 ,

(A.17)

(with similarly obtained expressions for l1 and l2 via a cyclic permutation of indices) where the stretched position and momentum coordinates are defined via the corresponding natural coordinates, qknat and pnat k , as follows, qk ≡ qknat (me ωk /~)1/2 =

a†k + ak √ , (k = 1, 2, 3), 2

(A.18)

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Fig. A.2. Ionization potentials of neutral KN clusters at three temperatures, T = 10, 300, and 500 K. Solid dots: theoretical FT-SE-SCM33 results. Open squares: experimental measurements.129 The best agreement between theory and experiment happens for T = 300 K (room temperature), indicating the importance of the electronic entropy in quenching the shell effects.

1/2 pk ≡ pnat =i k (1/~me ωk )

a†k − ak √ , (k = 1, 2, 3). 2

(A.19)

The stretched l2 is not a properly defined angular-momentum operator, but has the advantageous property that it does not mix deformed states which correspond to sherical major shells with different principal quantum number n = n1 + n2 + n3 (see, the appendix in Ref. 25 for the expression of the matrix elements of l2 ). The subtraction of the term < l2 >n = n(n + 3)/2, where < >n denotes the expectation value taken over the nth-major shell in spherical symmetry, guaranties that the average separation between major oscillator shells is not affected as a result of the lifting of the degeneracy. The oscillator frequencies can be related to the principal semi-axes a0 , b0 , and 0 c [see Eq. (A.8)] via the volume-conservation constraint and the requirement that the surface of the cluster is an equipotential one, namely ω1 a0 = ω2 b0 = ω3 c0 = ω0 R0 ,

(A.20)

where the frequency ω0 for the spherical shape (with radius R0 ) was taken according

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to Ref. 130 to be ~ω0 (N ) =

 −2 49 eV bohr2 t 1 + . rs2 N 1/3 rs N 1/3

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243

(A.21)

Since in this paper we consider solely monovalent elements, N in Eq. (A.21) is the number of atoms for the family of clusters MZ± N , rs is the Wigner-Seitz radius expressed in atomic units, and t denotes the electronic spillout for the neutral cluster according to Ref. 130. A.1.3. Averaging of single-particle spectra and semi-empirical shell correction esp [see Eqs. (A.1) and (A.2)] is calculated numerically.131 However, a Usually E variation of the numerical Strutinsky averaging method consists in using the semiclassical partition function and in expanding it in powers of ~2 . With this method, for the case of an anisotropic, fully triaxial oscillator, one finds43,132 an analytical result, namely133 osc esp = ~(ω1 ω2 ω3 )1/3 E   1 1 ω12 + ω22 + ω32 4/3 2/3 × (3Ne ) + (3Ne ) , 4 24 (ω1 ω2 ω3 )2/3

(A.22)

where Ne denotes the number of delocalized valence electrons in the cluster. In the present work, expression (A.22) (as modified below) will be substituted Pocc esp in Eq. (A.2), while the sum for the average part E εi will be calculated i numerically by specifying the occupied single-particle states of the modified Nilsson oscillator represented by the hamiltonian (A.15). e osc , In the case of an isotropic oscillator, not only the smooth contribution, E sp 43 but also the Strutinsky shell correction (A.2) can be specified analytically, with the result 1 Str ∆Esh,0 ~ω0 (3Ne )2/3 (−1 + 12x(1 − x)), (A.23) (x) = 24 where x is the fractional filling of the highest partially filled harmonic oscillator shell. 1 Str For a filled shell (x = 0), ∆Esh,0 (0) = − 24 ~ω0 (3Ne )2/3 , instead of the essentially vanishing value as in the case of the ETF-DFT defined shell correction (cf. Fig. 1 Str Str of Ref. 25). To adjust for this discrepancy, we add −∆Esh,0 (0) to ∆Esh calculated through Eq. (A.2) for the case of open-shell, as well as closed-shell clusters. A.1.4. Overall procedure We are now in a position to summarize the calculational procedure, which consists of the following steps: (1) Parametrize results of ETF-DFT calculations for spherical neutral jellia according to Eq. (A.6).

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Fig. A.3. Two-center-oscillator26,27 SE-SCM results for the asymmetric channel Na2+ 10 → + + + Na+ 7 +Na3 . The final configuration of Na3 is spherical. For the heavier fragment Na7 , we present results associated with three different final shape configurations, namely, oblate [(o,s); left], spherical [(s,s); middle], and prolate [(p,s); right]. The ratio of shorter over longer axis is 0.555 for the oblate case and 0.75 for the prolate case. Bottom panel: LDM energy (surface plus Coulomb, dashed curve) and total potential energy (LDM plus shell corrections, solid curve) as a function of fragment separation d. The empty vertical arrow marks the scission point. The zero of energy is taken at d = 0. A number (−1.58 eV or −0.98 eV), or a horizontal solid arrow, denotes the corresponding dissociation energy. Middle panel: Shell-correction contribution (solid curve), surface contribution (upper dashed curve), and Coulomb contribution (lower dashed curve) to the total energy, as a function of fragment separation d. Top panel: Single-particle spectra as a function of fragment separation d. The occupied (fully or partially) levels are denoted with solid lines. The unoccupied levels are denoted with dashed lines. On top of the figure, four snapshots of the evolving cluster shapes are displayed. The solid vertical arrows mark the corresponding fragment separations. Observe that the doorway molecular configurations correspond to the second snapshot from the left. Notice the change in energy scale for the middle and bottom panels, as one passes from (o,s) to (s,s) and (p,s) final configurations.

(2) Use above parametrization (assuming that parameters per differential element of volume, surface, and integrated curvature are shape independent) in Eq. (A.5) to calculate the liquid-drop energy associated with neutral clusters, and then add to it the charging energy according to Eq. (A.14) to determine the e total LDM energy E.

(3) Use Equations (A.15) and (A.16) for a given deformation [i.e., a0 , b0 , c0 , or equivalently ω1 , ω2 , ω3 , see Eq. (A.20)] to solve for the single-particle spectrum (εi ). esp , of the single-particle spectrum according to Eq. (4) Evaluate the average, E (A.22) and subsequent remarks. Str (5) Use the results of steps 3 and 4 above to calculate the shell correction ∆Esh according to Eq. (A.2).

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(6) Finally, calculate the total energy Esh as the sum of the liquid-drop contribution e + ∆E Str . (step 2) and the shell correction (step 5), namely Esh = E sh

The optimal ellipsoidal geometries for a given cluster MZ± N , neutral or charged, are determined by systematically varying the distortion (namely, the parameters a and b) in order to locate the global minimum of the total energy Esh (N, Z). A.2. Applications of SE-SCM to metal clusters As examples of applications of the SE-SCM, we present here three cases. In Fig. A.1, we show experimental electron affinities for doubly negatively charged silver clusters134 and compare them with theoretical calculations.36 In Fig. A.2, we compare FT-SE-SCM calculations for the IPs of neutral KN clusters with experimental results;33 such comparisons demonstrate the importance of electronic-entropy effects. Finally, in Fig. A.3, we display SE-SCM calculations for the fission barriers associ+ + 26,27 ated with the asymmetric channel Na2+ see caption for details. 10 → Na7 +Na3 ; The phenomenological binding potential as a function of fission-fragment separation is described via a two-center-oscillator model.26,27,135 A fourth application of the SE-SCM describing the IPs of triaxially deformed cold sodium clusters was already used in the introductory Sec. 7.1.2 [see Fig. 7.1(c)]. References 1. V. M. Strutinsky, Nucl. Phys. A 95, 420 (1967); Nucl. Phys. A 122, 1 (1968). 2. P. Ring and P. Schuck, The Nuclear Many-Body Problem, (Springer, New York, 1980). 3. C. Yannouleas and U. Landman, Phys. Rev. B 48, 8376 (1993). 4. L. W. Wang and M. P. Teter, Phys. Rev. B 45, 13196 (1992). 5. F. Perrot, J. Phys.: Cond. Matter 6, 431 (1994). 6. E. Smargiassi and P. A. Madden, Phys. Rev. B 49, 5220 (1994). 7. T. J. Frankcombe, G.-J. Kroes, N. I. Choly, and E. Kaxiras, J. Phys. Chem. B 109, 16554 (2005). 8. Y. A. Wang and E. A. Carter, in Theoretical Methods in Condensed Phase Chemistry, S. D. Schwartz (ed.), (Kluwer, Dordrecht, 2000), p. 117. 9. S. B. Trickey, V. V. Karasiev, and R. S. Jones, Int. J. Quantum Chem. 109, 2943 (2009). 10. M. Brack, in Atomic Clusters and Nanoparticles: Les Houches Session LXXIII 228 July 2000 , C. Guest, P. Hobza, F. Spiegelman, and F. David (eds.), (Springer, Berlin, 2001) p. 161. 11. D. Ullmo, T. Nagano, S. Tomsovic, and H. U. Baranger, Phys. Rev. B 63, 125339 (2001). 12. Ya. I. Delchev, A. I. Kuleff, T. Z. Mineva, F. Zahariev, and J. Maruani, Int. J. Quantum Chem. 99, 265 (2004). 13. W. Zhu, S. B. Trickey, Int. J. Quantum Chem. 100, 245 (2004). This paper studied a perturbative DFT approach in the context of the Harris functional.62 14. W. D. Knight, K. Clemenger, W. A. de Heer, W. A. Saunders, M. Y. Chou, and M. L. Cohen, Phys. Rev. Lett. 52, 2141 (1984).

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N. Van Giai and E. Lipparini, Z. Phys. D 27, 193 (1993). M. J. Puska and R. M. Nieminen, Phys. Rev. A 47, 1181 (1993). B. Grammaticos, Z. Phys. A 305, 257 (1982). G. A. Gallup, Chem. Phys. Lett. 187, 187 (1991). R. C. Haddon, L. E. Brus, and K. Raghavachari, Chem. Phys. Lett. 125, 459 (1986). M. Gerloch and R. C. Slade, Ligand field parameters, (Cambridge Univ. Press, London, 1973). A. Ros´en and B. W¨ astberg, J. Chem. Phys. 90, 2525 (1989); B. W¨ astberg and A. Ros´en, Physica Scripta 44, 276 (1991). L. Ye and A. J. Freeman, Chem. Phys. 160, 415 (1992). Due to the changing spill-out with excess charge z, the capacitance should be written as C + δ(z). For our purposes here the small correction δ(z) can be neglected. M. R. Pederson and A. A. Quong, Phys. Rev. B 46, 13584 (1992). Y. Wang, D. Tom´ anek, G. F. Bertsch, and R. S. Ruoff, Phys. Rev. B 47, 6711 (1993). M. Sai Baba, T. S. Lakshmi Narasimhan, R. Balasubramanian, and C. K. Mathews, Int. J. Mass Spectrom. Ion Processes 125, R1 (1993). R. L. Hettich, R. N. Compton, and R. H. Ritchie, Phys. Rev. Lett. 67, 1242 (1991). P. A. Limbach, L. Schweikhard, K. A. Cowen, M. T. McDermott, A. G. Marshall, and J. V. Coe, J. Am. Chem. Soc. 113, 6795 (1991). For certain systems, such as for example sodium clusters, an orbitally-averagedlike SIC treatment yielded highest-occupied-molecular-orbital (HOMO) energies for anions in adequate agreement with the calculated electron affinities (see Refs. 3,23). J. Cioslowski and K. Raghavachari, J. Chem. Phys. 98, 8734 (1993). A. I. Baz’, Y. B. Zel’dovich, and A. M. Perelomov, Scattering, reactions, and decay in nonrelativistic quantum mechanics, (Israel Program for Scientific Translations Ltd., Jerusalem, 1969). V. De Coulon, J. L. Martins, and F. Reuse, Phys. Rev. B 45, 13 671 (1992). U. Landman, W. D. Luedtke, N. Burnham, and R. J. Colton, Science 248, 454 (1990). U. Landman, W. D. Luedtke, B. E. Salisbury, and R. .L. Whetten, Phys. Rev. Lett. 77, 1362 (1996). U. Landman, W. D. Luedtke, and J. Gao, Langmuir 12, 4514 (1996). E. N. Bogachek, A. M. Zagoskin, and I. O. Kulik, Fiz. Nizk. Temp. 16, 1404 (1990) [Sov. J. Low Temp. Phys. 16, 796 (1990)]. J. I. Pascual, J. Mendez, J. Gomez-Herrero, J. M. Baro, N. Garcia, and V. T. Binh, Phys. Rev. Lett. 71, 1852 (1993). L. Olesen, E. Laegsgaard, I. Stensgaard, F. Besenbacher, J. Schiotz, P. Stoltze, K. W. Jacobsen, and J. N. Norskov, Phys. Rev. Lett. 72, 2251 (1994). J. I. Pascual, J. Mendez, J. Gomez-Herrero, J. M. Baro, N. Garcia, U. Landman, W. D. Luedtke, E. N. Bogachek, and H.-P. Cheng, Science 267, 1793 (1995). D. P. E. Smith, Science 269, 371 (1995). G. Rubio, N. Agrait, and S. Vieira, Phys. Rev. Lett. 76, 2302 (1996). A. Stalder and U. Durig, Appl. Phys. Lett. 68, 637 (1996). J. M. Krans, J. M. van Ruitenbeek, V. V. Fisun, I. K. Yanson, and L. J. de Jongh, Nature 375, 767 (1995). J. L. Costa-Kramer, N. Garcia, P. Garcia-Mochales, and P. A. Serena, Surface Science 342, 11144 (1995). R. N. Barnett and U. Landman, Nature 387, 788 (1997). For an axially symmetric nanowire with variable radius, see Ref. 41. A. G. Scherbakov, E. N. Bogachek, and U. Landman, Phys. Rev. B 53, 4054 (1996).

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119. E. N. Bogachek, A. G. Scherbakov, and U. Landman, Phys. Rev. B 56, 1065 (1997). 120. E. N. Bogachek, A. G. Scherbakov, and U. Landman, Phys. Rev. B 53, R13246 (1996). 121. A. Garcia-Martin, J. A. Torres, and J. J. Saenz, Phys. Rev. B 54, 13448 (1996). 122. R. A. Gherghescu, D. N. Poenaru, A. Solov’yov, and W. Greiner, Int. J. Mod. Phys. B 22, 4917 (2008). 123. W. A. Saunders, Phys. Rev. A 46, 7028 (1992). 124. Here, we consider clusters of monovalent elements (Na, K, and Cu). For polyvalent elements, N in Eq. (A.6) must be replaced by N v, where v is the valency. 125. I. S. Gradshteyn and I. M. Ryzhik, Table of integrals, series, and products, (Academic, New York, 1980) Ch. 8.11. 126. R. W. Hasse and W. D. Myers, Geometrical relationships of macroscopic nuclear physics, (Springer-Verlag, Berlin, 1988) Ch. 6.5. 127. S. G. Nilsson, K. Danske Vidensk. Selsk. Mat.-Fys. Medd. 29, No. 16 (1955). 128. K. L. Clemenger, Phys. Rev. B 32, 1359 (1985). 129. W. A. Saunders, Ph.D. dissertation, University of California, Berkeley (1986); W. A. Saunders, K. Clemenger, W. A. de Heer, and W. D. Knight, Phys. Rev. B 32, 1366 (1985). 130. K. L. Clemenger, Ph.D. dissertation, University of California, Berkeley (1985). 131. J. R. Nix, Annu. Rev. Nucl. Part. Sci. 22, 65 (1972). 132. R. K. Bhaduri and C. K. Ross, Phys. Rev. Lett. 27, 606 (1971). 133. The perturbation l2 − < l2 >n in the hamiltonian (A.15) influences the shell correcStr esp , of the single-particle spectrum, since U0 = 0 , but not the average, E tion ∆Esh for all shells with principal quantum number n higher than the minimum number required for accomodating Ne electrons (see, Ref. 43, p. 598 ff.). 134. A. Herlert, L. Schweikhard, and M. Vogel, Eur. Phys. J. D 16, 65 (2001). 135. J. Maruhn and W. Greiner, Z. Phys. 251, 431 (1972).

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Chapter 8 Finite Element Approximations in Orbital-Free Density Functional Theory Huajie Chen and Aihui Zhou LSEC, Institute of Computational Mathematics and Scientific/Engineering Computing Academy of Mathematics and Systems Science, Chinese Academy of Sciences Beijing 100190, P.R. China [email protected], [email protected] In this chapter, we give an introduction of the finite element method for orbitalfree density functional theory. In particular, we review several efficient finite element discretization approaches to associated eigenvalue problems. We also include numerical examples for illustration.

Contents 8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 Finite element discretizations for linear eigenvalue problems 8.2.1 Standard finite element discretization . . . . . . . . 8.2.2 Multi-scale based finite element discretizations . . . . 8.3 Finite element approximations of TF-HK equation . . . . . 8.3.1 SCF iteration for TF-HK equation . . . . . . . . . . 8.3.2 Adaptive finite element calculation . . . . . . . . . . 8.4 Finite element methods for direct minimization . . . . . . . 8.5 Numerical examples . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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8.1. Introduction The orbital-free (OF) density functional theory (DFT) may lead to a linear scaling method for electronic structure calculations.1–4 In the OF model, the total energy of an atomic or molecular system is expressed as a functional of electron density: E OF (ρ) = TvW (ρ) + Tc (ρ) + Ene (ρ) + Eee (ρ) + Exc (ρ),

(8.1.1)

of which the first two terms on the right hand side are the OF kinetic energy functionals, representing the von-Weizs¨acker term and the local/nonlocal correction term,1 and the last three terms are nuclei-electron attraction energy, electronelectron repulsion energy, and the exchange-correction energy, respectively. 251

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The ground state solution of the system can be obtained by minimizing total energy (8.1.1) under the normalization constraint: Z OF Min E (ρ) s.t. ρ = N, (8.1.2) R3

where N equals the number of valence electrons of the system when pseudopotential is used. The minimizer of the OF energy functional satisfies the Euler-Lagrange equation, which is also called the Thomas-Fermi-Hohenberg-Kohn (TF-HK) equation,1 of the form:  κ  OF − ∇2 + VT (ρ) + Vef (8.1.3) f (ρ) ψ(r) = µψ(r), 2 δTc (ρ) , κ is the parameter for the von-Weizs¨acker δρ OF term (and is chosen to be 1 afterwards), the effective potential Vef f is the sum of ionic pseudopotential, Hatree potential, and exchange-correction potential:


OF Vef (8.1.4) f ρ(r) = Vpseu (r) + VH ρ(r) + Vxc ρ(r) . where ψ 2 (r) = ρ(r), VT (ρ) =

Since (8.1.3) is a nonlinear eigenvalue problem, the self consistent field (SCF) iteration is usually applied in computations. Consequently, standard linear eigenvalue problems of the form   1 − ∇2 + V (r) ψ(r) = λψ(r) (8.1.5) 2

are solved repeatedly, where V (r) is the effective potential generated by the electron density from the previous steps. Various approximate methods have been successfully developed to solve minimizing problems and associated eigenvalue problems resulting from DFT. In the past decades, the methods using bases set such as plane waves (PW) and linear combination of atomic orbitals (LCAO) have been successful in a number of cases.5–8 However, there are several difficulties and limitations involved in such methods. For instance, the boundary condition does not correspond to that of an actual system; the extensive global communications in dealing with plane waves reduce the efficiency of massive parallelization, which is necessary for complex systems; and the generation of large supercell is needed for non-periodic system, which certainly increases the computational cost. In contrast, such kind of mathematical models can also be discretized numerically by real space methods, for example, finite difference (FD),6,9–15 finite element (FE),6,9,16–20 and wavelets6,18,21–26 methods, which do not involve the problems mentioned above, and have various advantages. The arbitrary boundary conditions can be easily incorporated; and the calculation including external field which breaks the translational symmetry can be easily performed; more importantly, the real-space methods do not use Fourier transforms, but produce sparse Hamiltonian

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matrices which are necessary for parallel implementation. The FE method is a typical real space method, using local piecewise polynomials bases functions. Although it uses much more degrees of freedoms than that of the traditional PW method, the strictly local basis functions produce well structured sparse matrices, which may yield O(N ) calculations. In addition, it is relatively straightforward for implementing adaptive refinement techniques to describe regions around nuclei or chemical bonds where the electron density varies rapidly, while treating the other zones with a coarser description. Therefore, the computational accuracy and efficiency can be well controlled by using adaptive meshes. There are a lot of works of using FE methods in electronic structure calculations.6,9,17,18,27–30 For instance, White et al.31 found that with uniform meshes as many as 105 bases functions per atom were required to achieve sufficient accuracy. To decrease the number of basis functions, Tsuchida and Tsukada32 had applied nonuniform hexahedron meshes for molecule H2 calculations. Beck9,33 studied the multigrid method based on such nonuniform hexahedron meshes. In these applications the grid can be made to vary logarithmically near the nuclei, but the smoothness of the wavefunction is not guaranteed for the nonconforming mesh. Afterward, Tsuchida and Tsukada34,35 proposed another approach with the adaptive curvilinear coordinates, which was recently applied for the calculations of ab initio molecular dynamics. All the above adaptive coordinates are based on hexahedron meshes. Zhang et al.36,37 proposed a parallel adaptive refinement approach to generate conforming tetrahedra meshes, which can be locally refined very flexible in any interested regions. In this chapter, we will give some associated fundamental knowledge of FE methods, show several efficient FE methods for solving eigenvalue problem (8.1.5), and apply the FE methods to electronic structure calculations under the framework of OF-DFT. 8.2. Finite element discretizations for linear eigenvalue problems In this section, we shall review the standard FE meth and some highly efficient multi-scale based FE methods for solving (8.1.5). Physically, the problem is posed in R3 , but in computation (8.1.3) and (8.1.5) are set in some bounded domain Ω, for example, a supercell for crystal or a large enough cuboid for finite system, which is reasonable since the solution of (8.1.3) usually decays exponentially due to the external potential.38–40 Throughout in this paper, for nonnegative integer s, we denote by H s (Ω) the standard Sobolev space of functions having square integrable derivatives of order up to s, and k · ks,Ω the associated norms,41 H01 (Ω) = {v ∈ H 1 (Ω) : v = 0 on ∂Ω}.

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8.2.1. Standard finite element discretization In FE calculation, Ω is divided into several regions called elements, in three dimensions, hexahedrons or tetrahedrons, which generate a FE mesh Th of Ω and a set of real space interpolation nodes. The parameter h, which tends to zero, is some measure of the size of elements in Th . In Fig. 8.2.1, we sketch a 3-dimensional mesh which is divided into several tetrahedra, and there are 10 interpolation nodes on each tetrahedron when quadratic basis functions are employed. The piecewise polynomial basis functions {ϕi } can be typically chosen in such a way that ϕi is 1 on the ith node while 0 on all the other nodes.

→

→

Fig. 8.2.1. Left: Partition of the domain. Middle: A cube consisting of 5 tetrahedra. Right: In the case of quadratic basis functions, the FE interpolation nodes of one tetrahedra occupy the vertices and midpoints of edges.

Associated with mesh Th , define a finite dimensional space S h (Ω) that is spanned by basis functions {ϕi }. Namely, any function ψ ∈ S h (Ω) can be expanded with bases {ϕi } ψ(r) =

n X

ui ϕi ,

(8.2.1)

i

where ui is just the value of the function itself on each node, and n is the number of degrees of freedom. Let S0h (Ω) = S h (Ω) ∩ H01 (Ω). We assume that {S h (Ω)} satisfy the following approximation properties as h tends to zero: inf

eh ∈S h (Ω) ψ

inf

eh ∈S h (Ω) ψ

kψ − ψeh k1,Ω → 0 as h → 0

kψ − ψeh k1,Ω ≤ Chm kψkm+1,Ω

∀ ψ ∈ H 1 (Ω),

(8.2.2)

∀ ψ ∈ H m+1 (Ω)

(8.2.3)

for some integer m ≥ 1. We may consult Ciarlet41 for conditions on FE meshes such that (8.2.2) and (8.2.3) are satisfied. To find an approximate solution, we discretize equation (8.1.5) within subspace S0h (Ω). The FE solution is then defined by: Find (λh , ψh ) ∈ R × S0h (Ω) satisfying

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Z

Ω

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ψh2 = N and 1 2

Z

Ω

∇ψh ∇ϕ +

Z

ψh V ϕ = λh

Ω

Z

Ω

ψh ϕ ∀ ϕ ∈ S0h (Ω).

Since (8.2.4) is equivalent to Z Z Z 1 ∇ψh ∇ϕi + ψh V ϕi = λh ψh ϕi 2 Ω Ω Ω

i = 1, 2, · · · , n,

(8.2.4)

(8.2.5)

eigenvalue problem (8.1.5) is reduced to a generalized matrix eigenvalue problem, determining the approximate lowest eigenvalue and the corresponding eigenfunction: Hφ = λh Sφ,

(8.2.6)

where H = (Hij )n×n is the discrete Hamilton matrix with its elements given by Z Z 1 (8.2.7) ∇ϕi ∇ϕj + ϕi V ϕj , Hij = 2 Ω Ω S = (Sij )n×n is the mass matrix with Sij =

Z

ϕi ϕj ,

(8.2.8)

Ω

and the vector φ represents the approximate solution by φ = (u1 , u2 , . . . , un )T

(8.2.9)

with ui the value of the associated eigenfunction on the ith node. This involves a generalized algebraic eigenvalue problem with large sparse matrix. Typical algebraic techniques such as PCG, Lanczos42,43 can be utilized to solve the large scale linear eigenvalue problem efficiently. It has been shown that (8.2.4) generates a sequence of convergent approximate solutions (as h → 0) under the minimal regularity condition.44 Moreover, optimal error estimates of FE eigenvalue λh and eigenfunction ψh corresponding to S0h (Ω) with piecewise kth order polynomial bases have been derived under certain conditions44–46 : |λh − λ| ≤ Ch2k ,

(8.2.10)

kψ − ψh k0,Ω + hkψ − ψh k1,Ω ≤ Chk+1 .

(8.2.11)

8.2.2. Multi-scale based finite element discretizations Although there are various advantages of FE methods for electronic structure calculations, the computational cost of the standard FE discretization is still limited by the large number of basis functions required to adequately describe the solution near nuclei, where the electron density can have cusps and oscillate rapidly.9,17,31,32

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Hence it is significant to design some efficient FE scheme for (8.1.5) when V is singular or vary rapidly. A basic two-scale FE discretization scheme and a combination based two-scale FE discretization scheme are shown in this subsection to reduce the computational cost. The approximations of the lowest eigenvalue λ and the corresponding eigenfunction u of (8.1.5) are considered, and the bilinear form a(u, v) is defined by Z Z 1 a(u, v) := ∇u∇v + V uv 2 Ω Ω for simplicity of notation afterwards. The basic two-scale FE discretization scheme has been studied for eigenvalue problems with smooth and non-smooth coefficients,47–53 and can be applied to electronic structure calculations.27,29,54–58 With the two-scale scheme, the solution of a linear eigenvalue problem with singular coefficients on a fine grid is reduced to the solution of an eigenvalue problem with singular coefficients on a much coarser gird and a solution of linear algebraic system associated with the Poisson equation on a fine grid. Let H  h and assume that S0H (Ω) ⊂ S0h (Ω), the two-scale discretization scheme for (8.1.3) is constructed as follows: Algorithm 8.2.1. Basic two-scale discretization scheme I R 1. Find (λH , ψH ) ∈ R × S0H (Ω) such that Ω |ψH |2 = N and Z a(ψH , v) = λH ψH v ∀ v ∈ S0H (Ω). Ω

h

2. Find ψ ∈

S0h (Ω) 1 2

Z

Ω

satisfying ∇ψ h ∇v = λH

Z

Ω

ψH v −

Z

V ψH v

Ω

∀ v ∈ S0h (Ω).

3. Compute the Rayleigh quotient: a(ψ h , ψ h ) λh = R h h . Ωψ ψ

For this two-scale scheme, the resulting approximation still maintains an optimal accuracy. Indeed, if λH is the FE eigenvalue satisfying |λH − λ| ≤ CH 2

(8.2.12)

when the piecewise linear bases are used, and (λh , ψ h ) is obtained from Algorithm 8.2.1, then55 |λ − λh | + hkψ − ψ h k1,Ω ≤ Ch2

(8.2.13)

holds when H = O(h1/2 ). In the above discretization scheme, it is noted that the singular eigenvalue problem is solved only on a relatively coarse grid and hence it would be significant

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in electronic structure computations. We may also obtain similar results for the following scheme:54–56 Algorithm 8.2.2. Basic two-scale discretization scheme II R 1. Find (λH , ψH ) ∈ R × S0H (Ω) such that Ω |ψH |2 = N and Z a(ψH , v) = λH ψH v ∀ v ∈ S0H (Ω). Ω

2. Find ψ h ∈ S0h (Ω) satisfying h

a(ψ , v) = λH

Z

Ω

ψH v

∀ v ∈ S0h (Ω).

3. Compute the Rayleigh quotient: a(ψ h , ψ h ) λh = R h h . Ωψ ψ

Furthermore, we are also able to design local and parallel FE algorithms for solving the TF-HK equations based on the basic two-scale discretizations 27,29,51,53 Note that a significant advantage of the FE method lies in its ability to place local refinements in regions where the electron density varies very rapidly near the nucleus. The main idea is that the more global component of a FE eigenfunction may be obtained in a relatively coarse FE space, and the rest of the computation can then be localized using a locally fine FE discretization (see, Fig. 8.2.2). This scheme may be viewed as a postprocessing of the standard FE discretization when the eigenfunctions are known to be oscillating in some local domain, for example, the region near the nuclei or chemical bonds.

Fig. 8.2.2.

Two-scale localization of the domain.

For eigenvalue problems having several isolated singular points, such as the TF-HK equation for a system that contains more than two atoms, we can further improve the efficiency by parallel versions of the algorithms designed above. Let m represent the number of isolated singular points or nuclei of the system. For the jth

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point, we choose a subdomain Ωj which contains it and satisfies that Ωi ∩ Ωj = ∅ if i 6= j (i, j = 1, 2, . . . , m) (see, Fig. 8.2.3). With the similar idea, an eigenvalue problem is solved on a global coarse grid, and boundary value problems are solved on locally refined subdomains Ωj (j = 1, 2, . . . , m) in parallel.

Ω1 Ω

Ω2

Ω4

Ω3

Fig. 8.2.3.

Local subdomains.

Note that it is not necessary that the union of the subdomains equals the whole domain ∪m j=1 Ωj = Ω, hence the localization and parallelization based two-scale algorithms are different from that in Algorithms 8.2.1 and 8.2.2. The local computational subdomains Ωj (j = 1, 2, . . . , m) may be much smaller, and the computational cost can be much reduced. Except for the basic two-scale discretization with its localization and parallelization version, we may apply a combination based two-scale FE discretization to reduce the computational complexity further over tensor product domains.50,57,58 Let S0h1 ,h2 ,h3 (Ω) ⊂ H01 (Ω) be the standard FE space associated with the FE mesh Th1 ,h2 ,h3 with mesh size h1 in x-direction, h2 in y-direction and h3 in z-direction, respectively. The main idea of the combination based two-scale FE method is to use a coarse grid TH,H,H to approximate low frequencies and combine univariate fine and coarse grids Th,H,H , TH,h,H , and TH,H,h to handle high frequencies in parallel (see, Fig. 8.2.4). One prototype scheme to discretize (8.1.5) is as follows: Algorithm 8.2.3. Combination based two-scale discretization 1. Solve (8.1.5) coarse grid: Find (λH,H,H , ψH,H,H ) ∈ R× S0H,H,H (Ω) R on a globally such that Ω |ψH,H,H |2 = N and a(ψH,H,H , v) = λH,H,H

Z

Ω

ψH,H,H v

∀v ∈ S0H,H,H (Ω).

2. Solve (8.1.5) on some partially fine grids in parallel:

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  +





259

Q s Q

A AU

Fig. 8.2.4. The solution on fine grid Th,h,h is asymptotically same as a combination of solutions on three univariate fine and coarse grids Th,H,H , TH,h,H , TH,H,h and a coarse grid TH,H,H .

R Find (λh,H,H , ψh,H,H ) ∈ R × S0h,H,H (Ω) such that Ω |ψh,H,H |2 = N and Z a(ψh,H,H , v) = λh,H,H ψh,H,H v ∀v ∈ S0h,H,H (Ω); Ω

Find (λH,h,H , ψH,h,H ) ∈ R ×

S0H,h,H (Ω)

a(ψH,h,H , v) = λH,h,H

Z

Ω

such that

ψH,h,H v

R

Ω

|ψH,h,H |2 = N and

∀v ∈ S0H,h,H (Ω);

R Find (λH,H,h , ψH,H,h ) ∈ R × S0H,H,h (Ω) such that Ω |ψH,H,h |2 = N and Z a(ψH,H,h , v) = λH,H,h ψH,H,h v ∀v ∈ S0H,H,h (Ω). Ω

3. Set

h ψH,H,H = ψh,H,H + ψH,h,H + ψH,H,h − 2ψH,H,H ,

λhH,H,H = λh,H,H + λH,h,H + λH,H,h − 2λH,H,H . The complexity of the standard FE solution uh,h,h is O(h−3 ) in three dimensional cases. With the same approximation accuracy, the degrees of freedom for getting the combination based two-scale FE approximation uhH,H,H is only of complexity O(h−2 ) when H = O(h1/2 ) is chosen for the corresponding univariate fine and coarse grids. This approach turns out to be advantageous in two respects. First, the possibility of using existing codes allows the straightforward application of the combination based two-scale discretization to large scale problems. Second, since the different subproblems can be solved fully in parallel, there is a very elegant and efficient inherent coarse-grain parallelism that makes the combination based two-scale discretization perfectly suitable for modern high-performance computers.

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8.3. Finite element approximations of TF-HK equation In this section, we shall apply the FE discretization methods to solve the nonlinear OF problem. 8.3.1. SCF iteration for TF-HK equation To obtain the ground state solution of OF problem, TF-HK equation (8.1.3) is solved, and the SCF iteration is used to deal with the nonlinearity, which can be sketched as follows: Algorithm 8.3.1. SCF iteration 1. Construct an initial electron density ρin . OF in 2. Compute VT (ρin ) and the effective potential Vef f (ρ ). 3. Solve linear eigenvalue problem   1 OF in − ∇2 + VT (ρin ) + Vef (ρ ) ψ(r) = λψ(r), f 2

(8.3.1)

and generate new electron density ρout . 4. If ρout is sufficiently close to ρin , then the self consistency has been reached; otherwise, a new ρin is constructed and go back to step 2. The self consistent solution requires repeatedly solving linear eigenvalue problem (8.3.1) until the self consistency is reached. The FE methods and various techniques described in the previous sections can be applied to solve (8.3.1) accurately and efficiently. In practice, a mixture of density is used to generate the new Hamiltonian, which is called charge mixing, and a naive implementation is not numerical stable with the so-called charge-sloshing effect.7 Furthermore, charge mixing is often combined with the use of the direct inversion of iterative subspace (DIIS) algorithm to accelerate the convergence of SCF iteration. δTx (ρ) It is noted that the kinetic energy functional term VT (ρ) = in OF probδρ lems should be paid special attention to when using FE discretizations. In the OF framework, the kinetic energy should be a functional of the electron density alone, free of orbitals, which always involves nonlocal correction terms, i.e., a family of convolution integrals are developed for kinetic energy functionals1,59–61 based on the linear response behavior1,62 of homogeneous electron gas, producing reasonable behaviors for metals with weak pseudopotential such as aluminum and sodium crystals. However, the efficiency of the FE method in the context of large-scale calculations derives from the strict locality of FE bases in real space, which is not efficiently applicable to such nonlocal terms. To evaluate these integrals, a real-space approach is proposed,59,63 which enables the FE representation for these nonlocal terms. Meanwhile, various formulations for the kinetic energy functionals without

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nonlocal terms were also developed in the past decades,64–69 including TT F + λTvW which was believed to be an appropriate functional for rapidly varying densities; F (N )TT F + TvW where N is the number of electrons and F (N ) is the N -dependent factor; and the most recently proposed hybridized functional aTT F + bTvW . These local kinetic energy functionals are well employed in OF computations by using FE discretizations. Practically, the FE approximation of the self consistent solution for ZTF-HK equation can be obtained as follows: Find (µh , ψh ) ∈ R × S0h (Ω) satisfying

N and

1 2

Z

Ω

∇ψh ∇ϕ +

Z

Ω

ψh V (ψh2 )ϕ = µh ψh

∀ ϕ ∈ S0h (Ω)

Ω

ψh2 =

(8.3.2)

and use the self consistent field iteration to solve discretization (8.3.2), where V (ρ) = OF 63,70–72 VT (ρ) + Vef that in some simplified cases of OF models, the f (ρ). It is shown above discrete approaches generate a sequence of convergent approximate solutions as h → 0. Moreover, some upper error bounds are also obtained71,72 even though there is not any optimal order of error estimate yet. 8.3.2. Adaptive finite element calculation In general, the approximation accuracy depends on the FE mesh density. And the accuracy of the numerical approximation in electronic structure calculations is usually deteriorated by local vibration of the electron density or the potentials. To obtain desired accurate approximations, we have to apply more grids to cover the regions around the atomic nuclei and between atoms of chemical bonds than the distant regions from atoms. Fortunately, this can be realized by the adaptive finite element computation, which has been extensively investigated.73–85 The adaptive algorithm for OF problems is sketched as follows: Algorithm 8.3.2. Adaptive FE algorithm for solving OF problems 1. Generate an initial mesh. 2. Minimize the total energy or solve the TF-HK equation to obtain the electron density on the current mesh. 3. Estimate the local error indicators. 4. If the global error estimator is not small enough, then refine the mesh by some strategy and go back to step 2. The basic idea of the algorithm is based upon the iteration of the following steps: Solve → Estimate → Mark → Refine which generate a sequence of approximations to the exact solution. We shall not discuss the step “Solve” here, which has been investigated in the previous section. The steps “Estimate” and “Mark” involve an a posteriori estima-

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tor, which is employed to guarantee the quality of the computed solutions and to design an adaptive mesh refinement scheme to improve their accuracy. Hence, the a posteriori estimator is an essential part of the adaptive FE computation. Note that each FE function ψh is associated with a corresponding FE mesh Th . For eigenvalue problem (8.1.5), which will be solved repeatedly based on the SCF iteration, there are generally three types of a posteriori estimators: 1. Residual type a posteriori error estimators. In the context of adaptive FE eigenvalue computations, a posteriori error estimates in terms of residuals have been investigated extensively.76,86–88 For each element τ ∈ Th with mesh size hτ , define the element residual Rτ (ψh ) and the jump residual Js (ψh ) as follows: Rτ (ψh ) := λh ψh + ∇ · (A∇ψh ) − V ψh , Js (ψh ) := −A∇ψh+ · ν + − A∇ψh− · ν − ,

where A = diag {1/2, 1/2, 1/2}, s is the common side of elements τ + and τ − with unit outward normals ν + and ν − , respectively. Then the local error indicator is defined by X ηh2 (ψh , τ ) = h2τ kRτ (ψh )k20,τ + hτ kJs (ψh )k20,s . (8.3.3) s∈∂τ

2. Average type a posteriori error estimators. This type of estimators, which is constructed by a simple postprocessing, has been proved by theory and practice to be efficient and reliable for solving eigenvalue problems29,56,89 not only for structured but also for irregular meshes. Define the local averaging operator Gh : S0h (Ω) → S h (Ω) × S h (Ω) by85,90–92 X (A∇v)z φz , ∀ v ∈ S0h (Ω), Gh v = (8.3.4) z∈∂ 2 Th

where (A∇v)z =

Jz X j=1

αjz (A∇v)τzj , and

Jz [

j=1

τzj =

[

τ,

z∈τ,τ ∈Th

Then the local error estimators ηh are formulated by

ηh (ψh , τ ) = kA−1/2 (Gh ψh − A∇ψh )k20,τ .

Jz X j=1

αjz = 1, αjz ≥ 0. (8.3.5)

3. Gradient of electron density type a posteriori error estimators. Although both of the above two error estimators are reasonable and have been proved to be efficient for eigenvalue problems, a more convenient and reflexible a posteriori error estimator can be used for electronic structure calculations. Note that the electron density is crucial in the whole computation, and a good FE mesh should be discretized according to the change of electron density. Therefore, a new a posteriori error estimator can be used based on the gradient of the electron density:27 ηh (ψh , τ ) = hτ k∇ρk0,τ .

(8.3.6)

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Although there is no any theoretical analysis for this kind of error indicators, it makes more physical sense. Moreover, (8.3.6) is more convenient to be implemented and a lot of computational cost and storage can be saved in practice. For minimization problem (8.1.2), the third type of the a posteriori error estimators and others27,79 can be used. Given the formulation of the local a posteriori error estimator ηh (ψh , τ ) for each element τ ∈ Th , we can define the global error estimator ηh (ψh , Ω) by ηh (ψh , Ω) =

X

τ ∈Th

!1/2

ηh2 (ψh , τ )

,

(8.3.7)

which is a criterion for convergence. The loop in Algorithm 8.3.2 will stop when the global error indicator (8.3.7) is smaller than some given tolerance. Then we shall address step 4 of Algorithm 8.3.2 in detail, which is consist of “Mark” strategy and “Refine” procedure. The marking strategy, which selects the elements in Th with large error indicators ηh (ψh , τ ), is used to enforce error reduction and can be defined as follows: Algorithm 8.3.3. Marking strategy Given a parameter 0 < θ < 1: 1. Construct a minimal subset Tˆh of Th by selecting some elements in Th such that X ηh2 (ψh , τ ) ≥ θηh2 (ψh , Ω). (8.3.8) τ ∈Tˆh

2. Mark all the elements in Tˆh . This marking strategy ensures that we choose sufficiently many elements {τ } such that their contributions {ηh (ψh , τ )} constitute a fixed proportion of the global error estimator ηh (ψh , Ω). Then the marking elements are refined to force the error reduction. For the step “Refine”, the mesh refinement approach is very important. The refinement algorithms of simplicial meshes have been applied in lots of work.78,93–97 Since the assignment of the FE nodes is applicable for interpolation during the calculations, we can use the simplicial bisection algorithm based on the newest vertex approach.78 The basic step in this refinement algorithm is the tetrahedral bisection, as shown in Fig. 8.3.1. In this algorithm, with the data structure named “marked tetrahedron”, the tetrahedra are classified into 5 types and the selection of refinement edge depends only on the type and the ordering of vertices for the tetrahedra.78 After generating a fine mesh, the initial electron density over the new mesh is determined by the interpolation from results on coarse mesh points. Therefore the SCF iteration can be carried out consecutively during the mesh refinement.

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⇒

Fig. 8.3.1.

A tetrahedron refinement using bisection.

Under certain marking strategies and refinement schemes, both convergence and optimal complexity of the adaptive FE algorithm are obtained for a closs of linear partial differential equations, not only boundary value problems but also eigenvalue problems.82,98–101 For OF problems, there is no theoretical analysis concerning the adaptive FE method yet. Anyway, it is shown numerically that adaptive FE methods are highly efficient for electronic structure calculations.27,29,36,37,56 Here we provide a self-adaptive mesh for molecule lithium-hydrogen which is generated by an iteration of the mesh refinement as shown in Fig. 8.3.2. With the a posteriori error estimation, the refinement is carried out, automatically paying special attention to the regions where the computed functions vary rapidly, especially near the nuclei. The shape regularity for tetrahedral meshes is preserved during the refinement process. It is shown that there are more dense nodes near the atoms and bonding areas, and much coarser for the regions distant from the nuclei. In this way, the computational accuracy can be controlled with very high efficiency and much computational cost is reduced.

⇒

Fig. 8.3.2. A slice of FE meshes for computation of molecular LiH. Left: The initio mesh. Right: A mesh generated by adaptive refinements.

The adaptive FE method can also be implemented on distributed parallel computation,27,29,36,37 which is a promising method for OF computation of large systems.

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We can further combine the adaptive FE method and the multi-scale discretization technique described in Section 8.2.2. The basic two-scale methods are believed to be a powerful technique in obtaining accurate and efficient approximations for large scale quantum eigenvalue problems. However, it is expensive to solve 3dimensional singular problems by using uniform grids when accurate approximate solutions are required. Thus, in real computations, we need to employ adaptive grids instead of uniform grids. The globally coarse grids that we shall use are some adaptive FE grids, which are constructed from the algorithms described above; while the fine grids are obtained from the globally coarse grids directly by using some tetrahedral bisection strategy. Combining the SCF iteration and the multiscale methods together, we obtain a complete scheme for solving TF-HK equation as shown in Fig. 8.3.3. Two nested iterations, the outer iteration and inner iteration are involved. The outer iteration is the mesh refinement for the accuracy required, and the inner loop deals with the SCF solutions for the nonlinear equation. The multi-scale corrections are added to raise the efficiency and accuracy of the FE scheme. 8.4. Finite element methods for direct minimization Several FE techniques have been presented for solving TF-HK equation in the previous sections. Although the basic SCF iteration works well for some problems, there is no theoretical guarantee that these SCF techniques will always work. In fact, there are cases in which these techniques fail. The difficulties encountered by the SCF algorithm can be attributed to the discontinuous changes in the Hamilonian from iteration to iteration. In the view of the optimization procedure, the SCF iteration is an optimization procedure that minimizes the total energy indirectly by minimizing a sequence of quadratic surrogate functions.102,103 But these instabilities would not be encountered if the total energy functional was minimized directly since it normally has a single well-defined energy minimum. A direct search for this energy minimum will not lead to instabilities in the evolution of the electronic configuration. The discrete FE approximation to the ground state solution of (8.1.2) can be formulated as follows: Z Min E OF (ψh2 ) s.t. ψh ∈ S0h (Ω) and ψh2 = N, (8.4.1) Ω

where the quasi-orbital ψ satisfying ψ 2 = ρ is chosen to be the basic variable. The energy minimization is thus reduced to a discrete problem, and the FE solution ψh can be represented by vector (u1 , u2 , . . . , un )T , where ui is the value of the function on the ith node. To minimize the total energy with respect to ψh , we present a conjugate-gradient optimization method here:4,7,104

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Create an initial mesh and electronic density ρ - ? Discretize the TF-HK equation and generate the algebraic problem Ax = λBx ? Solve the eigenvalue problem and obtain the new density ρout H ? HH   Self-consistent?H H No - Compute the  Refine the mesh H  HH self adaptively new density  HH 6 Yes ? Calculate a posteriori error estimator ? H  HH  No   H H convergent? H  H  HH   H Yes ? Multi-scale corrections ? Output: charge density, energy, forces, stress, . . .

Fig. 8.3.3.

The FE method for solving TF-HK equations.

Algorithm 8.4.1. CG algorithm for energy minimization 1. Initialize the electron density; 2. Calculate the steepest descent vector, and then construct the CG vector; 3. Update the optimization variable by moving along the CG vector direction for a certain distance that is determined either by an exact line search or by approximations, while the normalization constraint is imposed in each iteration for optimization. 4. If not convergent, generate a new density and go back to step 2. At each iteration, the initial vector is multiplied by the matrix S −1 H when calculating the steepest descent vector, where the discrete Hamilton matrix H and the mass matrix S have been given by (8.2.7) and (8.2.8). We will not discuss the de-

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tails of the CG formulations or line search technique here, which have been studied in many other works.105 Note that there are many other efficient numerical ways for minimization problems with constraints, such as Truncated-Newton methods and reliable region methods.106–108 Anyway, the discrete matrix H and S are necessarily involved in the minimizing procedure, which are calculated in the same way by the FE discretization. The multi-scale discretization schemes and the adaptive refinement techniques can be easily generalized to the direct minimization methods when solving (8.1.2) by the FE discretization. However, the convergence analysis and the error estimates are still open. 8.5. Numerical examples To show the efficiency of the FE discretization for the OF model, we provide some simulations of aluminum clusters. In practice, the computational domain is chosen to be a large cuboid and the zero boundary condition is posed. The GoodwinNeeds-Heine (GNH) pseudopotential109 is used for external potential, and the local density approximation (LDA)110 is adopted as the exchange-correction potential. The numerical experiments are carried out by a powerful parallel adaptive finite element toolbox PHG (Parallel Hierarchical Grid) on LSSC-II machine with processors up to 16. We model 172 aluminum atoms in the face-center-cubic (FCC) lattice with 3 × 3 × 3 unit cells as shown in the left hand side of of Fig. 8.5.1. The total energy of the system is directly minimized, and the gradient of electron density type a posteriori error estimator is used in the adaptive loop. The mesh generated by the adaptive FE method is presented in the right hand side of Fig. 8.5.1. As shown in the picture, the dark region shows that more refined meshes (nodes) are presented in the area where the nuclei are located. The energy per atom at different adaptive steps and the reduction of the error indicator with respect to the increasing number of degrees of freedom are presented in Fig. 8.5.2, which illustrate the convergence of our adaptive FE method. Acknowledgements The authors would like to thank Dr. Xiaoying Dai, Prof. Xingao Gong, Prof. Fang Liu, Prof. Lihua Shen, and Dr. Dier Zhang for their stimulating discussions and fruitful cooperations on electronic structure computations. This work was partially supported by the National Science Foundation of China under grants 10425105 and 10871198, the National Basic Research Program under grant 2005CB321704, and the National High Technology Research and Development Program of China under grant 2009AA01A134.

