Quantum Science. The Frontier of Physics and Chemistry 9789811944208, 9789811944215

Table of contents :
Preface
Contents
About the Editor
Quantum Electronic Structure
Quantum Spin Memory Using Inverse Copper Oxide Cluster—Spin Configurations Unpredicted from Ligand Field Theory
1 Quantum Mechanics in 3d Orbital
1.1 Quantum Expression of 3d Orbital
1.2 Gaussian-Type Quantum Wave Function
1.3 Ligand Field Theory
2 Molecular Orbital Analysis
2.1 Molecular Orbital Calculation
2.2 Chemical Bonding Rule
3 OCuO Cluster
3.1 3d3x2-r2 Type Spin Configuration
3.2 3dz2-y2 Type Spin Configuration
3.3 3d Orbital Energy-Splitting
4 Inverse Cu4O4 Cluster
4.1 3d3x2-r2 Type Spin Configuration
4.2 3dz2-x2 Type Spin Configuration
4.3 Comparison with Cu–O-Cu System
4.4 Comparison with Two Quintet Spin Configurations
4.5 Barrier Effect in O–Cu–O System
5 Conclusions and Future: Quantum Spin Memory
References
Characteristic Fermi Surface Properties in ff-Electron Systems
1 Introduction
2 Experimental Procedure
3 Combined Phenomenon Involving the Doniach Phase Diagram and Sharp Valence Crossover in CeRhInSubscript 55 and EuCuSubscript 22GeSubscript 22
3.1 CeRhInSubscript 55
3.2 EuCuSubscript 22GeSubscript 22
4 Noncentrosymmetric Compounds of CeIrSiSubscript 33 and EuPtSi
4.1 CeIrSiSubscript 33
4.2 EuPtSi
5 Summary
6 Outlook
References
Quantum Dynamics
Chemical Reaction Kinetics and Dynamics Re-Considered: Exploring Quantum Stereodynamics—From Line to Plane Reaction Pathways and Concerted Interactions
1 General Introduction
1.1 Arrhenius Equation Re-Considered
2 Rate Equations Re-Considered
3 Unimolecular Reaction Theory Re-Considered
4 Chemical Kinetics Toward Quantum Stereodynamics
5 Topical Examples for Current Status of Chemical Reaction Dynamics
6 Topic 1: UV Photodissociation of Halothane in a Focused Molecular Beam: Space-Speed Slice-Imaging of Competitive Bond Breaking Into Spin–Orbit Selected Chlorine and Bromine Atoms
6.1 Introduction
6.2 Experimental Method
6.3 Results and Discussion
6.4 Conclusions and Future Remarks
7 Topic 2: Roaming Dynamics in Some Cases of Carbonyl Compounds
7.1 Introduction
7.2 Methyl Formate: Roaming Via a Conical Intersection
7.3 Aliphatic Aldehydes: Roaming Through a Loose Complex Structure
7.4 Concluding Remarks
8 Topic 3: The Pathway Discrimination of Gas Phase Unimolecular Dissociation by Dynamical Model: The Recent Developments
8.1 Introduction
8.2 Computational Methods
8.3 Case Study: Photodissociation of Methyl Formate Under 248 nm Photolysis
8.4 Discussion
8.5 Conclusion
9 Topic 4: 2D Spatially Oriented Molecules and Chiral Discrimination Via Photodissociation Using 2D Imaging and Hexapole Techniques
9.1 Introduction
9.2 Theory
9.3 Experiments
9.4 Summary and Conclusions
10 Topic 5: Electrostatic Hexapoles in the Study of Stereo-Directed Dynamics Processes of Interest in Astrochemistry
10.1 Introduction
10.2 Background
10.3 Hexapole Orientation of Propylene Oxide
10.4 Conformer State-Selection on 2-Butanol
10.5 The Hexapolar Technique Applied to Astrochemistry
10.6 Final Remarks
11 Topic 6: Impact-Parameter-Dependent Analysis of the Trajectory Pattern for the H + H2 Exchange Reaction at T = 3 K and 300 K: Interconnecting the Trajectory Reactivity with the Time-Dependent Interaction Potential Energy and the Roaming-Like Libration Motion at Cold Temperature
11.1 Introduction
11.2 Methodology
11.3 Results and Discussion
12 Concluding Remarks
References
Quantum Theory
Solution for Lithium Problem from Supersymmetric Standard Model
1 Introduction
2 Review on Standard Big-Bang Nucleosynthesis
2.1 Thermal History of the Universe
2.2 Big-Bang Nucleosynthesis: Theory
2.3 Theoretical Prediction of the Abundances and the Dependence of Parameters
2.4 Big-Bang Nucleosynthesis: Observation
2.5 The Superscript 77Li Problem
3 Minimal Supersymmetric Standard Model (MSSM)
4 Long-Lived Charged Massive Particle
4.1 Long-Lived Slepton
4.2 Number Density of the NLSP Slepton at the BBN Era
5 Non-standard Nuclear Reactions
5.1 Internal Conversion
5.2 Spallation Reactions and Slepton Catalyzed Fusion
6 Numerical Result and Favored Parameter Space
7 Verification of the Scenario with Dark Matter Search
7.1 Formalism of the Effective Action for Non-relativistic Particle
7.2 S-Wave Annihilation Cross Section of Dark Matter
7.3 Results of the Dark Matter Research
8 Summary
References
Elements of Theory of Angular Moments as Applied to Diatomic Molecules and Molecular Spectroscopy
1 Introduction
2 Molecular Rotators
2.1 Prerequisites for Molecular Rotators
2.2 Rigid Molecular Rotators
2.3 Centrifugal Perturbations and Oscillatory–Rotational Interactions
2.4 “Oscillatory” Angular Moment in a Linear Molecule
2.5 Diatomic Molecule with Identical Nuclei
3 Diatomic Molecules with Electronic Angular Moments
3.1 Coordinates. General Integrals of Motion
3.2 Adiabatic Approximation
3.3 Electronic Wave Function Symmetry
3.4 The Angular Momentum of the Molecule. Accounting the Orbital Angular Momentum of Electrons
3.5 Equation for the Nuclear Wave Function
3.6 The Total Wave Function of a Molecule for the Case S = 0
3.7 Oscillatory and Rotational Energy of the Molecule
3.8 Symmetries of the Total Wave Function. Parity
3.9 Λ-Doubling
4 Application of Theory
4.1 Forces of Radiation Transition Lines in the Spectra of Rotators
4.2 Matrix Elements for Rotational Transitions of a Symmetric Rotator
4.3 Matrix Elements for Rotational Transitions of an Asymmetric Rotator
4.4 Matrix Elements for Rotational Transitions of a Linear Rotator
4.5 Line Strengths of Electrical Dipole Transitions Between Rotational Levels of Diatomic Molecules
5 Conclusions
References
Emission and Absorption of Photons in Quantum Transitions. Coherent States
1 Emission and Absorption of Photons at Transitions Between Discrete Energy Levels
1.1 General Definitions of the Probabilities of Quantum Transitions
1.2 Dipole Transitions
2 Coherent States
2.1 Hamiltonian of Coherent States
2.2 Determination of Coherent States
2.3 Orthogonality, Normalization and Completeness of Coherent States
2.4 Uncertainty Relation
2.5 Coordinate Representation of the Hamiltonian and Coherent States
2.6 Exact Solution of the Time-Dependent Problem of the Interaction of Radiation with Matter
2.7 Heisenberg Representation
References
Magnetic Ordering in a System of Identical Particles with Arbitrary Spin
1 Introduction
2 General Prerequisites and Provisions of the Theory of Angular Momentum
2.1 Turn Operator
2.2 Angular Momentum Conservation
2.3 Angular Momentum in Quantum Mechanics
2.4 Addition of Angular Momenta in Quantum Mechanics. The Wigner–Eckart Theorem
3 Spin Hamiltonian of Identical Particles with Arbitrary Spin
3.1 Interaction in a System of Two Identical Particles with Spin
3.2 Spin Hamiltonian of a System of Identical Particles
4 Applications of Spin Hamiltonians
4.1 Antiferromagnetic Ordering in a Chain of Identical Particles with Spin s = 1
4.2 Spin Waves and Spin Dark–Bright Soliton in the Spin-1 Chain
4.3 Ground States of an Isotropic Magnetic System of Particles with Spin s = 3/2
Appendix 1: Variational Heisenberg Model for the Spin-1/2 System
Appendix 2: Ising Model for the Spin-1/2 Chain
References
Quantum Computational Method
Basis Set Convergence and Extrapolation of Connected Triple Excitation Contributions (T) in Computational Thermochemistry: The W4-17 Benchmark with Up to k Functions
1 Introduction
1.1 Electron Correlation and the Correlation Energy
1.2 Coupled Cluster Theory
1.3 Broader Context of the Problem
1.4 Decomposition of the Total Atomization Energy
1.5 Gaussian Basis Sets
1.6 Basis Set Extrapolation
2 Computational Methods
3 Results and Discussion
3.1 Convergence in a Model System: Neon Atom
3.2 Convergence and Basis Set Extrapolation for W4-17
4 Conclusions
5 Future Outlook
References

Citation preview

Taku Onishi   Editor

Quantum Science The Frontier of Physics and Chemistry

Quantum Science

Taku Onishi Editor

Quantum Science The Frontier of Physics and Chemistry

Editor Taku Onishi Department of Chemistry University of Oslo Oslo, Norway Graduate School of Engineering Mie University Tsu, Japan

ISBN 978-981-19-4420-8 ISBN 978-981-19-4421-5 (eBook) https://doi.org/10.1007/978-981-19-4421-5 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Singapore Pte Ltd. The registered company address is: 152 Beach Road, #21-01/04 Gateway East, Singapore 189721, Singapore

Preface

This book focuses on recent topics of quantum science in physics and chemistry. Until now, quantum science has not been fully discussed from the interdisciplinary vantage points of both physics and chemistry. This book, however, is written not only for theoretical physicists and chemists, but also for experimentalists in the fields of physical chemistry and condensed matter physics, as collaboration and interplay between construction of quantum theory, and experimentation has become more important. This book consists of four parts: – – – –

Part I Quantum Electronic Structure Part II Quantum Dynamics Part III Quantum Theory Part IV Quantum Computational Method

Part I In Chap. 1, quantum spin configuration is investigated for inverse copper oxide cluster, by using quantum simulation based on density functional theory (DFT). In normal copper oxide as Cu-O-Cu system, the magnetic interaction is antiferromagnetic. However, spin configuration unpredicted from ligand field theory is given. Finally, application for quantum spin memory is discussed. In f electronic system, a variety of characteristic properties such as spin/charge ordering, spin/valence fluctuation, heavy fermion and anisotropic superconductivity are due to competitive phenomena between the RKKY interaction and the Kondo effect. In Chap. 2, in the cases of the Ce- and Eu-based compounds, quantum electronic structures and unique phenomena are explained from experimental approach.

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Preface

Part II Through developments of experimental technique and quantum simulation, chemical reaction can be now discussed from the viewpoint of quantum chemistry. In Chap. 3, several topics related to quantum dynamics, including photodissociation, roaming reaction, chirality and astrochemistry, are introduced.

Part III In quantum mechanics at electron and atomic levels, electron of atom is investigated as quantum particle. On the other hand, in astrophysics, other particles are treated in quantum manner. In Chap. 4, lithium problem in big bang nucleosynthesis scenario is discussed from supersymmetric standard model. Theoretical developments in atomic and molecular physics are introduced in Chaps. 5–7. For the purpose of calculating molecular spectra analytically, theory of angular momenta is explained in Chap. 5. Emission and adsorption of photons in quantum transitions are discussed in Chap. 6. Chapter 7 discusses the effects of magnetic ordering in systems of identical particles with arbitrary spin.

Part IV As explained in Quantum Computational Chemistry, Chap. 6 (Springer, 2018), configuration interaction based calculation methods sometimes predict wrong electronic structure for transition metal compounds. However, it has been widely accepted that coupled cluster method is accurate calculation method for organic molecular system. Recent developments are introduced in Chap. 8. Finally, I would like to thank Dr. Sinichi Koizumi and Springer staffs. This book will give you tips for starting new types of research projects thorough touching cutting-edge quantum science. Kobe, Japan A Happy Day 2022

Taku Onishi

Contents

Quantum Electronic Structure Quantum Spin Memory Using Inverse Copper Oxide Cluster—Spin Configurations Unpredicted from Ligand Field Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Taku Onishi Characteristic Fermi Surface Properties in f -Electron Systems . . . . . . . . ¯ Yoshichika Onuki, Rikio Settai, Yoshinori Haga, Tetsuya Takeuchi, Masato Hedo, and Takao Nakama

3 21

Quantum Dynamics Chemical Reaction Kinetics and Dynamics Re-Considered: Exploring Quantum Stereodynamics—From Line to Plane Reaction Pathways and Concerted Interactions . . . . . . . . . . . . . . . . . . . . . . Toshio Kasai, King-Chuen Lin, Po-Yu Tsai, Masaaki Nakamura, Dock-Chil Che, Federico Palazzetti, and Balaganesh Muthiah

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Quantum Theory Solution for Lithium Problem from Supersymmetric Standard Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159 Joe Sato, Yasutaka Takanishi, and Masato Yamanaka Elements of Theory of Angular Moments as Applied to Diatomic Molecules and Molecular Spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207 V. K. Khersonsky and E. V. Orlenko Emission and Absorption of Photons in Quantum Transitions. Coherent States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 349 E. V. Orlenko and V. K. Khersonsky

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Contents

Magnetic Ordering in a System of Identical Particles with Arbitrary Spin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 405 E. V. Orlenko, V. K. Khersonsky, and F. E. Orlenko Quantum Computational Method Basis Set Convergence and Extrapolation of Connected Triple Excitation Contributions (T) in Computational Thermochemistry: The W4-17 Benchmark with Up to k Functions . . . . . . . . . . . . . . . . . . . . . . . 467 Jan M. L. Martin

About the Editor

Taku Onishi was born in Kobe, Japan. He is quantum chemist who graduated from Department of Chemistry, Osaka University, Japan in 1998. He gained Master degree in 2000, and Ph.D. in 2003, both from Department of Chemistry, Osaka University, Japan. In 2003, he took up academic tenure as assistant professor in Graduate School of Engineering, Mie University, Japan. During 2010–2011, he stayed at Centre for Theoretical and Computational Chemistry (CTCC), University of Norway as Excellent Young Researcher Overseas Visit Program, Japan Society for the Promotion of Science. Since 2010, he has been a guest researcher of Department of Chemistry, University of Oslo, Norway. In 2021, the title of guest professor was given from University of the Ryukyus, Japan. He has served on international scientific activities: a Member of Royal Society of Chemistry; Chair of the Quantum Science (QS) Symposium; a position on the science committee of the International Conference of Computational Methods in Sciences and Engineering (ICCMSE); a member of the editorial board of Cogent Engineering, the Journal of Computational Methods in Sciences and Engineering (JCMSE). He has reviewed many international proceedings, books, and journals in various research fields of chemistry and physics. For example, AIP conference proceedings, Progress in Theoretical Chemistry and Physics, Cogent Chemistry, Physical Chemistry Chemical Physics, Molecular Physics, Dalton Transaction, The Journal of Physical Chemistry Letters, Journal of Computational Chemistry, International Journal of Quantum Chemistry, Journal of Solid State Chemistry, Solid State Ionics, Chemistry of Materials, Materials Chemistry and Physics, Chemical Engineering Journal, Chemical Physics, ACS catalysis, Applied Surface Science, Chemical Physics, Chemical Physics Letters, Chirality, Applied Surface Science, ACS catalysis, Biochimica et Biophysica Acta—Proteins and Proteomics, AIP advances, The Journal of Organic Chemistry, New Journal of Chemistry, etc.

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Quantum Electronic Structure

Quantum Spin Memory Using Inverse Copper Oxide Cluster—Spin Configurations Unpredicted from Ligand Field Theory Taku Onishi

Abstract In quantum mechanics, 3d orbital can be expressed using quantum wave function. Though five 3d orbitals have the same radial quantum wave function, they have different angular quantum wave functions. In molecular orbital (MO) calculation, Gaussian-type quantum wave function is applied, instead of Slater-type quantum wave function. 3dx 2 −y 2 and 3dz 2 orbitals are expressed by the superposition of x 2 , y 2 and z 2 components. Ligand field theory is constructed in semi-quantum manner, based on point charge. Though one specific spin configuration can be predicted, the correct orbital energy diagram cannot be predicted. For example, in OCuO cluster, ligand field theory predicts that highest orbital energy is given in 3d3x 2 −r 2 orbital. However, in fact, the orbital has lower orbital energy than other 3d orbitals. Two spin configurations: (1) 3d3x 2 −r 2 type and (2) 3dz 2 −x 2 type are here given. Next, we consider realistic model, inverse Cu4 O4 cluster, where oxygen atoms are allocated at edges of square, instead of copper atom. As same as OCuO cluster, two spin configurations are given. In 3dz 2 −x 2 type spin configuration, there is no orbital overlap between copper and oxygen atoms. In other words, 3dz 2 −x 2 type electron works as barrier to oxygen electron. The effect is called “Barrier Effect”. Further, in 3dz 2 −x 2 type spin configuration, quintet and singlet spin states are almost degenerated. When controlling the total energy difference, inverse Cu4 O4 cluster can be applied for quantum spin memory. Keywords 3d orbital · Quantum wave function · Ligand field theory · Molecular orbital · Chemical bonding rule · OCuO cluster · Inverse Cu4 O4 cluster · Superexchange rule · Barrier effect · Quantum spin memory

T. Onishi (B) Graduate School of Engineering, Mie University, Tsu, Japan e-mail: [email protected]; [email protected] Hylleraas Centre for Quantum Molecular Sciences, Department of Chemistry, University of Oslo, Oslo, Norway © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 T. Onishi (ed.), Quantum Science, https://doi.org/10.1007/978-981-19-4421-5_1

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T. Onishi

1 Quantum Mechanics in 3d Orbital 1.1 Quantum Expression of 3d Orbital In quantum chemistry, electronic structure of transition metal oxide is represented by using one-electron quantum wave functions. In principle, by solving Schrödinger equation, quantum wave function can be obtained. However, only in the case of neutral hydrogen atom having one electron, the quantum wave function is analytically solved. In general, orbital is categorised by three quantum numbers: principal quantum number (n), orbital angular momentum quantum number (l), magnetic quantum number (ml ). The following equations are satisfied: l = 0, 1, 2, · · ·, n − 1

(1)

m l = 0, ±1, ±2, · · ·, ±l.

(2)

and

It is known that the quantum wave function of hydrogen atom is expressed as the product of radial quantum wave function: Rn,l (r ) and angular quantum wave function (spherical harmonics): Yl,m (θ, φ). ψn,l,m l = Rn,l (r )Yl,m l (θ, φ),

(3)

where r denotes distance of atomic nucleus-electron, and θ and φ indicate angles of polar coordinates. When n = 3 and l = 2, it stands for five 3d orbitals: ψ3,2,m l = R3d (r )Y2,m l (θ, φ).

(4)

Note that R3,2 (r ) corresponds to R3d (r ). Though five 3d orbitals have the same radial quantum wave function, they have different angular wave function. When ml = 0, the quantum wave function is expressed without using imaginary number: / ψ3,2,0 = N

5 3z 2 − r 2 R3d ≡ ψ3dz2 , r2 16π

(5)

where N is normalisation constant. On the other hand, when {m l | ± 1, ±2}, angular quantum wave functions include imaginary number in φ component. Hence, other 3d orbitals are represented by the superposition of different spherical harmonics. It is allowed, due to mathematical feature of differential equation. Finally, other four 3d orbitals are expressed as follows.

Quantum Spin Memory Using Inverse Copper Oxide Cluster …

/ ψ3d yz = N / ψ3dx z = N

5

15 yz R3d 2π r 2

(6)

15 x z R3d 2π r 2

(7)

/

ψ3d 2

x −y

2

15 x 2 − y 2 R3d 32π r 2 / 15 x y R3d =N 8π r 2

=N

ψ3dx y

(8)

(9)

N is normalisation constant.

1.2 Gaussian-Type Quantum Wave Function Polar coordinates are applied for representing quantum wave function of hydrogen atom. On the other hand, in molecular system, Cartesian coordinates are used. Owing to advantage in calculating two-electron integral, it is normal to represent quantum wave function by using Gaussian function: N x l y m z n e−αr , 2

(10)

where N and α are constant, and r indicates distance of atomic nucleus-electron. In 3d orbital, the relation: l +m+n =2

(11)

is satisfied. Namely, Fml. (10) is rewritten as N (x 2 , y 2 , z 2 , x y, yz, x z)e−αr . 2

(12)

Note that Fml. (12) does not imply that the number of 3d orbitals is six. 3dx 2 −y 2 and 3dz 2 orbitals are expressed by the superposition of x 2 , y 2 and z 2 components.

1.3 Ligand Field Theory Ligand field theory is constructed in semi-quantum manner. It is because mathematical point called point charge is introduced when constructing the theory. Let

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T. Onishi

z

Fig. 1 Octahedral ligand field of transition metal surrounded by six point charges. Charge of each point charge is −Ze

-Ze

-Ze

Transition metal -Ze

-Ze

-Ze

x

y -Ze

us consider the case when transition metal is surrounded by six atoms allocated at octahedral edges (Fig. 1). In ligand field theory, the six atoms are replaced by negative point charges (−Ze). The potential energy produced by six point charges are expressed as

v(r) =

6 Σ i=1

Z e2 , |R i − r|

(13)

where |Ri | and |r| indicate distance of transition metal centre—i-th point charge, and transition metal centre—electron, respectively. Then, the Hamiltonian of this system is represented by Λ

H =−

ℏ 2 ∇ + u(r) + v(r), 2m

(14)

where u(r) indicates the Coulomb interaction between atomic nucleus and electron. In the right of Eq. (14), the first term of the Hamiltonian stands for operator of electron kinetic energy. Treating v(r) as perturbation term, it is known that the two orbital energies regarding 3d electron are given by (1) = ε3d + ε3d

6Z e2 + 6Dq a

(15)

Quantum Spin Memory Using Inverse Copper Oxide Cluster …

7

and (2) ε3d = ε3d +

6Z e2 − 4Dq, a

(16)

where a denotes distance between transition metal and point charge [1]. D and q are given by D=

35Z e 4a 5

(17)

and 2e q= 105

⌠∞ |R3d (r )|2 r 4r 2 dr ,

(18)

−∞

(1) respectively. In octahedral ligand field, ε3d indicates the orbital energies of 3dx 2 −y 2 (2) and 3d3z 2 −r 2 orbitals, while ε3d stands for the orbital energies of 3dx y , 3d yz and 3dzx orbitals.

2 Molecular Orbital Analysis 2.1 Molecular Orbital Calculation Consider octahedral cluster consisting of transition metal and surrounding six ligand atoms. In quantum chemistry, electrons of ligand atoms are expressed as quantum wave functions. Note that quantum wave function is expressed as the product of spatial orbital (ψ(r)) and spin function (α(ω) or β(ω)): ψ(r)α(ω) or ψ(r)β(ω).

(19)

Spin function is orthonormal. In Schrödinger equation, the Coulomb interaction between transition metal and ligand atom, and the electron–electron Coulomb interaction are more precisely incorporated than ligand field theory. In molecular orbital (MO) calculation using density functional theory (DFT) method, instead of Schrödinger equation, one-electron Kohn–Sham equation is numerically solved: Λ

KS

f i ψi (x i ) = εi ψi (x i ),

(20)

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T. Onishi Λ

where f K S , ψi (x i ) and εi denote Kohn–Sham operator, i-th spatial orbital and i-th orbital energy, respectively. In open shell system, there are two KS equations for α and β spins. Thus, electrons of surrounding ligand atom are treated as quantum particle in MO calculation. The calculation method employed here is BHHLYP, which is one of DFT methods, in combination with MINI basis set for copper atom, and 6-31G* basis set for oxygen atom.

2.2 Chemical Bonding Rule To characterise chemical bonding property, chemical bonding rule is useful. In MO, orbital overlap is represented by superposition of initial atomic orbitals (IAOs). Note that the coefficients of predetermined spatial orbitals, which correspond to IAOs, are numerically calculated by solving KS equation. Chemical bonding rule (Fig. 2) is summarised as follows: Chemical Bonding Rule For MOs including outer shell electrons, check whether orbital overlap exists or not. -With orbital overlap: Covalent -Without orbital overlap: Ionic For example, when there is orbital overlap between copper and oxygen atoms, it is determined that covalent bonding is formed. Note that ionic bonding coexists then.

3 OCuO Cluster Before the investigation of realistic system, we consider 3d orbital energy-splitting in simple OCuO model (allocated along x axis), where formal charges of copper and oxygen atoms are +2 and −2, respectively. From MO calculations, two spin configurations were given. The total energies of spin configuration I (SC(I)) and spin configuration II (SC(II)) were −48,706.45 and −48,706.04 eV, respectively.

Quantum Spin Memory Using Inverse Copper Oxide Cluster …

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Molecular orbitals with outer shell electrons

Molecular orbitals with outer shell electrons

(a)

(b)

Check orbital overlap Fig. 2 Chemical bonding rule: a closed shell system, b open shell system. Reprinted by permission from Springer Nature: Quantum Computational Chemistry—Modelling and Calculation for Functional Materials, Copyright Springer Nature Singapore Pte Ltd. 2018 [2]

3.1 3d 3x 2 −r 2 Type Spin Configuration Figure 3 depicts MOs related to copper 3d electrons in SC(I). 3dx y , 3d yz and 3dzx type α orbitals are paired with corresponding β orbitals: MO15α-MO14β, MO18αMO17β, MO16α-MO15β, while 3dz 2 −y 2 type α orbital (MO17α) is paired with corresponding β orbital (MO16β). On the other hand, 3d3x 2 −r 2 type α orbital (MO14α) has no paired β orbital. Hence, MO14α, where there is orbital overlap between copper 3d3x 2 −r 2 and oxygen 4s orbitals, is responsible for total spin of OCuO cluster. Spin density of copper atom is +1.20. Spin densities of 3x 2 , 3y 2 and 3z 2 components are +0.47, +0.17 and +0.17, respectively, while spin density of 4s component is +0.38. Note that 3dz 2 −y 2 and 3d3x 2 −r 2 orbitals are formed, instead of 3dx 2 −y 2 and 3d3z 2 −r 2 orbitals, because OCuO cluster is allocated along x axis.

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T. Onishi

Orbital Energy

Pair

MO17β (-0.011)

MO16β (-0.012) Pair

MO17α (-0.043)

MO18α (-0.042)

MO14β (-0.047)

MO15β (-0.047)

Pair Pair MO15α (-0.067)

MO16α (-0.067)

Spin Source MO14α (-0.089)

(a)

(b)

Fig. 3 Molecular Orbitals (MOs) related to copper 3d electrons in spin configuration I (SC(I)): a α MOs, b β MOs. Spin pair is shown with an arrow

3.2 3d z2 − y2 Type Spin Configuration Figure 4 depicts MOs related to copper 3d electrons in SC(II). Contrarily, 3d3x 2 −r 2 type α orbital (MO18α) is paired with corresponding β orbital (MO17β), while 3dz 2 −y 2 type α orbital (MO14α) has no paired β orbital. Hence, MO14α, where there is no orbital overlapped with oxygen atom, is responsible for total spin of OCuO cluster. Spin density of copper atom is +1.03. Spin densities of 3x 2 , 3y 2 and 3z 2 components are +0.01, +0.50 and +0.50, respectively, while spin density of 4s component is +0.01.

Quantum Spin Memory Using Inverse Copper Oxide Cluster …

11

Orbital Energy

MO17β (-0.063) Pair

MO18α (-0.102) Pair

MO15β (-0.106)

MO16β (-0.106) Pair

MO16α (-0.137)

MO17α (-0.137)

MO14β (-0.160)

Pair MO15α (-0.178) Spin Source MO14α (-0.223)

(a)

(b)

Fig. 4 Molecular Orbitals (MOs) related to copper 3d electrons in spin configuration II (SC(II)): a α MOs, b β MOs. Spin pair is shown with an arrow

3.3 3d Orbital Energy-Splitting In ligand field theory, specific pattern of 3d orbital energy-splitting is semiempirically predicted. In octahedral and square planar ligand fields, the patterns of 3d orbital energy-splitting are shown in Fig. 5. For example, in octahedral ligand field, highest orbital energy is given in two 3dx 2 −y 2 and 3d3z 2 −r 2 orbitals, while in square planar ligand field, highest orbital energy is given in 3dx 2 −y 2 orbital. Based on the facts, in OCuO cluster, it is considered that highest orbital energy is given in 3d3x 2 −r 2 orbital, due to electron repulsion along x axis. Figure 6a and 6b shows orbital energy diagrams of SC(I) and SC(II), respectively. Surprisingly, in both configurations, spin source α orbital has lowest orbital energy within orbitals related to copper 3d electrons, though ligand field theory predicts that

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T. Onishi Orbital Energy

Orbital Energy 3

2

3

2 2

3

2

2

2

3

3

3

3

3

3

Octahedron

3

2

2

Square Planar

(a)

(b)

Fig. 5 Patterns of 3d orbital energy-splitting in a octahedral ligand field, and b square planar ligand field. In the latter case, all atoms are allocated on xy plane

Orbital Energy

Orbital Energy

3 3 3 3 3

2

2

3 3

3

3 2

2

3 3

2

3

2

SC(I)

(a)

2

3

3

3

3 3

3 2

2

2

2

2

SC(II)

(b)

Fig. 6 Orbital energy diagrams of a spin configuration I (SP(I)) and b spin configuration II (SP(II))

spin source α orbital has highest orbital energy. This contradiction shows that ligand field theory cannot predict orbital energy correctly. Quantum chemists often try to associate ligand field theory with natural orbital. However, information of orbital energy disappears in natural orbital (Fig. 7). The direct association with natural orbital will lead scientific misunderstanding.

4 Inverse Cu4 O4 Cluster Previously, inverse Cu4 N4 system was investigated from the viewpoint of crystallography [3, 4]. Here, let us consider spin configuration in realistic system, inverse Cu4 O4 cluster, where oxygen atoms are allocated at edges of square (Fig. 8). Total

Quantum Spin Memory Using Inverse Copper Oxide Cluster …

Occupation Number 1.000

13

Occupation Number 1.000

SC(I)

SC(II)

(a)

(b)

Fig. 7 Natural orbitals with occupation number one: a spin configuration I (SP(I)) and b spin configuration II (SP(II))

O8

Fig. 8 Structure of inverse Cu4 O4 cluster y

Cu4

Cu3

O5 Cu1

O7

Cu2

O6

x

charge of inverse Cu4 O4 cluster is zero, as formal charges of copper and oxygen atoms are +2 and −2, respectively. Ligand field theory predicts one specific spin configuration, where highest orbital energy is given in 3d3x 2 −r 2 orbital, as same as OCuO cluster. However, from MO calculations, two different spin configurations were given in inverse Cu4 O4 cluster.

4.1 3d 3x 2 −r 2 Type Spin Configuration The total energy of quintet spin state was −186,681.5472 eV. Four unpaired α orbitals were given in MO45α, MO46α, MO47α and MO48α. They operate as spin source in the spin configuration. The wave functions (spatial orbitals) of MO45α, MO46α, MO47α and MO48α are expressed as follows.

14

ψ M O45α

ψ M O46α

ψ M O47α

ψ M O48α

T. Onishi

−0.19φCu1(3d x 2 ) + 0.47φCu1(3dy 2 ) − 0.30φCu1(3dz 2 ) − 0.47φCu2(3d x 2 ) +0.19φCu2(3dy 2 ) + 0.30φCu2(3dz 2 ) − 0.19φCu3(3d x 2 ) + 0.47φCu3(3dy 2 ) −0.30φCu3(3dz 2 ) − 0.47φCu4(3d x 2 ) + 0.19φCu4(3dy 2 ) + 0.30φCu4(3dz 2 ) = +0.05φ O5(2 px ' ) + 0.03φ O5(2 px ' ' ) − 0.05φ O5(2 py ' ) − 0.03φ O5(2 py ' ' ) +0.05φ O6(2 px ' ) + 0.03φ O6(2 px ' ' ) + 0.05φ O6(2 py ' ) + 0.03φ O6(2 py ' ' ) −0.05φ O7(2 px ' ) − 0.03φ O7(2 px ' ' ) + 0.05φ O7(2 py ' ) + 0.03φ O7(2 py ' ' ) −0.05φ O8(2 px ' ) − 0.03φ O8(2 px ' ' ) − 0.05φ O8(2 py ' ) − 0.03φ O8(2 py ' ' ) (21) 0.16φCu1(3d x 2 ) − 0.45φCu1(3dy 2 ) + 0.31φCu1(3dz 2 ) − 0.43φCu2(3d x 2 ) +0.15φCu2(3dy 2 ) + 0.29φCu2(3dz 2 ) − 0.16φCu3(3d x 2 ) + 0.45φCu3(3dy 2 ) −0.31φCu3(3dz 2 ) + 0.43φCu4(3d x 2 ) − 0.15φCu4(3dy 2 ) − 0.29φCu4(3dz 2 ) = −0.05φ O5(2 px ' ) − 0.04φ O5(2 px ' ' ) + 0.05φ O5(2 py ' ) + 0.04φ O5(2 py ' ' ) +0.07φ O6(2s ' ) + 0.08φ O6(2s ' ' ) − 0.05φ O7(2 px ' ) − 0.04φ O7(2 px ' ' ) +0.05φ O7(2 py ' ) + 0.04φ O7(2 py ' ' ) − 0.07φ O8(2s ' ) − 0.08φ O8(2s ' ' ) (22) −0.15φCu1(3d x 2 ) + 0.43φCu1(3dy 2 ) − 0.29φCu1(3dz 2 ) − 0.45φCu2(3d x 2 ) +0.16φCu2(3dy 2 ) + 0.31φCu2(3dz 2 ) + 0.15φCu3(3d x 2 ) − 0.43φCu3(3dy 2 ) +0.29φCu3(3dz 2 ) + 0.45φCu4(3d x 2 ) − 0.16φCu4(3dy 2 ) − 0.31φCu4(3dz 2 ) = −0.07φ O5(2s ' ) − 0.08φ O5(2s ' ' ) + 0.05φ O6(2 px ' ) + 0.04φ O6(2 px ' ' ) +0.05φ O6(2 py ' ) + 0.04φ O6(2 py ' ' ) + 0.07φ O7(2s ' ) + 0.08φ O7(2s ' ' ) +0.05φ O8(2 px ' ) + 0.03φ O8(2 px ' ' ) + 0.05φ O8(2 py ' ) + 0.03φ O8(2 py ' ' ) (23) −0.23φCu1(3d x 2 ) + 0.45φCu1(3dy 2 ) − 0.27φCu1(3dz 2 ) + 0.45φCu2(3d x 2 ) −0.23φCu2(3dy 2 ) − 0.27φCu2(3dz 2 ) − 0.23φCu3(3d x 2 ) + 0.45φCu3(3dy 2 ) = −0.27φCu3(3dz 2 ) + 0.45φCu4(3d x 2 ) − 0.23φCu4(3dy 2 ) − 0.27φCu4(3dz 2 ) −0.06φ O5(2s ' ) − 0.04φ O5(2s ' ' ) − 0.06φ O6(2s ' ) − 0.04φ O6(2s ' ' ) −0.06φ O7(2s ' ) − 0.04φ O7(2s ' ' ) − 0.06φ O8(2s ' ) − 0.04φ O8(2s ' ' ) (24)

From Eqs. (21)–(24), it is found that 3d3x 2 −r 2 and 3d3y 2 −r 2 orbitals are formed along x and y axes, respectively. It is difficult to form direct covalent bonding between copper 3d3x 2 −r 2 and 3d3y 2 −r 2 orbitals within inverse Cu4 O4 square. Hence, covalent bonding is formed via oxygen 2s or 2p orbitals. Then, the same oxygen 2s or 2p orbital is overlapped with two copper atoms. MO45α, MO46α, MO47α and MO48α are also depicted in Fig. 9. One titanium lobe interacts with one oxygen lobe. It is found that σ-type covalent bonding is formed.

