Recent Advances In Relativistic Molecular Theory 9789812794901, 9789812387097

Citation preview

Recent Advances in Computational Chemistry - Vol.

RECENT ADVANCES IN RELATIVISTIC MOLECULAR THEORY

edited by

Kimihiko Hirao & Yasuyuki Ishikawa World Scientific

RECENT ADVANCES IN RELATIVISTIC MOLECULAR THEORY

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 ed. K. Hirao

• ) ,,-,..*-.;

f\ .

:s ]f) Computational Chemistry - Vol, 3

RECENT ADVANCES IN RELATIVISTIC MOLECULAR THEORY

edited by

Kimiiiko Hirao Department of Applied Chemistry School of Engineering, The University of Tokyo, Tokyo

Yasuyuki Isiikawa Department of Chemistry University of Puerto Rico, USA

\H& World Scientific NEW JERSEY

• LONDON

• SINGAPORE

• SHANGHAI

• HONGKONG

• TAIPEI • BANGALORE

Published by World Scientific Publishing Co. Pte. Ltd. 5 Toh Tuck Link, Singapore 596224 USA office: Suite 202, 1060 Main Street, River Edge, NJ 07661 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE

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

RECENT ADVANCES IN RELATIVISTIC MOLECULAR THEORY Copyright © 2004 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.

For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher.

ISBN 981-238-709-9

Printed in Singapore by World Scientific Printers (S) Pte Ltd

PREFACE

The majority of theoretical treatments of electronic behavior are based upon the Schrodinger equation. The Schrodinger equation incorporates the major postulates of quantum mechanics, the assumption that energy states are quantized and the recognition that small particles have wave-like properties. It ignores effects explained by the other theory of non-classical mechanics, the theory of relativity, but these effects are minor when the electronic structures to be studied are those of molecules which contain only light atoms. The small relativistic effects which are important for light atoms, spectroscopic fine structure effects for example, may be incorporated with perturbation theory. This approach rapidly becomes unsatisfactory as heavy atoms and molecules which contain heavy atoms come under consideration. For heavy-atom species it is necessary to discard the Schrodinger equation in favor of the Dirac equation. Relativistic effects, minor in light atoms, increase in magnitude rapidly as atomic number increases. Such effects account for a significant fraction of the chemical bond energies in molecules which possess second row transition elements and larger. Relativistic effects are responsible for several well-known chemical phenomena. Among these are the lanthanide contraction; the disparity in porperties between Au and its smaller group members, Ag and Cu; the special stability of Hg2 "^ion; and the fact that Hg is a liquid under standard conditions. They all point up the fact that a general theory of atomic and molecular electronic structure, applicable uniformly to all elements of the periodic table, must be based upon the Dirac equation rather than the Schrodinger. The Dirac equation, like the Schrodinger, can be exactly solved only for one-electron problems. Atomic and molecular wave functions are approximate, and an approximation must have certain qualities in order for it to be a useful one. In the last two decades, a great deal of effort has been directed toward developing quasirelativistic and fully relativistic many-electron theory formulated in particular with discrete basis expansion methods. Construction of an effective many-body Hamiltonian that accurately accounts for both relativistic and electron correlation effects in many-electron systems is a challenge. It is only in the past 15-20 years that relativistic quantum V

vi

chemistry has emerged as a field of research in its own right. It seems certain that relativistic many-electron calculations of molecular properties will assume increasing importance in the years ahead as relativistic quantum chemistry finds wider range of application. The applications of relativistic quantum chemistry are indeed manifold. This volume is concerned with the determination of electronic structures for heavy atoms and molecules which contain heavy atoms where relativistic effects are not negligible. With the increasing use of relativistic quantum chemical techniques in chemistry, there is an obvious need to provide experts' reviews of the methods and algorithms. This volume has the ambitious aim of disseminating in a single volume aspects of relativistic many-electron theories and their exciting developments by practitioners. Together, the nine chapters provide an in-depth account of the most important aspects of contemporary research in relativistic quantum chemistry, ranging from quasirelativistic effective core potential methods to relativistic coupled cluster theory. In the first chapter of the present volume, M. D6lg and X. Cao provide a comprehensive review of the energy-consistent pseudopotential variant of the relativistic effective core potential method. In Chapter 2, E. Miyoshi, Y. Sakai, Y. Osanai, and T. Noro provide a detailed description of recent developments in relativistic model core potential methods, and outline how to determine effective core potential parameters and valence basis sets. In Chapter 3, A. B. Alekseyev, H.-P. Liebermann and Robert J. Buenker describe two computational methods for carrying out configuration interaction calculations in which spin-orbit and other relativistic effects are included in the Hamiltonian by means of effective core potentials. Various one and two electron spin-orbit coupling Hamiltonians are discussed, along with symmetry properties of their matrix elements by D. G. Fedorov, M. W. Schmidt, S. Koseki, and M. S. Gordon in Chapter 4. Chapter 4 covers both theoretical methods and chemical applications. A. Wolf, M. Reiher, and B. A. Hess provide in Chapter 5 a masterly review of the generalized Douglas-Kroll transformation, a transformation to twocomponent electron-only formulations by suitably chosen unitary transformations. R. Fukuda, M. Hada, and H. Nakatsuji review their generalized UHF theory in the framework of two-component quasirelativistic molecular orbital theory in Chapter 6. In Chapter 7, T. Nakajima, T. Yanai, Y Ishikawa and K. Hirao provide a detailed description of new

VII

four-component Dirac-Hartree-Fock and Dirac-Kohn-Sham methods for heavy-atom-containing polyatomics that employ highly efficient computational schemes utilizing generally contracted spherical harmonic Gaussian spinors for rapid integral evaluation. Two-component quasirelativistic Hamiltonains, RESC and higher-order Douglas-Kroll, are also described, and illustrative calculations are shown. O. Matsuoka and Y. Watanabe give a brief overview of their four-component relativistic atomic and molecular program suite, PROPHET4R, in Chapter 8. In Chapter 9, W. Liu, F. Wang, and L. Li give an overview of the Beijing Density Functional Program package that can perform nonrelativistic one-component, quasi-relativistic two-component, and four-component relativistic density functional calculations. Relativistic coupled cluster calculations are computationally demanding. Chapter 10 details such approaches by U. Kaldor, E. Eliav, and A. Landau, i.e. the solution of the matrix DF SCF equations by expansion in basis sets of Gaussian functions and its refinement by Fock-space relativistic coupled cluster (CC) theory. We owe a considerable debt to our publishers who have been helpful and understanding. In particular, we would like to thank Mr. Suwarno who has been most generous with his advice. February 2003

Kimihiko Hirao, Tokyo Yasuyuki Ishikawa, San Juan

This page is intentionally left blank

AUTHOR LIST A. B. Alekseyev, FB 9 - Theoretische Chemie, Bergische Universitat Wuppertal D-42097 Wuppertal, Germany R. J. Buenker, FB 9 - Theoretische Chemie, Bergische Universitat Wuppertal D-42097 Wuppertal, Germany X. Cao, Institut fur Theoretische Chemie, Universitat zu Koln, Greinstr.4, D-50939, Koln, Germany Michael Dolg, Institut fur Theoretische Chemie, Universitat zu Koln, Greinstr.4, D-50939, Koln, Germany E. Eliav, School of Chemistry, Tel Aviv University, 69978 Tel Aviv, Israel D. G. Fedorov, National Institute of Advanced Industrial Science and Technology, 1-1-1 Umezono, Tsukuba 305-6568, Ibaragi, Japan R. Fukuda, Department of Synthetic Chemistry and Biological Chemistry, Graduate School of Engineering, Kyoto University, Sakyo-ku, Kyoto 6068501,Japan M. S. Gordon, Ames Laboratory, US-DOE and Department of Chemistry, Iowa State University, Ames, IA 50011, USA M. Hada, Department of Synthetic Chemistry and Biological Chemistry, Graduate School of Engineering, Kyoto University, Sakyo-ku, Kyoto 6068501,Japan B. A. Hess, Lehrstuhl fur Theoretische Chemie, Universitat ErlangenNurnberg, Egerlandstrase, 3, D-91058 Erlangen, Germany K. Hirao, Department of Applied Chemistry, Graduate School of Engineering, The University of Tokyo, Tokyo 113-8656, Japan

X

Y. Ishikawa, Department of Chemistry and The Chemical Physics Program, University of Puerto Rico, Rio Piedras Campus, P. O. Box 23346, San Juan, PR 00931-3346, USA U. Kaldor, School of Chemistry, Tel Aviv University, 69978 Tel Aviv, Israel S. Koseki, Department of Material Science, Osaka Prefecture University, 1-1 Gakuen-cho, Sakai 599-8531, Osaka, Japan A. Landau, School of Chemistry, Tel Aviv University, 69978 Tel Aviv, Israel L. Li, Institute of Theoretical and Computational Chemistry and State Key Laboratory of Rare Earth Materials Chemistry and Applications College of Chemistry and Molecular Engineering Pecking University, Beijing 100871, P. R. China H.-P. Liebermann, FB 9 - Theoretische Chemie, Bergische Universitat Wuppertal D-42097 Wuppertal, Germany W. Liu, Institute of Theoretical and Computational Chemistry and State Key Laboratory of Rare Earth Materials Chemistry and Applications College of Chemistry and Molecular Engineering Pecking University, Beijing 100871, P. R.China O. Matsuoka, Department of Chemistry, Kyushu University, Fukuoka 8108560,Japan E. Miyoshi, Graduate School of Engineering Sciences, Kyushu University, 6-1 Kasuga Park, Fukuoka 816-8580, Japan T. Nakajima, Department of Applied Chemistry, Graduate School of Engineering, The University of Tokyo, Tokyo 113-8656, Japan and PREST, Japan Science and Technology Corporation (JST)

XI

H. Nakatsuji, Department of Synthetic Chemistry and Biological Chemistry, Graduate School of Engineering, Kyoto University, Sakyo-ku, Kyoto 606-8501, Japan T. Noro, Division of Chemistry, Graduate School of Science, Hokkaido University, Sapporo 060-0810, Japan Y. Osanai, Aomori University, Aomaori 030-0943, Japan M. Reiher, Lehrstuhl fur Theoretische Chemie, Universitat ErlangenNiirnberg, Egerlandstrase, 3, D-91058 Erlangen, Germany Y. Sakai, Graduate School of Engineering Sciences, Kyushu University, 61 Kasuga Park, Fukuoka 816-8580, Japan M. W. Schmidt, Ames Laboratory, US-DOE and Department of Chemistry, Iowa State University, Ames, IA 50011, USA Y. Watanabe, Department of Chemistry, Kyushu University, Fukuoka 8108560,Japan F. Wang, Institute of Theoretical and Computational Chemistry and State Key Laboratory of Rare Earth Materials Chemistry and Applications College of Chemistry and Molecular Engineering Pecking University, Beijing 100871, P. R. China A. Wolf, Lehrstuhl fur Theoretische Chemie, Universitat ErlangenNiirnberg, Egerlandstrase, 3, D-91058 Erlangen, Germany T. Yanai, Oak Ridge National Laboratory, P. O. Box 2008,MS-6414, Oak Ridge TN 37831-6414, USA

This page is intentionally left blank

CONTENTS

Preface

v

Author List

ix

The Relativistic Energy-Consistent ab initio Pseudopotential Approach and Its Application to Lanthanide and Actinide Compounds M. DolgandX. Cao

1

Recent Developments of Relativistic Model Core Potential Method 37 E. Miyoshi, Y. Sakai, Y. Osanai, and T. Now Spin-Orbit Multireference Configuration Interaction Method And Applications to Systems Containing Heavy Atoms A. B. Alekseyev, H.-P. Libermann, and R. J. Buenker

65

Spin-Orbit Coupling Methods and Applications to Chemistry D. G. Fedorov, M. W. Schmidt, S. Koseki, and M. S. Gordon

107

Transgressing Theory Boundaries: The Generalized Douglas-Kroll Transformation A. Wolf, M. Reiher, and B. A. Hess

137

Generalized-UHF Theory for Magnetic Properties with Quasi-Relativistic Hamiltonians R. Fukuda, M. Hada, and H. Nakatsuji

191

Recent Progress in Relativistic Electronic Structure Theory T. Yanai, T. Nakajima, Y. Ishikawa, and K. Hirao

221

PORPHET4R: Four-component Relativistic Atomic and Molecular Program Suite 0. Matsuoka and Y. Watanabe

247

XIII

xiv

Relativistic Density Functional Theory: The BDF Program Package W. Liu, F. Wang, and L. Li

257

Four-Component Relativistic Coupled Cluster — Method and Applications U. Kaldor, E. Eliav, and A. Landau

283

T H E RELATIVISTIC E N E R G Y - C O N S I S T E N T A B INITIO P S E U D O P O T E N T I A L A P P R O A C H A N D ITS A P P L I C A T I O N TO LANTHANIDE A N D ACTINIDE COMPOUNDS Michael Dolg and Xiaoyan Cao Institut fur Theoretische Chemie, Universitdt zu Koln, Greinstr. 4, D-50939 Koln, Germany E-mail: [email protected], [email protected]

The energy-consistent pseudopotential variant of the relativistic effective core potential method is briefly reviewed. The corresponding available valence model Hamiltonian parametrizations for lanthanide and actinide elements as well as related valence basis sets are discussed. Calibration studies for atoms and diatomic molecules assessing the reliability of the outlined approaches are summarized and selected examples for recent applications to problems of chemical interest are provided. Finally, future directions for the development of energy-consistent pseudopotentials for f-elements are mentioned.

1. I n t r o d u c t i o n T h e concept of valence and core electrons is well-known to chemists, cf., e.g., the periodic table of elements. In the field of electronic structure theory a restriction of the explicit calculations to t h e chemically relevant valence electrons is highly desirable in order to keep the computational effort within reasonable limits. However, q u a n t u m mechanics does not allow to perform a corresponding core-valence separation without introducing into the formalism far-reaching simplifications. Nevertheless, approximate valence-only (VO) methods like the effective core potential ( E C P ) approach have been proposed already in the early years of q u a n t u m mechanics 1 ' 2 and developed during the last decades to become the mostly used relativistic model Hamiltonians in first-principles q u a n t u m chemistry, i.e., s t a n d a r d ab initio wavefunction theory ( W F T ) as well as density functional theory (DFT) applied to problems of (mainly) chemical interest. Two major E C P approaches exist, i.e., the model potential (MP) and t h e pseudopotential ( P P ) tech-

1

2

niques. The MP approach generates valence orbitals with a nodal structure corresponding exactly to the one of the all-electron (AE) valence orbitals and shifts the (now unoccupied) core-like orbitals to the virtual space. As an additional approximation the P P scheme (formally) introduces the socalled pseudo-valence orbital transformation, i.e., atomic core and virtual orbitals are mixed into the valence orbitals in order to make these radially smooth and nodeless for the energetically lowest solution in each angular symmetry. Although the resulting so-called pseudo-valence-orbitals possess in the chemically inert core region a simplified nodal structure compared to the AE valence orbitals, their shapes in the chemically important valence region as well as their one-particle energies should be very similar to the AE case. Of course, the virtual space is also affected by the transformation. Among the PPs one may distinguish shape-consistent (SC) and energy-consistent (EC) PPs. Whereas the former rely on one-particle functions and energies as reference data for the adjustment of the VO model Hamiltonian, the latter use only quantum mechanical observables, i.e., total valence energies. Each scheme has advantages and disadvantages, some of which will be mentioned in this article. Benchmark studies reveal that modern MP and P P parametrizations are both able to yield results in close agreement with the more rigorous AE calculations, but at a significantly lower computational cost. Nowadays most of the modern quantum chemistry electronic structure packages support MP or/and P P Hamiltonians. The continuing success of ECP approaches in (valence) electronic structure calculations of atoms, molecules and solids results not only from the computational savings due to the chemically intuitive restriction of the explicit quantum mechanical treatment to the so-called valence electron system, but also from the possibility to include implicitly the most important relativistic contributions by means of a simple parameterization of the VO model Hamiltonian. If spin-dependent terms are averaged or neglected, i.e., a scalar-relativistic ECP formalism is applied, the calculations can be performed using the well-developed machinery of non-relativistic quantum chemistry. Changes are essentially necessary only in the integral codes of standard computer programs using finite basis sets or the differential equation solvers of finite difference programs. A number of more or less specialized reviews on ECPs and their application in chemistry and physics has been published during the last three decades. 3 ' 4 ' 5 ' 6 ' 7 ' 8 ' 9 ' 10 ' 11 ' 12 ' 13 ' 1 ^ excellent more general reviews on relativistic effects in chemistry and