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Fig. 8.5.1. Left: The structure of the aluminium system with 3 × 3 × 3 FCC unit cells. Right: The inner profile of an adaptive mesh generated from the calculation of this system.

Fig. 8.5.2. Left: The convergence curve of the energy per atom of the aluminum FCC lattice. Right: Reduction of the global error indicator.

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P. Morin, R. H. Nochetto, and K. Siebert, SIAM Rev. 44, 631 (2002). R. Schneider, Y. Xu, and A. Zhou, Adv. Comput. Math. 5, 259 (2006). J. Xu and A. Zhou, Math. Comp. 69, 881 (2000). N. Yan and A. Zhou, Comput. Methods Appl. Mech. Eng. 190, 4289 (2001). M. G. Larson, SIAM J. Numer. Anal. 38, 608 (2001). R. Becker and R. Rannacher, Acta Numer. 10, 1 (2001). V. Heuveline and R. Rannacher, Adv. Comput. Math. 15, 107 (2001). D. Mao, L. Shen, and A. Zhou, Adv. Comp. Math. 25, 135 (2005). M. Ainsworth and J. T. Oden, A Posteriori Error Estimation in Finite Element Analysis. (Wiley Interscience, New York, 2000). C. Carstensen, RAIRO M2AN. 33, 1187 (1999). O. C. Zienkiewicz and J. Z. Zhu, Int. J. Numer. Methods Engrg. 33, 1331 (1992). E. Bansch,Math. Comp. 66, 691 (1997). J. M. Maubach, SIAM J. Sci. Comput. 16, 210 (1995). R. Horst, Math. Comp. 66, 691 (1997). I. Kossaczky, J. Comput. Appl. Math. 55, 275 (1994). A. Liu and B. Joe, SIAM J. Sci. Comput. 16, 1269 (1995). W. Dorfler, SIAM J. Numer. Anal. 33, 1106 (1996). K. Mekchay and R. H. Nochetto, SIAM J. Numer. Anal. 43, 1803 (2005). R. Stevenson, Found. Comput. Math. 7, 245 (2007). X. Dai, J. Xu, and A. Zhou, Numer. Math. 110, 313 (2008). C. Yang, J. C. Meza, and L. W. Wang, J. Comp. Phys. 217, 709 (2006). C. Yang, J. C. Meza, and L. W. Wang,SIAM J. Sci. Comput. 29, 1854 (2007). M. P. Teter, M. C. Payne, and D. C. Allan, Phys. Rev. B 40, 255 (1989). J. Nocedal and S. J. Wright, Numerical Optimization, Springer Series in Operations Research. (Springer-Verlag, New York, 1999). C. J. Garcia-Cervera, Commun. Comput. Phys. 2, 334 (2007). C. Yang, J. C. Meza, and L. W. Wang, J. Comp. Phys. 217, 709 (2006). C. Yang, J. C. Meza, and L. W. Wang, SIAM J. Sci. Comput. 29, 1854 (2007). L. Goodwin, R. Needs, and V. Heine, J. Phys. Condens. Mat. 26, 351 (1990). J. P. Perdew and K. Burke, Int. J. Quant. Chem. 57, 309 (1996).

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The Functional for the Non-Additive Kinetic Energy and Its Applications in Numerical Simulations

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Chapter 9 Non-Additive Kinetic Energy and Potential in Analytically Solvable Systems and Their Approximated Counterparts Tomasz A. Wesolowski1 and Andreas Savin2 1

2

Department of Physical Chemistry, University of Geneva 30, quai Ernest-Ansermet CH-1211 Geneva 4, Switzerland [email protected]

Laboratoire de Chimie Theorique, CNRS and Universite Pierre et Marie Curie (Paris VI), Paris, France [email protected] The one-electron equation for orbitals embedded in frozen electron density (Eqs. 20-21 in [Wesolowski and Warshel, J. Phys. Chem, 97 (1993) 8050]) in its exact and approximated version is solved for an analytically solvable model system. The system is used to discuss the role of the embedding potential in preventing the collapse of a variationally obtained electron density onto the nucleus in the case when the frozen density is chosen to be that of the innermost shell. The approximated potential obtained from the second-order gradient expansion for the kinetic energy prevents such a collapse almost perfectly but this results from partial compensation of flaws of its components. It is also shown that that the quality of a semi-local approximation to the kinetic-energy functional, a quantity needed in orbital-free methods, is not related to the quality of the non-additive kinetic energy potential - a key component of the effective embedding potential in one-electron equations for embedded orbitals.

Contents 9.1 Introduction . . . . . . . . . . . . . . . . . . . . 9.2 Numerical results and discussion . . . . . . . . . 9.2.1 Eq. 9.1.3 with the exact potential vtnad (~ r) 9.2.2 Eq. 9.1.3 with the approximated potentials 9.3 Conclusions . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . .

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9.1. Introduction The basic formal results of Frozen-Density Embedding Theory (FDET), which is of the greatest relevance to any multi-level computational methods, is the fact that 275

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minimization of the Hohenberg-Kohn energy1 functional (E HK [n]) in the presence of the constraint n ≥ nB , where nB is some given in advance (frozen) electron density, can be achieved by means of a suitably chosen embedding potential, i.e., a multiplicative (local) operator.2–5 Moreover, the non-electrostatic component of this potential is the bi-functional depending on nB and nA , where nA is the embedded electron density, i.e., the density which is to be optimized. According to FDET, the exact embedding potential comprises a component which is the functional derivative of the bi-functional of the non-additive kinetic δTsnad [nA ,nB ] energy ( ). This component is of the greatest relevance for the present δnA volume. In principle, any approximation for the kinetic energy functional Ts [n] used in orbital-free DFT (OF-DFT) methods can be trivially used to approximate also Tsnad [nA , nB ] (see Eq. 9.1.5 below). Our numerical experience indicates, however, that the quality of any given approximation to Ts [n] is not correlated with that of the resulting approximation for Tsnad [nA , nB ].6,7 Turning back to FDET, its formal framework provides the basis for a large variety of computational methods aiming at quantitative predictions of the effect of environment on electronic structure of embedded species (see Ref. 14 or articles reviewed in chapters 10 and 11 in the present volume, for instance). The key steps resulting in the development of FDET-based computational methods and ther practical applications include: i) Introduction of the GGA97 approximation8 for the bi-functional of the non-additive kinetic potential and energy. The GGA97 approximation for Tsnad [nA , nB ] eliminates to large extend the spurious component of the non-additive kinetic potential occurring if this potential is obtained from the gradient expansion of Ts [n] .6 ii) Development of the NDSD approximation for Tsnad [nA , nB ], which enforces one of the exact properties of the bi-functional vtnad [nA , nB ] leading to an improvement of the embedding potential near the nuclei in the environment.9 iii) Combining the ground-state FDET with the linear-response time-dependent DFT strategy for electronic excitations10,11 . iv ) The proof that the original orbital-free effective embedding potential derived for embedding a reference system of non-interacting electrons is also the optimal effective embedding potential for embedding interacting systems3 or systems described by means of the first-order density matrix.4 These developments were matched by the improvements of the numerical implementation as well as techniques to generate the frozen electron density. Currently, the ADF code12 provides the most advanced and flexible implementation of FDET based methods. This implementation was made by Wesolowski (see for instance Refs. 11) and subsequently developed by others.13 For the most recent implementations, see Refs. 14 and 13. Several refinements and further advancements in both numerical implementation as well as in the theory of embedding such as an efficient method to account for dynamic response of the environment in evaluation of electronic excited levels (see chapter 11 of this volume) or explicit corrections for erroneous behavior of the embedding potential at dissociation,15 for instance, expanded the domain of applicability of the orbital-free embedding potential. Unfortunately, similarly to

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Kohn-Sham and orbital-free DFT formulation, the computational methods based on FDET suffer from the fact that they rely on approximations for the needed density functionals and only in a few cases the analytical expressions for quantities defined in FDET are available.9,15,16 The present chapter intends to fill the gap. We analyze in detail an analytically solvable case where all quantities defined in FDET can be evaluated analytically and compared with approximated quantities needed in practical simulations. The key object in FDET is the orbital-free embedding potential which depends on the quantum mechanical descriptor used for nA (~r) in order to optimize it. In the case of embedding a non-interacting reference system, one-particle reduced density matrix, and interacting wavefunction of the full Configuration Interaction form, this potential reads (see for instance Eq. 3 in Ref. 17): Z nB (~r0 ) 0 B B emb vKSCED [nA , nB , vext ](~r) = vext (~r) + d~r + (9.1.1) |~r0 − ~r| δExc [n] δExc [n] − + δn(~r) n(~r)=nA (~r)+nB (~r) δn(~r) n(~r)=nA (~r) δTs [n] δTs [n] − , δn(~r) n(~r)=nA (~r)+nB (~r) δn(~r) n(~r)=nA (~r)

where the functionals Exc [n] and Ts [n], are defined as in Kohn-Sham formulation of Density Functional Theory.18 In the present considerations and in practical applications of Eq. 9.1.1, the functional Ts [n] is considered as a density functional defined in the Levy-Lieb constrained search formulation19,20 because the KohnSham orbitals are not available for the total electron density. Eq. 9.1.1 shows clearly that explicit quantum mechanical descriptors - such orbitals - do not have to be known for the environment. The electron- and nuclear charge densities of the environment provide sufficient information to generate the embedding potential for any given nA (~r). For this reason, we refer to the potential defined in Eq. 9.1.1 as the orbital-free embedding potential. Eq. 9.1.1 provides the basis for various computational methods for multi-level simulations, such as the ones reviewed in Chapters 10 and 11, but constructing in practice good approximations to the potential defined in Eq. 9.1.1 represents the main challenge for the FrozenDensity Embedding Theory. If the embedded system is described by means of a reference system of non-interacting electrons, nA (~r) is constructed from embedded orbitals (ϕi ): nA (~r) =

N occ X

2|ϕi (~r)|2 ,

(9.1.2)

i

which are obtained from one-electron equations for embedded orbitals (Eqs. 20-21 in Ref. 2):

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Z

nA (~r0 ) 0 d~r + vxc [nA ](~r) |~r0 − ~r|  emb B A + vKSCED [nA , nB , vext ](~r) ϕA i = i ϕi

1 A − ∇2 + vext (~r) + 2

(9.1.3)

For the sake of simplicity, equations are given for spin-compensated electron densities: hence the factor 2 in Eq. 9.1.2. The acronym KSCED in Eqs. 9.1.1 and 9.1.3 stands for the Kohn-Sham Equations with Constrained Electron Density and is used to distinguish the effective potential and the orbitals in Eq. 9.1.3 from their counterparts in the Kohn-Sham equations. B Eq. 9.1.1 shows clearly that, except for vext (~r), the position dependency of each emb B other term in vKSCED [nA , nB , vext ](~r) is determined by the position dependency of nA (~r) and nB (~r). These components of the potential are, therefore, functionals of these two densities. In particular, the kinetic-energy functional dependent component of the potential given in Eq. 9.1.1, for which the following short notation will be used: δTs [n] δTs [n] nad − (9.1.4) vt [nA , nB ] = δn(~r) δn(~r) n(~ r )=nA (~ r)+nB (~ r)

n(~ r)=nA (~ r)

is also a functional nA (~r) and nB (~r). Throughout the present work, we apply the following convention for any local quantity which is defined as a functional, bi-functional, tri-functional, etc.: the symbol, v[f ](~r) or in short-hand notation v[f ] denotes the correspondence between the quantities in square brackets (the function f in this case) and the function v(~r), whereas the symbol v(~r) denotes not a functional but a function which might be evaluated for some particular choice of the arguments in the square brackets. Following this convention, vtnad [nA , nB ](~r) or vtnad [nA , nB ] denote the bi-functional of the non-additive kinetic potential, whereas vtnad (~r) denotes the non-additive kinetic potential evaluated for some particular pair of the densities nA (~r) and nB (~r). We make this distinction, to underline that knowing the potential vtnad (~r) for a particular pair of densities nA (~r) and nB (~r) does not imply that the corresponding bi-functional vtnad [nA , nB ] is known. The potential given in Eq. 9.1.4 is not only the component of the effective potential in Eq. 9.1.3, i.e., the case where the embedded system is described by means of the reference system of non-interacting electrons. It is also the component of the exact embedding potential in cases were the embedded density nA (~r) is obtained in other methods based on variational principle, in which the electron-electron interactions are treated as: i) the expectation value of the exact Hamiltonian calculated over a class of trial wavefunctions of the form ranging from Hartree-Fock to full CI wavefunction3 or ii) a functional depending on one-particle density matrix.4 In the context of FDET, it is convenient to introduce the bi-functional of the non-additive kinetic energy: Tsnad [nA , nB ] = Ts [nA + nB ] − Ts [nA ] − Ts [nB ]

(9.1.5)

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The bi-functional Tsnad [nA , nB ] represent one of the components of the total energy of the system, whereas its functional derivative with respect to nA (~r) is the bifunctional of the potential vtnad [nA , nB ](~r) which can be expressed alternatively as: δTsnad [n, nB ] nad vt [nA , nB ] = (9.1.6) δn(~r) n(~ r)=nA (~ r)

In practical applications of Eq. 9.1.3, vtnad [nA , nB ] needs to be approximated by means of some approximation depending explicitly on the pair of electron densities nA (~r) and nB (~r) (approximations are denoted with tildes in the present work). vtnad [nA , nB ] ≈ v˜tnad [nA , nB ]

(9.1.7)

The above approximation leads to errors in the effective potential and consequently to the deviation between the electron density obtained from Eq. 9.1.3 and the target density no (~r) − nB (~r), where no denotes the ground-state density of the system, even if the difference no (~r) − nB (~r) is pure-state non-interacting v-representable. Using such density in the evaluation of the total energy leads also to errors which add to the errors due to the approximation to Tsnad [nA , nB ]. It is worthwhile to underline here that due to the fact that computational methods based on FDET hinge on approximations to the differences of the kinetic energy. Approximations for the absolute value of neither the kinetic energy (T˜s [n]) nor the kinetic potential T˜s [n] (˜ vt [n](~r) = δδn(~ r ) ) are not needed in FDET. There is no reason, therefore, to expect a similar performance of a given approximation (T˜s [n]) in both OF-DFT and in FDET frameworks. Various methodologies based on the correspondence between the pair of electron densities and the embedding potential given in Eq. 9.1.1, which use an inexpensively calculated approximation for v˜tnad [nA , nB ], were recently developed and applied in numerical simulations of condensed matter (see Refs. 5,21–23 and chapters 10 and 11 of this volume). An obvious strategy is to use some approximation for Ts [n] and to obtain the corresponding vtnad [nA , nB ] from Eq. 9.1.4. It is convenient to call such approximations for Tsnad [nA , nB ] - decomposable. Unfortunately, one cannot count on fortuitous cancellation of errors in the approximated potentials v˜t [nA +nB ] and v˜t [nA ] needed to evaluate v˜tnad [nA , nB ]. Our previous dedicated studies6,7 and recent comprehensive benchmarking studies reported in Ref. 24 indicate, however, that improvements in approximating Ts [n] do not necessarily lead to improvements in vtnad [nA , nB ]. Therefore, one should rather consider the quest for a usable approximation for each of the two potentials, vtnad [nA , nB ] and vt [n], to represent independent tasks. The following strategies to approximate vtnad [nA , nB ] are, in principle, possible: • Decomposable approximations. For any approximation T˜s [n], for which the analytical expression for its funcT˜s [n] ˜tnad [nA , nB ] can be obtional derivative ( δδn(~ r ) ) is known, the corresponding v

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tained in a straightforward manner as the difference of two analytic expressions: δ T˜s [n] δ T˜s [n] nad nad vt [nA , nB ] ≈ v˜t [nA , nB ] = − (9.1.8) δn(~r) δn(~r) n(~ r)=nA (~ r )+nB (~ r)

n(~ r )=nA (~ r)

This is the most commonly used strategy in the literature because all the quantities derivable from a given approximation for Ts [n] are numerically available. • Decomposable approximations for potential only. s [n] For a given form of the potential v˜t [n] ≈ δT δn(~ r ) , such that the corresponding expression for neither T˜s [n] nor v˜tnad [nA , nB ] is known, one can construct trivially the corresponding approximation for v˜tnad [nA , nB ].

vtnad [nA , nB ] ≈ v˜t [nA + nB ] − v˜t [nA ]

(9.1.9)

The field of applicability of such approximations is, however, limited. In the absence of the corresponding analytic expression for T˜ nad [nA , nB ], it is not possible to evaluate the total energy self-consistently with v˜tnad [nA , nB ] used in Eq. 9.1.3. It is sufficient, though, for obtaining the ground-state density of the embedded system as well as the embedded orbitals. In the present work, we consider also such approximation to vtnad [nA , nB ] which is obtained from the Chai-Weeks approximation25 for vt [n]. • Non-decomposable approximations. This is a bottom-up approach starting from an approximation for vtnad [nA , nB ] and than constructing the corresponding approximation for Tsnad [nA , nB ]: δ T˜snad [n, nB ] nad nad vt [nA , nB ] ≈ v˜t [nA , nB ] = (9.1.10) δn(~r) n(~ r )=nA (~ r)

In this approach, the parent approximation for neither T˜s [n] nor its functional derivative is constructed. The latter quantities are not directly needed in the FDET-based methods because the approximations are used only for the differences of such quantities. The NDSD approximation9 for Tsnad [nA , nB ] was constructed following this strategy. • Non-decomposable approximations for the potential only. In principle, one can construct only v˜t [nA , nB ] without constructing the corresponding approximation to T˜snad [nA , nB ]. This would make possible to obtain the embedded density and orbitals but not the total energy. To our knowledge, no such constructions, were reported in the literature.

Since the accuracy of the used approximation v˜tnad [nA , nB ] is of key importance in FDET-based methods, this issue was subject of our previous studies based on the comparisons between the electron density obtained using the analyzed approximation with the target electron density obtained if the approximation would be exact.6–8 Such strategy to judge the adequacy of a given v˜tnad [nA , nB ] does not,

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however, involve the construction of the exact vtnad (~r). Moreover, the previous analyses focused on cases where the overlap between nA (~r) and nB (~r) is small because nA (~r) and nB (~r) were associated with different molecules in weak intermolecular complexes. This excluded such important case where nB (~r) corresponds to fully occupied inner shell whereas nA (~r) comprises contributions from the valence electrons and the potential vtnad [nA , nB ] is sufficiently repulsive to stop the valence electrons to collapse on the core region. The present work is intended to fill this gap. To this end, we use an artificial system for which the exact potential vtnad (~r) can be obtained analytically for particularly chosen nB (~r).16 The considered model system comprises four electrons with spin-compensated density, it is spherically symmetric, and its exact Kohn-Sham potential reads: 1 vKS (r) = − . (9.1.11) r Note that such a system does not correspond to any real atom. It is defined by its Kohn-Sham potential and not the external potential. Nevertheless, it can be used for the analysis of various density functionals. It is important to make the distinction between such an artificial system as the one considered here and the Be atom for which the Kohn-Sham potential comprises the nuclear attraction term −4/r, the repulsive Coulomb term, as well as the exchange-correlation component. It behaves as −1/r only far from the nucleus. For the considered system, the two doubly occupied Kohn-Sham orbitals are just hydrogenic functions: φ1s and φ2s and the ground-state electron density reads:
(9.1.12) no (~r) = 2 φ21s (~r) + φ22s (~r) ,

Concerning the choice for nB (~r), we limit the analysis to cases where no (~r) − nB (~r): i) is non-negative, ii) integrates to 2, and iii) is spin-compensated. Nonnegativity of no (~r) − nB (~r) is the necessary conditions that the density no (~r) − nB (~r) can be obtained from the ground-state wavefunction for some potential vs (~r) (Chapter 12 deals with the issue of the admissibility of the frozen density in more detail.) Owing to the other conditions the potential vs (~r) can be constructed by inverting analytically the Kohn-Sham equation associated with the density no (~r) − nB (~r).16   1 − ∇2 + vs (~r) ϕ(~r) = ϕ(~r) (9.1.13) 2 In the above equation vs (r) is such potential, for which the lowest-energy solution ϕ(~r) is such that 2|ϕ|2 (~r) = no (~r) − nB (~r). This potential reads thus: p 1 ∇2 no (~r) − nB (~r) p vs (~r) = + constant , (9.1.14) 2 no (~r) − nB (~r)

where the constant can be chosen such that the potential goes to 0 when r → ∞. Comparing Eqs. 9.1.3 and 9.1.13 provides a link between the potential vs (r) which can be obtained analytically and vtnad [nA , nB ](r) - the quantity of crucial

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interest in this study. To lead to the same density, the effective potential in Eq. 9.1.3 and vs (~r) must be equal (up to a constant). From that it follows that, up to a constant, 1 vtnad (r) = vs (r) + . (9.1.15) r The above analytic construction of vtnad (~r) can be made for any density no (~r) − nB (~r) comprising two spin-compensated electrons and is not restricted to R the case considered here (spherical symmetry and nB (~r)d~r = 2). Note also that the construction of the external potential vext (~r) in a real, i.e., interacting system with the same ground state density given in Eq. 9.1.12 is not involved in the above analytical construction of the exact vtnad (r) . The choices for nB (~r) in the above system considered in this work, are made in view of practical numerical simulations. If nB (~r) is chosen to be electron density of the core electrons, the potential vtnad (r) stops the density nA (~r) derived from Eq. 9.1.3 from collapsing into the core. Such cases have been reported9,15 for simulations using instead of the exact vtnad (r) some approximations to the functional vtnad [nA , nB ]. For more properties of the exact potential vtnad [nA , nB ](r) in the considered model system, see Ref. 16. The considered system, provides a simple illustration for the role of the potential vtnad [nA , nB ] in preventing such a collapse. If the electron density of the 1s shell is chosen as nB (i.e., nB (~r) = 2ϕ21s ), neglecting vtnad [n
A , nB ] in Eq. 9.1.3 leads
to wrong total electron density: n ˜ o (~r) = 2 ϕ21s + ϕ21s 6= no (~r) = 2 ϕ21s + ϕ22s . One way to stop the collapse into the core is to use the exact pseudopotential as prescribed by Phillips-Kleinman26 expressed with the projector operators. But such an operator is not a local potential and would require other than density descriptors for the environment. In our previous work, the local potential stopping such a collapse into the core was constructed and analyzed.16 The present work, focuses on such cases, but extends the analysis by: i) obtaining numerically the lowest-energy embedded orbital derived from numerical solution of Eq. 9.1.3, ii) obtaining numerical solutions of an approximated version of Eq. 9.1.3 in which instead of the exact potential vtnad (r) its approximated counterpart - v˜tnad [nA , nB ] - is used. The following approximations of v˜tnad [nA , nB ] are considered in the present work: nad(W )

i) v˜t [nA , nB ] obtained from Eq. 9.1.8 and the von Weizs¨acker functional for Ts [n]27 , nad(T F ) ii) v˜t [nA , nB ] obtained from Eq. 9.1.8 and the Thomas-Fermi functional for Ts [n],28,29 nad(T F W ) iii) v˜t [nA , nB ] obtained from Eq. 9.1.8 and the sum of the Thomas-Fermi and the von Weizs¨ acker functionals for Ts [n], nad(GEA2) iv) v˜t [nA , nB ] obtained from Eq. 9.1.8 and the second-order gradientexpansion functional for Ts [n],30 nad(CW ) v) v˜t [nA , nB ] obtained from Eq. 9.1.9 and the Chai and Weeks approximas [n] 25 tion for δTδn . Note that there exist no corresponding functional for Ts [n] in this case.

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Each among the considered approximations for the potential v˜tnad [nA , nB ] reflects some exact properties and does not involve any fitting to experimental data. Both the Thomas-Fermi- and the von Weizs¨acker functionals are frequently considered to be possible starting points for construction of approximations to Ts [n]. The former one yields the exact Ts [n] for uniform electron gas n(~r) =const, whereas the latter is exact for one-electron- or spin-compensated two electron systems, Our previous numerical studies on the accuracy of various v˜tnad [nA , nB ] for cases where the overlap between nA (~r) and nB (~r) is small showed that the Thomas-Fermi functional leads to a useful approximation for the potential vtnad [nA , nB ].6–8,31 Using the von Weizs¨ acker functional to approximate vtnad [nA , nB ] leads usually to qualitatively wrong embedded densities,6,7 worse than the one obtained using the just the Thomas-Fermi functional for this purpose. Interestingly, adding 91 of the von Weizs¨ acker term to the one derived from the zero-order gradient expansion worsens usually the approximation to vtnad [nA , nB ].6,7 nad(T F W ) The v˜t [nA , nB ] approximation was considered in the present work in view of the fact that the presence of the full von Weizs¨acker component (not divided by 9 as it is in gradient expansion) is indispensable to satisfy one of the exact conditions for vtnad [nA , nB ]9 (given also in Eq. 9.2.18 here). The full von Weizs¨acker term is also a key ingredient for a family of approximations32 for Ts [n]. The inclusion s [n] in the present analysis is motivated by of the Chai-Weeks approximation for δTδn its feature to describe correctly the density response of the atomic electron density. 9.2. Numerical results and discussion The numerical solutions of Eq. 9.1.3 using either the exact potential vtnad (r) or its approximate counterparts, were obtained using the solver of one-dimensional Schr¨ odinger equation33 implemented into Mathematica.34 We consider two choices for nB (~r) such that it is “almost” equal to 2ϕ21s and that no (~r) − nB (~r) is nonnegative. A small difference between 2ϕ21s and the considered nB (~r) assures that vtnad (r) is continuous. As shown in our previous work,16 the potential changes smoothly with changing the amount of the admixture provided it is small. The following two choices for nB (~r) are considered: • nB (~r) ≈ 2ϕ21s with “dumped valence density” admixture,
1 nB (r) = ϕ21s + ϕ22s · e−10r C and where 1/C is the normalization factor.

(9.2.16)

• nB (~r) ≈ 2ϕ21s with “core/valence” admixture, with w = 0.001.


nB (r) = 2 (1 − w)ϕ21s + wϕ22s ,

(9.2.17)

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Eq. 9.2.17 choice for nB (~r) was used in the analytical reconstruction of vtnad (~r) for the same model system16 and is considered here for the sake of completeness and for comparisons.

Fig. 9.2.1. Partitioning the total ground-state electron radial density 4πr 2 no (~ r ) in the model system. The white area represents the frozen radial density 4πr 2 nB (~ r ) and the shaded area represents the radial density 4πr 2 (no (~ r ) − nB (~ r )). Only the densities obtained with the Eq. 9.2.16 are shown because the alternative partitioning (Eq. 9.2.17) leads to densities indistinguishable on the scale of the picture.

The two above choices for nB (~r) assure that no (~r) − nB (~r) is N -representable. To obtain the density no (~r) − nB (~r) as the lowest-energy solution of Eq. 9.1.3, this density must be non-interacting pure-state v-representable.2,5

9.2.1. Eq. 9.1.3 with the exact potential vtnad (~ r) 9.2.1.1. The potential vtnad (~r) The total effective potential in a system of non-interacting electrons, which generates no (~r) − nB (~r) as its ground-state density comprises the −1/r and vtnad (~r) components. Since −1/r diverges at r = 0 and is known, only the vtnad (~r) is shown on the figures in this section. Figure 9.2.2 shows the reconstructed potential vtnad (~r) for the two considered choices for nB (~r). The potential vtnad (~r) varies rapidly in both cases. At r = 2, i.e., where the target radial density has minimum in both considered cases, the potential has a spike. For r = 2, vtnad = 204.82 and vtnad = 115.714 for the Eq. 9.2.16 and Eq. 9.2.17 case, respectively. The presence of such a spike

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vtnad 3.0

vtnad 2.0

2.5

1.5

2.0

1.0

1.5

0.5 r -0.5

285

2

4

6

8

1.0 0.5

-1.0 (a)

0

2

4

6

8

r

(b) Fig. 9.2.2. The exact vtnad (~ r ) (thick lines): a) nB (~ r ) given in Eq. 9.2.16, (b) nB (~ r ) given in Eq. 9.2.17. The frozen radial density 4πr 2 nB (~ r ) (dotted line) and the target density 4πr 2 (no (~ r ) − nB (~ r )) (solid line) also shown.

reflects the exact condition for vtnad [nA , nB ]:9 1 |∇nB |2 1 ∇2 nB vtnad [nA , nB ] nA −→0,R nB (~r)d~r=2 = − 8 n2B 4 nB

(9.2.18)

At these conditions, the exact potential vtnad [nA , nB ] is given by the functional derivative of the von acker functional evaluated at the density nB . We notice R Weizs¨ that the constraint nB (~r)d~r = 2 applies everywhere in the considered partitioning. The limit, nA −→ 0 occurs, however, in three regions: at r = 0, 2, or ∞. At r = 2, nA (~r)/nB (~r) → 0 for either choices for nB (~r). As a consequence of principally exponential behavior of nB (~r), Eq. 9.2.18 leads to vtnad [nA , nB ], which is strongly positive at r = 2, for either choices of nB (~r). At r = 0, however, the two choices for nB (~r) lead to qualitatively different behavior of vtnad [nA , nB ]. With the Eq. 9.2.16 choice for nB (~r), nA (~r)/nB (~r) → 0.0007 at r → 0 leading to a spike at r = 0 for similar reasons as at r = 2. With the Eq. 9.2.17 choice for nB (~r), nA (~r)/nB (~r) → 0.125 at r → 0. It is worthwhile to underline the the fact that, the potential vtnad (~r) is not the same in the case of nB (~r) chosen as in either Eq. 9.2.16 or 9.2.17 although the target densities are very similar in both cases (indistinguishable on the scale used in Figure 9.2.1). 9.2.1.2. Embedded orbital and embedded density Figure 9.2.3 shows the radial density obtained from the ground-state solution of Eq. 9.1.3 for nB (~r) given in Eq. 9.2.16 and the exact potential vtnad (~r) given in Eqs. 9.1.14 and 9.1.15. The density nA (r), which is derived from numerical solution of Eq. 9.1.3, matches perfectly the target density no (r) − nB (r) which is obtained analytically.

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no -nB ,n A 0.4 0.3 0.2 0.1

0

2

4

6

8

10

12

14

r

Fig. 9.2.3. Radial density 4πr 2 nA (~ r ) obtained from Eq. 9.1.3 with the exact vt (~ r ) (solid line) and the target radial electron density 4πr 2 (no (~ r ) − nB (~ r )) (dashed line). Only the densities obtained with the Eq. 9.2.16 are shown. The alternative partitioning (Eq. 9.2.17) leads to densities indistinguishable on the scale of the picture.

The minimum of the radial density 4πr2 (no (~r) − nB (~r)) at r = 2 is not zero (no (2) − nB (2) = 0.0003310 for nB (r) given in Eq. 9.2.16). It can be made vanishingly small with the exponential dumping factor in Eq. 9.2.16. This value is very accurately reproduced by the numerical solver of Eq. 9.1.3 for the exact potential (nA (2) = 0.0003317). For nB (~r) chosen as in Eq. 9.2.17, the behavior of the embedded density at r = 2 is very similar to the previously analyzed one. The radial density almost reaches zero for the chosen w = 0.001 and can be made vanishingly small by decreasing w which correspond to the increase of the hight of the maximum in vtnad (r).16 The corresponding numerical values are: no (2) − nB (2) = 0.0005861 and nA (2) = 0.0005864. This result shows robustness of the used numerical procedure to solve Eq. 9.1.3 (see also the discussion of the overlaps in the subsequent parts of the present work). The ground-state embedded orbitals obtained from Eq. 9.1.3 for the two considered choices for nB (~r) are shown on Figure 9.2.4. In each case, the orbital is nodeless with a maximum at about r = 5, a minimum at r = 2, and the secondary maximum near the nucleus r. The embedded orbital closely orbitals resembles |ϕ2s |. Compared to ϕ2s , the switch of the sign from negative to positive at r = 2 is the consequence of the barrier in vtnad (~r) which makes the embedded orbital nodeless. Most of the electron density is localized in the valence shell, i.e., at r > 2. It is worthwhile to underline that the considered choices for nB (~r) do not correspond to freezing the density 2ϕ21s because of the small admixture of the valence density in nB (~r). Our previous analyses of the same system16 show that vtnad (r) changes smoothly with decreasing amount of the admixture, i.e., if nB (~r) approaches

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Ψ

Ψ

0.4

0.4

0.2

0.2 r 2

4

6

8

10

12

r

14

2

-0.2

-0.2

-0.4

-0.4 (a)

4

6

8

10

12

14

(b)

Fig. 9.2.4. The radial orbital obtained from Eq. 9.1.3 with the exact vt (r): (a) 4πr 2 nB (~ r ) given in Eq. 9.2.16 (b) 4πr 2 nB (~ r ) given in Eq. 9.2.17. The radial hydrogenic ϕ2s function is shown for comparison (dotted line).

2ϕ21s . As the consequence, the embedded wavefunction obtained from Eq. 9.1.3 does not approach ϕ2s but rather |ϕ2s | as nB (~r) → 2ϕ21s (see Figure 9.2.4). The embedded wavefunction cannot be, therefore, orthogonal to the orbital used to construct nB (~r). This behavior of the embedded wavefunction illustrates the relation between the orbital-free embedding potential given in Eq. 9.1.1 and the exact pseudopotential as defined by Phillips and Kleinman26 frozen-density admissibility if the core orbitals would be chosen to be frozen. The exact Phillips-Kleinman pseudopotential and the orbital-free effective embedding potential achieve the same target density in a different manner. In the former case, the target density corresponds to the ground-state of some system for which the external potential (vext (~r)) was modified. In the latter one, the target density corresponds to the excited state of the non-modified system. The excited state becomes a ground state owing to projecting out the ground-state solution which involves non-local operators. In the Phillips-Kleinman pseudopotential case, the exact valence orbital has a node as it must be orthogonal to ϕ1s . The lowest-energy embedded orbital approaches a nodeless function if nB (~r) → 2ϕ21s and cannot be, therefore, orthogonal to ϕ1s . 9.2.1.3. The non-additive kinetic energy In the considered systems, the numerical value of Tsnad [no − nB , nB ] = Ts [no ] − Ts [no − nB ] − Ts [nB ] is available in a form of an analytic expression. The kinetic energy Ts [no ] is obtained as the expectation value of the kinetic operator calculated for the non-interacting wavefunction constructed from two known doubly occupied Kohn-Sham orbitals. For Ts [no − nB ] and Ts [nB ], the situation is even simpler as no −nB and nB represent doubly-occupied one-orbital systems. Tsnad [no −nB , nB ] is non-zero for both considered choices for nB (~r). Tsnad [no −nB , nB ] = 0.00629883 and Tsnad [no − nB , nB ] = 0.013866, for the Eq. 9.2.16 and Eq. 9.2.17 cases, respectively. Interestingly, the local behavior of the radial integrand tnad s [no − nB , nB ](r) =

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tnad 0.07

tnad

0.06

0.06

0.05 0.04

0.04 0.03

0.02 0.5

1.0

1.5

2.0

2.5

3.0

r

0.02 0.01

-0.02 (a)

0

1

2

3

4

r

(b) R∞

Fig. 9.2.5. The radial integrand tnad (r) in Tsnad [no − nB , nB ] = 0 tnad [no − nB , nB ](r)dr for s s two choices for nB : (a) nB (~ r ) given in Eq. 9.2.16, (b) nB (~ r ) given in Eq. 9.2.17.

R∞ ts [no −nB ](r)−ts [no ](r)−ts [nB ](r) used to obtain Tsnad [no −nB , nB ] = 0 tnad s [no − nB , nB ](r)dr behaves quite differently in both considered cases (see Figure 9.2.5). For the Eq. 9.2.17 case, also tnad s (r) is non-negative (see Figure 9.2.5), in line nad with a more general result that ts [nA , nB ] is non-negative for the partitioning of the total electron density based on mixing of orbital densities in any four-electron system.35 In the Eq. 9.2.16 case, although Tsnad [nA , nB ] is positive, tnad (r) changes sign from negative to positive at about r = 0.34. It is worthwhile to notice that neither the Thomas-Fermi- nor the von Weizs¨acker approximation to Ts [n] could yield such a behavior because the former leads to non-negative, whereas the latter one to non-positive tnad (r). It is worthwhile to underline that, although the integrand tnad (r) can be interpreted as the radial density of the non-additive kinetic energy, the latter quantity is not uniquely defined because any function integrating to zero will not affect Tsnad . 9.2.2. Eq. 9.1.3 with the approximated potentials v ˜tnad [nA , nB ] The considered approximated potentials v˜tnad [nA , nB ] are shown in Figures 9.2.6 and 9.2.7. None of them reproduces adequately the whole shape of vtnad (~r). nad(T F ) The potential v˜t [nA , nB ] decays monotonically differing qualitatively from the exact potential. In particular, the barrier at r = 2 occurring in both choices for nB (~r) is not reproduced. In the Eq. 9.2.16 case, instead of a narrow spike in vtnad (r) nad(T F ) at → 0, v˜tT F [nA , nB ] is significantly lower and wider. Moreover, v˜t [nA , nB ] is very similar for both choices for nB (~r) (either Eqs. 9.2.16 or 9.2.17, whereas the exact potential vtnad (r) at r → 0 differs qualitatively in these two cases. As far as the qualitative behavior of the exact potential is concerned, nad(W ) v˜t [nA , nB ] seems to be much better than v˜tT F [nA , nB ]. Up to about r = 2, nad(W ) v˜t [nA , nB ] and the exact potential are just shifted by almost a constant: from

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nA 2.0

vtnad 2.0 1.5 1.0 0.5

1.5 1.0 r

-0.5 -1.0 -1.5

1

2

3

4

5

0.5

0

2

(a) Fig. 9.2.6.

4

6

8

10

r

(b) nad(T F ) [nA , nB ] nad(T F W ) v˜t [nA , nB ]

The exact potential vtnad (~ r ) (bold), its approximated counterparts: v˜t

(bold dotted),

nad(W ) nad(GEA2) v˜t [nA , nB ] (bold dashed), v˜t [nA , nB ] nad(CW ) v˜t [nA , nB ] (solid), and the corresponding

(dotted),

(dashed), and ground-state densities of the embedded system obtained for the Eq. 9.2.16 choice for nB (~ r ): (a) potentials vtnad (~ r ) and v˜tnad [nA , nB ], (b) embedded radial densities 4πr 2 nA (~ r ) and 4πr 2 n ˜ A (~ r )). The dot-dashed line is the density obtained for v˜tnad [nA , nB ] = 0.

0.333463 at r → 0 to 0.490454 at r = 2 in the Eq. 9.2.16 case. The corresponding shifts for nB (~r) chosen as in Eq. 9.2.17 are 0.333335 and 0.49045. For the Eq. 9.2.16 choice for nB (~r), the maximum of the exact potential at r = 2 amounts to nad(W ) 204.82, whereas the corresponding value for v˜t [nA , nB ] is 204.329 (not shown in the figures). In the Eq. 9.2.17 case, the corresponding heights of the barrier are: nad(W ) vtnad (2) =115.714 and v˜t [nA , nB ](2) = 115.223. Beyond r = 2, the magnitude nad(W ) of the shift between v˜t [nA , nB ] and vtnad (r) diminishes to vanish at large r. nad(W ) As the result, v˜t [nA , nB ] is too negative for 2 < r < 4. nad(GEA2) The shape of v˜t [nA , nB ] reflects the strengths and weaknesses of its nad(T F ) nad(W ) v˜t [nA , nB ] and v˜t [nA , nB ] components (see Figures 9.2.6 and 9.2.7). On nad(GEA2) the scale of the pictures, v˜t [nA , nB ] is indistinguishable from the exact potential for r > 2.2. The barrier at r = 2 is about one order of magnitude too low due nad(GEA2) to the 1/9 factor in front of the von Weizs¨acker contribution to v˜t [nA , nB ]. nad(W ) Compared to v˜t [nA , nB ], the reduction of its contribution (negative) and the addition of the Thomas-Fermi component eliminates almost perfectly the artificial nad(W ) nad(T F ) negativity of v˜t [nA , nB ] at r > 2. At r < 1.7, the v˜t [nA , nB ] componad(T F ) nent of this approximated potential dominates. As a result, v˜t [nA , nB ] and nad(GEA2) nad(T F W ) v˜t [nA , nB ] are practically the same at small r. The v˜t [nA , nB ] potential reproduces adequately the barrier at small r but the weaknesses of its nad(W ) v˜t [nA , nB ] component become apparent beyond r = 2. nad(CW ) The potential v˜t [nA , nB ] reproduces the barrier at r = 2 which is, however, too wide and too low. Moreover, it is too negative at r > 2.5. The width of the barrier at r = 2 is significantly overestimated. At small r, deficiencies of the nad(CW ) Thomas-Fermi component of v˜t [nA , nB ] become apparent.

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vtnad 2.0

nA 2.0

1.5

1.5

1.0

1.0

0.5 0.5 1

2

3

-0.5 (a) Fig. 9.2.7.

4

5

r 0

2

4

6

8

10

r

(b) nad(T F ) [nA , nB ] nad(T F W ) v˜t [nA , nB ]

The exact potential vtnad (~ r ) (bold), its approximated counterparts: v˜t

(bold dotted),

nad(W ) nad(GEA2) v˜t [nA , nB ] (bold dashed), v˜t [nA , nB ] nad(CW ) v˜t [nA , nB ] (solid), and the corresponding

(dotted),

(dashed), and ground-state densities of the embedded system obtained for the Eq. 9.2.17 choice for nB (~ r ): (a) potentials vtnad (~ r ) and v˜tnad [nA , nB ], (b) embedded radial densities 4πr 2 nA (~ r ) and 4πr 2 n ˜ A (~ r )). The dot-dashed line is the density obtained for v˜tnad [nA , nB ] = 0.

Using the exact potential vtnad (~r) in Eq. 9.1.3 leads to the density nA (~r) which equals to the target density no (~r) − nB (~r). In all examples considered in the present work, the target density is given analytically as no (~r) − nB (~r). Replacing vtnad (~r) by some approximation v˜tnad [nA , nB ] leads to the embedded orbital and the embedded electron density denoted by n ˜ A (~r), which might differ from no (~r) − nB (~r). Similarly, the lowest energy embedded orbital in such a case might differ from p (no (~r) − nB (~r)) /2. Figures 9.2.6 and 9.2.7 show the densities (˜ nA ) obtained usnad ing the considered approximations v˜t [nA , nB ]. In practical calculations using Eq. 9.1.3, approximations are needed not directly for Ts [n] but for Tsnad [nA , nB ] (in the evaluation of energy) and for vtnad [nA , nB ] (in the evaluation of embedded orbitals). To discuss the quality of these quantities in the approximated case, Tables 9.1 and 9.2 collect the numerical vales of: Ts [no ], Tsnad [no − nB , nB ], and the overlap between the analytically obtained exact wavefunction: p ϕexact (~r) = (no (~r) − nB (~r)) /2 (9.2.19)

and the wavefunction obtained from Eq. 9.1.3: either that with the exact potential (ϕ(~r)) or with one of the considered approximations (ϕ(~ ˜ r )). The corresponding exact quantities available in the considered case are also given for comparison. Depending on which quantity is used as the accuracy criterion, the order of errors due to the approximations is different. For Ts [no ], the error in its approximate counterparts decreases in following the order: none > T F W > W > T F > GEA2.

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j

291

j

1.0

1.0

0.8

0.8

0.6

0.6

0.4

0.4

0.2

0.2 r 0

2

4

6

8

10

r 0

2

4

(a)

6

8

10

(b)

Fig. 9.2.8. The radial ground-state wavefunction obtained from Eq. 9.1.3 with the exact ponad(T F ) tential vtnad (~ r ) (bold), and its approximated counterparts: v˜t [nA , nB ] (bold dotted), nad(W )

v˜t

[nA , nB ] (bold dashed), nad(CW ) v˜t [nA , nB ] (solid). The

nad(GEA2)

v˜t

nad(T F W )

[nA , nB ] (dotted), v˜t

[nA , nB ] (dashed), and

dot-dashed line is the density obtained for v˜tnad [nA , nB ] = 0. (a) nB (~ r ) given in Eq. 9.2.16, (b) nB (~ r ) given in Eq. 9.2.17

Table 9.1. Exact and approximate quantities calculated for nB (~ r ) given in Eq. 9.2.16. Approximation

T˜s [no ]

T˜snad [no − nB , nB ]

hϕexact |ϕi ˜

none W TF GEA2 TFW CW

0 1.07179 1.10746 1.22655 2.17925 -

0 -0.171912 0.127427 0.108325 -0.0444851 -

0.375866 0.245701 0.864161 0.974925 0.846614 0.960765

exact

1.25

0.00629883

0.999992

Table 9.2. Exact and approximate quantities calculated for nB (~ r ) given in Eq. 9.2.17. Approximation

T˜s [no ]

T˜snad [no − nB , nB ]

hϕexact |ϕi ˜

none W TF GEA2 TFW CW

0 1.07179 1.10746 1.22655 2.17925 -

0 -0.164344 0.130186 0.111925 -0.0341583 -

0.37779 0.254705 0.863955 0.972546 0.848841 0.9597

exact

1.25

0.013866

0.99999

For T˜snad [no − nB , nB ], the errors are ordered differently: none > W > T F > GEA2 > T F W.