Quantum Spin Memory Using Inverse Copper Oxide Cluster …

15

Orbital Energy MO48α (-0.6245)

MO46α (-0.6314)

MO47α (-0.6314)

MO45α (-0.6407)

Fig. 9 Selected molecular orbitals (MOs) in 3d3x 2 −r 2 type spin configuration of inverse Cu4 O4 cluster

4.2 3d z2 −x 2 Type Spin Configuration The total energy of quintet spin state was −186,678.0288 eV. It is 3.5184 eV higher than 3d3x 2 −r 2 type spin configuration. Further, singlet spin state was also given. The total energy was −186,678.0326 eV. Total energy difference between different spin configurations was slight (0.0038 eV). Quintet Spin State In quintet spin state, four unpaired α orbitals were given in MO45α, MO46α, MO47α and MO48α. The wave functions (spatial orbitals) of MO45α, MO46α, MO47α and MO48α are expressed as follows. ψ M O45α =

0.46φCu1(3d x 2 ) − 0.40φCu1(3dz 2 ) + 0.46φCu2(3dy 2 ) − 0.40φCu2(3dz 2 )

+0.46φCu3(3d x 2 ) − 0.40φCu3(3dz 2 ) + 0.46φCu4(3dy 2 ) − 0.40φCu4(3dz 2 )

(25)

ψ M O46α = 0.64φCu1(3d x 2 ) − 0.57φCu1(3dz 2 ) − 0.64φCu3(3d x 2 ) + 0.57φCu3(3dz 2 ) (26) ψ M O47α = 0.64φCu2(3dy 2 ) − 0.58φCu2(3dz 2 ) − 0.64φCu4(3dy 2 ) + 0.58φCu4(3dz 2 ) (27)

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T. Onishi

ψ M O48α =

−0.45φCu1(3d x 2 ) + 0.41φCu1(3dz 2 ) + 0.45φCu2(3dy 2 ) − 0.41φCu2(3dz 2 ) −0.45φCu3(3d x 2 ) + 0.41φCu3(3dz 2 ) + 0.45φCu4(3dy 2 ) − 0.41φCu4(3dz 2 )

(28)

From Eqs. (25)–(28), it is found that 3dz 2 −x 2 and 3dz 2 −y 2 orbitals are formed along y and x axes, respectively. There is no orbital overlap between copper and oxygen atoms. It implies that direct covalent bonding between copper 3dz 2 −x 2 type orbitals is formed within inverse Cu4 O4 square. MO45α, MO46α, MO47α and MO48α are also depicted in Fig. 10. One titanium lobe interacts with one titanium lobe. It is found that σ-type covalent bonding is formed. Singlet Spin State In singlet spin state, unpaired α and β orbitals were given in MO45α, MO46α, MO45β and MO46β. The wave functions (spatial orbitals) of MO45α, MO46α, MO45β and MO46β are expressed as follows. ψ M O45α = 0.64φCu1(3d x 2 ) − 0.58φCu1(3dz 2 ) + 0.64φCu3(3d x 2 ) − 0.58φCu3(3dz 2 ) (29) ψ M O46α = −0.64φCu1(3d x 2 ) + 0.58φCu1(3dz 2 ) + 0.64φCu3(3d x 2 ) − 0.58φCu3(3dz 2 ) (30)

Orbital Energy MO48α (-0.7606)

MO46α (-0.7660)

MO47α (-0.7660)

MO45α (-0.7727)

Fig. 10 Selected molecular orbitals (MOs) in 3dz 2 −x 2 type spin configuration of inverse Cu4 O4 cluster (Quintet Spin State)

Quantum Spin Memory Using Inverse Copper Oxide Cluster …

17

ψ M O45β = 0.64φCu2(3dy 2 ) − 0.58φCu2(3dz 2 ) + 0.64φCu4(3dy 2 ) − 0.58φCu4(3dz 2 ) (31) ψ M O46β = −0.64φCu2(3dy 2 ) + 0.58φCu2(3dz 2 ) + 0.64φCu4(3dy 2 ) − 0.58φCu4(3dz 2 ) (32) From Eqs. (29)–(31), it is found that 3dz 2 −x 2 and 3dz 2 −y 2 orbitals are formed in α and β orbitals, respectively. There is no orbital overlap between copper and oxygen atoms. It implies that direct covalent bonding between copper 3dz 2 −x 2 type orbitals is formed within inverse Cu4 O4 square. MO45α and MO46α have no paired β orbital, while MO45β and MO46β have no paired α orbital. Hence, MO45α and MO46α are responsible for α spin of inverse Cu4 O4 cluster, while MO45β and MO46β are responsible for β spin of inverse Cu4 O4 cluster. MO45α, MO46α, MO45β and MO46β are also depicted in Fig. 11. One titanium lobe interacts with one titanium lobe. It is found that σ-type covalent bonding is formed. In comparison with quintet spin state, only two titanium atoms participate in covalent bond formation.

Orbital Energy

MO46α (-0.7658)

MO46β (-0.7658)

MO45α (-0.7667)

MO45β (-0.7667)

Fig. 11 Selected molecular orbitals (MOs) in 3dz 2 −x 2 type spin configuration of inverse Cu4 O4 cluster (Singlet Spin State)

18 Fig. 12 Schematic of superexchange rule in Cu–O–Cu system

T. Onishi

Overlap

Cu1

Overlap

O2

Cu3

4.3 Comparison with Cu–O-Cu System We reconsider antiferromagnetic interaction in simple Cu–O-Cu model (Fig. 12). When left Cu1 and right Cu3 have α and β spins, respectively, the antiferromagnetic interaction is explained by superexchange rule. 3dx 2 −y 2 α orbital of Cu1 is overlapped with 2p α orbital of O2, while 3dx 2 −y 2 β orbital of Cu3 is overlapped with 2p β orbital of O2. Though α and β spins of O2 are cancelled out, Cu1 α and Cu3 β spins remain as spin source. In 3d3x 2 −r 2 type spin configuration of inverse Cu4 O4 cluster, spins of oxygen atoms are not cancelled out, though there is orbital overlap between copper and oxygen atoms. Hence, superexchange rule cannot be applied in this case.

4.4 Comparison with Two Quintet Spin Configurations In inverse Cu4 O4 cluster, two types of quintet spin configurations: (1) 3d3x 2 −r 2 type and (2) 3dz 2 −x 2 type were given. In 3d3x 2 −r 2 type spin configuration, 3d3x 2 −r 2 type α orbital has no paired β orbital, while 3dz 2 −x 2 type α orbitals are paired with corresponding β orbitals. Compared with 3dz 2 −x 2 type orbital, 3d3x 2 −r 2 type orbital is stabilised. It implies that orbital energy of 3d3x 2 −r 2 type orbital is lower than that of 3dz 2 −x 2 type orbital (Fig. 13 (a)). MO45α, MO46α, MO47α and MO48α (−0.6245 ~ −0.6407 au) are 3d3x 2 −r 2 type orbitals, while MO57α, MO61α, MO62α and MO64α (−0.5740 ~ −0.5888 au) are 3dz 2 −x 2 type orbitals. On the other hand, in 3dz 2 −x 2 type spin configuration, 3dz 2 −x 2 type α orbital has no paired β orbital, while 3d3x 2 −r 2 type α orbitals are paired with corresponding β orbitals. Contrarily, orbital energy of 3dz 2 −x 2 type orbital is lower than that of 3d3x 2 −r 2 type orbital (Fig. 13 (b)). MO45α, MO46α, MO47α and MO48α (−0.7606 ~ −0.7727 au) are 3dz 2 −x 2 type orbitals, while MO61α–MO64α (−0.6447 ~ −0.6639 au) are 3d3x 2 −r 2 type orbitals. From the viewpoint of energetics, orbital energy of 3d3x 2 −r 2 type orbital is almost unchanged, even if one electron or two electrons are occupied. However, 3dz 2 −x 2 type orbital is stabilised, when it has doublet spin state.

Quantum Spin Memory Using Inverse Copper Oxide Cluster …

19

Orbital Energy 3 (3 3

2

2

2

2

2

)

2

3

3 (3

(a)

2

2

2

2

2

2

)

(b)

Fig. 13 Orbital energy diagrams in a 3d3x 2 −r 2 and b 3dz 2 −x 2 spin configurations

4.5 Barrier Effect in O–Cu–O System In 3dz 2 −x 2 type spin configurations, there is no orbital overlap between copper and oxygen atoms. In other words, direct interaction of copper 3dz 2 −x 2 type electron with oxygen electron is inhibited. Namely, copper 3dz 2 −x 2 type electron works as barrier to interact with oxygen electron. The effect is called “Barrier Effect”. It is considered that it is due to direct covalent bonding formation between copper atoms.

5 Conclusions and Future: Quantum Spin Memory Conclusions (1) Two spin configurations exist in copper atom of inverse Cu4 O4 cluster: (1) 3d3x2 −r2 type and (2) 3dz2 −x2 type. (2) 3d3x2 −r2 type spin configuration has lower total energy than 3dz2 −x2 type spin configuration. (3) For 3dz2 −x2 type orbital, orbital energy of doublet spin state (one electron allocation) is lower than that of singlet spin state (two electrons allocation). (4) In 3dz2 −x2 type spin configuration, quintet and singlet spin states are almost degenerated. (5) Due to direct covalent bonding between copper atoms, barrier effect to inhibit participation of oxygen electron in covalent bonding is observed.

20

T. Onishi

These results could expand to application in advanced memory: Quantum Spin Memory. It is because two spin configurations are switchable in inverse Cu4 O4 cluster. Two cases are expected. One is a switch between two quintet spin configurations (3d3x 2 −r 2 ⇔3dz 2 −x 2 ). There is a problem how to detect a spin configuration forthwith. Note that spin multiplicity is the same in both spin configurations. The other is a switch to detect different spin states in 3dz 2 −x 2 type spin configurations (quintet spin state⇔singlet spin state). When controlling total energy difference between quintet and singlet spin states, it can be applied for quantum spin memory. For example, coordination chemistry technique makes total energy difference larger.

References 1. Uemura, H, Sugano S, Tanabe Y (1969) Ligand field theory and its application, Chap. 2 (In Japanese). Shokabo 2. Onishi T (2018) Quantum computational chemistry, Chap. 5. Springer 3. James AM, Laxman RK, Fronczek FR, Maverick AW (1998) Inorg Chem 37:3785–3791 4. Robinson, TP, Price, RD, Davidson, MG, Foxb MA, Johnson AL (2015) Dalton Trans. 44, 5611–5619

Taku Onishi is the editor of ‘Quantum Science’. See also ‘About the Editor’.

Characteristic Fermi Surface Properties in f -Electron Systems ¯ Yoshichika Onuki, Rikio Settai, Yoshinori Haga, Tetsuya Takeuchi, Masato Hedo, and Takao Nakama

Abstract The 4 f -electrons in the rare-earth compounds are generally localized and order magnetically mediated by the Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction. The electronic states or the Fermi surface properties in the Ce- and Eubased compounds can be changed by decreasing temperature and/or applying pressure. Antiferromagnets CeRhIn5 and EuCu2 Ge2 are characteristic, revealing the firstorder phase transition under pressure, which is based on a combined phenomenon between the Kondo effect and the sharp valence crossover. Other antiferromagnets, CeIrSi3 with Rashba-type tetragonal structure and EuPtSi with chiral cubic structure, revealed huge upper critical fields in superconductivity for H ll [001] and the magnetic skyrmion lattice for H ll , respectively. The noncentrosymmetric crystal structure also brings distinctive properties to the electronic states. Keywords Fermi surface · de Haas-van Alphen effect · Pressure · Magnetism · Superconductivity · CeRhIn5 · CeIrSi3 · EuCu2 Ge2 · EuPtSi ¯ Y. Onuki (B) · M. Hedo · T. Nakama Faculty of Science, University of the Ryukyus, Nishihara, Okinawa 903-0213, Japan e-mail: [email protected]; [email protected] M. Hedo e-mail: [email protected] T. Nakama e-mail: [email protected] ¯ Y. Onuki RIKEN Center for Emergent Matter Science, Wako, Saitama 351-0198, Japan R. Settai Faculty of Science, Niigata University, Niigata 950-2181, Japan e-mail: [email protected] Y. Haga Advanced Science Research Center, Japan Atomic Energy Agency, Tokai-mura, Ibaraki 319-1195, Japan e-mail: [email protected] T. Takeuchi Low Temperature Center, Osaka University, Toyonaka, Osaka 560-0043, Japan e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 T. Onishi (ed.), Quantum Science, https://doi.org/10.1007/978-981-19-4421-5_2

21

¯ Y. Onuki et al.

22

1 Introduction In lanthanide and actinide compounds, f -electrons usually give rise to magnetism, but also various interesting phenomena such as heavy-fermion systems and anisotropic superconductivity. The 4 f (5 f )-electrons of the lanthanide (actinide) atom are confined deeply to the 5s 2 5 p 6 (6s 2 6 p 6 ) shell interior due to the strong centrifugal force potential l(l+1)/r 2 . Here, the f -electron has an azimuthal quantum number l = 3 and r is the radius of the orbital. This is the reason why f -electrons have an atomic character even in compounds. Thus, the localized f -electron picture is a good starting point of view for magnetism, especially in lanthanide compounds. On the other hand, the tail of the wave function in f -electron compounds extends outside the closed 5s 2 5 p 6 (6s 2 6 p 6 ) shell because it is strongly affected by potential energy, relativistic effect, the distance between lanthanide (actinide) atoms, and hybridization between f -electrons and conduction electrons. The f -electrons in cerium and uranium compounds cause a variety of characteristic properties, including heavy-fermions and anisotropic superconductivity, as mentioned above [1–5]. In these compounds, the Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction and the Kondo effect compete with each other. In the RKKY interaction, an indirect f – f interaction mediated by the spin polarization of the conduction electrons acts between the magnetic moments of the localized f -electrons, enhancing the long-range magnetic order. The strength of the RKKY interaction is proportional to the square of the magnetic exchange interaction Jcf between the f electron and the conduction electron. The corresponding characteristic temperature TRKKY is expressed as I I2 TRKKY ∝ D(εF ) I Jcf I ,

(1)

where D(εF ) is the density of electronic states at Fermi energy εF . Applying second-order perturbation theory based on s-d exchange interaction, Kondo indicated that electrical resistance diverges logarithmically with decreasing temperature and identified the cause of the long-standing resistance minimum problem observed in copper and gold with small amounts of iron impurities [6]. This was also the start of the Kondo problem. The Kondo effect I Iis characterized by the Kondo temperature TK , which depends exponentially on I Jcf I as follows: TK ∝ e−1/(D(εF )| Jcf |) .

(2)

In the simplest case, the localized spin ↑ (l) of the magnetic impurity is compensated by the spin ↓ (c) of the conduction electron at T = 0 K, forming the Kondo cloud. As a result, the spin-singlet state ↑ (l) ↓ (c) or ↓ (l) ↑ (c) with its binding energy kB TK is yielded [7]. The logarithmic divergence that appears in the perturbation theory thus changes to a constant in the singlet bound state at low temperatures. In Ce-based intermetallic compounds, the ground state is not a scattering state but a coherent

Characteristic Fermi Surface Properties in f -Electron Systems

23

100 X=

Cex La1-xCu6 J // a-axis

0.50 0.73 0.29 50 0.90 0.094 0.99 1.0 0.0 0 0.01

0.1

1 10 Temperature (K)

100

Fig. 1 Temperature dependences of the electrical resistivities in Cex La1−x Cu6 . The compound for x = 0.094 indicates a characteristic Kondo scattering and the unitarity limit, while CeCu6 (x = 1) shows a coherent Kondo state or a typical heavy-fermion behavior. The original data have been published in ref. [8]. ©[1986] The Physical Society of Japan (J. Phys. Soc. Jpn. [55, 1294].)

Kondo lattice state with an extremely large electronic specific heat coefficient γ ~ 1000 mJ/(K2 ·mol), unlike the case of Fe-3d electron Kondo impurities. Figure 1 shows the temperature dependences of electrical resistivities in Cex La1−x Cu6 . The electrical resistivity for x = 0.094 shows a minimum around 30K, a − log T dependence below that temperature, and at the lowest temperature a constant value, the so-called unitarity limit [8]. On the other hand, CeCu6 corresponding to x = 1 in Fig. 1 shows a minimum around room temperature and a − log T dependence below that temperature, followed by a peak around 15 K and then a T 2 -dependence, and finally has a residual resistivity. Such behavior is typical for coherent Kondo lattice systems or heavy-fermion systems. The Kondo scattering, the Kondo cloud (the spin-singlet state), and the 4 f -derived band in the periodic Kondo lattice are schematically shown in Fig. 2a, b, and c, respectively. We show in Fig. 3 the temperature dependences of the magnetic susceptibility of CeCu6 , CeRu2 Si2 , and UPt3 , UPd2 Al3 [5]. The magnetic susceptibility of each material increases with decreasing temperature following to the Curie-Weiss law at high temperatures, reaching a maximum at the characteristic temperature Tχ max , and below Tχ max , the magnetic susceptibility becomes almost temperature independent. In this state, the f -electron system changes to a new electronic state called the heavy electron or heavy-fermion state. Here, Tχ max approximately corresponds to the Kondo temperature TK . We can understand the heavy-fermion state in cerium compounds roughly as follows [9]. Figure 4 shows the level scheme of the 4 f -electrons for the trivalent cerium ion Ce3+ in a non-cubic crystalline electric field. Here the electron configuration of the Ce atom is [Kr core]4d 10 4 f 1 5s 2 5 p 6 5d 1 6s 2 and valence electrons 5d 1 6s 2 contribute to the band electrons in metallic compounds. The 4 f level of the cerium ion splits into J -multiplets (J =7/2, 5/2) due to spin-orbit interaction. Here, the 4 f -

¯ Y. Onuki et al.

24 Fig. 2 a Kondo scattering in the impurity system at T > TK , b the corresponding Kondo cloud at T < TK , and c the 4 f -derived band in the periodic Kondo lattice

(a) T > T K

(b) T < T K

(c) T 248 nm. Theoretical methods such as quasi-classical trajectory (QCT) calculations may help to determine the multiple dissociation pathways revealed by the ion image experiment, e.g., the origin of the slower component. We applied a 4D reduced QCT calculation with CH3 frozen as a unit in the calculation, which is based on the fact that the CH3 group may not decompose in the photodissociation. According to the

Chemical Reaction Kinetics and Dynamics Re-Considered ...

91

Fig. 10 JCO state-specific translational energy distributions from experiments (left and middle) and from quasi-classical trajectory simulations (right). The left panel shows the imaging results of the CO product, converted to the corresponding translational energy distributions in the c.m. frame. Separate contributions of different pathways are obtained by deconvolution from the imaging results: triple fragmentation (orange), roaming (blue), and molecular mechanisms (red). The translational energy distribution (green) shows a bimodal feature, consisting of a sharp peak in slower region and a broader profile in faster region. Despite that a reduced-dimensional model potential was adopted in simulations, the two data sets share a common feature

QCT results shown in Fig. 11, the translational energy distributions of CO from triple fragmentation tend to broaden and shift toward higher energy at shorter photolysis wavelengths, due to the increased internal energy of HCO fragment available for the CO translations after decomposition. The contribution of triple fragmentation decreases at longer photolysis wavelengths, since there is less residual energy in HCO

92

T. Kasai et al.

Fig. 11 Plots of the branching ratios, either obtained from the experiments (left) or simulations (right). The data refer to each rotationally state-resolved products, CO (J = 10, 14, 19, and 24, corresponding to 0.6, 1.2, 2.1, and 3.3 kcal/mol, respectively.). The slow component from experiments appears at energy threshold of about 115 kcal/mol, a value slightly higher than the one of the triple fragmentation channel in simulation

to cause sequential decomposition. In contrast, the roaming mechanism benefited from the suppressed decomposition of HCO and CH3 O products in slow motion. Such a channel competition occurs on the ground potential energy surface S0 to which the S1 surface is connected via conical intersection [73]. As shown in Fig. 11, the relative branching between the various pathways is presented as a function of excess energy. When the photolysis wavelength is shorter than 248 nm, one has to consider the competition among the pathways of conventional TS, roaming, and triple fragmentation. The CO component contributed by the triple fragmentation with extremely low translational energy decreases with JCO , for less energy deposited in HCO at larger JCO . After excluding the contribution from the triple fragmentation, the simulated translational energy distribution of CO still shows a bimodal feature: a narrow peak in the low-energy region from the roaming pathway and a broad component at higher energy due to the conventional TS route. Note that without the contribution of triple fragmentation, the primary molecular product of this molecule is to produce a close-shell molecular product in the two-body dissociation channel, CO, and CH3 OH. The CO contribution is mainly dominated by the roaming and the TS processes when photolysis wavelengths >248 nm, since the residual energy in HCO is insufficient for decomposition. The relative branching of the roaming process is enhanced when there is no contribution of triple fragmentation, but the roaming contribution also decreases with increasing JCO . The CO (v = 1) product was also investigated at three selected rotational levels, JCO = 2, 7, and 11 at 248 nm, while the signal-to-noise ratios for higher JCO levels and CO (v = 2) are too low to be detected [73]. The roaming fraction yields 0.29 ± 0.07, 0.20 ± 0.09, and 0.17 ± 0.04 for the above-mentioned three JCO levels, respectively, which decrease with the increase of J. The result of CO (v = 1, J) shares a similar trend with

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the CO (v = 0, J) case. The rotational Boltzmann fits to the CO rotational population from each pathway show the rotational temperatures of 200 K for the roaming route and 420 K for the TS pathway. The CO products from the roaming mechanism are rotationally cold with low translational energy, with the total roaming fraction of CO (v = 1) found to be 0.2 ± 0.06. In summary, the trend of photolysis energy dependence for the relative branching of each dissociation pathway observed is consistent with that from QCT simulations. Apart from the ion imaging experiments, the internal state distribution of CO (v = 1, J) can be detected by the Fourier-transform infrared (FTIR) emission spectroscopy. After the molecule absorbs one 248 nm photon, the emission spectra of CO product at 1900–2200 cm−1 can be resolved with a 0.25 cm−1 resolution in frequency and 5 μs resolution in time. The photolysis experiment was conducted in the presence of Ar gas with a pressure of about 3000 mTorr. Note that the collisions with Ar gas increase the radiationless transitions from the excited electronic state(s), which could be observed in the enhanced signal-to-noise ratio of the measurement [100–103]. A reliable reconstruction of the rotational population distributions is feasible only with the aid of forward spectral convolution, since the assignment of every single rotational line is obstructed by the spectral congestion among several vibrational states. The rovibrational emission spectra of CO from the measurement at 248 nm are shown in Fig. 12, along with the simulation. The time-dependent rotational distribution at each vibrational level can be extracted from these spectra. The rotational temperature of each vibrational level can be obtained by fitting the rotational population according to the definition of the Boltzmann plot: ln[N v,J /(2 J + 1)] versus E v,J ; where N v,J is the rovibrational population, and E v,J is the eigenvalues of corresponding quantum states. Then the time dependence of rotational temperature at each vibrational level can be determined accordingly (Fig. 12). In practice, the temperature profiles should be extrapolated to the zero-delay time to bypass the effect of the rotational relaxation by Ar collisions. Therefore, the step-scan FTIR emission spectroscopy provides information on time-dependent rovibrational state distribution. As shown in Fig. 12, we obtained a set of well-recognized rovibrational spectra of the CO (v = 1–4) from the 248 nm photolysis of methyl formate. The bimodal rotational distributions for CO (v = 1, 2) with the low and high rotational temperatures are produced from roaming and TS pathways, respectively. In contrast, the rotational populations of CO (v = 3–4) only show a single rotational temperature. The CO (v = 1) possesses the rotational temperature of 470 and 1080 K for the roaming and the TS pathways. The discrepancy in the rotational temperature from ion imaging results can be ascribed to the difference in the experimental conditions. The supersonic molecular beam condition supports the collisionless events, while the FTIR measurements are conducted at room temperature in presence of Ar colliders. The thermal condition with Ar-induced collisions could yield hotter and broader rovibrational energy disposals. Thus, the FTIR method observed the whole rotational profiles for at least v = 1–4, including the rotational bimodality at v = 1–2. In contrast, the supersonic beam experiment obtained the CO (v = 1) only with low rotational levels. The roaming/TS branching ratio can be determined by deconvoluting the relative contribution of cold and hot rotational populations for

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Fig. 12 Rovibrational spectra of CO at various time delays from photodissociation of methyl formate at 248 nm. The spectra were obtained using a 5 μs step-scan mode of time-resolved FTIR spectroscopy (left), with the comparison between measurement (7.5 μs) and simulation given on the right side. The v state-resolved rotational temperatures of CO (v = 1, 2, 3, and 4) were obtained by extrapolating to the zero delay time. The vibrational population is determined by summing the contribution from all the rotational levels at a given delay time

each vibrational state. The CO (v = 1) roaming fraction is evaluated to be 0.3 ± 0.04, in agreement with that by the ion imaging. The CO + CH3 OH products detected by both supersonic beam and FTIR experiments at 248 nm are produced through the S1 /So nonadiabatic transition via conical intersection, except that those products which carry more internal energy are detected in the FTIR emission experiment. Roaming signature of CO product is observed up to v = 2 for the FTIR experiments and up to v = 4 for the QCT calculations. Therefore, the FTIR emission spectroscopy can characterize the vibrational aspect of the roaming/TS branching ratio, whereas the ion imaging technique is suitable to visualize the rotationally state-resolved dependence of CO (v = 1). The complementary aspects of translational and internal energy characterization allow us to gain deeper insight into roaming dynamics.

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7.3 Aliphatic Aldehydes: Roaming Through a Loose Complex Structure 7.3.1

Photodissociation of Acetaldehyde

The bimodal rotational energy distribution of CO from the 308 nm photolysis of acetaldehyde has been first obtained by Houston and Kable, CH3 CHO, detected by using the vacuum ultraviolet (VUV) laser-induced fluorescence (LIF) method [85]. The low-JCO and high-JCO components are attributed to CH3 -roaming and conventional TS pathways, respectively. The relative branching of CH3 -roaming and TS pathways are about 13–87% at v = 1 and 0 to 100% at v = 2. Bañares and coworkers also showed that the roaming branching is about 20% at CO (v = 0) at 308 nm photolysis [104]. Heazlewood et al. reported a quite different roaming branching by analyzing the IR emission spectra of CH4 , 84 ± 10% [87]. In a sequential study by Kable and co-workers, the CO products were investigated in detail with a 2D REMPI/ion imaging technique in the range of 308–328 nm [105]. The low-JCO and high-JCO components of the bimodal distribution are re-ascribed to two roaming pathways: an H atom about a CH3 CO moiety (low-JCO ) and the other roaming of CH3 around CHO (high-JCO ), respectively. The relative branching ratios of CH3 roaming, H-roaming, and TS pathway were evaluated to be 71 ± 12%, 13 ± 3%, and 16 ± 10%, respectively [105], in which the tight TS contribution is derived from computational result [87]. We observed the bimodal feature in both the CO (v = 1) rotational state and CH4 vibrational state distributions by utilizing step-scan FTIR emission spectroscopy [106]. The bimodal feature of the rotational distribution of CO (v = 1) raises two rotational temperatures (1150 ± 80 and 200 ± 20 K), with the relative contribution of low-/high-JCO components to be (10 ± 2)/(90 ± 5)%, similar to the experimental value(13%/87%) [85]. The Δv3 = -1 C–H stretching emission of the CH4 co-product can be found in the range of 2500–3100 cm−1 . It was acquired at a delay time of 7 μs to decrease the spectral overlapping from the emission of HCO and CH3 . With the aid of the maximum entropy model [107] in the analysis, the vibrational distribution (Fig. 13) also shows a bimodal feature in which the branching ratio is about (25 ± 5)/(75 ± 13)% for the low-/high-energy components, respectively. The rotational part of available energy is assumed to be thermalized and deconvoluted from the vibrational degree of freedom in the analysis with the maximum entropy model. The model states the energy distribution which represents the current state of knowledge about the system is the most probable one with the largest informational entropy, in the context of precisely stated prior data. Figure 14 shows the relative energy of stationary points regarding the three pathways [87, 105], in which the tight TS is almost energetically equivalent to the CH3 roaming saddle point (SP); the energy of H-roaming SP is expected to be close to the low-lying vibrational levels of CO of the related radical products. This explains why the H-roaming signature disappears in CO (v ≥ 2): the excess energy of the H-roaming channel makes CO (v = 1) energy-accessible, but not sufficient for states with v > 1.

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Fig. 13 CH4 vibrational energy distribution (binned at 1500 cm−1 interval) extracted from the rovibrational spectrum observed

The relative contributions of H-roaming, CH3 -roaming, and TS pathway are evaluated to be (8 ± 3)%, (68 ± 10)%, and (25 ± 5)%, respectively, in agreement with the previous study [105]. The low-energy component of CH4 vibrational energy distribution matches the outcome of classical trajectories from the TS pathway, despite the small discrepancy which is probably due to the partial vibrational relaxation within a 7 μs delay in measurement [87]. The contribution of the conventional (tight) transition state was verified experimentally for the first time. The QCT results imply the origin of the CH4 high-energy component [87], which can be attributed to H- and CH3 -roaming pathways [105].

Fig. 14 The relative energy level scheme for the three CO-forming pathways: H or CH3 -roamings and also the tight SP (i.e., tight TS). The CO vibrational levels are labeled with the CO stretch mode (1800 cm−1 ) of CH3 CO structure, since the H + CH3 CO radicals which relate to H-roaming are expected to have similar energies as the CO vibrational level

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If the photolysis energy is high enough such as the 248 nm photoexcitation of CH3 CHO, the opening of a triple fragmentation channel, H + CO + CH3 , may compete with the aforementioned TS and roaming pathways. We conducted the step-scan FTIR emission experiment at this wavelength, to obtain the time-dependent state-resolved CO (v = 1–4) emission spectra by the photolysis of CH3 CHO in the presence of Ar at 3.6 Torr [108]. The rotational energy distribution of CO (v = 1) was found to be bimodal with two rotational temperatures of 1400 ± 110 and 540 ± 70 K from the Boltzmann fit. While for higher vibrational levels such as v = 2, 3, and 4, the rotational distributions can still be well fitted with a single temperature. The vibrational population of CO is Boltzmann-like with a vibrational temperature of 4900 ± 100 K. The relative contribution of the low-JCO and high-JCO components at v = 1 is about 0.19 and 0.81, respectively [108]. A bimodal vibrational energy feature in the C–H stretching emission spectra of CH4 (Δv3 = −1) was observed [108], similar to that at 308 nm [106], except the higher vibrational excitation can be achieved from the high excess energy. The CH4 vibrational population obtained by classical trajectories initiated from the TS structure shows higher energy than the observed low-energy component at around 20 kcal/mol, similar to the case under 308 nm photolysis. The addition of Ar gas enhances the collision-induced radiationless decay, as expected [102]. The low-v CH4 component is arguably due to the TS pathway [87, 106], consistently to the case at 308 nm [106]. When the classical trajectories are initiated from the global minimum (GM) of PES, the CH4 vibration energy distribution only has a single distribution in the high-energy region, which again shows the consistency with that of the experimental result. As shown in Fig. 15, the rotational energy distribution of CO by initiating the classical trajectory QCT at TS structure shows a peak at 20 kcal/mol and fades away at 40 kcal/mol. The calculated result has higher rotational excitation than the observed one from the experiment (about 1300 K), which implies that the CO (v = 1) rotational bimodality may not arise from the dissociation through the TS structure. With the help of the microcanonical prior statistical model [108, 109], the vibrational population of CO through the triple fragmentation is about 0.942: 0.057: 0.00079: 0 for v = 0–3. Therefore, the lower-JCO and high-JCO components of the bimodal rotational distribution are due to the three-body breakup and a CH3 -roaming, respectively. The branching fraction of triple fragmentation and the molecular channel (CO + CH4 ) are found to be 0.068 and 0.21, respectively, revealed by the classical trajectories initiated from GM. One then estimates the contribution of the triple fragmentation of the CO (v = 0–3), to be about 0.77, 0.18, 0.038, and 0.0065, respectively. Figure 15 shows the CO rotational energy distribution produced via various dissociation pathways. The same simulation gives the vibrational population of the CO products from the molecular channel (CO + CH4 ) to be 0.53, 0.27, 0.13, and 0.06 for v = 0, 1, 2, and 3, respectively. When the branching fractions 0.21 and 0.068 are adopted for the molecular and triple fragmentation channels listed above, the CO (v = 1) product has a branching ratio of 4.6 from each individual channel, a value in good agreement with the observed value of 0.81/0.19 or 4.3 in the above-mentioned rotational bimodality [108]. Such estimation allows us to conclude that the origin of the CO rotational

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Fig. 15 The rotational energy distribution of CO at rotational temperature of1300 K (black trace) from Boltzmann fit by averaging the outcome of CO (v = 1–4). The quasi-classical trajectories initiated from the transition state (red trace) and the global minimum (pink trace). The latter is contributed by the sum of two channels: three-body breakup (CO + H + CH3 , green trace) and the two-body molecular channel (CO + CH4 , blue trace)

bimodality should be ascribed to both triple fragmentation and the CH3 -roaming mechanism. In the aforementioned bimodal feature of CH4 , the fraction of the lower vibrational component (the TS pathway), is about 29% of the higher v component (the roaming pathway). The fraction of the low-JCO component (triple fragmentation) is found to be 0.23, given the low-/high-JCO population ratio at CO v = 1 is well known [108]. Thus the complete ratio of the triple fragmentation/roaming/TS pathways is 0.23: 1: 0.29. In conclusion, the photodissociation dynamics of CH3 CHO is quite different between 308 and 248 nm photolysis wavelength. The former proceeds via roaming (H atom or CH3 group) and TS routes, while the latter contains an extra triple fragmentation pathway.

7.3.2

Propionaldehyde and Larger Aldehydes

The FTIR emission spectroscopy is employed to obtain time-resolved infrared emission spectra of CO (v = 1–4) in the photodissociation of propionaldehyde with the presence of Ar at 3000 mTorr at 248 nm. The observed CO fragment is vibrationally hot and rotationally cold. The rotational population was found to be a single Boltzmann distribution at a vibrational level. The zero-time extrapolated rotational temperatures are found from 870 to 1390 K for CO (v = 1–4). The time-dependent vibrational populations are obtained to be 0.49 ± 0.05, 0.28 ± 0.03, 0.15 ± 0.02, and 0.08 ± 0.01 for v = 1–4, respectively, when the delay time is extrapolated to 0 μs. The (vibrational) state-averaged CO rotational energy disposal is evaluated to be 2.0 ± 0.2 kcal/mol [110, 111]. The vibrational temperature is fitted to be 5200 ± 100 K, a temperature which corresponds to a Boltzmann vibrational population

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(v = 0–4) 0.47:0.26:0.14:0.08:0.05. Thus, the CO products have average vibrational energy of 9.0 ± 0.6 kcal/mol with ZPE included. Figure 16a shows the dissociation pathways related to the channels of our interest. The energies of stationary points for reactants, TS, roaming SP, and products are obtained with ZPE being included, followed by the IRC calculation to confirm the connection between them. The CO fragments can be obtained from the following two pathways (Fig. 16a): (i) to form CO + CH3 CH3 via a TS at 75.5 (83.7) (or 76.7 (85.1)) kcal/mol from energized So state in cis-form (or in gauche-form), and (ii) triple fragmentation to form CH2 CH2 + H2 + CO via a TS at 68.4 (70.8) kcal/mol. Despite a lower barrier threshold of (ii), we do not find ethylene at emission frequency corresponding to the symmetric stretch (v9 ) mode (3105.5 cm−1 ). Further, the negligible contribution of CO coming from the three-body dissociation is confirmed by characterizing the vibrational energy of CO via a prior statistical model [112, 113]. The propionaldehyde exhibits the roaming saddle point with the typical feature of a loose transition state just like acetaldehyde and formaldehyde [114]; (Fig. 16b) shows a low imaginary frequency of 117.1i cm−1 as well as low frequencies in several transitional modes. The C(1)H3 C(2)H2 and C(3)HO moieties are almost parallel to each other at a large C(2)-C(3) distance of 3.93 Å at this roaming saddle point. The H atom of CHO may oscillate slowly between these two moieties, before the final abstraction by CH3 CH2 , resulting in a small torque to CO causing low rotational excitation. In contrast, CH3 CH2 is at a short C–C distance of 2.17 (2.20 for cis) Å from CHO for the TS structure (route (i)). The H abstraction leads to large translational and rotational energy partition in C2 H6 + CO. The direct dynamic simulations were initiated at each saddle point structure without excess energy, in order to characterize the dynamic signature for all the pathways. Figure 17a–c shows the time-dependent rovibrational coordinates of the products. The roaming products contain larger energy on vibrational modes of CO and C2 H6 but a lower CO rotational excitation. Figure 17d shows the fractions of energy disposal from each pathway: the roaming (tight SP) CO gains 17% (44%) of excess energies and distributes to the translational, vibrational, and rotational degree of freedoms as 4 (25), 6 (3), and 7 (16)%, respectively. C2 H6 obtains 79.2% (32.7%) of excess energy either to vibration (60.8% (29.4%)) or to rotation (18.4% (3.3%)). The CO product from the roaming pathway is both translationally and rotationally cold, whereas the co-product (C2 H6 ) attains high vibrational excitation. In contrast, the CO from tight TS is translationally and rotationally hot but vibrationally cold. The RRKM rate constant calculations [115, 116] were performed for each reaction channel. The roaming rate constant on the order of 1010 cm3 /molecule/s at 115.2 kcal/mol (or 248 nm) dominates over other dissociation pathways to form molecular products. Given the evidence from kinetic and dynamic aspects, the roaming mechanism can be more important with molecules of higher complexity, due to their variety of alternative pathways which cause the bypass of conventional MEP. A sequential study on isobutyraldehyde (CH3 CH3 CHCHO) at 248 nm confirmed the aforementioned trend. Thus, the family of aliphatic aldehydes can be considered as the first case with a specific functional group which demonstrates the predominance of the roaming mechanism.