3 Table 1.

Electronic ground states and configurations for the lanthanides L n n + (n = 0 — 4): M :TT

M La ~drP Ce fld}s2 Pr f3s2 Nd f4s2 P m f5s2 Sm f6s2 Eu fs2 Gd f7dls2 T b f9s2 Dy / 1 0 5 2 Ho fus2 Er f12s2 Tm / 1 3 5 2 Yb / 1 4 5 2 Lu f1Adla2

^D^T2 1

GA

%/2 5

l4

6

H5/2 F0

7

8S

9

7/2

D2

6W u 5

15/2 l8

4

Il5/2 3 H6 2 F7/2 S

2 D°

3/2

~d?

fld2

/V /V 1 f's

/ V1 fal

fd s

/ 1V(

/ V J 11 ! 1

/ ' V1 /"s

/1'4 V2 /

S

M^

-*FT~ 4

H7/2

5

l4

6

I7/2 7 H2 8 F1/2 9 S4 110 D5/2 7 H8 6T A

17/2

Jfe

Hl3/2 3 F4 2 Sl/2 'So

" d

1 -

f23 I4 f

I6s /

f

fd>

f9 /'°

/ 12 " f /

13

;

14

/"«'

w*+

^3/2 3

H4

%/2

5

U 6

H5/2 F0 8 S 7 /2 9 D2 6 Hl5/2 5 I8 4 I15/2 3 H6 2 F7/2 'So 2 Sl/2 7

^So^ /'2 23 F 5 / 2 f3 H 4 f 4I9/2

/ 45 /6 /

f8

/ 9 / 10 / / "2 /' /" /14

5

l4

6

H5/2

F0

I S7/2 7

F6

w+

-p

P6

/ 2' f3 f4 /5 / /6

f

6 wu

15/2 5 l8

f89 f

4

Il5/2 3 H6 2 F7/2

/

%

/'

10

"P3/2 'So 2 F5/2 3 H4 4 I 9 /2 S

H5/2 F0 8 S7/2 7

76 H

5

u

4

/' 32

3

/

l4

6

15/2 l8

Il5/2 H6 2 F7/2

f, d and s denote 4f, 5d and 6s shells.

physics also exist. 28 ' 29 ' 30 ' 31 ' 32 ' 33 ' 34 ' 35 ' 36 ' 37 ' 38 ' 39 ' 40 ' 41 ' 42 Recently two books with articles written by various authors covering most of the recent developments in the field have been published. 43 ' 44 In addition an extensive bibliography of relativistic electronic structure calculations is available, 45 ' 46 ' 47 with a nearly up-to-date electronic version on the world-wide-web.48 The scope of the present article is limited to the application of ECppg23,24,25,27 m q U a n t u m chemical calculations for f element systems, i.e., lanthanides (Ln) and actinides (An). Molecules (and solids) containing atoms/ions with open f shells pose significant challenges to first-principles approaches of electronic structure theory, since very large relativistic and electron correlation effects have to be accounted for with high accuracy in quantitative studies. Open shells with three different main and angular quantum numbers may be present ((n-2)f, (n-l)d, ns with n=6 for Ln and n=7 for An) and high angular momenta and spin-multiplicities are observed already for atomic/ionic ground states, 49 ' 50 cf., tables 1 and 2. Therefore it was only during the last two decades that f element compounds were not avoided any more by all but a few quantum chemists. The recent fast development in this field of research is manifested by several review articles. 40 ' 51 ' 52,53,54 The f elements are especially interesting for ECPs, since (at least) three different plausible ways to partition the electron system into core and valence exist.

4

Table 2.

Electronic ground states and configurations for the actinides A n n + (n = 0 — 4).

M

M^

T Ac '~WP "^ S^~ 3 Th d2S2 F2 2 d V 4F3/2 3 Pa !2d>s2 4 K 1 1 / 2 f2s2 H4 l 2 5 3 2 U f*d s K6 Z * 4I9/2 6 Np fAdlS2 Ln/2 fWs^U 6 2 7 Pu f S F0 Z6«' 8 F 1 / 2 8S Am fs2 / V 9S4 7/2 2 9 8 Cm fd}s D2 f7s2 S7/2 7 6H Bk fs2 / V H8 15/2 5 Cf f10s2 l8 Z10*1 6Ii7/2 4 Es flls2 Z'V 5 I 8 Il5/2 3 4 Fm f12s2 H6 Z'V H 13/2 13 2 13 1 3 2 Md f s f * F4 F7/2 No f14s2 'So Z14*1 2 S 1 / 2 1 4 2 Lr / p V P l / 2 Z14*2 ^ o

~ ^J

M3^

M^T T

1

"7

Z'd1 /2rf!

Z54 Z Z67 Z8 Z9 Z10 Zu Z

z 1213 z 14

z/ ' V

^77 'G 4

4

Ill/2

5

l4

6

H5/2 F0

7 8

7

S7/2

F6

6H

15/2

5

l8

4

Il5/2

3

H6 2 F7/2

%

2

s1/2

^s^

2 z 12 3HF45 / 2 z 3 4 I /2 z 4 5 l 49 z 6 z 56 7 FH0 5 / 2 z 7 8 S 7/2 z8 7F6 z 9 H 15/2 z 10 5I8 z 11 4 Il5/2 z 12 3H6 z 13 2 F 7 / 2 z 14 %

z

M4^

"?6

"P3/2

P 'So 1 2F5/2 2 3H4 3 4I9/2 4 5I4 5 6H5/2 6 7F0 7 8S7/2 8 7F6 9 6 Hi5/2 10 5 I 8 11 4 Il5/2 12 3 H 6 13 2 F 7 / 2

z z z z z z z z z z z z z

f, d and s denote 5f, 6d and 7s shells. Our present calculations find a P a 2 + 2 # n / 2 ground state.

2. Energy-consistent ab initio Pseudopotentials Relativistic ECP calculations should model accurately AE calculations, but at a significantly lower computational cost. The most accurate relativistic AE Hamiltonian available at present as well as the VO model Hamiltonian used in relativistic EC-PP theory is discussed in section 2.1. A very important issue for the accuracy of VO calculations is the core-valence separation, i.e., the choice of the P P core described in section 2.2. The energy-adjustment of PPs is described in section 2.3. Similar to AE work the results obtained in PP calculations highly depend on the choice of the valence basis sets, which is discussed in the final section 2.4. 2.1.

Valence-only

Model

Hamiltonian

Starting point of our discussion is a general configuration space Hamiltonian for n electrons and N nuclei

* = £ M O + £>•,;)+ £ ^ ,

a)

where the Born-Oppenheimer approximation is assumed to hold and external fields are neglected. The indices i and j refer to electrons, A and /1 to nuclei. Z\ represents the charge of the nucleus A. For the one- and two-particle operators h and g various expressions can be inserted (e.g., relativistic, quasirelativistic or nonrelativistic; all-electron or valence-only).

5

The most accurate Hamiltonian suitable for electronic structure calculations on atoms, molecules and solids is based on the Dirac (D) one-particle Hamiltonian

hD(i) = cctiPi + (ft - I 4 )c 2 + Y, v^ri\)

•

(2)

A

I4 denotes the 4 x 4 unit matrix, and pi = — iV% stands for the momentum operator for the z-th electron. a 2 is a three-component vector whose elements together with ft are the 4 x 4 Dirac matrices

and

(o-L)

5=

(*5)

(3)

These can be expressed in terms of the three-component vector of the 2 x 2 Pauli matrices (T,

''

=

ff =

( I O ) '

» (°~o)'

^

=

(o-°i) '

(4)

and the 2 x 2 unit matrix I2. In order to have the same zero of energy as in the nonrelativistic case, the rest energy c2 of the electron was subtracted from Eq. 2. V\(ri\) represents the electrostatic potential arising from the A-th nucleus at the position of the i-th. electron VA(riA) = - — -

(5)

In some cases finite nuclear models are used instead of the point charge approximation given above. In contrast to the Dirac one-particle Hamiltonian, which is correct to all orders in the fine-structure constant (a = 1/c), only approximate expressions are known for the two-particle terms. Approximations for g used for the generation of reference data for EC-PPs are either the nonrelativistic electrostatic Coulomb (C) interaction (leading to the Dirac-Coulomb (DC) Hamiltonian correct to 0 ( a 0 ) )

9c(iJ) = — , r

(6)

ij

or in addition the magnetic interaction and the retardation of the interaction due to the finite velocity of light, as it is accounted for in the frequency-independent Breit (B) interaction (yielding the Dirac-CoulombBreit (DCB) Hamiltonian correct to 0 ( a 2 ) ) 9CB(I,J)

= — ~ WT-[/[£f p - E?Ef) := min .

(27)

Based on AE reference data from the so-called Wood-Boring quasirelativistic HF approach corresponding small-core PPs have been derived for the 4dand 5d-transition metals 85 as well as the lanthanide 75 ' 76 and actinide 74,77 elements. In addition, large-core main group PPs using this approach have been generated. 86 In the most recent version of the EC-PP approach the reference data is derived from finite-difference AE MCDHF calculations based on the DC or DCB Hamiltonian. 87 2.4. Valence Basis

Sets

Recently (14sl3pl0d8f6g)/[6s6p5d4f3g] ANO basis sets using a generalized contraction scheme for relativistic small-core 4f-in-valence lanthanide and 5f-in-valence actinide PPs have been presented and extensively tested by Cao et al. 76 ' 77 For each element the contraction coefficients have been obtained from the diagonalization of a state-averaged density matrix obtained from configurations with different (n-2)f and (n-l)d occupations. In addition to an accurate reproduction of energy differences involving a change in these orbital occupations at the HF level and to a lesser extent also at the correlated level (cf. section 3.1), the newly derived basis sets also lead to quite small basis set superposition errors (BSSE) in molecular calculations (cf. section 3.2). The basis sets are of pVQZ quality. Smaller basis sets of pVTZ or pVDZ quality can be easily generated by omitting the (generalized) contractions corresponding to the smallest or two smallest ANO occupation numbers in each angular symmetry. When judging the quality of the basis sets at the correlated level for cases where changes of the f oc-

18 0.0

T

1

1

r

j

1

1

L

-0.5

> eT-1.0 »— o d>

-1.5

basis set quality (1/I3) Fig. 4. Errors in the fourth ionization potential IP4 of Ce and T h from basis set extrapolation studies at the CCSD(T) level including corrections for spin-orbit coupling. 9 1 ' 9 0 The largest applied basis sets were (16sl5pl2dl0f8g8h8i) for Ce and (14sl3pl0d8f6g6h6i) for Th. The labels f, g, h and i on the abscissa denote subsets containing basis functions including up to this angular symmetry.

cupation are of interest, the extremely slow convergence of the (standard) CI or CC expansions towards the basis set limit has to be remembered. Figure 4 illustrates the situation for the especially difficult cases of the fourth ionization potentials of Ce and Th. It has to be kept in mind for molecular studies of the relative stability of, e.g., Ce(III) and Ce(IV) compounds, that at the atomic level the relevant quantity IP4 cannot be obtained more accurately than 0.5 (0.25) eV with standard basis sets containing up to g (h) functions. Besides the pVQZ ANO generalized contracted basis sets also segmented contracted basis sets yielding results of similar quality were derived, i.e., (14sl3pl0d8f6g)/[10s8p5d4f3g] for lanthanides 88 and (14sl3pl0d8f6g)/[10s9p5d4f3g] for actinides. 89 The slightly larger s and p contractions are necessary to maintain the same accuracy for the first four ionization potentials in HF as well as correlated calculations. In case of the medium-core 4f-in-core lanthanide PPs the originally proposed (7s6p5d)/[5s4p3d] HF energy-optimized valence basis sets, augmented by one or two CI energy-optimized f correlation/polarization functions, are still in use. 78

13

La/Ac

Gd/Cm

Lu/Lr

Fig, 5. Differential relativistic (rel) and correlation (corr) effects in the third ionization potentials IP3 of the lanthanides and actinides. The relativistic contributions were determined from state-averaged AE M C D H F / D C and corresponding HP calculations, 7 3 whereas f tie correlation contributions stem from P P CASSCP/ACPP basis set extrapolation studies (cf, also t e x t ) , 9 0 , 9 1

3. C a l i b r a t i o n Studies Recently systematic calibration studies applying the small-core E O P P s and the newly de¥eloped generalized contracted ANO as well as segmented contracted valence basis sets were published. In addition, extrapolations to the basis set limit were performed for atoms at the correlated level. 76,77,88 ' 89,90 ' 91,92 A representative summary of the results on atoms and molecules is given in section 3.1 and 3,2, respectively. 3.1.

Atoms

The electronic ground states of the lanthanides and actinides and their up to fourfold positive ions are listed in tables 1 and 2, respectively. It is seen that for the third and fourth ionization potentials IP3 and IP4 in most cases the f occupation number changes by one electron. Thus, large differential relativistic and electron correlation effects are expected for these energy differences linking the chemically important di-, tri- and tetravalent forms of the ions. An estimate for the magnitude of these effects is displayed in Figures 5 and i for IP3 and IP4, respectively. Relativistic contributions for the actinides tend to be larger than for the lanthanides, whereas the opposite behavior is observed for electron correlation effects. Whereas the first

20

._i^

1 An res

" La/Ac~"~~

"

l^TjCm

"~~~~~"

^[ulr

Fig. 6. As Figure 5, but for the fourth ionization potential IP4.

observation is explained by the roughly « Z4 dependence of relati¥istic effects, it is most likely the more compact nature of the lanthanide 4f shell compared to the actinide 5f shell which causes the second fact. The compactness of the 4f shell is at least partially due to the missing orthogonality constraint onto inner shells with the same angular symmetry. In view of the high accuracy of the available relativistic Hamiltonians and the way slow convergence of CI and CC expansions with respect to the one- and many-particle basis sets, electron correlation seems to be the harder problem to sol¥e in heavy element chemistry. Therefore, from this point of Yiew, lanthanides are more difficult to treat at the ab initio level than actinides, Typical examples where an accuracy of better than 0,1 eV in the deterin i nation of energy differences between states with different f occupation numbers is required are calculations for the ground and low-lying excited states of cerocene Ce(CgHg)2 7 i , i 3 and ytterbium monoxide YbO. M The ionization potentials obtained with the generalized contracted ANO (14sl3pl0d8fig)/[6sip5d4f3g] basis sets at the P P CASSCP/ACPP level including corrections for spin-orbit effects (deriYed at the finite difference MCHF level) are listed in tables 4 and 5 for lanthanides and actinides, respecti¥ely. Only the (n-2)f (n-l)spd and ns shells were correlated. EelatiYely small mean absolute errors (m.a.e.) of 0.21 e¥ and 0.14 eV are observed for IPi and IP2, respectively, of the lanthanides. Due to the incomplete correlation treatment the m.a.e. for IP3 and IP4 of 0 JO e ¥ and 0.50 eV,

21 Table 4. First to fourth ionization potentials of the lanthanides (in eV). P P CASSCF/ACPF results, corrected for spin-orbit interaction, obtained with contracted (14sl3pl0d8f6g)/[6s6p5d4f3g] ANO standard basis sets (std.) and by extrapolation to the basis set limit (ext.) 7 6 , 9 1 are compared to experimental data (exp.). 4 9 The mean absolute errors (m.a.e.) are listed in the last line. IP std. La 5.48 Ce 5.85 Pr 5.30 Nd 5.36 Pm 5.41 Sm 5.45 Eu 5.48 Gd 6.00 Tb 5.67 5.72 Dy Ho 5.77 Er 5.83 Tm 5.89 Yb 5.91 Lu 5.26 m.a.e. 0.21

exp. 5.58 5.54 5.46 5.53 5.55 5.64 5.67 6.15 5.86 5.94 6.02 6.11 6.18 6.25 5.43

IP2 std. exp. 11.03 11.06 11.02 10.85 10.58 10.55 10.74 10.73 10.88 10.90 10.98 11.07 11.11 11.24 12.11 12.09 11.54 11.52 11.51 11.67 11.57 11.80 11.66 11.93 11.74 12.05 11.76 12.18 13.78 13.90 0.14

std. 18.89 19.42 21.01 21.49 21.84 23.09 24.51 20.56 20.93 22.13 21.97 21.81 22.82 24.20 20.70 0.60

IP3 ext. 18.82 20.05 21.54 21.98 22.30 23.44 24.58 20.65 21.42 22.53 22.41 22.28 22.73 24.37 20.82 0.30

exp. 19.18 20.20 21.62 22.10 22.30 23.40 24.92 20.63 21.91 22.80 22.84 22.74 23.68 25.05 20.96

std. 49.72 36.04 38.41 40.20 40.76 41.20 42.68 44.72 38.58 40.64 42.06 42.00 41.89 43.01 44.74 0.50