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Finally, the errors in the embedded orbital (and the embedded density) measured as the deviations of hϕexact |ϕi ˜ from one, which indicate errors in v˜tnad [nA , nB ], are ordered in yet another way: W > none > T F W > T F > CW > GEA2 nad(W ) v˜t [no

The flaws of − nB , nB ] result in the smallest overlap between the exact and approximated embedded wavefunctions. It is even smaller (about 0.25) than if this component of the embedding potential is neglected (about nad(T F ) 0.38)! v˜t [no − nB , nB ] is significantly better as the overlap between the approximated- and exact embedded wavefunctions increases to about 0.86. The Thomas-Fermi approximation stops the embedded density to collapse into the core nad(GEA2) but does it only partially (see Figure 9.2.8). v˜t [no − nB , nB ] is very efficiently blocking such a collapse into the core as the position of the maximum of the radial wavefunction is almost in the right place. As a result, the overlap with the exact orbital is close to 1. The deficiency of this approximation is most pronounced in the core region where the exact wavefunction still has a small secondary minimum whereas the GEA2 wave function is very small. This is probably the result of the nad(GEA2) nad(T F ) Thomas-Fermi component of v˜t [no − nB , nB ]. The T˜s [no − nB , nB ] nad(T F W ) and T˜s [no − nB , nB ] approximations lead to noticeable worse embedded nad(GEA2) nad(T F ) wavefunction than T˜s [no − nB , nB ] whereas T˜s [no − nB , nB ] is alnad(CW ) most as good as T˜s [no − nB , nB ]. These approximations lead, nevertheless, nad(W ) to significantly better embedded wavefunction then T˜s [no − nB , nB ]. Basically the same conclusions concerning the quality of the considered approximations can be drawn from the results for the Eq. 9.2.17 choice for nB (~r) (see Table 9.2). nad(GEA2) The superiority of the v˜t [nA , nB ] in preventing the collapse from valence into the core does not mean that this approximation is universally the best. The present work, concerns cases with a strong overlap between nA (~r) and nB (~r). Our previous studies for such pairs nA (~r) and nB (~r), which do not overlap signad(GEA2) nificantly,7,17 show that v˜t [nA , nB ] leads to erratic results as far as the nad(GEA2) embedded density is concerned. This flaw of v˜t [nA , nB ] was recently re24 confirmed by Gotz et al. Cutting off smoothly this term, as it is made in the nad(GEA2) GGA97 approximation for v˜t [nA , nB ]31 largely expands the area of applicability of Eq. 9.1.3.8 9.3. Conclusions The present work reports the complete application of one-electron equations for embedded orbitals in analytically solvable case. All the quantities related to the kinetic energy, which are approximated in practical calculations, i.e., i) the non-additive kinetic energy Tsnad [nA , nB ] together with its radial distribution tnad s [nA , nB ](r), ii)

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the potential vtnad (~r), and iii) the embedded orbital and iv) the embedded density are obtained analytically. These quantities are compared with their approximated counterparts. Concerning the exact calculations, the orbital-free embedding potential was constructed following the same steps as in our previous work16 in the same model system but for other choices for the frozen component nB (~r). The present work complements the previous studies by obtaining numerically the embedded orbitals and embedded electron density from the Eq. 9.1.3 for either the exact or approximated non-additive kinetic energy potential. We analyzed in particular the case related to possible collapse of the valence density onto the core in the absence of any explicit enforcement of the orthogonality. It is shown that the lowest-energy solution of Eq. 9.1.3 with the exact effective potential leads to the target density even in the case where this density is infinitesimally close to the valence density, i.e., the density which is not pure-state non-interacting v-representable. The exact non-additive kinetic potential makes a collapse of the optimized density onto the core impossible. In computational methods in common use, such a collapse is avoided by means of explicitly enforcing the orthogonality of the optimized orbital to the orbitals representing the frozen part. This involves a non-local component of the embedding operator operator such as the PhillipsKleinman pseudopotential. In FDET framework, however, such descriptors of the environment as the orbitals yielding nB (~r), are not used. It is shown that a local potential can be used for the same purpose although the obtained embedded orbital is quite different from the one which would be obtained by enforcing orthogonality. As a result, although the optimal orthogonal orbital approaches ϕ2s , the optimal embedded orbital obtained form Eq. 9.1.3 approaches rather |ϕ2s |. The difference is due to the fact that, although, the embedded orbital must yield the desired target density, it must still be nodeless. Concerning the non-additive kinetic energy, the radial distribution tnad s [nA , nB ] of the contributions to Tsnad [nA , nB ]) show that it can change sign in some cases. This indicates that the each of the two canonical approximations to Ts [n] (ThomasFermi- and von Weizs¨ acker) might be entirely inadequate in some regions in space as the Thomas-Fermi approximations leads always to non-additive tnad s [nA , nB ] whereas the latter one leads to always non-positive tnad [n , n ]. A B s Concerning the approximate potentials, their capacity to prevent the valence density to collapse into the core varies strongly from one approximation to another. The local density approximation leads to a monotonically decreasing repulsive potential, which prevents the collapse only partially. The exact functional for a close-shell two electron system (the von Weizs¨acker functional for Ts [n] leads to the embedding potential, which is even worse than such in which the non-additive kinetic energy potential is completely neglected. As far as semi-local approximations for Ts [n] are concerned, it is clearly demonstrated that improving the approximation for Ts [n] does not necessarily lead to improvements in the quantities derived from such approximations - the potential vtnad [nA , nB ] in particular. This indi-

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cates that the quest for a good approximations for vtnad [nA , nB ] is governed by its own rules. The challenge of accurate approximations for the functional Ts [n] and its functional derivative, i.e., quantities which are of key importance for OF-DFT methods overviewed in the present volume, is not necessarily directly related to the efforts in approximating Tsnad [nA , nB ] and its functional derivative, which are key ingredients in the FDET-based methods. The provided numerical examples provide also a good illustration for the highly non-local nature of the relation between the orbital-free embedding potential and embedded electron density. The three approximated potentials: nad(GEA2) nad(T F W ) nad(CW ) v˜t [nA , nB ], v˜t [nA , nB ], and v˜t [nA , nB ], although are quite different, lead to rather similar embedded electron densities. Among them nad(GEA2) v˜t [nA , nB ] is clearly the best one for the considered pairs nA (~r) and nB (~r). Concerning development of approximations for v˜tnad [nA , nB ], the present study indicates that the approximation should comprise the von Weizs¨acker term in some regions of space (as it is made in the NDSD approximation for the non-additive kinetic energy9 ) although it can be scaled down as it is in the second order gradient expansion for Ts [n]. Its full inclusion in some regions in space is, however, undesired. Finally, we note that this work concerned properties of the bi-functional Tsnad [nA , nB ] for a given nB (~r) as in typical FDET-based computations (frozen nB (~r)), the issue of approximating this bi-functional and its functional derivative by means of some explicit analytical expressions is also of key importance in fully variational calculations in which nB is also subject to optimization. Such fully variational calculations are based on the subsystem formulation of DFT formulated by Senatore and Subbaswamy36 and Cortona.37 In practice, fully variational calculations can be performed using any implementation of FDET by means of the “freeze-and-thaw” algorithm17 involving iterative solutions of the KSCED equations for each interacting subsystem. We can expect, therefore, that errors of any approximation for Tsnad [nA , nB ] considered in the present work would rather enhance than attenuate in fully variational calculations. Acknowledgments T.A.W. and A.S. acknowledge the support from the grants by Swiss National Research Foundation (Project 200020-134791) and ANR (Project 07-BLAN-0272), respectively. References 1. 2. 3. 4. 5.

P. Hohenberg and W. Kohn, Phys. Rev. 136, B864 (1964). T. A. Wesolowski and A. Warshel, J. Phys. Chem. 97, 8050 (1993). T. A. Wesolowski, Phys. Rev. A. 77, 012504 (2008). K. Pernal and T. A. Wesolowski, Int. J. Quantum Chem. 109, 2520 (2009). T. A. Wesolowski. One-electron equations for embedded electron density: challenge

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6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34. 35. 36. 37.

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for theory and practical payoffs in multi-level modelling of soft condensed matter. In ed. J. Leszczynski, Computational Chemistry: Reviews of Current Trends, vol. X, pp. 1–82. World Scientific, Singapore, (2006). T. A. Wesolowski and J. Weber, Int. J. Quantum Chem. 61, 303 (1997). Y. A. Bernard, M. Dulak, J. W. Kaminski, and T. A. Wesolowski, J. Phys. A-Math. Theor. 41, 055302 (2008). T. A. Wesolowski, J. Chem. Phys. 106, 8516 (1997). J. M. G. Lastra, J. W. Kaminski, and T. A. Wesolowski, J. Chem. Phys. 129, 074107 2008). M. E. Casida and T. A. Wesolowski, Int. J. Quantum Chem. 96, 577 (2004). T. A. Wesolowski, J. Am. Chem. Soc. 126, 11444 (2004). ADF2009 suite of programs Theoretical Chemistry Department, Vrije Universiteit, Amsterdam http://www.scm.com. 2009 C. R. Jacob, J. Neugebauer, and L. Visscher, J. Comput. Chem. 29, 1011 (2008). J .W. Kaminski, S. Gusarov, A. Kovalenko, and T. A. Wesolowski, J. Phys. Chem. A 114, 6082 (2010). C. R. Jacob, S. M. Beyhan, and L. Visscher, J. Chem. Phys. 126 , 234116 (2007). A. Savin and T. A. Wesolowski, Prog. Theor. Chem. Phys. 19, 327 (2009). T. A. Wesolowski and J. Weber, Chem. Phys. Lett. 248, 71 (1996). W. Kohn and L. J. Sham, Phys. Rev. 140, A1133 (1965). M. Levy, Proc. Natl. Acad. Sci. USA. 76, 6062 (1979); see also: M. Levy, Phys. Rev. A 26, 1200 (1982). E. Lieb, Int. J. Quantum Chem. 24, 243 (1983). N. Govind, Y. A. Wang, and E. A. Carter, J. Chem. Phys. 110, 7677 (1999). A. S. P. Gomes, C. R. Jacob, and L. Visscher, Phys. Chem. Chem. Phys. 10, 5353 (2008). M. Hodak, W. Lu, and J. Bernholc, J. Chem. Phys. 128, 014101 (2008). A. W. Gotz, S. M. Beyhan, and L. Visscher, J. Chem. Theor. & Comput. 5, 3161 (2009). J. D. Chai and J. A. Weeks, J. Phys. Chem. B 108, 6870 (2004). J. Phillips and L. Kleinman, Phys. Rev. 116, 287 (1959). C. F. von Weizs¨ acker, Z. Phys. 96, 431 (1935). L. H. Thomas, Proc. Cambridge Philos. Soc. 23, 542 (1927). E. Fermi, Z. Phys. 48, 73 (1928). D. A. Kirzhnits, Sov. Phys. JETP. 5, 64 (1957). T. A. Wesolowski, H. Chermette, and J. Weber, J. Chem. Phys. 105, 9182 (1996). P. K. Acharya, L. J. Bartolotti, S. B. Sears, and R. G. Parr, Proc. Natl. Acad. USA. 77, 6978 (1980). W. Lucha and F. Schoberl, Intl. J. Mod. Phys. C. 10, 607 (1999). S. Wolfram, The Mathematica book. 2003, fifth edition. ISBN 1-57955-022-3. T. A. Wesolowski, Mol. Phys. 103, 1165 ( 2005). G. Senatore and K. R. Subbaswamy, Phys. Rev. B 34, 5754 (1986). P. Cortona, Phys. Rev. B 44, 8454 (1991).

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Chapter 10 Towards the Description of Covalent Bonds in Subsystem Density-Functional Theory Christoph R. Jacob1 and Lucas Visscher2 1

Karlsruher Institut f¨ ur Technolgie (KIT) Center for Functional Nanostructures Wolfgang-Gaede-Str. 1a, 76131 Karlsruhe, Germany [email protected] 2 VU University Amsterdam Amsterdam Center for Multiscale Modeling De Boelelaan 1083, 1081 HV Amsterdam, The Netherlands [email protected] Current applications of frozen-density embedding (FDE)—or more generally subsystem density-functional theory (DFT) schemes—are limited to subsystems that are not connected by covalent bonds. This restriction is due to the insufficiencies of the available approximate kinetic-energy functionals, which are used to calculate the contribution of the nonadditive kinetic energy to the effective embedding potential. In this Chapter, we discuss two different approaches to overcome these limitations and to extend the applicability of the FDE scheme to subsystems connected by covalent bonds. First, we outline possibilities to improve the currently available approximations applied for the kinetic-energy component of the embedding potential. Second, we show how a generalized three-partition FDE scheme can be employed to circumvent the problems in the approximate kinetic-energy functionals by introducing capping groups, thus allowing for a subsystem DFT treatment of proteins.

Contents 10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2 Theoretical background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2.1 Frozen-density embedding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2.2 Subsystem DFT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2.3 Extension to many subsystems and multilevel simulations . . . . . . . . . . . . 10.2.4 Approximations to Tsnadd [ρI , ρII ] and to vT [ρI , ρII ] . . . . . . . . . . . . . . . . 10.3 Development of improved approximations to vT [ρI , ρII ] . . . . . . . . . . . . . . . . . . 10.3.1 Exact embedding potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.3.2 Embedding potential in the limit of a small electron density of the active subsystem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.3.3 Assessment of approximations to vT [ρI , ρII ] for the description of covalent bonds 10.4 Introduction of capping groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.4.1 Three-partition frozen-density embedding . . . . . . . . . . . . . . . . . . . . . 297

298 300 300 302 303 304 305 305 307 311 313 313

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10.4.2 Application to the description of proteins . . . . . . . . . . . . . . . . . . . . . 10.5 Conclusions and outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

315 317 319

10.1. Introduction The accurate quantum chemical treatment of large systems, such as biomolecules (e.g., proteins) or condensed phase systems (e.g., transition metal catalysts in solution, molecules interacting with surfaces, or impurities in crystals), presents a significant challenge for theoretical chemistry (see, e.g., Refs. 1–4). In particular, two main problems have to be overcome. First, the required computational effort increases with the size of the studied system, which puts a significant burden on an accurate treatment of large systems. This first problem can be addressed by using efficient computational methods, mostly based on density-functional theory (DFT), which show a linear scaling of the required computer time with the size of the system.5,6 However, a second problem remains. For full quantum-chemical calculations on large systems, a large amount of data is obtained, which hampers the interpretation of the results and makes it difficult to extract general conclusions from such calculations (see, e.g., Refs. 7 and 8, where this problem is discussed in the context of theoretical vibrational spectroscopy). Subsystem approaches, in which the full system is decomposed into its constituting fragments that are then each treated individually (for examples, see Refs. 9–15), offer a very attractive alternative to a treatment of the full system. First, subsystem methods are in general more efficient than a conventional treatment, since the computer time required for the calculation of one subsystem is usually independent of the size of the full system, so that one naturally obtains a linear scaling of the computational effort with the size of the system.9–11 Second, a partitioning into subsystems provides a more natural way for the interpretation of the results, since it offers a picture in terms of the chemical building blocks of the system such as, e.g., the individual molecules in a condensed phase system or the amino acid residues constituting a protein. Finally, subsystem approaches provide the possibility to focus on interesting parts of the system, such as the active site of an enzyme or of another catalyst, solute molecules in a liquid phase, absorbed molecules on surfaces, or impurities in crystals. Since the subsystems are treated individually, it is easily possible to employ a more accurate treatment only for one or a few selected subsystems of interest or to introduce additional approximations for subsystems that are less important.16–18 In particular, such a subsystem description makes it possible to apply a wave-function theory (WFT) description to one subsystem, while its environment is treated more efficiently using DFT.19–24 A very appealing subsystem approach is offered by the frozen-density embedding (FDE) scheme within DFT introduced by Wesolowski and Warshel.16,25 In this FDE scheme, the total electron density is partitioned into possibly overlapping electron densities of an active subsystem and of a frozen environment, and the

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electron density of the active subsystem is optimized in the presence of the frozen environment. The effect of the frozen environment on the active subsystem is included by the use of an effective embedding potential, that contains in addition to the electrostatic potential of the nuclei in the environment and the Coulomb potential of the electron density of the environment, also a component of the nonadditive exchange–correlation energy and of the nonadditive kinetic energy.16,25 The FDE scheme is based on a more general subsystem formulation of DFT by Cortona,9 in which the electron densities of an arbitrary number of subsystems are each optimized individually. Such a general subsystem DFT scheme makes it possible to employ different levels of approximations for different parts of the system.18 Compared to other subsystem approaches, the FDE scheme offers the advantages that it includes the effect of the environment in an accurate and improvable way,18,26,27 and that it provides a exact treatment in the exact functional limit.25 However, the FDE scheme relies on the use of an approximate functional for the nonadditive kinetic energy and the corresponding component of the effective embedding potential and its applicability is, therefore, limited by the quality of the available approximate functionals. While it has been shown that with the available generalized-gradient approximation (GGA) kinetic-energy functionals, in particular the PW91k functional,28 accurate results can be obtained for van der Waals complexes29–31 and hydrogen-bonded complexes,32–34 FDE currently cannot be applied to subsystems connected by covalent bonds. While for the description of liquid phase systems such as solvated molecules a partitioning into the individual solute and solvent molecules is possible and a description of covalent bonds between subsystems is not necessary, there are several important areas of application where a subsystem description would require the treatment of covalent bonds between subsystems. For instance, the theoretical modeling of biologically relevant systems, e.g., proteins or their active centers, naturally leads to a partitioning into individual amino acid residues (or of larger subunits) connected by covalent bonds. Similarly, a subsystem description of larger transition metal complexes in solution would benefit from the possibility to describe the active center and the ligands of the catalysts as separate subsystems. Therefore, further theoretical developments are required to make applications of FDE or more generally subsystem DFT possible for such systems. In this Chapter, we give an overview of recent work addressing these problems. In Sec. 10.2, we present the theoretical background of FDE and subsystem DFT. This is followed by the discussion of different approaches for the description of covalent bonds within FDE and subsystem DFT. In Sec. 10.3, we begin with reviewing some recent developments of improved approximations to the kinetic-energy component of the embedding potential and we discuss their relevance for the description of covalent bonds. In Sec. 10.4, we show how the insufficiencies in the available approximate kinetic-energy functionals can be circumvented by the introduction of

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capping groups in a three-partition FDE scheme, and how such a scheme can be applied to the description of proteins. Finally, concluding remarks are given and possible directions for future developments are outlined in Sec. 10.5. 10.2. Theoretical background 10.2.1. Frozen-density embedding The FDE scheme16,25 is based on a partitioning of the total electron density ρtot (r) into the electron densities of two subsystems, i.e., ρtot (r) is represented as the sum of two components ρI (r) and ρII (r), ρtot (r) = ρI (r) + ρII (r).

(10.2.1)

Except for the requirement that both subsystem densities integrate to an integer number of electrons, they are not subject to any further conditions. In particular, the subsystem densities are allowed to overlap. In addition to the electron density, the nuclear charges are partitioned accordingly. This partitioning of the density and of the nuclear charges defines two subsystems (subsystems I and II). Given this partitioning, the DFT total energy can (in the absence of any external fields) be expressed as a functional of ρI and ρII , Z

nuc E[ρI , ρII ] = ENN + ρI (r) + ρII (r) vInuc (r) + vII (r) dr

Z ρI (r) + ρII (r) ρI (r 0 ) + ρII (r 0 ) 1 (10.2.2) drdr 0 + 2 |r − r 0 | + Exc [ρI + ρII ] + Ts [ρI ] + Ts [ρII ] + Tsnadd [ρI , ρII ],

nuc where ENN is the nuclear repulsion energy, vInuc and vII are the electrostatic potentials of the nuclei in subsystems I and II, respectively, Exc is the exchange– correlation energy functional, Ts [ρ] is the kinetic energy of the noninteracting reference system, and Tsnadd [ρI , ρII ] is the nonadditive kinetic energy, which is defined as

Tsnadd[ρI , ρII ] = Ts [ρI + ρII ] − Ts [ρI ] − Ts [ρII ].

(10.2.3)

The densities ρI (r) and ρII (r) can be represented using the canonical Kohn– (n) Sham (KS) orbitals for the individual subsystems φi with ρn (r) = 2

Nn /2

X (n) 2 φi (r) ,

(10.2.4)

i=1

where n = I, II denotes the considered subsystem and NI and NII denote the number of electrons in subsystems I and II, respectively. For reasons of simplicity, only the closed-shell case with Nn /2 doubly occupied KS orbitals in each subsystem will be considered. A generalization to open-shell systems is possible in a straightforward

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way.35 Using the KS orbitals, it is possible to calculate the kinetic energy of the corresponding noninteracting reference system as Ts [ρn ] = −

Nn /2 Z

X

(n)

(n)

φi (r)∇2 φi (r) dr.

(10.2.5)

i=1

However, with the partitioning of the total electron density into ρI (r) and ρII (r) there is in general no representation of ρtot (r) in the canonical KS orbitals available, so that Ts [ρI + ρII ] cannot be calculated in this way. Therefore, in practical implementations Tsnadd [ρI , ρII ] has to be approximated (see Sec. 10.2.4). For a given frozen electron density ρII (r) in one of the subsystems (subsystem II) the electron density ρI (r) in the other subsystem (subsystem I) can be determined by minimizing the total energy bifunctional [Eq. (10.2.2)] with respect to ρI , while ρII (r) is kept frozen. If the complementary ρI (r) is positive, this will lead to the total density ρtot (r) = ρI (r) + ρII (r) that minimizes the total energy functional. This total density is, therefore, the same density that could also be obtained from a conventional DFT calculation on the total system. The minimization of the total energy E[ρI , ρII ] with respect to ρI , under the constraint that the number of electrons NI in subsystem I is conserved, leads to a set of Kohn–Sham-like equations for the KS orbitals of subsystem I (where it has to be assumed that the exact ρI is vs -representable),   ∇2 (I) (I) KSCED − + veff [ρI , ρII ](r) φi (r) = i φi (r); i = 1, . . . , NI /2, (10.2.6) 2 which are usually referred to as Kohn–Sham equations with constraint electron density (KSCED equations). In these equations, the KSCED effective potential is given by KSCED KS emb veff [ρI , ρII ](r) = veff [ρI ](r) + veff [ρI , ρII ](r),

(10.2.7)

KS where veff [ρI ](r) is the KS effective potential of the isolated subsystem I containing the usual terms of the nuclear potential, the Coulomb potential of the electrons, and the exchange–correlation potential, Z ρI (r 0 ) δExc [ρ] KS nuc 0 veff [ρI ](r) = vI (r) + dr + , (10.2.8) |r − r 0 | δρ ρ=ρI (r)

emb and the effective embedding potential veff [ρI , ρII ](r) describes the interaction of the subsystem I with the frozen density and nuclei of subsystem II and reads Z ρII (r 0 ) 0 emb nuc veff [ρI , ρII ](r) = vII (r) + dr |r − r 0 | δExc [ρ] δExc [ρ] + − + vT [ρI , ρII ](r). (10.2.9) δρ ρ=ρI +ρII δρ ρ=ρI

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In addition to the electrostatic potential of the nuclei and the electrons in the frozen subsystem, this effective embedding potential also contains an exchange– correlation component and a kinetic-energy component vT [ρI , ρII ] which is given as the functional derivative of the nonadditive kinetic-energy bifunctional, δT nadd[ρI , ρII ] δTs [ρ] δTs [ρ] vT [ρI , ρII ](r) = s − . (10.2.10) = δρI δρ ρ=ρtot (r) δρ ρ=ρI (r)

In practical applications of the FDE scheme, this kinetic-energy component vT has to be approximated (see Sec. 10.2.4). Note that for GGA exchange– correlation functionals no additional approximations have to be introduced for the exchange–correlation component of the embedding potential, but when using orbital-dependent exchange–correlation functionals, additional approximations have to be introduced in the exchange–correlation component of the embedding potential.31 For a given frozen density ρII (r), the density of the nonfrozen subsystem ρI (r) can be obtained by solving the above KSCED equations with the embedding potenemb tial veff as given in Eq. (10.2.9). If the initial assumption that the complementary ρI is positive and vs -representable is fulfilled, the solution of these equations will directly yield the exact ground-state electron density of the total system.25 In typical applications of the FDE scheme, the nonfrozen subsystem I is a small system of interest, which is embedded in a much larger environment. Especially for the calculation of molecular properties (e.g., electronic excitation energies, or nuclear magnetic resonance shieldings), this will be a very efficient scheme, since the property calculation generally has to be performed for the nonfrozen subsystem only. However, in these cases the construction of the electron density of the frozen environment becomes a bottleneck if the standard approach is used and the frozen density is obtained using a DFT calculation of the full environment. This problem can be overcome by applying approximations in the construction of the environment density, because Eq. (10.2.6) can be solved for any postulated electron density, so that ρII (r) may also be obtained from simpler considerations. Already in their initial papers, Wesolowski and Warshel proposed the use of such an approximate density to describe a water environment.16,36 In a study of solvent effects on excitation energies, Neugebauer et al. investigated the electronic absorption spectrum of acetone in water and tested different approximate descriptions of the frozen solvent environment. They found that compared to a full DFT calculation of the environment, the error introduced by using a superposition of densities of isolated water molecules is less than 0.01 eV for the n → π ∗ transition of interest. Subsequently, this strategy has been successfully applied in a number of studies of solvent effects on molecular properties.26,37–39 10.2.2. Subsystem DFT While the strategy to use fixed approximate densities as described in the previous

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section can be applied for large environments, the FDE formalism can also be used to determine the electron densities of both subsystems. In particular, for most approximate environment densities, the requirements that the complementary ρI is vs -representable and positive at any point in space will not be fulfilled. To correct for the errors introduced by these deficiencies of the approximate environment density, both the electron density in the nonfrozen subsystem and the environment density should be adjusted. This leads to the “subsystem DFT” formalism as it was initially proposed by Cortona,9 which provides an alternative to conventional KS-DFT. The starting point for this subsystem DFT formulation is again the total energy bifunctional of Eq. (10.2.2), but now this total energy is not only minimized with respect to the electron density ρI in one of the subsystems while the density ρII in the other subsystem is kept frozen, but it is minimized with respect to the electron densities in both subsystems. This leads to a set of two coupled sets of KSCED equations, which have to be solved self-consistently. This can be done by applying so-called “freeze-and-thaw” cycles,27 in which the roles of frozen and nonfrozen subsystem are interchanged until convergence is reached. 10.2.3. Extension to many subsystems and multilevel simulations The subsystem DFT scheme can be easily extended to an arbitrary number of subsystems by starting from the partitioning ρtot (r) =

M X

ρi (r),

(10.2.11)

i=1

where M is the number of subsystems. This leads to a formulation similar to the one presented above, except that a set of M coupled KSCED equations is obtained, in which the frozen density in the effective embedding potential is replaced by the sum of the densities of all frozen subsystems.9,10 This set of equations can be either solved iteratively using freeze-and-thaw cycles,27 or alternatively, the coupled KSCED equations can be solved simultaneously by updating all densities after each SCF cycle.10,11 This generalized subsystem DFT approach, in which the densities of all subsystems are optimized, can be used as an alternative to conventional KS-DFT calculation for large systems. By construction, it scales linearly with the number of subsystems. Initially, it has been applied by Cortona and co-workers for calculations on simple ionic crystals (e.g., alkali halides,40 alkali-earth oxides,41 and alkali-earth sulfides42 ) by determining the densities of the ions individually. While in the implementation of Cortona, these densities were constrained to be spherical, an extended scheme has been implemented by Mehl and co-workers. They allow deformations of the atomic densities, and studied alkali halides43 and corundum.44 This subsystem DFT approach has been implemented by Iannuzzi et al. in the CP2K program package.10 With their implementation molecular dynamics simulations can be performed, in which the individual molecules are treated as sub-

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systems. Another implementation has been presented by Shimojo et al., who also implemented a similar subsystem DFT scheme in combination with a numerical integration algorithm employing hierarchical real-space grids as an efficient alternative to standard KS-DFT calculations.11 They have applied their implementation to MD simulations of aluminum nanoparticles and of nanoindentation of ceramics materials.45 The implementation in the Adf program package18,37 supports the general subsystem DFT approach, in which the densities of an arbitrary number of subsystems are each optimized iteratively. On the other hand, it is also possible to optimize only the density of one active subsystem, while all other subsystems form a frozen environment, leading to the FDE scheme. Furthermore, the implementation also allows all kinds of intermediate setups, e.g., a number of subsystems can be fully optimized, while for other subsystems the gas-phase density is only polarized in one freeze-and-thaw cycle and while for the remaining subsystems the frozen density of the isolated molecule is used. In addition, a number of additional options can be specified for each fragment. This provides a very flexible framework for performing multilevel simulations, in which different levels of accuracy are employed for different subsystems.18,39,46 Such applications to large systems are further facilitated by the use of an efficient numerical integration scheme,37 that makes applications possible also in the case of large frozen environments. 10.2.4. Approximations to Tsnadd [ρI , ρII ] and to vT [ρI , ρII ] Both the total energy bifunctional and the effective embedding potential contain a nonadditive kinetic-energy component that usually cannot be calculated exactly. For the performance of the FDE scheme, the choice of the approximation which is used for this nonadditive kinetic-energy component is of great importance. Usually, the nonadditive kinetic energy is approximated in the form T˜snadd[ρI , ρII ] = T˜s [ρI + ρII ] − T˜s [ρI ] − T˜s [ρII ],

(10.2.12)

and the kinetic-energy component vT of the embedding potential is approximated as δ T˜s [ρ] δ T˜s [ρ] v˜T [ρI , ρII ](r) = − , (10.2.13) δρ δρ ρ=ρtot (r)

ρ=ρI (r)

where the tilde is used to label approximate quantities, and T˜s [ρ] refers to some approximate kinetic-energy functional. Approximation to Tsnadd[ρI , ρII ] and to vT [ρI , ρII ], that are of the form of Eq. (10.2.12) and Eq. (10.2.13) are denoted as decomposable approximations.47 An overview of different approximate kinetic-energy functionals that can be used to construct such decomposable approximations can be found, e.g., in Ref. 48. Here, only a brief overview of the approximate functionals that are commonly used in combination with the FDE scheme is given.

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The simplest approximation for the kinetic-energy functional, corresponding to the local-density approximation for exchange and correlation, is the Thomas–Fermi (TF) functional.49,50 For the construction of GGA kinetic-energy functionals, the suggestion of Lee, Lee, and Parr51 to use similar analytical forms for approximate kinetic-energy and exchange energy functionals can be applied. In a series of studies,32,52,53 Wesolowski and co-workers compared the accuracy of different approximate kinetic-energy functionals—including the TF functional and several GGA functionals based on the suggestion of Lee, Lee, and Parr—for different hydrogenbonded complexes. In particular, they investigated the hydrogen-bonded complexes FH· · · NCH (Ref. 52), HCN· · · H2 (Ref. 53) and a test set consisting of (H2 O)2 , (HF)2 , (HCl)2 , and HF· · · NCH (Ref. 32). By comparing results of subsystem DFT (freeze-and-thaw) calculations to supermolecular KS-DFT calculations it was found that the functional that yields the most accurate interaction energies for the investigated complexes is the GGA functional which has the same analytic form of the enhancement factor as the exchange functional of Perdew and Wang54 but should be reparametrized for the kinetic energy as described by Lembarki and Chermette. This functional is commonly dubbed PW91k. However, the PW91k functional is only applicable if the interaction between the subsystems is small. In this case, the nonadditive kinetic energy Tsnadd and the kinetic-energy component vT of the embedding potential are small compared to the other contributions, so that rather crude approximations can be applied. For weakly interacting or hydrogen-bonded systems, the PW91k functional, therefore, leads to total electron densities which are very similar to those obtained from supermolecular KS-DFT calculations, as was shown by Kiewisch et al. by means of a topological analysis of the resulting electron densities.34 This even holds for very strong hydrogen bonds, such as the one found in the complex F–H–F− . But when going to complexes where the bond between the subsystems has a larger covalent character, such as ammonia borane (NH3 –BH3 ), larger deviations in the electron densities occur, since the kinetic-energy component of the embedding potential is not sufficiently small anymore55 (see also the overview given below in Sec. 10.3.3). Therefore, improved approximations to vT have to be developed in order to describe covalent bonds (or even bonds with a significant covalent character) between subsystems adequately. 10.3. Development of improved approximations to vT [ρI , ρII ] 10.3.1. Exact embedding potential An ideal starting point for the development of improved approximations to the kinetic-energy component of the embedding potential vT [ρI , ρII ] is the knowledge of the exact vT [ρI , ρII ]—at least in specific limits or for certain systems. First, the exact behavior of vT [ρI , ρII ] in specific limiting cases can be used as guidance when

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constructing approximations by requiring that in these limits the exact vT [ρI , ρII ] is recovered by the approximation. Second, the exact vT [ρI , ρII ] can be used as a reference for assessing the quality and for identifying insufficiencies of a given approximation. It is important to notice that in the FDE scheme, the accuracy of the obtained electron density and of the KS orbitals and orbital energies, which in turn determine most molecular properties, are directly related to the quality of the approximation that is used for vT [ρI , ρII ], while an approximation to the nonadditive kinetic-energy Tsnadd[ρI , ρII ] is only required if the energy is needed. Since in general the quality of a certain approximation to Tsnadd[ρI , ρII ] is not related to the quality of the corresponding vT [ρI , ρII ] (which can be obtained from Tsnadd[ρI , ρII ] by taking the functional derivative),32 it is natural to directly consider vT [ρI , ρII ] instead of Tsnadd[ρI , ρII ] when developing improved approximations. In order to obtain vT [ρI , ρII ], one needs to evaluate the functional derivative of s [ρ] for two different densities, for ρtot = ρI + ρII the noninteracting kinetic energy δTδρ and for ρI . This functional derivative, which is often referred to as kinetic potential in the literature, is through the Euler–Lagrange equation of DFT56 related to the KS potential,57,58 δTs [ρ] = −vs [ρ](r) + µρ . δρ(r)

(10.3.14)

In this expression, vs [ρ] denotes the potential for which the density ρ is the ground state density. Such a potential exists by definition for any vs -representable density, and it is unique (up to a constant shift) according to the first Hohenberg-Kohn theorem. If the constant shift in the potential vs [ρ] is chosen such that it goes to zero at infinity, the constant µρ equals the chemical potential, which can be identified with the energy of the highest-occupied KS orbital.56,59 It should be noted that while Ts [ρ] can be defined for any N -representable density, its functional derivative is only defined for vs -representable densities.57 In practice, different algorithms exist which allow one to determine this potential vs [ρ] for a given (vs -representable) density.60–63 The exact kinetic potentials determined according to Eq. (10.3.14) have been used in a number of studies to assess the quality of approximate kineticenergy functionals in the context of orbital-free DFT.48,64,65 Using Eq. (10.3.14), the exact vT [ρI , ρII ] can be obtained from58 δTs [ρ] δTs [ρ] − vT [ρI , ρII ](r) = δρ δρ ρ=ρtot (r)

ρ=ρI (r)

= vs [ρI ](r) − vs [ρtot ](r) + ∆µ,

(10.3.15)

where vs [ρI ] denotes the potential for which ρI is the ground-state density, vs [ρtot ] is the potential for which ρtot is the ground-state density, and ∆µ = µρI − µρtot only leads to a constant shift of the potential. Equation (10.3.15) provides a recipe for the calculation of the exact vT [ρI , ρII ] for a given pair of electron densities by reconstructing the KS potentials yielding the

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densities ρI and ρtot . If one considers the density ρtot obtained from a supermolecular KS-DFT calculation, then vs [ρtot ](r) is known directly form this calculation. In this case, only vs [ρI ](r) has to be determined using a suitable algorithm for the reconstruction of the KS potential.66 However, the relation given in Eq. (10.3.15) only provides an implicit density functional, while an explicit density functional is needed in practical applications. Furthermore, the described procedure for the calculation of the exact vT [ρI , ρII ] will only be computationally feasible for a few benchmark systems. Nevertheless, it can provide useful reference potentials for the development of improved (explicit) density-functionals to approximate vT [ρI , ρII ]. Furthermore, such a procedure can be useful for the construction of accurate embedding potentials within WFT-in-DFT embedding schemes.67 It should further be noted that very similar approaches are used for the construction of local pseudopotentials, which are used in orbital-free DFT calculations.68–70 10.3.2. Embedding potential in the limit of a small electron density of the active subsystem In order to identify deficiencies in the currently available approximations to vT [ρI , ρII ], it can be very valuable to study the exact behavior of vT [ρI , ρII ] in certain limiting cases. One such case is the limit that the density of the active subsystem ρI is small. This situation commonly rises at the frozen subsystem, in particular if the distance between the two subsystems is large. This limit has recently been investigated in detail in Refs. 58 and 47. In this limit, vT [ρI , ρII ] simplifies to (see Appendix A in Ref. 47), δTs [ρ] lim vT [ρI , ρII ](r) = . (10.3.16) ρI (r)→0 δρ ρ=ρII (r)

Using Eq. (10.3.14) one obtains58

lim vT [ρI , ρII ](r) = −vs [ρII ](r) + µρII ,

ρI (r)→0

(10.3.17)

i.e., if ρI is small, the kinetic-energy component of the embedding potential is given by the KS potential that yields the frozen density ρII . To arrive at this expression, apart from the vs -representability of all the involved densities no further assumptions have to be made. If one additionally requires that the frozen density ρII is the ground-state density obtained for the isolated subsystem II, then vs [ρII ] is known and is given by the effective KS potential from the calculation on the isolated subsystem II. Therefore, one finds that in this case the kinetic-energy component cancels the other components of the embedding potentials, and the total embedding potential at the frozen subsystem is a constant,58 i.e., emb lim veff [ρI , ρII ](r) = µρII = constant.

ρI (r)→0

(10.3.18)

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In particular, near the nuclei of the frozen subsystem, vT [ρI , ρII ] compensates the large nuclear attraction as well as the other components of the effective embedding potential. However, this limit is not described correctly by all common approximations to vT [ρI , ρII ]. In particular, both the Thomas–Fermi (TF) approximation as well as the widely applied PW91k functional are not repulsive enough near the nuclei. This leads to a too attractive embedding potential near the nuclei of the frozen system, which can lead to severe problems in practical calculations. In particular, these problems show up when basis functions located on the frozen subsystem are included in the calculations (supermolecular basis set expansion32 ), which probe the embedding potential in regions where ρI should be small. In Ref. 58, the consequences of the wrong behavior of the TF and the PW91k approximations in the considered limit were demonstrated for the model system H2 O· · · Li+ at large separations, where the Li+ ion constitutes the frozen subsystem. It turned out that the incomplete compensation of the nuclear attraction of the lithium nucleus in the frozen subsystem leads to artificially low-lying orbitals on the frozen Li+ subsystem. At large separations, the orbital energy of one of these orbitals even drops below the one of the highest occupied molecular orbital of the active subsystem, so that the self-consistent field iterations only converge if a nonaufbau solution is enforced.58 Even if the artificial lowering of the energies of unoccupied orbitals on the environment is not so severe that it results in a non-aufbau solution, it leads to serious problems if excitation energies are considered. In Ref. 58, this was shown for a cluster of the dye molecule aminocoumarin C151 surrounded by 30 solvent water molecules, which are treated as frozen environment. For the active subsystem consisting of the dye molecule, spurious low-lying virtual orbitals appear if basis functions on the frozen environment are included (see Fig. 10.3.1a). These virtual orbitals are rather diffuse orbitals which are located on the solvent environment, as is shown in Fig. 10.3.1b, and excitations to these spurious virtual orbitals will lead to spurious excitation energies. One approach to address the incorrect behavior of common approximations to vT [ρI , ρII ] is to go beyond decomposable approximations, i.e., approximations that are of the form of Eq. (10.2.13), and to introduce a non-decomposable approximation to vT [ρI , ρII ], i.e., apply an approximation to vT [ρI , ρII ] that is not based on an approximate kinetic-energy functional (cf. Eq. (10.2.13)). a In Ref. 58, a correction was proposed that can be added on top of a given decomposable approximation. This correction is designed such that in regions where ρII is significantly larger than a Editors’

note: In the present chapter, no distinction is made between different mathematical quantities: the function vT (r) and the bifunctional vT [ρI , ρII ](r), i.e., unique mapping between the pair of functions ρI (r) and ρII (r) and the function vT (r) in 3D. For instance, RHS of Eq. 10.3.20 is not a bifunctional because it is not uniquely determined by ρI and ρII . It depends also on the nuc (r). Although making such distinction is not crucial in numerical practice it is third function vII essential in formal considerations.

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Fig. 10.3.1. Illustration of the consequences of the wrong behavior of the PW91k approximation at the frozen subsystem for the calculation of electronic excitation energies for a cluster of aminocoumarin C151 surrounded by 30 water molecules. (a) Orbitals energies (in eV) of the relevant orbitals. The orbital energies calculated for the isolated aminocoumarin C151 are shown as reference, together with those calculated using FDE (using the PW91k approximation for vT [ρI , ρII ]) not including [FDE(m)] and including basis functions on the frozen subsystem [FDE(s)]. Furthermore, the orbital energies calculated using FDE(s) and the correction proposed in Ref. 58 are shown [labeled FDE(s)-corr]. (b) Isosurface plots of the spurious virtual orbitals 62a to 65a obtained in the FDE(s) calculation. Reprinted with permission from Ref. 58. Copyright 2007 American Insitute of Physics.

ρI , i.e., at the frozen subsystem, the limit of Eq. (10.3.18) is explicitly enforced. For the case of the PW91k approximation, this leads to the non-decomposable approximation, v˜T [ρI , ρII ](r) = v˜TPW91k [ρI , ρII ](r)
(10.3.19) + f ρI (r), ρII (r) · v˜Tcorr [ρI , ρII ](r),
where f ρI (r), ρII (r) is a switching function that switches from 0 at the active subsystem to 1 at the frozen subsystem, and where the correction term is given by,  Z ρII (r 0 ) 0 nuc v˜Tcorr [ρI , ρII ](r) = − vII (r) + dr |r − r 0 | δExc [ρ] δExc [ρ] + − δρ ρ=ρI +ρII δρ ρ=ρI  + v˜TPW91k [ρI , ρII ](r) , (10.3.20) i.e., it is chosen such that it cancels the other components of the embedding potential and the total embedding potential is thus zero when this correction term is switched on. Using this long-distance corrected approximation to vT [ρI , ρII ], the problems with artificially low-lying unoccupied orbitals located on the frozen subsystem found for H2 O· · · Li+ at large separations can be overcome, and orbitals located on the frozen Li+ are shifted to higher orbital energies. Furthermore, the problem with spurious low-lying virtual orbitals on the frozen environment found for aminocoumarin C151 surrounded by water molecules disappears (see also Fig. 10.3.1a).

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However, it should be noted that the correction proposed in Ref. 58 has a number of disadvantages. First, the limit the correction enforces is only exact under the conditions given above for Eq. (10.3.18), i.e., if ρI (r) → 0 and if ρII is the ground-state density obtained for the isolated subsystem II. In particular the latter condition will not be satisfied if the frozen density has been obtained in freeze-and-thaw iterations. Second, the chemical potential of subsystem II, µρII , in Eq. (10.3.18) is set to zero, and therefore, the proposed correction will not be able to describe cases in which there should be a transfer of electron density from subsystem I to subsystem II, i.e., the partitioning of the electron density must a priori correspond to the correct dissociation limit. Third, the correction proposed in Ref. 58 is explicitly positiondependent, since the correction term in Eq. (10.3.20) contains the nuclear potential of subsystem II. Therefore, the approximation to vT [ρI , ρII ] given in Eq. (10.3.19) is no explicit density functional, even though the positions and charges of the nuclei can in principle be deduced from the electron density. Furthermore, it cannot be obtained as a functional derivative of an approximation to the nonadditive kineticenergy Tsnadd [ρI , ρII ]. The limit that the electron density ρI of subsystem I is small was also considered by Garcia Lastra et al. in Ref. 47. Starting from Eq. (10.3.16), they consider the case that ρII is a spin-compensated two-electron density. For such a system, the exact noninteracting kinetic energy is given by the von Weizs¨acker kinetic-energy functional,71 so that one obtains, 2 δ T˜svW [ρ] 1 ∇ρII (r) 1 ∇2 ρII (r) lim vT [ρI , ρII ](r) = = − , (10.3.21) 2 δρ 8 ρII (r) 4 ρII (r) ρI (r)→0 ρ=ρII (r) R for NII = ρII (r)dr = 2. The von Weizs¨acker functional also gives the correct limit near the nuclear cusps, where the electron density is dominated by a single 1s-type orbital. Therefore, the above expression should also be applicable near the nuclei of the frozen subsystem. Based on this exact limit given by Eq. (10.3.21), Garcia Lastra et al. develop a non-decomposable approximation (dubbed non-decomposable approximant using first and second derivatives of ρ, NDSD) to vT [ρI , ρII ] which is given by,47 v˜TNDSD [ρI , ρII ](r) = v˜TTF [ρI , ρII ](r)


δ T˜svW [ρ] + f ρII (r), ∇ρII (r) δρ

ρ=ρII (r)

,

(10.3.22)


where f ρII (r), ∇ρII (r) is a switching function depending on the reduced density gradient and the electron density of subsystem II that is designed such that it is 1 in regions close to the nuclei of the frozen subsystem, where Eq. (10.3.21) can be expected to hold, and that is 0 in region where this is not the case. In contrast to the correction proposed in Ref. 58, the NDSD approximant of Ref. 47 is an explicit density functional, and it can be obtained as the functional derivative of a corresponding approximation to the nonadditive kinetic energy

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Tsnadd[ρI , ρII ]. As is shown in Ref. 47 for a test set containing a diverse set of weakly interacting and hydrogen-bonded systems, the NDSD approximation leads for all considered molecules to more accurate complexation-induced dipole moments than both the TF and the PW91k approximations. However, also the NDSD approximation exhibits some limitations. For H2 O· · · Li+ at large separations, it is not able to completely remove the problem with too low-lying unoccupied orbitals located on the frozen Li+ subsystem, but it only shifts these orbitals to slightly higher energies so that the separation at which a non-aufbau solution is obtained is increased. This is because even though Eq. (10.3.21) is exact for the frozen two-electron system Li+ , this limit is only enforced near the nucleus and not in all regions where ρI is small. Furthermore, due to the form of the NDSD approximant, the zeroth-order TF term stays finite near the nuclei, although it should be switched off.47 10.3.3. Assessment of approximations to vT [ρI , ρII ] for the description of covalent bonds To assess the quality of a given approximation to vT [ρI , ρII ] one can compare the electron densities of a subsystem DFT calculation, in which the electron densities of both subsystems are determined iteratively in freeze-and-thaw iterations, to those from a supermolecular KS-DFT calculation.32 If in the subsystem DFT calculation the supermolecular basis set expansion is used, and if a GGA exchange–correlation functional is applied, all differences to the supermolecular KS-DFT results can be attributed to the approximations used for the kinetic-energy component of the embedding potential vT [ρI , ρII ]. Note that if the electron densities are compared, all differences can be attributed to the approximation used for vT [ρI , ρII ]. In contrast, if interaction energies are compared both the approximation used for vT [ρI , ρII ] and the one used for Tsnadd[ρI , ρII ] contribute to the observed errors. Therefore, it is preferable to investigate errors in the electron density, since such a strategy allows it to isolate the errors that originate from the approximation that is applied for vT [ρI , ρII ]. While most studies compare an integrated measure for the differences in the electron density, such as differences in the dipole moments31,32,47 or the integral of the absolute difference density,72 only few studies have performed a spatially-resolved comparison of the electron densities.34,52,55 A useful tool for the spatially-resolved comparison of the electron densities from subsystem DFT and supermolecular KSDFT calculations is a topological analysis according to the theory of atoms-inmolecules,73 that has first been applied to the analysis of electron densities obtained from subsystem DFT by Kiewisch et al.34 In Ref. 34, the electron densities obtained for HOH· · · F− , F–H–F− , and the nucleic acid base pair adenine-thymine were analyzed, and it was found that if the PW91k kinetic-energy functional is used to approximate vT [ρI , ρII ], accurate electron densities can be obtained for the considered hydrogen-bonded systems. This

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is true even for F–H–F− , which contains a very strong hydrogen bond. For this complex, subsystem DFT yields an almost symmetrical electron-density distribution, even though a non-symmetric partitioning into an HF and an F− subsystem is employed. A similar analysis was performed in Ref. 55, where coordination compounds in which the bonding has a significant covalent character were considered. First, ammonia borane NH3 BH3 was considered, divided into an NH3 and a BH3 subsystem, which are connected by a dative bond. This bond is actually weaker than the one found in F–H–F− , but due to its larger covalent character more difficult to describe in subsystem DFT. While subsystem DFT using the PW91k kinetic-energy functional to approximate vT [ρI , ρII ] yields overall a reasonable electron density, some of the features of the electron density obtained from a supermolecular KS-DFT calculation are not reproduced correctly.55 Second, the tetrahedral complex TiCl4 was considered, partitioned into a TiCl+ 3 and a Cl− subsystem. For this complex, large problems are observed in the subsystem DFT calculation if the PW91k approximation is employed. In the calculation in which Cl− is the active subsystem, a spurious charge transfer to the frozen TiCl+ 3 subsystem occurs, and in the subsequent calculation in which the TiCl+ 3 subsystem is active, a non-aufbau solution has to be enforced. These difficulties are due to the incorrect behavior of the available GGA approximations to vT [ρI , ρII ] at the frozen subsystem that was discussed in Sec. 10.3.2. If the correction suggested in Ref. 58 is applied, these problems disappear and the subsystem DFT calculations converge to aufbau solutions for both subsystems. Furthermore, the electron density obtained from subsystem DFT are in this case qualitatively correct, even though there are quantitative differences. Third, Ref. 55 considered the octahedral complex Cr(CO)6 , divided into a subsystem containing the chromium atom and a subsystem consisting of the CO ligands. In contrast to TiCl4 , the bonding between the ligands and the metal is to a large part of electrostatic nature, Cr(CO)6 is a prototypical example of a metal complex in which π-backdonation plays a significant role, i.e., it presents an even more challenging test case for the available approximations to vT [ρI , ρII ]. In the subsystem DFT calculations using the PW91k approximation, no aufbau solution could be obtained. This situation is not changed by applying the correction of Ref. 58, even though in this case the number of unoccupied orbitals that are too low in energy is decreased. If one considers the electron density corresponding to this non-aufbau solution, it is found that there are large qualitative deviations to the density obtained from a supermolecular KS-DFT calculation, i.e., the available approximations to vT [ρI , ρII ] are not able to describe the bonding in Cr(CO)6 . These results of Ref. 55 show that the limit investigated in Ref. 58 and 47 is—even though it was initially investigated in a very different context—of great relevance for a description of covalent bonds. In particular, the correction suggested in Ref. 58 turns out to be essential in order to be able to obtain an aufbau solution in

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Fig. 10.4.2. Partitioning using capping groups employed in the 3-FDE scheme illustrated for dialanine. The total electron density is obtained as ρtot (r) = ρI (r) + ρII (r) − ρcap (r). This partitioning can be easily generalized to larger capping groups and to a larger number of subsystems.

the case of TiCl4 , and is found to at least work in the correct direction for Cr(CO)6 . Similar results can be expected for the NDSD approximant of Ref. 47, even though it has not been tested for coordination compounds or other systems with a covalent bonds between subsystems yet. Nevertheless, none of the available approximations to vT [ρI , ρII ] is currently able to describe the considered coordination compounds adequately. A similar picture emerges from a study74 on the family of triatomic noble gas– goldhalides that feature a gold–noble gas bond of varying strengths.75,76 None of the currently available functionals was able to provide a quantitatively correct description of the magnitude of charge transfer from the noble gas to the goldhalide unit. Better approximations to vT [ρI , ρII ] that are able to utilize the subtle information od the atomic shell structure contained in the density of the heavy atom will be needed to tackle such cases. 10.4. Introduction of capping groups 10.4.1. Three-partition frozen-density embedding For the efficient computational treatment of biochemical processes a subsystem description of proteins, in which individual amino acid residues can be used as subsystems, is very desirable. However, as discussed above the currently available approximations to vT [ρI , ρII ] are not applicable to subsystems connected by covalent bonds, which is required for such a description. Even though the developments outlined in the previous section are promising, improved approximations cannot be expected to lead to a satisfactory description of covalent bonds between subsystems in the near future. Therefore, other approaches have to be developed. One possibility to allow for a description of subsystems connected by covalent bonds is the introduction of capping groups, in a similar way as it is done within combined quantum mechanics/molecular mechanics (QM/MM) schemes.77,78 In this way, it is possible to circumvent the insufficiencies of the available approximations to vT [ρI , ρII ] since it is no longer necessary to describe the covalent bonds connecting subsystems using an approximate kinetic-energy functional. Instead, these bonds are replaced by bonds to newly introduced capping groups, which are treated within the individual subsystems.