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Fig. 16 a Threshold energies (kcal/mol) and reaction pathways of radical and molecular channels on S0 state surface of C2 H5 CHO. b The roaming pathway and its IRC. The Geometry optimization and IRC calculation were performed via CASSCF(6,6)/6–311 + + G(d,p). The values with and without parentheses were obtained via CCSD(T)/6–311 + + G(d,p) and CASSCF(10,9)/CASPT2/6–311 + + G(d,p), respectively

7.4 Concluding Remarks The roaming mechanism was traditionally described as a process due to “the frustrated dissociation to form radical product” via a barrierless dissociation pathway,

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Fig. 17 Time-dependent geometric parameters and the product energy distribution obtained by the direct dynamic simulation from saddle points which relate to CO-forming pathways. a The CO bond distance variation, b the C’–H’ bond distance change on the CH3 C’H2 H’, and c the CO rotational angle variation of the CO-forming pathways via the two saddle points. Note that the rotational angle along the tight TS pathway can attain 0° or 180°, which implies a clear in-plane motion. d Bar charts of relative energy disposals from different pathways. All the outcomes are from the QCT results initiated on individual saddle point structure, except that “Tight 248 nm” is adopted from J. Phys. Chem. A, 2006, 110, 11,230, which denotes QCT results on the tight TS structure of C2 H5 CHO at 248 nm

since its first discovery by the Suits and Bowman groups in 2004. It is the weak, longrange attractive interaction between the two radical moieties which is responsible for the slow motion throughout the configurational space before the birth of molecular products, therefore the relative branching of the roaming pathway in H2 CO possesses a window along the photolysis energy. Extensive studies on roaming mechanisms of different molecules in the last decade, however, reveal the complexity in roaming dynamics and turn this traditional picture toward a diversity of interpretations. In this account, we focus on methyl formate, formic acid, and aliphatic aldehydes. The roaming mechanisms for the first two molecules rely on the radiationless decay of energetic precursors at the conical intersection of PES, then meet a bifurcation of transition state and roaming routes which is determined by additional dynamic

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constraint. It turns out that the roaming fraction is regulated by photolysis energy. For methyl formate, the photolysis wavelength on the verge of the energy threshold of triple fragmentation may maximize the extent of roaming. In contrast, the fraction of roaming progressively opens along the sequence of aliphatic aldehydes, when their fragmentation is depicted as two moieties weakly bound at a distance. The energy difference between transition state and roaming paths is indicative of the extent of the relative contribution of the roaming mechanism. Roaming paths are even more energy-accessible for large alkyl moieties such that their energies are increasingly lower than those of the conventional transition states, which decrease in importance already in propionaldehyde and as the size of the aliphatic aldehydes becomes larger. This finding for the aliphatic aldehydes is arguably general and may be expected in other families of organic compounds. From the experimental aspect, both the step-scan FT infrared emission spectroscopy and the velocity-mapping ion imaging methods complementarily provide internal and translational energy distributions of the products, respectively. The imaging method can obtain J-dependence of the roaming branching based on the translational energy release of fragments. The time-resolved FTIR emission spectroscopy offers a complementary aspect to determine the internal state distribution of roaming products. It helps to reveal the vibrational-state dependence of the roaming signature. Furthermore, the internal energy distribution of co-products may be an alternation to characterize the roaming signature. Another way to identify the roaming mechanism is by utilizing the vector correlation of products, though it is not widely applied so far. The developed multicenter impulsive model (MCIM) serves as a useful tool for predicting the roaming scalar and vector properties without involving the global energy surface. From this aspect, the impulsive model views the PED and fragment vector correlation via the roaming pathway as a result of a series of energy transfers and effective coupling between rotational excitation of products and parent vibrational frequency of transverse modes, respectively. A generalized kinematic model, which is inspired by the original idea of MCIM, is presented in Topic 3 of this chapter.

8 Topic 3: The Pathway Discrimination of Gas Phase Unimolecular Dissociation by Dynamical Model: The Recent Developments Po-Yu Tsai

8.1 Introduction Investigation of the photon-induced unimolecular processes, such as unimolecular dissociation or isomerization, permits us to understand the nature and characteristics of photochemical reactions, including the reaction pathway, product state

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propensity, and the steric effect. Though the ideas of the transition state (TS) and minimum energy reaction path have provided a clear view of the unimolecular dissociation dynamics of polyatomic molecules since the middle of the last century [117, 118], the true behavior of molecular trajectories may go beyond the scope of these formalisms. Such non-conventional behavior includes, for example, nonRRKM behavior [119], tunneling effect [120], multiple saddle points, reaction path bifurcation [121], isomerization-mediated processes [122], roaming mechanism [82, 123–125], and the nonadiabatic transition between multiple electronic states [126]. The determination of the reaction rate constants is one of the ultimate goals of chemical kinetics, and these non-conventional behaviors are vital factors in the theoretical prediction of state-to-state rate constants. In addition, further information about the reaction mechanism is necessary to clarify the detailed reaction mechanism. The product energy disposals and vector correlation usually provide comprehensive information about a chemical reaction, and these product state properties can be observed directly in measurements of microscopic scales nowadays. For the unimolecular dissociation processes, the nature of the dissociation mechanism can be characterized by analyzing the product state properties. Such investigation of the product states is especially useful when there are multiple pathways which lead to the same photoproduct. The molecules experience the local shape of the potential energy surface (PES) on different dissociation pathways, which may provide distinct characters to the product state distribution. Accordingly, the analysis of product state distribution is helpful for the discrimination of the multiple dissociation pathways, given that the state redistribution due to either interfragment or intrafragment mechanism is negligible. If possible, classical or quantum dynamical calculation on the full-dimensional PES can provide comprehensive information about the dissociation mechanism. Alternatively, it is straightforward to utilize simple dynamical models which not only help to distinguish the dissociation mechanism, but also reveal the dynamical origin of the features of product state distribution of each pathway. Further discussion about these theoretical models can be achieved by considering the following aspects. We will only consider the adiabatic dynamics on the single adiabatic PES in this section. To the best of our knowledge, there is no simple model that can predict both the electronic and nuclear-motional state distribution of the products which are dissociated from multiple adiabatic PESs. Therefore, only the adiabatic dynamics will be considered in this section. Once the state redistribution between multiple adiabatic PESs via nonadiabatic transitions is negligible, the molecular dissociation dynamic can be properly described within the framework of the Born–Oppenheimer approximation. Three molecular dissociation cases (Fig. 18) with different natures on the variation of nuclear potential energy along the dissociation coordinate are introduced here, which possess distinct situations on the energy redistribution among the nuclear degree of freedom (nuclear DOF). In each figure, the molecular potential energy (ordinate) at a certain electronic state is represented as the function of the dissociation coordinate (abscissa).

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Fig. 18 The potential energy curves of the three types of dissociation pathways which are commonly met in the studies of molecular photodissociation dynamics. See text for the details of these pathways

(a) Direct dissociation on a repulsive PES This type of dissociation (Fig. 18a) is usually found in the dissociation on the electronically excited state of the parent molecule, yielding the radical products via a one-bond-breaking event. The parent-molecular flux on the Franck– Condon region of excited state PES is prepared by vertical excitation of the parent molecules from the ground state PES via one or multiple photons. There is typically no time for redistribution of nuclear DOF, and the product state distribution can be predicted by considering the dynamics in sudden limit. The local shape of excited state PES, the initial state distribution on the ground state PES, and the coherence of electronic excitation can all affect the product state distribution. The parent molecules start to dissociate as soon as being excited by the repulsive PES; typically the dissociation time is too short to redistribute the available energy among nuclear DOF. Theoretical models for the prediction of product state distribution usually utilize the concept of sudden approximation, which connects the characters of the pathway to the products’ DOF by direct mapping. Conventional impulsive models, rigid or soft ones, were originally designed to predict the product energy of this case [127, 128]. These impulsive models are only suitable for the two-body breakup problem with one-bond fission, and thus can only provide a proper description of the radical-forming dissociation. Advanced treatments including the direct mapping of the product/parent-molecular state functions [129] were also developed to predict the photolysis-wavelength dependence of product energy distribution of triatomic molecular photodissociation. Models based on the direct mapping approach are rarely found for their application to a system with more than four atoms.

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(b) Dissociation pathway without an exit barrier (the barrierless pathway) This type of dissociation pathway (Fig. 18b) is frequently encountered in the production of the neutral radicals on the ground state PES after the parent molecules undergo one-bond fission, where the minimum in the figure corresponds to the equilibrium geometry of the parent molecule. There is no energy barrier on the reaction coordinate when considering the reverse bimolecular association of the products. When the parent molecules receive excess energy, either via thermal excitation or photon-excitation, the molecules have time to redistribute among the vibrational modes at the reactant well of PES, followed by accumulating enough energy on the local internal mode which will undergo the bond-fission process to form the product. Accordingly, the product state distributions of the barrierless dissociation pathway have to be described via models which allow energy redistribution among nuclear DOF. The barrierless reaction pathway is well known for its character of a “loose transition state” [124, 130– 133]. Theoretical prediction of its reaction rate is usually achieved by utilizing the variational variants of transition state theories [130], in order to locate the true dynamical bottleneck at the reaction coordinate around the asymptotic region toward the products. Similarly, the product state distribution can be predicted by considering the most probable distribution of product states, usually under given energy constraints and regardless of dynamical details that the molecules experienced on the dissociation path. Models with such statistical assumptions have been widely applied to barrierless dissociation, including various types of variants with extra constraints [109, 113, 130]. The microcanonical prior distribution [109, 130] assumed that the energy redistribution among nuclear DOF occurs so rapidly, thus the energy distribution of the molecular flux maintains its statistical nature throughout the dissociation. Accordingly, the product state distribution can be predicted by a microcanonical sampling of product states at given excess energy. The phase space theories [109, 130] include the conservation of linear and angular momenta as well as the total energy conservation, with some variants also including the effect of centrifugal barrier and long-range interaction potential between products. In more sophisticated treatments, one divides the nuclear DOF into different categories according to the nature of vibrational modes of the parent molecule. Energy redistribution between different groups is either forbidden or partially allowed. This can make part of nuclear DOF considered as adiabatic, i.e., the quantum numbers of certain vibrational modes are conserved throughout the whole dissociation pathway. Statistical methods [112, 130, 134, 135] developed by including the above strategies were found to be effective in the prediction of both product state distribution and reaction rate from the barrierless pathway, e.g., separated statistical ensemble (SSE), statistical adiabatic channel model (SACM), and also flexible transition state theories such as variable reaction coordinate transition state theory (VRC-TST). (c) Dissociation by crossing over an exit barrier This type of dissociation pathway (Fig. 18c) can be found on both the ground and excited electronic states. There is an energy barrier on the dissociation coordinate (the potential maximum in Fig. 18c), which separates either from

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the forward dissociation or from the reverse association to the parent molecule (the potential minimum in Fig. 18c). The exit barrier can be attributed to the variation of electronic structure with respect to nuclear DOF along the dissociation pathway. In the molecular orbital (MO) approximation of the electronic structure, the eigenvalues of a bonding/antibonding orbital pair vary with the change of nuclear coordinates. The energy ordering of the electronic configurations varies with the reordering of MOs at different geometries, and the adiabatic PESs show multiconfigurational behavior and possess a region of avoiding crossing. For the case if the nonadiabatic transition is negligible, the lower adiabatic PES shows an energy barrier. The dissociation pathways with such exit barrier usually are encountered in either the multiple-bond-breaking cases of molecular products on ground state PES, or one-bond fission cases of radical products on the excited PES. In the former case, the existence of the exit barrier implies the undulation of potential energy on the dissociation path, which corresponds to the simultaneous forming/breaking of chemical bonds on the single adiabatic PES; in the latter case, the energy barrier implies the multiconfigurational nature of the one-bond-fission process on excited state adiabatic PES. From the dynamical aspect, the exit barrier not only provides the dynamic bottleneck to the dissociation rate, but also serves as the origin of non-statistical behavior of product state distribution. If the dissociated products are all the neutral ones, then one may neglect the possibility of any potential well located at the post-barrier region of dissociation path. The conventional approach to treat this type of dissociation pathway is to separate the excess energy into two energy reservoirs: statistical energy reservoir and impulsive energy reservoir [136]. For the excess energy above the barrier top, i.e., the saddle point, intramolecular vibrational redistribution (IVR) is assumed to be completed rapidly at each stage of dissociation since the molecular flux departures from the reactant well. For the excess energy lower than the barrier top, i.e., the impulsive energy reservoir, this part of the energy is released after crossing the barrier top, in which the molecular flux arrives at the product side within a short period thereafter. The impulsive energy is quickly released with respect to the local nature of the “post-TS” region on the PES and the energy redistribution among nuclear DOF can be negligible. According to such a double-reservoir approach, several statistical/impulsive models were developed to predict the product energy distribution of the “post-TS” dissociation dynamics, e.g., the barrier impulsive model (BIM) [136, 137] and statistical adiabatic impulsive (SAI) model [137]. These hybrid dynamical models adopt the concepts of statistical models for the statistical energy reservoirs and treat the impulsive reservoir with a conventional impulsive model, thus again they are only suitable for the dissociation problem of one-bond fission. In the light of this fact, the multicenter impulsive model (MCIM) [138, 139] and the generalized multicenter impulsive model (GMCIM) [140, 141] are developed for the general N-body dissociation problem with multiple bond-breaking/forming events. These models are designed to overcome the limitation of conventional impulsive models, and thus are capable to predict product state distribution with minimal information on PES. This topic aims to review the recent development

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of a multicenter impulsive model, including the basic concepts and various applications. In order to make the relation between product state distribution and the nature of the dissociation path more transparent in the foregoing discussion, we only consider the concept of transition state in its classical form, i.e., a saddle point on the barrier top; or a multidimensional dividing surface in phase space which corresponds to a critical configuration along the 1D reaction coordinate, without any recrossing of molecular trajectories. Herein, we still have to emphasize that, in the framework of quantum mechanics, the transition state is rather delocalized in phase space, due to the position-momentum uncertainty. Besides, in order to fulfill the energy–time uncertainty relation, the molecular flux should spend a finite time interval to pass through the transition state, which implies that the dividing surface can never have a unique configuration on the 1D reaction coordinate. In addition, the quantization of intramolecular motion at the stationary points of PES leads to the consideration of zero-point energy and tunneling effect, which makes the actual activation energy deviate from an estimation based on the classical barrier height. In practice, the zeropoint energy can be included in an ad hoc manner during the sampling of the initial condition in a classical trajectory simulation. In the following content, we will review the concepts of MCIM and GMCIM. Then we will provide a brief summary of the recent application to the multiple COforming channels of methyl formate under 248 nm photolysis. The advantages and limitations of GMCIM will be discussed, followed by concluding remarks and future perspectives.

8.2 Computational Methods The conventional point of view of a chemical reaction relies on the concept of intrinsic reaction coordinate (IRC) [117, 118], which implies that the chemical reaction on the multidimensional PES can be simplified to an effective 1D problem: a minimum energy path and the remaining nuclear-motional coordinates which are locally perpendicular to it. An integration of such an effective picture into the chemical dynamics is the reaction path Hamiltonian (RPH) formalism [142–148]. Since the early 1980s, qualitative prediction of the product state excitation was attempted by considering the gradient vectors on the IRC and its projection on the products’ nuclear DOF [148–150]. It is well known that the direction of steepest descent path is along the principal axis of negative curvature on a saddle point [117, 151]. In the light of this fact, the product state excitation is usually predicted qualitatively by recognizing the direction of the imaginary normal mode of the saddle point as a gradient vector of impulsive energy, which can be obtained routinely by conventional electronic structure software. The multicenter impulsive model (MCIM) [138, 139] provides the quantitative prediction of product state distribution by utilizing the same Hessian information from the saddle point. MCIM also adopts the concept of

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statistical/impulsive energy reservoirs [136, 137], however, the model is more versatile than the previously developed ones due to the following features. MCIM [138] interprets the imaginary normal mode vector as the acceleration vector due to the impulse, thus it can calculate the product energies explicitly by simply utilizing the equation of motion at constant acceleration, subject to the constraint of impulsive energy. This is the main strategy that makes the prediction of multicenter impulse possible. A modified Franck–Condon mapping (FCM) method [138] is also adopted to evaluate the product vibrational population including the photolysis-wavelength dependence. Without the need to construct complicated potential energy surfaces, MCIM is able to characterize the dynamical feature of the conventional transition state (TS) and roaming pathways in the photodissociation of formaldehyde [138], H2 CO → CO + H2 In addition, MCIM can also predict the photofragment vector correlation (PVC). The result of PVC in diatomic photofragments is obtained for both the conventional and roaming pathways of the CO + H2 channel [139]. Despite these fruitful outcomes, the limitation of the multicenter impulsive model is obvious. MCIM adopts the Franck–Condon mapping of anharmonic wave functions of product states, which is easier to implement only for diatomic photofragments. Besides, the actual dissociation path could be non-straight on PES, but the curvature effect on the dissociation path is completely ignored in MCIM. The curvature effect plays a significant role in the dissociation path with a loose saddle point such as the roaming pathway of carbonyl compounds. The generalized multicenter impulsive model (GMCIM) [140] is developed to overcome these drawbacks. The development of the generalized multicenter impulsive model (GMCIM) is based on the ideas of intrinsic reaction coordinate and reaction path Hamiltonian, which make GMCIM the first impulsive model which considers the whole dissociation path instead of just the saddle point. GMCIM considers the molecular dissociation on multidimensional potential energy surfaces as a problem of 1D reaction coordinate (s) and multiple transverse vibrational modes, orthogonal to s [140, 142, 143, 147]. The model simplifies the original multidimensional problem to an effective 1D approach, by applying the sudden limit to the dissociation coordinate, assuming that the transverse modes have no time to evolve during the dissociation. In practice, only the gradient along the direction of s is included in the calculation, while the accelerations in transverse directions are completely ignored. The acceleration along the reaction coordinate was approximately obtained by extrapolating the imaginary mode vector of TS toward the product side in the original MCIM, which represents a good approximation as long as the dissociation path is straight. On the contrary, GMCIM solves the cases of curved dissociation paths by treating the curved reaction coordinate as successive events of constant acceleration motion. The acceleration along the reaction coordinate is calculated from the mass-weighted Cartesian gradient at each IRC point. The information required by GMCIM is the geometry and the gradient

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of each IRC point, and the nuclear Hessian at the saddle point. These data are available from typical quantum electronic structure calculations. After accounting for the acceleration to the velocity of each atom (in the molecule), the molecule moves to the following segment, where the constant acceleration motion is considered, except that the acceleration vector could point to different directions in different segments. This piecewise approach in mass-weighted Cartesian coordinates (MWCC) permits us to consider the curvature effect, without involving any coupling constants between reaction coordinate and transverse vibrations. Giving information about the initial conditions at the saddle point, the equations of the acceleration of MWCC velocity vector at each IRC segment and also the procedure to determine the product states can be found in Refs. [140, 141]. The ideas of GMCIM are summarized in Fig. 19. Though the original GMCIM only considers the impulsive energy reservoir at the beginning of dissociation, in the latter development of the model we suggested a unified treatment for the excess energy of both statistical and impulsive reservoirs by sudden approximation [141]. In such a case, the kinetic energies of the parent molecular modes, which are transverse to the dissociation coordinate s, are included in the initial velocity vectors that can interact with the gradient along the dissociation path. In contrast, the potential energy of the transverse modes can be treated separately as follows. Based on the nature of the dissociation path, the transverse normal modes at the saddle point of dissociation are divided into two types: (i) the conserved modes, i.e., the modes maintained till the end of the dissociation; (ii) the transitional modes, i.e., the modes that disappear at the end of the dissociation. In brief, conserved modes are vibrations of bonds within a fragment, while transitional modes are those vibrations that largely change the interfragment distances or twist the relative orientation between fragments. When performing normal mode analysis at all the IRC points with the assumption of vibrational adiabaticity [134, 135, 137, 142, 143, 152], one observes that the harmonic frequencies of transitional modes decay to a very low value at the product side of IRC, while the frequencies of the conserved modes correlate to the harmonic frequencies of fragments. Note that our approach to treating the potential energies of these transverse modes (which belong

Fig. 19 Schematic summary of the basic concepts of GMCIM

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to the “statistical reservoir”) is still based on the assumption in sudden approximation: the motions of transverse modes are much slower than the dissociation itself. Thus, if the displacement of the j-th conserved mode from its origin is Qj , then Qj remains constant as well as the potential energy during the dissociation, because the force constant of a conserved mode converges to a value corresponding to the product’s vibrational mode in the asymptotic limit. Thus, the potential energies of the conserved modes correlate to the corresponding mode of the vibration of products in sudden approximation. In contrast, the force constant of a transitional mode fades away at the product side of IRC, hence the transverse energy can be neither stored as potential, nor released in the transverse direction during such a short dissociation time (sudden approximation). Therefore, the potential energies of the transitional modes are released along the dissociation coordinate, i.e., mostly into the translation of products. According to the steps reported below, the potential energies of transitional and conserved modes are treated diversely in GMCIM. Details about the procedure to distinguish the transitional and conserved modes can be found in Ref. [141]. This can be done by picking out the components of a MWCC eigenvector that belong to each fragment. For those conserved modes, the distributed potential energy parts are assigned to the vibrations of the fragments. For those transitional modes, the potential energies are distributed to the translational degrees of freedom of each fragment, without generating extra angular momentum.

8.3 Case Study: Photodissociation of Methyl Formate Under 248 nm Photolysis 8.3.1

The CO-Forming Pathways

For the purpose of assessing GMCIM, we performed a direct dynamics investigation on various pathways [73, 83, 91, 98, 102, 103, 153] leading to the dissociation of CO on the S0 electronic ground state of methyl formate under photolysis of 248 nm. Details about the electronic structure calculation, GMCIM, and direct dynamic simulation can be found in Ref. [141]. There are four channels having as products methanol + CO, with distinct product energy disposals (denoted as TS1, TS4, TS5, and RTS), and a further channel forming H2 + CO + H2 CO (denoted as TS2). The optimization of the saddle point structures and the calculation of the Hessian/harmonic frequencies and the intrinsic reaction coordinates were performed via the GAMESS package. In a previous study [153], the saddle point structures of these dissociation pathways were reported by using several methods: MP2, CCSD, B3LYP, M06-2X, and WB97X functionals. The barrier height of each pathway calculated by CCSD(T) with geometries optimized by B3LYP and M06-2X functionals shows similar values to the geometries optimized with CCSD, except for a few saddle points, where B3LYP gives a better agreement with respect to M06-2X. In this work, the B3LYP functional was chosen for most of the direct dynamics calculations. We

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Fig. 20 The Cartesian vectors of imaginary mode on each saddle point of CO-forming channels. Adapted from Ref. [141]

employed the unrestricted M06-2X functional for the roaming saddle point, since this saddle point is not revealed by the B3LYP functional. The 6-311 g(d,p) basis set was utilized in both the saddle point geometries and direct dynamic simulation. Preliminary tests [141] have shown that diffuse functions play a minor role in the evaluation of the relative energy of the barrier heights (about 0.5 kcal/mol). The relative barrier heights are particularly sensitive to the increase of the split-valance from double to triple, resulting in an average of 2–3 kcal/mol. The direct dynamic simulations were performed by the VENUS/NWChem software package. To study the dissociation dynamics via each pathway, more than 1000 classical trajectories were initiated at each saddle point, with the initial condition being randomly sampled from a microcanonical ensemble under the constraint of constant total energy. At each time step, the VENUS program integrates the classical equation of motion via the velocity Verlet algorithm for the nucleus motion, with the required nuclear gradients at each point directly obtained from DFT calculation via NWchem. The criterion to end each trajectory is the relative distance between c.m. of the dissociated products, which is set to Å angstrom. When the separating distance of dissociation criterion are increased up to 8 Å in preliminary tests, there is no notable changes on the product energies and quantum numbers. The final state analysis of each trajectory, including the determination of translational, rotational, and vibrational energies of products, was done by the well-known treatments implemented in the VENUS program. The dynamics feature of the CO dissociating pathways can be speculated by following the mode related to the imaginary frequency on the saddle points [141, 153]. The vectors of these modes related to the five saddle points investigated in this work are shown in Fig. 20. TS1 is the conventional three-center transition state, known as “tight TS” [73, 83, 91, 98, 103, 141], which was considered as the predominant CO dissociation channel in aldehydes and acetone [85, 105, 114, 123, 131, 154, 155]. In the pathway from the saddle point toward the two-body products, the breaking of C–H and C–OCH3 bonds and the forming of the O–H bond occur concertedly. The recoil of the two products due to the bond interruption accelerates the movement of

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the fragment. The rotation of the CO product can be excited by torque due to the bond breaking. The potential energy released from the formation of the O–H bond correlates to the stretching excitation of the OH vibrational mode in the methanol product. From previous studies of aldehydes, one can conclude that the dynamic signature of this pathway could be ascribed to the fast-moving fragments, highly rotational excitation of CO, and medium vibrational excitation of the co-product of CO. TS2 corresponds to the concerted triple fragmentation mechanism. As shown in Fig. 20, the imaginary mode of TS2 concentrates on the separation of the three products along the whole dissociation path. The product energies are mainly released to the translational degree of freedom, with medium rotational excitation on the three fragments. TS4 and TS5 are is four-center transition states of “isomerization-mediated” channels [153]. The fragments are recoiled by breaking the two bonds simultaneously, thus a large amount of translational energy with low rotational excitation of the CO product is expected. The features of the roaming saddle point, RTS [153], and the related intrinsic reaction coordinate of methyl formate are quite similar to those found in aldehydes. The C–OCH3 bond-length increases until reaching a distance larger than 2.5 Å, where HCO moiety rotates along its CO axis and transfers the H atom to OCH3 . It is well known that the roaming mechanism can only produce CO with very low energy [82, 85, 123, 131, 138]. Most of the excess energy is taken by the co-product for vibrational excitation.

8.3.2

The Product State Distribution

The direct dynamic data of product energy disposals from each pathway are accumulated and exhibited by means of histograms, with optimal size and number of bins chosen to pursue the clarity and precision of each figure. We have focused on the translational, vibrational, and rotational energy of all the products as well as the corresponding quantum number distribution of diatomic products. Here, we report the results of the trajectory simulations related to the channels TS1, TS2, TS4, TS5, and RTS. For the channels TS1, TS2, TS4, and TS5, 1000 trajectories have been simulated, while for RTS we have simulated 1245 trajectories. Details on the trajectory simulations are reported in references. The outcome of GMCIM will be shown in the figures. The product energy disposals obtained from direct dynamics and GMCIM are summarized in Fig. 21a, b, respectively. In addition, a detailed comparison of results from direct dynamics and GMCIM is shown in Figs. 22, 23, 24, 25, 26, and 27.

8.4 Discussion The outcomes of the five pathways simulated by direct dynamics and GMCIM are compared in Figs. 22, 23, 24, 25, 26, and 27. In general, the results obtained from the two methods are in good agreement. It would be interesting to discuss the assumptions

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Fig. 21 The stacked column chart shows the relative contribution of the product energy in each pathway, which is obtained from a direct dynamics, b GMCIM. Note that ZPE of product has not been subtracted from vibrational energy yet. Et, Ev, and Er denote translational, vibrational, and rotational energy of the product, respectively. Adapted from Ref. [141]

made for the “statistical energy reservoir”, the excess energy that was traditionally treated via statistical methods, and compare the GMCIM with the conventional ones. The case study also aims to test the treatment of transverse vibration in GMCIM. For the conserved modes, it does not mean that these modes have no coupling with the reaction coordinate, especially when the reaction path is not a straight one. The only statistical assumption in GMCIM consists in the microcanonical sampling of

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Fig. 22 Product state distribution from TS1 pathway: comparison of direct dynamics and GMCIM. Translational energy (ET ), vibrational energy (EV ), rotational angular momentum (AM), and angular momentum of fragments’ orbiting motion (OAM) are shown in the figure. The ordinate counts the number of trajectories. Adapted from Ref. [141]

initial conditions, which means that although the parent molecule has enough time to build up a statistical distribution of transverse modes at the transition state, once the molecule starts to form the products, the duration time is still too short to redistribute the excess energy in the product states. Thus, for the barrier-crossing dissociations with a large amount of “statistical energy reservoir”, there is no assumption made by means of statistical or vibrationally adiabatic concepts in GMCIM. To refer to a statistical model, the transverse energy must be distributed in a way that maximizes the number of microstates; the assumption of vibrationally adiabaticity implies that the quantum numbers of the transverse modes are conserved during the dissociation, where the transverse motions must be fast if compared with the dissociation. Note that quantum number conservation of a mode is not equivalent to the conservation of the mode energy. Instead, the energy in the mode may change when the force constant of the mode varies with the IRC, even when the path is a straight one. For example, the energy of a transitional mode in its vibrational ground state decays to zero as the mode disappears in the asymptotic limit of dissociation. A transverse mode with a varying force constant can never be isolated from other DOF; there is no way to vary the force constant of a mode except if an external force is exerted on the mode. Accordingly, the force constants of the transitional modes have significant change during the dissociation, therefore it is natural to assume the reaction coordinate as the final one of their potential energies. On the other hand, the force constants of the

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Fig. 23 Product state distribution from TS4 pathway: comparison of direct dynamics and GMCIM. Adapted from Ref. [141]

Fig. 24 Product state distribution from TS5 pathway: comparison of direct dynamics and GMCIM. Adapted from Ref. [141]

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Fig. 25 Product state distribution from RTS pathway: comparison of direct dynamics and GMCIM. Adapted from Ref. [141]

Fig. 26 The internal state distribution of products from TS2 pathway: comparison of direct dynamics and GMCIM. Adapted from Ref. [141]

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Fig. 27 The translational state distribution from TS2 pathway: comparison of direct dynamics and GMCIM. Adapted from Ref. [141]

conserved modes do not change significantly during the dissociation, therefore their potential energies are conserved when the transverse vibrations are extremely slow if compared with the dissociation. A plausible question is why the excess energy of the barrier-crossing dissociation can be treated by both statistical treatment and sudden approximation. The assessments of the statistical-impulsive hybrid models [136, 137, 156] usually focus on radical-forming dissociations, which only involve one-bond fission without newly formed bond(s). Such dissociation paths usually possess minor curvatures, therefore the coupling between transverse modes and dissociation coordinate is weak. Neglecting such a curvature effect, both the statistical/adiabatic and the sudden approaches of excess energy tend to predict similar product energy excitation, i.e., the vibrational states of the polyatomic fragments. It donates preferentially most of the excess energy to the vibrational motions of the products, because most of the transverse modes (i.e., 3 N-7 modes) are conserved ones. In summary, the statistical/adiabatic treatments tend to distribute excess energy into the vibrations of the products, thus consistently with the outcome of GMCIM. The main difference is that the statistical-adiabatic approaches manipulate the mode energy directly, while GMCIM operates in terms of the velocity vectors and includes their coupling with the dissociation path.

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8.5 Conclusion Recently, we have extended the capability of GMCIM to include high excess energies and evaluate its performance by applying the model to all the CO-forming channels of methyl formate. GMCIM only requires the following information provided by electronic structure calculations: the relative energy of saddle points and products from the reactants, the equilibrium structures of saddle points and products, the IRC of the reaction path, and the Hessian matrices of the saddle points. Typically, the acquirement of this information is fulfilled as long as the IRC calculation has been done, therefore no extra electronic structure calculation is necessary. Under the same number of initial conditions, a simulation of the post-transition state dynamics by GMCIM is much faster than the one performed by direct dynamics. Hence, for studies of post-TS dynamics, GMCIM can easily provide a statistically significant number of results with a limited computational cost. In summary, the following points can be stated: (i)

GMCIM minimizes the requirement of electronic structure calculations: when a saddle point has been computationally investigated until the IRC is obtained, the information required by a simulation of GMCIM is also fulfilled; (ii) the post-transition state dynamics of polyatomic molecule is reduced by GMCIM to become an effective 1D (IRC) problem. The shape of the PES along the remaining 3 N-7 coordinates is approximated to a quadratic one at the saddle point, followed by simplifying the dynamics of the 3 N-7 degrees of freedom via sudden approximation; (iii) due to the reasons described in (i) and (ii), GMCIM can be implemented in array programming with high performances to generate a large amount of results. Limitations of GMCIM are also summarized here. GMCIM gives a single adiabatic description of the post-transition state dynamics of unimolecular dissociation with exit barriers (Fig. 18c). Any possible nonadiabatic coupling between adiabatic states at the asymptotic limit is ignored. The model cannot treat the dissociation pathways such as either Fig. 18a, b. Besides, GMCIM relies on the gradient evolution along the IRC, therefore the model inherently assumes that the molecular trajectories, although may deviate from the IRC, still remain in the harmonic potential valley about the reaction path. GMCIM cannot properly describe problems which possess bifurcation, bypass the saddle point, or deviate out of the reaction valley. Therefore, the outcome of GMCIM serves as a limiting case for dissociation channels which involve such non-IRC dynamics, assuming that the dynamical behavior is still predominated by the shape of the potential valley about the reaction path. Another limit of GMCIM is the lack of information about the non-statistical effect which involves before the parent molecules attain the transition state. Finally, GMCIM adopts harmonic approximation to the transverse modes, ignores the coupling between transverse modes, and freezes the motion on these modes completely.

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9 Topic 4: 2D Spatially Oriented Molecules and Chiral Discrimination Via Photodissociation Using 2D Imaging and Hexapole Techniques Masaaki Nakamura

9.1 Introduction Chirality is a structural property of molecules without an improper rotational symmetry. This property makes isomers that cannot be superimposed on each other with translational and rotational operations. The pair of isomers, called enantiomers, looked like mirror images. Generally, it is difficult to discriminate enantiomers since almost all the physical properties are common in a pair of isomers. Nevertheless, the bio-molecules, such as sugars and amino acids, have exceptionally high enantiomeric excess. Surprisingly, it is not yet clearly understood how the situation emerged. We have not gotten the commonly accepted explanation of the origin of the natural chiral selectivity. The scenario is still a big mystery for scientists. A number of spectroscopic chiral discrimination methods have been developed until now [157–159] and those enantio-selective techniques utilize specific interactions between target isomers and a circularly polarized light which is regarded as a “chiral” light. The differences between the responses of right- and left-circularly polarized irradiations allow us to discriminate molecular chirality due to the helicity of light. However, circularly polarized light sources hardly exist in nature. Linearly polarized light, in turn, is relatively more popular. It is frequently seen as an incident/reflected light at a flat boundary plane like a water surface. Now we came into a question; could photo-induced reactions with linearly polarized light have chiral selectivity? Since a linearly polarized light does not have helicity, the light itself does not have chiral selectivity. However, it will be shown in this topic that a photodissociation induced by a linearly polarized light could present different photofragment scattering distributions between enantiomers under a certain condition [160]. The key point is molecular orientation. The chiral discrimination could be achieved only if the target molecules are spatially oriented. The basic idea of chiral discrimination is discussed in the next subsection, and the results of ongoing experiments will be shown in the final subsection.