IP4 ext. 50.01 36.15 39.04 40.70 41.23 41.64 43.12 44.83 39.15 41.08 42.46 42.42 41.80 43.33 44.87 0.34

exp. 49.95 36.76 38.98 40.40 41.10 41.40 42.70 44.00 39.37 41.40 42.50 42.70 42.70 43.56 45.25

Electronic states cf. table 1.

respectively, are significantly larger. Using uncontracted basis sets up to (16sl5pl2dl0f8g8h8i), correlating all electrons in the P P valence space and linearly extrapolating the results for subsets including up to g, h and i functions over l / / 3 to the basis set limit yields improved m.a.e. of 0.30 eV and 0.34 eV. Virtually no improvement is observed for IPi and IP2, since here in most cases the 4f occupation number remains unchanged. A similar situation is expected to hold for the actinides, however, only for IPi a sufficiently complete set of experimental values exists for calibration. Here a m.a.e. of 0.25 eV and 0.24 eV are observed for the generalized contracted ANO standard basis sets and the basis set extrapolations based on uncontracted basis sets up to (14sl3pl0d8f6g6h6i). Unfortunately, almost no experimental data is available for the higher ionization potentials. The CASSCF/ACPF treatment for systems with open f shells is plagued by problems. Besides the open (n-2)f shell often also the (n-l)d and ns shells have to be included in the active space. In case of ground states of neutral atoms also np should be included in order to account for the important ns 2 ->> np 2 near-degeneracy excitations. The definition of larger active spaces is not obvious, however, comparison to corresponding CCSD(T) results for some single-reference dominated cases indicates that still some higher exci-

22 Table 5. First to fourth ionization potentials of the actinides (in eV). P P CASSCF/ACPF results, corrected for spin-orbit interaction, obtained with contracted (14sl3pl0d8f6g)/[6s6p5d4f3g] ANO standard basis sets (std.) and by extrapolation to the basis set limit (ext.) 7 7 ' 9 0 are compared to experimental data (exp.). 5 0 The mean absolute errors (m.a.e.) are listed in the last line. std. Ac 5.20 6.24 Th 5.81 Pa 5.98 U 6.09 Np Pu 5.74 5.73 Am Cm 5.70 Bk 5.93 5.98 Cf Es 6.07 Fm 6.16 Md 6.23 5.62 No Lr 4.79 m.a.e. 0.25

IPi ext. 5.17 6.25 5.81 6.06 5.98 5.71 5.71 5.68 5.90 5.96 6.07 6.18 6.25 6.33 4.78 0.24

exp. 5.17° 6.31 5.90 a 6.19 6.27 6.03 5.97 5.99 6.20 6.28 6.37 6.50 a 6.58 6.65

IP 2 std. ext. 11.58 11.60 12.40 12.11 11.96 11.96 11.97 11.63 11.54 11.35 11.49 11.50 11.69 11.71 12.63 12.17 11.95 11.96 12.02 12.03 12.16 12.20 12.33 12.38 12.43 12.47 12.54 12.58 14.22 14.25

IP 3 std. ext. 17.29 17.39 18.13 18.30 18.37 17.73 18.70 19.07 19.60 19.92 21.14 21.37 22.34 22.34 19.87 20.36 21.61 21.93 22.70 22.84 22.88 23.06 23.36 23.66 24.63 24.69 26.24 26.05 21.47 21.52

IP 4 std. ext. 44.72 44.99 28.15 28.45 30.58 31.24 32.62 33.17 33.84 34.27 35.07 35.43 36.96 37.26 38.96 39.06 36.01 36.52 37.82 38.12 39.32 39.52 39.87 40.16 40.42 40.60 41.99 41.96 44.39 44.12

Additional exp. values (in eV) I P 2 Ac 11.78±0.19 a , U 11.59±0.37; I P 3 Th 18.33±0.05, U 19.80±0.31; I P 4 Th 28.65±0.02, U 36.70±1.00. Electronic states cf. table 2. a Semiempirical estimates.

tations are missing in the CASSCF/ACPF treatment. The worst examples with respect to the convergence towards the correct result are IP4 of Ce and Th. It is seen from figure 4 that the spin-orbit corrected CCSD(T) results very slowly converge towards the experimental values. The linear behavior in l~3 of the results for up to g, h and i functions is also obvious. The few cases were HF/CCSD(T) could be applied for the higher lanthanide ionization potentials are summarized in table 6. With exception of IP4 of Gd, where the experimental error bar is quite large, all extrapolated results are within 0.1 eV of the experimental value. Similarly, for actinides a few HF/CCSD(T) results could be obtained for IPi and IP 2 . Table 7 lists the results which are slightly improved with respect to the CASSCF/ACPF values given in table 5. 3.2.

Molecules

Molecular calibration studies are essential for ECPs, since excellent performance in atomic test calculations does not necessarily guarantee the trans-

23 Table 6. Selected CCSD(T) results for I P 3 and I P 4 of the lanthanides (in eV). 9 1 Ln La La Ce Eu Gd Yb Lu Lu

IP3 IP4 IP4 IP3 IP4 IPs IP3 IP4

f 18.81 49.68 34.93 23.79 43.55 23.95 20.78 44.26

g 18.98 49.91 36.11 24.55 44.34 24.58 20.87 44.87

h 19.06 49.99 36.42 24.78 44.59 24.76 20.89 45.07

i 19.09 50.02 36.53 24.89 44.71 24.88 20.90 45.21

extr. 19.14 50.07 36.74 25.02 44.86 25.00 20.92 45.33

expt. 19.18 49.95±0.06 36.76±0.01 24.92±0.10 44.0±0.7 25.05±0.03 20.96 45.25±0.03

Basis sets (16sl5pl2dl0f)+(8g)+(8h)+(8i). Electronic states cf. table 1. Table 7. Selected values of I P i and IP2 from CCSD(T) calculations corrected for spin-orbit coupling using standard basis sets (std.) and extrapolation to the basis set limit (ext.). The mean absolute error (m.a.e. in eV) is given in the last line. 7 7

Ac Am Cm Md No Lr m.a.e.

IP]L CCSD(T) Exp. std. ext. 5.23 5.19 5.17±0.12 5.97 5.83 5.85 5.74 5.74 5.99 6.42 6.58 6.38 6.50 6.65 6.45 4.84 4.82 0.16 0.15

IP2 CCSD(T) Exp. std. ext. 11.65 11.73 11.78±0.19 11.77 11.81 12.23 12.60 12.58 12.53 12.64 12.69 14.41 14.35

Electronic states cf. table 2.

ferability to a molecular environment. Gasphase data of diatomic molecules, e.g., hydrides or oxides, provide useful references.95 In contrast to main group elements and d-transition metals accurate experimental data for calibrating electronic structure methods for lanthanides and especially for actinides are very scarce. Tables 8 and 9 present results of such a calibration for heteronuclear diatomic lanthanum and lutetium molecules.76 Relativistic small-core EC-PPs and segmented as well as generalized contracted basis sets were applied for La and Lu. Standard aug-ccpVQZ basis sets were used for H, O and F. An overall excellent agreement with available experimental data is observed. 95 ' 96,97,98 The counter-poise correction of the basis set superposition error is found to be nearly negligible at the SCF level, however, although it is relatively small, it has to be taken into account at the CCSD(T) level for accurate work. After the counter-poise correction both lanthanide basis sets yield essentially the

24

Table 8. Bond lengths R e (A), binding energies D c (eV) and vibrational constants ue ( c m - 1 ) of the monohydride, monoxide and monofluoride of lanthanum from energy-consistent pseudopotential (EC-PP) calculations using generalized (gen.) and segmented (seg.) contracted lanthanum valence basis sets 7 6 ' 8 8 in comparison to experimental d a t a 9 5 ' 9 6 ' 9 7 ' 9 8 and selected all-electron (AE) results. 9 9 The notation .../... refers to results without/with counterpoise correction of the basis set superposition error. molecule LaH * £ +

LaO

2

£+

LaF *£+

method EC-PP(WB,29),SCF,seg. EC-PP(WB,29),SCF,gen. EC-PP(WB,29),B3LYP,seg. EC-PP(WB,29),CCSD(T),seg. EC-PP(WB,29),CCSD(T),gen. Exp. EC-PP(WB,11),CISD+Q AE,DC,MP2 a EC-PP(WB,29),SCF,seg. EC-PP(WB,29),SCF,gen. EC-PP(WB,29),B3LYP,seg. EC-PP(WB,29),CCSD(T),seg. EC-PP(WB,29),CCSD(T),gen. Exp. EC-PP(WB,11),CISD+Q EC-PP(WB,29),SCF,seg. EC-PP(WB,29),SCF,gen. EC-PP(WB,29),B3LYP,seg. EC-PP(WB,29),CCSD(T),seg. EC-PP(WB,29),CCSD(T),gen. Exp. EC-PP(WB,11),CISD+Q AE,DC,MP2 a

Re 2.049/2.049 2.048/2.047 2.006/ 2.016/2.028 2.016/2.027 2.032 2.005/ 2.006/ 1.806/1.807 1.806/1.806 1.827/ 1.837/1.843 1.836/1.841 1.826 1.813/ 2.044/2.044 2.044/2.044 2.024/ 2.027/2.035 2.027/2.034 2.027 2.022/ 2.038/

De 2.11/2.10 2.10/2.09 2.91/ 2.98/2.89 2.97/2.88 3.01/ 5.27/5.27 5.27/5.26 8.32/ 8.28/8.12 8.30/8.13 8.296 8.02/ 5.13/5.11 5.09/5.09 6.86/ 6.96/6.85 6.92/6.83 6.906 6.55/

0Je

1454/1450 1456/1453 1461/ 1468/1447 1456/1447 1461/ 1500/ 864/863 864/864 824/ 813/806 814/807 813 847/ 567/567 567/567 576/ 580/574 578/574 570 571/ 571/

Basis sets: EC-PP(WB,29) La generalized contraction (14sl3pl0d8f6g)/[6s6p5d4f3g]; segmented contraction (14sl3pl0d8f6g)/[10s8p5d4f3g]; H, O, F aug-cc-pVQZ. EC-PP(WB,11) (8s7p6d6f); H (8s4p3d2f); O, F (13s7p4d3f2g). Inactive orbitals in CC: La 4s, 4p and O, F Is. a Fully relativistic calculation, roughly pVTZ basis set quality. 6 LaO: Dg; LaF: Dg with error bars of ± 0.14 eV, Dj] was converted to D e by accounting for zero-point vibration.

same results. Due to the large correlation effects within the compact occupied 4f shell, the counter-poise corrections are significantly larger for the Lu systems than for the La compounds. Freezing the 4f and other inner shells, decreases the BSSE, but it also deteriorates considerably the results and is not recommended. It has to be investigated if for these systems a combination of small-core PPs and large/medium-core CPPs provides an efficient strategy. The small-core PPs together with the segmented contracted lanthanide basis sets yield also reasonable results at the B3LYP DFT level. The present PP results are also in reasonable agreement with

25 Table 9. molecule LuH X E+

LuO

2

£+

LuF *£+

As Table 8, but for lutetium.

method EC-PP(WB,41),SCF,seg. EC-PP(WB,41),SCF,gen. EC-PP(WB,41),B3LYP,seg. EC-PP(WB,41),CCSD(T),seg. EC-PP(WB,41),CCSD(T),gen. Exp. EC-PP(WB,11),CISD+Q AE,DC,MP2 a EC-PP(WB,41),SCF,seg. EC-PP(WB,41),SCF,gen. EC-PP(WB,41),B3LYP,seg. EC-PP(WB,41),CCSD(T),seg. EC-PP(WB,41),CCSD(T),gen. Exp. EC-PP(WB,11),CISD+Q EC-PP(WB,41),SCF,seg. EC-PP(WB,41),SCF,gen. EC-PP(WB,41),B3LYP,seg. EC-PP(WB,41),CCSD(T),seg. EC-PP(WB,41),CCSD(T),gen. Exp. EC-PP(WB,11),CISD+Q AE,DC,MP2 a

Re 1.953/1.954 1.951/1.952 1.901/ 1.900/1.916 1.882/1.914 1.912 1.936/ 1.883/ 1.784/1.784 1.783/1.783 1.793/ 1.786/1.795 1.784/1.795 1.790 1.790/ 1.938/1.939 1.938/1.938 1.922/ 1.913/1.923 1.909/1.923 1.917 1.930/ 1.916/

De 2.73/2.72 2.69/2.68 3.16/ 3.53/3.37 3.64/3.35 «3.47 6 3.47/ 4.10/4.08 4.03/4.02 6.79/ 7.15/6.91 7.28/6.90 7.04 6 6.49/ 5.97/5.96 5.92/5.91 7.28/ 7.72/7.54 7.82/7.51 %5.94 6 7.28/

COe

1484/1483 1498/1496 1515/ 1520/1502 1577/1507 ^1520 c 1493/ 1540/ 893/893 892/892 846/ 848/840 857/840 842 861/ 601/601 600/600 605/ 612/605 620/604 612 599/ 604/

Basis sets: EC-PP(WB,41) Lu generalized contraction (14sl3pl0d8f6g)/[6s6p5d4f3g]; segmented contraction (14sl3pl0d8f6g)/[10s8p5d4f3g]; H, O, F aug-cc-pVQZ. E C - P P ( W B , H ) (8s7p6d6f); H (8s4p3d2f); O, F (13s7p4d3f2g). Inactive orbitals in CC: Lu 4s, 4p and O, F Is. a Fully relativistic calculation, roughly pVTZ basis set quality. 6 LuH: value for Dg of LuD; LuO: Dg; LuF: estimated Dg, Dg was converted to D e by accounting for zero-point vibration. c LuD: Exp. 1075 c m " 1 , CCSD(T) 1069 c m " 1 .

AE DHF/MP2 values, which were based on the DC Hamiltonian and basis sets of VTZ quality without BSSE corrections." ThO is a frequently investigated test molecule for the actinide series (Table io). 7 4 ' 7 7 ' 1 0 0 ' 1 0 1 The first reasonable results for the XE+ ground state were published by Marian et al., 100 who applied a Th MP derived from scalar-relativistic atomic calculations with a free particle no-pair Hamiltonian. The latter has the deficiency to overestimate the relativistic stabilization of s shells (by « 5 - 8 %). The MP follows the frozen-core ECP ansatz separating the wavefunction into an 'inner core' (Th ls-5s, 2p-5p, 3d-4d) replaced by the MP, an 'outer core' (Th 6s, 6p, 5d, 4f) frozen at the HF level and the variationally treated valence shell (Th 7s, 7p, 6d, 5f). However, since the 5f orbitals have a significant radial overlap with the frozen

26 Table 10. Bond lengths R e (A), vibrational constants ue ( c m - 1 ) and binding energies D e (eV) of T h O in the * £ + ground state from energy-consistent pseudopotential ( E C - P P ) , 7 4 ' 7 7 model potential ( M P ) 1 0 0 and ab initio model potential (AIMP) 1 0 1 calculations in comparison to experimental data. The values are without/with counter-poise correction of the basis set superposition error. EC-PP(WB,30), S C F a EC-PP(WB,30), SCF 6 AIMP(CG,12), SCF C EC-PP(WB,30), CASSCF a AIMP(CG,12), CASSCF C MP(no-pair, 36), CASSCF d EC-PP(WB,30), S C F / M R C I + S C C a

Re 1.829/ 1.817/1.817 1.819/ 1.882/ 1.886/ 1.928/ 1.861/

943/ 956/955 956/ 876/ 865/ 847/ 878/

EC-PP(WB,30), S C F / M R C I + S C C e

1.845/

902/

EC-PP(WB,30), S C F / C C S D ( T ) 6

1.839/1.845

898/891

MP(no-pair ,36), CASSCF/MRCH-SCC d Exp.