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Applying this strategy, a molecule is cut into two subsystems across some arbitrary covalent bond, and two capping groups are included in each of the two subsystems. These capping groups are chosen such that the electronic structure of the bond that is cut is preserved as well as possible. The two capping groups are joined to form a “capping molecule”, and the electron density of this capping molecule is subtracted from the densities of the subsystems to correct for the introduced capping groups. The total electron density is thus partitioned according to ρtot (r) = ρI (r) + ρII (r) − ρcap (r),

(10.4.23)

where ρI and ρII are the electron densities of subsystems I and II (including the corresponding capping groups), respectively, and ρcap is the electron density of the capping molecule. This partitioning is illustrated in Fig. 10.4.2 for the example of a dialanine molecule. The use of such a partitioning was first suggested for the description of proteins by Zhang and coworkers,12 who employed such a partitioning to calculate the electron density of proteins from that of the individual subsystems, which were each treated as isolated molecules (molecular fractionation with conjugate caps, MFCC scheme). However, this MFCC scheme does not include any effect of the neighboring amino acid residues and of the protein environment on the individual subsystems, so that it can only be considered as a first approximation to an adequate subsystem treatment of proteins. The use of the above partitioning of the total density in the FDE scheme was first proposed by Casida and Wesolowski.79 However, they did not present an implementation of this three-partition FDE (3-FDE) scheme, and their formalism did not ensure that the total electron density is positive. This positivity of the total electron density is ensured in the 3-FDE formalism presented in Ref. 80 by requiring that inside a suitably defined “cap region”, the density of the active subsystem I equals the density of the cap molecule. It is important to introduce such a constraint, since in partitioning of the total electron density given in Eq. (10.4.23) the density of the cap molecule is subtracted, so that regions of unphysical negative electron density could otherwise be obtained. Starting from the total DFT energy written as a functional of the densities of the three densities ρI , ρII , and ρcap ], E[ρI , ρII , ρcap ], one can derive a set of one-electron equations for the KS orbitals of subsystem I in the presence of a given frozen density ρII and a given cap density ρcap by minimizing E[ρI , ρII , ρcap ] with respect to ρI under the constraint that ρI (r) = ρcap (r) inside a cap region VIcap . This leads to the KSCED-like equations80   ∇2 (I) (I) KSCED − + veff [ρI , ρII , ρcap ](r) φi (r) = i φi (r); i = 1, . . . , NI /2, (10.4.24) 2 in which the effective potential is now given by  KS emb veff [ρI ](r) + veff [ρI , ρII ](r) for KSCED veff [ρI , ρII , ρcap ](r) = vcap [ρI , ρcap ](r) for

r∈ / VIcap (10.4.25) r ∈ VIcap .

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Outside the cap region VIcap , the effective potential contains—similar as in the case KS of the conventional two-partition FDE scheme—the KS potential veff [ρI ] of the isolated subsystem I as well as an effective embedding potential that reads, Z ρII (r 0 ) − ρcap (r 0 ) 0 emb nuc nuc veff [ρI , ρII , ρcap ](r) = vII (r) − vcap (r) + dr |r − r 0 | δExc [ρ] δExc [ρ] + − δρ δρ ρ=ρI ρ=ρI +ρII −ρcap + vT [ρI , ρII , ρcap ],

where the kinetic-energy component is given by, δTs [ρ] δTs [ρ] vT [ρI , ρII , ρcap ] = − . δρ ρ=ρI +ρII −ρcap δρ ρ=ρI

(10.4.26)

(10.4.27)

Inside the cap region VIcap , the effective potential is given by a cap potential vcap [ρI , ρcap ], which arises from the constraint that the electron density of the active subsystem I, ρI , should be equal to the density of the cap molecule ρcap . This cap potential has to be determined such that this constraint is satisfied. In practice, different algorithms can be applied to achieve this, and in Ref. 80 the algorithm proposed by van Leeuwen and Baerends60 has been employed, i.e., the cap potential is updated iteratively according to, new vcap (r) =

ρold I (r) old v (r), ρcap (r) cap

(10.4.28)

is the electron density of subsystem I obtained using the cap potential where ρold I old vcap in a certain iteration. To assess the accuracy of the 3-FDE scheme, one can compare the electron densities obtained from 3-FDE calculations, in which the densities of both subsystems have been optimized (possibly using a number of freeze-and-thaw iterations) to those from supermolecular KS-DFT calculations. So far, for the cap molecule the electron density calculated for the isolated molecule has always been used. Test calculations on different dipeptides show that the 3-FDE scheme can accurately model both the polarization of a subsystem due to its environment as well as the effects of hydrogen bonding.80 However, since the electron density in the cap region is constrained, the distribution of the electrons between the subsystems is also fixed, and a polarization of the bond between the subsystems cannot be accounted for. This problem could possibly be addressed by a more adequate choice of the density of the cap molecule that accounts for this polarization. 10.4.2. Application to the description of proteins The 3-FDE scheme can be easily generalized to an arbitrary number of subsystems.80 In the general case of nsub subsystems and ncap capping molecules, the

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total electron density is given by,

ρtot (r) =

n sub X i=1

ncap

ρi (r) −

X

ρcap j (r),

(10.4.29)

j=1

where ρi is the electron density of subsystem i (including the corresponding capping groups) and ρcap is the electron density of cap molecule j. This leads to a subsystem j DFT formulation that allows it to treat proteins, and in which the individual amino acid residues can be used as subsystems. In practical applications of such a subsystem DFT treatment, one faces the rather tedious task of defining the individual subsystems and of determining the atomic coordinates of the atoms in the capping groups. Furthermore, a large number of individual calculations have to be performed for each cap molecule and for each subsystem. To automate these tasks, the scripting framework PyAdf can be employed.81 To illustrate the accuracy of the proposed subsystem DFT treatment of a protein, Ref. 80 includes calculations performed for the protein ubiquitin [see Fig. 10.4.3(a)]. For this protein consisting of 76 amino acids, a full supermolecular KS-DFT calculation is still possible, so that the electron densities obtained from a subsystem treatment can be compared to the one from a supermolecular KS-DFT calculation. In the subsystem treatment, different levels of approximation can be applied. First, the electron densities obtained from calculations for the isolated molecules can be applied for all subsystems, corresponding to the MFCC scheme. Second, the electron density of each of the subsystems can be optimized in a 3-FDE calculation in which the embedding potential of the isolated fragments is included for each subsystem, and finally, the electron densities of all subsystems can be optimized iteratively in freeze-and-thaw cycles. The difference in the total electron density with respect to the supermolecular KS-DFT calculation is shown in Fig. 10.4.3 for calculations employing these different levels of approximations. It can be seen that already the simplest possible 3-FDE description improves significantly over the MFCC scheme, and that a very accurate electron density can be obtained if five freeze-and-thaw cycles are applied. The subsystem DFT description based on the 3-FDE scheme is particularly suited for applications where focus can be placed on a small part of a protein, such as, for instance, an active site of an enzyme, or for the calculation of rather localized molecular properties. In this case, only a few subsystems have to be treated accurately by employing 3-FDE with several freeze-and-thaw cycles, while for the electron density of subsystems further away from the region of interest, a simple approximation using the densities obtained from calculations for isolated molecules as it is used in the MFCC scheme can be used. Such a strategy is similar to the one chosen in applications of FDE for modeling solvent effects on molecular properties.17,26,37–39

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Fig. 10.4.3. (a) Cartoon representation of the secondary structure of ubiquitin. (b-d) Isosurface plots (contour value 0.002 a.u.) of the difference densities between (b) the MFCC calculation, in which all subsystems are treated as isolated fragments, (c) the 3-FDE(0) calculation, in which the embedding potential of the isolated fragments is included for each subsystem, and (d) the 3-FDE(5) calculation, in which five freeze-and-thaw iterations are performed, and the conventional supermolecular KS-DFT calculation for ubiquitin. Reprinted with permission from Ref. 80. Copyright 2008 American Insitute of Physics.

10.5. Conclusions and outlook For the computational treatment of a number of interesting chemical problems, in particular for studying biochemical processes and for investigating condensed phase chemistry, a subsystem DFT treatment is advantageous. Such a subsystem treatment not only allows an efficient computational treatment of large systems, but it also facilitates the analysis of the computational results by offering a picture in terms of the system’s chemical building blocks. Furthermore, a subsystem treatment makes it possible to focus on certain parts of a system, and to employ accurate (relativistic) wave-function based methods for certain parts of the system.46 A subsystem DFT treatment based on the FDE scheme introduced by Wesolowski and Warshel16 requires the use of approximate density-functionals for the nonadditive kinetic energy Tsnadd[ρI , ρII ] and the corresponding component of

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the embedding potential vT [ρI , ρII ]. So far, approximations based on GGA kinetic energy functionals have been used in practical applications of the FDE scheme. While those approximations are sufficiently accurate for weak interactions and hydrogen bonds, they are not adequate for the description of covalent bonds between subsystems. However, a treatment of subsystems connected by covalent bonds is essential for studying, e.g., large biological molecules or transition metal catalysts. Therefore, the development of subsystem DFT schemes applicable to such systems presents an important challenge. One possible approach to achieve this goal is the development of improved approximations to the kinetic-energy component of the embedding potential. A promising strategy for the development of improved approximations to vT [ρI , ρII ] is the investigation of exact limits, a route that has also been successful for the development of non-empirical approximations to the exchange–correlation functional (see, e.g., Ref. 82). One such exact limit for vT [ρI , ρII ] that has been studied recently is the limit that the electron density of the active subsystem ρI is small, and different ways to improve the description of this limit have been suggested.47,58 However, while these approximations improve the description in a number of cases, they both do not strictly incorporate the exact limit. Nevertheless, it has been shown that an improved description of the embedding potential at the frozen subsystem leads to an improved description of covalent bonds in some cases, even though there are still severe deficiencies, as has been shown for representative coordination compounds.55 Therefore, further improvements of the approximations to vT [ρI , ρII ] are necessary. From the recent developments reviewed here, some possible directions for future work emerge. First, it appears that the previously used decomposable approximations, which are derived from an approximate kinetic-energy functional, have reached their limits, and that non-decomposable approximations, which approximate Tsnadd [ρI , ρII ] or vT [ρI , ρII ] directly, are more promising. Second, for an accurate description of covalent bonds it might be necessary to go beyond the framework initially suggested by Wesolowski and Warshel,16 i.e., to abandon the restriction that an explicit density functional that only locally depends on the densities ρI and ρII should be used for approximating vT [ρI , ρII ]. An example of such an approximation is the position-dependent correction proposed in Ref. 58. In analogy to the development of approximate exchange–correlation functionals, one could imagine to climb the next rugs of Jacob’s ladder83 for approximations to vT [ρI , ρII ] by introducing dependencies on the occupied KS orbitals of subsystem I. A further step could be to introduce also a dependency on the KS orbitals of the frozen subsystem II, at the price of abandoning the idea of an “orbital-free” embedding scheme. A different line of development could arise by applying nonlocal approximate kinetic-energy functionals, such as the ones developed by Carter and coworkers,84–86 to approximate vT [ρI , ρII ]. These functionals are designed to describe the linear-

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response of the uniform electron gas correctly, and have successfully been applied in simulations of solid state systems using orbital-free DFT, even for materials with significant covalent bonding, such as silicon. In particular, these functional have recently been reformulated in real-space,87,88 which allows for an application in molecular DFT codes. However, since the development of improved approximations to vT [ρI , ρII ] is challenging, also other approaches that make a subsystem description possible in which subsystems are connected by covalent bonds are needed. Therefore, a generalization of the FDE scheme to three partitions has been developed that treats covalent bonds between subsystems by introducing capping groups. This allows it to treat subsystems connected by covalent bonds, even though additional approximations have to be introduced. Even though this generalization has been initially applied to proteins which are partitioned into individual amino acids, the method is applicable to arbitrary (bio)-molecules and arbitrary partitioning Altogether, significant progress has been made towards a description of covalent bonds within subsystem DFT, both on the side of the development of improved approximations to the kinetic-energy component of the embedding potential, and by developing ways to circumvent the problems in the currently available approximations. Acknowledgments The authors would like to acknowledge extensive and fruitful discussions with Tomasz Wesolowski (University of Geneva), in particular on the issues discussed in Sec. 10.3.2, and thank the Netherlands Organization for Scientific Research (NWO) for financial support via a Rubicon scholarship (C.R.J.) and the Vici program (L.V.). C.R.J. further thanks the DFG-Center for Functional Nanostructures. References 1. M. Reiher, Ed., Atomistic Approaches in Modern Biology. vol. 268, Topics in Current Chemistry. (Springer, Berlin, 2007). 2. R. R. Schrock, Angew. Chem., Int. Ed. 47, 5512 (2008). 3. G.-J. Kroes, Science 321, 794 (2008). 4. P. Huang and E. A. Carter, Annu. Rev. Phys. Chem. 59, 261 (2008). 5. J. VandeVondele, M. Krack, F. Mohamed, M. Parrinello, T. Chassaing, and J. Hutter, Comput. Phys. Commun. 167, 103 (2005). 6. C. Ochsenfeld, J. Kussmann, and D. S. Lambrecht. Linear-Scaling Methods in Quantum Chemistry, In Reviews in Computational Chemistry, vol. 23, pp. 1–82. WileyVCH, New York, (2007). 7. C. Herrmann, J. Neugebauer, and M. Reiher, New J. Chem. 1, 818 (2007). 8. Ch. R. Jacob and M. Reiher, J. Chem. Phys. 130, 084106 (2009). 9. P. Cortona, Phys. Rev. B 44, 8454 (1991). 10. M. Iannuzzi, B. Kirchner, and J. Hutter, Chem. Phys. Lett. 421, 16 (2006).

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11. F. Shimojo, R. K. Kalia, A. Nakano, and P. Vashishta, Comput. Phys. Commun. 167, 151 (2005). 12. D. W. Zhang and J. Z. H. Zhang, J. Chem. Phys. 119, 3599 (2003). 13. D. G. Fedorov and K. Kitaura, J. Phys. Chem. A. 111, 6904 (2007). 14. R. P. A. Bettens and A. M. Lee, J. Phys. Chem. A. 110, 8777 (2006). 15. H. M. Netzloff and M. A. Collins, J. Chem. Phys. 127, 134113 (2007). 16. T. A. Wesolowski and A. Warshel, J. Phys. Chem. 97, 8050 (1993). 17. J. Neugebauer, M. J. Louwerse, E. J. Baerends, and T. A. Wesolowski, J. Chem. Phys. 122, 094115 (2005). 18. Ch. R. Jacob, J. Neugebauer, and L. Visscher, J. Comput. Chem. 9, 1011 (2008). 19. N. Govind, Y. A. Wang, and E. A. Carter, J. Chem. Phys. 110, 7677 (1999). 20. T. Kl¨ uner, N. Govind, Y. A. Wang, and E. A. Carter, Phys. Rev. Lett. 86, 5954 (2001). 21. T. Kl¨ uner, N. Govind, Y. A. Wang, and E. A. Carter, J. Chem. Phys. 116, 42 (2002). 22. P. Huang and E. A. Carter, J. Chem. Phys. 125, 084102 (2006). 23. S. Sharifzadeh, P. Huang, and E. Carter, J. Phys. Chem. C. 112, 4649 (2008). 24. S. Sharifzadeh, P. Huang, and E. A. Carter, Chem. Phys. Lett. 470, 347 (2009). 25. T. A. Wesolowski. One-electron equations for embedded electron density: challenge for theory and practical payoffs in multilevel modelling of complex polyatomic systems. In ed. J. Leszczynski, Computational Chemistry: Reviews of Current Trends, vol. 10. (World Scientific, Singapore, 2006). 26. Ch. R. Jacob, J. Neugebauer, L. Jensen, and L. Visscher, Phys. Chem. Chem. Phys. 8, 2349 (2006). 27. T. A. Wesolowski and J. Weber, Chem. Phys. Lett. 248, 71 (1996). 28. A. Lembarki and H. Chermette, Phys. Rev. A. 50, 5328 (1994). 29. T. A. Wesolowski, Y. Ellinger, and J. Weber, J. Chem. Phys. 108, 6078 (1998). 30. T. A. Wesolowski and F. Tran, J. Chem. Phys. 118, 2072 (2003). 31. Ch. R. Jacob, T. A. Wesolowski, and L. Visscher, J. Chem. Phys. 123, 174104 (2005). 32. T. A. Wesolowski, J. Chem. Phys. 106, 8516 (1997). 33. T. A. Wesolowski, J. Am. Chem. Soc. 126, 11444 (2004). 34. K. Kiewisch, G. Eickerling, M. Reiher, and J. Neugebauer, J. Chem. Phys. 128, 044114 (2008). 35. T. A. Wesolowski, Chem. Phys. Lett. 311, 87 (1999). 36. T. Wesolowski and A. Warshel, J. Phys. Chem. 98, 5183 (1994). 37. J. Neugebauer, Ch. R. Jacob, T. A. Wesolowski, and E. J. Baerends, J. Phys. Chem. A 109, 7805 (2005). 38. J. Neugebauer, M. J. Louwerse, P. Belanzoni, T. A. Wesolowski, and E. J. Baerends, J. Chem. Phys. 123, 114101 (2005). 39. R. E. Bulo, Ch. R. Jacob, and L. Visscher, J. Phys. Chem. A 112, 2640 (2008). 40. P. Cortona, Phys. Rev. B 46, 2008 (1992). 41. P. Cortona and A. Villafiorita Monteleone, J. Phys.: Condens. Matter 8, 8983 (1996). 42. P. Cortona, A. Villafiorita Monteleone, and P. Becker, Int. J. Quantum Chem. 56, 831 (1995). 43. W. N. Mei, L. L. Boyer, M. J. Mehl, M. M. Ossowski, and H. T. Stokes, Phys. Rev. B 61, 11425 (2000). 44. M. M. Ossowski, L. L. Boyer, M. J. Mehl, and H. T. Stokes, Phys. Rev. B 66, 224302 (2002). 45. P. Vashishta, R. K. Kalia, and A. Nakano, J. Phys. Chem. B 110, 3727 (2006). 46. A. S. P. Gomes, Ch. R. Jacob, and L. Visscher, Phys. Chem. Chem. Phys. 10, 5353 (2008).

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82. J. P. Perdew, A. Ruzsinszky, J. Tao, V. N. Staroverov, G. E. Scuseria, and G. I. Csonka, J. Chem. Phys. 123, 062201 (2005). 83. J. P. Perdew and K. Schmidt. Jacob’s Ladder of density Functional Approximations for the Exchange-Correlation Energy, In eds. V. van Doren, C. van Alsenoy, and P. Geerlings, Density Functional Theory and Its Application to Materials, pp. 1–20. (American Institute of Physics, Melville, New York, 2001). 84. Y. A. Wang, N. Govind, and E. A. Carter, Phys. Rev. B 58, 13465 (1998). 85. Y. A. Wang, N. Govind, and E. A. Carter, Phys. Rev. B 60, 16350 (1999). 86. B. Zhou, V. L. Ligneres, and E. A. Carter, J. Chem. Phys. 122, 044103 (2005). 87. N. Choly and E. Kaxiras, Solid State Commun. 121, 281 (2002). 88. G. S. Ho, V. L. Ligneres, and E. A. Carter, Phys. Rev. B 78, 045105 (2008).

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Chapter 11 Orbital-Free Embedding Calculations of Electronic Spectra

Johannes Neugebauer Gorlaeus Laboratories, Leiden Institute of Chemistry, Leiden University P.O. Box 9502, 2300 RA Leiden, The Netherlands [email protected] Electronic spectra of molecules in condensed phase can be sensitive to a variety of factors arising from interactions with the surrounding solvent or more general environments including protein matrices, membranes, crystal environments or surfaces. If many chromophores are present, exciton-like interactions may lead to additional changes in the absorption properties. Orbital-free embedding calculations and related subsystem-density-functional theory approaches offer a convenient way of describing these effects in a consistent manner while exploiting the subsystem structure of the total system to keep the computational effort managable. The diabatic picture inherent in these approaches bridges the gap between exciton-like model theories and quantum chemical calculations for systems of coupled chromophores.

Contents 11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Theoretical background . . . . . . . . . . . . . . . . . . . . 11.2.1 The subsystem formulation of TDDFT . . . . . . . 11.2.2 Approximate solutions 1: Effective embedding kernel 11.2.3 Approximate solutions 2: Exciton-like treatment . . 11.2.4 Solvent effects on excitonic couplings . . . . . . . . 11.2.5 Computational details . . . . . . . . . . . . . . . . . 11.3 Subsystem TDDFT calculations of local excitations . . . . 11.4 FDEu as a quasi-diabatization scheme . . . . . . . . . . . 11.5 Excited-state couplings in hydrogen-bonded complexes . . 11.6 Solvated molecular dimers . . . . . . . . . . . . . . . . . . . 11.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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323 325 327 333 334 335 336 337 339 343 348 350 350

11.1. Introduction Quantum chemical calculations of electronic spectra often imply the assumption that the molecule under study is an isolated entity interacting with an external electromagnetic field. Since most experiments on absorption or electronic circular 323

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dichroism spectra are measured in condensed phase, such calculations can only be the first steps towards a more realistic description of experimental spectra. When trying to improve on results obtained for isolated molecules by including (parts of) the environment, several problems arise. Electronic-structure calculations become increasingly costly with the system size and are unfeasible for very large systems. In addition, the conformational dynamics of the system has to be taken into account. Furthermore, the analysis of electronic properties of a molecule in a certain environment may not be unambiguous. Electronic properties are only obtained for the total system, and any partitioning into “active subsystem” and “environmental” properties will introduce a certain arbitrariness, since the partitioning is a matter of definition. Therefore, embedding methods are very popular to study solvent or general environmental effects on molecular spectra and other properties. They include the environment only in an effective way, so that the properties obtained can directly be considered as the properties of the active subsystem, and no additional localization or partitioning step is necessary. In this context, QM/MM methods1–7 are a popular choice to study absorption properties when environmental effects are supposed to play a role, in particular in biological applications. Examples are theoretical investigations of the green fluorescent protein,8 the retinal chromophore in rhodopsin,9,10 and the photoactive yellow protein.11 For crystalline environments, embedded cluster models are frequently applied.12–16 Other popular environmental models are continuum solvation models,17–21 effective potential methods,22,23 including the socalled ab initio model potentials, which are based on the electron densities of the environment,24,25 or general multi-layer approaches such as the ONIOM model.26,27 In the context of density-functional theory (DFT), a partitioning into subsystems is possible on the basis of the electron density. This idea was exploited by Cortona in a subsystem approach to density-functional theory.28 Wesolowski and Warshel employed this idea to introduce an effective embedding approach.29 This approach corresponds to an orbital-free embedding, and is called frozen-density embedding (FDE) in the following. The electron density of the environment is considered frozen in FDE, and an effective embedding potential is derived from the density to obtain optimum molecular orbitals for the active subsystem under the influence of the environmental density. The assumption of a frozen environmental density can be relaxed by introducing so-called freeze-and-thaw cycles, in which the role of the active subsystem and the environment is exchanged.30 If both parts are fully relaxed, the distinction between active subsystem and environment becomes meaningless, and the frozen-density embedding turns into a subsystem DFT approach. The approach can easily be generalized to an arbitrary number of subsystems.31,32 Instead of using freeze-and-thaw cycles, it is also possible to perform the self-consistent-field iterations for all subsystems simultaneously by updating the electron density of the other (“environmental”) subsystems in each iteration.31 Also intermediate density update schemes are possible.32

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Concerning the nomenclature, it should be noted that other acronyms are in use for (different variants of) this approach, e.g., “frozen density functional theory” (FDFT),33 and “constrained density functional theory” (CDFT) (see, e.g., Ref. 34). The corresponding equations to be solved are called “Kohn–Sham equations with constrained electron density” (KSCED),30 so that also the expression “KSCED formalism” is employed.35 The acronym FDE can be found in a number of recent applications, in particular in the context of molecular property calculations (see, e.g., Refs. 32,36–39). It can be argued that this nomenclature is not precise in all cases, since “FDE” is sometimes used as an acronym in calculations with a completely frozen environment (FDFT), but also in cases with relaxed approximations for the environmental density (CDFT) or further partitionings of the environmental system (general subsystem DFT). The molecular orbitals of the subsystems give rise to a straightforward definition of subsystem properties as the properties obtained with the subsystem orbitals. However, a number of questions arise both concerning the fundamental basis of the formalism and the approximations that have to be introduced to make it applicable to target systems like molecules in solvents or in biological environments. E.g., problems are related to the behavior of common approximations in the embedding potentials at nuclear cusps in the environment40–42 or to the vs -representability conditions on the subsystem densities.43,44 Further formal concerns of the FDE method are discussed in Chapter 12 in this book. FDE and the underlying subsystem DFT have been extended to a timedependent density-functional theory (TDDFT) for response properties and excited states,45–48 which is a prerequisite to compute electronic spectra and related quantities like dispersion effects on polarizabilities and optical rotation. In the following, the theoretical background of orbital-free embedding and subsystem (TD)DFT calculations for excited states will be summarized. In Section 11.3, we will review previous applications of subsystem TDDFT for calculations of electronic spectra and response properties. Section 11.4 describes how FDE can be understood as a quasi-diabatization scheme for excited states, before we discuss environmental response effects on some hydrogen-bonded systems in Section 11.5. Section 11.6 deals with solvent screening effects on excitonic couplings, and conclusions are presented in Section 11.7.

11.2. Theoretical background The general idea of a subsystem DFT is to partition the ground-state electron density into contributions from all subsystem I according to ρ(r) =

X I

ρI (r),

(11.2.1)

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where the density of each subsystem I will be expressed in terms of (yet to be determined) subsystem orbitals φiI , X ρI (r) = |φiI (r)|2 . (11.2.2) i

Effective one-particle equations for the determination of the orbitals φiI can be derived by minimizing the energy functional w.r.t. the subsystem density ρI while keeping all other subsystem densities fixed (“frozen”).28,29,31,32,43 The minimization is carried out under the constraint that ρI correctly integrates to an pre-determined number of electrons NI , Z d3 rρI (r) = NI . (11.2.3)

The resulting one-particle equations (KSCED) for the subsystem orbitals φiI and orbital energies iI are similar to the conventional Kohn–Sham potential due to the nuclei in system I and the electron density ρI . But the effective potential vIsub for a certain subsystem I contains an embedding potential due to the other subsystems in addition to the Kohn–Sham effective potential (note that atomic units are used throughout, ~ = 1, e = 1, me = 1, 1/(4π0 ) = 1),   1 2 compl. sub − ∇ + v [ρI , ρI ](r) φiI = iI φiI (11.2.4) 2 or   1 − ∇2 + v KS [ρI ](r) + v emb [ρI , ρcompl. ](r) φiI = iI φiI , (11.2.5) I 2 with vIsub (r) := v sub [ρI , ρcompl. ](r) = v KS [ρI ](r) + v emb [ρI , ρcompl. ](r). The Kohn– I I Sham effective potential reads, Z X Z AI ρI (r0 ) 3 0 δExc [ρI ] v KS [ρI ](r) = − + d r + , (11.2.6) |r − RAI | |r0 − r| δρI (r) AI

and the embedding potential v emb is given as, v

emb

[ρI , ρcompl. ](r) I

=

X

AJ ,J6=I

+

Z AJ − + |r − RAJ |

Z

ρcompl. (r0 ) 3 0 I d r |r0 − r|

δExc [ρ] δExc [ρI ] δTs [ρ] δTs [ρI ] − + − . δρ(r) δρI (r) δρ(r) δρI (r)

(11.2.7)

The sum over AJ runs over all nuclei in the other subsystems, and we defined a density complementary to ρI as, X ρcompl. (r) = ρ(r) − ρI (r) = ρJ (r). (11.2.8) I J,J6=I

When solving the Kohn–Sham-like one-particle equations for system I with the embedding potential from Eq. (11.2.7), we can thus consider system I as the “active

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subsystem”, and the combination of all other subsystems as the “environment”. In Eqs. (11.2.6) and (11.2.7), Exc is the exchange–correlation potential and Ts [ρ] is the (single-particle) kinetic energy as defined in the context of Kohn–Sham density-functional theory.49 The exact single-particle kinetic energy functional in the Kohn–Sham approach requires the knowledge of the Kohn–Sham orbitals of the system, X Ts [{φ[ρ]}] = hφi | − ∇2 /2|φi i. (11.2.9) i

Since the idea of subsystem DFT is to avoid the calculation of the Kohn–Sham orbitals of the total system, the kinetic-energy functional and its functional derivatives are approximated by the sum of the single-particle kinetic energies of the subsystems plus a non-additive part of the kinetic energy, which is evaluated in terms of density-dependent expressions. This non-additive kinetic energy leads to the kinetic-energy-dependent terms in Eq. (11.2.7). Details on this formalism and its implementation can be found in Refs. 32,43. Examples of density-dependent kinetic-energy expressions are discussed in Refs. 50–56. A possible strategy in the context of orbital-free embedding calculations is to employ a non-decomposable approximation for the non-additive part of the kinetic-energy.41 Recent efforts have been made to investigate exact properties of the embedding potential,40,57 or to invert the procedure for the construction of effective embedding potentials.58 11.2.1. The subsystem formulation of TDDFT If the system described by the electron-density ρ(r) is exposed to a time-dependent perturbation in the potential, its electron-density will show a time-dependent density change δρ(r, t). The most common perturbations are (harmonic) oscillating electromagnetic fields, and typically one is interested in the Fourier components of the time-dependent density response at the frequency ω of the external perturbation, δρ(r, ω). In the following, we will briefly derive the equations to determine the frequency-dependent change in the electron density as well as the excitation energies of the subsystems and the total system. For brevity, we will skip the frequency dependence in the quantities below. Only closed-shell systems and singlet–singlet transitions are considered. In the derivations below, we use the labels i, j, . . . for occupied (spin) orbitals, and a, b, . . . for virtual (spin) orbitals. Following Refs. 45,47, we partition the total density response, δρ(r), into subsystem contributions δρ(r) =

X

δρI (r).

(11.2.10)

I

In analogy to the supermolecular case, the subsystem response densities can be expanded in terms of the occupied and virtual molecular orbitals of the respective

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subsystems (see Ref. 47), δρI (r) =

X

2δP(ia)I φiI (r)φaI (r),

(11.2.11)

(ia)∈I

where the index I labels the subsystems, and the summation is restricted to occupied–virtual pairs within subsystem I. In Ref. 47 it was mentioned that this is an approximation compared to the supermolecular case since it is assumed that the density response can be expanded in terms of intra-subsystem orbital transitions only. In a strict sense, this only applies if the subsystem orbitals are expanded into basis functions centered on the atoms of a certain subsystem only (“monomer expansion”), which is the most common type of application. If, however, in a calculation on subsystem I also basis functions centered at the nuclei of the other subsystems are provided (“supermolecular expansion”), it can be argued that (virtual) molecular orbitals located in the other subsystems may be obtained. Consequently, it should also be possible to describe charge-transfer-like excitations. A problem is, however, that most of the currently available approximations to the embedding potential have an incorrect structure far away from the active subsystem, which affects orbitals that are mainly localized in the environment.40,42 A restriction of the basis sets to the active subsystem only can be an advantage in practice, since no artificially low long-range CT excitations between different weakly interacting fragments will be obtained.36,59 An expression for the expansion coefficients δP(ia)I is obtained in analogy to the supermolecular case as,47 δP(ia)I = χs(ia)I δv(ai)I ,

(11.2.12)

2 − ω 2 ) and ω(ia)I = iI − aI . The matrix elements where χs(ia)I = ω(ia)I /(ω(ia) I of the perturbation in the potential, δv(ia)I , contain two contributions. The first contribution is the external perturbation applied to the system, and the second contribution is the change in the effective potential induced by the density change, which is assumed to react instantaneously to the external perturbation with the same frequency, ind ext , + δv(ia) δv(ia)I = δv(ia) I I

(11.2.13)

If the response of the effective one-electron potentials w.r.t. the density change is assumed to be linear, we obtain the following expression for the induced effective potential in system I,  sub  XZ δvI (r1 ) δvIind (r1 ) = d3 r2 δρJ (r2 ), (11.2.14) δρJ (r2 ) J

where the functional derivative has to be understood as a partial derivative in which all other subsystem densities ρK for K 6= J are kept fixed. The effective potential for a subsystem I is given in Eq. (11.2.5) and consequently, its derivative w.r.t. the

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density ρJ is obtained as δvIsub (r1 ) 1 δ 2 Exc [ρ] δ 2 Ts [ρ] δ 2 Ts [ρI ] = + + − δIJ , δρJ (r2 ) |r1 − r2 | δρ(r2 )δρ(r1 ) δρ(r2 )δρ(r1 ) δρI (r2 )δρI (r1 ) δ 2 Ts [ρI ] tot = fCoul (r1 , r2 ) + fxck (r1 , r2 ) − δIJ , (11.2.15) δρI (r2 )δρI (r1 ) δ 2 Ts [ρI ] tot = fCxck (r1 , r2 ) − δIJ , (11.2.16) δρI (r2 )δρI (r1 ) tot where fCoul is the Coulomb part of the kernel, fxck is given as, tot fxck (r1 , r2 ) =

δ 2 Ts [ρ] δ 2 Exc [ρ] + δρ(r2 )δρ(r1 ) δρ(r2 )δρ(r1 )

(11.2.17)

tot tot and fCxck = fCoul + fxck . Inserting this into Eq. (11.2.14), we obtain

δvIind (r1 )

  δ 2 Ts [ρI ] tot = d r2 fCxck (r1 , r2 )δρ(r2 ) − δρI (r2 ) δρI (r2 )δρI (r1 ) "   Z δ 2 Ts [ρI ] tot = d3 r2 fCxck (r1 , r2 ) − δρI (r2 ) δρI (r2 )δρI (r1 ) # X tot + fCxck (r1 , r2 ) δρJ (r2 ) . (11.2.18) Z

3

J,J6=I

The approximation introduced in Ref. 46 for the FDE case, which will be called “uncoupled FDE” (FDEu; synonyms are “local response approximation” or “neglect of dynamic response of the environment”) in the following, is to assume that only the response of the embedded system itself has to be taken into account for local excitations, δvIind,FDEu(r1 ) =

Z

d3 r2

 tot (r1 , r2 ) − fCxck

δ 2 Ts [ρI ] δρI (r2 )δρI (r1 )



 δρI (r2 ) ,

(11.2.19)

i.e., the density of the “environment” is assumed to be frozen even if an external perturbation is applied. It is important to note that even in the case of FDEu a change in the response part of a TDDFT calculation takes place because of the difference in the (intra-subsystem) response kernel. Environmental effects are thus not restricted to a change in the ground-state potential. The relative magnitudes of these two effects have been analyzed in Ref. 60. The matrix elements of the induced potential in the full expression, Eq. (11.2.18), are given as, X ind eff δv(jb) =2 K(jb) δP(ia)J , (11.2.20) I I ,(ia)J (ia)J

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where the sum runs over all orbital transitions (ia)J in all subsystems J and  Z eff 3 K(jb) = d r φjI (r1 )φbI (r1 )× 1 I ,(ia)J    Z δ 2 Ts [ρI ] tot d3 r2 fCxck (r1 , r2 ) − δIJ φiJ (r2 )φaJ (r2 ) . δρI (r2 )δρI (r1 ) (11.2.21) In contrast to the FDEu approximation, the full approach will be denoted as ind ind “coupled FDE” or FDEc in the following. Note that δv(jb) = δv(bj) for pure (nonI I ext hybrid) exchange–correlation functionals considered here. Furthermore δv(jb) = I ext δv(bj)I for real (e.g., electric-field) perturbations. Similar to the supermolecular case we arrive at   X ext eff +2 K(jb) δP(ia)J  . (11.2.22) δP(jb)I = χs(jb)I δv(jb) I ,(ia)J I (ia)J

Eq. (11.2.22) can be re-written in the following way, " 2 # X ω(ia) − ω2 J ext eff δv(jb)I = (11.2.23) δ(ia)J ,(jb)I − 2K(jb)I ,(ia)J δP(ia)J ω(ia)J (ia)J  X  δ(ia)J ,(jb)I eff = ω2 − ω(aiτ )J δ(iaτ )J ,(jbσ)I − 2K(jb) δP(ia)J I ,(ia)J ω(ai)J (ia)J

(11.2.24)

or, −1/2 ext S(jb)I ,(jb)I δv(jb) I

=

−1/2 S(jb)I ,(jb)I

X

(ia)J eff −2K(jb) I ,(ia)J

where the diagonal matrix S is given as,

i

S(ia)J ,(jb)I = We thus obtain, −1/2

ext S(jb)I ,(jb)I δv(jb) = I

ω2

δ(ia)J ,(jb)I − ω(ai)J δ(ia)J ,(jb)I ω(ai)J −1/2

1/2

× S(ia)I ,(ia)I S(ia)J ,(ia)J δP(ia)J (11.2.25)

δ(ia)J ,(jb)I . ω(ai)J

(11.2.26)

Xh −1/2 2 ω 2 − ω(ai) δ − 2S(jb)I ,(jb)I J (ia)J ,(jb)I

(ia)J

i −1/2 1/2 eff ×K(jb) S ,(ia) (ia)J ,(ia)J S(ia)J ,(ia)J δP(ia)J (11.2.27) I J X  1/2 (11.2.28) = ω 2 − Ω(jb)I ,(ia)J S(ia)J ,(ia)J δP(ia)J (ia)J

=

X

(ia)J

 ω 2 − Ω(jb)I ,(ia)J F(ia)J ,

(11.2.29)

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where the second line defines elements of the matrix Ω and the last line defines elements the vector F. This equation can be expressed in matrix form as,  2  ω − Ω F = S−1/2 δvext , (11.2.30)

ext where all components δv(jb) and δP(jb)I are stored in the vectors δvext and δP, I 1/2 respectively, and F = S δP. A formal solution for the perturbed density matrix is thus,  −1 −1/2 ext δP = S−1/2 ω 2 − Ω S δv , (11.2.31)

which can form the starting point for subsystem-based calculations of frequencydependent polarizabilities as explained in Ref. 48. Excitation energies can be identified from those frequencies ω for which the density response δP(ω), or alternatively F(ω), diverges even in the case of a vanishing external perturbation. I.e., we can set the right-hand-side of Eq. (11.2.30) to zero to find the excitation energies and thus have to solve the eigenvalue equation,   Ω − ων2 Fν = 0. (11.2.32)

In contrast to conventional TDDFT, the matrix Ω and the vectors Fν now have a subsystem structure. E.g., the matrix Ω can be divided into intra- (ΩII ) and intersubsystem blocks (ΩIJ ; for systems I, J = A, B, . . . , Z), so that we obtain, 

ΩAA ΩAB  ΩBA ΩBB   . ..  .. . ΩZA ΩZB

  1AA 0AB · · · ΩAZ  0BA 1BB · · · ΩBZ   2 .. .  − ων  .. ..  . . ..  . 0ZA 0ZB · · · ΩZZ

   A   0A Fν · · · 0AZ   B  · · · 0BZ    Fν   0B     .  =  .  . (11.2.33) .. .. . .   ..   ..  0Z FZ · · · 1ZZ ν

Compared to the eigenvalue equations for the isolated systems I, J, . . ., this equation contains three differences: (1) The inter-system coupling blocks ΩIJ are absent in isolated molecule calculations, (2) the coupling matrix elements carry the effective kernel that contains exchange–correlation and kinetic energy contributions from all subsystems, and (3) the orbitals and orbital energies employed in the calculation of the matrix elements are obtained from a ground-state FDE calculation. The approaches in Refs. 61,62 for calculating exciton-like couplings between local transitions starting from isolated monomers are perturbative and thus have to consider effects due to the non-orthogonality of the subsystem orbitals. In contrast to that, the effects arising from non-orthogonality are implicitly contained in our equations by means of the non-additive kinetic energy contributions, both to the groundstate potential and to the response kernel. Both approaches may be affected if non-orthogonality effects become to strong because either the perturbation theory breaks down, or because of the limitations in the approximate non-additive kinetic energy functionals currently in use. While the effects (2) and (3) mentioned above are also present in the FDEu approximation, the couplings to the environment are not. Assuming that we already

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know the eigenvectors of the subsystem matrices ΩII [from FDEu calculations in the approximate form of Eq. (11.2.19)], we can set up a unitary transformation matrix,   UA 0AB · · · 0AZ  0BA UB · · · 0BZ    (11.2.34) U= . .. . . ..  ,  .. . .  . 0ZA 0ZB · · · UZ where UI are the square matrices containing as columns all the eigenvectors of the subsystem matrices ΩII . Multiplying Eq. (11.2.32) from the left by U† and inserting UU† in front of Fν yields  †  U ΩU − ων2 U† Fν = 0 (11.2.35) i h 2 ˜ ν = 0, ˜ − ων F (11.2.36) Ω

˜ ν = U† Fν and Ω ˜ = U† ΩU. This transformation will bring Eq. (11.2.33) where F into the following structure, 

2 ωA,0  ˜  ΩBA  .  .  . ˜ ZA Ω

˜ AB Ω 2 ωB,0 .. . ˜ ZB Ω

··· ··· .. . ···

  ˜ AZ Ω 1AA 0AB ˜ BZ   0BA 1BB Ω  2 .. . ..   − ων   .. . .  2 0ZA 0ZB ωZ,0

   ˜ A   Fν 0A · · · 0AZ   ˜B   · · · 0BZ  Fν   0B      .  =  .  , (11.2.37)  .. ..  . .   ..   ..  ˜Z 0Z · · · 1ZZ F ν

2 where again a tilde denotes transformed quantities, and ωI,0 is a diagonal matrix containing the squared resonance frequencies of the subsystems in the absence of intersystem couplings. Transition dipole moments and oscillator strengths within TDDFT can be obtained from the solution vectors Fν ,63 which are obtained as

˜ν . Fν = UF

(11.2.38)

As is shown in Ref. 48 by a comparison to the sum-over-state expressions for electricdipole–electric-dipole and electric-dipole–magnetic-dipole polarizabilities, the components of the electric (µ) and magnetic (m) transition dipole moments for an excitation from the ground state Ψ0 to an excited state Ψν can be obtained as, s X (aJ − iJ ) µα,0ν = hΨ0 |ˆ µα |Ψν i = hφiJ |ˆ µα |φaJ i Fν,(ia)J , (11.2.39) ων (ia)J X r ων mα 0ν = hΨ0 |m ˆ α |Ψν i = hφiJ |m ˆ α |φaJ iFν,(ia)J , (11.2.40) aJ − iJ (ia)J

where µ ˆα and m ˆ α are the α-components of the electric and magnetic dipole operator, respectively. These expressions can be used to calculate oscillator and rotatory strengths.48

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11.2.2. Approximate solutions 1: Effective embedding kernel The solution of Eq. (11.2.33), or, alternatively, Eq. (11.2.37), is already simpler than the solution of the corresponding supermolecular response equation, since intersubsystem orbital transitions are excluded if monomer (subsystem) basis sets are employed. Assuming nocc and nvirt occupied and virtual orbitals, respectively, per subsystem in a supersystem with N identical fragments, this would mean that the dimension of the matrix Ω reduces from (N ×nocc )×(N ×nvirt ) to N ×(nocc ×nvirt ) only. Since almost all of the inter-subsystem transitions are not correctly described by conventional TDDFT and hardly couple to other transitions, not much information is lost. It was actually proposed in Ref. 64 to remove all orbital pairs with long-range charge-transfer character also from a conventional TDDFT procedure in case no correction can be applied for them. However, further simplifications are necessary to make this coupled response approach applicable to very large systems. Two different scenarios can be distinguished, for which different solution algorithms will be preferable. Of course, intermediate cases may require a combination of both. The first approach concerns cases in which one chromophore is embedded in an environment, and has briefly been discussed in Ref. 47. For simplicity, we will assume that there are only two subsystem: the active subsystem A, and the environment B. The environment may of course consist of several subsystems as well, so that further simplifications are possible. From Eq. (11.2.33) we get two sets of equations,   B ΩAA − ων2 FA (11.2.41) ν + ΩAB Fν = 0   2 B A ∧ ΩBB − ων Fν + ΩBA Fν = 0. (11.2.42)

The second equation can be solved for FB ν ,  −1 2 FB ΩBA FA ν = ων − ΩBB ν,

which can be inserted into Eq. (11.2.41), h i  −1 ΩAA + ΩAB ων2 − ΩBB ΩBA − ων2 FA ν = 0,

so that we can formally solve the equation,  eff  ΩAA − ων2 FA ν = 0,

where the effective response matrix is given as,  2 −1 Ωeff ΩBA . AA = ΩAA + ΩAB ων − ΩBB

(11.2.43)

(11.2.44)

(11.2.45)

(11.2.46)

Further approximations are possible, e.g., by treating ΩAA exactly and using approximations for [ων2 − ΩBB ]−1 (see also Ref. 65, pp. 46–48). As mentioned in Ref. 47, it will be more problematic to use Eq. (11.2.46) directly, since the effective matrix to be diagonalized depends on the sought-for eigenvalues, so that Eq. (11.2.45) has to be solved iteratively, and a different matrix has to be constructed and diagonalized for each eigenvalue. Moreover, numerical instabilities can occur if the

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diagonal elements of ΩBB are close to the eigenvalue ων2 . Eq. (11.2.46) can be regarded as a matrix analog to the effective kernel presented in Ref. 45. 11.2.3. Approximate solutions 2: Exciton-like treatment Another very important class of applications concerns systems of identical (or similar) pigment molecules, for which excitonic couplings can occur. This means that delocalized excitations arise from interactions between (near-)degenerate locally excited states of the monomers. In such a situation, it is advantageous to start from Eq. (11.2.37). Usually, a few low-lying, local excited states can easily be calculated with the FDEu approach employing a Davidson-type diagonalization66,67 of the ma˜ are obtained. trices ΩAA , ΩBB , etc., so that only selected diagonal elements of Ω We can then set up a truncated eigenvalue problem in a reduced basis of these local excitations only. Let us assume that we know a set {µA } of (uncoupled) transitions in system A and a set {νB } of transitions of subsystem B in terms of their eigenvectors stored in the columns νA,B of matrix U. Then we need the following off-diagonal matrix elements in order to set up the truncated eigenvalue problem, X ˜ µA νB = Ω U(ia)A µA Ω(ia)A (jb)B U(jb)B νB (11.2.47) (ia)∈A, (jb)∈B

X

=

√ √ eff 2U(ia)A µA ω(ai)A K(ia) ω(bj)B U(jb)B νB . (11.2.48) A ,(jb)B

(ia)∈A, (jb)∈B

Here, the sums run only over the orbital transitions within either subsystem A or B, respectively. This equation can be written more explicitly as, Z Z X √ 3 ˜ ΩµA νB = 2U(ia)A µA ω(ai)A d r1 d3 r2 × (ia)∈A, (jb)∈B

 √ tot φiA (r1 )φaA (r1 )fCxck φjB (r2 )φbB (r2 ) ω(bj)B U(jb)B νB Z Z X √ 3 tot = d r1 2U(ia)A µA ω(ai)A φiA (r1 )φaA (r1 ) d3 r2 fCxck 

×

(ia)∈A

X

(jb)∈B

 √ φjB (r2 )φbB (r2 ) ω(bj)B U(jb)B νB  .