9.2 Theory A molecular bonding dissociates when a molecule is excited to a repulsive state by absorbing enough energy of photons. In this subsection, it will be shown that the photofragments are scattered into 3D space with different directional scattering

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distributions for each enantiomer under a certain condition. To interpret the chiral discrimination in the photodissociation, it is convenient to adopt a semiclassical model regarding the molecular permanent dipole moment and the transition dipole moment as vectors. The photon absorption probability depends on the angle between the transition dipole moment and the light polarization vector; the former is fixed in the molecular fixed frame while the latter is fixed in the laboratory frame. Therefore, the spatial molecular orientation affects the photofragment scattering distribution. Usually, molecules in a gas phase are rotating freely and are oriented randomly. However, a molecular orientation technique such as hexapolar field state-selection allows to orient the permanent dipole moment of molecules with respect to an external electric field. The recoil velocity vector is also defined in the molecular-fixed frame, thus, the photofragment scattering distribution also depends on the relative orientation of those three vectors: the transition dipole moment, the permanent dipole moment, and the recoil velocity vector. The key point is that the relative orientation of the three vectors is expected to reflect the molecules’ chirality. Considering a pair of enantiomers, the magnitudes of those three vectors should be the same, however, the relative orientation is like mirror images, as shown in Fig. 28. If the orientations of those two vectors, the transition dipole moment and the permanent dipole moment, are constrained into the same directions, the rest one, the most probable recoil direction, would point different directions for each enantiomer. Hexapole state-selection combined with a static orienting field allows us to orient molecular permanent dipole moment with respect to an external electric field. Molecules naturally have their specific anisotropic orientational distribution depending on their rotational state. However, the orientations are usually averaged out in an ensemble of thermal state distribution. By passing through a hexapole, molecules are rotationally state-selected by the hexapolar inhomogeneous electric field and present the anisotropic orientation inside the subsequent orienting field. The molecular orientation around the orienting field is expressed as the linear combination of Legendre polynomials Fig. 28 Schematic image describing the relative orientation of the permanent dipole moment (d), the transition dipole moment (µ), and the recoil velocity vector (v) of each enantiomer

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2J Σ

ck Pk (cosθd O )

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(16)

k=0

where θ dO is the angle between the permanent dipole moment and the orientation field vector, Pk are the k-th order Legendre polynomial, and ck are those weighting factors. Since even and odd functions alternatively appear in the series of Legendre polynomials with k, the terms with odd k represent molecular orientation and the terms with even k do alignment; the former distinguish the head and tail of molecules but the latter do not. The coefficient c0 is always 0.5 due to the normalization condition. This method is a well-known and matured technique [161–163]. It has been adopted in a number of molecular collision experiments to study the steric effect in chemical reactions [164, 165] and photodissociation studies [166, 167]. The oriented target molecules are then irradiated by a photolysis laser. Since the photon absorption probability depends on the projection of the transition dipole moment onto the light polarization vector, the molecules are photo-dissociated selectively by the orientation of the transition dipole moment. Hence, it is possible to control the orientations of the permanent dipole moment and the transition dipole moment of photo-dissociated molecules by choosing a design of experimental setup. It is convenient to use the slice imaging technique for observing the scattered photofragments. With this method, the photofragments are projected onto a screen to give 2D map of the recoil velocities. The directional difference of recoil velocities of two enantiomers should appear on the angular distribution of the slice image. In the following discussion, we choose the photofragment ion imaging setup as shown in Fig. 29. The orienting field is parallel to the time-of-flight axis, and the photolysis laser polarization is tilted 45° from those axes. This is not only an available setup. However, it is easy to operate since the orienting field and the ion extraction field are identical. It is not required to switch orientation and ion extraction in time. The angular distribution of slice images of photofragments with parent-molecular orientation can be expressed as [74, 168]. I (θ ) = 1 + β10 P10 (cosθ ) + β20 P20 (cosθ ) + β11 P11 (cosθ ) + β21 P21 (cosθ ) Fig. 29 The geometry of laser polarization and the orienting field. The laser propagates along Y-axis, and the laser polarization axis, which interacts with the transition dipole moments, is in the ZX-plane. The orienting field interacting with the permanent dipole moment is parallel to the TOF axis (Z-axis)

(17)

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where Plm are the associated Legendre polynomials and the corresponding coefficients βlm are determined by blm /b00 using the following expressions: b00

( ) 1 3 = (1 − c2 P2 (cosα)) 1 − P2 (cosχ ) + c2 sin2 αsin2 χ cos2ϕμd 2 16

(18)

b10 = 3c1 sinαsinχ cosχ cosϕμd

(19)

3 b20 = (1 − c2 P2 (cosα))P2 (cosχ ) + c2 sin2 αsin2 χ cos2ϕμd 8

(20)

9 b11 = − c2 sin2 αsin2 χ sin2ϕμd 8

(21)

b21 = −c1 sinαsinχ cosχ sinϕμd

(22)

The coefficients c1 and c2 represent the spatial orientation of molecules as in Eq. 16. Here, the terms up to the second order (k = 2) were included for simplicity. The three angle parameters (α, χ, ϕ μd ) are shown in Fig. 30, the two angles from the recoil velocity vector to the permanent dipole moment and the transition dipole moment, and the azimuthal angle between the permanent dipole and the transition dipole around the recoil velocity, respectively. The magnitudes of the three angles are the same for the pair of enantiomers while only the sign of ϕμd is opposite. Figure 30 describes expected slice images of imaginary enantiomers, whose angular distribution follows Eq. 17. The circular images (a) and (b) correspond to the expected result of each enantiomer, and those angular distributions look shifted along the angle axis to opposite directions. To break the symmetry around θ = 0°, as demonstrated in Fig. 31, the contributions from odd functions of θ are required. According to Eqs. 18–21, the odd terms are only β11 P11 and β21 P21 in Eq. 17. The coefficients β11 depends only on c1 and the β21 similarly depends only on c2 . Hence, β11 P11 represents the chiral difference caused by the molecular orientation and β21 P21 does the discrimination by the molecular alignment.

9.3 Experiments The results of ongoing experiments are shown in this subsection. The schematic experimental setup is shown in Fig. 31 [74, 103]. A chiral molecule 2-bromobutane pure vapor was ejected into the vacuum chamber to form a molecular beam. The parent molecules were rotationally state-selected by a 70 cm long hexapole state selector with a 4 mm inscribed radius. The molecules were photolyzed by the 234 nm laser to emit the Br photofragments in the spin–orbit excited state. The photofragments were subsequently ionized by the same laser via the (2 + 1)

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Fig. 30 Simulated slice images of virtual enantiomers with (α, χ, ϕ μd ) = (45°, 45°, ± 150°). The angular distributions of the two images are shown in the lower panel

resonance-enhanced multiphoton ionization process to detect. The photofragment ions are projected onto the position-sensitive detector by keeping the velocitymapping condition and the coordinates on the screen and the time-of-flight (TOF) were recorded. As discussed in the previous subsections, it is necessary to orient 2-bromobutane molecules before photodissociation for chiral discrimination. However, the orientational control of asymmetric molecules had not been experimentally achieved yet. Additionally, it is practically difficult to evaluate the molecular orientation by calculations [61, 169] because of the condensed manifolds of the rotational states and the presence of several rotamers. However, in this case, we were able to estimate the degree of orientation with an experimental approach. The photofragments are emitted and accelerated to the detector. At that time, the photofragments recoiled straight to the detector observed earlier while those recoiled opposite to the detector fly along the initial recoil velocity and are pushed back to the detector to be detected later. Given that the dissociation undergoes fast enough compared to the molecular rotation, the recoil direction correlates with the molecular orientation. Therefore, it is possible to estimate molecular orientation by measuring the time and intensities of the observed photofragment ions. Figure 32 shows the TOF measurement of the fragment Br with and without the hexapole voltages. The light gray lines represent the experimental results, and the green lines are the results of the simulation. The red and blue lines are also the result of simulations but only of each bromine isotope; the bromine atom has two stable isotopes, 79 Br and 81 Br, and these natural abundances

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Fig. 31 Description of the experimental setup. The ~234 nm photolysis light was the frequencydoubled output of the excimer laser (Lambda Physik LPX-200) and pumped dye laser (Lambda Physik LPD-300). The 2-bromobutane molecules were emitted from the pulsed valve (PV) and passed through the hexapole (HP) for the rotational state-selection. Subsequently, the molecular beam was intersected with the laser at a right angle and the fragments are detected at the positionsensitive detector, an assembly of a micro-channel plate, and a phosphor screen (MCP & PS) gated by the gate pulse generator (GP). The light emission from the phosphor screen was captured by the CCD camera and the photomultiplier tube (PMT) and recorded on the PC and the oscilloscope (OS). The laser light was monitored by a photodiode detector to synchronize the laser and the molecular beam using two delay pulse generators (DG)

are almost even. Since the photofragment ions move with almost constant acceleration in the extraction field and with constant speed in the field-free region in front of the detector, it is easy to carry out the computations. For the photodissociation of 2-bromobutane at the present wavelength, the Br fragments are favorably emitted along the laser polarization axis; its anisotropy parameter is obtained as β ~1.85 [68]. Therefore, when the laser polarization was aligned parallel to the TOF axis, the forward and backward recoiling fragments were populated to result in the two peaks for each isotope. Without the hexapole voltage, the two peaks were equally distributed. However, when the hexapole voltage was raised up to 4 kV, the earlier peaks were clearly enhanced. The experimental results imply that the Br part of a molecule was oriented to the detector side. The c1 and c2 were obtained to be −0.35 and 0.5. The coefficient c1 can be directly translated to the degree of orientation in the conventional style: | | = 0.23. The spatial orientation of a chiral molecule was achieved and confirmed experimentally. The obtained sliced ion images of Br photofragments from spatially oriented 2-bromobutane are shown in Fig. 33a and the angular distributions are shown in Fig. 33b. The two angular distributions are almost matched within the experimental errors. Although the results suggest that the C–Br photolysis of 2-bromobutane at this wavelength is not appropriate for chiral discrimination, Eqs. 21 and 22 also explain the case without difference between enantiomers. The angle parameter (α, χ, ϕ μd ) dependencies on the angular distribution are the key to interpreting the experimental results, that is, β11 and β21 can be very small with a certain angle parameter. As

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Fig. 32 Time-of-flight spectra of Br (2 P1/2 ) photofragments from 2-bromobutane molecules at 234 nm. The light gray line is the experimental result. The red and blue lines are the simulated curve of 79 Br and 81 Br, respectively. The green line shows the summation of the two simulations. a The spectrum measured without adding voltage to the hexapole. b The spectrum measured by adding 4 kV

Fig. 33 a The slice images of R- and S-forms of 2-bromobutane. The purities of the chiral samples are approximately 85 and 92%, respectively. b The angular distributions of the above images

for the present observation, the anisotropy parameter of the Br fragment from 2bromobutane photodissociation at 234 nm is very close to the parallel limit. The C–Br bond cleavage undergoes very quickly at 234 nm excitation and thus χ ~0° or 180° for the present case. That condition made both β11 and β21 very small to minimize the difference in angular distributions.

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9.4 Summary and Conclusions We have carried out the experiments to explore the possibility of chiral discrimination with a linearly polarized light with 2-bromobutane molecules as the first example. The theory suggested that a pair of enantiomers could show different photofragment scattering distributions if parent molecules were spatially oriented. Despite those congested rotational manifolds, the spatial orientation of chiral molecules has been achieved by using a 70 cm long hexapole state selector combined with an orienting static electric field. The photodissociation of 2-bromobutane at 234 nm did not offer a significant difference with each enantiomer. However, the result can be understood with the theoretical analysis. The results implied that another transition for dissociation or another molecule might be required for the chiral discrimination. Especially, the difference would be pronounced when the fragment did not recoil axially, which could be caused by fragmentations coupled with a vibrational mode of parent or fragment molecules [166] and multiple fragmentations. In other words, the observation gives us plenty of dynamical information on the photodissociation process. The hexapole state selector has been utilized mainly for the collision experiments of small molecules, however, as shown in the present experiments it is also a powerful tool for photodissociation studies of complex molecules. Although the test molecule 2-bromobutane did not show clear differences, the theoretical implication is intriguing as an example of different behavior of enantiomers even in a simple chemical reaction, rather than just a static chiral discrimination method. The experimental requirements are now fulfilled. The subsequent experiments will clarify the role of chirality played in photodissociation reactions.

10 Topic 5: Electrostatic Hexapoles in the Study of Stereo-Directed Dynamics Processes of Interest in Astrochemistry Federico Palazzetti

10.1 Introduction Electrostatic hexapole is formed by six metal bars arranged in a hexagon, whose length is commonly included between 50 cm and 2 m. Alternate electric charges (positive and negative) are applied to the bars, and the effect is the production of a non-uniform electric field, which increases by moving from the axes of the hexapole to the surface of the bars. Electrostatic hexapoles have been widely employed to assist molecular beam experiments in the study of stereodynamics effects in photodissociation and collision processes [161, 170]. Non-uniform hexapolar electric fields to produce rotational state-selection, beam focusing, and molecular alignment. Since

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rotational-state-selected molecules must show a positive Stark energy, the main requisite of the molecule is to possess a non-zero permanent dipole moment. Hexapoles were initially employed for the rotational-state-selection of linear and symmetrictop molecules [63, 171–173]. More recently, this technique has been applied to asymmetric-top molecules of increasing complexity, where the dense rotational state manifold makes it impossible to select a single state [169, 174]. However, other features such as beam focusing, molecular alignment and orientation, and cluster selection have been exploited [175]. Molecular orientation is achieved by combining hexapoles with homogenous direct current electric fields [176]. This setup produces weak and stationary electric fields that permit the generation of polarized molecular beams, making the hexapole suitable for single or crossed molecular beam experiments [166]. Applications of this technique have gained considerable success in the stereodynamical studies of inelastic collisions [177], reactive scattering [178, 179], gas–surface reactions, electron transfer collisions [164, 165, 180, 181], and photodissociation [68, 71, 74, 182]. In addition, hexapole state selectors serve as the backbone in achieving several advances in experimental methods such as conformer separation [62]. The most common result obtained in experiments with hexapoles is the focusing curve, that plots the change of the molecular beam intensity as a function of the hexapole voltage. It goes without saying that the first information derived from the focusing curve is the suitability of the molecule to be studied by hexapole. There are several reasons for which the signal intensity does not change with the hexapole voltage. The most common reason is related to low permanent dipole moments; the rotation can also have the effect of “average” the electronic distribution, reducing the effective dipole moment. Such averaging effects have been observed in acetic acid, where the internal rotation of the -CO2 H group makes this asymmetric-top molecule an effective symmetric-top [183]. Explicit consideration of the Stark energy and gradient of angular momentum state in classical trajectories simulation of the molecular ensemble under well-defined rotational and velocity distributions permits to reproduce the focusing curves for given experimental conditions, allowing one to determine the quantum composition of the molecular beam, as well as the statistical variables like the rotational temperature. Evaluation of the Stark effect is thus a crucial point in the characterization not only of the molecular beam, but also of the aligning process and, if contemplated, of the orientational process. Symmetric- and asymmetric-top molecules are usually considered to give first- and second-order Stark effects, respectively. However, in the case of hexapoles, where electric fields are more intense than those usually employed in spectroscopy, this rule is often violated. For symmetric-top molecules, it was demonstrated that the Stark effect of order higher than one can alter the peak positions in focusing curves. In addition, miscellaneous effects can occur, such as nuclear quadruple and magnetic dipole interactions [184]. For triatomic linear top molecules, see also Ref. [26], while for a more general treatment, see Ref. [61]. In this section, we discuss possible applications of the hexapolar technique to studies of dynamics in processes of interest in astrochemistry. An increasing number of molecules, of increasing complexity, has been detected in the last decades in the

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interstellar medium (see, for example, Ref. [185]). Many of these molecules have been already studied by hexapoles and many of them are suitable to be investigated using this technique. Among these molecules, CO is one of the fundamental molecules of interest in astrochemistry, being a column density tracer in molecular clouds [186]. Here, we will review two applications of the hexapoles that are of interest in astrochemistry. The first application is the alignment and rotational state-selection of propylene oxide; this molecule was found for the first time in the interstellar medium a few years ago [187]. It is the first organic chiral molecule found in space, and this fact raised interest around this molecule, especially for the issues related to the study and the debate on the origin of chiral selectivity in nature [188]. The second application of hexapole concerns the potential capability of this device to select the conformer composition of molecular beams. Developments in this application could open the way to the study of conformer-dependent dynamics. This aspect has remarkable implications in astrobiology and could constitute a prototypical study of the dynamics of those biomolecules where the different conformers play a basic role, e.g., proteins [189].

10.2 Background In Fig. 34, we report a typical molecular beam apparatus with an electrostatic hexapole. The gas is expended from a nozzle, selected in direction by a skimmer, and selected in velocity by a chopper. The beam is thus collimated in and out with respect to the hexapole. Finally, a detector measures the beam intensity. The length of the hexapole is commonly between 50 cm and 2 m, and the diameter, i.e., the smallest distance between two opposite rods is of the order of 4–6 mm. This experimental setup can be implemented in studies of collisions between crossed molecular beams, or photo-initiated processes, involving oriented molecules. The additional orienting field is placed in the collision region between the hexapole and the detector. Molecular alignment is maintained in the passage of the beam from the hexapolar electric field to the orienting field by a guiding field. The Stark effect. The application of an electric field to a molecule with a permanent dipole moment produces a change in the rotational spectrum and removal of the degeneracy (for rotational quantum numbers, see Ref. [184]). The variation of energy of the rotational states is known as Stark energy and depends on the applied electric field. This dependence is usually linear for symmetric-top molecules and quadratic for asymmetric-top molecules. Usually, this behavior is respected in the case of weak electric fields and small molecules, while a more rigorous approach is necessary for general cases, especially for asymmetric-top molecules (see Ref. [61]). Focusing curve and trajectory simulations. Focusing curves provide a macroscopic view of the number of molecules that are selected by the hexapolar field and reach the detector. They are a sum of the contributions of the selected rotational states to the beam intensity and depend on the velocity distribution of the molecular beam, on the geometrical properties of the experimental setup and on the Stark effect on the

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Fig. 34 Experimental setup of a hexapole in a molecular beam apparatus. The gas is expended from the nozzle and passes through the skimmer, the chopper, and a first collimator, before entering the hexapole. A second collimator is placed between the hexapole and the detector. In the upper right side of the figure, we show a profile of the six rods of the hexapole, r is the diameter of the hexapole, and ρ the diameter of the rods

rotational states of the molecule. Trajectory simulation is carried out to obtain the distribution of the selected rotational state. It consists in simulating the molecular trajectories in the experimental apparatus for a given rotational state, according to the rotational distribution. The electric field exerted by the hexapole is given by E = 3V0

r2 R3

where V 0 is the hexapole voltage, r is the distance of the molecule from the hexapole axis, and R is the radius of the hexapole.

10.3 Hexapole Orientation of Propylene Oxide Propylene oxide was detected in space in 2016 [187]. The discovery of this chiral molecule, already well known by the spectroscopy community, has meant that propylene oxide became a molecule of interest in astrochemistry. The origin of homochirality in the biosphere, that is the phenomenon for which proteins are formed only by left-rotatory amino acids, while sugars in nature are only dextrorotatory is a still open debate. Propylene oxide has been widely characterized by the hexapolar technique, both for the selection of rotational states and for its orientation in order to carry out studies

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on the observation of chiral effects in collisional processes. Here, we report the issues related to the rotational state-selection and orientation of this molecule. In Fig. 34, we show the molecule, oriented with respect to an orienting field. Propylene oxide is an asymmetric-top molecule, having three different components of the moment of inertia. This aspect, combined with its molecular weight, 58 a. m. u., makes the rotational state manifold much denser than other molecules previously investigated by the hexapolar technique (Fig. 35). For this reason, the selection of a single state is not possible; the hexapolar field can at least select a distribution of the selected rotational states as a function of the voltage. The hexapole turns out to be important for the following purposes: (i) the focusing effect permits to obtain an intense beam, suitable for crossed beam experiments; (ii) the hexapole aligned molecules can be oriented by using less intense field, with respect to other techniques (see, for example, brute-force techniques, Ref. [176]). In Fig. 36a, we report a focusing curve of a supersonic beam of pure propylene oxide, for which the distribution of the rotational states according to the J quantum number, corresponding to the total angular momentum, is given in Fig. 36b. In Fig. 36c, we report the orientation probability distribution function, opdf , as a function of cos θ and the intensity of the orienting field, OF. Asymmetric profiles of the opdf show a preferred orientation of the molecule at a given value of the orienting field. For example, at HV = 10 kV and OF = 10 kV/cm the orientation probability distribution function is higher for cos θ = 1, than for cos θ = −1. Fig. 35 The propylene oxide molecule, C3 H6 O; in gray we reported the carbon, in red the oxygen, and in white the hydrogen atoms. The axes labeled with a, b, and c are the principal axes of inertia, the blue arrow is the permanent dipole moment (μ), the green arrow is the external orienting field (OF), and finally θ is the angle between the projection of μ on the b axis and OF

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Fig. 36 a The focusing curve of propylene oxide pure beam within a hexapole voltage range from 0 to 15 kV. Black dots are the experimental values of the beam intensity, while the blue line represents the simulated focusing curve. b The distribution of the selected rotational states according to the J quantum numbers, for the pure propylene oxide beam, as a function of the hexapole voltage. c The orientation probability distribution function at 15 kV of the propylene oxide beam, with respect to cos θ, as a function of the orienting field.

10.4 Conformer State-Selection on 2-Butanol In determinate conditions, the hexapolar field can act as a conformer selector. This feature turns out to be of interest for astrochemistry, more specifically in prebiotic chemistry, if one plans to perform collision or photodissociation experiments on specific conformers, prototypical of the more complex biomolecules, where torsion motions play a fundamental role. 2-butanol presents torsion motions around the carbon–oxygen bond and around the bond joining the second and the third carbon. We use the symbols T, G+ , and G− . To describe the rotamers related to the C–C bond, and the symbols t, g+ , and g− , for the C–O bond. T and t indicate the trans conformations, while G+ , G− , g+ , and g− denote the gauche conformations (see [62] and references therein). In Fig. 37, we report the focusing curve of the 2-butanol molecular beam seeded in argon (Fig. 37a) and seeded in helium (Fig. 37b). The insets in each plot show the conformer composition of the beams, calculated according to the Boltzmann distribution. Depending on the beam conditions, it is possible to relax the vibrational states of the molecule and change the conformer population. In this regard, 2-butanol seeded in 95% of Ar determines a more drastic relaxation; the vibrational temperature

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is 50 K, while for 2-butanol seeded in 95% of He the vibrational temperature is 200 K. The vibrational temperature, as well as the rotational temperature, are inferred from the best fitting between the experimental and simulated focusing curves. Trajectory simulations show that conformers can focus in different ways (Fig. 39a). Basically, there are two kinds of focusing curves, namely σ-type (Fig. 39b) and λ-type (Fig. 39c), based on the beam response to the change of the hexapole voltage. The λ-type is characterized by a slow increase of the beam intensity as the hexapole voltage increases, while for the σ-type there is increase of the beam intensity even at low hexapole voltage. Analysis of experimental and theoretical data shows that those conformers where the charge is mostly distributed along the largest of the principal axes of inertia are characterized by σ-type focusing curves, while different distributions are more attributable to λ-type focusing curves (Table 1).

Fig. 37 The focusing curve of 2-butanol seeded in Ar and seeded in He, a and b, respectively. Dots indicate the experimental value of the beam intensity as a function of the hexapole voltage, while lines indicate the simulated focusing curves. Insets show the intensity of the most populated conformers as a function of the hexapole voltage Table 1 The energy in cm−1 and the components of the dipole moments μa , μb , and μc , in debye, of the nine conformers of 2-butanol Conformers

Energy, cm−1

μa , debye

μb , debye

μc , debye

1.17

−0.74

1.01

T, g−

42

−1.37

−0.88

0.86

T, t

121

−0.22

−1.53

0.91

T,

g+

0

G+ , g+

173

0.06

1.49

0.85

G+ , g−

135

1.63

−0.53

0.67

G+ , t

268

−1.23

0.89

1.08

G− , g+

266

1.10

1.19

0.55

G− , g−

248

0.82

−0.09

1.52

G− , t

437

0.95

1.57

−0.39

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Fig. 38 a Focusing curves of the nine conformers of 2-butanol. b A prototypical σ-type focusing curve, typical of molecules having distribution of charge parallel (||) to the larger of the principal axes of inertia. c A prototypical λ-type focusing curve, typical of molecules having distribution of charge perpendicular ( ) to the larger of the principal axes of inertia

In 2-butanol, a combination of seeding effect and hexapole field permits modulating the conformer composition of the beam. Further developments in this technique could permit selecting a single conformer. The specific design of the experiment would make the selected conformer suitable for collision and photodissociation experiments.

10.5 The Hexapolar Technique Applied to Astrochemistry An interstellar cloud is a region of the interstellar medium where the density of matter is higher than the average. Molecular clouds are a type of interstellar cloud, where the density is high enough to permit the formation of molecules. The first molecule found in such a context was methylidyne (CH) (see Ref. [190] and references therein). Here, we report a selection of the 200 molecules detected in the interstellar medium, based on the census made by McGuire [190], we will focus on molecules suitable to be investigated by hexapole, having a non-negligible dipole moment and vapor pressure. Among the simplest molecules, i.e., those formed by two or three atoms, there are (reference citation is reported for the molecules already investigated by hexapolar technique) CH, OH [162, 171], CN, NH, CO [179], CS, NO [176], HCl [175], HF, as well as H2 O, HCN, OCS [34], HNC, H2 S, HCO, and N2 O [191]. Among the molecules with four atoms, there are NH3 [192] formaldehyde (H2 CO) [193], phosphine (PH3 ), which has never been investigated by hexapole; it is of interest in astrobiology, as a possible marker for the presence of O2 in exoplanet atmospheres [194], and finally hydrogen peroxide, H2 O2 , the first chiral molecule found in the interstellar medium [195]. More complex molecules detected in space are formic acid, HCOOH, methanimine CH2 NH, cyanamide NH2 CN, cyanomethyl radical CH2 CN, methanol CH3 OH, acetaldehyde CH3 CHO, methylamine CH3 NH2 , acetic acid CH3 COOH

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Fig. 39 Some of the molecules found in the interstellar medium, suitable to be studied by electrostatic hexapolar technique, applying the conformer selection. In white the hydrogen atoms, in gray the carbon, in gray the oxygen, and in blue the nitrogen. Curved blue arrows indicate the bonds involved in torsion motions

[183], ethanol CH3 CH2 OH, acetamide CH3 CONH2 , propylene CH2 CHCH3 , acetone CH3 COCH3 , ethylene glycol HO(CH2 )2 OH, propanaldehyde CH3 CH2 CHO, propylene oxide C3 H6 O (see the previous section), ethyl formate CH3 CH2 OCHO, methyl acetate CH3 COOCH3 , benzene C6 H6 [173], and benzonitrile C6 H5 CN. The list of molecules of interest in astrochemistry, and astrobiology, must be completed by those biomolecules found in meteorites and comets, such as ribose in meteorites [196] and glycine in comets [197]. As mentioned in Sect. 10.4, the study on selected conformers could be prototypical for the investigation of molecules of interest in biological processes. In Fig. 39, we report a scheme of the abovementioned molecules, suitable to be studied by the hexapolar technique, applying the conformer selection, i.e., hydrogen peroxide, formic acid, methanol, acetaldehyde, methylamine, acetic acid, ethanol, ethylene glycol, propanal, ethyl formate, and methyl acetate.

10.6 Final Remarks During the years, electrostatic hexapole demonstrated to be a valid tool for the study of stereo-directed dynamics of collisional and photodissociation processes. Initially, it was employed as a rotational state selector permitting to determine, under welldefined rotational and velocity distributions, the quantum composition of the molecular beam, as well as the statistical variables like the rotational temperature. In the

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last years, it demonstrated to possess other important features, such as the possibility to orient molecules, if combined with a homogeneous field, and in particular cases to select conformers. These characteristics might be exploited to characterize the dynamics involving molecules of interest in astrochemistry. This is a multidisciplinary area that nowadays has exceeded the scope of chemistry and involves other disciplines like biology. Part of the astrochemical investigations are performed in laboratory, where processes are reproduced to characterize processes that occur in the interstellar medium or in planetary atmospheres. This latter aspect is often related to prebiotic chemistry issues. Hexapole can assist experiments on stereo-directed processes by exploiting its ability to align and eventually orient molecules, covering a wide range of molecules found in the interstellar medium. The hexapolar technique must be further developed toward the realization of conformer-selected photodissociation and collision experiments. This aspect is of interest in astrobiology, where conformers play a basic role. Nevertheless, hexapolar technique can be applied to molecules of interest in astrochemistry, both for those found in molecular clouds and those found in comets and meteorites, providing information on the reactions involving the formation of molecules, especially those of interest in life processes.

11 Topic 6: Impact-Parameter-Dependent Analysis of the Trajectory Pattern for the H + H2 Exchange Reaction at T = 3 K and 300 K: Interconnecting the Trajectory Reactivity with the Time-Dependent Interaction Potential Energy and the Roaming-Like Libration Motion at Cold Temperature Muthiah Balaganesh

11.1 Introduction Rapid advancement in modern science and technology allows chemists to control chemical reactions, especially producing a high yield of desired products [67, 198– 205], and to pursue scientific curiosities, for instance, how chemical reactions proceed in cold and less dense interstellar clouds in the Universe. So far, organic molecules such as methanol, formaldehyde, and vinyl alcohol are observed in the interstellar clouds, which are generally synthesized at higher pressures and temperatures on the Earth. Chemical reactions could perhaps occur faster than we may expect [185, 206–213]. To lower reactant temperature has basic importance in view of reaction dynamics for prolonging reaction time, that is the alternative way of femtochemistry which probes intermediate complexes and/or the transition state species during chemical reaction at extremely short time scale. A good example of such cold chemistry is

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the direct observation of the reaction intermediates and the products in the bimolecular KRb + KRb → K2 Rb2 * K2 reaction in the ultracold regime, and such a state-ofthe-art experiment, however, is extremely difficult [214]. Computationally, it would be easy to prepare the cold molecules that we show in this topic, where we simply reduce the initial temperature of reactant molecules in the trajectory calculation. The second advantage of the trajectory calculation is to study the impact-parameter dependence of the reaction mechanism from which we obtain detailed dynamics. To control the impact parameter is hardly possible by experiment. This is a reason why we employ the QCT trajectory calculation in the present study [215–222]. Studies on kinetics and dynamics in cold temperatures are exciting for testing the transition from quantum to classical behaviors of chemical reactions [223]. We show that quantum tunneling would be small at T = 3 K in this study. The LEPS potential energy function is constructed based on the valence bond (VB) theory, and this theory enables us to understand the principle of the chemical bond and that the atomic orbital picture indeed complements the molecular orbital theories. It is known that even the simple LEPS energy potential could provide us a more accurate picture of collision dynamics oftentimes, especially when bonds are broken and formed during the course of a chemical reaction [224]. In the previous studies, Roaming pathways which bypass the transition states (TS), are discussed in detail [83, 85, 86, 98, 104, 122, 123, 131, 225–232]. The Polanyi rule provides us with a very significant guiding principle on reaction dynamics [233, 234]. Strictly speaking, this rule is only valid for the collinear collision configuration. The rotational degree of freedom due to the orbital angular momentum originated by nonzero impact-parameter collisions would lead to collision dynamics more complex and fruitful [71, 72, 74, 235, 236]. In this topic, we focus on the detailed trajectory pattern analysis with the aid of Lagrangian mechanics and the time-dependent interaction potential as a function of the impact parameter b at T = 3&300 K, and we emphasize the coupling of the rotational and the vibrational motion (libration) which plays a major role in the three-body system for enhancing reactivity via efficient energy transfer. There have been few studies on impact-parameter b-dependent trajectory pattern analysis to clarify how the trajectory pattern is interconnected with the amplitude/phase of the time-dependent interaction potential [237–244]. Though it will be time-consuming to check each trajectory pattern in thousands of sampled trajectories with naked eyes, it is definitely worthy carrying out.

11.2 Methodology London and Heitler established a clear-cut quantum mechanical treatment for the formation of the hydrogen molecule. According to their computational treatment, the allowed energies for the H2 molecule are the sum and differences of two integrals given as E = Q ± J, where Q is the Coulombic integral and J represents the exchange integral. The Coulombic energy is roughly equivalent to the classical energy of the system for the H2 molecule consisting of two nuclei and two electrons and the system

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energy is given by the charged particle interaction energy. However, the exchange energy appears in quantum mechanics and represents the energy due to the exchange of indiscernible particles. On the basis of the London equation, Eyring and Polanyi calculated the potential energy surface for the triatomic A + BC system, which is known as London–Eyring–Polanyi (LEP) surface. In this treatment, the Coulombic integral Q and exchange integral J for a diatomic molecule have been assumed to be the constant fractions of the total energy E for all internuclear distances. The assumption of constant Coulombic and exchange terms in the LEP method was adjusted by Sato by introducing the fractional constant S AB , S BC , and S AC [245, 246]. The equations of LEPS and the Morse are given by Eqs. (23) and (24), respectively

V (r AB , r AC , r BC ) =

Q AB Q BC Q AC + + 1 + S AB 1 + S BC 1 + S AC

⎧ ⎡ ( )2 ⎤⎫ 21 J BC ⎪ ⎪ J AB ⎪ ⎪ − ⎪ ⎪ 1+S AB 1+S BC ⎪ ⎪ ⎨1⎢ )2 ⎥ ⎢ ( ⎥⎬ J J ⎢+ ⎥ BC AC − − ⎥ 1+S BC 1+S AC ⎪2⎢ ⎪ ⎪ ⎣ ( ) 2 ⎦⎪ ⎪ ⎪ ⎪ ⎪ J AC J AB ⎭ ⎩ + 1+S AC − 1+S AB

┐ ┌ E = D e−2β (r −req ) − 2e−β (r −req )

(23) (24)

where D is the dissociation energy, β a constant of a molecule, and r eq the equilibrium internuclear distance, respectively. We performed quasi-classical trajectory (QCT) calculations as implemented in the VENUS96 code [247, 248]. To describe the H + H2 interaction, we employed the London–Eyring–Polanyi–Sato (LEPS) potential energy surface [245, 246, 249, 250], with corresponding experimental spectroscopic constants (e.g., Do , β, and r eq ) which were taken from the book, “Spectroscopic constants of diatomic molecules by Huber, K. P., & Herzberg, G. (1979)” for the H2 molecule. How the individual trajectory pattern depends on the impact parameter b (0.0–5.0 Å) at temperatures T = was analyzed carefully. A total of 6600 trajectories involves sampling of 300 trajectories per each impact parameter b was used for the trajectory pattern analysis. A propensity rule was proposed from the analysis of interaction potential and its Lagrangian of all the time-dependent trajectories. Every trajectory starts at reactants H and H2 with a separation of 5 Å and is completed when the separation of product H–H2 is 9.0 Å for all impact parameters. The initial orientations of the H2 molecule with respect to its center of mass and approaching point of the H atom are randomized. The initial H2 bond-length ranges from 0.65 to 0.85 Å, with the 0.05 Å broadening. The equilibrium bond-length of H2 is 0.75 Å, and the variation of the initial H2 bond-length reflects the timing or the phase of the H atom and the H2 molecule encounter and the threebody interaction. The vibrational and rotational energies were set to be zero for the H2 molecule, whereas the velocity and translational energy were changed by setting the desired temperatures of the system, i.e., 3 K or 300 K. For each and every impact parameter, we have calculated 300 trajectories where the samples are orientated in a random fashion. The effect of the initial configuration of randomly oriented

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Fig. 40 The schematic view of non-collinear configuration at non-zero impact-parameter b collision. RAB , RBC, and RAC are the distances between the pairs of atoms (A & B), (B & C). and (A & C), respectively

samples was checked against collinear orientation. The trajectory time progressed with step size (s) of 0.1 fs. The change in the interatomic distances between three atoms are denoted as RAB , RBC , and RAC as shown in Fig. 40. This allows us to include explicitly rotational motion in the collision system [230]. From the impactparameter-dependent collisions which involve energy transfer and reactivity, one can understand the interaction forces including amplitude and phase and their importance. This chapter explains the importance of impact-parameter-dependent pattern analysis to understand the collision dynamics based on the QCT trajectory calculation.

11.3 Results and Discussion 11.3.1

Impact-Parameter b Dependence of the Fraction of Reactive Trajectories to the Total Trajectories at Temperatures T = 3 K and 300 K

The number of reactive trajectories was calculated for every impact parameter and at temperatures 3 and 300 K which are given in Table 2. Ensembles of 100 and 300 trajectories were used for the QCT calculation for all the impact parameters. The total number of reactive trajectories and their fraction out of 300 trajectories at 3 and 300 K were given in column 2 to column 5 of Table 2, respectively. The fractional value present in the parenthesis of column 3 and column 5 corresponds to the 100 ensemble trajectories. A total of 3300 trajectories of 300 ensembles and 1100 trajectories of 100 ensembles were utilized in this calculation. The impact-parameter

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Table 2 Total number of reactive trajectories and their corresponding fractions for every impact parameter at temperatures 3 and 300 K [From [251]. Copyright © 2022 by John Wiley Sons, Inc. Reprinted by permission of John Wiley & Sons, Inc.] Initial impact parameter b (Å)

Temperature (3 K) Number of reactive trajectories

Temperature (300 K) Reactive fraction

Number of reactive trajectories

Reactive fraction

0

194

0.65 (0.66)

105

0.35 (0.33)

0.5

188

0.63 (0.66)

99

0.33 (0.32)

1

200

0.67 (0.57)

95

0.32 (0.34)

1.5

191

0.64 (0.68)

97

0.32 (0.32)

2

202

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values were varied from 0.0 to 5.0 Å with a step size of 0.5 Å. The results obtained from 300 ensembles of trajectories and 100 ensembles of trajectories are in very good agreement with each other. The reactive trajectory was classified if the bond exchange happened during a collision and the hydrogens were labeled as HAB , HBC , and HAC in the QCT calculation. It is not possible to discern if a collision is reactive or non-reactive in experiment, for three hydrogen atoms are identical. A plot of the ratio of reactive trajectories between total trajectories at temperatures of 3 K and 300 K against the impact parameter varying from 0.0 to 5.0 Å with the interval of 0.5 Å is shown in Fig. 41. It is interesting to observe from the figure that about 0.66 fractions of reactive trajectories were observed in all cases of impact parameter at 3 K, whereas the fraction is 0.33 for b 0.0–1.5 Å and it drastically decreases to zero for the b > 1.5 Å. at 300 K, which contains useful information on reaction dynamics we discuss later.