1.923/ 1.840

852/ 896

UJe

De (Do') 6.07/ 6.26/6.24 5.99/ 8.92/ 9.15/ 8.67/ (8.25/) 8.87/ (8.45/) 9.58/9.38 (9.16/8.96) 7.85/ (9.00±0.09) (8.87±0.15) (8.79±0.13)

a

E C - P P for Th with [ 36 Kr] 4d 1 0 4f14 core; basis sets Th (12sllpl0d8f)/[8s7p6d4f], O (10s6pld)/[5s3pld]; Th 5s,5p,5d,6s,6p and O Is frozen in MRCI. 6 as a, but basis sets Th (14sl3pl0d8f6g)/[6s6p5d4f3g] ANO, O aug-cc-pVQZ (spdfg); Th 5s,5p and O Is frozen in CCSD(T). C AIMP for Th with [ 54 Xe] 4f14 5d 1 0 and for O with [2He] core; basis sets Th (14sl0plld9f)/[6s5p5d4fj, O (5s6pld)/[3s4pld]. d M P for Th with [54Xe] core, 4f,5d,6s,6p outer core frozen at all levels; basis sets Th (12sl0p9d6f)/[6s3p6d3f], O (9s5pld)/[5s4pld]. e as a, but including two g-functions on Th. 'theoretical D e values have been corrected for molecular (0.03 eV) and atomic (Th 0.38 eV, O 0.01 eV) spin-orbit energy lowerings; the zero-point energy (0.06 eV) was subtracted to obtain Do- 74

6s, 6p and 5d shells (Figure 2) large frozen-core errors result when these orbitals are involved in chemical bonding. Due to the strong relativistic destabilization of the 5f shell (Figure 2), low-lying electronic states with occupied 5f orbitals were not found. In agreement with experimental findings Marian et al. obtained a cr%s7p lY,+ (0 + ) ground state. This fact and the Th [seRn] 6d2 7s2 3 F 2 ground state should not lead to the conclusion that Th is a 6d transition metal rather than a 5f element: the Th 5f shell seems to be extremely important for chemical bonding in ThO, i.e., the neglect of f functions on Th leads to a 1.44 eV lower binding energy.74 The small-core EC-PP results of Kiichle et al. 74 were obtained with a O basis sets of similar quality and nearly identical MRCI correlation treatments,

27

but a more flexible Th f basis set, and are closer to the experimental values than the MP results. Medium-core AIMPs for lanthanides ([36Kr] 4d 10 core) and actinides ([54Xe] 4f14 5d 10 core) were recently published by Seijo and coworkers, who also presented results for ThO. 1 0 1 The spectroscopic constants obtained at the HF and CASSCF level are similar to those from EC-PP calculations. It remains to be seen how well these AIMPs perform for systems with partially occupied 4f and 5f shells, especially when electronic states with different f occupation number play a role. Related test calculations for CeO indicate a good transferability of the AIMP from the atom to the molecule,102 however in both Ce ( ^ X e ] 4fx 5dx 6s2 ^ 4 ground state) and CeO (ip\f CFQSJQP 3 $4 ground state) a 4fx subconfiguration on Ce is predominant. 103 A more demanding application for a Ce ECP might be cerocene Ce(C 8 H 8 )2, where Ce appears to be a mixed configuration with significant 4f° contributions besides the leading 4fx subconfiguration. 79,93

4. Selected Applications In this chapter two selected investigations using small-core 4f-in-valence and medium-core 4f-in-core EC-PPs are briefly summarized. The homonuclear lanthanide dimers proved to be quite difficult cases for both theoreticians as well as experimentalists. The most curious molecule seems to be Gd 2 for which a 4f 4f cr2 cr1 cr1 7r2 1 9 £+ ground state was theoretically predicted using 4f-in-core EC-PPs 1 0 7 and later experimentally confirmed.104 The theoretical description was refined in recent 4f-in-valence EC-PPs calculations. 105,106 At present, the best theoretical estimates obtained at the small-core EC-PP CCSD(T) level with extended basis sets and including spin-orbit corrections (Re = 2.88 ± 0.02 A, D e = 1.38 ± 0.18 eV, ue = 149 ± 2 c m - 1 ) are in fair agreement with the available experimental values (D c - 1.78 ± 0.35 eV, ue = 138.7 ± 0.4 cm" 1 ). 1 0 6 In case of the presumably simpler homonuclear dimers with empty and filled 4f shell, i.e., La 2 and Lu 2 , the originally derived spectroscopic constants 107 were in clear disagreement to the experimental values, especially the vibrational frequencies determined recently by Raman spectroscopy in Ar matrix. 108,109 A theoretical reassignment of the ground states and a determination of the molecular constants by small-core ECP P large-scale CCSD(T) and MRCI calculations yielded values in better agreement with experimental data (Tables 11 and 12). 110 It is noteworthy, however, that the calculated vibrational constant of La 2 is ^50 c m - 1 too

28 Table 11.

Spectroscopic constants of La2

method EC-PP(WB,29),CCSD/CCSD(T) EC-PP(WB,29),CCSD/CCSD(T), C P C EC-PP(WB,29),MRCI/MRCI+Q EC-PP(WB,11),CPP, M R C I / M R C I + Q Est. Exp. EC-PP(WB,29),DFT(BP)

Re (A) 2.655/2.671 2.673/2.694 2.734/2.701 2.710/2.677 2.70±0.03 2.606

*2*i %+D e (eV) 2.12/2.59 1.89/2.34 2.50/2.60 2.52/2.62 2.31±0.13 2.52±0.22 2.75

uje ( c m - 1 ) 211/198 202/188 174/183 181/191 186±13 236±1 229

Cf. Ref. 1 1 0 for details and references. Est.: theoretical estimate based on the best CCSD(T) and MRCI results including counter-poise and spin-orbit corrections. Estimated/calculated matrix effect: La2 —> Ar-La2: Acj e = 22 c m - 1 . Table 12.

Spectroscopic constants of Lu2 crg^n^u

method EC-PP(WB,41),CCSD/CCSD(T) EC-PP(WB,41),CCSD/CCSD(T), C P C EC-PP(WB,41),MRCI/MRCI+Q EC-PP(WB,11),CPP, M R C I / M R C I + Q Est. Exp. EC-PP(WB,29),DFT(BP)

Re (A) 3.043/3.034 3.064/3.058 3.108/3.072 3.094/3.083 3.07±0.03 3.123

3

^ Ar-Lu2: Aoje = 3 c m - 1 .

low, whereas for Lu2 an almost perfect agreement with the experimental values is observed. Model calculations on Ar-La2 complexes using the La medium-core EC-PP point to a quite large matrix shift which could explain about 50 % of the error. A similar behavior is found in Ce2 and Pr2, whereas due to different ground state configurations no significant matrix shift is observed for Gd 2 and Lu 2 . 106 ' 110 The lanthanide 4f-in-core EC-PPs combined with DFT allow also the theoretical study of larger metal-organic complexes. An example is an investigation of the structures, stabilities and excitation spectra of lanthanide texaphyrins (Ln-Tex 2+ , Ln = La, Gd, Lu), 111 which play an important role in such diverse and potentially beneficial areas as X-ray radiation therapy, photodynamic therapy, photoangioplasty, or magnetic resonance imaging. 112 ' 113 Texaphyrins are tripyrrolic, pentaaza macrocycles (Figure 7) that have a strong resemblance to porphyrins and other naturally occurring tetrapyrrolic groups, but their « 20 % larger core allows the formation

29

~l 2+

Fig. 7. Structure of lanthanide (III) texaphyrin ( R = 0 ( C H 2 ) 3 0 H ) and motexaphyrin ( R = 0 ( C H 2 C H 2 0 ) 3 C H 3 ) complexes.

of stable 1:1 complexes with a range of larger metal cations, such as lanthanide trications. Ten years after their first synthesis, numerous systems with varying side chains have been investigated, some of them reaching even the stage of clinical trials (Gd-Tex, XCYTRIN; Lu-Tex, LUTRIN). The lanthanide texaphyrins are found to accumulate in cancerous lesions and the Gd compound can be easily localized by magnetic resonance imaging techniques. It is believed that Gd-Tex + + captures electrons formed as a result of the interaction of X-rays with water at nearly diffusion rates and thus, in the absence of oxygen, leads to an augmented concentration of hydroxyl radicals, which are important and well-known cytotoxins in X-ray radiation therapy. On the other hand, in the presence of oxygen, the electron capture product Gd-Tex + reacts with oxygen to form superoxide anion as a reactive oxygen species via a fast equilibrium. It is hypothesized that Gd texaphyrins in combination with X-rays leads in vivo to a cascade-like cell killing process. DFT calculations using the B3LYP functional yield quite high electron affinities of Ln-Tex2+ (La 7.54 eV, Gd 7.56 eV, Lu 7.58 eV). The nearly metal-independent value is in accordance with the fact, that the Ln-Tex 2+ LUMO is mainly localized on the macrocyclic ring, not on the metal. Gd, Tb, Ce, La and Lu were found in the calculations to form the most stable complexes. The Ln-Ns-plane distances reflect the lanthanide

30

400

500 600 Wavelength (nm)

700

800

Fig. 8. Absorption spectrum of gadolinium (III) texaphyrin from time-dependent D F T calculations. 1 1 1 The bars correspond to the calculated transition energies and intensities, the solid line was generated from a superposition of Gaussians applying some artificial broadening. The dashed curve was derived from the experimental A m a x (loge) values for the system in methanol in the same manner.

contraction (calculated gas phase (experimental X-ray) structure: La 0.73 (0.91) A, Gd 0.34 (0.61) A, Lu 0.02 (0.27) A). An interesting property of texaphyrins is their electronic excitation spectrum, which is dominated by two characteristic bands at « 480 mn and « 730 nm (Figure 8). Especially the absorption in the far-red portion of the visible spectrum, where blood and bodily tissues are most transparent (> 700 nm) is ideal for an effective photosensitizer. Time-dependent DFT yields maxima of « 454 - 462 mn and « 681 - 686 nm for the La, Gd and Lu texaphyrins. Whereas the high-energy part of the spectrum is quite insensitive with respect to the ligation of the complex, the important low-energy part is not. Since the actual ligation of Ln-Tex 2+ in cancerous tissue is not known, model calculations with one bidentate NO^~ and two monodentate C H 3 0 ~ counter ions yielding the experimentally observed 9-fold coordination of Gd(III) in Gd-TexL^* were performed. The low-energy maximum of the spectrum is shifted by « 150 nm. The in vivo situation of texaphyrin gadolinium (III) is probably intermediate to the limiting situations Gd-Tex 2+ and Gd-TexL^", thus explaining the far-red absorption.

31

5. Conclusions and Outlook The ab initio EC-PP approach for lanthanides (n=6) and actinides (n=7) briefly summarized here proved to be quite successful in various applications in the past. Nevertheless, the present status of the small-core (valence shells (n-2)spdf and higher) f-in-valence approach is not completely satisfactory. Future parametrizations of these PPs should be based exclusively on MCDHF AE reference data obtained with the DCB Hamiltonian. Although the programs for such an adjustment already exist, the actual fitting is a tremendous effort due to the very large number of J levels resulting for many of the configurations. It has to be investigated if all J levels should enter with equal weight or if the lowest J levels of each configuration should have higher importance. Hereby also higher ionized reference configurations ( M m + , m=2,3,4 and possibly m=-l) than considered in the present parameterization (M and M + ) should be taken into account. Core-valence correlation is important for very accurate calculations. Effective medium-core CPPs attributing the (n-2)spd shells to the core should be derived for small-core PPs in order to decrease the computational effort in the correlation treatment. In addition effective medium-core (valence shells (n-2)f (n-l)spd and higher) and/or even large-core (valence shells (n-2)f (n-l)d and nsp and higher) spin-orbit operators should be adjusted together with scalar-quasirelativistic EC-PPs. Here, especially for actinides, separate spin-orbit operators suitable for a perturbative and a variational treatment should be provided in order to account properly for the significant relaxation of the radial functions under the spin-orbit operator. The above mentioned readjustment is not necessary for the lanthanide f-in-core PPs due to their approximate nature. Similar PPs should also be constructed for the heavier actinides, where the 5f shells are sufficiently core-like. Similar to the corresponding lanthanide PPs effective CPPs should be provided to account for the large polarizability of the 5f shell. For all PPs ANO basis sets of different quality should be generated in order to allow for basis set extrapolations also for molecules in terms of the cardinal quantum number. Especially for the small-core PPs it should be investigated if removing the singularity —Q/r of the potential near the nucleus will limit the magnitude of the required basis set exponents.

Acknowledgments The authors are grateful to Fonds der Chemischen Industrie for support.

32

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.

H. Hellmann, J. Chem. Phys. 3, 61 (1935). P. Gombas, Z. Phys. 94, 473 (1935). J. D. Weeks, A. Hazi and S. A. Rice, Adv. Quant. Chem. 16, 283 (1969). J. N. Bardsley, Case studies in atomic physics 4, 299 (1974). R. N. Dixon and I. L.Robertson, Spec. Period. Rep., Theor. Chem., Vol. 3, (The Chemical Society, London, 1978), p. 100. A. Hibbert, Adv. Atom. Molec. Phys. 18, 309 (1982). K. S. Pitzer, Int. J. Quant. Chem. 25, 131 (1984). L. R. Kahn, Int. J. Quant. Chem. 25, 149 (1984). M. Krauss and W. J, Stevens, Ann. Rev. Phys. Chem. 35, 357 (1984). P. A. Christiansen, W. C. Ermler and K. S. Pitzer, Ann. Rev. Phys. Chem. 36, 407 (1985). W. C. Ermler, R. B. Ross and P. A. Christiansen, Adv. Quantum Chem. 19, 139 (1988). C. Laughlin and G. A. Victor, Adv. At. Mol. Phys. 25, 163 (1988). O. Gropen, in: Methods in Computational Chemistry, Vol. 2, Ed. S. Wilson (Plenum, New York, 1988), p. 109. W.E. Pickett, Comput. Phys. Rep. 9, 115 (1989). S. Huzinaga, J. Mol. Struct. (Theochem) 234, 51 (1991). J. R. Chelikowsky and M. L. Cohen, in: Handbook on Semiconductors, Vol. 1, Ed. P. T. Landsberg (Elsevier, Amsterdam, 1992), p. 59. S. Huzinaga, Can. J. Chem. 73, 619 (1995). G. Prenking, I. Antes, M. Bohme, S. Dapprich, A.W. Ehlers, V. Jonas, A. Neuhaus, M. Otto, R. Stegmann, A. Veldkamp and S. F. Vyboishchikov, Rev. Comp. Chem. 8, 63 (1996). T. R. Cundari, M. T. Benson, M. L. Lutz and S. O. Sommerer, Rev. Comp. Chem. 8, 145 (1996). K. Balasubramanian, in: Encyclopedia of Computational Chemistry, Eds. P. v. R. Schleyer, N. L. Allinger, T. Clark, J. Gasteiger, P. A. Kollman, H. F. Schaefer III and P. R. Schreiner (Wiley, Chichester, 1998), p. 2471. P. Pyykko and H. Stoll, in: R.S.C. Spec. Period. Rep., Chemical Modelling, Applications and Theory, Vol. 1, 2000, p. 239. L. Seijo and Z. Barandiaran, in: Computational Chemistry: Reviews of Current Trends, Vol. 4, Ed. J. Leszczynski (World Scientific, Singapore, 1999), p. 55. M. Dolg, in: Modern Methods and Algorithms of Quantum Chemistry, Ed. J. Grotendorst, NIC Series, Vol. 1 (John Neumann Institute for Computing, Jiilich, 2000), p. 479; Vol. 3 (Julich, 2000), p. 507. H. Stoll, B. Metz and M. Dolg, J. Comput. Chem. 23, 767 (2002). P. Schwerdtfeger, in: Progress in Theoretical Chemistry and Physics: Theoretical chemistry and physics of heavy and superheavy elements, Eds. U. Kaldor and S. Wilson (Kluwer, 2003, in press). B. A. Hefi and M. Dolg, in: Relativistic Effects in Heavy-Element Chemistry and Physics, Ed. B. A. Hefi (Wiley, Chichester, 2002), p. 89.