(11.2.49)

Our implementation of this algorithm in the Amsterdam Density Functional (Adf) package68,69 employs density fitting techniques similar to the supermolecular case70 for the whole term in square brackets in the last line of the above equation, i.e., X X √ φjB (r2 )φbB (r2 ) ω(bj)B U(jb)B νB ≈ aλ fλ (r2 ), (11.2.50) δρνB = (jb)B

λ

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so that only matrix elements of an induced potential caused by this fitted density have to be calculated. With this fitting procedure, the potential induced in subsysind tem A by transition νB , δvA,ν , can be calculated analytically,47 so that the matrix B element can be obtained by numerical integration as Z X √ ind ˜ ΩµA νB = d3 r1 2U(ia)A µA ω(ai)A φiA (r1 )δvA,ν (r1 )φaA (r1 ). (11.2.51) B (ia)A

Hence, only the eigenvectors for the transitions {µA , νB }, which describe the local subsystem transition densities, are needed in order to set up the truncated eigenvalue problem. It is usually possible to estimate which monomer transitions will couple significantly. A first approximation is to evaluate the coupling between the monomer transitions in terms of F¨ orster-type transition-dipole interactions.71 This already indicates that the largest effects in the coupled FDE (FDEc) approach can be expected for intense transitions (in the absence of strong short-range couplings due to exchange or charge-transfer effects). Also transition monopole interactions in terms of atomic partial transition charges,72,73 or a real-space partitioning of the transition densities and a subsequent numerical integration of the Coulombic interactions as in the transition density cube method74 have been employed to evaluate couplings between local excitations in cases where the dipole approximation is not accurate enough. In most of these approaches, it is assumed that only certain excited states couple, which are selected right from the beginning. The perturbative methods in Refs. 61,62,75,76, however, are formulated for more general cases. The FDEc protocoll can be employed to assess the approximation that only certain couplings are required to adequately describe the excitations in the supersystem. In principle, all subsystem excitations can be calculated, and all couplings can be included. The effect of increasing the number of coupled excitations has been tested in Ref. 47, and it was found that the effect of additional transitions is usually weak if there is an intense monomer transition in a dimer of identical molecules. Such analyses allow to understand why excitonic coupling models work for systems like photosynthetic antenna complexes,77,78 and can justify their application. 11.2.4. Solvent effects on excitonic couplings Since FDE was shown to be an efficient model for solvent effects on electronic excitations when applied within the local response approximation (FDEu), it may be worthwhile to employ it also to consider solvent effects on excitonic couplings and excitation-energy transfer rates. From a fundamental point of view, such an approach would be very appealing since it is based entirely on density functional theory. However, such a simulation also causes some additional difficulties in comparison to continuum models: Since it is an explicit model, it requires to consider a representative set of snapshots of solvent configurations instead of just one structure.36 Moreover, the size of the solvation shell must be sufficiently large, which

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can involve hundreds of solvent molecules even for medium-sized molecules.37 It was proposed47 to include the effect of a surrounding medium on a system of coupled chromophores in the following way: To calculate the subsystem excitations of a given chromophore, the entire surrounding system (i.e., other chromophores and solvent) can be included according to Eqs. (11.2.7) and (11.2.19), while couplings are afterwards only included in a selective manner between the chromophores. This would assume that the solvent effect mainly changes the effective embedding potentials and the local response of the solvated chromophores, but that the environmental response is less important. The study in Ref. 60 discusses this neglect of solvent response by a comparison to a polarizable classical solvent model. The importance of solvent response effects on excitation-energy transfer couplings has been investigated in terms of continuum solvation models,61,62,75,76 but also with polarizable force fields.79 For the FDEc approach as outlined above, this is a challenge since a large number of excited states may have to be coupled, because the dynamic polarization of the environment as a reaction to the perturbation has to be modeled in a sum-over-states like manner. One approximation that can be applied, and that is related to approximations for [ων2 − ΩBB ]−1 in Eq. (11.2.46), is to neglect all inter-subsystem couplings in the environmental part, and only include coupling matrix elements among the solvated chromophores and between the chromophores and the solvent molecules. The effect of such an approximation will be tested in Section 11.6. 11.2.5. Computational details The subsystem TDDFT formalism is implemented47,80 in a development version of the Adf program,68,69 which was employed for all calculations presented in the following. It makes use of the Adf implementation32 of the frozen-density embedding scheme proposed by Wesolowski and Warshel,29 which is based on a previous pilot implementation in Adf.46 The subsystem TDDFT implementation makes use of density-fitting techniques, prescreening of solution vector elements, and subsystem integration grids as explained in Ref. 80 for improved efficiency. The calculations presented in the following in general used the same technical settings as applied in the corresponding calculations in earlier work. They were carried out with the Becke–Perdew functional, dubbed BP86,81,82 with the “statistical averaging of (model) orbital potentials” (SAOP) potential,83–85 or with the Perdew–Wang exchange–correlation functional,86 which is denoted as PW91. The TZP or ETpVQZ basis sets from the Adf basis set library have been employed. As a default for the non-additive kinetic energy contribution, the so-called GGA97 generalizedgradient approximation (GGA) to the kinetic-energy functional was employed.51 It has the same functional form for the enhancement factor F (s) as the exchange functional of Perdew and Wang,86 and has therefore sometimes also been denoted as PW91k in previous work. It was parameterized for the kinetic energy by Lembarki and Chermette.87 The exchange–correlation component of the embedding poten-

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tial is evaluated with the same exchange–correlation functional that is used for the non-embedding part. In case of the orbital-dependent SAOP potential, we use the potential derived from the Becke exchange-energy functional81 and the Perdew– Wang correlation functional,86 which is denoted as BPW91. The adiabatic local density approximation (ALDA) was employed for the exchange–correlation kernel in TDDFT calculations. In subsystem TDDFT calculations, this also applies to the kinetic-energy component of the response kernel. A kernel contribution derived from the Thomas–Fermi kinetic-energy functional88,89 is used in that case. Graphics of molecular structures have been prepared with Vmd.90

11.3. Subsystem TDDFT calculations of local excitations Frozen-density embedding has been employed in a number of previous studies on absorption spectra. The initial study in Ref. 46 dealt with the effect of hydrogen bonding in DNA base pairs on the excitation energies and presented the first application of FDEu (in the terminology employed here) for excited states. The shifts of the excitation energies induced by the hydrogen bonds could be obtained to good accuracy compared to supermolecular calculations, and allowed for a straightforward analysis of (ground-state) polarization effects. This could be achieved by a comparison of calculations with and without relaxation of the “environmental” electron density. Subsequent studies showed that solvent effects on absorption spectra of acetone36 or the aminocoumarin C15137 could be modeled by a combination of molecular dynamics calculations and FDEu calculations on solute–solvent clusters extracted from the trajectories. In these studies, it was shown that large environmental systems can be described by this methodology if simplifications are introduced in the construction of the environmental density,36 e.g., by employing a superposition of molecular fragment densities. Technical improvements in the calculation of the embedding potential and shell-like approaches for the construction of the environmental density have been implemented to further enhance the efficiency of FDE computer implementations.37 With the current design in the Adf program, it is possible to automatically construct an “environmental” density as a superposition of fragment densities, which can individually be relaxed in terms of a general subsystem DFT.32 The study in Ref. 36 also indicated a major advantage of FDE calculations in comparison to full TDDFT calculations on solvated systems. The failure of TDDFT for long-range charge-transfer excitations is severe for solvated molecules because there are a number of well-separated subsystems giving rise to a multitude of excitations that appear at energies at which only low-lying valence transitions would be expected. This problem is avoided by construction in FDE, since chargetransfer excitations cannot occur if monomer basis sets are employed. Conventional TDDFT calculations on solvated systems can become prohibitively demanding in terms of memory and CPU time requirements. Several hundreds of excited states

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may have to be optimized to find the valence excitations of interest among the charge-transfer transitions.36 An alternative is to use simple correction schemes, which can detect and correct long-range charge-transfer excitations on the basis of the vanishing orbital density overlap64,91 (note that a very similar diagnostic has later also been presented in Ref. 92). The excited states of a water molecule in water have been studied with subsystem-DFT methods in Ref. 60. A considerable blueshift for the lowest excitations due to the neighboring water molecules was found in that study. By a comparison to a polarizable classical model it was found that the additional response contributions in FDEu calculations lead to a further increase in the excitation energies. In contrast to that, the polarizable classical model lead to a stabilization of the excited states by means of induced dipole moments in the environment as a reaction to the density change in the embedded system. Cooperative hydrogen-bonding effects on the absorption properties of 7-hydroxyquinoline, microsolvated with H2 O, NH3 , or CH2 OH were investigated in Ref. 93. The differences in the spectral shifts for these molecules could directly be compared to shifts available from experiment, and an overall good agreement was found. A great advantage of orbital-free embedding calculations lies in the fact that environments with (almost) arbitrarily complicated structure can be treated. The only requirement is that an approximate electron density can be constructed for them. Many other models require a careful parameterization, which makes it difficult to transfer them to new types of environments. In contrast to that, orbital-free embedding calculations have been employed to study diverse phenomena and systems. Examples are investigations of the optical properties of lanthanide complexes in a crystal environment in terms of a non-empirical ligand-field like approach,94,95 or of protein–pigment interaction effects on the absorption properties of bacteriochlorophyll a (BChl a) molecules in the light-harvesting complex LH2 of the photosynthetic purple bacterium Rhodopseudomonas acidophila.80 Orbital-free embedding calculations have also been extended to circular dichroism spectroscopy and were employed to study effects of induced chirality on circular dichroism spectra.59 Induced circular dichroism (ICD) is the effect that the circular dichroism (CD) signals of two compounds are non-additive, i.e., the CD spectrum of a mixture of the two compounds (often in solution) differs from the sum of the CD spectra of the individual compounds. A common reason for this is complex formation between a chiral molecule and a non-chiral partner. In Ref. 59 it was shown that ICD can be modeled in many cases on the basis of FDEu, i.e., without considering inter-subsystem couplings. Examples in that study were complexes of benzoylbenzoic acid and amphetamine or an artificial amino acid receptor based on ferrocenecarboxylic acid conjugated with a crown ether moiety, which shows ICD when interacting with protonated amino acids. However, that study also presented some prototypical systems for which FDEu fails, since excited-state couplings are essential. Prominent examples are molecular dimers or cyclodextrin inclusion com-

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pounds. The failures for these types of systems necessitated the development of the coupled FDE approach in Ref. 47, and it was shown there that pathological cases for the FDEu approach can be treated well with FDEc. Environmental response effects can play a big role in chiroptical spectroscopies, as was recently demonstrated for the optical rotation of methyloxirane in benzene.96 Such studies are complicated by the fact that many snapshots of solvated molecules have to be evaluated, so that embedding methods able to capture environmental response effects are highly desirable.97 The reason for the rather accurate results in most of the early applications of FDE in TDDFT calculations in spite of the missing dynamic environmental response lies in the fact that in these applications the environmental potential has a much larger effect on the excitation energies than the response. Nevertheless, this environmental response may lead to additional changes in the spectra, which will be tested for some examples in the following sections. It should be mentioned that similar embedding techniques have also been applied in the context of wavefunction/DFT hybrid schemes (see, e.g., the work in Refs. 98, 99). A formal theoretical analysis of wavefunction-in-DFT embedding approaches is given in Ref. 100, and a recent extension to the density-matrix functional theory case can be found in Ref. 101. The importance of embedding approaches for simulations of materials and nanostructures is discussed in Refs. 102,103. An approach for the treatment of excited states of clusters described by configuration interaction or complete active space self-consistent field wavefunctions, embedded in an infinite periodic crystal described in terms of density-functional theory was proposed in Refs. 104,105 (see also Refs. 106,107). One problem in the application of this embedding scheme for excited states is that the embedding potential should, in principle, be state-dependent. In Ref. 105, ground-state-density-based embedding operators were applied for the excited states as well, and it was proposed to apply a state-averaged density of the active subsystem for the construction of an embedding potential in CASSCF calculations. A similar approach was used by Visscher and co-workers, though not under periodic boundary conditions, in which additional simplifications were introduced by employing a fixed embedding potential.108

11.4. FDEu as a quasi-diabatization scheme FDEu calculations allow to extract the hypothetical excitation energy of a chromophore-containing molecule in the presence of other subsystems under the assumption that there is no excited-state interaction with the environment. I.e., it is assumed that a density change upon excitation only takes place in the active subsystem. The same approximation is made whenever point-charge models are applied for the simulation of environmental effects on absorption properties. In contrast to such models, however, FDEu includes the Coulomb interaction with the environment exactly (within the chosen quantum chemical model and basis set),

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and additionally also covers non-electrostatic effects. If response couplings with the environment are included, we go from the diabatic picture of localized excitations to an adiabatic picture, in which excitations can be delocalized over several subsystems. If these response couplings are weak, the diabatic and adiabatic picture will be very similar. But if they are strong, the resulting excited-state energies and transition densities may be very different from the FDEu results. 5.2

iso

FDEu

FDEc

super

iso (coupl.)

5.0 B

E / eV

4.8

A

4.6 A

4.4 B B

4.2 A

Fig. 11.4.1. Left: Structures of the benzaldehyde dimer with a head-to-tail arrangement. Right: Excitation energies of the dimer (BP86/TZP) between 4 and 5 eV. Note that inter-monomer charge-transfer excitations are not shown for the supermolecular calculation.

This effect is investigated for a dimer of benzaldehyde molecules in a head-totail arrangement (which is sometimes called a J-type aggregate) as shown on the left-hand side of Figure 11.4.1. The monomers in this aggregate are TZP/BP86 optimized ¡indexOptimization structures, and in the dimer they have been displaced by 8.23 ˚ A along the C−CHO bond. The diagram on the right-hand-side of Figure 11.4.1 shows the excitation energies of the isolated benzaldehyde molecule as obtained with TZP/BP86 in the energy range between 4 and 5 eV. Also the corresponding energies for the dimer are presented. A full list of low-lying excitation energies is given in Table 11.1. With FDEu, only the effects of the environment on the ground-state potential and on the intra-subsystem response kernel are included. The embedding potential is not symmetric for monomers A and B in the dimer because of the head-to-tail arrangement. Therefore, the shifts in the excitation energies for monomer A are larger than those for monomer B, although the shifts for A and B always have the same sign (i.e., they are either both bathochromic or both hypsochromic). When the coupling between the local response results for the monomers is included by means of FDEc, we see that the low-lying excitations hardly change in energy. The excitation at 4.88 eV in the isolated monomer, however, shows an additional splitting due to the excited-state interaction.

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Orbital-Free Embedding Calculations of Electronic Spectra Table 11.1. Excitation energies (BP86/TZP; in units of eV) for the benzaldehyde dimer shown in Figure 11.4.1; shown are results for isolated molecules (iso), coupled excitation energies based on transition densities of the isolated monomers [iso(coupl.)], FDEu, FDEc, and supermolecular calculations. To identify charge-transfer excitations in the supermolecular case, we also report orbital energy differences of the dominant orbital transitions from the supermolecular calculation (∆super ). no. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17

iso

3.26

iso(coupl.)

FDEu

FDEc

3.26

3.30(B)

3.30

3.26

3.39(A)

3.39

4.25

4.25 4.25

4.17(A) 4.23(B)

4.17 4.23

4.32

4.32

4.38(B)

4.38

4.32

4.53(A)

4.53

4.84 4.93

4.84(A) 4.85(B)

4.80 4.89

4.88

super

∆super

2.53 3.24 3.30 3.30 3.37 3.74 3.87 4.17 4.22 4.34 4.39 4.42 4.52 4.59 4.65 4.79 4.88

2.53 3.24 3.10 3.30 3.17 3.74 3.87 3.77 3.88 4.34 4.38 4.42 4.51 4.59 4.65 3.85 3.82

From the data in Table 11.1 it can be seen that the agreement between the supermolecular calculation and FDEc is very good, with errors of the order of 0.01 to 0.02 eV. However, there are a number of additional states in the supermolecular calculation (which are not shown in Figure 11.4.1 for clarity). Inspection of the dominant orbital contributions revealed that these are all inter-subsystem charge-transfer excitations, which are not correctly described by the ALDA kernel.64,91,109–112 This can also be proven by comparing the calculated excitation energies with the orbital energy differences for the dominant orbital transition. The orbital-energy differences for these additional states agree within less than 0.001 eV with the TDDFT excitation energies, which is a clear indication of the long-range charge-transfer character in TDDFT.64 The calculated energy splitting between states 16 and 17 in the supermolecular calculation is 751 cm−1 , and it is only slighly lower (723 cm−1 ) in the FDEc calculation. As discussed in Section 11.2.3, many phenomenological approaches to excitonic couplings try to calculate the energy splitting on the basis of the transition densities of isolated molecules, and consider only (approximate) Coulomb or Coulomb and exchange–correlation contributions. If this is done in the present case, i.e., if the FDEc coupling scheme is used (excluding the small kinetic-energy contribution to the response kernel) on the basis of transition densities obtained for

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isolated molecules, we obtain a splitting energy of 692 cm−1 . This is still of the same order of magnitude as the splitting in the supermolecular calculation, but the deviation is almost doubled when compared to the FDEc result. These results indicate that FDEc gives access to adiabatic states as obtained in supermolecular calculations, whereas FDEu results can be regarded as the underlying (quasi-)diabatic states. In the context of excitonic coupling models or energy-transfer phenomena, the FDEu energies can be identified with the site energies of pigments in a certain environment, whereas FDEc yields the excitonic states. Note, however, that site-energies are non-observable, so that there is a freedom of definition of these quantities. The coupling constants that can be extracted from the FDEc calculations in terms of energy differences correspond to the excitonic couplings. This has been employed in a study on the absorption properties of the LH2 complex of Rhodopseudomonas acidophila that was already mentioned above. The antenna complex is shown in Figure 11.4.2. It consists of two rings of pigment molecules, which are denoted as B800 and B850, embedded in a protein matrix. The names are chosen according the absorption maxima of these subunits, which occur at 800 and 850 nm. Both rings consist of BChl a molecules. In the B850 subunit, there are 18 BChl a molecules in two different conformations (9 so-called α- and 9 β-BChl a molecules), whereas the B800 subunit consists of 9 BChl a molecules. The antenna is augmented by 9 carotenoid molecules. More details on the structure are given in Ref. 80 and references therein. Crystal structures have been determined in Refs. 113–115. The different positions of the absorption maxima in the B800 and B850 subunits are due to a stronger excitonic coupling in the B850 unit, which could be reproduced in terms of an FDEc calculation. The resulting spectra for the low-lying Qy excitations of α- and β-BChl a molecules in the B850 ring, a coupled (α, β)-dimer, and the fully coupled B850 system are shown on the left-hand side of Figure 11.4.3. The intensities in the coupled FDE calculations in that figure differ slightly from those in Ref. 80, since it turned out that the oscillator strength in that references were subject to an inconsistent energy weighting. A Gaussian broadening of 0.01 eV has been applied to the peaks in the calculated absorption spectra. The spectrum of the B850 unit is compared to that of the combined B800 + B850 system on the right-hand side of Figure 11.4.3. It can clearly be seen that for the combined system (about 3800 atoms) two intense transitions in the QY region, i.e., around 800 nm, can clearly be distinguished. By comparison to the B850-only spectrum, it is apparent that the longer-wavelength peak must be attributed to the B850 ring, and that the B850 spectrum is hardly changed by the additional couplings to the B800 system. This application demonstrates the ability of FDE to identify both site energies and excitonic coupling effects in natural photosynthetic systems, which is unfeasible for conventional TDDFT methods.

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Structure of the LH2 complex of Rhodopseudomonas acidophila.

e+0

5

Fig. 11.4.2.

1.5

3×

10

2×

10

αB850

ε / m mol

βB850

2

05

-1

-1

full B850 (scaled by 0.1)

6

1×

5e+

Qy

Qx 6

04

2

1e+

ε / m mol

B850

6

(α,β)B850

B850 + B800

10

0

(Qy band only) 760

780

800

wavelength / nm

820

840

0 600

700

800

wavelength / nm

Fig. 11.4.3. Left: Qy contributions to the absorption spectra (SAOP/TZP) of (subunits of) the B850 ring of LH2 (Rhodopseudomonas acidophila). The spectrum of the full B850 unit was scaled by a factor of 0.1 for better comparability. Right: Absorption spectra (SAOP/TZP) of the B850 ring and the combined B850 and B800 rings. Note that the oscillator strengths in Ref. 80 for the FDEc calculations are slightly different due to an inconsistent energy weighting of the transition moments.

11.5. Excited-state couplings in hydrogen-bonded complexes The examples in previous excited-state studies based on subsystem TDDFT including response couplings considered identical chromophores,47,80 for which monomer

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energies are very close, so that even small couplings can lead to non-negligible energy splittings. This, in turn, can have significant effects on absorption or circular dichroism spectra. In the following, we present two examples of systems with inter-subsystem hydrogen bonds, which have been studied before by FDE excluding inter-subsystem response couplings. A good agreement with supermolecular reference calculations could be obtained, so that apparently coupled response effects are of little importance. The first example is the adenine–thymine complex studied in Ref. 46 (see Figure 11.5.4; coordinates can be found in the Supporting Information to Ref. 46). The second is the snapshot of an aminocoumarin C151 molecule with five explicit water molecules shown in Figure 4 of Ref. 37.

Fig. 11.5.4. Structure of the adenine–thymine complex. Coordinates can be found in the Supporting Information to Ref. 46.

For the adenine–thymine complex, we first reproduced the PW91/ET-pVQZ excitation energies for the free monomers, the supermolecule complex, and the results of the embedding calculations given in Ref. 46. All excitation energies calculated here are shown in Table 11.2 and agree with those of Ref. 46. Small deviations of about 0.01 eV occur for the 1A00 state of adenine in the FDEu (“embedding” in Ref. 46) and supermolecule calculation. These may be due to slightly different technical settings in combination with roundoff effects. Another deviation can be seen for the A0 states in the supermolecular calculation. In our calculation, we found three excited states of 1A0 symmetry between 4.65 and 4.70 eV. We assigned these states to the lowest A0 states of the monomers on the basis of a comparison of (modified) transition densities as explained in Ref. 116, which has been utilized before for similar purposes, see, e.g., Ref. 36. The lowest one at 4.65 eV was assigned to the 1A0 state of adenine, and the second one at 4.66 eV to the 1A0 state of thymine. A third state appeared at 4.70 eV and could not be uniquely identified on the basis of monomer transitions, so that its major contributions are expected to arise from intermolecular charge-transfer excitations. Apparently, it was this state that was assigned to the adenine 1A0 state in Ref. 46, which was slightly lower in energy in that study (4.69 eV). It should be noted that the assignment made here is by no means unambiguous, since the transition-density overlap between the adenine 1A0 state in the isolated monomer and the supermolecule state at 4.65 eV is 0.72, whereas the overlap with the supermolecule state at 4.70 eV is 0.65. This heavy

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mixing is confirmed by an analysis of the major orbital contributions to this excitation. The dimer orbitals of a00 symmetry around the HOMO–LUMO gap are shown in Figure 11.5.5. The main contribution to the excitation at 4.70 eV arise from the 10 a00 → 13 a00 (44 %) orbital transition, which is of charge-transfer type, and from the 12 a00 → 14 a00 (33 %) orbital transitions on the adenine subsystem. For the excitation at 4.65 eV, the contributions of these orbital transitions are 47 % and 39 %, which makes the assignment cumbersome. Again, it must be emphasized that this mixing with long-range charge-transfer-like orbital transitions is an artifact because of the too low excitation energies.36,64 Therefore, an analysis in terms of FDE may be better suited for this system.

Fig. 11.5.5. dimer.

Isosurface plots of the a00 frontier orbitals (PW91/ET-pVQZ) in the adenine–thymine

Excitonic effects could be expected to play a role, since the monomers both show an A0 excitation of π → π ∗ type with an excitation energy of 4.67 ± 0.01 eV. In contrast to the states of A00 symmetry, the A0 states also show a non-negligible oscillator strength, which is a prerequisite for significant excitonic couplings (in the absence of strong short-range and higher-multipole Coulomb couplings). Therefore, we also calculated the FDEc energies of the dimer. As can be seen from the data in Table 11.2, the energy changes due to the response couplings are small and amount to only about 0.01 eV for the state that is predominantly of thymine 1A0 character. This shows that the energies from the embedding calculations presented in Ref. 46 (which were recalculated here as FDEu energies) are a very good approximation to the fully coupled results. When investigating the character of the adiabatic states as obtained from FDEc, however, we see that the excitation at 4.65 eV, which is predominantly (82 %) of thymine 1A0 type, contains a 18 % admixture of the adenine 1A0 state. Similarly, the state at 4.68 eV consist of a 82 % adenine 1A0 and a 18 % thymine 1A0 con-

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J. Neugebauer Table 11.2. Lowest excitation energies and oscillator strengths (PW91/ET-pVQZ; energies in units of eV) of the adenine–thymine complex shown in Figure 11.5.4; shown are results for isolated monomers (iso), and results for the complex obtained with FDEu, FDEc, and supermolecular calculations. Excitation energies base sym. iso

FDEu

FDEc

super

4.71 4.19 4.73

4.66 4.31 4.68

4.65 4.31 4.68

4.31

4.53

4.53

4.66 4.29 4.65 4.70 4.47

Oscillator strengths base sym. iso

T T A A A

T T A A A

A0

A00 A0 A0 A00

A0 A00 A0 A0 A00

FDEu

FDEc

super

0.071 0.000 0.107

0.061 0.000 0.116

0.088 0.000 0.087

0.000

0.000

0.000

0.066 0.000 0.058 0.071 0.000

tribution. Such a mixing of the local excited states can of course not be described in the uncoupled FDE approach. This mixture of local excited states also becomes apparent in the oscillator strengths, which are listed in Table 11.2. For the isolated monomers, the intensity is about 1.5 times higher in the adenine 1A0 state than in the thymine 1A0 state. This effect is even more pronounced in the FDEu calculation (1.9 times higher). In contrast to that, the FDEc calculation results in almost identical oscillator strengths for the two transitions, which are now allowed to mix. The supermolecular calculation confirms this picture, although the situation is even more difficult there due to the additional excited state at 4.70 eV with an oscillator strength of 0.071. This example indicates that the full subsystem TDDFT formalism with coupled subsystem response becomes mandatory if the character of the excited states shall be analyzed in detail. For the second example, the aminocoumarin dye from Ref. 37 with five explicit water molecules, response couplings may be important for a different reason. There are no low-lying excitations of the water molecules that could lead to excitonic splittings. However, the water molecules could show a polarization in the excited state, which could lead to an energy lowering. It could thus at least partly explain the remaining deficiency of 0.09 eV between the supermolecular result and the best FDEu result obtained with a monomer basis set in Ref. 37 (the effect of a supermolecular basis set in the FDEu calculations observed in that reference was very small and will thus not be considered here). In the subsystem TDDFT formalism, such a polarization may be modeled by including many excited states of the water molecules in a sum-over-states-like manner.

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Table 11.3. Excitation energies (in units of eV) for the aminocoumarion C151 snapshot with five water molecules from Ref. 37. The data for the isolated aminocoumarin and the supermolecular results are taken from that reference. Note that different exchange–correlation (XC) functionals and basis sets may have been used for C151 and the environment (env.). calc. iso FDEu FDEc FDEu FDEc super

XC/basis C151

XC/basis env.

Eex

SAOP/TZP SAOP/TZP SAOP/TZP SAOP/TZP SAOP/TZP SAOP/TZP

— LDA/DZP LDA/DZP SAOP/TZP SAOP/TZP SAOP/TZP

3.248 2.998 2.985 2.995 2.982 2.910

To test the influence of response couplings on the excitation energy of the lowest π → π ∗ excitation of the aminocoumarin dye, we performed an FDEc calculation with the same functionals and basis sets as employed in Ref. 37. The calculations on the aminocoumarin C151 were carried out with the SAOP potential and a TZP basis set, whereas the density of the fragment molecules was constructed with a DZP basis set within the local density approximation. The more approximate treatment of the environment was applied to be able to treat very large environmental systems efficiently. The excitation energies for this system are presented in Table 11.3. They differ slightly (about 0.001 eV) from the data in Ref. 37 for two reasons. First, the parameterization of the enhancement factor of the non-additive kinetic-energy functional has been slightly adapted (see the discussion in Ref. 42). Second, the environmental system in the original work was considered as one subsystem in the freeze-and-thaw calculations, whereas here the environment consists of five separate (but fully relaxed) subsystems. Since the water molecules are not strongly interacting in this system, this has only a minor influence on the excitation energies. However, a decomposition of the environment into five separate subsystems has the major advantage that inter-subsystem charge-transfer transitions do not occur in the calculation, and consequently do not have to be coupled explicitly in the FDEc calculation. The FDEc calculation, in which 100 excited states per subsystem were coupled, reduces the excitation energy by about 0.013 eV compared to FDEu. It thus improves the FDEu result, but accounts for only about 15 % of the deviation between FDEu and the supermolecular calculation. Switching from LDA/DZP to SAOP/TZP in the description of the subsystems has only a minor effect on the excitation energies. The major part of the discrepancies between FDEu and the supermolecular calculation must thus be attributed to deficiences in the non-additive kinetic-energy functional and the corresponding potential. Also contributions of charge-transfer-like contributions could have an effect in

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the supermolecular calculation. But as becomes clear from the FDEc results, response couplings are of minor importance in this example, which justifies the use of the local response approximation in Ref. 37.

11.6. Solvated molecular dimers The electronic coupling between two excited states of a donor and an acceptor molecule is related to the rate of energy transfer between them.77 A topic of great current research interest is the question how such couplings are modified by a solvent or a more general environment.61,62,76,79,117,118 Supermolecular calculations are only of limited value in this context because of the large number of long-range charge-transfer excitations, which may hamper the identification of the excitonic states. Subsystem TDDFT in the FDEc variant, however, allows for a clear identification of excited states of an aggregate of molecules in terms of localized excitations of the subunits. This is demonstrated in the following for the dimer of benzaldehyde molecules shown in Figure 11.6.6, which are parallel displaced by 7 ˚ A and are solvated explicitly by 20 water molecules. The structure was obtained by keeping the positions of the benzaldehyde molecules fixed and optimizing the coordinates of the water molecules. The calculations were performed in the following way: We first calculated the 100 lowest FDEu energies of the fully relaxed subsystems (benzaldehyde and water) in a BP86/TZP calculation. The excitation energies of the lowest intense π → π ∗ excitations of the benzaldehyde molecules changed from 4.880 eV in the isolated molecules to 4.842 and 4.848 eV in the FDEu calculation. The slight differences reflect the influence of the non-symmetric arrangement of the water molecules. Subsequently, a response coupling step was performed in which this π −→ π ∗ excitation was chosen as a reference state, and excited states of the other subsystems were included in the calculation if their FDEu energy was within a certain energy threshold with respect to this state. This threshold was varied from 0.01 eV (only the two π −→ π ∗ excitations of the monomers coupled) to 1000 eV (all excitations of all subsystems included). The resulting energies of the excitonic states are shown in Table 11.4. Coupling the two intense π −→ π ∗ transitions (coupling threshold = 0.01 eV) results in a splitting of 322 cm−1 . The upper excitonic state, which assembles almost the entire intensity in this arrangement, appears at 4.865 eV. It can be seen that the energy splitting becomes slighlty larger with increasing threshold, i.e., with increasing number of coupled states. When all 100 states calculated per subsystem in the FDEu step are coupled, the splitting energy is 375 cm−1 . The energies of both excitonic states systematically decrease when more excited states are coupled. The oscillator strengths, however, behave differently. For the lower excitonic state, we observe a slight increase, whereas there is a significant decrease for the upper excitonic state. Note that the step from a threshold of 100 to 1000 eV does not have a large effect in

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Table 11.4. Excitation energies E (BP86/TZP; in units of eV) and oscillator strengths f of the solvated benzaldehyde dimer shown in Figure 11.6.6; shown are results for isolated molecules (iso), FDEu, and FDEc with different coupling thresholds (given in units of eV). The indices l and u refer to the lower and upper excitonic state, respectively. calc. iso FDEu FDEc, FDEc, FDEc, FDEc,

0.01 10 100 1000

El

fl

Eu

fu

∆E / cm−1

4.880 4.842 4.825 4.816 4.792 4.792

0.240 0.237 0.001 0.003 0.009 0.009

4.880 4.847 4.864 4.857 4.838 4.838

0.240 0.258 0.494 0.444 0.388 0.388

0 42 320 336 374 375

this calculation, because most of the 100 excited states calculated for the individual water molecules are already captured with the 100 eV threshold. The efficiency of the FDEc calculations in this approach to include environmental response couplings is not very high, since the success of subsystem TDDFT calculations for large systems is mainly due to the fact that only a few excitations have to be coupled. One approximation that has been employed in the calculations presented here is that only couplings between the solvent subsystems and the solute subsystems have been included in addition to the couplings in the benzaldehyde dimer (see Section 11.2.4). Couplings between different solvent molecules are thus neglected in all energies reported in Table 11.4. We tested the effect of the additional inter-solvent molecule couplings for a coupling threshold of 10 and 100 eV. The changes in the excitation energies compared to the approximate treatment were smaller than 0.001 eV, and the change in the splitting energy was about 2 cm−1 in case of the 10 eV threshold. With the 100 eV threshold, deviations in the excitation energies were somewhat larger, but still below 0.002 eV. The splitting energy changed by 19 cm−1 or 5 % of the total splitting.

Fig. 11.6.6. Structure of the solvated benzaldehyde dimer model. The positions of the water molecules were optimized (BP86/TZP), whereas the benzaldehyde molecules were held fixed at a parallel displacement of 7 ˚ A. The benzaldehyde monomer structure was taken from a BP86/TZP optimization.

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11.7. Conclusions In this work, we have reviewed a subsystem approach to time-dependent density functional theory in the linear response regime, which is related to orbital-free embedding and subsystem-density functional theory approaches. We have outlined efficient ways for the calculation of excitation energies for systems composed of several subunits. It was demonstrated that (quasi-)diabatic and adiabatic excited states of chromophores in larger aggregates are accessible by switching the excitedstate couplings between different subunits on or off. Several previous studies employing frozen-density embedding for excited states have been reviewed, and partly re-examined in the light of the coupled FDE approach. It was shown that the effect on excitation energies in systems like the adenine–thymine base pair, but also in solvated systems like the aminocoumarin C151 with five water molecules, is typically small, and the local response approximation (FDEu) often leads to good results for excitation energies. Exceptions are certainly systems of identical or similar chromophores, like the BChl a molecules in the LH2 complex. In turn, the missing environmental response could only partly explain the remaining deviation between supermolecular and subsystem TDDFT results for the aminocoumarin system. For oscillator strengths, excited-state couplings may be more important, as was shown for the base-pair system. Finally, we have demonstrated that the environmental response as described by FDEc can have a non-negligible effect on the excitation energies and excitonic coupling constants for molecular dimers, as was shown for the benzaldehyde dimer surrounded by 20 water molecules. The additional approximation of neglecting inter-solvent response couplings was found to have a small effect. This indicates that subsystem TDDFT approaches may be an alternative to continuum solvation or QM/MM solvation studies in the context of screening effects on energy-transfer phenomena, or in more general applications requiring models capable of describing the dynamic environmental response. Acknowledgment This work was supported by a TOP grant of the Dutch Research Council (NWO). References 1. 2. 3. 4. 5. 6. 7.

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Chapter 12 On the Principal Difference Between the Exact and Approximate Frozen-Density Embedding Theory O. V. Gritsenko Afdeling Theoretische Chemie, Vrije Universiteit De Boelelaan 1083, 1081 HV Amsterdam, The Netherlands [email protected] A principal difference is revealed between the exact and approximate frozendensity embedding (FDE) theory with respect to the incoming frozen density ρf . In the exact FDE the resultant total density does not depend on ρf , which has to belong to a restricted set of admissible densities. The FDE “freeze-and-thaw” (FT) procedure cannot alter the outcome of the exact FDE. On the other hand, in approximate FDE Kohn-Sham equations the effective “external” potential could be self-consistently adjusted to a given ρf , which offers a greater flexibility at the approximate level. In this case FT is characterized as a self-consistent procedure, which at a given iteration searches for a stationary point of the FT functional specified with the solution from the previous iteration. A condition is obtained for the FT saturation point. A generalization of the frozen-core procedure is presented in the context of frozen-orbital embedding theory.

Contents 12.1 Introduction . . . . . . . . . . . . . . . . . . . . 12.2 Exact FDE . . . . . . . . . . . . . . . . . . . . . 12.3 FDE with a non-additive kinetic approximation . 12.4 Freeze-and-thaw procedure in approximate FDE 12.5 Frozen orbital embedding . . . . . . . . . . . . . 12.6 Conclusions . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . .

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12.1. Introduction Any theory formulated exactly, which also has applications at various approximate levels, should, ideally, possess an internal consistency towards the refinement of its approximations. This means, that the exact and approximate levels share certain basic features and a consistent refinement of approximations should smoothly bring better results. One could attempt to apply this consistency criterion to frozendensity embedding (FDE) theory, which has been formulation in Refs.1–5 and enjoyed several applications at various approximate levels.6–8 355

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The main feature of FDE reflected in its name is the partitioning of the total electron density ρ in functionals of density functional theory (DFT) into two nonnegative functions, the variable subdensityρv integrating to N electrons and the frozen subdensity ρf integrating to N ’ electrons ρ (r1 ) = ρv (r1 ) + ρf (r1 )

(12.1.1)

The optimized subdensityρv is obtained from the solution of the one-electron FDE Kohn-Sham (FDE-KS) equations for the subsystem V with the effective local potential vsv , which includes contributions from the subsystem F with ρf 5 (see Section 12.2 for the corresponding formulas). Application of FDE shows that often the solution of approximate FDE-KS depends on ρf chosen, i.e. different ρv are obtained with different ρf . We will call this feature a flexibility of approximate FDE towards the choice of ρf . In order to refine the density obtained, the so-called “freeze-and-thaw” (FT)2,4,9 procedure is employed, in which densities of both subsystems are used in turns as the frozen density at the subsequent FT cycles (see Section 12.4 for the characterization of the FT procedure). A choice of ρf in FDE creates the problem of its admissibility. What is the set of “admissible” frozen densities, for which FDE-KS is able to provide a regular stationary-point solution? Does the subsequent FT always provide an improvement of the calculated density? In this paper flexibility of FDE, admissibility of frozen density, and the prospects of FT are analyzed at the exact and approximate FDE levels. In Section 12.2 the exact FDE is considered. The energy is represented as a functional Eρf [ρv ] of the variable densityρv specified with the frozen density ρf . A necessary condition for a stationary point of this functional is that ρf has to be a part of the exact total density, ρf (r1 ) < ρe (r1 ), so that only those ρf could be admissible frozen densities. The exact FDE is a rigid construction towards ρf in the sense, that the resultant total density ρe does not depend on ρf from a restricted set of admissible densities, while other frozen densities have to be discarded. As a consequence, the FT procedure cannot alter the outcome of the exact FDE. In Section 12.3 FDE with approximate non-additive kinetic functional is considered. In this case the potential of the FDE-KS equations is represented as the exact potential plus an effective correction to the external potential from the non-additive approximation. Such a potential could be self-consistently adjusted to a given frozen density, which explains the abovementioned flexibility of approximate FDE. In Section 12.4 FT in approximate FDE is characterized as a self-consistent procedure, which at a given iteration searches   for a minimum (stationary point) of the approximate energy functional E˜ρI/II ρII/I specified with the stationary-point solution ρI/II from the previous FT iteration. A condition is obtained for the FT saturation point. As follows from this condition, FT might not necessarily lead to the improvement of the calculated density. The frozen-density admissibility problem might well be alleviated with a generalization of a frozen-core procedure presented in Section 12.5.

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In this case application of the FT procedure is expected to produce canonical orbitals of the supermolecule. In Section 12.6 the consequences of the present results for FDE are discussed and the conclusions are drawn. 12.2. Exact FDE We start with the conventional DFT expression for the functional of the total electronic energy Z E [ρ] = ρ (r1 ) vext (r1 ) dr1 + Eee [ρ] + Ts [ρ] (12.2.2) where vext is the external potential, Eee is the electron-electron interaction energy functional, and Ts is the non-interacting kinetic energy functional. A usual assumption in DFT is that E [ρ] possesses a single stationary point, the minimum with the exact density ρe , which is determined from the Euler-Lagrange equation δE [ρ] δEee [ρ] δTs [ρ] = vext (r1 ) + + =µ δρ (r1 ) δρ (r1 ) δρ (r1 )

(12.2.3)

In FDE (12.2.2) can be rewritten as a functional of the variable part ρv of (12.1.1) Z Eρf [ρv ] = [ρv (r1 ) + ρf (r1 )] vext (r1 ) dr1 + Eee [ρv + ρf ] (12.2.4) + Ts [ρv + ρf ]

specified with the constant function, the frozen density ρf . A stationary point of the functional (12.2.4) is determined from the corresponding Euler-Lagrange equation Z δEρf [ρv ] δρ (r3 ) δρ (r2 ) = vext (r3 ) dr2 dr3 (12.2.5) δρv (r1 ) δρ (r2 ) δρv (r1 ) Z Z δTs [ρ] δρ (r2 ) δEee [ρ] δρ (r2 ) + dr2 + dr2 = µ δρ (r2 ) δρv (r1 ) δρ (r2 ) δρv (r1 ) where the chain rule differentiation is used to express the derivatives with respect to ρv . The derivatives of ρ with respect to ρ(v ) are the delta-functions, so that Eq.(12.2.4) turns to the Euler-Lagrange equation (12.2.3) of the supermolecule δEρf [ρv ] δEee [ρ] δTs [ρ] δE [ρ] = vext (r1 ) + + = δρv (r1 ) δρ (r1 ) δρ (r1 ) δρ (r1 )

(12.2.6)

This should be so, since FDE of (12.2.4) and (12.2.5) is merely a special case of the conventional DFT of (12.2.2) and (12.2.3), in which all variations of the total density ρ are confined in its part ρv . From the established equivalence (12.2.6) of (12.2.3) and (12.2.5) follows the necessary condition for a stationary point of the FDE functional (12.2.4) that ρf ,

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with which it is specified, is a part of the exact total density ρe ρf (r1 ) < ρe (r1 )

(12.2.7)

The inequality (12.2.7) is required for all domains of the non-zero measure. A possible equality of ρe and ρf is only admitted in domains of zero measure (See the corresponding discussion below). Then, the optimal subdensity ρev , the solution of (12.2.5), (12.2.6) is simply ρev (r1 ) = ρe (r1 ) − ρf (r1 )

(12.2.8)

The FDE functional (12.2.4) has at ρev the same value as that of the corresponding conventional DFT functional (12.2.2) at ρe Eρf [ρev ] = E [ρe ]

(12.2.9)

On the other hand, if the frozen density ρf violates the condition (12.2.7), no nonnegativeρv could be found to satisfy (12.2.8). From this follows, that the FDE functional (12.2.4) specified with this ρf is a deficient functional in the sense, that it has no stationary point. This should be so, otherwise, as follows from (12.2.3), (12.2.5) and (12.2.6), the exact functional (12.2.2) would have a stationary point at the density ρv +ρf , which differs from ρe , and that would contradict its definition given in the beginning of this section. A distinguished element of FDE is an auxiliary non-interacting system, which yields (a non-interacting vs -representable) ρv through the one-electron FDE-KS equations (closed-shell systems are considered)   1 2 v − ∇ + vs (r1 ) φvi (r1 ) = εvi φvi (r1 ) (12.2.10) 2 ρv (r1 ) = 2

N/2 X i=1

|φvi (r1 )|

2

(12.2.11)

where fiv are the KS orbitals with the energies vi . The effective potential in (12.2.10) is determined through the Euler-Lagrange equation for the non-interacting system δTs [ρ] + vsv (r1 ) = µv (12.2.12) δρ (r1 ) ρ=ρv

Since Eqs.(12.2.5), (12.2.6) and (12.2.12) are solved for the same density ρv , one obtains from the comparison of these equations the following expression for vsv δEee [ρ] δTs [ρ] δTs [ρ] v vs (r1 ) = vext (r1 ) + + − (12.2.13) δρ (r1 ) ρ=ρv +ρf δρ (r1 ) ρ=ρv +ρf δρ (r1 ) ρ=ρv Eqs. (12.2.4), (12.2.5), (12.2.10)-(12.2.13) specify FDE-KS theory.5 For a frozen density ρf , which satisfies (12.2.7), the FDE-KS equations (12.2.10), (12.2.11), (12.2.13) yield the optimal density ρev of (12.2.8). However, for ρf , which

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violates (12.2.7), the FDE-KS equations become a deficient construction, since they lack a normal stationary-point solution. This should be so, otherwise, the presence of a stationary point for the non-interacting system with the potential (12.2.13) would mean through (12.2.6) and (12.2.12), that the exact functional (12.2.2) would have a stationary point at the density ρv +ρf , which differs from ρe . This, however, would contradict the conventional DFT assumption (adopted in the beginning of this section) of a single stationary point of the exact energy functional. Naturally, a direct minimization of the FDE functional (12.2.4), which does not employ the KS construction (12.2.10), (12.2.13) does attain a minimum even for ρf , which violates (12.2.7). This minimum, however, will be higher than that for the corresponding DFT functional (12.2.2). Furthermore, since this minimum is, most likely, not a stationary point, the correspondingρv might well vanish in the region where ρf ≥ ρe . Note, that for a density, which vanishes in a spatial region with a non-zero measure, neither the Hohenberg-Kohn theorem, nor the KS theory are applicable, since they cannot accommodate the corresponding singularity (infinite wall) of the potential in this region.10 Because of this, only normal stationary-point solutions are considered in the conventional KS theory. Then, in order to keep the FDE-KS theory at the same well-defined level, one should consider only frozen densities ρf , which satisfy (12.2.7), as possible admissible densities. All ρf , which violate (12.2.7), should be excluded as inadmissible at the exact FDE-KS level. It appears, in addition, that the FT procedure is pointless in the exact FDE-KS. Indeed, in the case of an admissible ρf the FT procedure cannot alter the results, since the insertion of the obtained ρev as a new frozen density within FT would yield just the initial admissible ρf . On the other hand, in the case of an inadmissible ρf no normal stationary-state solution ρv is available in the exact FDE-KS to start a well-defined FT procedure similar to that formulated for approximate FDE in Section 12.4.