11.3.2

Impact-Parameter b Dependence of the Fraction of Direct Collision Pattern Trajectories to the Total Trajectories at T = 3 K and 300 K

The contact time between H atom and H2 molecule during a collision is very short for many trajectories, and those trajectories were considered as direct collision. We calculated the ratio of direct collision trajectories with the total number of trajectories at 3 and 300 K. It is surprising to notice that almost 98% of the trajectories were found to be a direct collision at 300 K, whereas it is 72% in the case of 3 K. The ratio

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Fig. 41 The ratio of reactive trajectories between total trajectories at temperatures of 3 K (blue curve) and 300 K (red curve) against the impact parameter varies from 0.0 to 5.0 Å with the interval of 0.5 Å [From [251] Copyright © 2022 by John Wiley Sons, Inc. Reprinted by permission of John Wiley & Sons, Inc.]

of direct collision trajectories with a total number of trajectories was calculated for all b values and plotted as shown in Fig. 42. It is surprising that the direct collision pattern trajectory pattern fully dominates 0.98 at temperature T = 300 K throughout the whole impact-parameter range, while the blue curve for T = 3 K gives the average fraction only 0.72. Figure 42 tells us that the direct collision trajectories at b larger than 2.5 Å are all non-reactive at T = 300 K, so this result could be explained using the hard-sphere model. On the other hand, at T = 3 K, a considerable fraction of trajectories, i.e., 0.28, are the trajectory with intermediate H + H2 complex formation. The reason behind the high fraction

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Fig. 42 The fraction of the direct collision pattern trajectories as a function of the impact parameter from b = 0.0 to 5.0 Å in steps of 0.5 Å. The blue curve is obtained at T = 3 K and the red curve at T = 300 K. The average value of the fraction is 0.98 at T = 300 K, and 0.72 at T = 3 K, respectively [From [251] Copyright © 2022 by John Wiley Sons, Inc. Reprinted by permission of John Wiley & Sons, Inc.]

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of 0.66 for the whole impact parameters at cold temperature T = 3 K was due to the formation of a collision complex which enhances both energy transfer and reactivity. It is important to point out that the impact-parameter dependence of the fraction of reactive trajectories and the impact-parameter dependence of the fraction of direct trajectories at two temperatures T = 3 K and 300 K provide us an indication that the quantum tunneling could be negligible at T = 3 K in the H + H2 exchange reaction, because of the following two main reasons. First, the rate of quantum tunneling is independent of temperature, while Fig. 42 tells us that the reactive fraction 0.66 is big enough at T = 3 K which is double 0.33 for T = 300 K. Second, the origin of quantum tunneling is the penetration of the wave function through the energy barrier. Assuming either the partial waves or the plane wave penetrates through the barrier in the H + H2 exchange reaction, it is unlikely for T = 3 K that the reactivity is kept as high as 0.66 at the larger impact parameters b = 2.5–5.0 Å. An alternative interpretation would be reactive orbiting in the long-range attractive force region. The trajectory pattern was analyzed and categorized in the following sections, and a unique propensity rule is proposed based on the interaction potential and its correlation with trajectory reactivity.

11.3.3

Impact-Parameter-Dependent Trajectory Pattern Analysis at T = 3 K and 300 K: A Characteristic Propensity Between the Trajectory Reactivity Versus the Time-Dependent Interaction Potential

The proposed propensity rule would be very useful in classifying the reactive and non-reactive trajectories in a simple manner. We compare the amplitude/phase of interaction potential with the trajectory pattern. Out of 40 characteristic trajectories, only two of them for each temperature were selected as representative trajectories for the detailed discussion. We mainly classified all the 3300 trajectories into two broad categories: one is direct collision and another one is intermediate complex formation. We particularly pay attention to the local collision dynamics around the contact point in terms of the time-dependent interaction potential.

A Characteristic Propensity Between the Trajectory Reactivity versus the Time-Dependent Interaction Potential at Temperature T = 300 K As discussed in the previous section, the reactive trajectories were calculated to be 98% at 300 K and within the short range of b from 0.0 to 2.0 Å. Thus, one example is a non-reactive and the other is a reactive trajectory of the direct collision pattern at b = 0.0 Å, respectively. A representative non-reactive and direct collision trajectory at b = 0.0 Å is shown in the top panel of Fig. 43. The blue line corresponds to the vibrational oscillation of H2 named as RAB . During the collision of H and H2 at 120 fs, the energy transfer causes the translational and rotational excitation labeled as RBC (gray in

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Fig. 43 An example of non-reactive trajectory resulting from direct collision at b = 0.0 Å and T = 300 K. [Top panel] Three interatomic distances/bond-lengths [Å] are given with RBC (blue curve), RAB (red curve), and RAC (gray curve). [Bottom panel] The enlarged view of the time window of t = 80 ~180 fs, where the time-dependent interaction potential E ABC [kcal/mol] (green curve) and the corresponding Lagrangian (red curve) are shown, respectively [From [251] Copyright © 2022 by John Wiley Sons, Inc. Reprinted by permission of John Wiley & Sons, Inc.]

color) and RAC (red in color). We observe the rotational excitation even at b = 0.0, due to the non-collinear random orientation of H and H2 molecules applicable to all our QCT calculations. This rotation is due to the torque exerted by the three-body interaction. As can be seen from the bottom panel, there is no notable change in the amplitude/phase of interaction potential (in green), whereas we observed interesting changes in the Lagrangian (in red) before and after the collision which explains the sensitivity of Lagrangian function and local collision dynamics due to the energy transfer. In Fig. 44, we have shown a rotational excitation of reactive trajectory due to direct collision at zero impact parameter. We have observed significant and interesting changes at collision 130 fs in the case of reactive trajectory, in contrast to non-reactive trajectory. In the case of interaction potential, there is a clear dump of amplitude at collision, whereas in the case of Lagrangian function we observed a perturbed feature which might be due to the rotational excitation of product H2 .

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Fig. 44 An example of reactive trajectory with rotational excitation resulting from direct collision at b = 0.0 Å at T = 300 K. [Top panel] Three interatomic distances/bond-lengths [Å] are given with RBC (blue curve), RAB (purple curve), and RAC (brown curve). [Lower panel] The enlarged view of the time window t = 80 ~180 fs near the contact point. The interaction potential E ABC [kcal/mol] (in green) and the corresponding Lagrangian (in red) are given, respectively [From [251] Copyright © 2022 by John Wiley Sons, Inc. Reprinted by permission of John Wiley & Sons, Inc.]

So far, we only show two trajectories of the direct collision pattern, but we similarly can confirm the characteristic propensity of the amplitude dump of the timedependent interaction potential for other reactive trajectories either the direct pattern or the complex formation pattern at T = 300 K. Thus, we may expect that this characteristic propensity rule in the time-dependent interaction potential is a useful measure for judging if a collision is reactive or non-reactive and also, we can monitor the energy transfer process through Lagrangian for clarifying local reaction dynamics. In the next section, we shall find out whether or not this propensity rule holds at low temperature T = 3 K.

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A Characteristic Propensity Between the Trajectory Reactivity versus the Time-Dependent Interaction Potential at T = 3 K As mentioned before, 28% of the trajectories were observed to have intermediate complex formation with a high fraction of reactivity of 0.66 for all the b values and at 3 K. Out of 28% of those trajectories, we selected two trajectories for the discussion at b = 2.0 and 2.5 Å. The first one is a non-reactive trajectory with a complex formation pattern at b = 2.5 Å and the other one is a direct and reactive trajectory with three contact points at b = 2.0 Å. We note that an intermediate complex formation pattern trajectory has two contact points, i.e., at the entrance and the exit of the complex formation process and they are either reactive or non-reactive collision. As a result, the final reactivity depends upon the combination of the two collision events at the contact points. Figure 45 shows a non-reactive with complex formation trajectory at b = 2.5 Å. It has two contact points corresponding to the entrance at 65 fs and exit channels at 225 fs. As mentioned before, interaction potential has two clear dumps at contact points of 65 fs (top panel) and 225 fs (bottom panel) respectively where the chemical bond exchange occurs. The interesting observation is that, though there is chemical bond exchange occurring at contact points, the final trajectory is a non-reactive trajectory. We can conclude from the Lagrangian in the bottom panel that vibrational energy transferred to the translation and rotation degrees of freedom of product H2 . We choose another interesting trajectory which is reactive and contains three contact points at T = 3 K and b = 2.0 Å as shown in Fig. 46. There is no sign of dump in amplitude at the entrance and middle contact points (t = 61 and 87 fs, respectively) as can be seen from their corresponding second and third panels. However, we observed a smaller perturbation at adjacent amplitudes due to prolonged contact time. Interestingly, we observe the dump in interaction potential at the exit contact point (t = 112 fs) due to bond exchange. The vibrational energy transfer might cause the amplitude dump after the collision. Thus, we come to conclusion and propose a propensity rule that the amplitude dump occurs at contact points if the trajectory is reactive; conversely, non-reactive trajectory won’t have amplitude dump at contact points at both the temperatures 3 and 300 K. So far, we have confirmed by the trajectory pattern analysis that there is a clear characteristic propensity in the behavior of the amplitude of the time-dependent interaction potential, with which we can judge easily if a trajectory is reactive or non-reactive. Additionally, it appears to be a close correlation between the behavior of Lagrangian and the energy transfer processes, and it could be a significant clue to analyzing the detailed local dynamics in a collision. It is also noteworthy that this characteristic propensity in the time-dependent interaction potential is independent of collision temperature and the trajectory pattern (i.e., whether it is a direct collision pattern or one via intermediate complex formation). In the next section, we touch on the coupling between rotational and vibrational motion, so-called roaming-like libration motion at a cold temperature for non-collinear collision configurations.

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Fig. 45 An example of non-reactive trajectory resulting from the complex formation pattern at b = 2.5 Å. [Top panel] Three interatomic distances/bond-lengths [Å] are given with RBC (blue curve), RAB (red curve), and RAC (gray curve). [Middle panel] Enlarged view of the time window t = 60–70 fs centered at the contact point (t = 65 fs) at the entrance of the intermediate complex formation process. [Bottom panel] Enlarged view of the time window t = 220–230 fs centered at the contact point (t = 225 fs) at the exit of the intermediate complex formation process. The trajectory in the top panel manifests rotational motion during the complex formation process and also the translational and slight rotational excitation of the product H2 molecule. The interaction potential E ABC [kcal/mol] and its corresponding Lagrangian are shown in green and red, respectively [From [251] Copyright © 2022 by John Wiley Sons, Inc. Reprinted by permission of John Wiley & Sons, Inc.]

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Fig. 46 An example of reactive trajectory resulting from the three-step direct collision trajectory pattern with the energy transfer to translation at b = 2.0 Å. [Top panel] Three interatomic distances/bond-lengths [Å] are given with RBC (blue curve), RAB (red curve), and RAC (gray curve). [second panel] Enlarged view of the time window t = 57–67 fs centered at the contact point (t = 61 fs) at the entrance contact. [third panel] Enlarged view of the time window t = 84-94 fs centered at the contact point (t = 87 fs). [Bottom panel] Enlarge view of the time window t = 107–117 fs at the exit contact point centered at contact point (t = 111 fs). The interaction potential E ABC [kcal/mol] and its corresponding Lagrangian are shown in green and red, respectively [From [251] Copyright © 2022 by John Wiley Sons, Inc. Reprinted by permission of John Wiley & Sons, Inc.]

11.3.4

A Roaming-Like Libration Motion at Cold Temperature

It is known that the concerted motion (i.e., two bondssynchronization in A-B-C) could enhance reactivity in the A + BC reaction, especially when A = B = C as in the H + H2 exchange reaction. In the collisions at non-zero impact parameters (b /= 0), the orbital angular momentum is involved, therefore, the coupling between rotational

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Chemical Reaction Kinetics and Dynamics Re-Considered ...

Fig. 46 (continued)

Fig. 47 A trajectory of roaming-like libration motion at b = 3.0 Å at cold temperature T = 3 K. During the time range of 60–200 fs, the libration motion continues and then is followed by rotational excitation. Three interatomic distances/bond-lengths [Å] are given with RBC (blue curve), RAB (red curve), and RAC (gray curve) [From [251] Copyright © 2022 by John Wiley Sons, Inc. Reprinted by permission of John Wiley & Sons, Inc.]

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and vibrational degrees-of-freedoms results in the libration motion. We may accept this similar idea of the concerted libration motion in non-collinear collisions for promoting reactivity, especially at cold temperatures as T = 3 K. We call such case the roaming-like libration motion, for it is known that the rainbow orbiting occurs at low translational energy due to the long-range attractive force in the elastic collision scattering [252]. It is surprising that the high fraction of 0.66 of reactive trajectories was observed for the whole range of impact parameters b at T = 3 K. In addition, 28% of the trajectories manifest the trajectory pattern via the H + H2 intermediate complex formation. As a result, we may expect to see a trajectory of the roaming-like libration as is seen in Fig. 47 at cold temperatures. It is noticeable in the figure that the prolonged three-body interaction is occurring accompanying rotational and vibrational motions which is followed by rotational excitation and the energy transfer to translation after the contact point. The time scale of collision will decide the reactivity and energy transfer among translational, vibrational, and rotational degrees of freedom. A very small amount of translational energy can enhance the collision times at cold temperatures. The collision takes place as a nonadiabatic limit associated with b > 0. This kind of phase sensitivity reactivity can be understood from the angle-action diagram in analytical mechanics [253, 254]. A detailed discussion of adiabatic and nonadiabatic limits can be found for further understanding of Collision energy transfer of the Landau– Teller model and unimolecular reactions of Slater’s harmonic theory [255, 256]. As mentioned, the rainbow orbiting occurs at low translational energy due to the longrange attractive force in the elastic collision scattering [252]. A significant number of H + H2 complex formation trajectories were observed for larger b where the long-range attractive force would be dominant. This long-range force plays a major role in cold temperature dynamics and mainly proceeds via roaming-like libration. There were many such trajectories with the complex formation, and a representative trajectory is shown in Fig. 47. This roaming-like libration motion enhances the reactivity and energy exchange significantly among the vibration, translational, and rotational degrees of freedom, especially in a cold temperature atmosphere such as interstellar clouds in the Universe. This is a challenging future objective of the research.

11.3.5

Conclusions

In this chapter we carried out QCT calculation of H + H2 exchange reaction using LEPS potential energy surface at temperatures of 3 and 300 K for the impact parameter from 0.0 to 5.0 Å with step size of 0.5 Å. We have carefully performed the pattern analysis of 3300 total trajectories (300 trajectories per impact parameter). We calculated the fractions of reactive trajectories of 0.66 for all ‘b’ values at 3 K whereas at 300 K this fraction is 0.33 for b = 0.0~1.5 Å, and decreases to zero for b = 1.5~5.0 Å. Another interesting result is at 300 K ~98% of the trajectories were observed to be direct collisions where as it was 72% at 3 K and the remaining trajectories were the complex formation of H + H2 intermediate. These results suggest that

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quantum tunneling would be negligible at T = 3 K in the H + H2 reaction. We also observed a unique propensity rule in the interaction potential amplitude via trajectory pattern analysis. This propensity rule is very useful not only in predicting the collision which is reactive or not but also in predicting whether the collision is direct or via complex formation. Furthermore, there seems to be a correlation between the amplitude/phase pattern of Lagrangian and how the energy transfer takes place during collision. Therefore, the Lagrangian provides an additional information in analyzing local dynamics during collisions. The concerted libration motion influences the energy transfer and reactivity of collision for b greater than zero. Also, this concerted libration motion leads to the roaming like motion which was expected for the H + H2 exchange reaction at cold temperatures. We expect this study would be useful in understanding the interstellar chemistry.

12 Concluding Remarks Thus far, we have re-considered chemical reaction kinetics and dynamics, and proposed a way of exploring stereo quantum dynamics—from line to plane reaction pathways and concerted interactions. We are not certain if the proposal presented in this chapter convinces the reader or not, but it seems clear by the experimental and theoretical evidences that the traditional chemical reaction scheme of the transitionstate barrier along one dimensional reaction coordinate is no more sufficient to explain the diversity of chemical reactions which have exit channels with a variety of energy proposal into different degree of freedoms of the same reaction products and so on. In this review, we have tried to understand a new reaction scheme by using the flux of excited reaction intermediate complex on a two-dimensional plane or more than two reaction coordinates. Roaming mechanisms are typical such environment. Either the ground or excited electronic states are involved in the reaction, where conical intersection might be present and non-adiabatic coupling may play a key role. Topics 1 on UV-photodissociation of halothane with competitive bond breaking, Topic 2 on roaming dynamics in carbonyl compound photodissociations, and Topic 3 on the unimolecular dissociation by dynamical model are such examples, where the flux of reaction intermediate species should be treated under non-equilibrium steady-state condition. Topic 4 on spatially oriented molecules and chiral discrimination via photodissociation and Topic 5 on electrostatic hexapoles in the study of stereo-directed dynamics processes in astrochemistry are exactly relevant to stero quantum dynamics. We may find future perspective of reaction dynamics in astrochemistry at cold temperature, like in the interstellar clouds. In Topic 6, we presented the impact-parameter dependent trajectory pattern analysis as a powerful tool for obtaining fruitful information on non-collinear reaction dynamics. The treatment is based on the quasi classical mechanics, but the trajectory pattern analysis reflects the phase/amplitude-sensitive dynamics through the time-dependent interaction potential. Because of this reason we may regard it as dynamics of concerted

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interaction. We hope this review article is of some help for fining out future directions and perspectives about exciting reaction kinetics and dynamics toward quantum stereodynamics.

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Author Biographies Toshio Kasai received Dr. Sci. (Ph.D) degree from the Graduate School of Science, Osaka University in 1979. He became a full professor of Osaka University in 1995. He is now an emeritus and invited professor of Osaka University, and Fellow of Royal Society of Chemistry. He has been a visiting professor of National Taiwan University since 2010. He received the Richard Bernstein Award of Stereodynamics 2016 and the Chemical Society of Japan Award for Creative Work 2002. His main research interest is quantum stereodynamical aspects in chemical reactions and reaction dynamics at the molecular level by using oriented molecular and laser beams with the aids of theory. His research interest extends to physicochemical clarification of cellular respiration and Respiratory Quotient dynamics. He also puts his effort on interconnecting science and arts. King-Chuen Lin is a Distinguished Professor of the Department of Chemistry at National Taiwan University and a Distinguished Research Fellow of National Science Council, Taiwan. He received his B.S. degree in Chemistry from National Taiwan University, Taiwan, his PhD in Chemistry from Michigan State University, USA, and his postdoctoral career at Cornell University. His research interests are photodissociation and reaction dynamics in gas and condensed phases, atmospheric chemistry, materials designed for sensors and catalysts, and single molecule spectroscopy. He received Academic Award of Ministry of Education, Taiwan, in 2014, Richard B. Bernstein Award in International Conference on Stereodynamics in 2018, and Chair Professorship Award from Ministry of Education, Taiwan, in 2019. He has served as an Associate Editor for J. Chin. Chem. Soc. (Taipei) until 2020, and now a member of Editorial Board for Scientific Reports and a Guest Editor for this Journal. He has published more than 237 peer-reviewed papers and edited one book on reaction dynamics and chemical kinetics.

Quantum Theory

Solution for Lithium Problem from Supersymmetric Standard Model Joe Sato, Yasutaka Takanishi, and Masato Yamanaka

Abstract A review on a non-standard Big-Bang nucleosynthesis (BBN) scenario within the minimal supersymmetric standard model is given to explain our solution for the 7 Li and 6 Li problems. There is a well-known discrepancy between the predicted abundance in the standard BBN and the observed one. In the solution, the lightest slepton mainly consisting of stau, a supersymmetric partner of the tau lepton, plays a crucial role. It is a long-lived charged particle and becomes long-lived when it degenerates in mass with the lightest supersymmetric particle. According to coannihilation mechanism, then the lightest supersymmetric particle will be a good candidate for dark matter. The long-lived “stau” forms a bound state with a nucleus and provides non-standard nuclear reactions. Among them, the internal conversion process is most important to destruct 7 Be and 7 Li, leading a solution to the 7 Li problem. Note that, in addition, the catalyzed process which is another exotic process caused by a bound state of stau and 4 He can solve the 6 Li problem. In general, the lifetime of particles is determined by the degeneracy of the masses and also controlled by the strength of lepton flavor violation. If stau is a long-lived particle, then it can contribute to the exotic process for nucleosynthesis. The internal conversion process must be effective while other exotic processes must be ineffective. For these requests, the parameter space of stau is strictly constrained, however. Therefore, we need to study carefully the stau-4 He bound state for solving the 6 Li problem. The scenario of the long-lived stau simultaneously and successfully fits the abundances of light elements (D, T, 3 He, 4 He, 6 Li, and 7 Li) and the neutralino dark matter to the experimental data. Consequently, we can determine the parameter space of the J. Sato (B) · Y. Takanishi Physics Department, Saitama University, 255 Shimo-Okubo, Sakura-ku, Saitama, Saitama 338-8570, Japan e-mail: [email protected] Y. Takanishi e-mail: [email protected] M. Yamanaka Nambu Yoichiro Institute of Theoretical and Experimental Physics (NITEP), Osaka City University, Osaka 558-8585, Japan e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 T. Onishi (ed.), Quantum Science, https://doi.org/10.1007/978-981-19-4421-5_4

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stau and the neutralino with excellent accuracy. We address that, in this parameter space, the mechanism of Sommerfeld enhancement plays a crucial role, and then, as a result, the dark matter signal becomes large enough to be observed by the current sensitivity of indirect experiments. Keywords Big Bang · Nucleosynthesis · Early universe · Li problem · Standard model of particle physics · Supersymmetry · Dark matter

1 Introduction Lithium problem [1] is one of the long-standing questions. It is the discrepancy of the Lithium primordial abundance between the observed values and the theoretical prediction, which is evaluated in the framework of standard big-bang nucleosynthesis (SBBN). The observed value of 7 Li/H is set as (1 − 2) × 10−10 , [2–7] are inferred from metal-poor stars while the SBBN prediction is (5.24 ± 0.7) × 10−10 [8, 9]. There is a large difference greater than 4σ level. It is called 7 Li problem. There is another tension between the observed and predicted values for 6 Li primordial abundance: the observed value is about 1000 times higher than the theoretical prediction (see Eq. (3)) [10], though it is still controversial [11]. These theoretical (SBBN) values are obtained by using the standard model (SM) of particle physics. SM describes three out of four fundamental forces of nature, i.e., the electromagnetic, weak, and strong forces. This model also offers the interactions of the elementary particles that are displayed in Fig. 1. The SM has been tested

Fig. 1 Particle content of the Standard Model ©Akimoto Yuki@higgstan

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by many different kinds of experiments: confirmation of the existence of quarks in nucleus and discovery of W - and Z -bosons in 70th, and discovery of top quark in 1995, and tau neutrino in 2000. Finally, it is completed by the discovery of the Higgs boson particle in 2012. Though there are lots of astrophysical explanation for the Li problems [12], we take it as a smoking gun to extend the SM. Since the big-bang nucleosynthesis (BBN) took place from 1 to 1000 sec, after the big bang, it means that we have to add a long-lived particle. Indeed, for example, long-lived charged massive particles (socalled long-lived CHAMPs) induce non-standard nuclear reactions in the BBN and drastically change light element abundances [13–35]. Besides the Li problems, a large number of questions are still leftover and suggests the presence of a more fundamental theory beyond the SM. Among those, it is one of the main goals of particle physics to understand the nature of dark matter (DM) and here it could guide us to a compelling solution for the Li problems. One of the most prominent candidates for the DM is the weakly interacting massive particle (WIMP) [36–39]. Indeed, the supersymmetric (SUSY) extension of the SM provides a stable exotic particle, the lightest supersymmetric particle (LSP), if the R-parity is conserved. Among the LSP candidates, the lightest neutralino χ, ˜ a linear combination of neutral fermions which are the supersymmetric partners of hypercharge gauge boson (bino), weak one (wino), and Higgs bosons (higgsinos), is a most suitable for non-baryonic dark matter since its nature fits the observed evidences of DM [40, 41]. Bino-like neutralino LSP is most extensively studied. In this case, the next-tolightest supersymmetric particle (NLSP) is required to be degenerate in mass to predict the observed dark matter density employing coannihilation mechanism [42]. A scalar partner of tau lepton, Stau τ˜ , is the most likely candidate for the NLSP [43]. It must be an admixture of other scalar partner of leptons (electron and muon) due to LFV. To explain the Lithium problems that we will discuss in the later sections, it is better to have a tiny mixing than the flavor conserving state, pure stau. Though there is a tiny mixing of lepton flavors, we will call the lightest scalar lepton stau. It is well known that to predict observed dark matter abundance their masses must degenerate strongly, that is, (m τ˜ − m χ˜ )/m χ˜ < a few % where m τ˜ is the NLSP stau mass and m χ˜ is the LSP neutralino mass. When δm = m τ˜ − m χ˜ is smaller than tau mass m τ , the stau cannot decay into two body and as a result is very long-lived particle [44, 45]. The SBBN prediction can be altered by long-lived stau It can effectively destroy 7 Be which becomes 7 Li by electron capture reaction after the BBN era. Here the electron capture is a process in which a nucleus absorbs an orbital electron and converts into a nucleus with one lower atomic number, e.g., 7 Be+e− →7 Li+νe , where νe is the electron neutrino. Since at the BBN era would be 7 Li exists as 7 Be, to destruct 7 Be effectively means to reduce 7 Li primordial abundance. Such a long-lived stau with degenerate mass can offer the solution to 7 Li problem [20–22, 46–49]. Here this idea and astrophysical prediction of this scenario [50, 51] are reviewed. First a brief review on SBBN is given, especially highlighting the current status on the 7 Li problem. In Sect. 3, we give an introductory review of the SM of particle physics and the minimal supersymmetric SM, which is the most extensively studied

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framework for physics beyond the standard model. Then, in Sect. 4, we study the nature of the long-lived CHAMPs, in particular, its aspects related with the BBN; an origin of its longevity, the parameter dependence on the lifetime, and the number density at the BBN era. Next, in Sect. 5, we expound on non-standard nuclear reactions caused by the bound state formation of nuclei and the long-lived CHAMP: (i) Internal conversion. (ii) Catalyzed fusion. (iii) 4 He spallation processes. In Sect. 6, we numerically summarize a favored parameter region wherein predicted abundances of both the DM and light elements including lithium7 fit with the current observed ones. We apply our model to predict the DM mass and also the annihilation cross section, then we can compare our results with current cosmic ray measurements in Sect. 7. Finally, in Sect. 8, we summarize this review. Note that, in this review, we will use the “natural unit” where Planck constant ℏ and the speed of light c are equal to “1”. Therefore, all dimensionfull quantities are expressed by energy eV (electron volt). Further we also set the Boltzmann constant kB = 1 so that we use the absolute temperature in this review.

2 Review on Standard Big-Bang Nucleosynthesis In this chapter, we review the BBN within the standard model of particle physics. We discuss not only the theoretical studies of BBN but also the experimental analyses of the BBN. It becomes clear that there exists trouble on the primordial abundance of 7 Li. As we will see later, the theory beyond the SM can solve this issue.

2.1 Thermal History of the Universe In this section, we briefly overview the history of the universe. The history of the universe is described by either its age or the change of the temperature. The latter is very convenient since the temperature indicates the energy scale of physics from which we understand what kind of interactions are relevant. Our universe accrued with a primitive fireball so-called “Big Bang”. Immediately after the Big Bang (t = 10−43 s), the temperature, T , of the universe was as high as the Planck scale (T = 1019 GeV). Then the temperature decreases along with the expansion of the universe. • t ∼ (10−37 − 10−33 ) s, i.e., T ∼ (1016 − 1014 ) GeV. If the grand unified theory (GUT) is correct theory, a spontaneous symmetry breaking occurs which breaks the simple GUT group. The particles of the SM appear at this time. • t ∼ 10−10 s, i.e., T ∼ 300 GeV. The electroweak symmetry occurs at this time: photons and electromagnetic interactions appear.

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• t ∼ 10−4 s, i.e., T ∼ 100 MeV. The chiral symmetry is broken at this time. The color confinement occurs, and protons and neutrons are formed. • t ∼ (1 − 103 ) s, i.e., T ∼ (1 − 0.01) MeV. Light nuclei are synthesized in the universe. The abundances are precisely explained by the big-bang nucleosynthesis. The main theme of this article is a problem occurring at this timescale. • t ∼ 1012 s, i.e., T ∼ 1 eV. The dominant component of the universe is changed from radiation to matter. At this time the structure formation begins. • t ∼ 1013 s, i.e., T ∼ 0.1 eV. Matter and radiation are decoupled in this era because the small number density of the electron cannot maintain the thermal equilibrium. The formation of atoms also occurs in this era. The fog has lifted due to these phenomena, and then the universe is finally cleared. Thus photons are able to travel straightforward.

2.2 Big-Bang Nucleosynthesis: Theory In the following sections, we will mainly consider a time region, t ∼ (1 − 103 ) s, equivalently T ∼ (1 − 0.01) MeV. Nucleosynthesis proceeds at this time range and their abundances are well explained by the BBN. In this section, we see the BBN and its prediction. Then, in the next section, we will discuss the observed value of the abundances. The result is displayed in Fig. 2, Ref. [52], which shows the time and thermal evolution of the mass fractions of n, p, D, T+3 He, 4 He, 6 Li, 7 Li, 7 Be, respectively. The temperature varies between 1011 K and 108 K, which corresponds to the mass fraction between 10 MeV and 10 keV. Note that in this figure the baryon-to-photon ratio η is set to be as 5.1 × 10−10 . It is very important since it determines the abundances. We have to begin with T ≃ 1 MeV, where the neutron-to-proton ratio is fixed, to understand the prediction of nucleosynthesis, though the nucleosynthesis itself starts at T ≃ 0.1 MeV. This ratio is very crucial for precise predictions.

2.2.1

Synthesis Processes

1. T ≃ 1 MeV; freeze-out of neutron-to-proton ratio At this time, the following processes, n + e+ ↔ p + ν¯ e ,

(1)

n + νe ↔ p + e − , n ↔ p + e− + ν¯ e ,

(2) (3)

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1

1/60

1

5

15

60

10

p n

10

Mass Fraction

Fig. 2 The time (thermal) evolution of the mass fractions of n, p, D, T+3 He, 4 He, 6 Li, 7 Li, 7 Be. Here the baryon-to-photon ratio η = 5.1 × 10−10 . Reprinted from [52], ©IOP Publishing. Reproduced with permission. All rights reserved

10

−4

−9

D 10

10

−14

4

6

He

7

Li

Li

−19

7

Be

3

3

H, He

−24

10

2

10

10

1

0

10

−1

10

9

Temperature (10 K)

freeze-out. They are the weal process by which proton and neutron interconvert. The neutron-to-proton ratio at the time of freeze-out is given by the value of the nuclear statistical equilibrium, ( ) ( ) Q 1 n = exp − , ≃ p freeze-out TF 6

(4)

where Q is the mass difference between proton and neutron, and TF is the temperature of the universe at the freeze-out time. Exactly speaking, the ratio at the freezing time (temperature) is determined by solving the Boltzmann equation for the processes Eqs. (1)–(3). After the freeze-out until the beginning of nucleosynthesis, the neutron-to-proton ratio changes due to the spontaneous decay of the neutrons. As a result, the value at the beginning of the nucleosynthesis becomes about 1/7. This value plays a crucial role for BBN and is a very important parameter because its precise value is determined by the lifetime of the neutron. 2. T ≃ 0.1 MeV; nucleosynthesis Nucleosynthesis begins at this time. We see the synthesis processes of each nucleus; D, 4 He, 3 He, T, 7 Li, and 6 Li. For precise prediction, we have to solve the set of Boltzmann equations; however, we just give a qualitative argument here. a. Deuteron Firstly deuteron synthesis starts. That is, all the other elements are synthesized from deuteron. Its dominant creation process is n+ p →D+γ .

(5)

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We note here that two-body reactions are dominant for not only producing process of deuteron but also those of other nuclei. It is because the density has become too low at this time for the three or more body processes to be effective. Though the binding energy of deuteron is 2.22 MeV, the backward process, D + γ → n + p,

(6)

is effective at temperature higher than 0.1 MeV. It is because the baryon-tophoton ratio η is quite small (order of 10−10 ) and hence there are a sufficient number of photons with higher energy than 2.22 MeV until the temperature falls down to 0.1 MeV. Thus the nucleation of deuteron starts at T ∼ 0.1 MeV. This is understood as follows. The photon number density with energy higher than 2.22 MeV is given by ( ) 2.22 MeV , n γ-2.22MeV = n γ exp − T n D = η · n γ /2 ,

(7) (8)

where n γ and n D are the photon number density and the deuteron number density, respectively. The production process (5) exceeds the reduction process (6) as long as the condition n γ-2.22MeV = n D

(9)

is satisfied. It means when T ≲ 0.1 MeV, the creation process dominates the destruction process and as a result syntheses of other nuclei start. b. Helium 4 The synthesis Helium 4 (4 He) starts after the deuteron production via the following processes: D + p → 3 He + γ,

(10)

D + n → T + γ,

(11)

D + D → 4 He + γ,

(12)

T + p → He + γ,

(13)

4

3

He + n → He + γ, 4

(14)

D + D → He + n,

(15)

D + D → T + p,

(16)

3

He + n → T + p,

(17)

T + D → He + n,

(18)

3

4

3

He + D → He + p. 4

(19)

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Here in the processes (10)–(14) both the strong and the electromagnetic interactions are needed, while the processes (15)–(19) are induced by only the strong interaction. As 4 He is the most stable nucleus, almost all neutrons are captured in 4 He via these processes. Therefore, the neutron-to-proton ratio almost determines the 4 He abundance as follows:

Y ≡ X 4 He =

2(n n /n p ) ≃ 0.25, 1 + (n n /n p )

(20)

where we use n n /n p = 1/7 that is the neutron-to-proton ratio at the stating time of nucleosynthesis. c. Helium 3 and Triton Helium 3 (3 He) and Triton (T) are also synthesized via processes (10), (11), and (15)–(17). As 4 He is more stable than 3 He and T, they are used to create 4 He via the processes (13), (14), (17), and (18). Therefore, the abundance of 3 He and T is much lower than that of 4 He. Furthermore, since triton is unstable and decays into Helium 3 with half-life 12.32 years, 3 He abundance observed today is the total abundance of 3 He and T synthesized at the early universe. d. Lithium 6, 7 and Beryllium 7 Finally, we consider the synthesis of 6 Li, 7 Li, and 7 Be. There are two obstacles to the syntheses of nuclei heavier than 4 He. First one is absence of a stable nucleus with mass number five and eight. They are understood in the following way. Note that the structure of nuclei with mass number five is described as (i) a core of 4 He and orbital nucleon or (ii) a deuteron and three neutrons or protons or (iii) five neutrons or protons. In case (i), interactions among nucleons in 4 He are much stronger than those between 4 He and orbital nucleon. It means the orbital nucleon easily is unbounded from 4 He core. In case (ii), the deuteron and neutrons are also weakly interacted and easily go to pieces. Also in case (iii), the interactions between five neutrons or protons are weak, and hence they fall apart. Thus we have no stable state with mass number five. We also note that the structure of nuclei with mass number eight is described as (i) two cores of 4 He, or (ii) one 4 He, one deuteron and two neutrons or protons, or (iii) one 4 He and four neutrons or protons, or (iv) one deuteron and six neutrons or protons, or (v) eight neutrons or protons. In case (i), the two cores are weakly bound and are easily separated from each other. Apparently, cases (ii)–(v) are less stable than case (i) and thus they are not stable. This means that there is also no stable nucleus with mass number eight. The second obstacle is the Coulomb barrier. At the formation era of heavy nuclei, the effect of the Coulomb barrier is effective, since the kinetic energy of heavier nuclei at its formation time is smaller than the Coulomb barrier. In addition, heavier nuclei have larger electric charge and stronger Coulomb barrier. Due to these obstacles lithium and beryllium are synthesized only slightly. Incidentally, it also means synthesis of the nuclei heavier than beryllium is practically impossible.

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Production processes for 6 Li, 7 Li, and 7 Be are as follows: 4

He + D → 6 Li + γ,

(21)

3

He + T → Li + γ,

(22)

4

He + T → Li + γ,

(23)

4

He + He → Be + γ.

(24)

6

7

3

7

Note that the 7 Be is unstable and becomes 7 Li due to the electron capture, 7

electron capture

Be −−−−−−−−→ 7 Li . 53.22 days

(25)

Its half-life is 53.22 days and hence it occurs after BBN era. Therefore, socalled 7 Li abundance today is total synthesized abundance of 7 Li and 7 Be. The primordial abundance of these nuclei is much more sensitive to the baryon-tophoton ratio η.

2.2.2

Calculation of the Big-Bang Nucleosynthesis

The prediction of the light element abundances is given numerically. By solving a nuclear reaction network as shown in Fig. 3, we simulate the nucleosynthesis in

Fig. 3 Nuclear reactions network. Reprinted from [53], Copyright 2006, with permission from Elsevier

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the early universe. Calculation codes have been developed continuously by many authors: A code for the 4 He abundance was first written by Alpher, Follin, and Herman [54] in 1953. In 1966, Peebles gave a very simple code to follow 4 He synthesis [55]. In 1967, Wagoner, Fowler, and Hoyle completed a very detailed reaction network to follow primordial nucleosynthesis [56]. Wagoner, in 1973 [57], published the “standard code” for primordial nucleosynthesis. The code has been successively updated by correcting nuclear reaction rates, the effect of finite temperature, the procedure of calculation, and so on. Now so-called “Kawano code” [58] and its updated version is standard.