33 27. M. Dolg, in: Relativistic Electronic Structure Theory, Part 1, Fundamentals, Ed. P. Schwerdtfeger (Elsevier, Amsterdam, 2002), p. 793. 28. K. S. Pitzer, Ace. Chem. Res. 12, 271 (1979). 29. P. Pyykko and J.-P. Desclaux, Ace. Chem. Res. 12, 276 (1979). 30. P. Pyykko, Adv. Quant. Chem. 11, 353 (1978). 31. W. H. E. Schwarz, Phys. Scr. 36, 403 (1987). 32. W. Kutzelnigg, Phys. Scr. 36, 416 (1987). 33. K. Balasubramanian and K. S. Pitzer, in: Ab initio Methods in Quantum Chemistry, Vol. 1, Ed. K. P. Lawley (Wiley, New York, 1987), p. 287. 34. P. Pyykko, Chem. Rev. 88, 563 (1988). 35. G. Malli, in: Molecules in Physics, Chemistry and Biology, Vol. 2, Ed. J. Maruani (Kluwer, Dordrecht, 1988), p. 85. 36. K. Balasubramanian, J. Phys. Chem. 93, 6585 (1989). 37. W. H. E. Schwarz, in: Theoretical Models of Chemical Bonding, Vol. 2, The Concept of the Chemical BondEd. Z. B. Maksic (Springer, Berlin, 1990), p. 593. 38. B. A. Hefi, C. M. Marian and S. Peyerimhoff, in: Advanced Series in Physical Chemistry: Modern Electronic Structure Theory, Vol. 2, Ed. D. R. Yarkony (World Scientific, 1995), p. 152. 39. J. Almlof and O. Gropen, in: Reviews in Computational Chemistry, Vol. 8, Eds. K. B. Lipkowitz and B. D. Boyd (VCH Publishers, New York, 1996). 40. M. Dolg and H. Stoll, in: Handbook on the Physics and Chemistry of Rare Earths, Vol. 22, Eds. K. A. Gschneidner Jr. and L. Eyring (Elsevier, Amsterdam, 1996), p. 607. 41. B. A. Hefi, Ber. Bunsenges. 101, 1 (1997). 42. B. A. Hefi, in: The Encyclopedia of Computational Chemistry, Eds. P. v. R. Schleyer, N. L. Allinger, T. Clark, J. Gasteiger, P. A. Kollman, H. F. Schaefer III, P. R. Schreiner (Wiley, Chichester, 1998), p. 2499. 43. B. A. Hefi, Ed., Relativistic Effects in Heavy-Element Chemistry and Physics (Wiley, Chichester, 2002). 44. P. Schwerdtfeger, Ed., Relativistic Electronic Structure Theory, Part 1, Fundamentals (Elsevier, Amsterdam, 2002). 45. P. Pyykko, Relativistic Theory of Atoms and Molecules. A Bibliography 1916 -1985, in: Lecture Notes in Chemistry, Vol. 41 (Springer, Berlin, 1986). 46. P. Pyykko, Relativistic Theory of Atoms and Molecules II. A Bibliography 1986 -1992, in: Lecture Notes in Chemistry, Vol. 60 (Springer, Berlin, 1993). 47. P. Pyykko, Relativistic Theory of Atoms and Molecules HI. A Bibliography 1993 - 1999, in: Lecture Notes in Chemistry, Vol. 76 (Springer, Berlin, 2000). 48. P. Pyykko, Relativistic Theory of Atoms and Molecules (RTAM) database 1915 - 2001, http://www.csc.fi/lul/rtam/rtamquery.html 49. W. C. Martin, R. Zalubas and L. Hagan, Atomic Energy Levels - The Rare Earth Elements, NSRDS-NBS 60 (U.S. Dept. of Commerce, Washington DC, 1978). 50. J. Blaise and J.-F. Wyart, Energy Levels and Atomic Spectra of Actinides, in: International Tables of Selected Constants, Vol. 20, (CNRS, Paris, 1992). 51. P. Pyykkoe, Inorg. Chim. Acta 139, 243 (1987).

34 52. M. Pepper and B. Bursten, Chem. Rev. 91, 719 (1991). 53. M. Dolg, in: The Encyclopedia of Computational Chemistry, Eds. P. v. R. Schleyer, N. L. Allinger, T. Clark, J. Gasteiger, P. A. Kollman, H. F. Schaefer III, P. R. Schreiner (Wiley, Chichester, 1998), p. 1478. 54. G. Schreckenbach, P. J. Hay and R. L. Martin, J. Comput. Chem. 20, 70 (1999). 55. L. R. Kahn, P. Baybutt and D. G. Truhlar, J. Chem. Phys. 65, 3826 (1976). 56. L. E. McMurchie and E. R. Davidson, J. Comput. Chem. 4, 289 (1981). 57. R. M. Pitzer and N. W. Winter, J. Phys. Chem. 92, 3061 (1988). 58. R. M. Pitzer and N. W. Winter, Int. J. Quant. Chem. 40, 773 (1991). 59. C.-K. Skylaris, L. Gagliardi, N. C. Handy, A. G. Ioannou, S. Spencer, A. Willetts and A. M. Simper, Chem. Phys. Lett. 296, 445 (1998). 60. M. Pelissier, N. Komiha and J.-R Daudey, J. Comput. Chem. 9, 298 (1988). 61. Y. Wang and M. Dolg, Theor. Chem. Ace. 100, 124 (1998). 62. W. Miiller, J. Flesch and W. Meyer, J. Chem. Phys. 80, 3297 (1984). 63. P. Fuentealba, H. Preuss, H. Stoll and L.v. Szentpaly, Chem. Phys. Lett. 89, 418 (1982). 64. P. Fuentealba, J. Phys. B: At. Mol. Phys. 15, L555 (1982). 65. L. V. Szentpaly, P. Fuentealba, H. Preuss and H. Stoll, Chem. Phys. Lett. 93, 555 (1982). 66. J. Flad, G. Igel, M. Dolg, H. Stoll and H. Preufi, Chem. Phys. 75, 331 (1983). 67. A. Savin, M. Dolg, H. Stoll, H. Preufi and J. Flesch, Chem. Phys. Lett. 100, 455 (1983). 68. H. Stoll, P. Fuentealba, M. Dolg, J. Flad, L. v. Szentpaly and H. Preufi, J. Chem. Phys. 79, 5532 (1983). 69. G. Igel, U. Wedig, M. Dolg, P. Fuentealba, H. Preufi, H. Stoll and R. Frey, J. Chem. Phys. 8 1 , 2737 (1984). 70. P. Schwerdtfeger and H. Silberbach, Phys. Rev. A 37, 2834 (1988), erratum: ibidem, 42, 665 (1990). 71. M. J. Smit, Int. J. Quant. Chem. 73, 403 (1999). 72. U. Wedig, M. Dolg and H. Stoll, in: Quantum Chemistry: The Challenge of Transition Metals and Coordination Chemistry, Ed. A. Veillard, NATO ASI Series, Series C, Mathematical and Physical Sciences, Vol. 176 (Reidel, Dordrecht, 1986), p. 79. 73. atomic structure code GRASP; K. G. Dyall, I. P. Grant, C. T. Johnson, F. A. Parpia and E. P. Plummer, Comput. Phys. Commun. 55, 425 (1989). 74. W. Kiichle, M. Dolg, H. Stoll and H. Preuss, J. Chem. Phys. 100, 7535 (1994). 75. M. Dolg, H. Stoll and H. Preufi, J. Chem. Phys. 90, 1730 (1989). 76. X. Cao and M. Dolg, J. Chem. Phys. 115, 7348 (2001). 77. X. Cao, M. Dolg and H. Stoll J. Chem. Phys. 118, 487 (2003). 78. M. Dolg, H. Stoll, A. Savin and H. Preufi, Theor. Chim. Acta 75, 173 (1989). 79. M. Dolg, P. Fulde, W. Kiichle, C.-S. Neumann and H. Stoll, J. Chem. Phys. 94, 3011 (1991). 80. M. Dolg, H. Stoll and H. Preuss, Theor. Chim. Acta 85, 441 (1993).

35 81. 82. 83. 84. 85. 86. 87. 88. 89. 90. 91. 92. 93. 94. 95.

96. 97. 98. 99. 100. 101. 102. 103. 104. 105. 106. 107. 108. 109. 110. 111. 112. 113.

M. Dolg, J. Chem. Phys. 104, 4061 (1996). M. Dolg, Chem. Phys. Lett. 250, 75 (1996). J. Flad, H. Stoll and H. Preuss, J. Chem. Phys. 71, 3042 (1979). M. Dolg, U. Wedig, H. Stoll and H. Preuss, J. Chem. Phys. 86, 866 (1987). D. Andrae, U. Haufiermann, M. Dolg, H. Stoll and H. Preufi, Theor. Chim. Acta 77, 123 (1990). A. Bergner, M. Dolg, W. Kiichle, H. Stoll and H. Preuss, Mol. Phys. 80, 1431 (1993). M. Dolg, H. Stoll, H. Preuss and R. M. Pitzer, J. Phys. Chem. 97, 5852 (1993). X. Cao and M. Dolg, J. Molec. Struct. (THEOCHEM) 581, 139 (2002). X. Cao and M. Dolg, to be published (2003). X. Cao and M. Dolg, Mol. Phys. (Judd-Ofelt Theory issue), in press (2003). X. Cao and M. Dolg, Chem. Phys. Lett. 349, 489 (2001). X. Cao, J. Chin. Chem. Soc, submitted (2003). M. Dolg, P. Fulde, H. Stoll, H. Preufi, R. M. Pitzer and A. Chang, Chem. Phys. 195, 71 (1995). M. Dolg, H. Stoll, H.-J. Flad and H. Preufi, J. Chem. Phys. 97, 1162 (1992). K. P. Huber and G. Herzberg, Molecular Spectra and Molecular Structure, Vol. 4, Constants of Diatomic Molecules (Van Nostrand Reinhold, New York, 1979). M. Dulick, E. Murad and R. F. Barrow, J. Chem. Phys. 85, 385 (1986). R. S. Ram and P. F. Bernath, J. Chem. Phys. 104, 6444 (1996). D. L. Hildenbrand and K. H. Lau, J. Chem. Phys. 102, 3769 (1995). J. K. Laerdahl, K. Faegri, L. Visscher and T. Saue, J. Chem. Phys. 109, 10806 (1998). C. M. Marian, U. Wahlgren, O. Gropen and P. Pyykko, J. Mol. Struct. (Theochem) 169, 339 (1988). L. Seijo, Z. Barandiaran and E. Harguindey, J. Chem. Phys. 114, 118 (2001). S. Diaz-Megias and L. Seijo, Chem. Phys. Lett. 299, 613 (1999). M. Dolg, H. Stoll and H. Preufi, J. Mol. Struct. (Theochem) 231, 243 (1991). R. J. van Zee, S. Li and W. Weltner, J. Chem. Phys. 100, 994 (1994). M. Dolg, W. Liu and S. Kalvoda, Int. J. Quant. Chem. 76, 359 (2000). X. Cao and M. Dolg, Mol. Phys. (B. O. Roos issue), accepted (2003). M. Dolg, H. Stoll and H. Preufi, J. Mol. Struct. (Theochem) 277, 239 (1992). Y. Liu, L. Fang, X. Shen, X. Chen, J. R. Lombardi and D. M. Lindsay, Chem. Phys. 262, 25 (2000). L. Fang, X. Chen, X. Shen and J. R. Lombardi, J. Chem. Phys. 113, 10202 (2000). X. Cao and M. Dolg, Theor. Chem. Ace. 108, 143 (2002). X. Cao and M. Dolg, Mol. Phys. (W. G. Richards issue), accepted (2003). J. L. Sessler, G. Hemmi, T. D. Mody, T. Murai, A. Burrell and S. W. Young, Ace. Chem. Res. 27, 43 (1994). T. D. Mody and J. L. Sessler, in: Supramolecular Materials and Technologies, Vol. 4, Ed. D. N. Reinhoudt (Wiley, Chichester, 1999), p. 245.

This page is intentionally left blank

RECENT DEVELOPMENTS OF RELATIVISTIC MODEL CORE POTENTIAL METHOD

E. MYOSHI AND Y. SAKAI Graduate School of Engineering Sciences Kyushu University 6-1 Kasuga Park, Fukuoka 816-8580, Japan E-mail: miyoshi@asem. kyushu-u. ac.jp Y. OSANAI Aomori University Aomori 030-0943, Japan E-mail: [email protected] T.NORO Division of Chemistry Graduate School of Science Hokkaido University Sapporo 060-0810, Japan E-mail: tashi@sci. hokudai. ac.jp

We present in this paper some recent developments of relativistic model core potential (MCP) method. We briefly outline the determination procedure of the MCP parameters and valence basis sets. Some characteristics of MCPs and valence basis sets for the lanthanide elements recently developed are discussed and the applications to the ground and low-lying excited states of diatomic molecules containing lanthanide elements are reported. The calculated results show that the MCP method can predict quantitatively not only the ground state properties but also the term values of low-lying excited states. We also outline the development of generally contracted Gaussian-type function sets for relativistic correlating functions of Ga-Kr, In-Xe, and Tl-Rn atoms. The correlating functions are constructed using atomic natural orbital approach based on scalar relativistic CI calculations with MCPs and are applied to several atoms and the T1H molecule. The spectroscopic constants of the T1H molecule yielded by relativistic MCP calculations including spin-orbit effects are in excellent agreement with the experimental results. It is demonstrated that the MCP method can properly evaluate the valence correlation owing to the nodal structures of the associated valence orbitals, and can appropriately include the relativistic effects.

37

38

1.

Introduction

The model potential method 18 proposed by Huzinaga and co-workers is unique among various effective core potential (ECP) methods in that it is capable of producing valence orbitals with nodal structures and thus suitable to accurately describe the correlation effects of valence electrons. In the last decade, we (YS and EM) have developed model potentials for various elements,9'12 while Seijo and co-workers 6 and Katsuki and co-workers 7 developed model potentials respectively. We have designated our model potential method "Model Core Potential (MCP)" instead of MP, to eliminate any confusion with M0llerPlesset (MP) perturbation theory,11 and Seijo and co-workers designated their method "ab initio model potential (AMP)". Our model potentials have been used in a variety of molecular calculations, and it has been confirmed that the method yield reliable results constantly.13"31 We have developed MCPs for main group elements treating only the valence ns and np electrons explicitly, except for alkali and alkaline-earth metal elements, for which the (n-l)p electrons were also treated explicitly in addition to the ns electrons.10 In some cases, however, it is important to take into account the correlation of inner-shell electrons for accurate calculations of spectroscopic constants of molecules: equilibrium distances, vibrational frequencies, and binding energies. Thus, we developed advanced relativistic model core potentials (dsp-MCPs) for the fourth-, fifth-, and sixth-row main group elements (Ga~Kr , In~Xe, and Tl~Rn), where the outermost core (n-\)d electrons were explicitly treated in addition to the ns and np valence electrons and the remaining core electrons were replaced by MCPs.11 The major relativistic effects were incorporated in the MCPs at the level of Cowan and Griffin's quasi-relativistic Hartree-Fock (QRHF) method.32 For transition-metal atoms, two types of MCPs have been developed: one is li

M2

Ai(2)

A2(2)

6

Bi

Bi(l)

1

0

v^(0

v^(0

1

1

0

0

1

1

0

1

0

5

o i(0

i(0 o

i

i(0

i(0

0

I

1(0

1(0

V2

0

0

B2,B2(l)

Mi

B2(l)

Ai(0)

o

Ai,Ai(2)

Bi5Bi(l)

B2

0

M2,A2(2)

5

5

y/T/3 -7173 V*73(0

°

- > / 3 ( r ) >/3(r)

0

The Wigner-Eckart factors are calculated with respect to matrix elements between spin functions with Ms = S. Exceptions are cases marked by (I) and (r), indicating spinfunctions (left or right) to which the S- operator should be applied to obtain the reference matrix element (see text).

with —Ms. For example, the E\ subspace for sextets has Ms values of 5/2, 1/2 and - 3 / 2 for the A1 and A2 functions and 3/2, - 1 / 2 and - 5 / 2 for those of Bi and B2 symmetry. The opposite association is made for the E2 subspace. The underlying reason for this division of basis functions can be found in Table 3. The lz operator only connects A\ with A2 functions and Bi with B2. Since sz has a AM5 = 0 selection rule, it is not possible to have non-zero spin-orbit matrix elements coupling Ai and Bi (i = 1,2) functions when they have the same Ms value. On the other hand, in order to have non-zero x or y spin-orbit matrix elements, it is necessary to have one function of Ai type and one of Bi type, again as shown in Table 3. Since sx and sy both are characterized by AMS = ± 1 , this means again that the Ai function must have the opposite Ms value as its Bi counterpart in order for there to be a non-zero result. The result of employing this division of basis functions is to define two CI secular equations which always lead to the same set of energy eigenvalues. If transition moments are to be calculated, it is nonetheless necessary to solve both secular problems to obtain the corresponding distinct sets of eigenvectors, as will be discussed in Sec. 3. The calculated Wigner-Eckart factors for doublet and quartet states for both the Ei and E2 subspaces are given in Table 5 and for sextets and septets in the Appendix.