12.3. FDE with a non-additive kinetic approximation From the analysis of the previous section follows, that the exact FDE is a rigid construction towards the choice of frozen densities. Indeed, the resultant total density ρe does not depend on ρf . Furthermore, the latter has to belong to a restricted set of admissible densities, while other frozen densities have to be discarded. As a consequence, the FT procedure cannot alter the outcome of the exact FDE. This is not a favorable conclusion for FDE, since admissible frozen densities cannot be, in principle, specified without that same all-electron supermolecule calculation, which yields the exact total density ρe and which FDE is designed to avoid. As we will argue below, the situation in approximate FDE is rather different. For the simplicity, we use only a common FDE approximation to the non-additive

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part of the KS kinetic functional n o Tsappr [ρv + ρf ] = Ts [ρv ] + Ts [ρf ] + T˜s [ρv + ρf ] − T˜s [ρv ] − T˜s [ρf ]

(12.3.14)

In (12.3.14) T˜s is a certain approximate kinetic functional, for instance, a direct functional of the electron density, which is employed to approximate the non-additive part of Ts in the figure brackets. Inserting (12.3.14) in (12.2.4), one obtains the expression for the approximate energy functional Z ˜ρ [ρv ] = [ρv (r1 ) + ρf (r1 )] vext (r1 ) dr1 + Eee [ρv + ρf ] E (12.3.15) f n o + Ts [ρv ] + Ts [ρf ] + T˜s [ρv + ρf ] − T˜s [ρv ] − T˜s [ρf ] Eq. (12.3.15) can be rewritten as the exact functional (12.2.4) plus/minus the differences between the approximate and exact kinetic functionals of the same functional argument Z ˜ Eρf [ρv ] = [ρv (r1 ) + ρf (r1 )] vext (r1 ) dr1 + Eee [ρv + ρf ] (12.3.16) n o + Ts [ρv + ρf ] + T˜s [ρv + ρf ] − Ts [ρv + ρf ] n o n o + Ts [ρv ] − T˜s [ρv ] + Ts [ρf ] − T˜s [ρf ] = Eρf [ρv ] + ∆T˜s [ρv + ρf ] − ∆T˜s [ρv ] − ∆T˜s [ρf ]

In a complete analogy with the exact FDE of the previous section, the stationary point of the approximate functional (12.3.16) is determined from the Euler-Lagrange equation ˜ρ [ρv ] δE δEee [ρ] δTs [ρ] f ef f = vext (r1 ) + + =µ δρv (r1 ) δρ (r1 ) δρ (r1 )

(12.3.17)

The second and third terms in the right-hand side of (12.3.17) are the derivatives of the exact functionals, while the first term is the external potential plus a correction δ∆T˜s [ρ] δ∆T˜s [ρ] ef f vext (r1 ) = vext (r1 ) + − (12.3.18) δρ (r1 ) δρ (r1 ) ρ=ρv +ρf

ρ=ρv

We denote (12.3.18) as an effective external potential. In approximate FDE-KS theory the non-interactive system is introduced with the one-electron equations   1 − ∇2 + v˜sv (r1 ) φ˜vi (r1 ) = ε˜vi φ˜vi (r1 ) (12.3.19) 2

and the Euler-Lagrange equation for the stationary-point ρv δTs [ρ] + v˜sv (r1 ) = µ ˜v δρ (r1 ) ρ=ρv

(12.3.20)

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From the comparison of (12.3.17) and (12.3.20) for the potential follows δEee [ρ] δTs [ρ] δTs [ρ] ef f v v˜s (r1 ) = vext (r1 ) + + − (12.3.21) δρ (r1 ) ρ=ρv +ρf δρ (r1 ) ρ=ρv +ρf δρ (r1 ) ρ=ρv

Approximate FDE-KS equations (12.3.19), (12.3.21) can be viewed as the exact equations with the effective external potential (12.3.18). The latter, with its dependence on ρv and ρf due to the approximation made, acquires flexibility towards the choice of the frozen density. This means, that the iterative KS procedure (12.3.19) would change ρv and, as a consequence, the form of the potential (12.3.18) until a ef f given ρf would become a part (12.2.7) of the exact total density ρe [vext ] correspondef f ing to the resultant vext . Unlike in the rigid exact FDE of the previous section, the ef f ef f resultant ρe [vext ] would, in general, depend on ρf chosen. The resultant ρe [vext ] can be further refined with the FT procedure, which will be characterized in the next section. 12.4. Freeze-and-thaw procedure in approximate FDE

The dependence of the results of approximate FDE on the chosen ρf can be reduced by application of the FT procedure. FT starts with the density (ρI + ρII ) where ρII = ρf , while the part ρI = ρv yields the minimum (stationary point) of the ˜ρII [ρI ] obtained from the solution of the FDE-KS approximate FDE functional E equations   1 2 I (12.4.22) − ∇ + v˜s (r1 ) φ˜Ii (r1 ) = ε˜Ii φ˜Ii (r1 ) 2 with the potential (12.3.21), which can be rewritten in a somewhat simpler form δEee [ρ] δ T˜s [ρ] δ T˜s [ρ] I v˜s (r1 ) = vext (r1 ) + (12.4.23) + − δρ (r1 ) ρ=ρI +ρII δρ (r1 ) δρ (r1 ) ρ=ρI +ρII

ρ=ρI

This constitutes the first FT iteration. In the next iteration, the roles of ρI and ρII are reversed. The density ρI calculated in the first iteration is frozen, while the optimal ρII for the functional ˜ρI [ρII ] is obtained from the solution of the equations E   1 ˜II − ∇2 + v˜sII (r1 ) φ˜II ˜II (12.4.24) i (r1 ) = ε i φi (r1 ) 2 with the potential v˜sII

δEee [ρ] δ T˜s [ρ] (r1 ) = vext (r1 ) + + δρ (r1 ) ρ=ρI +ρII δρ (r1 )

ρ=ρI +ρII

δ T˜s [ρ] − δρ (r1 )

(12.4.25) ρ=ρII

Thus, the FT procedure goes through the succession of the minima of the functionals ˜ρI [ρII ] and E˜ρII [ρI ] specified with the frozen densities ρI and ρII , respectively, E

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from the previous iteration. FT continues, until it reaches a saturation point, at which both functionals share the same total density. From the difference of the corresponding Euler-Lagrange equations n o ˜s [ρ] δ T [ρ] − T ˜ s δEee [ρ] δ Ts [ρ] vext (r1 ) + + + = µI δρ (r1 ) ρ=ρI +ρII δρ (r1 ) δρ (r1 ) ρ=ρI +ρII

ρ=ρI

(12.4.26)

and δEee [ρ] δ T˜s [ρ] vext (r1 ) + + δρ (r1 ) ρ=ρI +ρII δρ (r1 )

ρ=ρI +ρII

n o δ Ts [ρ] − T˜s [ρ] + δρ (r1 )

= µII

ρ=ρII

(12.4.27)

which are valid for the same ρI and ρII , the following FT saturation condition emerges δ∆T˜s [ρ] δ∆T˜s [ρ] = µI − µII − (12.4.28) δρ (r1 ) δρ (r1 ) ρ=ρII

ρ=ρI

It is instructive to compare (12.4.28) with an optimal density condition, which one can derive from the comparison of the exact (12.2.10), (12.2.13) and approximate (12.3.19), (12.3.21) FDE-KS equations. One can expect, that the smaller will be a deviation of the effective external potential (12.3.18) of approximate FDE from the external potential, the closer will be the calculated approximate density to the exact one. Then, one can search for ρI and ρII , which yield the following minimum ˜ δ∆T˜s [ρ] δ∆Ts [ρ] min − (12.4.29) δρ (r1 ) δρ (r1 ) ρ=ρI +ρII

ρ=ρI /ρII

on a certain grid {r1}. Adding and subtracting the derivative with respect to the total density in (12.4.29) to the terms in the left-hand side of (12.4.28), one obtains      δ∆T˜ [ρ]    ˜ ˜ ˜ δ∆Ts [ρ] δ∆Ts [ρ] δ∆Ts [ρ] s − − −  δρ (r1 )   δρ (r1 )  δρ (r1 ) δρ (r1 ) ρ=ρI +ρII

= µI − µII

ρ=ρI

ρ=ρI +ρII

ρ=ρII

(12.4.30)

Evidently, the fulfillment of the condition (12.4.30) does not necessarily mean that the condition (12.4.29) is also satisfied. This means, that, in general, the FT procedure would not necessarily drive the FDE-FT density towards the exact one.

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12.5. Frozen orbital embedding The frozen density problem might be alleviated with a generalization of the frozencore procedure considered in this section. Suppose, that we can express the electronic energy of the supermolecule as a functional of the occupied orbitals E [{φj }] = 2

0 /2 (N +N X) Z

j=1

2

|φj (r1 )| vext (r1 ) dr1

(12.5.31)

+ Eee [{φj }] + Ts [{φj }] In a complete analogy with FDE of the previous sections, one can consider a functional of the orbitals of the subsystem V specified with the frozen orbitals of the system F  occ Z 2  X 2 f v  v  vext (r1 ) φj (r1 ) + φj (r1 ) dr1 E{φf } φj = 2 j

j

hn oi n oi   φvj , φfj + Ts φvj + Ts φfj h n oi + Tsnad φvj , φfj + Eee

h

The requirement of the orbital orthonormality is included in the corresponding Lagrangian     L{φf } φvj = E{φf } φvj (12.5.32) j j ! V X V V X F F X V X X X − + + µlk (hφk | φl i − δkl ) k

l

k

l

k

l

Minimization of (12.5.32) leads to the coupled one-electron equations h n oi   1 δE φvj , φfj ee 1 ef f 2 − ∇ + vext (r1 ) + v (12.5.33)  2 φi (r1 ) δφv∗ i (r1 ) h n oi  nad V δT φvj , φfj  X s 1 v + v φ (r1 ) = µji φvj (r1 )  i φi (r1 ) δφv∗ i (r1 ) j

where the term with the products of the Lagrange multipliers and frozen orbitals is included in the effective state-dependent external potential ef f vext (r1 ) = vext (r1 ) −

F X

µji φfj (r1 )

(12.5.34)

j

The procedure of Eqs. (12.5.32)-(12.5.34) is expected to be a rather flexible towards the choice of the frozen orbitals. Indeed, the solution (12.5.33) apparently depends on them through the effective external potential (12.5.34). The second

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term in the right-hand side of (12.5.34) plays, together with the last kinetic term in the figure brackets of (12.5.33), the role of a pseudopotential in the conventional pseudopotential theory. Note, that the energy (12.5.32) obtained with given frozen orbitals φfj is, in general, higher than the minimal energy (12.5.31) obtained with the canonical orbitals. However, in order to attain this minimum, one can apply the FT procedure. One can expect, that the successive FT cycles would produce more delocalized orbitals. Eventually, the subsets φvj and φfj would be transformed with the FT procedure into the single set φj of the canonical orbitals. The conclusions of this section are expected to be valid at both approximate and exact levels, so that the present procedure appears to meet the criterion of an internal consistency, which was mentioned in the first paragraph of this paper.

12.6. Conclusions In this paper an important for FDE problem of a choice of the frozen density is analyzed. This analysis reveals a principal difference in this respect between the exact FDE and that with an approximation for the non-additive kinetic functional. The exact FDE is a rigid construction towards the choice of ρf . The resultant total density ρe does not depend on ρf , which has to be chosen from a restricted set (12.2.7) of potentially admissible densities. This precludes application of the “freeze-and-thaw” procedure in the exact FDE. Since the admissibility of ρf cannot be, in general, determined beforehand without a full supermolecule calculation, one can conclude that FDE is, in a certain sense, not fully justified at the exact level. Unlike to this, approximate FDE might well be a rather flexible theory. Its resultant total density depends on ρf , while the complimentary variable density ρv is obtained as the stationary-point solution of the FDE Euler-Lagrange equation (12.3.17) or from the stationary-point solution of the FDE-KS equations (12.3.19), (12.3.21) with an effective adjustable external potential (12.5.34). The approximate total density could be altered with the FT procedure, which passes through a succession of stationary-point solutions for alternating subdensities towards a saturation point specified with the condition (12.4.28), (12.4.30). The FT alteration might not necessarily lead to the improvement of the calculated density. The abovementioned rigidity of FDE at the exact level stems from its very definition. With the equivalence (12.2.6) of the DFT and FDE Euler-Lagrange equations, FDE offers a special case of the conventional DFT functional, in which all variation of the total density is confined in its part. With this, the only possible solution of the exact FDE is ρe and the subdensities ρf andρv have to be non-negative parts of ρe , so that “inadmissible” densities violating (12.2.7) cannot serve as frozen densities. Unlike to this, with an approximate non-additive kinetic part, the FDE energy (12.3.15) becomes an “emergent” functional with the optimal density, which can accommodate a chosen frozen density as its part.

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This feature of FDE makes problematic a consistent refinement of its approximations. Indeed, accommodation of a chosen ρf might well require a substantial deviation of an approximate kinetic functionals in (12.3.14)-(12.3.16) from the exact one. This makes the potential (12.3.18) flexible enough to yield the resultant density, which has a given ρf as its non-negative part. On the other hand, one can expect that a consistent refinement of an approximate functional would reduce the set of admissible frozen densities and with a sufficiently close approximation FDE would acquire the undesirable rigidity of its exact level. Apparently, an internally consistent embedding theory should allow approximate solution for certain frozen densities even at its exact level. The procedure of Section 12.5 gives an example of a possible theoretical development in this direction. With its requirement of the orthogonality to the frozen orbitals, it represents a generalization of the frozen-core procedure. Acknowledgement The author gratefully acknowledges Evert Jan Baerends for the instructive discussions and insightful ideas. Support of the Netherlands Foundation for Research (NWO) through its Chemistry Division (Chemische Wetenschappen) is gratefully acknowledged. References 1. 2. 3. 4. 5. 6. 7.

G. Senatore and K. R. Subbaswamy, Phys. Rev. B 34, 5754 (1986). M. D. Johnson, K. R. Subbaswamy, and G. Senatore, Phys. Rev. B 36, 9202 (1987). P. Cortona, Phys. Rev. B 44, 8454 (1991). P. Cortona, Phys. Rev. B 46, 2008 (1992). T. A. Wesolowski and A. Warshel, J. Phys. Chem. 97, 8050 (1993). T. A. Wesolowski, H. Chermette, and J. Weber, J. Chem. Phys. 105, 9182 (1996). J. Neugebauer, C. R. Jacob, T. A. Wesolowski, and E. J. Baerends, J. Phys. Chem. A 109, 7805 (2005). 8. C. R. Jacob and L. Visscher, J. Chem. Phys. 128, 155102 (2008). 9. T. A. Wesolowski and J. Weber, Chem. Phys. Lett. 248, 71 (1996). 10. R. M. Dreizler and E. K. U. Gross, Density Functional Theory: An Approach to the Many-Body Problem (Springer-Verlag, Berlin, 1990).

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PART 3

Kinetic Energy Functional and Information Theory

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Chapter 13 Analytic Approach and Monte Carlo Sampling for Electron Correlations Luca M. Ghiringhelli and Luigi Delle Site Max-Planck-Institute for Polymer Research Ackermannweg 10, D 55021 Mainz, Germany [email protected] By combining rigorous mathematical conditions and a computational approach for sampling the electronic configurations in space, we build a procedure for the design of kinetic functionals for electrons which is internally consistent. We then apply it to the test case of the almost uniform interacting electron gas and derive the corresponding kinetic functional. Interestingly the functional form obtained has strong similarities with the Shannon Entropy within the Theory of Information.

Contents 13.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.2 From the Hohenberg-Kohn Theorem to a reformulation of the Levy Constrained Search Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.3 Levy Constrained Search Principle reformulated . . . . . . . . . . . . . . . . . . . . . 13.3.1 The novelty of the function f in this context . . . . . . . . . . . . . . . . . . . . 13.3.2 A practical way to design f . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.3.3 The parametric exponential form of f . . . . . . . . . . . . . . . . . . . . . . . 13.4 Monte Carlo evaluation of the non local part of the Kinetic Functional . . . . . . . . . 13.4.1 Scaling properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.4.2 Connection to information theory . . . . . . . . . . . . . . . . . . . . . . . . . . 13.5 Introduction of (statistical) spin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

369 370 372 374 374 375 377 381 383 384 385

13.1. Introduction Bridging length and time scales in a systematic way is becoming the central theme and aspiration of modern physics which has found in computer simulations a powerful tool to this aim. The current tendency in the field is that of searching for flexible theoretical and computational approaches which optimally balance the accuracy of the calculations and the computational costs. Coming to the study of electronic properties of matter, the success, both conceptual and applicative (due to computational affordability), of Density Functional Theory (DFT) is not a matter of discussion, however the raising question is how to improve its computational 369

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efficiency yet retaining a satisfying level of accuracy. The standard Kohn-Sham (KS) approach is rather robust but still, for an extended class of systems, by far too expensive, and thus the search for a simplified computational scheme is a topic of high interest. An alternative (or a complementary method) to the KS-DFT, which has been emerging in the latest years with a certain vigor, is the Orbital Free DFT (OFDFT).1–6 OFDFT is a linear-scaling real-space kinetic energy functional method, where the kinetic energy is calculated as a functional of the electron density only. Since, differently from the Kohn-Sham method, neither the diagonalization of the electronic Hamiltonian nor the reciprocal space sampling are required, such a method allows for studies of relatively large systems compared to those treatable with the standard methods. In facts, rather than scaling like in the KS method as the number of electrons cube, N 3 , by construction it scales as N . Moreover, being a real space method it is ideal for multiscale approaches as an interface to classical simulation schemes.5 Conceptually, the crucial point of OFDFT, is that it is in principle exact, provided that the kinetic functional is rigorous. This means that the design of rigorous kinetic density functionals would allow to explore the full power of the Hohenberg-Kohn theorem for extended systems. In the light of the discussion above, the search for a rigorous kinetic functional becomes the primary goal for a valid application of this method. The aim of this chapter is to report on theoretical developments in the design of a robust kinetic functional based on solid physical principles and rigorous mathematical prescriptions. To do so, we start the analysis from the very basic principles of DFT and reformulate the variational problem to the end of obtaining a criterion for basic optimal design of a kinetic functional. Thereafter, we review the recently proposed7,8 approach to design kinetic functionals combining mathematical requirements and the basic principles of electron correlations. This approach, complemented with the Monte Carlo (MC) sampling of electronic configurations in real space, leads to a functional form for the correlation term of the kinetic part.9 We continue by showing how the known formal properties of a kinetic functional apply to the numerical one we have obtained.10 Interestingly, this analysis allows us to make a connection between our functional and the key quantities of the information theory, namely the Fisher functional and the Shannon Entropy.11 While the Fisher term comes out analytically, the MC approach give us a term proportional to the Shannon entropy which we present as a first order approximation of a possible, formally rigorous kinetic functional. We conclude by outlining our most recent work, concerning the introduction into our formalism of spin interactions, in a statistical sense.

13.2. From the Hohenberg-Kohn Theorem to a reformulation of the Levy Constrained Search Principle The Hohenberg-Kohn (HK) theorems laid the basis of a rigorous formulation of the DFT and at the same time opened new perspectives to the calculations of

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electronic properties of condensed matter. The problem of the ground state of a 3N-dimensional electron system : X X X 1 1 HN ψ(r1 , . . . , r2 ) = E0 ψ(r1 , . . . , r2 ); HN = (− ∇2i ) + v(ri ) + 2 r ij6=1 e−γEh (r,rn )−βEh (ri ,rj ) dr2 . . . drN

(13.3.19)

Here γ and β are two free parameters. As can be easily verified, this expression of f satisfies all the requirements of Eq.13.3.10. The meaning of f as expressed in Eq.13.3.18 is that the probability of finding a certain configuration for the N − 1 particles, having fixed particle r1 = r, depends not only on the fixed particle and its interaction with the N − 1 other particles as before, but also on the mutual arrangements of the N − 1 particles (it has also to be kept in mind that using the particle indistinguishability the formalism can be applied to any ri as a fixed particle). The parameters γ and β express how important the N − 1 mutual interactions are with respect to the interactions with r. Being now f a biparametric function, one can use the Levy-Lieb constrained search in our formulation and find the optimal values for γ and β. This practical example shows two different aspects of our formulation; basically we have shown that indeed it is possible to build a function f and actually it can be chosen in a way that its optimal expression can be determined via the constrained-search formulation. It must be noticed that this form of f is still rather simple since as said before the spins are not explicitly considered when constructing the function. This means that one cannot distinguish between the exchange and the correlation part of the electron-electron interaction as it is done in standard Density Functional Theory. As a consequence one should expect only an overall average description of these two terms which are here incorporated into the global correlation. However, we have anticipated that the construction of a more complete expression of f , which takes care of the effects of the spins, is outlined at the end of the chapter. Next we briefly show how the actual calculation of E[ρ] is carried on, for spinless particles.

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13.4. Monte Carlo evaluation of the non local part of the Kinetic Functional As highlighted before, apart from the Weizs¨acker term and the external potential term, Inonloc [ρ(r1 )] and the electron-electron Coulomb term are multidimensional integrals. Here we show how to deal with the multidimensional integration and thus how to reduce them to a local functional, then to be used in practice for the variational problem w.r.t. ρ. Let us remind how the energy functional is written in terms of f : Z 2 1 |∇ρ(r1 )| E[Ψ] = hΨ|T + Vee |Ψi = E[ρ] = dr1 (13.4.20) 8 ρ(r1 ) Z Z 2 |∇f | 1 + dr1 ρ(r1 ) drN −1 8 f ΩN −1 Z Z N −1 f + dr1 ρ(r1 ) drN −1 2 |r1 − r2 | ΩN −1 Here, r1 is the variable which is not integrated out when calculating the density from ψ, previously indicated for simplicity as r. Now, for a practical aim this other formalism becomes convenient. However, we must keep in mind that due to the particles’ indistinguishability, any other variable, r2 , . . . , rN , could have been equally chosen. We define I[ρ(r1 )] and C[ρ(r1 )] as the inner integrals in the second and last terms at the r.h.s. of Eq. 13.4.20, respectively; for the purpose of numerical evaluation, we rewrite them into an equivalent form: " Z # 2 1 |∇f | I[ρ(r1 )] = drN −1 (13.4.21) 8 ΩN −1 f ρ(r1 )   Z X X 1 f  C[ρ(r1 )] =  drN −1 (13.4.22) N |r − rj | i j>i ΩN −1 i=1,N

ρ(r1 )

so that :

E[ρ] =

1 8

Z

2

dr1

|∇ρ(r1 )| + ρ(r1 )

Z

dr1 ρ(r1 ) {I[ρ(r1 )] + C[ρ(r1 )]} (13.4.23)

We want to evaluate the multidimensional integral I[ρ(r1 )] + C[ρ(r1 )] at a fixed r1 . To the purpose we write: " Z ∇1 f 2 1 I[ρ(r1 )] + C[ρ(r1 )] = drN −1 f 8 ΩN −1 f  Z 1 X X f  + drN −1 N |r − r | i j Ω N −1 j>i i=1,N

ρ(r1 )

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When written as the r.h.s. of the equation above, the integrals are suitable for a Monte Carlo evaluation (sampling) so that, repeated for a large number of values of r1 , we can have local functionals. We write: * + 2 1 ∇1 f 1 X X 1 I[ρ(r1 )] + C[ρ(r1 )] = + 8 f N |ri − rj | j>i i=1,N

f, ρ(r1 )=const

Where h. . .if, ρ(r1 )=const stays for ensemble average by sampling the density f , with the additional constraint of constant ρ(r1 ). The sampling of f is insured by importance sampling, i.e. constructing a sequence of configurations as a Markov chain. Given a configuration r1 , r2 , . . . , rN , the next configuration in the Markov chain is produced by tentatively displacing one electron selected at random. The moved is accepted according to the acceptance rule:   fnew a(old → new) = min 1, (13.4.24) fold Y new old new old fnew = e−γ (Eh (r1 ,rk )−Eh (r1 ,rk )) e−β (Eh (ri ,rk )−Eh (ri ,rk )) fold

(13.4.25)

i6=1,k

where k labels the attempted moved electron. In practice, the move is always accepted when fnew > fold ; when fnew < fold then the move is accepted with probability fnew /fold. The latter prescription is accomplished by extracting a (pseudo -)random number from a uniform distribution between 0 and 1. If fnew /fold is bigger than the extracted number the move is accepted, rejected in the contrary case. If the acceptance rule is satisfied, the trial configuration is accepted, i.e. it becomes the next element of the Markov chain. If the move is rejected, the next configuration is again the starting configuration. The process is iterated until the average quantity in Eq. 13.4.24 converges within a prefixed tolerance. Coming back to the evaluation of I[ρ(r1 )] and C[ρ(r1 )], 2 X ∇1 f 2 = ∇1 E(r1 ) − γ ∇ E (r , r ) (13.4.26) 1 h 1 n f n=2,N 2 = ∇1 E(r1 ) + (13.4.27)   X − 2γ∇1 E(r1 ) ·  ∇1 Eh (r1 , rn ) + (13.4.28) + γ2

XX i6=1 j6=1

n=2,N

[∇1 Eh (r1 , ri ) · ∇1 Eh (r1 , rj )]

(13.4.29)

All the quantities in the above equation are functions of the coordinates and can be sampled during the MC procedure. In facts: ! ∇1 ρ(r1 ) (r1 − rn ) ρ(r1 ) ∇1 Eh (r1 , rn ) = ρ(rn ) − (13.4.30) |r1 − rn | |r1 − rn |3

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and ∇1 E(r1 ) =

*

X

+

γ∇1 Eh (r1 , rn )

n=2,N

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379

(13.4.31) f

In summary: 2 γ 2 D ~ E γ2 Q + hP if 8 8 f C[ρ(r1 )] = hRif I[ρ(r1 )] = −

(13.4.32)

where . X ~ = Q ~qn

(13.4.33)

n=2,N

. XX P = [~qi · ~qj ]

(13.4.34)

i6=1 j6=1

. ~qn = ∇1 Eh (r1 , rn ) 1 . 1 X X R= N |r − rj | i j>i

(13.4.35) (13.4.36)

i=1,N

The Monte Carlo procedure for a given topology of density (uniform, spherical, high, low, intermediate and possible combinations thereof) optimized w.r.t. the γ and β parameters, allows to numerically evaluate Inonloc [ρ(r1 )]. The multidimensional integral is evaluated and thus all the non local effects are averaged out by the integration, which means we end up with a (numerical) local expression for Inonloc [ρ(r1 )]. In turn the numerical form of Inonloc [ρ(r1 )] can be fitted to an analytic function of ρ(r1 ) and employed in existing OFDFT codes. Of course for situations which heavily deviate from one of the categories of ρ given above, the analytic expressions can be used only in an approximative way. In fact ρ(r1 ) enters into the expression of f but it is at the same time the variable of the variational problem of Eq.13.2.6. This problem would disappear by so lving Eq.13.2.6 in a self-consistent iterative way, where the Monte Carlo optimization would add up at each step to the optimization of ρ. In principle this would represent a novel approach to the design of a fully consistent OFDFT algorithm. Given the plausibility of the physical hypothesis for constructing f the remaining part of the procedure is rigorous. MC settings for the (quasi)uniform electron gas We modelled a uniform distribution of electrons by means of a system of N electron, with N ranging from 10 to 500, in a cubic box. In theory the number of electrons considered in a uniformly distributed gas should be infinite, but, for the sake of numerical evaluation, they ought to be a finite number. Care has to be used in order

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either to obtain a size independent result or design a strategy for extrapolating the finite sized result towards the infinite size limit. The MC scheme works as follows: one electron is selected randomly and a trial move is attempted. We adopt a trial move as a uniformly distributed (isotropic) displacement of a randomly selected electron. The acceptance of the move is decided on the basis of Eq. 13.4.25. The maximum displacement is chose such that the accepted moves are around 50 %, when possible (if the density is very low practically each move is accepted, whatever the maximum displacement). The electron in position r1 is kept fixed, so that the evaluation of the integral I[ρ(r1 )] is referred to the position of such fixed electron. In the case we consider here, i.e. (quasi) uniform density, the acceptance rule given in Eq. 13.4.25 automatically keeps on sampling configuratons at constant, uniform density. In fact, one can see the adopted Eh (ri , rj ) term as a EH (ri , rj ) term, with uniform (r independent) density in the numerator, where the actual value of the density is absorbed into γ. Here we stress that the “quasi” specification we add to “uniform”, means that with a finite number of electrons the uniformity is conserved only in an average sense. I.e., taking few configurations along the chain, the average distribution of the particels is indeed uniformly distributed in the simulation box. Obviously, every displacement from a lattice-wise rigorously uniform distribution of the electrons is a “wiggle” in the uniformity of the distribution itself. It is actaully in this wiggles that the correlation information we are sampling is hidden. In case of non-uniform density (where an extrernal field is acting, e.g. atomic-like distributions generated by the nuclear electric field), a more complicated acceptance rule must explicitely fix (on average) the imposed density. In order to tackle finite sizeness, the standard strategy of imposing periodic boundary conditions and minimum image convention was adopted.41 In practice, each displaced electron was in turn the center of the box and only one instance per particle was used in evaluating the quantities in equations 13.4.25, 13.4.30, and 13.4.31. Periodic replicas of the system would be necessary for the evaluation of the slowly decaying “Coulomb” integral (Eq. 13.4.32); in contrast, for the evaluation of I[ρ(r1 )], it would be physically wrong to count correlations of periodic replicas (therein including spurious self-correlations between the displaced electron and its images): each electron should contribute once to the integral. This choice leads to an obvious strong finite size effect for the term C[ρ(r1 )]. Concerning the minimization w.r.t f (Eq. 13.3.11), we perform the search numerically at each density. We impose γ = β in order to perform the search with only one parameter. A physical motivation of this choice relies in the indistinguishability of the electrons. At each density we calculate the value of I[ρ(r1 )] + C[ρ(r1 )], varying the parameter γ and looking for the minimum of I[ρ(r1 )] + C[ρ(r1 )]. In Ref. 9 we showed the outcome of our calculations. Here we report in Figs 13.4.1 and 13.4.2 the main result. It turns out the variational problem is, at least

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numerically, well posed, in the sense that for each density, the functional I[ρ(r1 )] + C[ρ(r1 )] is convex and has a minimum at a certain value γ¯. Furthermore, the numerical results show that:9 Z Inonloc [ρ(r)] ∝ ρ ln ρdr. (13.4.37) Thus the total kinetic functional we found is: Z Z 1 |∇ρ(r)|2 K[ρ] = dr + B ρ(r) ln ρ(r)dr 8 ρ(r)

(13.4.38)

where B is a constant. The numerical result is characterized by two notable characteristics. The value of γ that minimizes the functional at each density (see Fig. 13.4.1), does not depend on the number of electrons in the box. Likewise, the value of I[ρ(r1 )] at each density does not depend on the number of electrons, within numerical accuracy. This a posteriori size independence is a robust property of our numerical scheme.

I(ρ)+C(ρ)

ρ=1.0

0

0.5

ρ=0.2

0

50

γ

ρ=0.04

0

2500

Fig. 13.4.1. The three panels show the values of γ that minimize the functional I[ρ(r1 )]+C[ρ(r1 )] at three representative densities (for N=100 electrons). Densities are expressed in atomic units.

A logarithmic form of the kinetic functional does not resemble any of the functionals employed so far. For this reason its validation goes through the ability of answering two questions: (a) Does it reproduce known formal properties of the kinetic functional? (b) What is its physical interpretation? In Ref. 10 one of us addressed these points. Here we summarize the outcome. 13.4.1. Scaling properties The conditions of validity for the functional above (or equivalently, to have the correct interpretation of this numerical result), can be determined by checking the formal properties of Inonloc [ρ(r)] under a linear uniform scaling of the coordinates (see e.g. Refs. 42,43). Given a scaling parameter λ > 0, the uniformly scaled wavefunction is defined as: ψλ (r1 , . . . , rN ) = λ3N/2 ψ(λr1 , . . . , λrN )

(13.4.39)

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I(ρ)

L. M. Ghiringhelli & L. Delle Site

_

50 e _ 100 e_ 250 e 0

0.5

ρ

1

1.5

Fig. 13.4.2. Calculated values of I[ρ(r1 )] for N = 50, 100, and 250 electrons. The continuous line is the fit we propose (see text).

this leads to hψλ |ψλ i = hψ|ψi = 1.

(13.4.40)

With the above expression for the density ρ(r) one has: ρλ (r) = λ3 ρ(λr)

(13.4.41)

and the number of particles is conserved: Z Z 0 0 ρλ (r)dr = ρ(r )dr = N

(13.4.42)

0

with r = λr.42,43 For the kinetic energy under the same coordinate scaling, one obtains:



 ψλ |∇2 |ψλ = λ2 ψ|∇2 |ψ . (13.4.43)

Deriving the scaling of the correlation part of the kinetic functional Inonloc [ρ(r)] is more involved. Levy and Perdew have elegantly shown44 that (in the original paper, our Inonloc is named Tc ): Inonloc [ρλ (r)] < λ2 Inonloc [ρ(r)], λ > 1

(13.4.44)

2

Inonloc [ρλ (r)] > λ Inonloc [ρ(r)], λ < 1 (13.4.45) R At this point, the question is what happens to ρ ln ρ under this scaling process. The process of coordinate scaling in f corresponds to vary the distance between each −γ

1

pair of particles |ri − rj |, uniformly in the expression e λ|ri −rj | (linearly in λ), that is |ri − rj | → λ|ri − rj |. For an almost uniform density (i.e. the approximation in which we work) λ|ri − rj | corresponds to varying the lattice parameter of a grid,

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that is to have the mapping ρ → ρλ = λ3 ρ. This, in turn, means applying the Monte Carlo procedure for

R

ΩN −1

2 ∇ 0 f (r0 ,...,r0 |r0 ) 2 N 1 r1 0

0

0

f (r2 ,...,rN |r1 )

0

0

dr2 . . . drN in Inonloc [ρ(rλ )] at

density ρλ . But, as shown by the numerical results in Ref. 9, this sampling still gives a point that lies on the ln ρ curve. 13.4.2. Connection to information theory It is interesting to observe that the first term on the r.h.s. of Eq.13.4.38 is the exact Fisher functional, while the logarithmic term in the same equation is proportional to the Shannon entropy, i.e. the two key terms of information theory.45 Recently, in several publications, this theory became of interest in the field of electron density functionals;46–49 in particular, the Shannon entropy has been linked to some properly defined measure of the electron correlations (see e.g.46,50–54 ). More in detail, Nagy,47,54 and Romero and Dehesa46 discuss the ratio of the Fisher functional and the Shannon entropy in terms of localizations versus delocalization . This means that, for a given a system of electrons, the ground state properties are expressed via its electron density which carries, in a local form, also the non local propert ies of the system. These latter are the direct expressions of the strength of the correlations among electrons that are far apart. Since the Fisher functional appears as an exact term of the kinetic functional, an intuitive argument would suggest that something related to the Shannon entropy would be another term of this functional. This, because the balance between locality and non-locality expresses the basic principle of equilibrium for a system of electrons. For example, for a manyatoms system, one would consider the balance between the localization of ρ around an atomic nucleus and the delocalization, i.e. the spreading of ρ in space, due to the electron-electron interactions among particles belonging to different atoms. The functional of Eq.13.4.38 has got both terms, but, as we have explained above, the scaling properties of the Shannon term are not correct. The solution to this problem moves from the work of Refs. 46,54. They define the following quantity: Jσ =

1 2 Sσ e3 2π

(13.4.46)

R with σ = Nρ 46,54 and Sσ = − σ ln σ. Jσ scales as λ−2 . While in these references the interest is about the ratio between the Fisher and the Shannon, for our pourposes we are interested to consider Jσ−1 , a quantity that would express the correlation measure for the electrons and have a scaling as λ2 , which is not the correct scaling for the correlation part of the kinetic functional Inonloc , though.10 In order to make a connection between Inonloc and Jσ−1 , we have first to make the latter consistent with the conditions of Eqs. 13.4.44 and 13.4.45. One can still use the same construction of Eq. 13.4.46 and generalize it: n

Jσ−1 = ξe− 3 Sσ

(13.4.47)

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where ξ is a suitable factor and the parameter n must be such that 0 < n < 2. In fact, as one can easily verify, Jσ−1 = λn Jσ−1 λ

(13.4.48)

where Jσ−1 is Jσ−1 with the spatial coordinates r lineraly scaled by a factor λ. The λ scaling properties of Eqs. 13.4.44 and 13.4.45 are preserved because for 0 < n < 2, one has λn < λ2 for λ > 1 and λn > λ2 for λ < 1.10 The question now, is whether one can connect our numerical results and the functional form of Jσ−1 . This may clarify the meaning of the numerical approach and, at the same time, the role of the Shannon term in writing a kinetic functional. To this aim we suggest the following interpretation: in order to have the correct scaling for the kinetic functional, K[ρ] may be written as: Z 1 |∇ρ|2 n K[ρ] = dr + ξe− 3 Sσ (13.4.49) 8 ρ The relation between Eq. 13.4.49 and our numerical result: Z Z 1 |∇ρ|2 K[ρ] = dr + B ρ ln ρdr 8 ρ

(13.4.50)

n

becomes clear, if we expand e−R3 Sσ in a Taylor series around S = 0; at the first order we get a term proportional to ρ ln ρ dr, thus our numerical results are proportional to the first order expansion of (the generalized version of) Jσ−1 . This in turn suggests that Jσ−1 may represent a high order Rform of Inonloc [ρ(r)]. The question that one must still address is whether the term ρ ln ρ is a first order approximation because of the statistical/numerical character of the Monte Carlo procedure or because the f proposed does not contains enough explicit correlations so that a functional form as the Jσ−1 can be obtained. The problem in addressing this question is that accounting for higher order correlations terms, in the method proposed, would mean; (a) to find a proper form of f with higher correlation terms without the loss of a direct physical interpretation of such a function; (b) this form may be rather complex and would inevitable induce a further complexity in the MC approach which would require substantial technical improvements in the sampling procedure. A possible, indirect answer to this question would be that of employing such a functional for calculating material properties and compare the outcoming results with those of currently used functionals, however this goes beyond the aim of this contribution and here should be intended as a suggestion for a practical evaluation of the functional. 13.5. Introduction of (statistical) spin As a final remark, we point out another problem our approach presents, and the strategy we are adopting in order to solve it. The kinetic functional we proposed does not satisfy the Lieb-Thirring22 inequality . This is not surprising, since it does not converge to the Thomas-Fermi functional, in the limit of non-interacting

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particles (i.e. if the Coulomb interactions are set to zero in f ). In order to overcome this shortcoming, we devised a new formulation of f which, by construction, yields the Thomas-Fermi kinetic functional if the Coulomb interaction is switched off. To the purpose we introduced physically sounding “spin interactions” between our electrons, by defining what we call “Pauli pairs”.11 The idea is that each electron interacts (only) with its closest neighbour with an extra energy term that varies linearly with the distance between these two electrons (the linear dependence is an exact result, if one wants to obtain Thomas-Fermi from an exponential-like f , in the limit of uncharged particles). We call “Pauli pair” the pair of these closest neighbours. In the same time all the electrons still repel each other via 1/r Coulomb interactions. Thus, spin is introduced in a statistical sense (the Pauli pairs can recombine during the sampling), in the same spirit as it is seen in the ThomasFermi theory. Results that we presented in detail in Ref. 11 show that for a (quasi) uniform electron gas, this modification does not change the ρ log(ρ) functional form of the numerically evaluated Inonloc [ρ(r)].

References 1. Y. A. Wang and E. A. Carter, “Orbital-Free Kinetic-Energy Density Functional Theory,” in “Theoretical Methods in Condensed Phase Chemistry,” edited by S. D. Schwartz (Kluwer, Dordrecht, 2000), Chap. 5, pp. 117-184. 2. N. Choly, E. Kaxiras, Solid State Communications 121, 281, (2002). 3. R. Pino, A.J. Markvoort, R.A. van Santen and P.A.J. Hilbers, Physica B 339, 119 (2003). 4. J.D. Chai, J.A. Weeks, J. Phys. Chem. B 108, 6870 (2004). 5. R.L. Hayes, M. Fago, M. Ortiz and E.A. Carter, Multisc. Mod. Sim. 4, 359 (2005) 6. G. Lu and E. Kaxiras, in Handbook of Theoretical and Computational Nanotechnology, Michael Rieth and Wolfram Schommers Eds. Vol. X, pp. 1-3 (American Scientific Publishers, 2005). 7. L. Delle Site, J. Phys. A 39, 3047 (2006). 8. L. Delle Site, J. Phys. A 40, 2787 (2007). 9. L.M. Ghiringhelli and L. Delle Site, Phys. Rev. B 77, 073104 (2008). 10. L. Delle Site, Eur. Phys. Lett. 86, 40004 (2009). Erratum Eur. Phys. Lett. 88, 19901 (2009) 11. L.M. Ghiringhelli, I.P. Hamilton and L. Delle Site, J. Chem. Phys. 132, 014106 (2010). 12. M. Levy, Proc. Natl. Acad. Sci. U.S.A. 76, 6062 (1979); see also: M. Levy, Phys. Rev. A 26, 1200 (1982). 13. E. Lieb, Int. Jour. Quant. Chem. 24, 243-277 (1983). An expanded version appears in Density Functional Methods in Physics, R. Dreizler and J. da Providencia eds., Plenum Nato ASI Series 123, 31-80 (1985). 14. L. Delle Site, J. Phys. A 38, 7893 (2005). 15. R.A. Mosna, I.P. Hamilton, and L. Delle Site, J. Phys. A 38, 3869 (2005). 16. R.A. Mosna, I.P. Hamilton, and L. Delle Site, J. Phys. A 39, L229 (2006). 17. R.A. Mosna, I.P. Hamilton, and L. Delle Site, Th. Chem. Acc. 118, 407 (2007). 18. S.B. Sears, R.G. Parr, and U. Dinur, Israel. J. Chem. 19, 165 (1980). 19. P.W. Ayers, J. Math. Phys. 46, 062107 (2005).

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20. R.G. Parr ans W. Yang, Density Functional Theory of Atoms and Molecules, Clarendon Press Oxford 1989. 21. N.H. March and W.H. Young, Proc. Phys. Soc. London 72, 182 (1958). 22. E.H.Lieb and W.E.Thirring, Phys. Rev. Lett. 35, 687 (1975); Errata, Phys. Rev. Lett. 35 1116 (1975). 23. E.H. Lieb, Rev. Mod. Phys. 48, 553 (1976). 24. E.H. Lieb, in Mathematical Problems in Theoretical Physics, K.Osterwalder Ed., Lecture Notes in Physics, Vol.116 (Springer-Verlag, Berlin) (1980). 25. E.H. Lieb, Rev. Mod. Phys. 53, 603 (1981). 26. W. Macke, Phys. Rev. 100 992 (1955). 27. P.K. Acharya, L.J. Bartolotti, S.B. Sears, and R.G. Parr, Proc. Natl. Acad. Sci. U.S.A. 77 6978 (1980). 28. J.E.Harriman, Phys. Rev. A, 24 680 (1981). 29. J.L. Gazquez and J. Robles, J. Chem. Phys. 76, 1467 (1982). 30. R.K. Pathak and S.R. Gadre, Phys. Rev. A 25 3426 (1982). 31. I. Daubechies, Comm. Math. Phys. 90, 511 (1983). See also P. Blanchard and J. Stubbe, Rev. Math. Phys. 8, 503 (1996). 32. G. Zumbach, Phys. Rev. A 31 1922, (1985). 33. L. Spruch, Rev. Mod. Phys. 63, 151 (1991). 34. V.V. Karasiev, E.V. Lude˜ na and A.N. Artemyen, Phys. Rev. A 62 062510 (2000) 35. E.V. Lude˜ na, V.V. Karasiev, and P. Nieto, Theor. Chem. Acc. 110, 395 (2003). 36. E.V. Lude˜ na, V.V. Karasiev and L. Echevarria, Int. J. Quant. Chem. 91, 94 (2003). 37. E.V. Lude˜ na, D. Gomez, V. Karasiev, P. Nieto, Int. J. Quant. Chem. 99, 297 (2004). 38. N.H. March, Int. J. Quant. Chem. 101, 494 (2005). 39. E. Wigner and F. Seitz, Phys. Rev. 46, 509 (1934). 40. E. Wigner, Phys. Rev. 46, 1002 (1934). 41. B.J. Smit and D. Frenkel, Understanding Molecular Simulation, Academic Press Inc. U.S. (1996). 42. A Primer in Density Functional Theory, C.Fiolhais, F.Nogueira and M.Marques (Eds.), Lecture Notes in Physics, 620, Springer-Verlag, Berlin Heidelberg 2003. 43. G.F.Giuliani and G.Vignale, Quantum Theory of Electron Liquid, Cambridge University Press (2005). 44. M. Levy and J.P. Perdew, Phys. Rev. A 32, 2010 (1985). 45. L.M. Ghiringhelli, L. Delle Site, R.A. Mosna and I.P. Hamilton, J. Math. Chem. (2010) in press. 46. E. Romera and J.S. Dehesa, J. Chem. Phys. 120, 8906 (2004). 47. A. Nagy, J. Chem. Phys. 119, 9401 (2003). 48. R.F. Nalewajski, Advances in Quantum Chemistry 43, 119, (2003). 49. R.F. Nalewajski, Chem. Phys. Lett. 386, 265 (2004). 50. P. Ziesche, Int. J. Quant. Chem., 56, 363 (1995). 51. N.L. Guevara, R.P. Sagar and R.O. Esquivel, Phys. Rev. A 67, 012507 (2003). 52. A.D. Gottlieb and N.J. Mauser, Phys. Rev. Lett. 95, 123003 (2005). 53. K.D. Sen, J. Antol´in and J.C. Angulo, Phys. Rev. A 76, 032502 (2007). ´ Nagy, Phys. Lett. A 372 2428 (2008). 54. J.B. Szab´ o, K.D. Sen and A.

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Chapter 14 Kinetic Energy and Fisher Information

´ Agnes Nagy Department of Theoretical Physics, University of Debrecen H–4010 Debrecen, Hungary [email protected] The relationship between the kinetic energy and the Fisher information is discussed. It is argued that the sum of the one-electron Fisher information, called modified Fisher information, is the original Fisher information plus a sum of differences of quantum and classical variances. A generalization of the Stam’s inequality is presented. It is shown how to derive the Euler equation of the density functional theory from the principle of minimum Fisher information. Making use of the differential virial theorem of Nagy and March, a first-order differential equation for the functional derivative of the kinetic energy functional is derived for spherically symmetric systems.

Contents 14.1 14.2 14.3 14.4 14.5 14.6

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fisher information . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Relationship between the Shannon and Fisher information, the local wave-vector . The modified Fisher information . . . . . . . . . . . . . . . . . . . . . . . . . . . . Pauli energy in the density functional theory . . . . . . . . . . . . . . . . . . . . . The Euler equation of the density functional theory from the principle of extreme physical information . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.7 The functional derivative of the kinetic energy functional . . . . . . . . . . . . . . 14.8 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . .

. . . . .

387 388 389 390 392

. . . .

. . . .

394 395 398 399

14.1. Introduction Nowadays, density functional calculations are mainly performed by solving the Kohn-Sham equations. There is, however, a growing interest in orbital-free calculations, too. These are inexpensive, but inaccurate because of the lack of accurate approximation for the kinetic energy functional. The search for approximate kinetic energy functional has turned to the information theory, too. Since the fundamental paper of Sears, Parr and Dinur1 it is known that there exist a relationship between the quantum mechanical kinetic energy functional and the Fisher information.2 In this chapter this relationship is detailed. 387

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After presenting the definition and the main properties of the Fisher information, it is shown that the local wave-vector establishes a relationship between the Shannon3 and Fisher information.4 Then, a modified Fisher information,5 the sum of the one-electron Fisher information, is introduced. It is shown that the modified Fisher information is the original Fisher information plus a sum of differences of quantum and classical variances. A generalization of the Stam’s inequality6 is derived, too.5 Using the definition of the Pauli energy,7–10 the Euler equation of the density functional theory is derived from the principle of extreme physical information. Making use of the differential virial theorem of Nagy and March,11 a first-order differential equation for the functional derivative of the kinetic energy functional is derived for spherically symmetric systems.12 14.2. Fisher information Fisher information2 is a measure of the ability to estimate a parameter and is a measure of the state of disorder of a system or phenomenon. The Fisher informational functional2 is defined as 2  Z Z 2 [p0 (x|θ)] ∂lnp(x|θ) dx = dx . (14.2.1) IF (θ) = p(x|θ) ∂θ p(x|θ) p(x|θ) is a probability density function, obeying proper regularity conditions and depending on a parameter θ. Take θ to be a parameter of locality: p(x|θ) = p(x + θ) = p(γ) .

(14.2.2)

∂p(x|θ) ∂p(x + θ) ∂p(γ) = = . ∂θ ∂(x + θ) ∂γ

(14.2.3)

Then

In this case Eq. (14.2.1) has the form 2 Z  ∂p(x + θ) IF (θ) = /p(x + θ)dx. ∂(x + θ)

(14.2.4)

This is Fisher information per observation with respect to the locality parameter θ. As the expression does not depend on θ, we may set the locality at zero: Z [p0 (x)]2 IF (θ = 0) = dx . (14.2.5) p(x) This Fisher information for locality is called intrinsic accuracy. It measures the ‘narrowness’ of a distribution. For the variance of x holds the Cramer-Rao inequality13 V arx ≥ I −1 .

(14.2.6)

For the normal distribution the Fisher information is equal to the inverse variance. In that case relation (14.2.6) is an equality showing that a narrower distribution

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Kinetic Energy and Fisher Information

has a larger Fisher information. In general x is vector-valued and the expression (14.2.5) has the form Z [∇p(r)]2 IF (θ = 0) = dr . (14.2.7) p(r) To establish a relationship between the kinetic energy and the Fisher information, we have to recall that the full Weizs¨acker kinetic energy 14 Tw is defined as: Z 1 |∇n|2 Tw = dr. (14.2.8) 8 n

A comparison of Eqs. (14.2.8) and (14.2.5) leads to the relation that the Weizs¨acker kinetic energy14 is proportional to the Fisher information: N Tw = I , (14.2.9) 8 where the number of electrons N arising from the fact that the electron density n is normalized to N , while the probability density function is normalized to 1. Fisher information has proved to be very useful in studying several systems. E. g. Fisher information of single-particle systems with a central potential was determined15 and that of a two-electron ‘entangled artificial’ atom proposed by Moshinsky was studied.16 In a recent paper atomic Fisher information17 has also been investigated. The relationship between the densities of Shannon and Fisher information has been studied18,19 and ionization processes have been analyzed in the Fisher-Shannon plane.20 Even phase-space Fisher information21 has been introduced. 14.3. Relationship between the Shannon and Fisher information, the local wave-vector Recently, a simple relationship between the Shannon and Fisher information has been explored.4 Nagy and March22 introduced the ratio of the density gradient to the electron density as a local wave-number to characterize the ground state of atoms and molecules. Independently, Kohout, Savin and Preuss23 also investigated the role of the quantity |∇n/n| in the shell structure of atoms. The local wave-vector q(r) = −

∇n(r) , n(r)

(14.3.10)

has the dimension of wave-number. The local wave-vector q gives a simple relationship between the Shannon and Fisher information. The Shannon information3 is defined as Z Z S = − p(r) ln p(r)dr = s(r)dr. (14.3.11)

It has been shown18 that the Fisher information can also be written as Z I = − ∇2 p(r) ln p(r)dr.