2.3 Theoretical Prediction of the Abundances and the Dependence of Parameters 2.3.1

Baryon-to-Photon Ratio η

As an example, we can see from Eq. (8), the primordial abundance of nuclei depends on strongly the baryon-to-photon ratio η. If η is a large value ( i.e., the baryon number density is large), the synthesis of nuclei begins earlier. It means that the syntheses of D and T occur earlier, and thus the synthesis of heavier nuclei also begins earlier which makes their final abundances smaller. But the abundance of He changes slightly because its abundance is determined by the neutron-to-proton ratio. Similarly, complicated dependence can be found also in the abundance of 7 Li. It is not monotonically dependent on η. Together with the increase of η, the 7 Li abundance decreases monotonically for η ≲ 3 × 10−9 , while it increases monotonically for η ≳ 3 × 10−9 . This is because of a replacement of dominant 7 Li (7 Be) synthesis processes. The dominant process for η ≲ 3 × 10−9 is 4 He + T → 7 Li + γ , Eq. (23), while for η ≳ 3 × 10−9 is 4 He + 3 He → 7 Be + γ , Eq. (24). The backprocess for 7 Li, Eq. (23), turns out stronger for a larger value of η, since this process begins in higher temperature and hence there is more energetic photons. Conversely, the synthesis for 7 Be, Eq. (24), becomes more effective because this process occurs at higher temperature, while its backward process stays ineffective due to the necessity of higher energy photon. The baryon density is extracted from the measurement of the Cosmic Microwave Background (CMB) [59] and the most precise value of η can be found in Ref. [60] η = (6.104 ± 0.058) × 10−10 .

(26)

Figure 4 from Ref. [61] shows the η dependence of the abundances of 4 He, D, 3 He+T, and 7 Li. The width of the bands presents the 95% C.L. range, and the observed light element abundances are displayed by the boxes (smaller boxes: ±2σ statistical errors; larger boxes: ±2σ statistical and systematic errors). The narrow vertical band indicates baryon-to-photon ratio from CMB measurement [59], while the wider band

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shows the BBN concordance range (both at 95% C.L.). The abundances of 4 He, D, 3 He, and 7 Li at the observed η are given by Ref. [62] as follows: Y p = 0.24691 ± 0.00018,

(27) −5

D/H = (2.57 ± +0.13) × 10 , 3 7

2.3.2

−5

He/H = (1.003 ± 0.090) × 10 , Li/H = (4.72 ± 0.72) × 10

−10

.

(28) (29) (30)

Relativistic Degree of Freedom g∗

The relativistic degree of freedom g∗ affects the freeze-out time of nuclear thermal equilibrium and the beginning time of nucleosynthesis because of its dependence of the expansion rate of the universe. This degree of freedom g∗ is a function of the temperature of the universe and determined by the number of relativistic particles1 and as well as its spin. g∗ ≡

Σ i=bosons

( gi

Ti T

)4 +

( )4 7 Σ Ti gi . 8 i=fermions T

(31)

In the BBN era, photons and neutrinos of three generations in the standard model of elementary particle physics are relativistic particles. If there exist furthermore other relativistic particles, g∗ becomes larger because of the additional degrees of freedom of these particles. A larger value of g∗ leads larger expansion rate of the √ universe since this rate is proportional to g∗ , and therefore earlier breaking of nuclear statistical equilibrium and nucleosynthesis. We use this fact as one of the constraints of the generation number of neutrino. In this review, we consider only standard model particles and their superpartners, and we consider that there exist only three generations.

2.3.3

Neutron Lifetime τn

The weak interaction rates of the processes, Eqs. (1)–(3), are proportional to the inverse neutron lifetime τn−1 . It is clear that larger τn leads to smaller weak interaction rate, and so that earlier freeze-out of neutron-to-proton ratio, and therefore this fact leads a larger mass fraction of 4 He. The lifetime is well investigated by experiments and its uncertainty has been reduced. The lifetime of Neuron, τn = (879.4 ± 0.6) s, is used in Ref. [60] (see Fig. 4).

1

The relativistic particle means here that its momentum is much larger than own mass.

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Fig. 4 The η dependence of the abundances of 4 He, D, 3 He+T, and 7 Li. The width of the bands shows the 95% C.L. range. Boxes indicate the observed light element abundances (smaller boxes: ±2σ statistical errors; larger boxes: ±2σ statistical and systematic errors). The narrow vertical band indicates the CMB measure of the baryon density, while the wider band indicates the BBN concordance range (both at 95% C.L.). Reprinted from [61], Copyright 2020, with permission from the Physical Society of Japan

2.4 Big-Bang Nucleosynthesis: Observation In this section, we discuss the current status of observed primordial abundances. First, we consider general conditions for the observed objects before presenting details. The main point is that nuclear fusion in stars, supernovae, and photonuclear reaction processes modifies the primordial mass fractions. Thus, we must extract the value before these changing processes to determine the true observed primordial abundance. Metallicity is a good criterion for deciding whether primordial or not. An object whose inside is with low metallicity, we can find the primordial value because metals are created in not the early universe, but by stars and supernovae. Note that we parametrize the metallicity by the logarithm of the mass fraction of Fe normalized by solar metallicity. ( [Fe/H] ≡ log10

NFe NH

) object

( − log10

NFe NH

) .

(32)

sun

Here NFe and NH are the number densities of Fe and p, respectively. Subscript, objects and sun, means the value of the observed object and sun, respectively.

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171

Deuteron

The deuteron is the most precisely observed in high-redshift, low-metallicity quasar absorption system, via its isotope-shifted Lyman-α absorption. Elements in gas absorb light with specific energies, i.e., wavelengths in the language of quantum mechanics, from quasars. For this reason, we can prove the existence of elements in the gas by absorption. The isotope-shifted Lyman lines are absorbed by deuterons. The energy of these lines is determined by subtracting the energy of an electron in the ground state from the n-th orbital state. Further, the Lyman lines of hydrogen are identified by substituting the mass of deuteron m D with the mass of hydrogen m p . It should be mentioned that the energy of the Lyman-α line of the deuteron is 0.025% larger than that of hydrogen. So that we can evaluate the primordial value of the deuteronto-hydrogen ratio by comparing the strength of the Lyman-α lines of deuteron and hydrogen [61]: [D/H]p = (2.547 ± 0.025) × 10−5 ,

(33)

where the subscript p is the primordial value.

2.4.2

3

He

The primordial abundance of [3 He/H] is measured only in galactic HII regions [63]. The value of the primordial 3 He is reported in Ref. [63] as [3 He/H]p = (1.1 ± 0.2) × 10−5 .

(34)

It is clear that this value is, however, unreliable because no one can measure it in a direct way. We can obtain the upper limit of the primordial 3 He abundance by considering a ratio [3 He/D]. The ratio increases monotonically with the progress of nuclear reactions in astronomical objects. Therefore, the observed value of [3 He/D] is larger than the primordial value and becomes an upper limit which is investigated in Ref. [64] which is [3 He/D]p < 0.83 + 0.27 .

(35)

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He

In this subsection, the observation of 4 He in recombination line from extragalactic HII regions is considered where there are not only H+ but also He+ . They recombine and ionize in the following way: H+ + e − ↔ H + γ , +

−

He + e ↔ He + γ .

(36) (37)

Here we use a notation that H and He are not nuclei but atoms. Note that it is different from the previous sections. The emitted γ is called the recombination line in the above processes and has a specific wavelength. In addition, the wavelengths of the recombination line of these ions are not the same as in the previous sections. The ratio of the primordial abundances of helium 4 and hydrogen [4 He/H] is evaluated by comparing the strength of the recombination lines of H+ and He+ , Eqs. (36) and (37), since almost all hydrogen and helium 4 are in H+ and He+ , respectively. The typical temperature of the HII region is 1 eV and this is smaller than the ionization energy of He+ , 24.6 eV. Thus, the number of photons with 24.6 eV is strongly suppressed by the Boltzmann factor e−E/T ∼ 10−11 . However, the total number of photons is larger than the number of 4 He which is a quarter of the baryon number because baryon-to-photon ratio η is of order 10−10 . Therefore, the number of photons with 24.6 eV n γ |24.6eV is the same order of the number of helium 4 n γ |24.6eV ∼ n γ e−E/T ∼

4n 4 He −11 ∼ n 4 He . 10 η

(38)

Under the same consideration, we can see that He2+ is not in the HII region since the number of photons with the ionization energy of He2+ , 54.4 eV, is suppressed by the Boltzmann factor 10−24 . The nuclear fusion process creates Helium 4 in stars, and thus the observed helium-to-hydrogen ratio may not be a primordial one. We measure metallicity for the identification of the ratio being primordial. Helium 4 and metals are created together in stars and it is natural that the created values are positively correlated. Therefore, in observing metal-poor HII region, we may find primordial helium-tohydrogen ratio by determining the correlation and extrapolating to zero metallicity. The most metal-poor HII regions are in distant blue compact galaxies. The observed primordial 4 He abundance given by Ref. [61] is Yp = 0.245 ± 0.003.

(39)

Solution for Lithium Problem from Supersymmetric Standard Model

2.4.4

7

173

Li

The 7 Li abundance is observed in the absorption line from metal-poor Pop II stars in our galaxy. The Pop II star is the oldest star in observed ones and a parent generational star of the sun. Its metallicity is at least of the order of 10−4 and perhaps going down to 10−5 . It turns out that the 7 Li abundance does not vary significantly with metallicity and temperature in Pop II stars. This feature is called “Spite plateau” [65]. We display the plateau in Fig. 5 which is from Ref. [4]. We see on the upper panel ALi of the Spite plateau stars with temperature Teff > 6000 K as a function of [Fe/H]. The logarithm of ALi is given as a function of the value of [7 Li/H] being defined as ALi ≡ log10 ([7 Li/H]) + 12 .

(40)

Fig. 5 Upper panel: ALi of the Spite plateau stars (Teff > 6000 K) as a function of [Fe/H]. The dotted line indicates the mean 7 Li abundance of the plateau stars, and the solid line represents the lower limit imposed by the WMAP constraint. The error bars are the predicted error, σpred (= 100.05 ≃ 1.12), and 3σpred (= 100.15 ≃ 1.41). Lower panel: ALi as a function of Teff for stars with [Fe/H] ≤ −1.5 (filled triangles) and [Fe/H] > −1.5 (open triangles). The star (HD106038) with the highest Li abundance (open triangle inside the open circle) is a star with peculiar abundances [67]. Reprinted from [4], Copyright 2004, with permission from IOP Publishing

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The dotted line shows the mean value of 7 Li abundance of the plateau stars, and the solid one represents the lower limit given by the measurement of Wilkinson Microwave Anisotropy Probe (WMAP) [66]. The error bars are the predicted errors, σpred (= 100.05 ≃ 1.12), and 3σpred (= 100.15 ≃ 1.41). The lower panel shows ALi as a function of Teff for stars with [Fe/H] ≤ −1.5 (filled triangles) and [Fe/H] > −1.5 (open triangles). The star (HD106038) with the highest Li abundance (open triangle inside the open circle) is a star with peculiar abundances [67]. There is tiny correlation between the abundance and the metallicity. It can be understood as the result of 7 Li production from galactic cosmic ray [68]. By extrapolating the result to zero metallicity, a primordial value [69], [7 Li/H]p = (1.23 ± 0.06) × 10−10 ,

(41)

is derived. There are many different values derived, e.g., [7 Li/H]p = (2.19 ± 0.28) × 10−10 ,

(42)

[ Li/H]p = (1.86 ± 0.28) × 10

−10

,

(43)

[ Li/H]p = (1.26 ± 0.26) × 10

−10

.

(44)

7 7

Equations (42), (43), and (44) are from Refs. [3, 70, 71], respectively. These results depend on (1) a physical technique to determine the temperature of the stellar atmosphere in which the 7 Li absorption line is formed and (2) the uncertain inner structure of stars. The difference of the treatment for them makes the difference between these results. In Fig. 4, relatively wide error range is taken [7 Li/H]p = (1.6 ± 0.3) × 10−10 .

(45)

We will, however, use a more strictly constrained value (44) in the following chapters.

2.4.5

6

Li

Observation for 6 Li is made in the same source of 7 Li. However, it is more difficult than that of 7 Li. It is because the absorption line of 6 Li is almost hidden in 7 Li since the difference of wavelengths of the lines is only 0.16Å and the strength of 6 Li line is at least 20 times weaker than that of 7 Li. 6 Li is detected only by slight extra depression of the red wing of the unresolved 7 Li feature (see Fig. 6 which is from Ref. [10]). In Fig. 6, the squares indicate the observation for a star LP 815–43 which has atmospheric parameters Teff = 6400K and [Fe/H] = −2.74. The solid, dotted, and dashed lines correspond to the best fitting 3D local thermodynamic equilibrium profile from a χ2 analysis for 6 Li/7 Li = +0.00, +0.05, and +0.10, respectively. We thus obtain the upper bound for primordial abundance of 6 Li,

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Fig. 6 The squares indicate the observed data for a star LP 815–43 which has atmospheric parameters Teff = 6400K and [Fe/H] = −2.74. The solid, dotted, and dashed line correspond to the best fitting 3D local thermodynamic equilibrium profile from a χ2 analysis for 6 Li/7 Li = +0.00, +0.05, and +0.10, respectively. Reprinted from [10], Copyright 2005, with permission from IOP Publishing 6

Li/7 Li < 0.046 ± 0.022 .

(46)

2.5 The 7 Li Problem From Fig. 4, it is clear that the theoretically predicted abundances with the baryon-tophoton ratio determined by the WMAP experiment [66] agree with the observed data, except for 7 Li. The predicted 7 Li abundance2 is (4.72 ± 0.72) × 10−10 as shown in Eq. (30), while observed one is (1.6 ± 0.3) × 10−10 as in Eq. (44). Thus, the BBN prediction gives three times larger abundance than the experimental data. One might consider that the BBN was wrong at one glance. However, the BBN is believed to be reliable since it explains well the origin of the abundances of 4 He and D. The abundance of 4 He changes with metallicity very little and does not fall below about 23 % as shown in Fig. 7, Ref. [53]. In particular, even in the systems with an extremely low value of oxygen, which traces stellar activity, the abundance of 4 He is nearly constant. This is very different from all other elements, e.g., nitrogen. The abundance of nitrogen goes to zero, as oxygen goes to zero (see Fig. 8 which is from −10 for the theoretically predicted value of the lithium After Sect. 3, we will use 4.15+0.49 −0.45 × 10 −10 abundance, and (1.26 ± 0.26) × 10 for the observed value because we made these analyses with this slightly old value.

2

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Fig. 7 The change of the abundance of 4 He with the abundance of O. The abundance of 4 He is nearly constant and does not fall below about 23%. Reprinted from [53], Copyright 2006, with permission from Elsevier Fig. 8 The change of the abundance of N with the abundance of O. The abundance of N goes to zero according to O. Reprinted from [53], Copyright 2006, with permission from Elsevier

[53]). The abundance of D cannot be produced by any stellar source. There does not exist astrophysical mechanism to destroy the production of significant amounts of deuterium [72–74], we think that stars destroy deuterium, and we conclude the abundance of D must be made by the BBN. The inconsistency between observed and predicted values might be caused by poor knowledge of the internal constitution of stars. The quoted observed value is obtained under an assumption that the Li abundance in the stellar sample reflects the primordial abundance. However, the value might have been affected by convection and/or diffusion in stars. 7 Li on the surface of stars are carried into a deeper part of the stars by its convection. One believes normally that the convection for the hot (T ≳ 5700 K) stars does not affect since for the hotter stars (i.e., larger stars) the

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log NFe / NH +12

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Li 6000 5500 effective temperature Teff [K]

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Fig. 9 Temperature change of iron and lithium of the observed stars compared to the model predictions. The gray crosses are the individual measurements, while the blue circles are the group averages. The solid lines are the predictions of the diffusion model [12], with the original abundance given by the dashed line. In lower panel, the shaded area around the dotted line indicates the 1 σ confidence interval of CMB + BBN [66]: log(NLi /NH ) + 12 = 2.64 ± 0.03. For iron, the error bars are the line-to-line scatter of FeI and FeII (propagated into the mean for the group averages), whereas for the absolute lithium abundances 0.10 is adopted. The 1 σ confidence interval around the inferred primordial lithium abundance (log(NLi /NH ) + 12 = 2.54 ± 0.10) is indicated by the light gray area. Reprinted by permission from Nature [75], Copyright (2006)

convection layer is relatively thin, and hence in stars with temperature T ≳ 5700 K convection layer does not reach the center of the stars. Thus, it cannot alter the 7 Li abundance. It turns out that if we adopt the unusual model of the internal constitution of stars the convection layer can reach the center and the 7 Li are reduced. Moreover, the 7 Li abundance on the surface of stars might be reduced by diffusion. The heavy element is diffused at the center of the star. Diffusion more actively occurs in hotter stars as we can find from the case of Fe (see upper panel of Fig. 9 which is from [75]). The 7 Li abundance is maximum for the stars with T ∼ 5700 K, and decreases for T > 5700 K (see lower panel of Fig. 9 which is from [75]). Therefore, the abundances for ≳ 6000 K are not a primordial one but also a reduced one. However, the effects of convection and diffusion are quite dependent on features of each star. For example, the rotation and magnetic field of stars suppress the convection

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since the tension of the magnetic field line and the repulsion force of the vortex line hold the shape of each line and the convection [76]. Thus, the effect of the convection is strongly dependent on each star. It is a conflict with the observed abundance converging to a value (see Fig. 5). There is another idea that model uncertainty of the stellar atmosphere might explain the discrepancy. The essence of the idea comes from the fact that we do not observe 7 Li directly but the absorption line of the stellar atmosphere. The determination of the strength of the line depends on models in which the surface temperature via some physical parameters, e.g., surface gravity are important. The uncertainty of the surface temperature is up to 150–200K and can lead to an underestimation of up to 100.09 ≃ 1.23. Unfortunately, this kind of modification is not enough to explain the discrepancy. Another possible explanation is the uncertainty of nuclear reaction rates. The uncertainty of nuclear reactions rate of 7 Li reduction processes, Li + D → n + 4 He + 4 He,

(47)

Be + D → p + He + He,

(48)

7 7

4

4

is bigger than that of D, 3 He and 4 He. Nevertheless, the uncertainty is not enough to resolve the discrepancy, see Ref. [62]. In addition, the uncertainty of 7 Li reduction processes 3

He + 4 He → γ + 7 Be

(49)

is strictly constrained by the combination of standard solar model and neutrino experiment [77]. Thus, we have to conclude that the 7 Li problem remains a puzzle in the SM. An interesting approach to this problem is given by effects induced by new physics beyond the SM. Exotic particles which have a long lifetime and interact with nuclei might give a solution to the 7 Li problem since they survive until the BBN era and open new channels to reduce the nuclei. In the minimal supersymmetric standard model (MSSM) with the conservation of R-parity, such a long-lived and interacting particle can appear and give the solution. The candidate particle is the lightest scalar partner of charged lepton. Often it mainly consists of a superpartner of the tau lepton, namely, stau. Though there must be a tiny mixing from other lepton flavors. It is obvious that in the sense this particle is not an exact stau, but we will call it stau in the following sections. In the next chapter, we see the nature of stau and check the possibility to solve the 7 Li problem.

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3 Minimal Supersymmetric Standard Model (MSSM) Our current established theory in particle physics is the SM as have mentioned in introduction that explains all interactions except gravity. Thus, we consider the SM is being “correct” theory of particle physics. However, as we have already discussed in the previous section, the 7 Li problem stands an anomaly in the SM of particle physics. The most extensively studied framework for physics beyond the SM is represented by SUSY. This model is an extension to the SM in the particle physics, which could resolve the cosmic problem as well as the subatomic puzzles, e.g., the DM in the universe, the origin of masses of elementary particles [78]. The supersymmetry transformation turns a boson state into a fermion state [79], and vice versa, which calls a supersymmetric particle for each SM field, for example, the scalar electron is a bosonic partner of the electron and the neutralino is a fermionic partner of the photon.3 The particles and superpartners of MSSM are displayed in Fig. 10. The simplest supersymmetric model consistent with the SM is the MSSM with the R-parity that is defined as follows: the assignment of the charge for standard model particles is to be R = +1, and for supersymmetric particles—superpartners—is to be R = −1. For instance, a phenomenological giving from the R-parity is the stability of proton. The R-parity conservation prohibited the baryon and lepton number violating interactions and makes the proton lifetime much longer than the age of universe. Additionally, the R-parity conservation ensures that the LSP cannot decay into any SM particles and hence is stable. The existence of DM has been confirmed by many different cosmological measurements and the total abundance of DM has been measured with very high

Fig. 10 Particle content of the minimal supersymmetric standard model (MSSM). ©Akimoto Yuki@higgstan 3

In general, the neutralino is a linear combination of the fermionic partners of the photon, Z boson, and Higgs bosons. There are four neutralinos in the MSSM.

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precision by the WMAP collaboration [80] during the last few decades. The fact that we must consider a kind of particle that does not exist in the SM. If LSP in the MSSM with R-parity is electromagnetically neutral, it is a leading DM candidate, because it is massive and stable (for reviews see, e.g., [81]).

4 Long-Lived Charged Massive Particle A variety of particle physics models beyond the SM calls long-lived CHAMPs [82]. Thermally produced long-lived CHAMPs form a bound state with nuclei at the BBN era and have a big impact on the primordial abundances of light nuclei.

4.1 Long-Lived Slepton In a feasible scenario of the MSSM, the lightest neutralino χ ˜ is the LSP and is a good DM candidate; it is classified as WIMPs. The calculation for the current abundance of this class of DM is well established. The abundance ΩDM increases with the DM mass mχ˜ , ΩDM ∝ m 3χ˜ . Here ΩDM = ρDM /ρc is the so-called density parameter, which governs the expansion of the universe; ρDM and ρc represent the DM energy density and the critical energy density, respectively. Direct search for supersymmetric particles at the Large Hadron Collider (LHC) experiment shows no evidence so far, which indicates the neutralino DM of ≳ O(100) GeV [83, 84]. With such direct search bound, the basic calculation of abundance brings the overabundant neutralino DM, which leads to an overclosure of the universe. The coannihilation mechanism successfully resolves such a DM overabundance problem [42]. A key ingredient for the mechanism is the degeneracy in mass between the neutralino DM and the NLSP. In the MSSM scenario, we consider the stau τ˜ is the NLSP, which is a supersymmetric partner of tau lepton. The mass degeneracy for the coannihilation mechanism to work is (m τ˜ − m χ˜ )/m χ˜ ≲ O(%). Due to the degeneracy, in the early universe, the neutralino DM and the NLSP stau are almost equally thermalized via the frequent conversion of each other. Then the effective annihilation rate of the neutralino DM is enhanced through the stau-stau and stauneutralino annihilation channels. Eventually the predicted DM abundance will be consistent with the observed one. Such a tight degeneracy makes the stau to be long lived by kinematical suppression in its decay [44, 45]. Especially, important and interesting parameter space is where the mass difference of the stau and the neutralino, δm = m τ˜ − m χ˜ , is smaller than tau lepton mass.4 In such a parameter space, the two-body decay channel (diagram (a) in Fig. 11) is kinematically forbidden and allowed decay channels are three- and four4

Indeed, in a large part of parameter space wherein the Higgs mass is consistent with the reported one, the mass difference between the LSP neutralino and the NLSP stau is smaller than the tau

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Fig. 11 Decay channels of stau: a τ˜ → χ˜ + τ , b τ˜ → χ˜ + ντ + π, c τ˜ → χ˜ + ℓ + ντ + νℓ . Here ℓ ∋ {e, μ}. Reprinted figure with permission from [44], Copyright 2006 by the American Physical Society

body ones (diagram (b) and (c) in Fig. 11, respectively). The lifetime ττ˜ is given by ⌈ ⌉−1 the inverse of total decay width, thus ττ˜ = Γ(τ˜ → χν ˜ τ π) + Γ(τ˜ → χν ˜ τ ℓi ν¯ ℓi ) . Here π, ντ , ℓi , and ν¯ ℓi represent the pion, tau neutrino, charged lepton (i = e for electron, i = μ for muon), and its anti-neutrino, respectively. The mass difference ˜ π) ∝ dependences for three- and four-body decay channel widths are Γ(τ˜ → χν ) ( ) τ ( 3 2 5/2 2 2 5/2 2 /m τ˜ and Γ(τ˜ → χν ˜ τ ℓi ν¯ ℓi ) ∝ (δm) (δm) − m ℓi /m τ˜ . (δm) (δm) − m π Analytic calculation and detailed analysis of the lifetime ττ˜ are given in Ref. [44]. We find that the stau lifetime ττ˜ is controlled by the mass difference. Another key ingredient for the lifetime is the slepton mixing. The measurements of neutrino oscillation reveal the existence of nonzero neutrino masses. Neither the SM nor the simplest MSSM can address it. The seesaw mechanism is a well-motivated scenario to understand the neutrino masses of ≲ O(eV) [95, 96], which extends the MSSM by introducing three right-handed neutrinos. The right-handed neutrinos introduce the additional interactions to the MSSM, and these give rise to the slepton mixing through the renormalization group Equations [97]. Then the NLSP turns into ˜ of muon (μ), ˜ and of tau the mixing state of supersymmetric partners of electron (e), lepton (τ˜ ). We call this state the NLSP slepton in this review, and is written as ℓ˜ = ce e˜ + cμ μ˜ + cτ τ˜ .

(50)

The slepton mixing ce , cμ , and cτ are obtained from the parameters of right-handed neutrinos, which satisfies ce2 + cμ2 + cτ2 = 1. Additional two-body decay channels via the slepton mixing take place in the NLSP slepton decay, ℓ˜ → χe and ℓ˜ → χμ. The partial decay rates of these channels are proportional to the slepton mixing, Γ(ℓ˜ → χe) ∝ ce2 and Γ(ℓ˜ → χμ) ∝ cμ2 . Consequently, the lifetime of NLSP slepton is controlled by the mass difference δm and the slepton mixing ce , cμ , and cτ . The dependences on these parameters is shown in Fig. 12. The additional two-body decays could be a dominant channel even lepton mass [49, 85, 86]. Collider phenomenology in such parameter region is extensively studied in Refs. [49, 87–94].

Fig. 12 Dependences on the slepton mixing of slepton lifetime. Solid line shows the result in the absence of slepton mixing. Reprinted figure with permission from [48], Copyright 2012 by the American Physical Society

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Lifetime [s]

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though the slepton mixing is tiny, without tight suppression for the final state phase space unlike the three- and four-body decay channels. A parameter space in δm and the slepton mixing lead to a solution of the 7 Li problem, and simultaneously accounts for the observed abundances of both DM and other primordial light elements.

4.2 Number Density of the NLSP Slepton at the BBN Era The number density of the NLSP slepton is determined in two steps as shown in Fig. 13. In the early universe of temperature T > m χ˜ , the neutralino DM and the NLSP slepton are chemically and kinematically thermalized with SM particles. Since these number densities follow the Maxwell–Boltzmann distribution, f ∝ exp[−E/ T ], as the universe cools down to T ≲ m χ˜ , the detailed balance between “pair” annihilations of superparticles into SM particles and its inverse processes breaks down due to the energy loss of background SM particles, and sum of number densities of the neutralino DM and the NLSP slepton freezes out. Even after the freeze-out of sum of number densities, as long as the equilibrium with the SM thermal bath is maintained, the neutralino DM and the NLSP slepton

Fig. 13 Schematic picture of evolutions of number densities. The size of the circle represents the ˜ and the long-lived slepton ℓ˜± . Here ± stands for sum of number densities of the LSP neutralino χ the electromagnetic charge

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convert each other via the scattering with SM particles, e.g., ℓ˜± + γ ↔ χ˜ + τ ± , ℓ˜± + τ ∓ ↔ χ˜ + γ, and so on. Then, applying the Boltzmann distributions to the neutralino DM and the NLSP slepton, the number density of the negative-charged slepton is given in terms of sum of number densities n sum as follows: n ℓ˜− =

n ℓ˜− n χ˜ 1 ) n sum . n sum = e−δm/T ( n χ˜ n sum 2 1 + e−δm/T

(51)

Here n i represents the number density of a particle species i. Even though the slepton mixings ce and cμ are quite small, as well as above conversion processes involving tau lepton, conversion processes involving electron and muon could be leading, e.g., ℓ˜± + γ ↔ χ˜ + μ± , ℓ˜± + e∓ ↔ χ˜ + γ, and so on. This is because, at the temperature T ≲ m τ (m τ stands for the tau lepton mass), number densities of electron and muon are much larger than that of tau lepton, which exponentially damps as n τ ∝ exp(−m τ /T ). For example, at T = 70 MeV, the ratio of conversion rates involving and not involving tau lepton is 〈σv〉ℓe→ ˜ χγ ˜ ne 〈σv〉ℓτ ˜ →χγ ˜ nτ 〈σv〉ℓμ→ ˜ χγ ˜ nμ 〈σv〉ℓτ ˜ →χγ ˜ nτ

≃ 1.08 × 109 ce2 ,

(52)

≃ 9.93 × 107 cμ2 .

(53)

Here 〈σv〉process represents the thermally averaged cross section, which is the statistical-averaged cross section integrating over the initial state momenta weighted by the Maxwell–Boltzmann distribution. n e , n μ , and n τ are number densities of electron, muon, and tau lepton at T = 70 MeV, respectively. As long as ce ≳ 3.2 × 10−5 (cμ ≳ 1.0 × 10−4 ), the conversion involving electron (muon) holds the long-lived slepton in the equilibrium and makes the slepton density smaller according to Eq. (51). The detailed evolution of the slepton number density is calculated by a coupled set of evolution equations of the slepton and the neutralino DM, so-called Boltzmann equation (Eqs. (5)–(7) in Ref. [48]). The slepton mixing dependence on the evolution of slepton yield value Yℓ˜ ≡ n ℓ˜/s is shown in Fig. 14. Here s stands for the entropy density in the universe. In the upper, middle, and lower panels, we take δm = 20 MeV, 60 MeV, and 100 MeV, respectively. In the shaded regions in each panel, too small number density of the long-lived slepton to solve the 7 Li problem. In other words, the successful BBN prediction consistent with observations derives the upper bound on the slepton mixing cμ .

184 _ 1 / 10 12

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Fig. 14 cμ dependence on the evolution of slepton yield values. cμ are attached on corresponding curves. Curves tagged by “stau” are yield values for cμ = ce = 0. Curves tagged by “Equilibrium” are yield values of the slepton in equilibrium. In shaded region, the slepton number density is insufficient for solving the 7 Li problem. Reprinted figure with permission from [48], Copyright 2012 by the American Physical Society

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5 Non-standard Nuclear Reactions We briefly review non-standard nuclear reactions caused by the bound state formation of the long-lived slepton and nuclei.

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5.1 Internal Conversion In the early stage of BBN, the bound states of the long-lived slepton and 7 Be (7 Li) nucleus are formed (Fig. 15). These bound states give rise to a non-standard nuclear reaction, so-called internal conversion processes (Fig. 16) [15, 20], (7 Be ∼ ℓ− ) → ∼ χ + ντ + 7 Li, (7 Li ∼ ℓ− ) → ∼ χ + ντ + 7 He.

(54a) (54b)

The daughter 7 Li in (54a) is destructed either by an energetic proton or the process (54b). The daughter 7 He in (54b) immediately decays into 6 He and neutron. The 6 He beta-decay produces 6 Li nucleus and may lead to the overproduction of 6 Li. Actually, since most of 7 Li is destructed by a background proton into harmless nuclei, the chain reaction with (54b) produces negligible amount of 6 Li. Hence, the chain reactions (54a) and (54b) could successfully lead the smaller 7 Be and 7 Li abundances than those in the SBBN, which is desired situation for solving the 7 Li problem (see Introduction). The reaction rate of the internal conversion processes is given as follows:

Bound Ratio

(a)

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(c)

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=5.4x10–12

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=1.3x10–20

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Temperature / [MeV]

10–4 10–1

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Temperature / [MeV]

10–4 10–1

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10–3

10–4

Temperature / [MeV]

Fig. 15 Evolutions of the bound state ratio for the nuclei 4 He,7 Be, and 7 Li [21]. We take the NLSP slepton density at the time of bound state formation from 10−10 to 10−16 . The dotted lines in the panel (A) show the results using the Saha equation for reference. Reprinted figure with permission from [21], Copyright 2008 by the American Physical Society

Fig. 16 Internal conversion processes in the bound state of NLSP slepton ℓ˜ and a nucleus; ˜ → 7 Li + χ˜ + ντ and (7 Li ℓ) ˜ → 7 He + χ˜ + ντ (7 Be ℓ)

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ΓIC = |ψ|2 · (σv)IC .

(55)

Here |ψ|2 represents the overlap of wave functions in the bound state of initial state. Taking a hydrogen-like 1S state for the bound state, we obtain the overlap as 3 ), where anucl = (1.2 × A1/3 ) fm is the nucleus radius and A is its |ψ|2 = 1/(πanucl mass number. (σv)IC stands for the cross section of each process. For the process (7 Be ℓ˜− ) → χ˜ + ντ + 7 Li, ⌠ d 3 pχ˜ d 3 pν d 3 pLi 1 (σv)IC = (2E ℓ˜)(2E Be ) (2π)3 2E ν (2π)3 2E χ˜ (2π)3 2E Li | (7 )|2 × |M ( Be ℓ˜− ) → χν ˜ τ 7 Li | (2π)4 δ (4) ( pℓ˜ + pBe − pχ˜ − pν − pLi ).

(56)

Here pi and E i are the momentum and the energy of a particle species i, respectively. We decompose the amplitude into the leptonic and the hadronic part as ) ( ˜ τ 7 Li = 〈7 Li|J μ |7 Be〉 〈χ˜ ντ | jμ |τ˜ 〉. M (7 Be τ˜ − ) → χν

(57)

The leptonic part is computed in a standard manner. We evaluate the matrix element of nuclear conversion by applying the f t value derived from β decay measurements. The f t value for 7 Li ↔ 7 Be is experimentally measured, but not for 7 Li ↔ 7 He. We assume that the two processes have the same f t value. When we focus on the ground state of 7 Li and 7 He, it is expected that the leading channel is a Gamow–Teller transition. This is because the quantum numbers of ground state of 7 Li and 7 He are similar with that of 6 He and 6 Li which convert to other nuclei via the Gamow–Teller transition. The Gamow–Teller transition is classified as superallowed and possesses a similar f t value to the Fermi transition such as 7 Li → 7 Be. Figure 17 displays the timescale of the internal conversion, (54a) and (54b). Due to the kinematical threshold δm = m 7 He − m 7 Li = 11.7 MeV, the timescale of (7 Li ℓ˜− ) → χ˜ + νℓ + 7 He diverges around there. The timescale of the internal conversion, ∼10−3 s, is much shorter than the Hubble time at the BBN era, ∼10 s. This implies that a parent nucleus is converted into another nucleus immediately once the bound state is formed. The effective interaction between the slepton and a nucleus becomes more efficient by forming the bound state due to (i) the large overlap of wave functions of the slepton and a nucleus because of confinement in a small space, (ii) the enhancement of hadronic current exchange even if δm < m π because of the short distance between the slepton and a nucleus.

5.2 Spallation Reactions and Slepton Catalyzed Fusion At a late stage of the BBN, bound states of the long-lived slepton and a 4 He nucleus are formed. The bound state could cause a fatal effect on other light element abundances. In order to achieve the successful BBN, we need to investigate the effect of the

Solution for Lithium Problem from Supersymmetric Standard Model mμ mπ

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1

mμ m π

4

10

-1

10-2

lifetime(s)

lifetime(s)

10

187

10-3

102 1

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-4

10

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10-4

10

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10

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10

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0.01

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0.1

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Fig. 17 Timescales of internal conversions as a function of δm. Left panel: (7 Be ℓ˜− ) → χ˜ + ντ + 7 Li, right panel: (7 Li ℓ˜− ) → χ ˜ + ντ + 7 He. Reprinted figure with permission from [20], Copyright 2007 by the American Physical Society

slepton-4 He bound state. This bound state gives rise to two types of non-standard reactions: (1) Slepton-catalyzed fusion (4 He ℓ˜− ) + D → ℓ˜ + 6 Li.

(58)

(2) Spallation of the 4 He nucleus (Fig. 18) (4 He ℓ˜− ) → χ˜ + ντ + T + n, (4 He ℓ˜− ) → χ˜ + ντ + D + n + n, (4 He ℓ˜− ) → χ˜ + ντ + p + n + n + n.