Table 5. Wigner-Eckart factors for spin-orbit matrix elements between d State

2

2

Ai

±1/2 2

A2(±l/2)

2

Bi(=Fl/2)

Bi

±1/2

if 1/2

±1

0

Tl(r)

l(r)

0

2

1(0

Tl(r)

Tl

MI(=F3/2)

0

0

(±1/2)

0

A 2 (=f3/2)

0

B2(Tl/2)

4

4

B2

=fl/2

4

A2

Ml ^3/2

±1/2

=F3/2

1

Tl

0

l(i)

073

±\/!73

0

0

=pl

1

0

/

y/T/3

Tl

0 0

0

±1/3

0

1(0

Bi(Tl/2)

yi/3

T\A73

0

1(0

1

±1

0

0

0

±1(0

y/T/3

1(0

0

1(0

v / V3(0

Tl(0

0

0

1(0

0

#2(Tl/2) (±3/2)

±

0

(±1/2)

(±3/2) 4

2

0

^4i(±l/2)

2

2

A2

T y ^ ±

1

1

±v V3

0

0

qFl(0 =F>/4/3(0

1(0

y/

0

±v^ ±

The Wigner-Eckart factors are calculated with respect to matrix elements between spin cases marked by (0 and (r), indicating spin-functions (left or right), to which the Sreference matrix element. The upper and lower signs refer to the E\ and E2 subspaces,

76

It should be emphasized that the Wigner-Eckart factors used in the present study are determined with respect to standard reference matrix elements (between Ms = S functions, if possible) rather than in terms of reduced matrix elements {Ms independent), as is commonly the case. The reason for this, as already mentioned, is that the Ms = S functions are employed in the conventional CI procedure. The Wigner-Eckart factors are tabulated in this form in the programs and used to obtain the final ( | if s o I ) matrix elements. The computational techniques employed to calculate the spin-orbit Hamiltonian matrix elements make use of a phase convention introduced by Esser, Butscher and Schwarz13 in order to guarantee that only real values are obtained (see also Ref. [6]). Accordingly, the Ax and Bx spatial functions are assumed to be the same as in the A — S calculations, whereas those belonging to the A2 and B2 IRs are multiplied with the imaginary unit i (i2 = — 1). Only the y component of the spin-orbit operator is able to couple A{ and Bi functions within each set, that is, A\ with Bi and A2 with B2. Because the Pauli matrix for sy is imaginary, the corresponding component of the spin-orbit operator is real. By the above construction, the product of any two basis functions which gives a non-zero matrix element is real, so that the value itself is also real. On the other hand, the x and z components of the / • s spin-orbit operator are imaginary and they only lead to non-zero results when the two basis functions come from different sets, so that their product is imaginary as well, again as can be seen from Table 3. Table 6. Number of secular equations and character of spin-orbit matrix elements Hso f ° r various double groups. Group

Number of sec. eqs.

< \HSO\

>

Number of electrons even D

2k

r' r" c$> c2 C'i

c[

,D>2

8 4 2 2 1

odd 4 2 1 2 1

real real real complex complex

A similar approach is possible for the D2h and D'2 point groups, whereby in the former case one simply defines two subsets of four IRs of the same parity in each case. The fact that / is an even operator prevents any mixing between spatial basis functions of different parity. When the molecule

77

of interest belongs to either of the C8 and C2 groups, a variation of the above procedure can also be used which results in exclusively real matrix elements. In the former case, for example, one can choose the xz plane as the symmetry element. In that case, the A\ and B\ IRs of Ch(=F3/2)

6

^3/2

±1/2

±5/2

=F3/5

0

0

0

(±1/2)

0

±1/5

0

0

0

(±5/2)

0

0

±1

0

0

0

=Fl

0

0

1

0

0

B!(=F5/2)

F

(=Fl/2)

6

6

e

State

/

(±3/2)

0

±v 8?5

=Fl

B2(T5/2)

i

o

o

(=Fl/2)

v/8/5

y/9jb

0

(±3/2)

0

y/8/5

1

r

=f5/2

r

0

y/S/b

y/9/5

0

0

0

y/8/5

1

0

=Fl

0

0

F

=Fl 0

0

±y/8/5

=Fl

=p

T

0

The Wigner-Eckart factors are calculated with respect to matrix elements between spi are blocks marked by (r), indicating that the S- operator should be applied to the (r)igh matrix elements. The upper and lower signs refer to the E\ and E2 subspaces, respecti

Table C. Wigner-Eckart factors for spin-orbit matrix elements between quintet and septet C2v double group. 5

State, Spin

Mi

sym.(|M 5 |)

Ai(2)

A2(2)

£i,£i(3)

1

1

0

B2:B2(3)

1

1

0

0

i(0

y/2jZ

i(0

0

Vyz

7 7

7

AuAl{2)

7

A2,A2(2)

7

Bi,Bi(l)

7

B2,B2(1)

7

A 2 ,^ 2 (0)

A2

V^l/15 -y/l/lb -v/1/15 0

y/TJlb 0

5

Bi

Bi(l)

0

y/*Jb(l)

~Vy$

5

B2

5

A,

7

B,

7

B2

Mi

B2(3)

Ai(2

Ai(0)

Bi(3)

0

0

0

0

0

1

0

\/2?3

0

i(0

i(0

0

V^/3

0

KO

1(0

2/3

V^/5

0

0

yfi/S

0

0

y/9/5(l)

0

0

B 2 (l)

^§75(0 0

y/2/5

-y/5/3

VV3 0

The Wigner-Eckart factors are calculated with respect to matrix elements between spin-fu cases marked by (0 and (r), indicating spin-functions (left or right), to which the 5_ ope reference matrix element (see text).

This page is intentionally left blank

SPIN ORBIT COUPLING METHODS AND APPLICATIONS TO CHEMISTRY

D. G. Fedorov*'1, M. W. Schmidt2, S. Koseki3, M. S. Gordon2 corresponding author, Email: [email protected] 1

National Institute ofAdvanced Industrial Science and Technology, 1-1-1 Umezono, Tsukuba, 305-6568 Ibaraki, Japan. 2 Ames Laboratory, US-DOE and Department of Chemistry, Iowa State University, Ames, IA 50011, USA. 3 Department ofMaterial Science, Osaka Prefecture University, 1-1 Gakuen-cho, Sakai, 599-8531 Osaka, Japan.

Abstract Various one and two electron spin-orbit coupling Hamiltonians are discussed, along with symmetry properties of their matrix elements. Alternatives to full all-electron treatments are given in some detail, most importantly, the effective charge approach, an approach based on model core potentials and the partial two-electron method. Assessment of the applicability of these methods is based on numerous tests on atoms and molecules. Spin-orbit multi-configuration quasi-degenerate theory (SO-MCQDPT) is briefly introduced and several applications illustrating its promising performance are given. The vibrational dependence of spin-orbit coupling constants is investigated in some diatomics. I.

Introduction

The present chapter consists of a review of our recent work on the inclusion of spin-orbit coupling (SOC) effects via perturbational treatment of the Breit-Pauli operator. Our approach typically involves either the variational treatment of spin-free relativistic effects (mass-velocity, Darwin correction) when using all electron basis sets, or their implicit inclusion through effective core potential operators. The present chapter covers both theoretical methods in Section II, and chemical applications in Section III. Some of the material overlaps with a recently submitted comprehensive review of spin-orbit coupling computations by the present authors1. That review contains a much lengthier discussion of other group's work using 2 or 4 component wavefunctions, which is a more sophisticated procedure for treating spin-orbit coupling. It also considers the work of other groups, as well as our own, on various operators useful in more commonplace single component programs, again in more detail than is given below. For readers interested in the physical origin of the spin-orbit coupling effect, the review of Marian2 contains an excellent presentation, and also covers computational procedures well. Additional reviews of spin-orbit coupling due to Hess3, Marian4, Yarkony5, and Ermler, Ross, and Christiansen6 are available. Section II briefly summarizes possible spin-orbit operators before focusing on the Breit-Pauli operator that we and many others1 have used. The use of symmetry in reducing the computational expense of spin-orbit coupling is described in some detail. This is followed by a 107

108 discussion of some effective one-electron operators used by others, and the Zeff operator suggested by the one-electron term of the Breit-Pauli operator, which we have previously explored for nearly all elements. The present chapter contains less information about the Zeff approach than has been given elsewhere1, as our more recent implementations of the full operator, as well as a judicious approximation to the two-electron term now make it feasible to work with the two-electron term included. A summary of our partial two-electron method is given and this is shown to be nearly as accurate as the full evaluation of the operator, at considerable time savings. The use of the full operator together with model core potentials, which permit evaluation of core orbital contributions to SOC, is also discussed. Section II then turns to the inclusion of spin-orbit coupling in computations that also include electron correlation beyond the complete active space (CAS) level. In the present chapter we give considerably more detail than appears elsewhere1 regarding the inclusion of the spin-orbit (SO) coupling operator as an additional perturbation on multi-configurational quasi-degenerate perturbation theory (SOMCQDPT). Our previous review1 begins with a set of experimental and theoretical examples showing the importance of SOC in chemistry for systems ranging from light to medium to heavy atoms, for main group or transition metals alike. Taking it for granted that SOC is relevant in chemistry, Section III therefore consists primarily of our own group's applications of SOC. It includes results obtained recently for transition metal hydrides using the Zeff approximation, and then turns to more accurate calculations using the full Breit-Pauli operator. Spin-orbit coupling effects are easily observed spectroscopically, even for light elements, and our work on predicting the observed vibrational state dependence of the SOC in 02+ and CO+ is summarized. The SOMCQDPT method is shown to give good results for the complex atomic spectrum of the very heavy elements Os and U. The computations for both U and the diatomic UF are reviewed in detail here. Development of our program for spin-orbit coupling has therefore produced a code capable of good results for both light and heavy elements, which means that considerable future use of this program can be expected for polyatomic cases. The applications section therefore concludes with a short discussion of some of the interesting dynamical effects of SOC, and a summary of a few of our early applications to polyatomics: heavy methylene analogs, the gas phase reaction of Ti+ with ethane, organic iodides, and binuclear couplings in bridged dititanium compounds. The great majority of the methods described herein are included in the publicly distributed quantum chemistry program GAMES S7, and it is hoped that reading this chapter allows potential users to see additional applications for the SOC modules in this code. II.

Theory and Methods

1.

general remarks A) SOC operators

In relativistic quantum chemistry the basic equation is the Dirac equation that involves a fourcomponent wavefunction. Two components called large correspond to an electron with a and p spin and the other two called small correspond to a positron, also with a and p spin. The Dirac

109 equation explicitly couples both electron and positron components, as well as a and p spins, thus it explicitly contains spin-orbit coupling. This equation when used for many-electron systems with the addition of the Coulomb potential is known as the Dirac-Coulomb equation. It is possible to manipulate the four equations to eliminate the two small component spinors corresponding to positrons, producing a two component equation. There are a number of transformations available. A few are those due to Foldy and Wouthuysen8, Douglas and Kroll (DK)910,11, Dyall12, and Nakajima and Hirao13. The Dirac Hamiltonian thus transformed usually splits into additive spin-free and spin-dependent parts. We note in passing that there are spin-dependent interactions other than spin-orbit coupling such as spin-spin coupling that we do not cover here. The spin-dependent interaction can be treated in two ways, namely variational and perturbation. Not all two-component Hamiltonians (both spin-free and spin-dependent) are variationally stable (i.e., not bound from below), meaning variational collapse can occur if both components are included into variational procedures, such as SCF orbital optimization. However, even the perturbational approach, in which the variational part is performed without spin-orbit coupling and only the matrix elements of the latter are computed, is capable of reasonably describing most chemical systems, excluding perhaps only those containing very heavy atoms. In fact such a perturbation approach is much simpler and faster computationally than the variational approach, and the former can be relatively simply added to most available programs that solve the non-relativistic Schrodinger equation. Let us now consider existing spin-orbit Hamiltonians. The simplest and the most widely used one is obtained with the Foldy-Wouthuysen transformation of the Dirac-Coulomb equations, leading to the Breit-Pauli Hamiltonian H80 operator14 given below (in a.u.):

ir-y*E E V ' - - « ' - Z E - T M '

; +2

Y

'ia

Y

i

J

»

(i)

'ij

which contains both one and two-electron terms. In these formulae, 1 and s are space and spin angular momentum operators, lia = (Y{ -Ra)x p., / = (Y{ - Y})x p., Roman and Greek subscripts refer to electrons and nuclei, respectively, and the other symbols have their usual meanings. The Breit-Pauli Hamiltonian is variationally unstable and thus must be used as a perturbation only (Hess et al.3). Matrix elements over the Breit-Pauli 7/50 operator introduce new integrals to be evaluated, related to derivatives of ordinary integrals (see, e.g., King and Furlani15, Bearpark^tf/.16). Other Hamiltonians in fact look very similar, and for example, in case of the Douglas-Kroll or relativistic elimination of small component (RESC13) Hamiltonians, they can be written as HSO

= A]HSOA;,

(2)

where A\ and A2 are operators known as kinematic multiplicative factors (A\ is equal to A2 in case of Douglas-Kroll and not equal for RESC, in the latter case the Hamiltonian is symmetrised by adding its complex conjugate). The Douglas-Kroll Hamiltonian is considered variationally stable, whereas the RESC Hamiltonian was found to be variationally unstable17. The above kinematic

110 factors tend to reduce the magnitude of spin-orbit coupling. Since they are diagonal in the momentum representation they have a simple scaling effect and do not change important properties of the Hamiltonian, such as symmetry (an exception is variational stability, which they directly affect). Kinematic factors can be constructed at various orders in the original transformation. So far practical implementations of these SOC Hamiltonians have often been based on a lower order than the spin-free Hamiltonian. This is justified due to small numerical difference (unless very heavy atoms are considered) compared to significant work to keep the same order in the spin-free and spin-orbit terms. In case of Douglas-Kroll one often uses the kinematic factors from the 1st order DK transformation. In case of RESC or Normalised elimination of small components (NESC12) the full 2nd order operator is implemented into GAMESS, however, only for the one-electron terms. Let us now consider properties of the spin-orbit coupling Hamiltonian, which are common to all Hamiltonians considered. Symmetry properties for both spin and orbital rotations are considered below in Section IB. The Breit-Pauli spin-orbit coupling operator defined in Eq. 1 contains one and two-electron terms. The computational expense of evaluating these two operators is considerably different, by one order of magnitude. In most cases, the contributions due to the one and two-electron SOC have opposite sign. An exception was found by Fedorov et a/.18 in the A2I1U state of 0 2 + , where both contributions are small and there is a region of the internuclear distance in which both have the same sign. A very important question is how the magnitude of spin-orbit coupling changes with the nuclear charge. The two-electron term of the Breit-Pauli operator grows slowly as the nuclear charge Z increases. In contrast, the one-electron term contains Z both explicitly, and implicitly through the ria3 reciprocal, and thus grows rapidly with Z. This suggests an approximation in which the two-electron terms are omitted, and compensating for this by treating Z as an adjustable parameter Zeff. One-electron approximations are considered in some detail in Section 1C. It may be desirable for the molecular orbitals (MOs) used to construct the bra and ket to be different, in case these two states have different physical natures, for example, one might be a Rydberg state and another a valence state. In this case, the two sets of orbitals can be made biorthogonal to reduce computational costs. The use of such corresponding orbitals (King et a/.19) with identical core orbitals (doubly occupied space in all configurations) does not appreciably complicate matrix element evaluation for the case of CASSCF (or full configuration interaction (CI), FCI) wavefunctions (Lengsfield et al.20). It is possible to use two separately and fully optimized orbital sets for the RASSCF generalization of CASSCF without too much extra work (Malmqvist et al.21), again using biorthogonal orbitals. General CI states should be limited to a common set of orbitals in order to avoid the great increase in time needed to deal with nonorthogonal orbitals. B) Symmetry in SOC A thorough review of the symmetry properties of if0 was given by Fedorov and Gordon22. In atoms, instead of orbital (L) and spin (S) angular momenta, the total momentum J=L+S is conserved. The symmetry group of atoms is Kh with or without SOC. In the case of molecules, however, the symmetry is described by either point groups (integer spin) or double groups (halfinteger spin). More details on the use of double groups in SOC calculations can be found in Marian2.