(14.3.12)

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Introducing the local Shannon s(r) R and the local Fisher information i(r) with the definitions Eq. (14.3.11) and I = i(r)dr one can immediatelly see that  2   2 ∇p −s i 2 = = [∇(ln p)] = ∇ (14.3.13) p p p As ∇(ln p) = we arrive at the result

∇p = −q, p

  s q=∇ p

(14.3.14)

(14.3.15)

and q2 =

i . p

(14.3.16)

That is, the local wave-vector q is the gradient of the Shannon information per particle and the square of the local wave-vector is the Fisher information per particle. Consequently, the local wave-vector gives the connection between the Shannon and Fisher information. 14.4. The modified Fisher information Hall24 showed that Fisher information of a quantum observable is proportional to the difference of a quantum and a classical variance:   (14.4.17) I = 4 hP 2 iΨ − hPcl2 iΨ ,

where P is the momentum observable conjugate to X and Pcl is the classical momentum observable corresponding to the state Ψ. Pcl is defined as

1 [Ψ0 (x)/Ψ(x) − Ψ∗0 (x)/Ψ∗ (x)] = [arg Ψ(x)] . (14.4.18) 2i Hall argued that this definition is supported by the facts that the probability density |Ψ|2 satisfies the classical continuity equation and holds the identity Pcl =

hP iΨ = hPcl iΨ .

(14.4.19)

Thus the position Fisher information is proportional to the nonclassical variance of the conjugate momentum: I = 4(∆Pnc )2 .

(14.4.20)

Analogous relations can be obtained in momentum space. Note that Pcl disappears if we have a real wave function. The method of Hall was generalized to more than one particle.5 Consider a system of N independent electrons, that is a Kohn-Sham scheme. Here we have

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N occupied orbitals uj , j = 1, ..., N in the ground state. Define a modified Fisher information as a sum of one-particle Fisher information X I˜ = Ij , (14.4.21) j

where Ij =

Z

nj



∇nj nj

2

dr

(14.4.22)

and nj = |uj |2

(14.4.23)

is the one-particle density. Utilizing the method of Hall the one-particle Fisher information can be rewritten as   (14.4.24) Ij = 4 hP 2 iuj − hPcl2 iuj ,

or

j 2 Ij = 4(∆Pnc ) .

(14.4.25)

Consequently, the one-particle Fisher information of a quantum observable is proportional to the difference of a quantum and a classical variance of the momentum taken with the orbital uj . To derive a relationship between the modified Fisher information and the Fisher information Z [∇n(r)]2 1 I= dr (14.4.26) N n(r) introduce the functions χj with the following definition: 1

u j = n 2 χj .

(14.4.27)

As it can be seen from this definition (14.4.27) the functions χj are not all independent: X 1= |χj |2 . (14.4.28) j

Taking the gradient of this equation we obtain X 0= (χ∗j ∇χj + ∇χj χ∗j ) .

(14.4.29)

j

Combining Eqs. (14.4.21), (14.4.24), (14.4.27), (14.4.28) and (14.4.29) we are led to the important result ! Z X Z X ∗ 2 ∇χ ∇χ j j I˜ = I + 4 n |∇χj |2 + n |χj |2 − ∗ (14.4.30) χj χj j j

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that can also be written as I˜ = I + 4

X j

 hP 2 iχj − hPcl2 iχj .

(14.4.31)

The modified Fisher information is the sum of the Fisher information and a term which is proportional to the sum of differences of one-paricle quantum and classical variances of the momentum taken with the functions χj : X j 2 I˜ = I + 4 (∆Pnc ) . (14.4.32) j

Keeping in mind that the non-interacting kinetic energy can be written as Z 1 X Ts = |∇uj |2 dr (14.4.33) 2 j and using Eqs. (14.4.27) and (14.4.18) we readily obtain that 1X 2 N Ts = I˜ + hPcl iχj . 8 2 j

(14.4.34)

This is a relationship between the modified Fisher information and the noninteracting kinetic energy. (It has been known that the kinetic energy can be expressed with the variables χj and n since the works of Macke25 and Harriman.26 ) The immediate consequence of Eq. (14.4.34) is that N˜ I. 8

Ts ≥

(14.4.35)

Equality is for real one-particle wave functions, that is, for real one-particle wave functions the modified Fisher information is proportional to the non-interacting kinetic energy. As from Eq. (14.4.32) follows that I˜ ≥ I,

(14.4.36)

X

(14.4.37)

we are led to the inequality I ≤4

j

hPi2 i.

This is the generalization of Stam’s inequality.6 We note in passing that the relationship between the kinetic energy and the Fisher information has been studied by Nalewajski27 and Delle Site.28 14.5. Pauli energy in the density functional theory The total kinetic energy of the density functional theory can be separated into two terms: Ts = Tw + Tp ,

(14.5.38)

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(14.5.39)

and the term Tp is called the Pauli energy.7–10 The functional derivatives are 2   1 2 δTw 1 ∇n 1 ∇2 n −1/2 − ∇ n1/2 (14.5.40) = − =n δn 8 n 4 n 2 and

vp =

δTp , δn

(14.5.41)

where vp is the Pauli potential. Tp and vp embody all the effects of the Pauli principle (the antisymmetry requirement for the wave function). The Euler-equation has the form δTw + vp + vKS = µ , δn

(14.5.42)

where µ is the Lagrange multiplier arising from the condition that the number of electrons is kept fixed. The Euler-equation can also be written as a single Schr¨ odinger-like equation for n1/2   1 2 − ∇ + vp + vKS n1/2 = µn1/2 . (14.5.43) 2 As the density is the same in the non-interacting and the interacting systems Eq. (14.5.43) holds in both systems. Levy et al.8 derived the potential vef f = vp + vKS in the interacting system from the Schr¨odinger equation. In the non-iteracting system the Pauli potential can be written as10,29 tp 1X + vp = λj (εM − εj )nj , (14.5.44) n n j where tp , εj , λj and nj are the Pauli kinetic energy density, the orbital energies, the occupation numbers and orbital energy densities, respectively. (Double occupation is assumed: M = N/2.) Sears, Parr and Dinur1 established a precise relationship between the quantum mechanical kinetic energy functional and the Fisher information. They derived the identity Z Z N |∇n|2 N T = dr + n(r)IFf (r)dr, (14.5.45) 8 n 8 where the first term is proportional to Fisher information and in the second term IFf is a Fisher information Z [∇1 f (2, ..., N |1)]2 f IF (1) = d2...dN (14.5.46) f (2, ..., N |1)

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associated with the conditional density f (2, ..., N |1) = |Ψ|2 /n(r). They normalized the one-particle density to 1. In this paper n is normalized to N as is usually done in density functional theory. So there is a difference in factor between Eqs. (14.5.45) and (14.5.38). 14.6. The Euler equation of the density functional theory from the principle of extreme physical information The principle of extreme physical information is a variational principle. Frieden13 showed that it can be used to derive major physical laws. The principle states that the ‘physical information’ K of the system is extremum K = I − J = extrem.,

(14.6.47)

where I is the fixed form of ‘intrinsic’ information defined above (Eqs. (14.2.5) and (14.2.7)). J is the bound Fisher information that embodies all constraints that are imposed by the physical phenomenon under measurement. Based on the work of Brillouin30 Frieden showed that J ≥ I,

(14.6.48)

as a consequence of the second law of thermodynamics. Frieden presented the unifying aspect of the principle of extreme physical information by applying it to the major fields of physics: quantum mechanics, classical electromagnetic theory, statistical mechanics, etc. In accordance with the principle of extreme physical information we minimize the Fisher information (14.2.5) with conditions. It is more convenient to minimize Tw instead of IF . The minimization is done under the conditions: 1. The wave function is antisymmetric. This requirement is taken into account by a local potential w(r). Earlier Flores and Keller31 already used the idea of including the internal symmetry of the system by a local potential in the variational principle for the energy functional. 2. The density is kept fixed. This requirement is taken into account by a local potential vKS (r). This constraint is used in the adiabatic connection to ensure that the density of the non-interacting system be equal to that of the interacting one. 3. The electron density is normalized to N . A Lagrange multiplicator µ is introduced. Now the Euler-Lagrange equation coming from the minimization of the Fisher information  Z  Z Z Z δ 1 |∇n|2 dr + nwdr + nvKS dr − µ ndr = 0 (14.6.49) δn 8 n

takes the form:

δTw + w + vKS = µ . δn

(14.6.50)

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Comparing Eq. (14.6.50) with the Euler equation (14.5.42) we see that w is the Pauli potential: w(r) = vp (r) .

(14.6.51)

This is the formal derivation of the Euler equation of the density functional theory from the principle of extreme physical information. This is formal as it does not result in an explicit expression for the Pauli energy density tp and the Pauli potential vp . A comparison of Eqs. (14.5.44) and (14.6.51) leads to X tp (r) = w(r)n(r) − λj (εM − εj )nj . (14.6.52) j

The time-dependent Euler equation of the density functional theory has also been derived using this principle32 and the time-independent Kohn-Sham equations have recently been derived,33 too. 14.7. The functional derivative of the kinetic energy functional In spherically symmetric systems the Kohn-Sham equations 1 − ∇2 uk + vKS uk = k uk , 2

(14.7.53)

take the form −

1 d2 Pk lk (lk + 1) + Pk + vKS Pk = k Pk , 2 2 dr 2r2

(14.7.54)

where vKS , uk , k and Pk = rRk (r) are the Kohn-Sham potential, the orbitals, the orbital energies and the radial wave functions, respectively. The differential form of the virial theorem for spherically symmetric Kohn–Sham potential vKS (r)11 is 1 1 0 q˜0 q˜ τ 0 = − %000 − %vKS + 2− 3 , 8 2 r r

(14.7.55)

q˜ = 4r2 πq

(14.7.56)

where

and q=

1X λk lk (lk + 1)nk . 2

(14.7.57)

k

The radial density and the radial kinetic energy density are given by % = 4r2 πn and

(14.7.58)

1 ∂ 2 %(r0 , r) 1X %k (r) τ (r) = − + λk lk (lk + 1) 2 , 2 2 ∂r 2 r r 0 =r k

(14.7.59)

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respectively. %k = Pk2

(14.7.60)

are the one-partical radial densities. For particles having zero angular momentum i.e. for s electrons the differential virial theorem reduces to the special form of March and Young34 1 1 0 . τ 0 = − %000 − %vKS 8 2 The Euler equation of the density functional can be written as δTs + vKS = µ, δn

(14.7.61)

(14.7.62)

where µ is the chemical potential.35–37 The radial kinetic energy density can also be written as τ (r) = 4r2 πt(r),

(14.7.63)

where t(r) = −

1X λk u∗k ∇2 uk . 2

(14.7.64)

k

Combining Eqs. (14.7.53), (14.7.62) and (14.7.64) we arrive at the relation: δTs X t(r) = n + (k − µ)nk . (14.7.65) δn k

From Eqs. (14.7.63) and (14.7.65) we are led to the relation τ =%

δTs − µ% + g˜, δn

(14.7.66)

where g˜ = 4r2 πg(r)

(14.7.67)

X

(14.7.68)

and g(r) =

λk k nk .

k

(For closed shells the functions q and g are spherically symmetric.) Differentiating Eq. (14.5.40) with respect to r and substituting it into the differential virial theorem (14.7.55) we obtain  0 1 δTs δTs % + %0 = f˜, (14.7.69) 2 δn δn where 1 q˜0 q˜ f˜ = − %000 − g˜0 + µ%0 + 2 − 3 . 8 r r

(14.7.70)

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Partitioning the total non-interacting kinetic energy as a sum of the full Weizs¨ acker kinetic energy14 Tw and the Pauli energy Tp ((14.5.38)) and substituting Eqs. (14.5.40)-(14.5.41) into Eq. (14.7.69) we arrive at a first-order differential equation for the functional derivative of the Pauli energy, that is, for the Pauli potential vp :  0 1 δTp δTp % + %0 = f, (14.7.71) 2 δn δn where f = −˜ g 0 + µ%0 +

q˜ q˜0 − 3. 2 r r

(14.7.72)

In the knowledge of the functions %(r), g(r) and q(r) the differential equation (14.7.72) can be solved and the Pauli potential can be written as Z r 2 %(r1 )f (r1 )dr1 . (14.7.73) vp = 2 % ∞ The solution of this equation and Eq. (14.5.40) gives the functional derivative of the kinetic energy. Then the kinetic energy density τ can be obtained from Eq. (14.7.66) and the integration of τ leads to the total kinetic energy. As an example consider a three-level atom (e. g. Ne) and isoelectronic ions. It is instructive to introduce the functions ϑ and ϕ with the Dawson–March transformation38 1 P1 = √ %1/2 sin ϑ cos ϕ λ1

(14.7.74)

1 P2 = √ %1/2 sin ϑ sin ϕ λ2

(14.7.75)

1 P2 = √ %1/2 cos ϑ. λ3

(14.7.76)

Eqs. (14.7.74), (14.7.75) and (14.7.76) lead to the relations between the radial electron density and the phases ϑ and ϕ: ϕ00 +

Γ0 0 ϕ − 2ζ sin(2ϕ) = 0 Γ

 %0 0 1 ϑ + ϑ − sin(2ϑ) (ϕ0 )2 + % 2    1 2 3 − 1 − 2 + 2ζ sin ϕ = 0 , r

(14.7.77)

00

(14.7.78)

where Γ = % sin2 ϑ

(14.7.79)

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and 1 (1 − 2 ) . (14.7.80) 2 After some mathematics Eq. (14.7.73) gives  2  2 1 dϑ 1 dϕ vp = + sin2 ϑ + sin2 ϑ(3 − 1 ) 2 dr 2 dr cos2 ϑ . (14.7.81) + 2ζ sin2 ϕ sin2 ϑ + r2 In the knowledge of the density, the differential equations (14.7.77) and (14.7.78) can be numerically solved and the relation (14.7.81) gives the Pauli potential. ζ=

14.8. Discussion In quantum mechanics the variational method based on the variational principle was worked out. Considering the energy as a functional of the wave function E[Ψ] an approximation can be obtained for Ψ by minimizing E. An approximate Ψ of better quality gives deeper energy. It has been observed32 that Fisher information can also be used as a measure of the quality of an approximate wave function. It has been numerically demonstrated that Fisher information is a measure of the density quality.32 The ground-state electron density is a fundamental quantity. It determines every property of the electron system. Other quantities have also been turned out to be descriptors of a Coulomb system. For example, the shape function, or density per particle, reactivity indicators as Fukui function, local softness, softness kernel, electrostatic potential, local kinetic energy and local temperature have been shown39–41,41 to provide a full description of a Coulomb system. Recently, a broader family of descriptors of Coulomb systems is presented.12 It was proved that the quantities q and g are descriptors of a Coulomb system, that is, are capable of fully determining every property of a Coulomb system. In the ground state the density not only determines every property of an electronic system but there also exists a variational principle. The quantities q and g are descriptors of a Coulomb system, though they do not obey a variational principle. We would like to mention that the local ionization potential ε˜(r) = P 43,44 is closely related to the funck εk nk (r)/n(r) introduced by Politzer et al. tion g: ε˜(r)n(r) = g(r). The local ionization potential is a measure of chemical reactivity and is linked to the local temperature, and thus to the local kinetic energy.42 So there is a very intimite relationship between the function g and the local kinetic energy t. A further study of this relationship can help finding accurate approximation for the function g and would point toward practical applications in orbital-free DFT. In the differential equation (14.7.71) derived for the Pauli potential the knowledge of the functions %, q and g are needed to obtain a solution. Of course, in

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principle one of them is enough, as the other two quantities can be given as a functional of the selected descriptor. To find relations between them will be the subject of further studies. Whether the method presented in the previous section provides an efficient and accurate way to solve the Euler equation depends on how accurately the functions q and g are approximated using the density. It might turn out that it is more appropriate to use the descriptor q or g instead of the density.

Acknowledgments The work is supported by the TAMOP 4.2.1/B-09/1/KONV-2010-0007 project. The project is co-financed by the European Union and the European Social Fund. Grant OTKA No. K 67923 is also gratefully acknowledged.

References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28.

S. B. Sears, R. G. Parr and U. Dinur, Israel J. Chem. 19, 165 (1980). R. A. Fisher, Proc. Cambridge Philos. Soc. 22, 700 (1925). C. E. Shannon, Bell Syst. Tech. J. 27, 623 (1948). ´ Nagy and S. B. Liu, Phys. Lett. A 372, 1654 (2008). A. ´ Nagy, Chem. Phys. Lett. 449, 212 (2007). A. A. J. Stam, Inform. Control. 2, 101 (1959). N. H. March, Phys. Lett. A 113, 476 (1986). M. Levy, J. P. Perdew, and V. Sahni, Phys. Rev. A 30, 2745 (1984). S. B. Liu, J. Chem. Phys. 126, 244103 (2007). ´ Nagy, Acta Phys. Hung. 70, 321 (1991) and references therein. A. ´ Nagy and N. H. March, Phys. Rev. A 40, 554 (1989). A. ´ Nagy, Chem. Phys. Lett. 460, 343 (2008). A. B. R. Frieden, Physics from Fisher Information.A unification. (Cambridge, U. P., 1998). C. F. Weizs¨ acker, Z. Phys. 96, 431 (1935). E. Romera, P. S´ anchez-Morena and J. S. Dehesa, Chem. Phys. Lett. 414, 468 (2005). ´ Nagy, Chem. Phys. Lett. 425, 157 (2006). A. ´ Nagy and K. D. Sen, Phys. Lett. A 360, 291 (2006). A. S. B. Liu, J. Chem. Phys. 126, 191107 (2007). ´ Nagy, Phys. Lett. A 372, 2428 (2008). J. B. Szab´ o, K. D. Sen and A. K. D. Sen, J. Antolin, and J. C. Angulo, Phys. Rev. A 76, 032502 (2007). ´ Nagy, Chem. Phys. Lett. 437, 132 (2007). I. Horny´ ak and A. ´ Nagy and N. H. March, Mol. Phys. 90, 271 (1997). A. M. Kohout, A. Savin and H. Preuss, J. Chem. Phys. 95, 1928 (1997). M. J. W. Hall, Phys. Rev. A 62, 012107 (2000). W. Macke, Phys. Rev. 100, 992 (1955). J. E. Harriman, Phys. Rev. A 24, 680 (1982). R. Nalewajski, Chem. Phys. Lett. 367, 414 (2003); Int. J. Quantum. Chem. 108, 2230 (2008) L. Delle Site, J. Phys. A 39, 3047 (2006); R. A. Mosna, I. P. Hamilton and L. Delle

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29. 30. 31. 32. 33. 34. 35. 36. 37. 38. 39. 40. 41. 42. 43. 44.

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Site, J. Phys. A 39, L229 (2006); I. P. Hamilton, R. A. Mosna and L. Delle Site, Theor. Chem. Acc. 118, 407 (2007). M. Levy and H. Ou–Yang, Phys. Rev. A 38, 625 (1988) and references therein. L. Brillouin, Science and Information Theory. (Academic Press, New York, 1956). J. A. Flores and J. Keller, Phys. Rev. A 45, 6259 (1992). ´ Nagy, J. Chem. Phys. 119, 9401 (2003). A. R. Nalewajski, Chem. Phys. Lett. 372, 28 (2003). N. H. March and W. H. Young, Nucl. Phys. 12, 237 (1959). I. Lindgren and S. Salomonson, Phys. Rev. A 67, 056501(2003). R. van Leeuwen, Adv. Quantum Chem. 43, 24 (2003). S. B. Liu and P.W. Ayers, Phys. Rev. A 70, 022501 (2004). K. A. Dawson and N. H. March, J. Chem. Phys. 81, 5850 (1984). P. W. Ayers, Proc. Natl. Acad. Sci. 97, 1959 (2000). P. W. Ayers, Chem. Phys. Lett. 438, 148 (2007). ´ Nagy, J. Chem. Phys. 126, 144108 (2007). P. W. Ayers and A. ´ Nagy, Int. J. Quantum Chem. 90, 309 (2002). P. W. Ayers, R. G. Parr and A. P. Politzer, J. S. Murray, M. E. Grice, T. Brinck and S. Ranganathan, J. Chem. Phys. 95, 6699 (1991). J. S. Murray, J. M. Seminario, P. Politzer and P. Sjoberg, Int. J. Quantum Chem. 24, 645 (1990).

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Chapter 15 Quantum Fluctuations, Dequantization, Information Theory and Kinetic-Energy Functionals I. P. Hamilton1 , R. A. Mosna2 , and L. Delle Site3 1

2

Department of Chemistry, Wilfrid Laurier University Waterloo, Canada N2L 3C5 [email protected]

Instituto de Matem´ atica, Estat´ıstica e Computa¸c˜ ao Cient´ıfica Universidade Estadual de Campinas CP 6065, 13081-970, Campinas, SP, Brazil [email protected] 3

Max-Planck-Institute for Polymer Research Ackermannweg 10, D 55021 Mainz, Germany [email protected] Density functional theory, which employs the one-electron density as the fundamental variable, is extensively employed for the calculation of atomic and molecular properties. Achieving greater chemical accuracy or computational efficiency is highly desirable and this has motivated attempts to construct improved kineticenergy functionals. We hope that our results will ultimately aid in this endeavor but our aim in this chapter is conceptual rather than practical. We employ a dequantization procedure to obtain an exact expression for the kinetic energy of an N -electron system as the sum of an N -electron classical kinetic energy and an N -electron purely quantum kinetic energy arising from the quantum fluctuations that turn the classical momentum into the quantum momentum as in Nelson’s stochastic approach to quantum mechanics. We show that the N -electron purely quantum kinetic energy can be written as the sum of the Weizs¨ acker term, which is directly proportional to the Fisher information, and a purely quantum kinetic correlation term. We further show that the Weizs¨ acker term results from local quantum fluctuations while the purely quantum kinetic correlation term results from nonlocal quantum fluctuations. The N -electron classical kinetic energy is seen to be explicitly dependent on the phase of the N -electron wavefunction and this has implications for the development of accurate orbital-free kinetic-energy functionals. For stationary one-electron systems we show that there is a direct connection between the classical kinetic energy and the angular momentum. For nonstationary one-electron systems we show that the classical kinetic energy is complementary to the purely quantum kinetic energy which is directly proportional to the Fisher information. In this sense, the classical kinetic energy plays a role analogous to that of the Shannon information. 401

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Contents 15.1 Introduction . . . . . . . . . . . . . . . . . . . . . 15.2 Quantum-classical correspondence . . . . . . . . . 15.2.1 Equations of motion . . . . . . . . . . . . . 15.2.2 Momentum and kinetic energy operators . . 15.3 Dequantization . . . . . . . . . . . . . . . . . . . . 15.3.1 Witten deformation approach . . . . . . . . 15.3.2 Variational approach . . . . . . . . . . . . . 15.4 Information theory and kinetic-energy functionals 15.4.1 Information theory . . . . . . . . . . . . . . 15.4.2 Orbital-free kinetic-energy functionals . . . 15.4.3 Kinetic energy decomposition . . . . . . . . 15.4.4 Noninteracting kinetic energy . . . . . . . . 15.4.5 Results for one-electron systems . . . . . . . 15.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . .

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402 403 404 405 407 408 410 412 413 413 414 417 417 421 423

15.1. Introduction Density functional theory has developed into an extremely successful procedure for the calculation of atomic and molecular properties.1–3 In this approach, the electron density (of a single electron), ρone (r), is the fundamental variable and properties such as the energy are obtained from ρone rather than from the N electron density, ρN or the N -electron wavefunction, ψ(r1 , . . . , rN ). Here ρN = R R |ψ(r, . . . , rN )|2 and ρone (r) = N ρN d3 r2 . . . d3 rN , so that ρone (r) d3 r = N . The motivation for density functional theory is clear — if properties such as the energy can be obtained from ρone then calculations on systems with a large number of electrons are, in principle, no more difficult than those on systems with a small number of electrons. However, this depends on having accurate energy functionals which, in practice, is a serious problem. For multi-electron atoms, the energy can be partitioned into kinetic and potential terms and a clear zerothorder choice functional is the classical electrostatic enR for the potential-energy 2 R R ρ one (r1 )ρone (r2 ) 3 ergy −Ze2 ρoner (r) d3 r + e2 d r1 d3 r2 . However, for multi-electron r12 atoms, there is no correspondingly clear zeroth-order choice for the kinetic-energy functional. One of the key aspects of quantum mechanics is that one cannot simultaneously ascribe well-defined (sharp) values for the position and momentum of an electron (or any other particle). Motivated by this, quantization procedures have been proposed in which the quantum regime is obtained from the classical regime by adding a stochastic term to the classical equations of motion. In particular, Nelson4 has shown that the time-independent Schr¨odinger equation can be derived from Newtonian mechanics via the assumption that electrons are subjected to Brownian motion with a real diffusion coefficient. The Brownian motion results in an osmotic momentum and adding this term to the classical momentum results in the quantum momentum. We recently proposed a dequantization approach whereby the clas-

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sical regime is obtained from the quantum regime by stripping these “quantum fluctuations” from the quantum momentum, resulting in the classical momentum. This was first done via a Witten deformation5 and subsequently via a variational procedure.6 The latter formulation introduced a deformation of the momentum operator which corresponds to generic fluctuations of the quantum momentum and these lead to a deformed kinetic energy, which quantifies the amount of “fuzziness” caused by these fluctuations. We showed that the deformed kinetic energy possesses a unique minimum, which is seen to be the classical kinetic energy. In this way, a rigorous procedure determines the unique deformation that has the effect of suppressing the quantum fluctuations, resulting in dequantization of the system. From this variational procedure we obtain a term (essentially identical to Nelson’s osmotic momentum term4 ) which, when added to the classical momentum, results in the quantum momentum. In our dequantization approach the kinetic energy of an N -electron system is written as the sum of an N -electron classical kinetic energy and an N -electron purely quantum kinetic energy arising from the quantum fluctuations that turn the classical momentum into the quantum momentum. We show that the N -electron purely quantum kinetic energy can be decomposed as the sum of two terms. The first term results from local quantum fluctuations and is a functional of the electron density. It is the Weizs¨ acker term which is directly proportional to the Fisher information. The second term results from the nonlocal quantum fluctuations and is a functional of the N -1 conditional electron density. It can be evaluated using a rigorous quantum Monte Carlo procedure to obtain a functional of the electron density which is well-approximated by a Shannon information expression. The Fisher information and Shannon information are cornerstones of information theory and we believe that these connections are of significant conceptual value in showing the content that should be incorporated in any kinetic-energy functional.

15.2. Quantum-classical correspondence Since the origins of quantum mechanics there has been interest in the quantumclassical correspondence.7 Schr¨odinger in his 1926 paper8 began with the classical Hamilton-Jacobi equation and then wrote down what is now known as the Schr¨ odinger equation without making an explicit connection between the two. Subsequently Van Vleck in his 1928 paper9 modified the classical Hamilton-Jacobi equation to obtain a quantum-like formulation of classical mechanics. In the other direction, Madelung in his 1928 paper10 began with the wavefunction in polar form and wrote down hydrodynamic equations to obtain a classical-like formulation of quantum mechanics. This approach was extended by Bohm11 who explicitly introduced the quantum potential, Q. One can think of the quantum-classical correspondence as “switching off” the quantum potential term in the modified Hamilton-Jacobi equation12 and this approach has been explicitly explored.13 Also, the Q → 0 and

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(more usual) ~ → 0 approaches to the quantum-classical correspondence have been explicitly considered.14 15.2.1. Equations of motion We approach the quantum-classical correspondence at the level of the equations of motion of an N-electron system. Recall that, by expressing the wavefunction in polar form, the Schr¨ odinger equation can be transformed into two equations:10,15 a modified Hamilton-Jacobi equation in which the quantum potential, Q, appears in addition to the external potential, and a continuity equation. In this context, one formally obtains the quantum equations of motion by “switching on” the quantum potential term in the classical Hamilton-Jacobi equation. Similarly one formally obtains the classical equations of motion by “switching on” the quantum potential term in the Schr¨ odinger equation.12 Either way, Q is incorporated in the potential term in these approaches. We adopt an alternate approach, in which Q is incorporated in the kinetic term. This different perspective results in the definition of a deformed kinetic energy operator which naturally motivates the search for a deformed momentum operator. In the hydrodynamic formulation of quantum mechanics, the quantum potential appears as an additional potential term in a modified Hamilton-Jacobi equation which defines a classical-like description of quantum mechanics. In a complementary way, the quantum potential also appears as an additional term in a modified Schr¨ odinger equation which defines a quantum-like description of classical mechanics.12,13 Consider a quantum N-electron system described by a wavefunction ψ = ψ(r1 , ..., rN , t) satisfying the Schr¨odinger equation N

i~

X ~2 ∂ψ =− ∇2 ψ + V ψ, ∂t 2m k

(15.2.1)

k=1

where m is the electron mass and V includes internal (electron-electron) and exter√ i nal potentials. Writing ψ = ρN e ~ S , and expressing the Schr¨odinger equation in terms of ρN and S yields10,12 N

∂S X (∇k S)2 + + V + Q = 0, ∂t 2m k=1   N ∂ρN X ∇k S + ∇k · ρN = 0, ∂t m k=1

where Q is the quantum potential √ N X ~2 ∇k2 ρN Q=− . √ 2m ρN k=1

(15.2.2a)

(15.2.2b)

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The system of coupled equations (15.2.2) comprises a modified Hamilton-Jacobi equation — in which Q appears in addition to the classical potential V — and a continuity equation. In this way, it is the quantum potential that introduces, in the context of Eq. (15.2.2a), all the non-classical effects of quantum mechanics, such as superposition, interference and entanglement.13,15 We now consider a classical N -electron system whose action function S is govPN (∇k S)2 16 erned by the usual Hamilton-Jacobi equation ∂S The assok=1 ∂t + 2m +V = 0. drk ciated (local) momentum is given by pk = m dt = ∇k S. Then ρN defines an ensem
PN ∇k S N ble of trajectories and satisfies the continuity equation ∂ρ = k=1 ∇k · ρN m ∂t + 0. One can then introduce the so-called “classical wavefunction”12,13 √ ψCl = ρN eiS/~ , (15.2.3) which can be shown to satisfy the modified Schr¨odinger equation ! N X ~2 2 ∂ψCl i~ = − ∇ + V ψCl − QψCl . ∂t 2m k

(15.2.4)

k=1

Note that, because of the last term, Eq. (15.2.4) is nonlinear. In this way, the quantum potential completely eliminates the quantum effects of the linear Schr¨ odinger equation, giving rise to completely classical behavior. 15.2.2. Momentum and kinetic energy operators The interpretation of Q as an additional potential term in the electron’s equation of motion has far-reaching consequences, as the hydrodynamical formulation of quantum mechanics shows. Notwithstanding, both Eqs. (15.2.2a) and (15.2.4) allow an alternative interpretation of Q as a deformation of the kinetic term in the corresponding equation of motion. In particular, one can readily interpret the additional term in Eq. (15.2.4) as a deformation of the kinetic energy operator in quantum mechanics, −

N N X X ~2 2 ~2 2 ∇k → − ∇ − Q. 2m 2m k k=1

k=1

This different perspective results in the definition of a classical version of the kinetic energy operator as KCl = −

N X ~2 2 ∇ − Q. 2m k

(15.2.5)

k=1

This naturally motivates the search for a classical version of the momentum operator and we now develop this idea. In the spirit of the factorization method,17 we write Eq. (15.2.4) as ! N N X X ~2 2 1 − ∇k − Q ψ = (−i~∇k − gk ) · (−i~∇k + gk )ψ, (15.2.6) 2m 2m k=1

k=1

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where gk is a function to be determined. Expanding this expression yields  √  N X ∇k2 ρN 1 i~∇k · gk + gk2 + ~2 √ ψ = 0. m ρN k=1

This suggests that we choose a purely imaginary gk , of the form gk = iαk (with αk real) which yields  √  N X ∇k2 ρN 1 ~∇k · αk + α2k − ~2 √ ψ = 0. m ρN k=1    √ √
√  Now we note that √1ρN ∇k2 ρN = ∇k · √1ρN ∇k ρN − ∇k √1ρN · ∇k ρN =     2 √ √ ∇k · √ρ1N ∇k ρN + √1ρN ∇k ρN . Substituting in the above equation yields the following condition on αk : "   2 #  N X ~ 1 √ √ ~ 2 ψ = 0. ~∇k · αk + αk − ~∇k · √ ∇k ρN − √ ∇k ρN m ρN ρN k=1

This is immediately fulfilled by the choice √ ∇k ρN ~ ∇k ρN αk = ~ √ = , ρN 2 ρN

with k = 1, . . . , N . Bearing in mind Eq. (15.2.6), the above discussion motivates the definition of a classical version of the momentum operator as ~ ∇ρN , 2 ρN ~ ∇ρN = P−i , 2 ρN

PCl = P + i

(15.2.7a)

† PCl

(15.2.7b)

where P = −i~∇ is the usual N -electron momentum operator, with P = (P1 , . . . , PN ) and ∇ = (∇1 , . . . , ∇N ). It is crucial to note that ρN in Eqs. (15.2.7) is that associated with the wavefunction of the system (so that ρN = ψ ∗ ψ), regard† less of the function on which PCl and PCl operate. Therefore, PCl is a functional of the N -electron density and the action of PCl on the wavefunction of the system is given by   1 ∇(ψ ∗ ψ) ψ, PCl ψ = −i~ ∇ − 2 ψ∗ ψ and it is clear that PCl is nonlinear. The above discussion shows that the classical version of the kinetic energy operator of Eq. (15.2.5) can be expressed as KCl =

N X 1 (P† )k · (PCl )k . 2mk Cl

k=1

(15.2.8)

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In the limiting case of a one-electron system (N = 1), this simplifies to KCl =

1 † P · PCl . 2m Cl

Note that PCl is a classical version of the momentum operator in the sense that it is associated with the kinetic term KCl appearing in the modified Schr¨odinger equation (15.2.4). Since Eq. (15.2.4) is nonlinear, it is necessary that PCl is nonlinear. A straightforward calculation shows that the action of PCl on the wavefunction of the system can be expressed as   ∇ψ PCl ψ = ~ Im ψ. (15.2.9) ψ √ Therefore, with the “classical wavefunction” ψCl = ρN eiS/~ , PCl ψCl = ∇S ψCl

(15.2.10)

and the quantity ∇S can be interpreted as the local momentum associated with an ensemble of trajectories. This reinforces our interpretation of PCl as a classical version of the momentum operator. We now compare the action of P and PCl for some particular one-dimensional i examples. For a plane wave ψ = e ~ p0 x (with p0 constant), we see that the actions of PCl and P are identical, and PCl ψ = Pψ = p0 ψ. On the other hand, it follows from Eqs. (15.2.7) that any ψ with a nontrivial probability distribution will lead to different results under the action of P and PCl . Consider, for example, a Gaus2 2 i sian wave packet ψ(x) = e−(x−x0 ) /(2∆x) e ~ p0 x which corresponds to a state that minimizes the position-momentum uncertainty. Note that, although ψ is as close as possible to a quantum state with definite position x0 and momentum p0 , it is not an eigenfunction of P and therefore does not correspond to a state of definite momentum. On the other hand, we readily see that PCl ψ = p0 ψ so ψ is an eigenfunction of PCl . 15.3. Dequantization uantum mechanics is an extremely successful theory for the description of atomic and molecular systems. However, because of the success of classical mechanics in its domain of validity, there is continued interest in dequantization procedures whereby the classical regime is obtained from the quantum regime. By dequantization we do not mean the procedure of obtaining a semiclassical limit of a quantum system, as in the WKB approximation. Rather, we mean “a set of rules which turn quantum mechanics into classical mechanics”.18 An insightful step towards dequantization is the introduction of operator based formulations of classical mechanics and the earliest such formulation is that of Koopman19 and von Neumann.20 This work was the foundation of more recent

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path integral formulations of classical mechanics21 and the related dequantization procedure of Abrikosov, Gozzi and Mauro.18 On the other hand, there has been interest in quantization procedures formulated in a quasi-classical language, whereby a stochastic term is added to the equations of classical mechanics. In particular, Nelson4 and earlier work of F´enyes22 and Weizel23 showed that the Schr¨odinger equation can be derived from Newtonian mechanics via the assumption that a classical particle is subjected to Brownian motion with a real diffusion coefficient. Also, Hall and Reginatto24 have shown that the Schr¨odinger equation can be derived from the classical equations of motion by adding generic fluctuations which obey an exact Heisenberg-type relation to the classical momentum. In a similar vein, Reginnato25 has shown that the Schr¨ odinger equation can be derived by minimization of the Fisher information.26 We present a dequantization procedure whereby classical mechanics is derived from quantum mechanics by suppressing the effects of such “quantum fluctuations”. To develop this approach within a consistent mathematical framework, we introduce local deformations of the momentum operator, which correspond to generic fluctuations of the quantum momentum. These naturally induce an associated deformed kinetic energy, which quantifies the amount of “fuzzyness” caused by these fluctuations. Considered as a functional of such deformations, the deformed kinetic energy is shown to possess a unique minimum which is seen to be the classical kinetic energy. Moreover, the minimizing deformation automatically determines an expression for the quantum fluctuations (essentially identical to Nelson’s osmotic momentum) which, when added to the classical momentum, results in the quantum momentum. For a classical system described by a probability density there is uncertainty in the position (and momentum). For the corresponding quantum system there is additional uncertainty. Our dequantization method removes the additional part of the uncertainty that is quantum leaving only the uncertainty that is classical and it does this in a “minimalist” way — without introducing any artifacts — through a deformation procedure based on a variational principle. As a result of the dequantization method the quantum fluctuations are suppressed and, in this sense, the momentum-space localization of the system is increased. However, the spatial localization of the system is unchanged as this quantity is determined by ρN which is unaffected by the dequantization process. For a system which is more spatially localized both the purely quantum kinetic energy and the Fisher information are larger and in the limit of extreme spatial localization both become infinite but the classical kinetic energy can remain finite.

15.3.1. Witten deformation approach Some time ago, Witten formulated a connection between supersymmetric quantum mechanics and Morse theory27 which has been very influential. Central to Witten’s

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approach is the deformation of the exterior derivative, d, d → dλ = e−λf deλf ,

λ ∈ R.

(15.3.11)

We note that, by choosing f = − 21 ln ρN , the action of dλ on scalar functions is given by −i~∇(λ) = −i~∇ + i~

λ ∇ρN , 2 ρN

(15.3.12)

where we employ a vector notation for the outcome of d and dλ when applied to a scalar function. This suggests (cf Eqs. (15.2.7)) that we define a deformed momentum operator, Pλ , by ~ ∇ρN , 2 ρN ~ ∇ρN P†λ = P − iλ . 2 ρN Pλ = P + iλ

(15.3.13a) (15.3.13b)

Again, it is crucial to note that ρN in Eqs. (15.3.13) is that associated with the wavefunction of the system, regardless of the function on which Pλ and P†λ operate, and that the same observations made for PCl concerning its nonlinearity also apply to Pλ . For λ = 0, Pλ recovers the usual quantum momentum while for λ = 1, Pλ recovers the classical version of the momentum operator, PCl , of the previous section. As λ increases from 0 to 1, one can envisage a scenario in which the quantum mechanics increasingly assumes classical effects. Also central to Witten’s approach27 is the deformation of the Laplacian, L, L → Lλ = (dλ + δλ )2 .

(15.3.14)

Here Lλ is the natural Laplacian corresponding to dλ and its associated coderivative ~2 1 Lλ scalar = 2m P†λ · δλ = d†λ and, when restricted to scalar functions, Lλ satisfies 2m functions Pλ . This suggests that we define a deformed kinetic energy operator, Kλ , by Kλ =

N X 1 (P† )k · (Pλ )k . 2mk λ

(15.3.15)

k=1

In the limiting case of a one-electron system (N = 1), this simplifies to Kλ =

1 † P · Pλ . 2m λ

For λ = 0, Kλ recovers the usual (quantum) kinetic energy operator, while for λ = 1, Kλ recovers the classical version of the kinetic energy operator, KCl , of the previous section.

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15.3.2. Variational approach We begin by considering a local deformation P → Pw of the (quantum) momentum operator P = −i~∇ for a one-electron system (the generalization to many-electron systems is straightforward), with Pw ψ = (P − w) ψ,

(15.3.16)

where w is a position-dependent (complex) vector field. Since our aim is to dequantize the system, there is no a priori reason to assume that Pw is Hermitian when w 6= 0. Writing w = v + iu, where v and u are respectively the real and imaginary parts of w, we see that the term v in Pw ψ = −(i~∇ + v)ψ − iuψ acts in the same way as an electromagnetic field A, which is known to change the momentum operator −i~∇ to −i~∇ + κA, where κ is a constant. Therefore, in what follows we restrict the deformations in Eq. (15.3.16) to those corresponding to imaginary w so that w = iu (with u real) and Pu ψ = (P − iu) ψ. Let

(15.3.17)

1 2m

Z

(P ψ)∗ (P ψ)dτ

(15.3.18)

1 Tu = 2m

Z

(Pu ψ)∗ (Pu ψ)dτ

(15.3.19)

T = and

be the kinetic energies arising from P and Pu , respectively, where dτ denotes the associated volume element. Integration by parts shows that one can alternatively write Z 1 T = ψ ∗ P 2 ψdτ, (15.3.18a) 2m and

Tu = where

1 2m

Z

ψ ∗ Pu† Pu ψdτ,

(15.3.19a)

Pu† ψ = (P + iu) ψ is the adjoint of Pu . Note that although Pu and Pu† are, in general, not Hermitian operators, Pu† Pu is an Hermitian operator so that Tu (like T ) is necessarily real. We then have Z
1 Tu = T + ρN −~∇ · u + kuk2 dτ, (15.3.20) 2m Note that Tu = Tu [ψ, u] is a functional of both ψ and u. Therefore, the full-fledged variational principle associated with Tu should involve minimization with respect to both ψ- and u-variations.

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A straightforward calculation shows that variation of Tu [ψ, u] with respect to u yields δTu 1 = (2ρN u + ~∇ρN ). δu 2m

(15.3.21)

Therefore, extremization of Tu with respect to u-variations leads to the critical value uc = −

~ ∇ρN . 2 ρN

(15.3.22)

This corresponds to the deformed momentum operator Pc = P +

i~ ∇ρN , 2 ρN

(15.3.23)

which is identical to the classical version of the momentum operator, PCl , of the previous section. In the quantization procedure of Nelson,4 a classical particle is subjected to Brownian motion. In addition to its classical velocity, a Brownian particle has a velocity due to the osmotic force that Nelson terms the osmotic velocity (which is half the difference between the forward and backward drift velocities). From Einstein’s theory, the osmotic velocity is given by ν∇ρ/ρ where ν is the diffusion coefficient. Since macroscopic bodies do not appear to be subjected to Brownian motion, Nelson assumes that ν is inversely proportional to the particle mass, m, and makes the ansatz ν = ~/2m. Then the corresponding osmotic momentum, which is the term added to the classical momentum to give the quantum momentum, is (~/2)∇ρ/ρ. This expression is seen to be identical in magnitude but opposite in sign to our uc of Eq. (15.3.22). This is no coincidence and can be qualitatively understood as follows. In Nelson’s stochastic approach, quantum fluctuations are explicitly added to PCl , thereby resulting in the quantum momentum. In our dequantization approach, the quantum fluctuations latent in P are stripped off, thereby resulting in the classical momentum. In this process, our dequantization approach automatically identifies the expression for uc . Expanding Tu around the critical point yields Z 1 (15.3.24) Tuc +δu = Tuc + ρN kδuk2 dτ, 2m which shows that the deformation uc of Eq. (15.3.22) leads to the unique minimum of Tu , given by T uc = T −

~2 IN , 8m

where IN is the N -electron Fisher information,26 Z 2 (∇ρN ) IN = dτ. ρN

(15.3.25)

(15.3.26)

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Thus we have shown that there is a unique solution to the deformation parameter u which minimizes the deformed kinetic energy Tu under u-variations. A straightforward calculation shows that the action of Pc on the wavefunction √ ψ = ρN eiS/~ is given by Pc ψ = ∇S ψ,

(15.3.27)

so that, from Eq. (15.3.19), T uc =

1 2m

Z

ρN k∇Sk2 dτ.

(15.3.28)

This is the mean kinetic energy of an ensemble of trajectories defined by ρN and momentum field ∇S.12,16 We therefore refer to Tuc as the classical kinetic energy, TCl . 15.4. Information theory and kinetic-energy functionals Density functional theory was placed on a solid foundation by the work of Hohenberg and Kohn28 who proved that the total energy can be obtained as a functional of ρone . Their proof also applies to the kinetic energy but they could provide no prescription for constructing the exact kinetic-energy functional. Kohn and Sham29 subsequently provided a prescription for calculating the noninteracting kinetic energy by adapting aspects of Hartree-Fock theory. In Hartree-Fock theory the wavefunction is approximated as the product of N one-electron orbitals (antisymmetrized to ensure that electron exchange is incorporated exactly for the approximate wavefunction). In constructing these orbitals the effect of the other electrons is included only in an average way (through the use of an effective potential) and electron correlation is neglected. Calculations scale as N 3 and post Hartree-Fock approaches which incorporate electron correlation (usually required for chemical accuracy) typically scale as N 5 or N 7 . Kohn and Sham employed the orbital approximation but chose the effective potential such that for the one-electron orbitals, φi , the resulting electron density is equal to ρoneR. From these orbitals they obtained the noninteracting PN ~2 2 3 kinetic energy as Ts = 2m i=1 |∇φi | d r rather than as a direct functional of ρone . As in Hartree-Fock theory, electron exchange is incorporated exactly (for the approximate wavefunction) and electron correlation is neglected. Complete calculations employ an exchange-correlation functional for the difference between Ts and the exact kinetic energy (and also the difference between the classical electrostatic energy and the exact potential energy). As for Hartree-Fock theory, the KohnSham expression is order N 3 but, as high-quality exchange-correlation functionals have been developed, chemical accuracy can be realized and it is in this form that density functional theory has been most successful for the calculation of atomic and molecular properties.