(59a) (59b) (59c)

Since the overproduction of light elements strongly depends on the origin of longevity of the slepton, it is necessary to carefully investigate the evolution of the slepton-4 He bound state for each non-standard nuclear reaction. The catalyzed fusion process could cause the 6 Li overproduction [32]. This process is a type of photon-less nuclear transfer reaction, in which rate is evaluated by using the astrophysical S-factor as S(0) ∼ 102 keV b [98]. References [18, 23] derive the thermally averaged cross section of the catalyzed fusion, which is much

Fig. 18

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larger than the 6 Li production in the SBBN, 4 He + D → 6 Li + γ, by six to seven orders of magnitude. Since the standard 6 Li production 4 He + D → 6 Li + γ is an E1 forbidden transition, its rate is so small and the magnitude is evaluated by the astrophysical S-factor as S(0) ∼ 2 × 10−6 keV b [98]. The lifetime and the number density of long-lived CHAMPs in the BBN era are stringently constrained from the overproduction of 6 Li by the catalyzed fusion process [32]. The non-standard spallation processes of the 4 He (59) produce a triton, a deuteron, and neutrons. We formulate the reaction rate of a spallation process (4 He ℓ˜− ) → χ˜ + νℓ + T + n. The reaction rate is given by the combination of overlap of initial wave functions and the cross section as follows: ) ( (60) Γ (4 He ℓ˜− ) → χ˜ + ντ + T + n = |ψ|2 · σvTn . Following the same treatment with the internal conversion process, the overlap of initial wave functions in the bound state is obtained by |ψ|2 = (Z αm He )3 /π, where Z is the atomic number of nucleus and )α is the fine structure constant. The cross ( ˜ τ Tn is calculated as follows: section σvTn ≡ σv (4 He ℓ˜− ) → χν σvTn =

1 2E ℓ˜

⌠

)|2 d 3 pχ˜ d 3 q n d 3 q T | ( 4 d 3 pν |M ( He τ˜ − ) → χν ˜ τ Tn | 3 3 3 3 (2π) 2E ν (2π) 2E χ˜ (2π) (2π)

× (2π)4 δ (4) ( pℓ˜ + pHe − pν − qT − qn ).

(61)

We decompose the amplitude into leptonic and hadronic part as ) ( ˜ ˜ τ Tn = 〈Tn|J μ |4 He〉 〈χ˜ ντ | jμ |ℓ〉. M (4 He ℓ˜− ) → χν

(62)

The hadronic current J μ consists of a vector current Vμ and an axial vector current Aμ as Jμ = Vμ + gA Aμ , where gA is the axial coupling constant, which is experimentally measured, gA ≃ 1.26. V 0 and Ai (i = 1, 2, 3) contribute the spallation are | components | | elements by| these | | processes, |and| the hadronic matrix √ 〈Tn |V 0 |4 He〉 = 〈Tn | A+ |4 He〉 = −〈Tn | A− |4 He〉 = −〈Tn | A3 |4 He〉 = 2MTn . √ Here A± = ( A1 ± i A2 )/ 2. The wave functions of 4 He, T, and n to evaluate the hadronic amplitude are derived in Appendix A of Ref. [22]. Finally, we obtain the hadronic matrix element as follows: (

⌉⎫ ⌉ ⌈ (q T + q n )2 q 2T q 2n − . exp − − exp − MTn = 3aHe 3aHe 6(aHe + aT ) (63) Here q T and q n are spacial momenta of the triton and the neutron, respectively. The mean square matter radius Rmat is related with aHe and aT by aHe = 9/(16(Rmat )2He ) and aT = 1/(2(Rmat )2T ), respectively. The matter radii are listed in Table 1. The leptonic part is calculated in a standard manner as 128π aHe aT2 3 (aHe + aT )4

)3/4 ⎧

⌈

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Table 1 Matter radius Rmat for D, T, and 4 He, the magnetic radius Rmag for p and n, nucleus mass m X , excess energy Δ X for the nucleus X , and each reference Nucleus Rmat(mag) m X [GeV] Δ X [GeV] [fm]/[GeV−1 ] p n D T 4 He

0.876 / 4.439 [99] 0.873 / 4.424 [102] 1.966 / 9.962 [103] 1.928 / 9.770 [104] 1.49 / 7.55 [105]

0.9383 [100] 0.9396 [100] 1.876 [101] 2.809 [101] 3.728 [101]

6.778 × 10−3 8.071 × 10−3 1.314 × 10−2 1.495 × 10−2 2.425 × 10−3

[101] [101] [101] [101] [101]

Fig. 19 δm = m ℓ˜ − m χ˜ dependence of ITn . The analytic formula is given by Eq. (17) in Ref. [22]

|2 | |2 | | ˜ || = ||〈χν ˜ || = 4G 2F |gR |2 cτ2 m χ˜ E ν , ˜ τ | j0 | ℓ〉 ˜ τ | jz | ℓ〉 |〈χν m 2τ ( |2 | z ) | ˜ || = 4G 2F |gR |2 cτ2 m χ˜ E ν 1 ∓ pν , ˜ τ | j± | ℓ〉 |〈χν m 2τ Eν

(64)

where E ν and pνz are the energy and the momentum z-component of tau neutrino, respectively. Consequently, the cross section is obtained as follows: ( ) 3/2 8 32 3/2 2 2 aHe aT3 2 2 2 4 mTmn g tan θ sin θ (1 + 3g )G Δ ITn , W τ A F Tn π 2 3π m τ˜ m 2τ (aHe + aT )5 (65) where θW represents the Weinberg angle, one of the parameter in the SM of particle physics, and sin2 θW = 0.23. The coupling constant g for the interaction between slepton and the LSP neutralino is related with the electromagnetic coupling constant e using the Weinberg angle as g = e/ sin θW . ITn is a dimensionless integral, which includes kinematical information of the reaction. The numerical result of ITn is shown in Fig. 19, and it is analytically calculated by Eq. (17) in Ref. √ √ [22]. We define ΔTn ≡ δm + ΔHe − ΔT − Δn − E b , kT ≡ 2m T ΔTn , and kn ≡ 2m n ΔTn . Here E b is the binding energy of (4 He τ˜ − ) system, and Δ X is the excess energy of the nucleus X . The reaction rate of spallation and slepton-catalyzed fusion is shown in Fig. 20. The catalyzed fusion rate strongly depends on the temperature [18]. We take the σvTn =

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10 6 10 4 10 2 1

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stau tn

Timescale (s)

10 8

dn

n

lifeti me

pnn

n

10-2

0

0.05

0.1

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Fig. 20 Timescale of spallation processes [22] and slepton-catalyzed fusion at the temperature T = 30 keV [18]. The slepton lifetime for ce = cμ = 0 is also depicted (solid line). Lines labeled by tn, dnn, and pnnn show timescale of each processes of Eqs. 59a, 59b, and 59c, respectively. Reprinted figure with permission from [22], Copyright 2011 by the American Physical Society

reference temperature to 30 keV for the comparison with the spallation processes. Around the temperature 30 keV (corresponds to cosmic time of 103 s), the long-lived slepton begins to form a bound state with 4 He. Figure 20 shows the δm dependence of rates of spallation and catalyzed fusion processes. The slepton lifetime for ce = cμ = 0 is also shown by a solid line. Once the bound state (4 He ℓ˜− ) is formed, as long as the spallation processes are kinematically allowed, i.e., δm ≳ 0.026 GeV, the reaction rates of these processes are much bigger than both the slepton decay and the catalyzed fusion process. Thus, the slepton parameters are constrained to evade the overproduction of d and/or t. When the spallation processes are kinematically closed, i.e., δm ≲ 0.026 GeV, the catalyzed fusion dominates over other processes. In this case, the slepton parameters are constrained by the overproduction of 6 Li (the excluded space is displayed by light gray region).

6 Numerical Result and Favored Parameter Space The primordial abundances of light elements are so sensitive to the slepton number density at the BBN era, and the slepton number density depends on both the mass difference between the slepton and the neutralino dark matter (δm = m ℓ˜ − m χ˜ ) and the slepton mixing (ce and cμ ). Figure 21 exhibits the parameter region wherein the slepton solves the 7 Li problem, and wherein the non-standard nuclear reactions do not spoil the successful BBN. We choose the slepton number density n ℓ˜ normalized by the entropy density s (the so-called yield value Yℓ˜ = n ℓ˜/s) at the freeze-out of the ˜ χ˜ ratio (Fig. 14) and the mass difference δm as free parameters. We qualitatively ℓ/ explain our results. For more detailed discussion, see Refs. [20, 22, 28, 48].

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Fig. 21 Parameter region wherein the 7 Li problem is solved in δm-Yl˜− plane for the slepton mixing ce = 5 × 10−11 and cμ = 0. The constraints are shown from the measurements of D/H (dark gray), 3 He/D (light gray), and 6 Li/7 Li (blue curves) at 2σ C.L. The favored region for 7 Li/H is the area enclosed by purple dotted (solid) curve at 2σ C.L. (3σ C.L.). The slepton yield value calculated using a set of model parameters is shown by the solid orange line

In the region enclosed by purple solid (dashed) lines, the 7 Li primordial abundance is consistent with the observed one within 3σ (2σ) confidence level [4]. In this region, 7 Be abundance is efficiently reduced through the internal conversion process (54a) with either a following standard reaction 7 Li + p → 4 He + 4 He or the internal conversion (54b). The orange curve represents the slepton yield value calculated using a set of model parameters, whose daughter particle (LSP neutralino) accounts for the dark matter. The cosmologically interesting region is below the line. Note that the slepton density becomes smaller with ce (Fig. 14), while the efficiency of internal conversions is independent on ce as long as ce ≲ O(0.1), because, in addition to cτ ≃ 1, it is regardless with the densities of electron and muon. Consequently, in the overlap area of the region enclosed by the purple curve and the region below the orange curve, our scenario successfully could solve the 7 Li problem (dashed box in Fig. 21). Successful BBN could be spoiled in the most of space of Fig. 21. D and T are overproduced by the 4 He spallation processes Eq. (59) with the subsequent standard reactions [48]. Thus, the overproductions of D and T exclude the gray region. Finally, we leave a comment on the 6 Li problem. If we seriously take this problem, the allowed region is constrained on the white band within two lines indicated 6 Li/7 Li. For δm ≲ 50 MeV, the spallation process rates are smaller than the catalyzed fusion one due to the phase space suppressions (Fig. 20), and the catalyzed fusion Eq. (58) efficiently works to generate 6 Li. In the green region, the overenhancement of 6 Li production occurs, because the slepton decay rate is much smaller than the catalyzed fusion rate due to tight phase space suppression. Consequently, this region

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is excluded. Near future once the 6 Li abundance is measured definitely, we can pin down the allowed region within the white band. Thus, in this case, both the 6 Li and the 7 Li problems are solved.

7 Verification of the Scenario with Dark Matter Search We are mainly forcing on finding the answer to the 7 Li problem in the framework of an extension of the SM of particle physics. In fact that this model—MSSM—can find not only the the solution to the 7 Li problem but also predicts the properties of DM particle in a certain region of parameter space of the MSSM. As we have mentioned above, we are interested in models which may successfully solve the 7 Li problem, the allowed parameter space is fortunately tiny. For this reason, we can easily predict, for example, the mass of DM [50] and can further investigate the properties of DM concerning the direct measurements [51].

7.1 Formalism of the Effective Action for Non-relativistic Particle In this subsection, we will review the scenario that has been studied in the framework of MSSM in Ref. [51] in which the parameter space is called the neutralino–slepton coannihilation region, i.e., we assume that DM is LSP Bino-like neutralino and NLSP is the lightest slepton5 which is required very tight mass degeneracy with LSP neutralino. The main contributions of Lagrangian in the mass basis which appear in the below diagram are (for reader interest in all interaction, see appendices of Ref. [51]) L = Lkinetic term + Linteraction ( ) 1 / i j − (m e )i j e j − ∼ ∼ ∼ τ ∗ (∂ 2 + m∼2τ )∼ τ χ¯ (i ∂/ − m) ∼ χ + e¯ i i ∂δ 2 1 1 + Z μ (∂ 2 + m 2Z )Z μ + Aμ ∂ 2 Aμ + Lgauge + · · · , 2 2

(66)

where i, j = 1, 2, 3 and (m e )i j = diag(m e , m μ , m τ ). Z μ and Aμ represent Z-boson and photon, respectively. Lgauge has the following form:

ℓ1 almost consists of We use terminology in this review that the lightest slepton is “stau” because ∼ right-handed stau in the flavor base.

5

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Fig. 22 DM annihilation ladder diagram. Z , γ and f are taken as an example of φ

(

)

1 ← → ← → Lgauge = i.e. Aμ∼ τ ∗ ∂ μ∼ τ − igz Z μ sw2 − N∼l 1 i N∼l† i 1 ∼ τ τ ∗ ∂ μ∼ 2

√ ( 2 ← → ← → ) g Wμ+∼ ν ∗i ∂ μ N∼l† i 1∼ τ + Wμ−∼ τ ∗ N∼l 1 i ∂ μ∼ νi 2 ( )2 ( ) 1 1 + e2 A2 |∼ τ |2 + gz2 sw2 − N∼l 1 i N∼l† i 1 Z 2 |∼ τ |2 − 2egz sw2 − N∼l 1 i N∼l† i 1 Aμ Z μ |∼ τ |2 2 2 −i

+

g2 N∼1 i N∼l† i 1 Wμ+ W −μ |∼ τ |2 , 2 l

(67)

where the sum of i is taken from 1 to 3, and we have used the following definition: ← → τ =∼ τ ∗ ∂ μ∼ τ − (∂ μ∼ τ ∗ )∼ τ . ∼ τ ∗ ∂ μ∼

(68)

Here we note g and g ' as SU (2) and U (1) coupling, respectively, and g = cw gz . To calculate the cross sections of our interest, we apply the method which is developed by authors of Ref. [106]. This method is based on non-relativistic twobody effective action: (i) We integrate out the fields except for the neutralino ∼ χ and the stau ∼ τ . Thus, we obtain one-loop effective action. (ii) We integrate out large momentum mode of ∼ χ and ∼ τ . This means that we integrate out relativistic modes. (iii) The non-relativistic action obtained in the last step is expanded by DM velocity. (iv) Finally, we introduce auxiliary fields that represent a two-body state and integrate out all fields except these auxiliary fields. In order to understand these steps, we consider integrating out the photon field Aμ as an example, however, without going detail of calculation techniques which can be found in Ref. [107]. By this calculation we can represent the one-loop interaction shown in Fig. 23. Let us follow our calculation setups: All fields except ∼ χ,∼ τ will be integrated out, the effective action becomes

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Fig. 23 One-loop interaction diagram mediated by photons

⌈

⌠ Seff =

d4x

⌉ 1 τ + ··· ∼ χ¯ (i/ ∂ − m) ∼ χ −∼ τ ∗ (∂ 2 + m∼2τ )∼ 2

Next step, we integrate out the large momentum modes of ∼ χ,∼ τ . Here we divide the fields into two parts, namely, relativistic part and non-relativistic part. For the case of ∼ χ, it will be χ(x) N R , ∼ χ(x) =∼ χ(x) R + ∼ ⌠ 4 d q 0 ∼ χ(x) R = φ (q)e−iq x , (2π)4 R ⌠ d 4q 0 φ (q)e−iq x , ∼ χ(x) N R = (2π)4

(69)

NR

where φ0 is the Fourier coefficient of the DM field. After this division, we integrate τ , and in the result we obtain out ∼ χ R . The same operation is done for ∼ ⌈ ⌉ ⌠ 1 4 ∗ 2 2 ¯ SN R = d x χN R − ∼ τ N R (∂ + m∼τ )∼ τN R ∼ χ (i ∂/ − m)∼ 2 NR χ N R ,∼ τ N R ) + S I m (∼ χ N R ,∼ τ N R ). (70) + S Pot (∼ Here we note S Pot is the real part except the kinematic part, and S I m is the imaginary part. In the following, we omit the subscript NR for ∼ χ and ∼ τ. Then, we expand this action by DM velocity. For this expansion, we use twocomponent spinors of neutralino and stau. These spinor fields are defined in the following form: ( χ¯ =

− → ∇ ·σ c ζ 2m − → −imt ∇ ·σ ie ζ 2m

e−imt ζ + ieimt e

ζ −

imt c

) ,

1 1 ∼ τ = √ ηe−imt + √ ξeimt , 2m 2m

(71)

T

where ζ c = −iσ 2 ζ † . In this form S N R = S K T + S Pot + S I m with the kinetic term ⌠ SK T =

⌈

) ) ( ( ∇2 ∇2 ∗ d x ζ i∂0 + ζ + η i∂0 + − δm η 2m 2m ) ⌉ ( ∇2 ∗ −ξ i∂0 − + δm ξ , 2m 4

†

(72)

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where the potential part S Pot is the real part except the kinematic part, and S I m is the imaginary part as before.

7.2 S-Wave Annihilation Cross Section of Dark Matter The S-wave annihilation cross section of DM to electroweak boson pair can be written in the term of the two-by-two matrix form of Green function which satisfies the Schwinger–Dyson equation ⌈

∇2 δ(r ) ∇r2 + i∂x 0 + x − V(r) + iΓ m 4m 4πr

⌉

〈0|T Φ(x, r)Φ† (y, r' )|0〉 = iδ(x − y)δ(r − r' ),

(73)

where V(r) is ⎛

⎞ −m Z r √ α e−m W r 2 e − 2α2 ⎜ 2δm − r − α2 cW r r ⎟ V(r) = ⎝ ⎠. √ e−m W r 0 − 2α2 r

(74)

By defining the radial component of Green function as G(E,l) , Eq. (73) becomes the following form: ⌈

⌉ δ(r ) δ(r − r ' ) 1 d2 (E,0) ' r + V(r) − iΓ (r, r ) = . G −E − mr dr 2 4πr r2

(75)

To determine G(E,0) , Eq. (75) must be solved in a proper boundary condition. For this purpose, we consider the terms involved Γ as in perturbation series and by using variable transformation g(r, r ' ) = rr ' Gii(E,0) (r, r ' ). The leading order’s solution g0 (r, r ' ) satisfies the equation −

1 d2 g0 (r, r ' ) + V(r)g0 (r, r ' ) − Eg0 (r, r ' ) = δ(r − r ' ). m dr 2

(76)

We get g0 (r, r ' ) in the following form: g0 (r, r ' ) = mg> (r )gT< (r ' )θ(r − r ' ) + mg< (r )gT> (r ' )θ(r ' − r ) . The solutions g> (r ) and g< (r ) satisfy the below boundary conditions: d g< (0) = 1. dr (ii) g> (0) = 1 , g> (r ) has only outgoing wave at r → ∞. (i) g< (0) = 0 ,

(77)

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(iii) Additionally, in the case of E < 2δm when the limit as r approaches infinity, √ i m Er | = δi2 d2 j (E)e must stau ∼ τ does not exit, thus the condition [g> (r )]i j | r →∞

be fulfilled.

By using the first-order perturbation for calculation of g> (r ) we can write down the cross section for annihilation channel f as ∗ Γ f ]11 d21 (mv 2 /4)d21 (mv 2 /4) . σ2(S) v| f = [∼

In addition, when we write the sum of each annihilation channel f as Γ = total cross section satisfies ∗ (mv 2 /4) . σ2(S) v = Γ11 d21 (mv 2 /4)d21

(78) Σ f

∼ Γf,

(79)

It is clear that it is necessary to solve the equation about g> (r ) to determine d2 j and cross section of our interest.

7.3 Results of the Dark Matter Research We have calculated the cross sections for several annihilation channels. The reader can find our input parameters of dimensionless such as the gauge coupling constants and the relevant mixing angles, also the dimensionfull parameters for our setup in Ref. [51]. The dependence of δm on the cross sections of ∼ χ∼ χ → γγ channel is shown in Fig. 24. There exist resonance peaks, and annihilation cross sections reach the height point at different DM mass. Further, we notice that the cross sections decrease as its δm increases. The dotted horizontal line in Fig. 24 presents the unitarity bound calculated for the short-range scattering problem of the effective quantum mechanics [108], so that above this line the unitary of the cross section breaks down, i.e., only below this line the cross section has physical meaning. Taking this effect into account, the cross sections become about seven to eight orders of magnitude smaller. The cross section of ∼ χ∼ χ → Z 0 Z 0 is displayed in the left panel of Fig. 25 and the + − cross section of ∼ χ∼ χ → W W can be found in the right panel of Fig. 25, respectively. It is quite important to note that these cross sections increase significantly—very sharp peaks—in these figures, this is fact that the general future of the mechanism of Sommerfeld enhancement. We compare our result to HESS experimental data of χχ → γγ focusing on coannihilation region. The graph representing this is shown in Fig. 26. As we can see from this figure cross section invades the prohibited area, and therefore we can constrain the parameter in our model. We also show the sensitivity projected by

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annihilation cross section 1x10

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Fig. 24 Annihilation cross section to photons per δm. Each graph name represents δm in MeV units. The dotted horizontal line represents the unitarity bound for the annihilation cross section

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-

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Fig. 25 The left panel shows the DM annihilation cross section into Z 0 Z 0 , the right panel shows the DM annihilation cross section into W + W − . We set both cases δm = 3 MeV

CTA [110]. Using this restricted parameter, we are able to constrain the parameters even more than now. In conclusion, the cross section calculated by our model has already reached an observable range. Thus, we can find signals from DM and solve the problems such as DM and 7 Li problems in the near future.

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annihilation cross section 1x10-20 HESS CTA γγ

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Fig. 26 A comparison of the cross section for δm = 3 MeV in the coannihilation region with the HESS result [109] and projected CTA sensitivity [110]. The blue solid line shows the calculation result and the purple region shows the HESS result. The green solid line shows the CTA sensitivity

8 Summary In this review, we expound on our idea to fit the primordial abundance of 7 Li to the observed value and give an idea to verify it. We extend the SM in particle physics to the MSSM, which is a most promising and extensively studied particle physics model beyond the SM. What we did is that we incorporate the long-lived CHAMP into the BBN. The neutralino DM in the MSSM calls for the coannihilation mechanism to account for the observed DM abundance, and the mass degeneracy between the neutralino DM and the slepton for this mechanism to work causes the longevity of the slepton. The slepton ℓ˜ is in general the mixing state of supersymmetric particles of ˜ of muon μ, ˜ and of tau lepton τ˜ . In our scenario, it consists of a mainly scalar electron e, partner of the tau lepton. It can be so long lived that it affects the BBN prediction. With appropriate parameters, we can nicely fit the abundances both of 7 Li and the neutralino DM to the observed ones. With such a long-lived slepton we have to consider the following three processes in addition to the processes in the standard BBN: (a) The destruction process of 7 Be and 7 Li in the bound state with the stau: (7 Be ℓ˜− ) → 7 Li + χ ˜ + ντ and (7 Li ℓ˜− ) → 7 He + χ ˜ + ντ (Sect. 3.1). It is often called “internal conversion”. This reaction is important to reduce the primordial 7 Li abundance. More reactions make the prediction better. ˜ (b) The 4 He spallation processes in the bound state of 4 He and ℓ: (4 He ℓ˜− ) → χ ˜ + ντ + T + n, (4 He ℓ˜− ) → χ ˜ + ντ + D + n + n, and (4 He ℓ˜− ) → χ ˜ + ντ + p + n + n + n (Sec. 3.2).

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These processes give extra light elements such as D, 3 He. Lower reaction rates are better. To evade the overproductions of these light elements, the stau must not be long-lived particle so that it decays before (4 He τ˜ − ) is formed or δm must be large enough as shown in Fig. 20. (c) The 6 Li production process called catalyzed fusion, ˜ + 6 Li (Sec. 3.2). (4 He ℓ˜− ) + D → χ 6 If we take the Li problem seriously, an appropriate reaction rate of the catalyzed fusion would be needed. From this requirement, we can pin down the parameter space if necessary. The key property is the small mass difference between the slepton and the neutralino DM. It gives the following important features: (1) Very long lifetime of O(1000) s since the phase space is quite narrow, and the open decay channel is only the four- body decay (in Fig. 11c). (2) The slepton in (7 Be ℓ˜− ) very efficiently destructs 7 Be, because the phase space becomes much wider in the bound state than that of the free slepton decay. This is shown in Fig. 16. Comparing it with the Fig. 11b, it is understood as the opening of the threshold of the three-body decay. Therefore, the timescale of the destruction is of order 1 s or less. We emphasize that the first two processes (a) and (b) are specific to our scenario, or more exactly to the small mass difference scenario. These do not work in the scenarios wherein the origin of longevity is Planck suppressed coupling. On the contrary, the third process (c) can happen in all scenarios with long-lived CHAMPs. We also evaluate the slepton number density at the BBN era. Larger slepton density makes the 7 Li abundance lower. The number density strongly depends on both the slepton mixing and the mass difference between the neutralino DM and the slepton; either the smaller mass difference and/or the small slepton mixing leads to a larger slepton density. As long as the tau flavor is conserved exactly the process (b) offers a disastrous creation of light elements D and 3 He. Fortunately due to the small mixing of other flavors than tau one, we can evade the constraints from this process keeping the longevity of slepton. Thus, all the rates for the processes (a)–(c) and the slepton number density are the function of only the mass difference δm and slepton mixing parameter. Taking into account all processes, we have calculated the primordial abundance and we have shown that it may be a possible solution to the 7 Li problem. We emphasize that, upon solving the 7 Li problem, neither unknown particles nor unnatural degrees of freedom are introduced. Consequently, this excellent scenario casts light on the observed abundances of both the DM and primordial light elements all at once and is in harmony with our universe. Such an extended theory can be tested soon. Indeed we show some of the parameter space for 7 Li problem can be surveyed in near future by indirect DM searches, that is, searching line spectrum of high-energy γ ray.

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Acknowledgements This work was supported by JSPS KAKENHI Grants No. JP18H01210 (J.S.), No. 20H05852 (M.Y.), and MEXT KAKENHI Grant No. JP18H05543 (J.S.). This work was supported by MEXT Joint Usage/Research Center on Mathematics and Theoretical Physics JPMXP0619217849 (M.Y.). This work was partly supported by the FJPPL Mu-04 (J.S. and M.Y.).

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Author Biographies Joe Sato received his Ph.D. in 1996 from Kyoto University. He is a theoretical physicist. He studies particle physics and cosmology. He also joins High energy experiment COMET. He was a postdoctoral researcher at Institute for Cosmic Ray Research (1997), JSPS fellow at University of Tokyo (1998-1999), an assistant professor at Kyushu University (1999-2002), an associate professor at Saitama University (2002-2021) and professor at Yokohama National University (2021-). Yasutaka Takanishi received his Diploma degree in physics in 1999 from University of Hamburg, Germany, and Ph.D. degree in physics in 2002 from The Niels Bohr Institute, University of Copenhagen, Denmark. He was a postdoctoral researcher at the International Centre of Theoretical Physics (ICTP), at the Scuola Internazionale Superiore di Studi Avanzati (SISSA), at the Technische Universitt Munich (TUM) and at the Max-Planck-Institut fur Kernphysik (MPIK) in Heidelberg. Since 2015 he is working at the University of Saitama, Japan. His research topic is theoretical physics including elementary particle physics and cosmology. Masato Yamanaka is a theoretical physicist. His interest is particle physics, cosmology, and astronomy. He received his Ph.D. in 2008 from Saitama University. He was a postdoctoral researcher at the university of Tokyo (2009), Maskawa Institute (2010, 2016-2017), KEK (20112012), and Nagoya University (2013-2015). He was an assistant in Kyushu Sangyo University in 2018. Since 2019, he is an assistant professor in Nambu Yoichiro Institute of Theoretical and Experimental Physics, Osaka City University.

Elements of Theory of Angular Moments as Applied to Diatomic Molecules and Molecular Spectroscopy V. K. Khersonsky and E. V. Orlenko

Abstract The Chapter is dedicated to the mathematical apparatus of the theory of angular momenta as applied to calculating the spectra of molecules and the probabilities of transitions between molecular levels. The apparatus is used in relatively simple problems of molecular spectroscopy, and we will mainly discuss the rotational structure of molecular spectra or the structure of electronic spectra (mainly diatomic molecules). It is associated with the presence at the electron shell of the orbital and/or spin moments of electrons. We take into account an analysis of vibrational-rotational interactions and its influence on to rotational spectra of molecules, since these interactions affect the parameters that determine the positions of rotational energy states. The vibrational motion of atoms in a rotating molecule leads to the Coriolis interaction, which manifests itself in the rotational bands of vibrational spectra. We consider also the model of Quantum rotators as they simulate rotational properties molecules and atomic nuclei, allowing the study of their rotational spectra. The main goal of this Chapter is to demonstrate a possibility of analytical calculation and consideration of the physical quantities in molecular spectroscopy. Keywords Molecular rotations · Oscillatory–rotational interactions · Quantum rotators · “Oscillatory” angular moment · Diatomic molecules with electronic angle moments · The total wave function of a molecule · Parity · /\-doubling · Hund’s schema · Matrix elements for rotational transitions · Forces of radiation transition lines · Spectra of rotators

1 Introduction The mathematical apparatus of the theory of angular momenta plays a fundamental role in the theory of rotational and electronic spectra of molecules, allowing one to calculate the positions of energy states, classify these states according to the types of V. K. Khersonsky · E. V. Orlenko (B) Theoretical Physic Department, Institute of Physics and Mechanics (PhysMech), Peter the Great Saint-Petersburg Polytechnic University, Polytechnicheskaya, 29, Saint-Petersburg 195251, Russia e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 T. Onishi (ed.), Quantum Science, https://doi.org/10.1007/978-981-19-4421-5_5

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symmetry, and determine the probabilities of transitions between them. Despite the fact that molecular spectroscopy is one of the very well-developed areas of knowledge in both theoretical and experimental aspects, most of the methodological elements related to the use of the theory of angular momenta in theoretical study are placed in separate articles. It makes difficult to understand these methods, especially for those who do not use them systematically or are acquainted with them. Therefore, in this chapter, without pretending to be comprehensive, we will demonstrate how the methods under consideration allow one to obtain solutions to some selected simple problems, focusing our attention on the details of calculating the spectra of molecules and the probabilities of transitions between molecular levels. Since our aim is to illustrate how the methods of the quantum theory of angular momentum are used in relatively simple problems of molecular spectroscopy, we will mainly discuss the rotational structure of molecular spectra or the structure of electronic spectra (mainly diatomic molecules). It is associated with the presence at the electron shell of the orbital and/or spin moments of electrons. We note right away that, strictly speaking, even a discussion of purely rotational spectra of molecules cannot be carried out without an analysis of vibrational–rotational interactions, since these interactions affect the parameters that determine the positions of rotational energy states. For example, the vibrational motion of atoms in a rotating molecule leads to the Coriolis interaction, which manifests itself in the rotational bands of vibrational spectra. However, if we consider only rotational states and transitions between them without changing the vibrational state, the contribution of the Coriolis interaction usually turns out to be negligible and commonly be neglected. Another important effect caused by the vibrations of atoms in molecules is that, in vibrations corresponding to large quantum numbers, the atoms, on average, spend a longer time at large distances from each other, which effectively increases the moment of inertia and decreases the rotational constants. These and other effects are taken into account using the adiabatic perturbation theory and, methodically, in most cases, are not directly related to the methods of the theory of angular momenta. Therefore, here, keeping in mind the discussion of these methods using simple but illustrative examples and not aiming to illuminate all aspects of molecular spectroscopy, we will not touch upon the problems associated with complex and rather cumbersome calculations of vibrational-rotational interactions. We refer the interested reader to special monographs devoted to these questions (see, for example [1–9]). An exception will be made in a few cases where it will be necessary and the presentation will avoid large deviations from the context of this chapter.

2 Molecular Rotators A quantum rotator is a model of a micro-object that does not have any internal or external properties other than a certain distribution of mass in it. This mass distribution is described by the tensor of inertia. The motions of this object include rotations

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around the coordinate axes and are quantized in the sense that the integrals of motion, that is, the rotational energy and angular momentum, can only take on certain values. The set of allowed energy values forms the energy spectrum of the rotator, while the rotations themselves are described by stationary wave functions. Quantum rotators play an important role in quantum theory as they simulate rotational properties molecules and atomic nuclei, allowing the study of their rotational spectra.

2.1 Prerequisites for Molecular Rotators The concept of a solid is based on a model that considers a system of particles rigidly connected to each other. Although such a system as a whole can experience various types of movements, the distances between its internal parts do not change and the system always retains its configuration. In turn, a rotator is generally a rotating rigid body. Obviously, the rotator is an idealized system. In many cases, the rotation is so fast that it begins to affect the bond lengths in the body, changing the distances between its various parts. For example, the rapid rotation of molecules or atomic nuclei leads to a change in the geometric properties of these systems and can be observed spectroscopically. However, these effects are not directly related to those aspects in the description of rotations that define the context of the fundamental properties of rotators. Despite the fact that this chapter is devoted to the quantum theory of angular momentum, methodically, in the context of discussing the applications of this theory to the description of a rotator, it would be useful to consider its foundations in classical mechanics, since the main fundamental results in the quantum field find a direct correspondence in the transition to classical mechanics. Therefore, in the first section of this chapter, we will look at the classic rotator.

2.1.1

Classic Rotator

The quantum rotator will be discussed in the next section. In classical mechanics, the tensor of inertia in the center of mass system, I, is defined as a second-rank tensor upon rotations of the coordinate system, and its components Iij are transformed as components of this tensor [

I=

Iij ei ⊗ ej ,

i,j=x,y,z

/

Iij =

/ / ρ(r) r 2 δij − xi xj dV .

V

Angular momentum of a body in the same system,

(1)

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J = I · ω, Ji =

⎛

[

ωx

⎞

)⎜ ⎟ ( Iij ωj = Iix Iiy Iiz ⎝ ωy ⎠, i,j=x,y,z ωz

(2)

where is the internal product I · ω of the inertia tensor and the angular velocity vector. The columnar representation of the angular velocity vector used here coincides with the generally accepted representation of the vector in the Cartesian basis. The kinetic energy of a volume element in the laboratory coordinate system is ⎛

Ixx Ixy Ixz

⎞⎛

ωx

⎞

)⎜ 1( 1 ⎟⎜ ⎟ E = M V 0 · V 0 + ωx ωy ωz ⎝ Iyx Iyy Iyz ⎠⎝ ωy ⎠ 2 2 Izx Izy Izz ωz 1 ω · I · ω, =Et + ~ 2

(3)

The first term, Et , represents the kinetic energy of the translational motion of the body as a whole relative to the laboratory coordinate system. Second, Erot there is ω means the transposed representation of the kinetic energy of rotation of the body, ~ the angular velocity vector, that is, the component column is replaced by the string. Like any tensor of the second rank, the tensor of inertia can be reduced to diagonal form. And since the tensor of inertia is symmetric with respect to the permutation of indices, all three diagonal components, k (λ) (λ = 1, 2, 3), are real numbers that can be found as the roots of the characteristic polynomial. The quantities k (λ) (λ = 1, 2, 3) are the principal values of the tensor of inertia or simply the principal moments of inertia and are usually denoted as I1 = k (1) , I2 = k (2) ,

(4)

(3)

I3 = k . The quantities k (λ) (λ = 1, 2, 3) satisfy the following three equations k (1) + k (2) + k (3) = g1 , k (1) k (2) + k (1) k (3) + k (2) k (3) = g2 , (1) (2) (3)

k k k

(5)

= g3 .

Here we introduced invariants of the tensor of inertia, g1 , g2, and g3 (these quantities are invariant with respect to the rotation of the coordinate system),

Elements of Theory of Angular Moments as Applied to Diatomic Molecules …

g1 = I11 + I22 + I33 , I I I I I I I I22 I32 I I I11 I21 I I I11 I31 I I I I I I, I + + g2 = I I23 I33 I I I12 I22 I I I13 I33 I I I I I11 I12 I13 I I I g3 = II I21 I22 I23 II. II I I I 31 32 33

211

(6)

I1 = k (1) , I2 = k (2) , I3 = k (3) . In a geometric sense, diagonalization of a tensor means rotating the coordinate system to new axes, so that the projections of the tensor onto these new axes coincide with the principal values of the tensor. The directions of these new axes can be defined as the directions of the eigenvectors of the diagonalized tensor matrix. Let’s designate these vectors as u(1) , u(2) , u(3) . Therefore, it is possible to introduce three orthonormal vectors that determine the three main directions of the tensor of inertia, u(1) u(2) u(3) eˆ 1 = I (1) I , eˆ 2 = I (2) I , eˆ 3 = I (3) I . Iu I Iu I Iu I

(7)

The axes of the coordinate system coinciding with these directions are called the principal axes of the tensor. If the axes of the system of the center of inertia coincide with the directions of the principal axes of the tensor of inertia, then the expressions for the angular momentum of a rigid body and its rotational kinetic energy are simplified, Ji = Ii ωi , ) 1( Erot = I1 ω12 + I2 ω22 + I3 ω32 . 2

(8)

If all three main moments of inertia are different, that is I1 /= I2 /= I3 , the rotating body is called an asymmetric rotator or an asymmetric top. In the case when two of the three main moments of inertia are equal to each other, for example, I1 = I2 /= I3 the rotator (or top) is called symmetric. In this case, the z-axis of the coordinate system coincides with the main direction eˆ 3 , while the choice of the directions of the x and y axes is arbitrary in the (x, y) plane, while, of course, they remain mutually orthogonal. Finally, if all three main moments are equal, I1 = I2 = I3 , the case of a spherical rotator (top) takes place. In this case, the choice of three mutually orthogonal axes x, y, and z is arbitrary. In this section, we will consider a freely rotating rigid body, which, according to the classification introduced above, is generally an asymmetric rotator, the main moments of inertia of which are not equal to each other I1 /= I2 /= I3 . Particular

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cases of an asymmetric rotator are symmetric, I1 = I2 /= I3 or, I1 = I3 /= I2 or I3 = I2 /= I1 , spherical I1 = I2 = I3 , and linear rotators, I1 = I2 = 0, I3 /= 0 or I1 = I3 = 0, I2 /= 0 or I1 /= 0, I3 = I2 = 0. We assume that the axes of the center of mass system are rigidly connected with the rotating body and are directed along the main directions of the inertia tensor. Thus, the angular momentum and rotational energy of a body can be represented by formulas (8) in which the quantities are the projections of the angular velocity of rotation on the axis of the center of mass system. In the absence of external forces, a rigid body (freely) rotating around the center of mass is a conservative system, the angular momentum of which must satisfy the condition dJ = {H , J} dt

(9)

where the curly brackets {H , J} denote the Poisson brackets, and H is the Hamiltonian function, which, in the absence of external potential fields, is simply the rotational energy of the body. Taking into account the properties of the Poisson brackets for vector functions, the last equation has the form dJ = −ω × J, dt dJi = −εijk ωj Jk , dt

(10)

or, taking into account that Ji = Ii ωi , this system of equations can be rewritten as Ii ωi + εijk ωj ωk Ik = 0.