111 For practical purposes it is convenient to formulate selection rules for the SOC matrix elements. The rules differ somewhat, depending on how one constructs the bra and ket states. The most convenient approach is to take the "ZS" states coming from computations with the spin-free Hamiltonian as the basis states. Then one can easily add code for computing their SOC matrix elements to a conventional quantum chemistry program. There is some advantage in the alternative of taking linear combinations of the "LS" terms, for example, by constructing such combinations that transform as the irreducible representations of the double groups. This allows block diagonalization of the SOC Hamiltonian matrix that can be useful for large spin-orbit CI (SOCI) calculations. Another choice is to make the Hamiltonian matrix real by taking linear combinations that have definite time reversal symmetry (not possible for some point groups). In any case, such linear combinations reduce the diagonalization cost but ultimately one still has to compute matrix elements for the "LS" terms. The general SOC matrix element can be written as: ,pt

» » «

# - GA

T R A N S G R E S S I N G THEORY B O U N D A R I E S : T H E GENERALIZED D O U G L A S - K R O L L T R A N S F O R M A T I O N

Alexander Wolf, Markus Reiher, Bernd Artur Hess Lehrstuhl fur Theoretische Chemie, Universitdt Erlangen-Niirnberg, Egerlandstrafie 3, D-91058 Erlangen, Germany { alexander. wolf, markus.reiher,hess] @chemie.uni- erlangen. de

The fundamental theory underlying relativistic quantum chemistry is quantum electrodynamics (QED). For almost all chemical situations, however, electronic interactions with the radiation field or with the positronic degrees of freedom, which are inherently included in QED, are of negligible importance. Relativistic effects are hence very satisfactorily described by the traditional 4-component Dirac equation, which is currently computationally still too expensive for most systems of chemical interest, i.e., for molecules with more than two heavy nuclei. It is thus highly desirable to further reduce the computational requirements. This may most conveniently be achieved by a transition to two-component formulations, either by elimination techniques for the small component or by suitably chosen unitary transformations which decouple the Hamiltonian. Only these further simplifications of the theoretical framework enable a transgression of traditional theory boundaries, opening relativistic quantum chemistry to a far larger class of systems by application of 2-component and 1-component electron-only theories. After a brief discussion of the most important reduction techniques for the Dirac equation, we focus on the generalized Douglas-Kroll (DK) transformation. Its central idea is the expansion of the Hamiltonian in even terms of definite order in the external potential, achieved by the application of a sequence of unitary transformations, which eliminate the odd terms of the Hamiltonian step by step. For this purpose, the most general parametrization of the unitary matrices is used, and subsequently applied in order to derive the fifth-order approximation. While DKH2 - DKH4 are independent of the parametrization of the unitary matrices, DKH5 turns out to be dependent on the third expansion coefficient of the innermost unitary transformation which is carried out after the initial free-particle Foldy-Wouthuysen transformation. The freedom in the choice of this expansion coefficient may be employed to seek for

137

138

an optimum unitary transformation. Various DK approximations, DKH1 to DKH5, are applied to both one-electron hydrogen-like ions and many-electron atomic systems in order to investigate their numerical performance. Whilst the higher-order approximations yield remarkable accuracy for the one-electron systems as compared to the exact Dirac ground state energy, the deviations of the energy from Dirac-Fock results is systematically increasing for many-electron calculations. This behavior is due to the complete neglect of spin-dependent terms and of the transformation of the two-electron terms in the standard protocol of the DK transformation. 1. Introduction Almost a century ago, Albert Einstein discovered the special theory of relativity, which enabled a unified description of Newtonian mechanics and electrodynamics. For high-energy phenomena, e.g., in the the case of fastmoving particles, predictions due to the special theory of relativity deviate significantly and in many cases even qualitatively from those based on classical, i.e., non-relativistic mechanics. In 1928, only four years after the triumphal birth of quantum mechanics, Dirac merged the basic principles of quantum theory and special relativity and constituted relativistic quantum mechanics 1 . In the following two decades the quantization of the Dirac theory and a series of subsequent developments lead to a consistent quantum theory of the radiation field interacting with matter, which is nowadays commonly dubbed quantum electrodynamics (QED). An equally exciting and instructive survey over the history of its genesis as well as a collection of amusing anecdotes about the pioneering scientists who developed it may be found in the book of Schweber2. Though relativity has been an everyday's concept for physicists since those early days, it was not before the seventies of the last century that the importance of relativistic effects for atomic and molecular systems, the natural fields of interest for chemists, was realized. There are at least two plausible reasons for this delay: On the one hand this might be due to the fact that most organic compounds only contain elements of the first two rows of the periodic table, and are hence sufficiently well characterized by a non-relativistic description based on the Schrodinger equation. Nonetheless, even for these cases there are some subtle effects requiring a relativistic treatment, which may be important for the interpretation of highly accurate spectroscopic data. On the other hand, relativistic treatments require huge computational resources as compared to non-relativistic methods, and neither the computational techniques nor the resources were available be-

139

fore the late seventies. Heavy-element compounds and highly charged ionic systems, however, are strongly governed by relativistic effects, and must necessarily be described by relativistic quantum theory even for qualitative predictions. The shift of the main absorption bands of gold, which is in consequence responsible for its golden color, is one of the most famous examples of the importance of relativity in chemistry. Contemporary chemistry is aware of a large number of such intrinsically relativistic systems, whose theoretical description requires an extension of the framework of nonrelativist ic quantum mechanics. Meanwhile, relativistic quantum chemistry is a well-established part of theoretical chemistry, which is also documented by the availability of excellent and detailed reviews about this field3'4'5'6'7. Furthermore, an exhaustive overview over the literature dealing with relativistic quantum chemistry has been provided by Pyykko 8 ' 9 ' 10 , and can also be found online on the world-wide web 11 . Relativistic quantum chemistry is based on the fundamental theory of QED. From today's point of view it is fair to say that QED appears to be the final theory for all electromagnetic processes, for which other fundamental interactions, i.e., weak, strong, and gravitational forces, may be neglected. This perspective is supported by innumerable low- and high-energy scattering experiments of charged particles and photons and spectral analyses. It can account for nearly all observed phenomena from macroscopic scales down to about 10~ 13 cm, and is based on the classical Lagrangian density £ Q E D — ^rad.

+

= -^F^Ff>»

int.

+ V # < $ - rhc2)^

- l^A"

+ A***').

(1)

The first term (£ ra d.) describes the electromagnetic degrees of freedom based on the four-potential AM, the second (£ m a t.) the Dirac matter field, i.e., the electrons (ip), and the last (£i n t.) the interaction between the former two. The gauge field A* = (0, A) comprises the scalar and vector potentials and defines the antisymmetric electromagnetic field tensor

Fnv

=

QHAU _ QVAH

=

/ 0 -Ex -Ey -Ez\ Ex 0 — Bz By Ey Bz 0 — Bx \EZ -By Bx 0 /

(2)

where Einstein's summation convention is consequently employed throughout this chapter, i.e., Greek indices occuring twice, once as (covariant) subscript and once as (contravariant) superscript, are summed from 0 to 3.

140

Corresponding Latin indices are summed from 1 to 3. The standard notation (cf. Ref. 12) for the relativistic notation of 4-vectors has been employed, i.e., the contravariant position 4-vector is given by x = x» = ( x 0 , ^ ) = (ct,r),

(3)

and the covariant derivative 4-vector is defined by

d = 8, = (do,dt) = ( ~ , v ) .

(4)

Aext comprises external, i.e., non-dynamical electromagnetic potentials in addition to the radiation field, describing, e.g., the electric fields due to the atomic nuclei in a molecular system. In a more familiar notation the dynamical electric and magnetic fields are given as 18A = - V 0 - - — , B = VxA. c ot The coupling between the four components of the Dirac spinor E

US(r),

(5)

(6)

\Mr)J

is accomplished by the mathematical structure of a four-dimensional Clifford algebra (cf. Ref. 13) defined by the anticommutation relation {7^5 lu] — 2. Both the mass m and the charge e < 0 of the electron are not the physically observable quantities but the bare parameters that are still subject to a renormalization procedure according to the rules of quantum field theory. We will consequently use Gaussian units, i.e., Aneo = 1> and explicitly write down the fundamental constants of quantum theory and relativity, h and c, for all theoretical developments throughout this chapter. The connection of this Gaussian system of units to the SI system of units is established by the relation As/Vm = 1, i.e., both electric (E) and magnetic (B) field strengths are given in V/m. The Lagrangian £ Q E D describes both electromagnetic and fermionic degrees of freedom and the electromagnetic interaction between photons and matter simultaneously, i.e., the photon and the Dirac field are treated as dynamical variables. It has all symmetry properties necessary for a fundamental physical theory: it is Lorentz covariant as well as gauge invariant, since local gauge transformations of the form A^x) ^{x)

—> 4 ( x ) - A^x)

+ d^X(x),

_ ^ ^ ( r c ) = exp[ - £ A ( . T ) ] 1>{X) ,

(9) (10)

with X(x) being an arbitrary gauge function, leave the Lagrangian in Eq. (1) invariant. The special form of the gauge transformation of the spinor ^ , which simply describes rotations in the complex plane, is the reason for calling QED an Abelian £7(1) gauge theory. The Hamiltonian principle of least action, SS — c~l Sfd4xCQED = 0 under arbitrary infinitesimal variations of the dynamical variables A1* and i/>, yields the coupled equations of motions for the electromagnetic and Dirac fields, dpF'"' = —j" , c (ihrf - fhc2)^ = e(4 + 4 e x t ) il>.

(11) (12)

Eq. (11) is the most general covariant form of the inhomogeneous Maxwell equations, which immediately implies the continuity equation d*f

= Q + divj

= 0,

(13)

142

and Eq. (12) is the covariant time-dependent Dirac equation in the presence of external electric and magnetic fields. In this context it is very important to note that the Dirac equation is interpreted as a classical Euler-Lagrange equation for the spinor field ip rather than as a quantum-mechanical wave equation. The transition to a Hamiltonian formulation of this field theory requires the definition of the conjugate momenta dip

dA»

4TTC

M

and is achieved by a Legendre transformation. After some tedious algebraic manipulations the final Hamiltonian density is given as %QED

= UpA!1 + Trip = ^-{E2 07T

CQED

+ B2) + ^-E

• Vcf> - ${ihc~fkdk - mc2)iP

47T

+ efa»1>(A» + AF).

(15)

Although this expression is no longer manifestly Lorentz or gauge invariant, all physical observables like energies, field strengths, transition amplitudes, etc., which might be deduced from this Hamiltonian are Lorentz and gauge invariant. A subsequent integration over all space yields the classical Hamiltonian #QED =

/d 3 r / H Q E D

= U\ I ±-(E2 + B2) + ^ [cot • ( - iftV --cA+ f3mc2 + ecjf

>}•

V ,

1 A ext ) (16)

where partial integration of the second term of Eq. (15) and Gauss' law {v = 0 in Eq. (11)) have been employed; for later convenience we have introduced the computational standard notation of the Dirac matrices, i.e.,

It has to be emphasized that so far no special gauge has been chosen and no quantum theory has been developed yet. Eq. (16) is just the classical Hamiltonian of the U(l) gauge field theory of electrodynamics interacting with a dynamical fermionic field.

143

Unfortunately, quantization of this gauge theory is not as straightforward as the development of the theory to this point. As it is reflected by the mere existence of a gauge symmetry transformation, gauge theories necessarily comprise redundant degrees of freedom, which have to be removed by a suitable gauge fixing procedure before consistent quantization may be achieved. This is most easily seen by an application of the so-called Coulomb, radiation, or transverse gauge, divA = 0. With this choice of gauge the Gauss' law, i.e., the 0-component of Eq. (11), may be simplified to A0 =

-4TTQ

= -Anei/jty,

(18)

which is just the familiar Poisson equation of electrostatics. Since it is explicitly solved by the instantaneous Coulomb potential

Hr,t)

= A\r,t)

= / d V f ^ J \r-r'\

=

J

Url^ir',t)Hr>,t) \r-r'\

the scalar potential 4> is no longer an independent dynamical variable, but uniquely determined by the charge distribution g. The remaining spatial components of the vector potential are subject to the Coulomb gauge condition. Hence only two independent field components survive, reflecting the fact that real photons possess only two transversal polarization states. Taking all this into account and omitting the technical difficulties, consistent quantization of this field theory may be achieved, e.g., by a constrained canonical procedure 14 or the manifestly covariant Gupta-Bleuler formalism, which employs an indefinite metric at the expense of an easy physical interpretation 15 ' 16 . The resulting quantum field theory of QED properly describes the interaction of electrons and positrons with photons, the quanta of the electromagnetic field, and the external potential. The quantum-mechanical Hamiltonian operator of QED has exactly the same form as in Eq. (16), but all fields have now been upgraded to field operators acting on occupation number vectors in a suitable Fock space. Furthermore, in this canonical formalism all products of field operators have to be expressed in normal-ordered form, normalizing the vacuum energy to zero. Application of this time-honored formalism of second quantization has several major advantages. The total number of particles is no longer fixed and electron-positron pair-creation processes are naturally included. Also the number of photons may be increased or decreased, depending on the details of the electromagnetic interaction. The problem of negative-energy states has completely been removed, since both electrons and positrons feature strictly positive energies due to the process of normal ordering. One might

144

summarize, that a well-defined quantum theory of electromagnetic interactions may only be established by taking advantage of the benefits of second quantization. From the computational point of view of quantum chemistry, however, there are some serious drawbacks. The radiative corrections due to the electromagnetic field, e.g., the Lamb shift, are very small as compared to the typical energies occuring in chemical bonds. Furthermore, the energies necessary for pair-creation processes are magnitudes larger than the energy scale of the valence shell. It is therefore highly desirable for most chemical applications to integrate out those degrees of freedom from the very beginning in order to arrive at a theory for a fixed number of electrons only. Technically, projection operators onto suitable Hilbert subspaces of the original Fock space arise from this procedure. As a consequence, this simplification does not only make the theoretical framework fit much better to most chemical questions focused on the electronic structure of molecules, but it is also an essential prerequisite for a computationally feasible approach, since even calculations taking explicitly care of only some of the simplest QED effects are hardly capable for an atom with more than 3 electrons 17 ' 18 . Furthermore, employing the techniques of second quantization prohibits the direct interpretation of the field operators ip as usual quantum mechanical wave functions, since superpositions of states with variable numbers of particles are not compatible with the simple probabilistic interpretation of the wave function. In order to restore this feature and to get rid of the above mentioned projection operators, it has become the standard procedure of quantum chemistry to return to a first quantized formulation based on a suitable generalization of the original Dirac equation Eq. (12). One therefore has to specify the number of electrons explictly from the very beginning and to formulate the theory in terms of operators acting on individual particles. A molecular system containing n electrons and M nuclei is thus described by the electronic Hamiltonian n

n

H = X)ifo(*) + X>(M),

(20)

where the one-electron Dirac operator HD = cct-(p--A-Aext) + (/? - l)mc 2 + V, (21) V c c J consists just of the the one-electron parts of Eq. (16). In order to compare resulting energies to the non-relativistic values due to the Schrodinger equation the energy scale has been shifted by mc2. Since field-theoretic effects

145

are not explicitly considered at this stage, e and m are now the physically observable charge and mass of the electrons, and the external potential V is given by

V = V(r) = e^

( 22 )

= EuTZlh' M=1

\r

*ifi\

The derivation of the two-electron interaction is by far more subtle. We first decompose the electric field E into longitudinal (irrotational) and transverse (solenoidal) components, 1BA Et = — - a - • (23) c ot Consequently, the complete electromagnetic field energy, which comprises all3, two-electron effects, may be written as E = Et + Eu

^ / d 3 r (E2 + B2)

Ei = - V 0 ,

= i|d'rBf

+ ^|d

3

r(^+B

2

),

(24)

where the mixed term E\ • Et vanishes in Coulomb gauge, which is seen after a further partial integration. The first term on the right hand side of Eq. (24) represents the instantaneous Coulomb interaction between the charge distributions, ^|d3r£;f(r)(1=8)I|d3r i | d 3 7 - d V ^ ( r ^ V ) - ! ! — ^ ) ^ ' ) ,

(25)

where partial integration and the Coulomb gauge condition have been employed. It is thus only the longitudinal component of the electric field which mediates the instantaneous Coulomb interaction. Magnetic interactions, retardation effects due to the finite speed of light, and radiation corrections are contained in the second term on the right hand side of Eq. (24). Needless to say, a similar analysis can be carried out for A e x t . In Coulomb gauge, a T h i s clearly demonstrates that electromagnetic interactions between two charged particles are mediated by the exchange of (virtual) photons, i.e., the quanta of the electromagnetic field.

146

its expansion in powers of 1/c yields the leading terms

Summing all this up, we arrive after transition to the configuration space formulation at the following expression for the two-electron interaction,

«(p)

(28)

may be convenient, which is obtained by a Fourier transformation of the original configuration-space Dirac equation. The external potential is now no longer a local (multiplicative) operator, but acts as an integral operator

VrP(p) = / ( ^ 3 ^ ( P - P ' ) V - ( P ' )

(29)

148

on the spinor tp. The kernel of V is the Fourier transform of V(r), i.e.,

V(p-p')

= ld3rexp[-^(p-p')-r}v(r)

4.7rZe2/i2

= ~\

*n

(30)

for a Coulomb potential V(r) — —Ze2/r. Thus, the four coupled differential equations (12) were transformed into four coupled integral equations (28), which are often a more convenient starting point for further approximations, especially if the operators are represented in a basis of single particle wave functions. The Dirac equation is often formulated in the so-called split notation, i.e., (2 x 2)-matrix equations for the large and small components. We shall mention these equations for the sake of completeness: (V -E)I/JL

ca'pipL

+ ca-p^s

= 0,

(31)

+ (V - 2mc2 - E) ips = 0.