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15.4.1. Information theory The Fisher information,26,30 which is a cornerstone of information theory is given by Z |∇p(r)|2 3 I= d r, (15.4.29) p(r) R where p(r) = p(r1 ) = |ψ(r1 , . . . , rN )|2 d3 r2 . . . d3 rN is the one-electron (probability) density. The electron density is related to the one-electron (probability) density by ρone (r) = N p(r). Thus I is a functional of the electron density and the greater the localization of ρone (r) the greater the value of the Fisher information. The Shannon information, 31 which is another cornerstone of information theory, is given by Z IS = − ρone (r) ln(ρone (r))d3 r. (15.4.30) Thus IS is a functional of the electron density and the greater the delocalization of ρone (r) the greater the value of the Shannon information. Thus the Fisher information and the Shannon information are complementary quantities and they have been used in conjunction to analyze electron correlation and other properties.32,33 15.4.2. Orbital-free kinetic-energy functionals We begin by considering some previously proposed kinetic-energy functionals whereby the kinetic energy is obtained from the electron density, ρone . A wellknown functional for the kinetic energy is the Thomas-Fermi term,34,35 Z 3~2 (3π 2 )2/3 ρone (r)5/3 d3 r. (15.4.31) TT F = 10m

This expression is exact for the uniform electron gas (an N = ∞ system) for which the reduced gradient (|∇ρone |/2kf ρone with kf = (3π 2 ρone )1/3 ) is zero. Another well-known functional for the kinetic energy is the Weizs¨acker term,36 Z ~2 |∇ρone (r)|2 3 TW = d r. (15.4.32) 8m ρone (r) This expression is exact for the ground state of the hydrogen atom (a one-electron system) but not for the ground states of multi-electron atoms. Comparison of Eq. (15.4.29) and Eq. (15.4.32) shows that the information content of the Fisher information and the Weizs¨ acker term is the same and these quantities are essentially ~2 identical (with TW = N8m I). We generally employ the Weizs¨acker term as the connection to the kinetic energy is more direct. For atomic and molecular systems it might be hoped that an accurate kineticenergy functional could be obtained via some combination of TT F and TW and, in fact, Weizs¨ acker originally proposed TT F + TW . Other researchers subsequently

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proposed either a smaller coefficient for TT F 37–41 or, more commonly, TW ( secondorder gradient expansion of the electron density for a nonuniform electron gas leads to the coefficient 19 ).42–44 Other expressions for the kinetic energy have been developed and, of particular relevance to our approach, Herring45 proposes Tθ + TW where he terms Tθ the relative-phase energy. In our expression for the kinetic energy the relative-phase energy is replaced by the classical kinetic energy. For large Z atoms, the electron density is slowly varying for the bulk of the electrons in the intermediate r region, a second-order gradient expansion is valid, and the expression TT F + 91 TW (with the Dirac exchange functional46 ) is accurate. However, this expression is not accurate for the large r region which is responsible for chemical bonding and the Thomas-Fermi term cannot describe molecular systems. Despite the success of the Kohn-Sham approach, there has been continued interest in developing orbital-free kinetic-energy functionals which obtain the noninteracting kinetic energy, Ts , as a direct functional of ρone . The practical motivation is that these expressions could be order N and much larger systems would therefore be tractable but chemical accuracy has not yet been realized. A recent study47 carefully analyzed kinetic-energy functionals of the TT F + λTW form while other recent studies48,49 considered the accuracy of various kinetic-energy functionals which combine TT F , TW and higher-order gradient expansion terms in more complicated ways. The development of orbital-free kinetic-energy functionals continues to be an active area of research50–55 as shown by the present volume. 15.4.3. Kinetic energy decomposition The N -electron kinetic energy can be expressed, from Eq. (15.3.25), as56 ~2 IN . (15.4.33) 8m This is the sum of the N -electron classical kinetic energy and an N -electron purely quantum kinetic energy which is essentially given by the N -electron Fisher information, IN , although (as our approach is restricted to scalar particles) effects due to electron spin are not explicitly included and our expressions are valid only for single-spin systems. We first consider the N -electron classical kinetic energy of Eq. (15.4.33). It immediately follows from Eq. (15.3.28) that TCl,N =0 if and only if the N -electron phase is constant. Since a constant N -electron phase can always be redefined to be zero, this is the case if and only if the wavefunction is real. We now consider the purely quantum kinetic energy of Eq. (15.4.33) and decompose the N -electron density as57 TN = TCl,N +

pN (r1 , . . . , rN ) = p(r1 )f (r2 , . . . , rN |r1 ), where p(r1 ) =

Z

pN (r1 , . . . , rN )d3 r2 · · · d3 rN

(15.4.34)

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and f (r2 , . . . , rN |r1 ) =

pN (r1 , . . . , rN ) . p(r1 )

Here f (r2 , . . . , rN |r1 ) is the N -1 conditional electron density. That is, the electron density associated with a set of values for r2 , . . . , rN given a fixed value for r1 . Here p and f satisfy the normalization conditions Z Z p(r1 )d3 r1 = 1 and f (r2 , . . . , rN |r1 )d3 r2 · · · d3 rN = 1 ∀ r1 .

(15.4.35) This immediately yields an expression for the quantum fluctuations (cf Eq. (15.3.22)) as ! N ~ ∇r1 ρone (r1 ) X ∇ri f (r2 , . . . , rN |r1 ) −uc = + , (15.4.36) 2 ρone (r1 ) f (r2 , . . . , rN |r1 ) i=2 where the relation ρone (r) = N p(r) was used. In this way it is possible to distinguish ∇ ρ (r1 ) a local part of the quantum fluctuations, ~2 rρ1oneone (r1 ) , corresponding to fluctuations of the electron density (in the arbitrary but fixed variable r1 ) and a nonlocal part P ∇ri f (r2 ,...,rN |r1 ) of the quantum fluctuations, ~2 N i=2 f (r2 ,...,rN |r1 ) , corresponding to fluctuations of the N -1 conditional electron density. The N -electron Fisher information (cf Eq. (15.3.26)) can be written as Z 2 [∇r1 pN (r1 , . . . , rN )] 3 IN = N d r1 · · · d3 rN . pN (r1 , . . . , rN )

The decomposition for pN in Eq. (15.4.34) can then be used to express this quantity as Z [∇r1 p(r1 )f (r2 , . . . , rN |r1 ) + p(r1 )∇r1 f (r2 , . . . , rN |r1 )]2 3 IN = N d r1 · · · d3 rN p(r1 )f (r2 , . . . , rN |r1 ) Z Z 2 2 [∇r1 p(r1 )] 3 [∇r1 f (r2 , . . . , rN |r1 )] 3 =N d r1 + N p(r1 ) d r1 · · · d3 rN , p(r1 ) f (r2 , . . . , rN |r1 ) (15.4.37) where Eq. (15.4.35) was used to simplify the first term and cancel the mixed term. We then have Z |∇ρone (r)|2 3 IN = d r + I corr (15.4.38) ρone (r) where

I corr = with f Ione (r)

. f = Ione (r1 ) =

Z

Z

f ρone (r)Ione (r)dr

|∇r1 f (r2 , . . . , rN |r1 )|2 dr2 . . . drN . f (r2 , . . . , rN |r1 )

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Thus Eq. (15.4.38) decomposes the N -electron Fisher information as a sum of two terms. The first is local, and is N times I (cf Eq. (15.4.29)), and the second is f nonlocal and comprises many-electron effects through Ione . From Eqs. (15.4.32), (15.4.33) and (15.4.38), we obtain the N -electron kinetic energy as corr TN = TCl,N + TW + TW

(15.4.39)

2

~ corr where TW = 8m I corr . Each term of Eq. (15.4.39) adds an independent nonnegative contribution to the kinetic energy and this equation agrees with the decomposition of Sears et al.57 when the N -electron phase is constant (since, as discussed above, TCl,N is zero in this case). Thus we see that the classical term in Eq. (15.4.39) improves the lower bound for the general case in which the N -electron phase is not constant. In Eq. (15.4.39) TW contributes to the noninteracting purely quantum kinetic corr energy and TW contributes to the effects of electron correlation on the purely quantum kinetic energy. We now assume that the N -electron classical kinetic energy, TCl,N , can be decomposed as the sum of a term, TCl , which contributes to the corr noninteracting classical kinetic energy, and a term, TCl , which contributes to the effects of electron correlation on the classical kinetic energy. Terms that contribute to the noninteracting kinetic energy can be estimated by employing the orbital ap√ proximation. If the one-electron orbital is written as φi = p eiSi /~ where Si (r) is R PN 1 the electron phase then TCl = 2m p(r) i=1 |∇Si (r)|2 d3 r but we have no explicit corr expression for TCl . From Eq. (15.4.39), we then obtain the (one-electron) kinetic energy as corr corr T = TCl + TCl + TW + TW .

(15.4.40)

corr In Eq. (15.4.39) the purely quantum terms, TW and TW , comprise the N electron Weizs¨ acker term and, as discussed above, arise in our approach from the quantum fluctuations that turn the classical momentum into the quantum momentum, as in Nelson’s stochastic approach to quantum mechanics.4 Many decompositions of the N -electron Weizs¨ acker term are possible58,59 and, as noted above, a decomposition similar to ours has previously been proposed.57 The novelty of our decomposition is that, from the calculation leading to Eq. (15.4.37), we can unequivocally identify TW as resulting from the local part of the quantum fluctuations, and corr TW as resulting from the nonlocal part of the quantum fluctuations. As noted above, TW has been universally utilized to construct kinetic-energy functionals. corr However, the connection between TW and the nonlocal quantum fluctuations provides a new rationale for the need to incorporate this term in exchange-correlation functionals in order to capture the complete range of many-electron effects. We corr have shown64 that, as discussed in detail in Chapter 13 of the present volume,TW can be evaluated using a rigorous quantum Monte Carlo procedure to obtain a functional of ρone which is well-approximated by a Shannon information expression.

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15.4.4. Noninteracting kinetic energy Omitting the kinetic correlation terms in Eq. (15.4.39), we obtain the noninteracting kinetic energy as Ts = TCl + TW .

(15.4.41)

There are two limiting cases for which this expression can be obtained analytically. For the ground state of the hydrogen atom (an N = 1 system), the electron phase is zero, so TCl = 0. Therefore, Ts = TW which is the correct result for this limit. For the uniform electron gas (an N = ∞ system) ρone is uniform so TW = 0. Therefore Ts = TCl which can be calculated by adding up the kinetic energies of one-electron orbitals approximated as local plane waves. This results in the Thomas-Fermi term34,35,44 which is the correct result for this limit. In the orbital approximation expression for the noninteracting R PN the standard ~2 2 3 kinetic energy is Ts = 2m |∇φ | d r. Using this expression, Herring45 dei i=1 fines angular variables representing points on the surface of an N -dimensional unit 1/2 sphere as (in our notation) ui (r) = φi /ρone . He then expresses the noninteracting kinetic energy as Ts = Tθ + TW where he terms Tθ , which is dependent on the ui , the relative-phase energy. Comparison of our Eq. (15.4.41) and Herring’s Eq. (28) shows that (in the orbital approximation) TCl and Tθ are equivalent. Herring interprets the relative-phase energy as the additional kinetic energy resulting from the exclusion principle which requires the N -electron phase to vary with position (when there is more than one electron with the same spin). His results for a variety of one-dimensional potentials show that Tθ is usually a significant fraction of the kinetic energy and that Tθ generally becomes larger relative to TW as Z increases.45 The contribution of the electron phase to the kinetic energy, which is implicit in hydrodynamic formulations of quantum mechanics,65 has been noted in other contexts.45,66,67 For hydrogenic orbitals there is an explicit relationship between the electron phase and (as shown below) the angular momentum and for hydrogenic orbitals with nonzero angular momentum, TCl is a significant fraction of the kinetic energy. If hydrogenic orbitals are used as basis functions for the ground states of multi-electron atoms then, as Z increases, the exclusion principle will force electrons into orbitals with higher angular momentum and the number of electrons with a given angular momentum will increase in a stepwise fashion. We note that this behavior has been demonstrated for the Thomas-Fermi electron density68,69 and there have been several approaches which include angular momentum effects in ThomasFermi theory.70,71 In the work of Englert and Schwinger,72,73 angular momentum effects are included for the express purpose of correcting the Thomas-Fermi electron density for large r. 15.4.5. Results for one-electron systems For a one-electron system the noninteracting kinetic energy is simply the kinetic energy and Eq. (15.4.41) becomes T = TCl + TW . We note that the integrands of

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TCl and TW (TCl and TW ) are never negative and correspondingly, TCl and TW are never negative. Thus both TCl and TW are lower bounds to the kinetic energy. In the next two subsections we explicitly show that our expression for the kinetic energy is correct for both stationary and nonstationary states. Furthermore we show that T = TCl + TW . That is, the integrand of T is equal to the sum of the integrands of TCl and TW . This is the case for all values of the position at each value of the time.

15.4.5.1. Stationary systems To gain insight into the nature of TCl , we now examine our decomposition of the kinetic energy for basis functions that are the product of radial functions and spherical harmonics. These basis functions are typically used to represent one-electron orbitals for the ground states of multi-electron atoms. For practical reasons it is more common to employ Slater orbitals but, for simplicity, we present results for hydrogenic orbitals. We explicitly show that, for these basis functions, our expression for the kinetic energy is correct and, furthermore, that it is correct for the radial distributions of the integrands of T , TCl and TW . That is that, for all values of r, T is equal to the sum of TCl and TW . The hydrogenic orbitals, ψ(n, l, m), are dependent on the principal quantum number n, the angular momentum quantum number l and the magnetic quantum number m but the energy is dependent only on n and (in atomic units) the energy is E = -1/2n2. Then, from the virial theorem, the kinetic energy is T = -E = 1/2n2 . The radial distribution for TCl is dependent on n, l and |m| but TCl is dependent 3 only on n and |m| and the classical kinetic energy is TCl = |m| n T = |m|/2n . Corn−|m| respondingly, the purely quantum kinetic energy is TW = n T = (n − |m|)/2n3 and both the classical kinetic energy and the purely quantum kinetic energy are constant for n and |m| fixed. We note that whereas TCl can equal zero, TW cannot since (for a normalizable state) ρone cannot be uniformly constant and therefore ∇ρone cannot be identically zero. The fact that the purely quantum kinetic energy cannot equal zero is in accord with the position-momentum uncertainty principle. We previously56,74 presented results for small n values, n = 2 and n = 3. Here we present results for a relatively large n value, n = 11. Figure 15.4.1 shows radial distributions (integrated over the angular variables) of the integrands of TCl (dashed curve), TW (dotted curve) and T (solid curve) for hydrogenic orbitals with n = 11, l = 10 and (a) |m| = 10, (b) |m| = 6 and (c) |m| = 1. It may be seen that, for these cases in which the l value is large, the integrand of T has a broad maximum at a large value of r and it is smoothly varying without oscillations. When |m| is large, Fig. 15.4.1(a), the integrand of TW is close to zero and the integrand of T is well-approximated by that of TCl . On the other hand, when |m| is small, Fig. 15.4.1(c), the integrand of TCl is close to zero and the

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integrand of T is well-approximated by that of TW . Thus as |m| decreases, the kinetic energy goes from almost (but not entirely) classical (for |m| = l) to entirely purely quantum (for |m| = 0). Hn,l,mL=H11,10,10L 0.00006

Hn,l,mL=H11,10,6L 0.00006

HaL

Hn,l,mL=H11,10,1L 0.00006

HbL

0.00005

0.00005

0.00005

0.00004

0.00004

0.00004

0.00003

0.00003

0.00003

0.00002

0.00002

0.00002

0.00001

0.00001

0

0.00001

0 0

50

100

150

200

250

300

HcL

0 0

50

100

150

200

250

300

0

50

100

150

200

250

300

Fig. 15.4.1. Radial distributions (integrated over the angular variables) of TCl (dashed curve), TW (dotted curve) and T (solid curve) for hydrogenic orbitals with n = 11, l = 10 and (a) |m| = 10; (b) |m| = 6; (c) |m| = 1. The horizontal axis is the radial variable, r, in atomic units.

Figure 15.4.2 shows these same radial distributions with n = 11 and (a) l = 6, |m| = 6 and (b) l = 1, |m| = 1. For the case in which the l value is intermediate, Fig. 15.4.2(a), the integrand of T has a global maximum at an intermediate value of r and it is fairly smoothly varying. Around this maximum, the integrand of T is well-approximated by the integrand of TCl . At larger values of r the integrands of both TCl and TW are highly oscillatory but are largely out of phase, with the maxima of one close to the minima of the other. For the case in which the l value is small, Fig. 15.4.2(b), the integrand of T has a sharp global maximum at a small value of r and it is highly oscillatory. Around this maximum, the integrand of T is well-approximated by the integrand of TCl . At larger values of r the integrand of TCl is close to zero and the integrand of T is well-approximated by that of TW which is highly oscillatory. Hn,l,mL=H11,6,6L

Hn,l,mL=H11,1,1L

0.00008

HaL

HbL

0.00015

0.00006 0.00010 0.00004

0.00005 0.00002

0

0.00000 0

50

100

150

200

250

300

0

50

100

150

200

250

300

Fig. 15.4.2. Radial distributions (integrated over the angular variables) of TCl (dashed curve), TW (dotted curve) and T (solid curve) for hydrogenic orbitals with n = 11 and (a) l = 6, |m| = 6; (b) l = 1, |m| = 1. The horizontal axis is in atomic units.

The results for these stationary states support our expression for the kinetic energy. Furthermore, it is clear from Figs. 15.4.1 and 15.4.2 that the integrand of

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T is equal to the sum of the integrands of TCl and TW for all values of the position. For the ground states of multi-electron atoms we expect that TCl will be greater than zero but smaller than TT F (when the reduced gradient is small TT F has been shown45,67 to be an upper bound to TCl ) and, across a row of the periodic table, TCl generally increases as Z increases. For example, the one-electron orbital for the ground state of the C atom will have a larger l = 1 contribution than will that for the ground state of the Be atom. Correspondingly, Ts for the C atom will have a larger TCl component than will that for the Be atom. However, we have no algorithm for optimizing the TCl component of the one-electron orbital and, since TCl is dependent on the m value for each basis function, this algorithm is expected to be (at best) order N 3 and would therefore have no practical advantage over the standard Kohn-Sham algorithm.

15.4.5.2. Nonstationary systems We previously examined our decomposition of the kinetic energy for a free particle and a particle in a box74 and here we present results for a particle in a quadratic potential. For the one-dimensional harmonic oscillator V = 12 kx2 and, for convenience, 2 we choose k = 1. Then the ground state eigenfunction is φ0 (x) = π −1/4 e−x /2 and (in atomic units) the energy is E0 = 12 while, from the virial theorem, the kinetic energy is T0 = 14 . For the first excited state the eigenfunction is φ1 (x) = 2 π −1/4 21/2 xe−x /2 and the energy is E1 = 23 while the kinetic energy is T1 = 34 . For the ground state and all excited states the classical kinetic energy is zero and the kinetic energy is equal to the Weizs¨acker term. For nonstationary states, φ(x), whereas TCl = 0 at t = 0, this is generally not the case at later times. 2 For a nonstationary state that is initially φ(x) = π −1/4 e−αx /2 with α 6= 1, the classical kinetic energy remains equal to its initial (zero) value at all times while the purely quantum kinetic energy exhibits harmonic oscillations. For a nonstationary 2 state that is initially φ(x) = π −1/4 e−(x−β) /2 with β 6= 0, the classical kinetic energy exhibits harmonic oscillations while the purely quantum kinetic energy remains equal to its initial (nonzero) value at all times. Clearly, for a nonstationary state 2 that is initially φ(x) = π −1/4 e−α(x−β) /2 with α 6= 1 and β 6= 0, the classical kinetic energy and the purely quantum kinetic energy both exhibit harmonic oscillations. We now consider a nonstationary state that is initially φ(x) = 2−1/2 (φ0 (x) + φ1 (x)). Figure 15.4.3 shows (upper panel) the probability distribution and (lower panel) the integrands for TCl (dashed curve), TW (dotted curve) and T (solid curve) at (a) t = 0, (b) t = 1 and (c) t = 2. It may be seen that, whereas the integrand of TCl is identically zero at t = 0, this is not the case at the later times. Note that at t = 1 there is a peak on the right side of the probability distribution where it is rapidly changing and a shoulder on the left side where the probability distribution is relatively constant. In the vicinity of the peak the integrand of TCl is small and the integrand of T is well-approximated by that of TW , On the other hand, in the

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vicinity of the shoulder the integrand of TW is small and the integrand of T is well-approximated by that of TCl . At t = 2 there is a peak on the left side of the probability distribution where it is rapidly changing and a shoulder on the right side where the probability distribution is relatively constant. Again, the integrand of TW is large (and that of TCl is small) where the probability distribution is rapidly changing while the integrand of TCl is large (and that of TW is small) where the the probability distribution is relatively constant. These results for the harmonic oscillator, which are similar to those for the particle in a box,74 support our expression for the kinetic energy. Furthermore, it is clear from Fig. 15.4.3 that the integrand of T is equal to the sum of the integrands of TCl and TW for all values of the position at each value of the time. These results also show that the classical kinetic energy is complementary to the purely quantum kinetic energy which is directly proportional to the Fisher information. In this sense, the classical kinetic energy plays a role analogous to that of the Shannon information. As noted previously, the Fisher information and the Shannon information are complementary quantities and have been used in conjunction to analyze electron correlation and other properties.32,33 t=0

t=1

t=2

Fig. 15.4.3. One-dimensional harmonic oscillator state that is initially φ(x) = 2−1/2 (φ0 (x) + φ1 (x)) where φ0 (x) and φ1 (x) are the first two harmonic oscillator eigenfunctions. Distributions shown at t = 0; t = 1; t = 2: Probability distributions (upper panel); Distributions of TCl (dashed curve), TW (dotted curve) and T (solid curve). (lower panel)

15.5. Conclusions Quantization procedures have been proposed in which the quantum regime is obtained from the classical regime by adding a stochastic term to the classical equations of motion. In Nelson’s quantization approach an osmotic momentum term is added to the classical momentum resulting in the quantum momentum. This term represents the quantum fluctuations that are an essential part of quantum mechanics in accord with the position-momentum uncertainty principle. We have proposed a dequantization procedure in which the osmotic momentum term is removed from the quantum momentum resulting in the classical momentum.

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We obtained this term via a Witten deformation approach and variational approach in which the deformed kinetic energy is minimized with respect to variations of a deformation parameter. We showed that the critical value of this parameter which minimizes the deformed kinetic energy is directly related to the osmotic momentum term. From our dequantization procedure we obtained a decomposition of the kinetic energy which is applicable to all kinetic-energy functionals. In this expression the kinetic energy of an N -electron system is written as the sum of the N -electron classical kinetic energy and the N -electron purely quantum kinetic energy arising from the quantum fluctuations that turn the classical momentum into the quantum momentum. We obtained an expression for the N -electron purely quantum kinetic energy as the sum of two terms. The first term results from local quantum fluctuations and is a functional of ρone . It is the Weizs¨acker term which is directly proportional to the Fisher information. The second term results from the nonlocal quantum fluctuations and is a functional of the N -1 conditional electron density. It can be evaluated using a rigorous quantum Monte Carlo procedure to obtain a functional of ρone which is well-approximated by a Shannon information expression. We have thereby established a stronger connection between information theory and kinetic-energy functionals. Omitting the kinetic correlation terms, we obtained an expression for the noninteracting kinetic energy as the sum of the classical kinetic energy and the Weizs¨acker term. The Weizs¨ acker term is well-known and the classical kinetic energy is related to the Thomas-Fermi term which is also well-known. However, we believe that our derivation, which obtains both these terms within a single theoretical framework, is novel. Our expression for the noninteracting kinetic energy is exact and it correctly reduces to the Thomas-Fermi term for the uniform electron gas and to the Weizs¨ acker term for the hydrogen atom. However, the classical kinetic energy (unlike the Thomas-Fermi term) is explicitly dependent on the electron phase. Consequently, Ts = TCl + TW is expected to be at best order N 3 and would therefore have no practical advantage over the standard Kohn-Sham expression. To gain insight into the nature of TCl , we examined our decomposition of the noninteracting kinetic energy for eigenfunctions that are the product of radial functions and spherical harmonics and established a direct connection between the classical kinetic energy and the angular momentum. We believe that this intrinsic connection between the angular momentum and a component of the noninteracting kinetic energy is of significant conceptual value in showing the information that should be incorporated in any kinetic-energy functional. For small and intermediate Z atoms, the basic problem with the expression Ts = TT F + λTW (or λTT F + TW ) is that TW incorporates exactly a part of the noninteracting kinetic energy that is also incorporated approximately in TT F .38 This component of TT F should be removed and that is why simply optimizing λ offers only limited improvement.47 The expression Ts = TCl +TW is a significant improve-

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ment in this regard as TCl and TW are orthogonal. However, as the classical kinetic energy is explicitly dependent on the electron phase, our expression is manifestly not orbital-free. As all explicit information regarding the electron phase is lost in constructing the electron density it seems clear that any direct functional of ρone which embodies this information must be highly nonlocal.45,75–78 Reconstructing this information from the electron density represents a significant challenge for the development of accurate orbital-free kinetic-energy functionals. For large Z atoms, the electron density is slowly varying for the bulk of the electrons in the intermediate r region and a second-order gradient expansion is accurate. However, this expression is not accurate for large r. Unfortunately, the large r region is (by virtue of the valence electrons) responsible for chemical bonding and Thomas-Fermi theory cannot describe molecular systems. Our expression for the noninteracting kinetic energy is equally valid for intermediate and large r but it is much more difficult to evaluate. For large Z atoms (where the order N 3 aspect is of greatest concern) it might, in principle, be possible to develop a hybrid approach in which TT F + 91 TW is employed for the bulk of the electrons in the intermediate r region and corrected for large r by evaluating TCl + TW for the valence electrons only. References 1. R.G. Parr and W. Yang, Density Functional Theory of Atoms and Molecules, Clarendon Press (1989). 2. R. M. Dreizler and E. K. U. Gross, Density Functional Theory: An Approach to the Quantum Many Body Problem, (Springer-Verlag, Berlin, 1990). 3. W. Koch and M. C. Holthausen, A Chemist’s Guide to Density Functional Theory, (Wiley-VCH, Weinheim, 2000). 4. E. Nelson, Phys. Rev. 150, 1079 (1966); E. Nelson, Dynamical Theories of Brownian Motion (Princeton Univ. Press, Princeton, 1967). 5. R. A. Mosna, I. P. Hamilton and L. Delle Site, J. Phys. A 38, 3869 (2005). 6. R. A. Mosna, I. P. Hamilton and L. Delle Site, J. Phys. A 39, L229 (2006). 7. A.O. Bolivar Quantum-Classical Correspondence: Dynamical Quantization and the Classical Limit, Springer-Verlag (2004). 8. E. Schr¨ odinger, Annalen der Phys. Leipzig 79, 361 (1926). 9. J.H. VanVleck, Proc. Natl. Acad. Sci. USA 14, 178 (1928). 10. E. Madelung, Zeit. F. Phys. 40, 322 (1927). 11. D. Bohm and B.J. Hiley, The Undivided Universe, Routledge & Chapman & Hall (1993). 12. P.R. Holland, The Quantum Theory of Motion, Cambridge University Press (1993). 13. P. Ghose, Found. Phys. 32, 871, (2002); P. Ghose and M.K. Samal, Found. Phys. 32, 893, (2002). 14. A.O. Bolivar, Can. J. Phys. 81, 971 (2003). 15. S.K. Gosh and B.M. Deb, Phys. Rep. 92, 1 (1982). 16. H. Goldstein, Classical Mechanics, 2nd edition, Addison-Wesley (1980). 17. L. Infeld and T.E. Hull, Rev. Mod. Phys. 23, 21 (1951); H. C. Rosu, in Proc. “Symmetries in Quantum Mechanics and Quantum Optics”, Burgos, Spain, 1999, pp. 301-315;

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18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34. 35. 36. 37. 38. 39. 40. 41. 42. 43. 44. 45. 46. 47. 48. 49. 50. 51. 52. 53. 54. 55. 56. 57. 58. 59. 60. 61. 62. 63. 64.

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B. Mielnik and O. Rosas-Ortiz J. Phys. A: Math. Gen. 37 10007 (2004). A. A. Abrikosov Jr, E. Gozzi and D. Mauro, Ann. Phys. 317, 24 (2005). B. O. Koopman, Proc. Nat. Acad. Sci. USA 17, 315 (1931). J. von Neumann, Ann. Math. 33, 587 (1932). E. Gozzi, M. Reuter and W.D. Thacker, Phys. Rev. D 40, 3363 (1989); 46, 757 (1992). I. F´enyes, Z. Physik 132, 81 (1952). W. Weizel, Z. Physik 134, 264 (1953); 135, 270 1953; 136, 582 (1954). M. J. W. Hall and M. Reginatto, J. Phys. A 35, 3289 (2002). M. Reginatto, Phys. Rev. A 58, 1775 (1998). R. A. Fisher, Proc. Cambridge Philos. Soc. 22, 700 (1925). E. Witten, J. Diff. Geo. 17, 661 (1982). P. Hohenberg and W. Kohn, Phys. Rev. 136, B864 (1964). W. Kohn and L. J. Sham, Phys. Rev. 140, A1133 (1965). A. Nagy, J. Chem. Phys. 119, 9401 (2003). C. E. Shannon, Bell Syst. Tech. J. 27, 379 (1948); 27 623 (1948). E. Romera and J. S. Dehesa, J. Chem. Phys. 120, 8906 (2004). K. D. Sen, J. Antol´ın and J. C. Angulo, Phys. Rev. A 76, 032502 (2007). L. H. Thomas, Proc. Camb. Phil. Soc. 23, 542 (1927). E. Fermi, Rend. Accad. Lincei 6, 602 (1927). C. F. v Weizs¨ acker, Z. Phys. 96, 431 (1935). N. H. March and W. H. Young, Proc. Phys. Soc. 72, 182 (1958). P. K. Acharya, L. J. Bartolotti, S. B. Sears, and R. G. Parr, Proc. Nat. Acad. Sci. USA 77, 6978 (1980). J. L. G´ azquez and E. V. Lude˜ na, Chem. Phys. Lett. 83, 145 (1981). J. L. G´ azquez and J. Robles, J. Chem. Phys. 76, 1467 (1982). P. K. Acharya, J. Chem. Phys. 78, 2101 (1983). A. S. Kompaneets and E. S. Pavlovski, Sov. Phys.-JETP 4, 328 (1957). P. A. Kirzhnits, Sov. Phys.-JETP 5, 64 (1957). W. Yang, Phys. Rev. A 34, 4575 (1986). C. Herring, Phys. Rev. A 34, 2614 (1986). P. A. M. Dirac, Proc. Cambridge Philos. Soc. 26, 376 (1930). G. K. Chan, A. J. Cohen and N. C. Handy, J. Chem. Phys. 114, 631 (2001). S. S. Iyengar, M. Ernzerhof, S. N. Maximoff and G. E. Scuseria, Phys. Rev. A 63, 052508 (2001). F. Tran and T. A. Wesolowski, Chem. Phys. Lett. 360, 209 (2002). E. Sim, J. Larkin, K. Burke, C. W. Bock, J. Chem. Phys. 118, 8140 (2003). H. Jiang and W. T. Yang, J. Chem. Phys. 121, 2030 (2004). J. D. Chai and J. A. Weeks, J. Phys. Chem. B 108, 6870 (2004). X. Blanc X and E. Cances, J. Chem. Phys. 122, 214106 (2005). I. V. Ovchinnikov and D. Neuhauser, J. Chem. Phys. 124, 024105 (2006). B. Zhou and Y. A. Wang, J. Chem. Phys. 124, 081107 (2006). I. P. Hamilton, R. A. Mosna and L. Delle Site, Theor. Chem. Acct. 118, 407 (2007). S. B. Sears, R. G. Parr and U. Dinur, Isr. J. Chem. 19, 165 (1980). M. S. Miao, J. Phys. A 34, 8171 (2001). P. W. Ayers, J. Math. Phys. 46, 062107 (2005). S. R. Gadre, Adv. Quantum Chem. 22, 1 (1991). E. Romera and J. S. Dehesa, Phys. Rev. A 50, 256 (1994). E. Romera and J. S. Dehesa, J. Chem. Phys. 120, 8906 (2004). R. P. Sagar and N. L. Guevara, J. Chem. Phys. 123, 044108 (2005). L. M. Ghiringhelli, L. Delle Site, R. A. Mosna and I. P. Hamilton, J. Math. Chem.

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65. 66. 67. 68. 69. 70. 71. 72. 73. 74. 75. 76. 77. 78.

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PART 4

Appendix

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Chapter 16 Semilocal Approximations for the Kinetic Energy

Fabien Tran1 and Tomasz A. Wesolowski2 1

Institute of Materials Chemistry, Vienna University of Technology Getreidemarkt 9/165-TC, A-1060 Vienna, Austria [email protected] 2

Department of Physical Chemistry, University of Geneva 30, quai Ernest-Ansermet, CH-1211 Geneva 4, Switzerland [email protected] Approximations to the non-interacting kinetic energy Ts [ρ], which take the form of semilocal analytic expressions are collected. They are grouped according to the quantities on which they explicitly depend. Additionally, the approximations for quantities related to Ts [ρ] (kinetic potential and non-additive kinetic energy), for which the analytic expressions for the “parent” approximation for the functional Ts [ρ] are unknown, are also given.

Contents 16.1 Notation and conventions . . . . . . . . . . . 16.2 Known exact functionals . . . . . . . . . . . . 16.3 Local density approximation — LDA . . . . . 16.4 Gradient expansion approximation — GEAn 16.5 Generalized gradient approximation — GGA 16.6 Other semilocal approximations . . . . . . . . 16.7 N - and r-dependent approximations . . . . . 16.8 Miscellaneous . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . .

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429 430 430 431 432 437 439 440 441

16.1. Notation and conventions Atomic units [me = e = ~ = 1/(4πε0 ) = 1] are used in all formulas. The notation for functions and constants that will be used throughout the text is the following: • ρ = ρ(r) – the electron density.
1/3 4/3  • s = |∇ρ| / 2 3π 2 ρ – the dimensionless reduced density gradient. • x = (5/27) s2 – a quantity proportional to the square of s.
2/3 • CF = (3/10) 3π 2 ' 2.8712 – the Thomas-Fermi constant. 429

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1/3 • b = 2 6π 2 – a conversion factor. • Capital T – integrated kinetic energy. • Small t – kinetic-energy density per volume unit. The analytic expressions of the non-interacting kinetic-energy functionals Ts [ρ] will be given for the spin-compensated case (ρ↑ = ρ↓ ). For spin-polarized electron densities, the corresponding expression for Ts [ρ↑ , ρ↓ ] can be easily obtained by applying the extension formula of Oliver and Perdew:1 Ts [ρ↑ , ρ↓ ] =

1 (Ts [2ρ↑ ] + Ts [2ρ↓ ]) 2

(16.1.1)

The labels used for the approximations reflect the name given by the authors, the most common convention used in the literature, or the names of the authors. 16.2. Known exact functionals For two types of systems, the exact analytic form of Ts [ρ] is known: • Thomas and Fermi:2,3 TsTF [ρ] = CF

Z

ρ5/3 (r)d3 r

(16.2.2)

The Thomas-Fermi functional2,3 is exact for the homogeneous electron gas. Applying it for inhomogeneous systems leads to an approximation known as the Thomas-Fermi functional or the local density approximation (LDA) functional. • von Weizs¨ acker:4 TsW [ρ] =

1 8

Z

2

|∇ρ(r)| 3 d r ρ(r)

(16.2.3)

This functional is exact for one-electron and spin-compensated two-electron systems. Applying it for other systems leads to an approximation known as the von Weizs¨ acker functional. 16.3. Local density approximation — LDA The label LDA is sometimes used in a more general way for any approximation which depends solely on the electron density like the TF functional [Eq. (16.2.2)]. In addition to TF, a few such functionals were proposed in the literature. GaussianLDA

Lee and Parr:5 TsGaussianLDA [ρ] =

3π 25/3

Z

ρ5/3 (r)d3 r

(16.3.4)

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Note that the coefficient 3π/25/3 ' 2.9686 is about 3% larger than the coefficient CF of the TF functional [Eq. (16.2.2)]. Fuentealba and Reyes:6    1/3 Z ρ(r)   TsZLP [ρ] = c1 ρ5/3 (r) 1 − c2 ln 1 + 2

ZLP

c2



1 ρ(r) 2



 3 1/3  d r

(16.3.5)

where c1 = 3.2372 and c2 = 0.00196. It was constructed following the “conjointness conjecture”7 applied to the ZLP8 approximation for the exchange-correlation energy. Note that in Eq. (9) in Ref. 6 the symbol ρ should be replaced by ρσ for the given coefficients. For the numerical verification see Ref. 9. LP97

Liu and Parr:10 TsLP97 [ρ]

= 3.26422

Z

+0.000498

ρ

5/3

Z

3

(r)d r − 0.02631 ρ11/9 (r)d3 r

3

Z

ρ

4/3

3

(r)d r

2 (16.3.6)

Note that the coefficient in front of the third term was given incorrectly in Ref. 10. 16.4. Gradient expansion approximation — GEAn The gradient expansion approximation (GEA) until the nth order: n n Z X X GEAn Ts [ρ] = Ts,i [ρ] = ti (ρ(r), ∇ρ(r), . . .) d3 r i=0

(16.4.7)

i=0

where only the terms for i even are non-zero. The analytical form of the terms up to i = 6 have been derived: t0 = tTF = CF ρ5/3

(16.4.8) 2

t2 =

1 W 1 |∇ρ| t = 9 72 ρ

(16.4.9) !

 2 2 2 4 ∇ ρ (3π 2 )−2/3 1/3 9 ∇2 ρ |∇ρ| 1 |∇ρ| ρ − + (16.4.10) 540 ρ 8 ρ ρ2 3 ρ4
−4/3
 3 2 ∇ ∇2 ρ 2 3π 2 2575 ∇2 ρ 249 |∇ρ| ∇4 ρ −1/3 13 t6 = ρ + + 45360 ρ2 144 ρ 16 ρ2 ρ
2  2 2 2 2 1499 |∇ρ| ∇ ρ 1307 |∇ρ| ∇ρ · ∇ ∇ ρ + − 2 18 ρ ρ 36 ρ2 ρ2 !  2 4 6 8341 ∇2 ρ |∇ρ| 1600495 |∇ρ| 343 ∇ρ · ∇∇ρ + + − (16.4.11) 18 ρ2 72 ρ ρ4 2592 ρ6 t4 =

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These terms were obtained by various authors: t0 by Thomas and Fermi,2,3 t2 by Kompaneets and Pavlovskii11 and by Kirzhnits,12 t4 by Hodges,13 and t6 by Murphy.14 t2 and t4 were obtained after integration by part of 15,16 2

1 |∇ρ| 1 + ∇2 ρ (16.4.12) 72 ρ 6    2
 2 2 2 ∇2 |∇ρ| 2 −2/3 4 ∇ρ · ∇ ∇ ρ (3π ) ∇ ρ ∇ ρ ρ1/3 12 − 30 − 14 −7 tJ4 = 4320 ρ ρ2 ρ ρ2    2 4 2 2 ∇ρ · ∇ |∇ρ| 140 |∇ρ| ∇ ρ 92 |∇ρ| + − 48 4  (16.4.13) + 3 ρ3 3 ρ3 ρ

tJ2 =

respectively. Choosing n = 0, 2, and 4 in Eq. (16.4.7) leads to approximations with the following labels: GEA0 which is just the TF functional [Eq. (16.2.2)], GEA2 which is also denoted with TF 19 W, and GEA4. 16.5. Generalized gradient approximation — GGA The general form of GGA functionals reads Z Z TsGGA [ρ] = f (ρ(r), |∇ρ(r)|) d3 r = CF ρ5/3 (r)F (s(r))d3 r

(16.5.14)

where F (s) is the so-called enhancement factor. The enhancement factor of the von Weizs¨ acker functional [Eq. (16.2.3)] is given by 5 2 s (16.5.15) 3 while for the GEA truncated at the 0th and 2nd orders [Eqs. (16.4.7)-(16.4.9)], the enhancement factors are F W (s) =

F GEA0 (s) = 1 F GEA2 (s) = 1 +

5 2 s 27

(16.5.16) (16.5.17)

respectively. A large group of approximations in the GGA family were constructed following the “conjointness conjecture” of Lee, Lee, and Parr7 according to which the enhancement factor of an exchange GGA functional can be used (with possible reoptimization of the free coefficients) for the kinetic energy. The following convention is used for labeling the conjoint functionals: if not only the analytic form but also the free coefficients are the same as in the exchange functional, then the functional is called conjoint X, where X stands for the name of the “parent” exchange functional. In the case of reoptimization of the coefficients, the standard convention applies (see Sec. 16.1).

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The functionals in this family differ only in the value of the constant λ: TsTFλW [ρ] = TsTF [ρ] + λTsW [ρ]

(16.5.18)

GEA0 and GEA2 correspond to λ = 0 and 1/9, respectively. Other values of λ were proposed in the literature, e.g., λ = 1.290/9 (modified 2nd order gradient expansion proposed by Lee et al.17 ). See Ref. 18 for a compilation of the different values of λ proposed in the literature. The enhancement factor of the TFλW functional is

P82

5 F TFλW (s) = 1 + λ s2 3

(16.5.19)

5 s2 27 1 + s6

(16.5.20)

Pearson:19 F P82 (s) = 1 +

DKPad´ e DePristo and Kress:20 F DKPad´e(x) = LLP

1 + 0.95x + 14.28111x2 − 19.57962x3 + 26.64765x4 1 − 0.05x + 9.99802x2 + 2.96085x3

(16.5.21)

Lee, Lee, and Parr:7 F LLP (s) = 1 +

0.0044188b2s2 1 + 0.0253bs arcsinh(bs)

(16.5.22)

This is the enhancement factor of the exchange functional B88 of Becke21 refitted for the kinetic energy. OL1 and OL2

Ou-Yang and Levy:22 F OL1 (s) = 1 +

F

OL2


−1/3 5 2 20 s + 0.00677 3π 2 s 27 3


1/3 s 5 2 0.0887 2 3π 2 (s) = 1 + s + 27 CF 1 + 8 (3π 2 )1/3 s

(16.5.23)

(16.5.24)

The coefficients in front of the third terms in OL1 and OL2 were inverted in the original paper of Ou-Yang and Levy (caption of Table I in Ref. 22). The correctness of the coefficients given here was confirmed by numerical values of Ts [ρ] .23–25 The correct scaling properties of Ts [ρ] require that the coefficient in front of the last terms in OL1 and OL2 are not free but must be non-negative. P92

Perdew:26 F P92 (s) =

1 + 88.396s2 + 16.3683s4 1 + 88.2108s2

(16.5.25)

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Thakkar:23 F T92 (s) = 1 +

LC94

0.0055b2s2 0.072bs − 1 + 0.0253bs arcsinh(bs) 1 + 25/3 bs

(16.5.26)

Lembarki and Chermette:27

F LC94 (s) =

  2 1 + 0.093907s arcsinh(76.32s) + 0.26608 − 0.0809615e−100s s2 1 + 0.093907s arcsinh(76.32s) + 0.57767 · 10−4 s4

(16.5.27) This is the enhancement factor of the exchange functional PW91 of Perdew and Wang28 refitted for the kinetic energy. FR95A Fuentealba and Reyes:6 F FR95A (s) = 1 +

0.004596b2s2 1 + 0.02774bs arcsinh(bs)

(16.5.28)

This is the enhancement factor of the exchange functional B88 of Becke21 refitted for the kinetic energy. FR95B Fuentealba and Reyes:6 F FR95B (s) = 1 + 2.208s2 + 9.27s4 + 0.2s6


1/15

(16.5.29)

This is the enhancement factor of the exchange functional PW86 of Perdew and Wang29 refitted for the kinetic energy. Vitos, Skriver, and Koll´ar:30

VSK98

1 + 0.95x + 3.564x3 1 − 0.05x + 0.396x2

(16.5.30)

1 + 0.8944s2 − 0.0431s6 1 + 0.6511s2 + 0.0431s4

(16.5.31)

F VSK98 (x) = VJKS00

Vitos et al.:31 F VJKS00 (s) =

E00 Ernzerhof:32 F E00 (s) = TW02

135 + 28s2 + 5s4 135 + 3s2

(16.5.32)

κ 1 + µκ s2

(16.5.33)

Tran and Wesolowski:25 F TW02 (s) = 1 + κ −

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where κ = 0.8438 and µ = 0.2319. This is the enhancement factor of the exchange functional PBE of Perdew, Burke, and Ernzerhof 33,34 refitted for the kinetic energy. PBEn Karasiev, Trickey, and Harris:35 F PBEn (s) = 1 +

n−1 X

(n)

Ci

i=1



s2 1 + a(n) s2

i

(16.5.34)

This is the enhancement factor of the exchange functional mPBE of Adamo and Barone36 refitted for the kinetic energy. Three approximations (n = 2, 3, and 4) of (n) the above general form were considered in Ref. 35. The coefficients Ci and a(n) are given in Table 16.1. Table 16.1. Coefficients of the enhancement factor of PBEn [Eq. (16.5.34)]. Functional PBE2 PBE3 PBE4

exp4

(n)

C1

2.0309 −3.7425 −7.2333

(n)

C2

50.258 61.645

(n)

C3

−93.683

a(n) 0.2942 4.1355 1.7107

Karasiev, Trickey, and Harris:35     2 4 F exp4 (s) = C1 1 − e−a1 s + C2 1 − e−a2 s

(16.5.35)

where C1 = 0.8524, C2 = 1.2264, a1 = 199.81, and a2 = 4.3476. CR

Constantin and Ruzsinszky:37
2 5 1 + a1 + 27 s + a2 s4 + a3 s6 − a4 s8 CR F (s) = 3 1 + a1 s2 + a5 s4 + 40β−5 a4 s6

(16.5.36)

Three approximations (corresponding to β = 1/5, 1/6, and 0.185) of the above general form were considered in Ref. 37. The corresponding sets of coefficients ai are given in Table 16.2. Table 16.2. Coefficients of the enhancement factor of CR [Eq. (16.5.36)]. β

1/5

1/6

0.185

a1 a2 a3 a4 a5

1.122609 0.900085 −0.227373 0.014177 0.731298

1.301786 3.715282 0.343244 0.032663 2.393929

1.293576 2.161116 −0.144896 0.025505 1.444659

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MGE2 Constantin et al.:38 F MGE2 (s) = 1 + κ −

κ 1 + µκ s2

(16.5.37)

where κ = 0.804 and µ = 0.23889. This is the enhancement factor of the exchange functional PBE of Perdew, Burke, and Ernzerhof.33,34 Conjoint B86A Lacks and Gordon:39 F B86A (s) = 1 + 0.00387

b 2 s2 1 + 0.004b2s2

(16.5.38)

This is the enhancement factor of the exchange functional B86A of Becke.40 Conjoint PW86 Lacks and Gordon:39 F PW86 (s) = 1 + 1.296s2 + 14s4 + 0.2s6


1/15

(16.5.39)

This is the enhancement factor of the PW86 exchange functional of Perdew and Wang.29 Conjoint B86B Lacks and Gordon:39 F B86B (s) = 1 + 0.00403

b 2 s2 4/5

(1 + 0.007b2s2 )

(16.5.40)

This is the enhancement factor of the exchange functional B86B of Becke.41 Conjoint DK87 Lacks and Gordon:39 F DK87 (s) = 1 +

7b2 s2 1 + 0.861504bs . 324(36π 4 )1/3 1 + 0.044286b2s2

(16.5.41)

This is the enhancement factor of the exchange functional DK87 of DePristo and Kress.42 Conjoint B88 Tran and Wesolowski:25 F B88 (s) = 1 + 1/3

where Cx = (3/4) (3/π) tional B88 of Becke.21

0.0042 b 2 s2 1/3 2 Cx 1 + 0.0252bs arcsinh(bs)

(16.5.42)

. This is the enhancement factor of the exchange func-

Conjoint PW91 Lacks and Gordon:39 F PW91 (s) =

  2 1 + 0.19645s arcsinh(7.7956s) + 0.2743 − 0.1508e−100s s2 1 + 0.19645s arcsinh(7.7956s) + 0.004s4

(16.5.43)

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This is the enhancement factor of the exchange functional PW91 of Perdew and Wang.28 Conjoint xFit Lacks and Gordon:39  1 10−8 + 0.1234 2 F xFit (s) = 1 + s + 29.790s4 + 22.417s6 1 + 10−8 s2 0.024974
0.024974 +12.119s8 + 1570.1s10 + 55.944s12 (16.5.44)

This is the enhancement factor of the exchange functional xFit of Lacks and Gordon.43 Conjoint PBE Perdew et al.:44 F PBE (s) = 1 + κ −

κ 1 + µκ s2

(16.5.45)

where κ = 0.804 and µ = 0.21951. This is the enhancement factor of the exchange functional PBE of Perdew, Burke, and Ernzerhof.33,34 16.6. Other semilocal approximations TB78

Tal and Bader:45 TsTB78 [ρ] = TsTF [ρs ] + TsW [ρs ] +

M X

TsW [ρr,A ]

(16.6.46)

A=1

where ρr,A (r) = ρ(RA )e−2ZA |r−RA |

(16.6.47)

and ρs (r) = ρ(r) −

M X

ρr,A (r)

(16.6.48)

A=1

M , RA , and ZA are the number, positions, and charges of the nuclei, respectively. CN84 Cummins and Nordholm:46 TsCN84 [ρ] =

Z

tCN84 (r)d3 r

(16.6.49)

where
tCN84 (r) = max tTF (r), tW (r)

(16.6.50)

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PG85

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Pearson and Gordon:47 TsPG85 [ρ]

where t

PG85

 Pn−1

=

Z

tPG85 (r)d3 r

t2i (r) + 21 t2n (r)

(16.6.51)

if t2 (r) 6 t0 (r) if t2 (r) > t0 (r)

(16.6.52)

1 1 TsmGEA4 [ρ] = TsTF [ρ] + TsW [ρ] + Ts,4 [ρ] 9 2

(16.6.53)

(r) =

i=0

t0 (r)

mGEA4 Allan et al.:48

PP88

Plindov and Pogrebnya:49 1 TsPP88 [ρ] = TsTF [ρ] + TsW [ρ] + 9

Z

t4 (r) 1+

1 t4 (r) 8 t2 (r)

d3 r

(16.6.54)

MGGA Perdew and Constantin:50

where



F MGGA = F W + F GE4−M − F W fab F GE4−M − F W

(16.6.55)

0

F

GE4−M

F GEA4 = r  2 1 + 1+F54s2

(16.6.56)

3

and fab (z) =

 0    

1

1+ea/(a−z) ea/z +ea/(a−z)

b

z60 0