(32)

In the following overview, the theory of two-component methods in relativistic quantum chemistry is reviewed. The relationship between the large and small component is analyzed in section 2, followed by a brief survey of possible methods to decouple the Dirac Hamiltonian. The basic principles of both elimination and transformation techniques are presented with special emphasis on the discussion of their validity, accuracy, and computational cost. Special attention will be paid to the generalized Douglas-Kroll (DK) method, which is discussed comprehensively in section 3, where the underlying assumptions and approximations are studied in detail. First, the generalization of the parametrization of the employed unitary transformations is reviewed, and afterwards the the DK Hamiltonians up to fifth order in the external potential are presented. The section closes with a discussion of the DK transformation of the two-electron terms of the Hamiltonian and a detailed presentation of technical aspects to be considered by an implementation of the DK method into existing computer program packages. In section 4 numerical results obtained with the fifth order Douglas-Kroll Hamiltonian (DKH5) are given, and the binding energies of hydrogenlike atoms for the whole periodic table are compared to the exact values obtained from the Dirac equation. Furthermore, all-electron SCF calculations with DKH5 for noble gas atomic systems up to element 118 (Eka-Rn) are presented and compared to the numerical four-component Dirac-Fock benchmark results. The presentation closes with conclusions and further perspectives for the field of two-component methods in quantum chemistry.

149

2. Two-component relativistic quantum chemistry 2.1.

Basic properties

of Dirac

4-spinors

Since all problems accompanying relativistic quantum chemistry are related to the off-diagonal terms occuring in the one-electron Dirac Hamiltonian, we focus on the one-electron theory in the following. A useful prerequisite for the reduction of the Dirac equation to two-component form is the analysis of the relationship between the large and small components of an exact eigensolution of the Dirac equation. For every Dirac 4-spinor ip of the form given by Eq. (6), which has not necessarily to be an eigenspinor of ifc, this relationship may be formalized by 28 il>s = X*l>L,

(33)

where X is a yet undetermined (2 x 2)-matrix operator, whose properties are subject of investigation in this subsection. If and only if ip is an exact solution of the Dirac equation (28), an apparently simple expression for X can immediately be given in closed form by employing Eq. (32), X = X{E)

= (E-V

+ 2mc2)~1c(T-p.

(34)

However, a serious problem is connected with this energy-dependent expression for X. As E decreases towards the positronic continuum, the inverse operator on the right hand side will inevitably cross its singularity at E — V — 2mc2. As a consequence, one unique and universal X-operator valid for the whole energy range describing both electronic and positronic solutions does not exist. Nevertheless, a direct substitution of ips — XipL into Eq. (31) yields a two-component equation for the large component only, which will be discussed in the next subsection. The explicit energy dependence of the operator X is undesirable, since it gives rise to non-hermitean operators and non-orthogonal orbitals. In order to arrive at an energy-independent expression for X, Eq. (31) has to be multiplied with X from the left and Eqs. (32) and (33) have to be utilized. This yields XVifiL

+ XccrpX^

= c(T'pi)L

+ VXiPL - 2mc2X^L.

(35)

The operator X must to satisfy this equation for all possible choices for the large component ipL, and is thus determined by the non-linear operator identity X = -^\ca-p

- [X,V] - XcapX}.

(36

150

The solution of this equation is as complex as the solution of the Dirac equation itself, and thus only for a restricted class of potentials closed-form solutions for X are known 29 . It is important to realize that Eq. (36) is a quadratic equation for X and has thus always two independent solutions X+ and X_, corresponding to electronic (E>—2mc2) and positronic [E 0

(38)

is the familiar square root operator reflecting the relativistic energymomentum relation. We note in passing that X+=0 defines the exact freeparticle Foldy-Wouthuysen (fpFW) transformation to be discussed in detail later on. The action of the operator X is most conveniently studied in momentum space, where the inverse operator may be applied in closed form without expanding the square root. Thus the four normalized free-particle Dirac eigenspinors with the eigenvalues E±- •mc2 can compactly be given as (s = 1,2)

IP±AP^)

=

I E± + mc2 2E+

Xs exp - (pr A

> ±

-

E±t)

(39)

Xs

u±,»(P) with Xi =

(40)

X2 =

We have chosen the standard notation of quantum chemistry and have normalized the spinors to unity instead of Ep/mc2. After some elementary algebraic manipulations the standard form of the spinors u± is obtained as / W

+,5

I Ep + mc2 2Ev

\

(.

ccr-p X, \ y/2Ep(Ep + mc2)

=

cap \ y/2Ep{Ep. + mc2)

jEp + mc2 As

I

2Er>

151

In the literature the relation ips = XipL with X defined by Eq. (36) is sometimes called exact kinetic balance. Recently, an important consequence of the fulfillment of exact kinetic balance was realized by Kutzelnigg 30 : As long as exact kinetic balance is guaranteed for all Dirac spinors under consideration, i.e., all spinors within the space of test functions T^var, a variational scheme is established. Variational means, that the energy expectation value (HD)^ is bounded from below by the exact electronic ground state energy E0, {HDU s

= MffoM

> Eo,

V^G^var.

(41)

L

If the relation i/; — Xip is, however, only satisfied approximately for at least some Dirac spinors in the variational domain, only a variationally stable approach is achieved. I.e., the energy expectation value is still bounded from below, but not necessarily by EQ, but by a different bound which may be above or below EQ. The various two-component theories known from the literature satisfy the kinetic balance relation only to certain degrees of accuracy and hence establish only variationally stable but not variational approaches. The simplest approximation to exact kinetic balance may be obtained in the nonrelativistic limit of Eqs. (34) or (36), • —-, X_ • -2mc (a-p)-l. 42 2rac The introduction of the operator X leads to a modified normalization description for the Dirac spinor -0, X+

(V|V> = + ( V - V ) = (VL|i + xtx|^ L > =L l,

(43)

which can now be expressed in terms of the large component only and which must be taken into account when dealing with a two-component theory. 2.2. Elimination

techniques

The central idea of all elimination methods for the small component is to employ relations (33) and (34) and to substitute ijjs in Eq. (31) by an expression for the large component only. This yields a two-component equation for the latter only, which can be written as (V - E) ^L + ^

[(c vp) u (c a -p)] ^L

= 0,

(44)

with V -E 2mc2

= E(^l). 2mc 2

x

k=0

(«)

152

where we have utilized the properties of the geometric series in the second step of Eq. (45). Note that the non-relativistic Schrodinger equation is recovered by restricting this expansion to zeroth order, i.e., using the nonrelativistic limit c —> oo, which gives LJ = 1. The historically first reduction of the Dirac equation to two-component form is the Pauli approximation, which can be obtained from Eq. (44) by truncating the series expansion for UJ after the first two terms, and eliminating the energy dependence by means of a systematic expansion in c~2. The result is the famous Pauli Hamiltonian

n*» = f ^ - ^

+ ^ ' ^ ' + i ^ ! ^ * * ) '

where the mass-velocity term —p4/8m3c2, the Darwin term h2AV/8m2c2, and the spin-orbit coupling term proportional to a • [(VV) x p] describe relativistic corrections correct up to 0(c~2). However, several problems are connected with the Pauli Hamiltonian which prohibit its use within a variational procedure. These insurmountable problems cannot be remedied by going to higher orders 31 . Therefore the Pauli Hamiltonian and all other operators based on simple expansions of u in powers of c~2 are in general singular and may not be used for variational procedures. They are applicable only within perturbation theory to lowest order. The Pauli Hamiltonian yields satisfactory relativistic corrections to the energy up to the first and second transition metal row. Since a; is a scalar operator, even this simple energy-dependent elimination of the small component permits an exact separation of the spin-free and spin-dependent terms of the Dirac Hamiltonian by applying Dirac's relation (cr-p) oj((T'p)

— pup

+ icr(p

x up)

(47)

to the two-component operator of Eq. (44). This is still valid in an improved version of the original Pauli approximation, where the small component was eliminated in an exact way by avoiding any truncation of the expansion given in Eq. (45)32,33,34,35,36,37,38,39 T h e r i j E q ( 4 4 ) r e p r 0 ( i U ces exactly both the positive and negative eigenvalues of the Dirac Hamiltonian, and has been successfully applied in atomic many-body calculations 40 ' 41 . However, all operators of this kind are non-hermitean and energy-dependent and thus plagued by not mutually orthogonal orbitals. In the mid-eighties another method to eliminate the small component has been developed in order to arrive at regular expansions for the Hamiltonian 28 ' 42 . These regular approximations are based on the general

153

theory of effective Hamiltonians 43 ' 44 , where the full problem under consideration is projected onto a smaller, suitably chosen model space with an effective Hamiltonian, which comprises all desired properties of the problem sufficiently well. In the case of the Dirac Hamiltonian the basic idea is to rewrite the expression for u in the form UJ

2rac2 r E "l~1 2mc2 - V 1 + 2rac2 - V

=

(48)

and to choose the new expansion parameter E/{2mc2 — V), which is the starting point for the so-called regular approximations developed by the Amsterdam group 45 ' 46 ' 47 to a workable and successful method of electronic structure theory. A truncation of this expansion for u defines the zerothand first-order regular approximation (ZORA, FORA) 48 . Excellent agreement of orbital energies and other valence shell properties with the results from the Dirac equation is obtained, and can even be improved by transition to the scaled ZORA variant 49 . The analysis 50 shows that in regions of high potential the ZORA Hamiltonian reproduces relativistic energies up to an error of order —E2/c2. On the other hand, in regions of small potential but high kinetic energy of the particle, it does not provide any relativistic correction. The main disadvantage of the method is its gauge dependence, i.e., a constant shift of the electrostatic potential does not lead to a constant shift in the energy, because the potential enters non-linearly in the denominator of the Hamiltonian. This deficiency can, however, be approximately remedied by suitable means 49 ' 51 . The latest major achievement in the field of elimination techniques for the small component is due to Dyall and has been worked out to an efficient computational tool for quantum chemistry within the last few years 52 ' 53 ' 54 ' 55 . This method is commonly dubbed normalized elimination of the small component (NESC) and is based on the modified Dirac equation 56,57 , where the small component ips of the 4-spinor ip is replaced by a pseudolarge component (j)L defined by the relation ca-p(j)L

= 2mcV5.

(49)

An insertion of this relation into the split-form of the Dirac equation, Eqs. (31) and (32), yields the modified Dirac equation (V -E)^L

T^L +

1

4m2c2

(*.p)(V-E)(*.p)

+ TL = 0, -T F = 0,

(50) (51)

154

where T is the non-relativistic kinetic energy operator. By exploiting the special features of the matrix representation of the modified Dirac equation it is possible to preserve the proper normalization of the large component during the elimination of the small component L. This normalized elimination procedure results in energy eigenvalues which deviate only in order c - 4 from the correct Dirac eigenvalues, whereas the standard (unnormalized) elimination techniques (UESC) are only correct up to order c~ 2 . In addition, the NESC method is free from the singularities which plague the UESC methods, and can systematically be simplified by a sequence of approximations 53 ' 58 ' 55 , which reduce the computational cost significantly. 2.3. Transformation

techniques

The second possibility to reduce the four-component Dirac spinor to twocomponent Pauli form is to decouple the Dirac equation, i.e., to transform the Dirac Hamiltonian to block-diagonal form by a suitably chosen unitary transformation U, Hbd =UHDU^

=

,

(52)

with UW — 1. In the following, we shall call a transformation of the Dirac Hamiltonian Ho of this kind an exact Foldy-Wouthuysen (EFW) transformation, and the corresponding transformed Dirac spinor a FoldyWouthuysen (FW) spinor. Similar to the Hamiltonian, also the spinor is simplified by this unitary transformation and has only one non-vanishing 2-spinor component, i.e.,

u

* = * =(£)

^

with ips = 0 for electronic solutions and ipL = 0 for positronic solutions. Note that the two 'effective' operators h+ and h- reproduce the entire energy spectrum of the Dirac operator HE>. Employing the most general ansatz for the unitary matrix U,

fu11u12\ U =

,

(54)

\U2iU22J s

the requirements ip = 0 for electronic solutions and ipL = 0 for positronic solutions together with the unitarity constraint UW = 1 yield the most

155

general form of the unitary matrix U that achieves a block-diagonalization of the Dirac Hamiltonian ifo 5 9 ,

(i + xtx)"1/2

(

(i + xtx) _1/2 xt -

U = U(X) =

(55) \-e^(l

+ XX^y1/2X

e^(l +

XX^)~1/2

Here

. It has thus become the standard protocol of the DK transformation to focus on the block-diagonalization of the one-electron terms Ho(i) of the DFC Hamiltonian # D F C according to Eq. (78) and to neglect the complexity introduced by the effect of this transformation on the originally diagonal two-electron terms. We will therefore restrict our discussion to the DK transformation of the one-particle Dirac operator Ho in this subsection and postpone the adequate treatment of the two-electron terms to subsection 3.3 After these prelimineries, the sequence of unitary transformations defined by Eq. (78) is set up and the block-diagonal Hamiltonian Hbd is constructed step by step. In order to investigate a potential dependence of the DK Hamiltonians HBKHU on the coefficients a{ k, we do not restrict the derivation to the optimal parametrization of the transformations U{ derived in the last subsection, but apply the most general parametrization of U{ with the coefficients a{ k satisfying the unitarity conditions Eqs. (82) only. The first subscript of the coefficients aik characterizes the corresponding unitary matrix U%. For later convenience the odd and antihermitean expansion parameter is denoted by W[ instead of W\ for the moment. The transformation of the fpFW Hamiltonian H\ with U\ yields H2 = f/i Hx U\ p

OO

-i

r

CO

= k , 0 l + $ > i , * W i ' 1 (£o + £ i + 0 i ) k o l + £ ( - ! ) * ai.fcWf k=l

k=l

166 oo

= £o + Si+ 0[2) + f2 + 0i2) + £3 + 0{2) + £ (42) + ^i 2 ) ) • (87) The subscript attached to each term of the Hamiltonian denotes its order in the external potential, whereas the superscript in parentheses indicates that the corresponding term belongs to the intermediate, partially transformed Hamiltonian with the same number. It is only relevant for the following steps and will affect higher-order terms. Only those even terms, which will not be affected by the succeeding unitary transformations Ui, (i — 2 , 3 , . . . ) bear no superscript and may already be identified with the corresponding terms in the expansion of HM given by Eq. (78). It is a consequence of the so-called (2n+l)-rule, that £2 and £3 are already completely determined after the first unitary DK transformation U\. Hence, Hbd is already defined up (2) to third order in the external potential although the second order term 02 is still present and will be eliminated in the next transformation step. The terms £ 0 , £1, and 0\ (cf. Eqs. (65)-(67)) are independent of W[ and thus completely determined from the very beginning. In general, the first 2 n + l even terms of H\,d depend only on the n lowest matrices W[, W2, • • •, W^, i.e., they are independent in particular of all succeeding unitary transformations. This remarkable property of the even terms originates from the central idea of the DK method to choose the latest odd operator W[ always in such a way, that the lowest of the remaining odd terms is eliminated. The explicit form of the other terms of H2 is given in Ref. 72 in detail, and we thus recall only the expression for the lowest-order odd term 0[2)

= 01 + a 1 | 0 a 1 | 1 [ W 1 / , £ 0 ] .

(88)

W[ is chosen in order to annihilate 0[ ', and thus the following condition for W[ is obtained, [Wl,£0]

= - ^ O i , a

(89)

i,i

which is satisfied if and only if the kernel of W[ is given by

Wl(iJ) = ^f>pM-.

(90)

This choice of W[ satisfies all constraints, namely that it is an odd and antihermitean operator of first order in V. Note that W[ depends on the

167

beforehand arbitrarily chosen coefficients a10 and alx, i.e., it is linear in a i,o/°i,i- We therefore introduce the modified operator W\ defined by Wi(iJ)

= /? OiiU) Ei -f- Ej

= ali0altlW{(i,j)

(91)

which is manifestly independent of the coefficients ax k. With this choice of W\ and by utilizing relation (89) most terms occuring in H2 are simplified to a large extent, and can be found in Ref. 72. The next unitary transformation U2 is applied in order to eliminate the odd term of second order, H3 = U2H2U% =

«2,ol + E

a

2,kW2k

H2 k

01

+ £ ( - 1 ) * a 2|jb W k=i

oo

= £ ^ + k=6 E^ fc=0

3)

+ k=2£