(11)

This system is called the Euler equations for a free asymmetric rotator. Usually, it is presented in component form, I1 ω1 − ω2 ω3 (I2 − I3 ) = 0, I2 ω2 − ω3 ω1 (I3 − I1 ) = 0,

(12)

I3 ω3 − ω1 ω2 (I1 − I2 ) = 0. When solving this system, it is necessary to take into account that the angular velocities included in these equations satisfy two additional equations representing the integrals of motion for energy and angular momentum J 2 = I1 ω12 + I2 ω22 + I3 ω32 , ( ) 2Erot = I1 ω12 + I2 ω22 + I3 ω32 .

(13)

In order to obtain equations that give a quantum-mechanical description of the rotator, it is convenient to transform equations for angular momentum (2) and energy

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(3) to a form that allows one to directly use the generalized momentum pi quantization rule, pi = −ih

∂ ∂qi

(14)

where qi is the generalized coordinate ⎛ ⎞ α ⎜ 2⎟ ⎜ ⎟ q = ⎝ q ⎠ ≡ ⎝ β ⎠, ⇒ ⎛

q1

⎞

γ ⎛ ⎞ q˙ α˙ ⎜ 2⎟ ⎜ ˙ ⎟ q˙ = ⎝ q˙ ⎠ ≡ ⎝ β ⎠, ⇒ ⎛

q3

⎛

1⎞

q˙ 3 Ω1

⎞

γ˙

(15)

⎜ ⎟ ˙ ω = ⎝ Ω2 ⎠ = U (α, β, γ ) · q, Ω3 ⎛ ⎞ ω1 ⎜ ⎟ ˙ ωc = ⎝ ω2 ⎠ = B(α, β, γ ) · q, ω3 where we introduced vector designations for the angular velocity represented by the projections on the axis of the laboratory coordinate system, ω and, for the same angular velocity represented by the projections on the axis of the center of mass ωc , where matrices U (α, β, γ ) and B(α, β, γ ) ⎛

⎞ 0 − sin α cos α sin β U (α, β, γ ) = ⎝ 0 cos α sin α sin β ⎠, 1 0 cos β ⎛ ⎞ − sin β cos γ sin γ 0 B(α, β, γ ) = ⎝ sin β sin γ cos γ 0 ⎠. cos β 0 1

(16)

Thus, the kinetic energy of rotation (3) can be represented as ( )T 2Erot = ~ ω · I · ω = ωc · I c · ωc ˙ = q˙ T · U T · I · U · q˙ = q˙ T · B T · I c · B · q˙ = q˙ T · G · q.

(17)

The quantity G is the metric tensor of the quadratic form 2E, which defines the classical generalized momentum in the form

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V. K. Khersonsky and E. V. Orlenko

[ ∂Erot ˙ = Gij q˙ j , ⇔ p = G · q, i ∂ q˙ j ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ α˙ α˙ p1 ⎜ ⎟ ⎜ ˙⎟ ⎜ ˙⎟ T T c ⎝ p2 ⎠ = U · I · U ⎝ β ⎠ = B · I · B ⎝ β ⎠. pi =

p3

γ˙

(18)

γ˙

We represent the rotational energy and angular momentum in the following form, taking into account (2) ( )−1 ( )−1 2Erot = q˙ T · G · q˙ = pT · U T · I · U · p = pT · B T · I c · B · p, ( T )−1 J = I · ω = I · U · q˙ = I · U · U · I · U ·p ( ( ) ) −1 −1 = I · U · U −1 · I −1 · U T ·p= UT · p. p = U T J, pT = J T U .

(19)

and, similarly, for the center of inertia system ( )−1 ( )−1 J c = I c · ωc = I c · B · q˙ = I c · B · B T · I c · B · p = BT · p, ⇒ p = BT J c , ( )T pT = J c B.

(20)

The formulas obtained make it possible to directly quantize the energy and angular momentum using rule (14).

2.1.2

Quantum Rotator

Let us first quantize the angular momentum. We start with the angular momentum vector represented by the projections on axis of the center of mass system,

Elements of Theory of Angular Moments as Applied to Diatomic Molecules …

⎞ ⎛ ⎞ p1 J1c ( T )−1 ⎜ ⎟ ⎜ c ⎟ ( T )−1 c J = ⎝ J2 ⎠ = B ·p= B ⎝ p2 ⎠, ⇒ J3c p3 ⎛ ⎞ ∂ −ih 1 ⎞ ⎛ c ⎜ ⎟ ∂q hJˆ1 ⎜ ⎟ ⎜ ⎟ ⎜ ( ) ∂ ⎟ ⎟, ˆ2c ⎟ = B T −1 ⎜ −ih hJ c = ⎜ h J ⎜ ⎠ ⎝ ∂q2 ⎟ ⎜ ⎟ ⎝ hJˆ3c ∂ ⎠ −ih 3 ∂q

215

⎛

/\

(21)

∂ ⎞ ⎛ ⎞⎜ ∂α ⎟ Jˆ1c ⎟ ⎜ − cos γ sin β sin γ cos β cos γ ⎟ ⎜ ⎜ ∂ ⎟ 1 ⎝ sin γ sin β cos γ − cos β sin γ ⎠⎜ −i ⎟. ˆ2c ⎟ = Jˆ c = ⎜ J ⎜ ∂β ⎟ ⎝ ⎠ sin β ⎟ ⎜ 0 0 sin β ˆJ3c ⎝ ∂ ⎠ −i ∂γ ⎛

⎞

⎛

−i

Similarly, for the projections of the angular momentum on the axis of the laboratory coordinate system, we obtain )−1 ( J= UT ·p

⎛

∂ ⎞ ⎛ ⎞⎜ ∂α ⎟ Jˆ1 ⎟ − cos α cos β − sin α sin β cos α ⎜ ⎜ ⎜ ⎟ ∂ ⎟ ˆJ = ⎜ Jˆ2 ⎟ = ⎝ − sin α cos β cos α sin β sin α ⎠⎜ −i ⎟. ⎜ ∂β ⎟ ⎝ ⎠ ⎜ ⎟ sin β 0 0 ⎝ Jˆ3 ∂ ⎠ −i ∂γ ⎛

⎞

−i

(22)

Thus, we obtain two sets of operators for the projections of the angular momentum vector on the axis of the center of mass system and on the axis of the laboratory system

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V. K. Khersonsky and E. V. Orlenko

) ( ∂ cos α ∂ ∂ + sin α − , Jˆ1 = JˆX = i cos α cot β ∂α ∂β sin β ∂γ ) ( ∂ sin α ∂ ∂ − cos α − , Jˆ2 = JˆY = i sin α cot β ∂α ∂β sin β ∂γ ∂ Jˆ3 = JˆZ = −i , ∂α ( ) ∂ ∂ ∂ 1 c c cos γ − sin β sin γ − cos β cos γ , Jˆ1 = Jˆx = i sin β ∂α ∂β ∂γ ( ) ∂ ∂ ∂ 1 − sin γ − sin β cos γ + cos β sin γ , Jˆ2c = Jˆyc = i sin β ∂α ∂β ∂γ ∂ Jˆ3c = Jˆzc = −i . ∂γ

(23)

Operator of the square of the total angular momentum of the rotator is /\

/\/\

/\/\

/\/\

J 2 =Jxc Jxc + Jyc Jyc + Jzc Jzc = JˆX JˆX + JˆY JˆY + JˆZ JˆZ ]] ( ) [ 2 [ ∂ 1 ∂2 ∂ 1 ∂ ∂2 . sin β + + =− − 2 cos β sin β ∂β ∂β ∂α∂γ ∂γ 2 sin2 β ∂α 2

(24)

To obtain the energy operator or the Hamiltonian of the asymmetric rotator, which we denote as, we can use Eq. (17), (19) substituting the operators of angular momentum (23) into it. We will accept, without loss of generality, that the axes of the center of mass system coincide with the principal axes of the tensor of inertia. Then, ⎞⎛ c ⎞ Jx 0 0 ⎜ ⎟ h2 ( c c c ) ⎜ ⎟ 1 c⎟ J J J ⎝ 0 Iyc 0 ⎠⎜ = ⎝ Jy ⎠ 2 x y z 1 0 0 Ic Jzc z ⎛

Hˆ rot

/\/\/\

/\

1 Ixc

/\

/\

=

h2 c c h2 c c h2 c c J J + J J . J + J 2Iyc y y 2Izc z z 2Ixc x x /\/\

/\/\

/\/\

(25)

Substituting here the expressions for the projections of the angular momenta, we can obtain the required expression for the Hamiltonian. However, this expression will be cumbersome and inconvenient for further analysis. Therefore, we will consider several other (but equivalent) forms of the Hamiltonian. In accordance with generally accepted practice, we will write the Hamiltonian in the form /\/\

/\/\

/\/\

Hˆ rot = AJac Jac + BJbc Jbc + CJcc Jcc , A=

h2 h2 h2 , B = , C = , A ≥ B ≥ C. c 2Iac 2Ib 2Icc

(26)

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217

where the quantities A, B, and C are called rotational constants and indices a, b, and c can take the values 1, 2, 3 or x, y, z. Note also that the factor is included in the values of the rotational constants and we will not repeat it when writing projections of angular momenta or energy. The inequality of the rotational constants is conventional and generally accepted in spectroscopy for convenience—the constant A is always determined by the smallest principal moment of inertia and the constant C by the largest. Using this notation, the Hamiltonian can also be represented by the following two expressions ] [ 1 1 2 ˆ ˆ Hrot = (A + B)J + C − (A + B) Jcc Jcc 2 2 [( )2 ] )2 ( 1 , + (A − B) Jac + iJbc + Jac − iJbc 2 ] [ 1 1 Hˆ rot = (B + C)Jˆ 2 + A − (B + C) Jac Jac 2 2 [( )2 ( )2 ] 1 + (B − C) Jbc + iJcc + Jbc − iJcc . 2 /\/\

/\

/\

/\

/\

/\/\

/\

/\

/\

/\

(27)

These expressions are especially useful when an asymmetric rotator is close to an oblate (A → B) or elongated (B → C) symmetric rotator. Then it is convenient to represent the Hamiltonian of an asymmetric rotator as the sum of two terms—the Hamiltonian Hˆs of a symmetric rotator and a small addition that breaks the symmetry Aˆ. Hˆ rot = Hˆs + ε(A, B, C)Aˆ

(28)

where ε(A, B, C) is a small parameter, depending on A, B, C. For an oblate rotator, we can write that /\

/\/\

Hˆs ≡ Hˆso = lim Hˆ rot = BJ 2 + (C − B)Jcc Jcc , A→B

1 (A − B), 2 )2 ( )2 ( Aˆ ≡ Aˆao = Jac + iJbc + Jac − iJbc ≈ 2Jac Jac . ε(A, B, C) =

/\

/\

/\

/\

(29)

/\/\

For a prolate rotator, you can write a similar decomposition, /\/\

Hˆ ≡ Hˆsp = lim Hˆ rot = BJˆ 2 + (A − B)Jac Jac , C→B

1 (B − C), [2( )2 ( )2 ] c c c c ˆ ˆ ≈ 2Jbc Jbc . A = Aap = Jb + iJc + Jb − iJc ε(A, B, C) =

/\

/\

/\

/\

/\/\

(30)

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V. K. Khersonsky and E. V. Orlenko

These formulas are easy to express in terms of Euler angles. In this case, the correspondence of the indices a, b, c to the axes of the axes of the center of mass x, y, z should be chosen so that the projections of the angular momenta Jcc in the case of an oblate rotator or Jac in the case of an elongated rotator correspond to the z-axis (or axis 3). Then you can get the following. If a = 1, b = 2, c = 3, then for an oblate rotator we obtain that /\

/\

h2 h2 h2 ,B = ,C = , A ≥ B ≥ C ⇒ I1 ≤ I 2 ≤ I 3 . 2I1 2I2 2I3 ( ) ] [ 2 [ ∂ 1 ∂ 1 ∂ ∂2 ˆ sin β + Hso = −B − 2 cos β sin β ∂β ∂β ∂α∂γ sin2 β ∂α 2 ( ) 2 ] ∂ C + cot 2 β + , B ∂γ 2 )2 )2 ( ( Aˆao = Jac + iJbc + Jac − iLcb ≈ 2Jxc Jxc A=

/\

/\

/\

/\

(31)

/\/\

Similarly, for a prolate rotator, a = 3, b = 1, c = 2, we have h2 h2 h2 ,B = ,C = , A ≥ B ≥ C ⇒ I1 ≤ I 2 ≤ I 3 . 2I3 2I1 2I2 ( ) ] [ 2 [ ∂ 1 ∂ ∂2 1 ∂ ˆ sin β + Hsp = −C − 2 cos β sin β ∂β ∂β ∂α∂γ sin2 β ∂α 2 ( ) 2 ] ∂ A + cot 2 β + , C ∂γ 2 )2 )2 ( ( Aˆap = Jac + iJbc + Jac − iJbc ≈ 2Jxc Jxc A=

/\

/\

/\

/\

(32)

/\/\

The main idea of such expansions is that the part of the Hamiltonian of an asymmetric rotator represented by the operators Aˆao at A → B or Aˆap at C → B can be regarded as a perturbation, thus allowing perturbation theory to be used to determine the eigenvalues and eigenfunctions of slightly asymmetric rotators. Another form of representation of the Hamiltonian of an asymmetric rotator used in spectroscopy contains the asymmetry parameter κ, 1 1 Hˆ rot = (A + C)Jˆ 2 + (A − C)Eˆ (κ), 2 2 Eˆ (κ) = Jac Jac + κJbc Jbc − Jcc Jcc , 2B − A − C . κ= A−C /\/\

/\/\

/\/\

(33)

This parameter changes in the range [−1, 1] at Ic ≤ Ib ≤ Ia . It should immediately be noted here that the model of an asymmetric quantum rotator, as well as its modifications of symmetric rotators, flattened, elongated and

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219

linear, can serve as a good approximation in describing both the rotations of complex molecules and heavy atomic nuclei, to which the weight distribution, mass distribution has a complex configuration. In this case, the rotational spectra of both molecules and rapidly rotating nuclei, calculated within the framework of these models, quite adequately describe the experimentally obtained spectral characteristics of rotational states, such as frequency, line strength, and spectral line width.

2.1.3

Symmetrical, Spherical, and Linear Rotators

A quantum rotator model is used to describe the rotation of molecules as a whole. Below we will use these results here to calculate the probabilities of quantum transitions between the rotational levels of molecules and to discuss some of the details in the rotational spectra of molecules. It is convenient to start the discussion of the energy spectrum and wave functions of quantum rotators with an analysis of the properties symmetrical and spherical rotator. The energy spectrum of a quantum rotator is the spectrum of the eigenvalues of the Hamiltonian, and the wave functions are the eigenfunctions of the Hamiltonian [10– 13]. In addition, the wave functions are eigenfunctions of the operators of the square /\

/\

of the angular momentum J 2 and its projections JZ onto the Z-axis of the laboratory coordinate system and Jzc onto the z-axis of the center of mass system. Considering the fact that the Hamiltonian is also expressed in terms of the square of the total angular momentum, we can conclude that the functions that are eigenvalues for all these operators simultaneously must depend on at least three quantum numbers. The first is the rotational quantum number J (often called simply the angular momentum). The second is the quantum number of the projection of the moment on the z-axis, K (called simply the projection of the moment on the quantization axis in the center of mass system). And, the third is the quantum number of the projection of the moment on the Z-axis, M (also called the projection of the moment on the quantization axis in the laboratory coordinate system). We will designate this function as ψJMK . Thus, for an symmetric rotator, we have /\

/\

J 2 ψJMK = J (J + 1)ψJMK , ____J = 0, 1, 2, . . . /\

JZ ψJMK = M ψJMK , ____M = −J , −J + 1, . . . 0 . . . J − 1, J ,

/\

Jzc ψJMK = KψJMK , ____K = −J , −J + 1, . . . 0 . . . J − 1, J ,

(34)

/\

HS ψJMK = EJK ψJMK ,

where the quantities EJK are the eigenvalues of the Hamiltonian, called the energy or energy spectrum of the rotator [14]. We note the following important fact about the energy spectrum. Since the Hamiltonian does not explicitly depend on the derivatives with respect to the angle describing the rotation around the Z-axis of the laboratory coordinate system, the energy spectrum of the rotator does not depend on the quantum

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number M, that is, the rotation energy is degenerate in this quantum number. The soluJ tion to the first three differential equations is DMK (α, β, γ ) the Wigner D-function, which we normalize so that the integral of the square of the modulus of the rotator wave function over the full range of angles is equal to unity. Thus, one can write that / ψJMK (α, β, γ ) = |JMK> = /2π

/π dα

0

/2π sin βd β

0

2J + 1 J D (α, β, γ ), 8π 2 MK (35)

ψJ∗' M ' K ' (α, β, γ )ψJMK (α, β, γ )d γ = δJ ' J δM ' M δK ' K .

0

The wave functions commonly used in spectroscopy, SJMK (φ, θ, χ ), satisfy the same first three equations and have the form (see, for example, [3]) ] |M −K| [ ] |M +K| [ 2 2 1 − cos θ 1 + cos θ × SJMK (φ, θ, χ ) = NJMK eiM φ eiKχ 2 2 ) ( |M − K| + |M + K| + 2 |M − K| + |M + K| 1 − cos θ ,J + ; 1 + |M − K|; , F −J + 2 2 2

(36) where F is the hypergeometric function, and the normalization factor NJMK is defined as

NJMK

( ⎡ J+ 2J + 1 1 ⎣ ( = 2 |M − K|! 8π J−

)( ! J− )( |M +K|+|M −K| ! J+ 2

|M +K|+|M −K| 2

) ⎤ 21 ! ) ⎦ . (37) |M +K|−|M −K| ! 2

|M +K|−|M −K| 2

Angles, φ, θ, and χ are Euler angles, which are defined as follows: ϕ—the angle is measured from the new position of the x-axis and not the y, as is usually described in the definition of rotation angles. The following relations between the angles relate to the two descriptions under consideration are π + α, 2 θ = β, π χ =γ − . 2

φ=

(38)

Substituting the eigenfunction ψJMK into the fourth equation, we find that, in order to satisfy this equation, the energy spectrum EJK = ESOJK of a symmetric oblate (SO) rotator must be represented in the following form, ESOJK = BJ (J + 1) − (B − C)K 2

(39)

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221

In the case of a symmetric prolate (SP) rotator, the energy spectrum EJK = ESPJK will be ESPJK = BJ (J + 1) + (A − B)K 2 .

(40)

In the most general case, molecules of the asymmetric rotator type, characterized 2 2 by three rotational constants A, B, and C (A > B > C), (quantities A = 2Ih A , B = 2Ih B ,

and C = 2IhC are called rotational constants), the wave function. Formulas (39) and (40) show that the energy spectrum of a symmetric rotator undergoes a twofold degeneracy of rotational levels with respect to the quantum number K—the rotational energy has the same values for two mutually opposite directions of the angular momentum relative to the rotator axis. Technically, this is due to the fact that the Hamiltonian depends only on the second derivative with respect to the angle and is invariant with respect to reflections in the plane passing through the axis of symmetry of the rotator. While we are interested in purely rotational spectra of the symmetric rotator, the considered degeneracy in K cannot be eliminated, and the wave functions Eq. (35) adequately describe quantum states. However, if the motion of the rotator is somehow perturbed, this degeneracy can be lifted and the energies of the levels will depend on the sign of the quantum number K. To obtain the correct wave functions to describe these split states, it is necessary to indicate an operator that would mix states with different signs of K, and require that the sought wave functions be eigenfunctions of this operator. It is easy to understand how this operator should act on the wave functions (35), if we take into account [2] that the reflection in the plane passing through the symmetry axis of the molecule changes the sign of K to −K. That is, the required operator is the reflection operator, Pν , in the plane passing through the axis of the molecule, which is determined by the relation, 2

/\

/\

Pν ψJMK (α, β, γ ) = ψJM −K (α, β, γ ).

(41)

The repeated action of the operator on this result is reduced to the identical transformation, that is, /\

/\/\

Pν Pν ψJMK (α, β, γ ) = Pν ψJM −K (α, β, γ ) = ψJMK (α, β, γ ).

(42)

It is now clear that the correct functions of the zero approximation, which should be used in calculating the splitting of the levels corresponding to the values ±K in the presence of some perturbation, should be designed in such a way that they are not only eigenfunctions of the operators listed in (34), but also eigenfunctions of the reflection operator Pν . In other words, the newly constructed functions—let us denote them ℘JMKπν (α, β, γ ) by showing the dependence on the new quantum number πν (eigenvalue of the operator Pν )—must satisfy the equation /\

/\

/\

Pν ℘JMKπν (α, β, γ ) = πν ℘JMKπν (α, β, γ )

(43)

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A double application of the reflection operator to this equation should bring the wave function to its initial form, /\/\

/\

Pν Pν ℘JMKπν (α, β, γ ) = πν Pν ℘JMKπν (α, β, γ ) = πν2 ℘JMKπν ≡ ℘JMKπν , then πν2 = 1 and πν = ±1. A double application of the reflection operator to this equation should bring the wave function to its initial form, ] [ 1 ψJMK (α, β, γ ) + πν Pν ψJMK (α, β, γ ) , (44) ℘JMKπν (α, β, γ ) = √ 2(1 + πν δK0 ) /\

It is easy to see that these functions satisfy Eqs. (34) and (41). In this case, we say that the wave functions have a certain parity with respect to reflections in the plane passing through the z-axis of the rotator. For K = 0, there is only one type of function for which we take πν = +1, / ℘JMKπν (α, β, γ ) =

1 + πν ψJM 0 (α, β, γ ) = iM 2

/

2J + 1 J ∗ D (α, β, γ ). 8π 2 M 0

(45)

In the case of a spherical rotator, A = B = C, the Hamiltonian takes on an extremely simple form—the energy operator turns out to be proportional to the operator of the square of the angular momentum, that is, the wave functions satisfy Eq. (34), in which the last equation repeats the first one up to a numerical factor B. Therefore, wave functions can be represented in the form / ψJMK (α, β, γ ) ≡ |JMK> = i

M −K

2J + 1 J ∗ D (α, β, γ ) 8π 2 MK

(46)

and rotational energies can be described as EJ = BJ (J + 1).

(47)

The rotational spectrum of the spherical rotator is degenerate with multiplicity (2J + 1)2 , that is, each eigenvalue of energy (47) corresponds to functions (46) with different values of M and K. A linear rotator corresponds to the case when one of the moments of inertia, for example, I2 = 0, is equal to zero and the other two are therefore equal to each other, I1 = I3 . In this case, the angular momentum is perpendicular to the rotator axis, that is, K = 0. As in the case of a spherical rotator, the Hamiltonian is proportional to the square of the angular momentum operator, and the wave functions are eigenfunctions of the angular momentum squared operator and the angular momentum projection onto the axis of the laboratory coordinate system,

Elements of Theory of Angular Moments as Applied to Diatomic Molecules …

223

/\

J 2 ψJM 0 = J (J + 1)ψJM 0 , /\

JZ' ψJM 0 = M ψJM 0 ,

(48)

Functions satisfying these equations are spherical functions, and in normalized form can be obtained directly from (46) by setting K = 0, / ψJM 0 (α, β, γ ) = i

M

2J + 1 J ∗ iM D YJM (α, β). β, γ = √ (α, ) 8π 2 M 0 2π

(49)

Here the factor disappears in the normalization integration over the angle, on which the function does not depend. The energy spectrum of the rotator is again described by formula (47), in which the constant B is determined by a nonzero moment of inertia, I, that is, however, in this case, degeneration takes place only in the quantum number M with multiplicity 2J + 1.

2.1.4

Asymmetric Rotator

In the case when all three main moments of inertia are different, the set of equations describing the rotational quantum states of the rotator includes three equations. All of them express the already discussed requirements—the wave function corresponding to a given rotational state must be an eigenfunction of the operator of the square of the angular momentum and its projection onto the Z-axis (with our choice of the quantization axis) of the laboratory coordinate system and the eigenfunction of the Hamiltonian operator of the asymmetric rotator. However, in this case, it should be taken into account that the asymmetry of the rotator not only completely removes the degeneracy in the projection of the angular momentum onto the rotator axis (the z-axis for our choice), but also makes it impossible for stationary states of rotation to have certain values of this projection. Therefore, there must be some other quantum number or numbers that make it possible to distinguish (2J + 1) levels with a given value of the rotational quantum number J. Let us temporarily denote this number as λ. Without knowing in advance what values this number can take, nevertheless, one can see that when the asymmetry parameter (33), κ → −1, that is, C → B, or, in other words, the asymmetric rotator tends to be a symmetric prolate rotator. The rotational level with given J becomes the rotational level of this prolate rotator, characterized by the quantum numbers J and K−1 , where we have designated the quantum number K of the corresponding prolate rotator as K−1 , as corresponding to the case κ → −1. You can also say that in this case λ → K−1 . Note that due to the already mentioned sign degeneracy of the quantum number K in the spectra of symmetric rotators, although K takes values in the interval [−J, J], we can consider only positive values of K, that is, we will assume that K−1 ∈ [0, J ]. The same rotational level is converted to the rotational level of a symmetrical flattened rotator, if κ → +1, or A → B. Therefore λ → K+1 , we can say that where K+1 denotes the quantum number K of the projection of the angular momentum

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onto the quantization axis of the corresponding symmetric oblate rotator (again, the subscript+1 means that κ → +1) and K+1 ∈ [0, J ]. Summarizing what has been said, we can see that each rotational level of an asymmetric rotator can be described by a rotational quantum number J and a pair of numbers K−1 , K+1 , which are the quantum numbers of the angular momentum projections on the quantization axis of the limiting prolate and oblate symmetric rotators. Moreover, due to the continuity of the transition between these limiting cases, such a description uniquely determines the rotational levels—each of the 2J + 1 rotational levels with a given J is characterized by “its own” pair of numbers K−1 , K+1 and there are no identical pairs in the complete set of these pairs. Based on this method of describing the quantum states of an asymmetric rotator, there is another equivalent method in which, instead of a pair of numbers K−1 , K+1 , their difference τ = K−1 − K+1 is determined. Since this new quantum number changes in the interval τ =∈ [−J , +J ]. We will qualitatively illustrate the behavior of several lower levels of the asymmetric rotator depending on the asymmetry parameter in the diagram shown in Fig. 1. The energy of the rotational state is denoted as EJK−1 ,K+1 or EJ τ . Both of these designations appear in the diagram. Note that the lines on this diagram, corresponding to a given value of J, never intersect. However, the lines corresponding to different values of J can intersect if, for example, the level of an asymmetric flattened rotator with a given J and K = 0 lies higher than the level characterized by the rotational quantum numbers J + 1 and K = J + 1. This situation is shown for the levels of a symmetric oblate rotator characterized by quantum numbers J = 2, K = 0 and J = 3, K = 3. The fact that the rotational states of an asymmetric rotator strictly correlate with the states of limiting symmetric rotators suggests that the wave functions of an asymmetric rotator can be constructed as linear combinations of the wave functions of a symmetric rotator, that is, ΨJM τ ≡ ΨJMK−1 K1 ≡ |JM τ > ≡ |JMK−1 K1 > =

J [

gτJK ΨJMK (α, β, γ )

K=−J

/ =

J 2J + 1 [ J M −K J ∗ g i DMK (α, β, γ ), 8π 2 K=−J τ K

(50)

where the quantities gτJK are the coefficients of the given expansion and can be determined by the method of diagonalization of the Hamiltonian (27). The complete system of equations for the wave functions can be represented as

225

Prolate rotator

Oblate rotator

Elements of Theory of Angular Moments as Applied to Diatomic Molecules …

Fig. 1 Schematic diagram of the correspondence between the rotational energy levels of an asymmetric rotator and the levels of the energies of the limiting elongated (left) and oblate (right) rotators. The horizontal axis shows the change in the asymmetry parameter. Transition lines between levels are shown in simplified straight lines /\

J 2 ΨJM τ = J (J + 1)ΨJM τ , /\

Jz ΨJM τ = M ΨJM τ ,

(51)

Hˆ rot ΨJM τ = EJ τ ΨJM τ . Since the wave function (50) is a linear combination of the Wigner D-functions, which correspond to certain values of the quantum numbers J and M, the first and second equations in system (51) are satisfied automatically. Solving the third equation and determining the energy levels is reduced to determining the expansion coefficients. This equation can be represented as

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V. K. Khersonsky and E. V. Orlenko

/

J ] 2J + 1 [ J M −K [ ˆ J ∗ J∗ gτ K i H DMK (α, β, γ ) − EJ τ DMK (α, β, γ ) = 0 2 8π K=−J

/

2J +1 M −K ' J i DMK ' (α, β, γ ) and integrate over 8π 2

If we now multiply this equation by all angles, we get J [

(52)

[< ] I gτJK JMK ' IHˆ |JMK>∗ − EJ τ δKK ' = 0,

K=−J

I < JMK ' IHˆ |JMK>∗ '

= (−1)K −K

(53)

2J + 1 8π 2

/2π

/π dα

0

/2π d β sin β

0

J∗ ˆ J d γ DMK ' (α, β, γ )H DMK (α, β, γ ).

0

Thus, we obtain a system of equations for determining the eigenvalues and eigenfunctions of the Hamiltonian of the asymmetric rotator for each value of the rotational quantum number J. Since it takes 2J + 1 values, = -J, -J + 1, … J − 1, J, this system contains 2J + 1 equations for 2J + 1 eigenvectors gJτ , each of which has 2J + 1 components gτJK . In addition, since this is a system of homogeneous equations, the eigenvectors gJτ can be determined only up to a certain constant factor, for the determination of which one additional inhomogeneous equation is required, which connects all components of the components of each eigenvector. As such an additional equation, the condition of completeness of eigenfunctions is usually considered, which can be written in the form /2π

/π dα

0

/2π d γ |ΨJM τ |2 = 1, ⇒

d β sin β 0

0

J [ I J I2 Ig I = 1. τK

(54)

K=−J

Accordingly, the matrix elements of the angular momentum projection operators are expressed by the following formulas, I i/ JMK ' IJxc |JMK> = − J (J + 1) − K(K − 1)δK ' K−1 2 i/ + J (J + 1) − K(K + 1)δK ' K+1 , 2 I < 1/ JMK ' IJyc |JMK> = J (J + 1) − K(K − 1)δK ' K−1 2 1/ + J (J + 1) − K(K + 1)δK ' K+1 , 2


0 consists of several substates corresponding to different values of the vibrational angular momentum, in the harmonic oscillator approximation it remains degenerate in the values of these moments—the energy depends only on the vibrational quantum number ν. If we take into account the anharmonicity of the oscillations, then the degeneracy in magnitude |l| is lifted. Indeed, let the real potential curve that determines the nature of the oscillations is not strictly parabolic (that is, kρ 2 /2), but contains an additional small term (that is, κρ 3 the next term in the expansion of the potential curve relative to the equilibrium position ρ = 0). The correction to the energy of the level with the data ν and l can be calculated in the first order of the perturbation theory as κ 2π

ΔEνl = κ = (

h κ = Nνl2 2 μω

)5/2 / 0

/2π 0

∞

(

eilγ

)∗ ( ilγ ) e dγ

/∞ Fνl2 (ρ)ρ 3 ρd ρ 0

[ ]2 |l| z |l|+3/2 e−z L(ν−|l|)/2 (z) dz

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V. K. Khersonsky and E. V. Orlenko

) − |l|2 − 23 ( ) [( ) ]2/3 |l|! ν2 − |l|2 ! ν2 + |l|2 ! ( ) |l| |l| ν 5 5 5 ν × F2 − + , |l| + , ; |l| + 1, − + ; 1 . 2 2 2 2 2 2 2 (

h =κ 2μω

)3/2

3(2|l| − 1)!! 22|l|+1/2

T

(

ν 2

(93)

( ) Since ν2 − |l|2 there is a positive integer or zero, the hypergeometric series in this formula turns into a finite sum, the terms of which are expressed through the G-functions of the given parameters. This result shows that the obtained correction to the energy depends not on l, but on |l|. This means that the degeneracy in the value of l is lifted—now −Σ and ∏-states, Δ- and −Φ states, etc., have different energies. However, the degeneracy in the sign of the quantity l still remains. In general, the splittings obtained with allowance for anharmonicity can be described by the following rules. The (v + 1)—fold degenerate triatomic molecule splits, depending on the ) ( level ) of( a linear sublevels. If ν even, then the state ν splits into parity of ν, into ν2 + 1 or ν+1 2 ν/2 doubly degenerate sublevels, which are characterized by the quantum number of the vibrational angular momentum l = ±ν, ±(ν − 2), ±(ν − 4) . . .(± 2,)and one doubly nondegenerate sublevel l = 0. If ν is odd, then the state ν splits into ν+1 2 degenerate sublevels l = ±ν, ±(ν − 2), ±(ν − 4) . . . ± 1. Removal of double degeneracy is associated with calculations in the second order of the perturbation theory and is physically explained by the fact that rotation interacts with two degenerate components of the bending vibration in different ways. The Coriolis force arising from the vibrational motion of atoms perpendicular to the angular momentum of rotation (the first of the degenerate components) makes a significant contribution to the interaction energy of vibrations and rotations, and this contribution increases with increasing rotational quantum number. On the other hand, the vibrational motion parallel to the angular momentum of rotation (the second of the degenerate components) does not generate the Coriolis force and, thus, does not make any contribution to the energy of the vibrational-rotational interaction. Taking these differences into account turns out to be sufficient to remove the degeneracy between the levels from ±l. We will not repeat here the cumbersome calculations in the second order of the perturbation theory that illustrate what has been said. The interested reader can refer to the classical work [18], which sets out the main ideas related to these calculations, as well as to numerous monographs devoted to issues related to molecular spectroscopy. So, we have determined the vibrational angular momentum and its properties. Now it is necessary to answer the question of how the rotational spectrum of a linear molecule will look like taking into account the vibrational angular momentum. Based on the fact that the total angular momentum J is the sum of the purely rotational moment N directed perpendicular to the axis of the molecule and the vibrational moment l directed along the axis of the molecule, we can conclude that the total moment cannot be less than the vibrational moment and must take values. In general,

Elements of Theory of Angular Moments as Applied to Diatomic Molecules …

239

this situation exactly coincides with that which occurs in molecules of the type of an elongated symmetric rotator with angular momentum K (the projection of the total momentum J onto the axis of symmetry of the rotator), which allows us to write an expression for the vibrational-rotational energy in the form, [ ] Eνl = hω(ν + 1) + Bν J (J + 1) − l 2

(94)

where the role of the moment K is played by the vibrational angular momentum l. The wave function describing the rotation of the molecule and its degenerate vibrations can now be represented in the form iM ΨνlJM (ρ, α, β, γ ) = √ Fνl (ρ)YJM (α, β)eilγ 2π

(95)

Taking into account the centrifugal stretching of molecules and the contribution of nondegenerate vibrations, we have Eν1 ν2 νRl = hω1 (ν1 + 1/2) + hω2 (ν2 + 1/2) + hω(ν + 1)+ [ ] [ ]2 +Bν J (J + 1) − l 2 − Dν J (J + 1) − l 2

(96)

where the quantities Bν and Dν are determined using formulas (72)–(75), taking into account all possible oscillations. Note that in molecules of the symmetric rotator type with linear chains along the axis of symmetry, the effect under consideration can lift the degeneracy in K and l. To illustrate this qualitatively [4], we write the Hamiltonian of a rotating molecule of the type of an elongated symmetric rotator with excited degenerate oscillation of atoms about the symmetry axis, which generates an additional vibrational angular momentum l (in the coordinate system rigidly connected with the molecule), )2 ( Hˆ rot = BJ 2 − BJˆz2' + A Jˆz' − ˆlz' = BJ 2 + (A − B)Jˆz2' − 2AJˆz' ˆlz' + Aˆlz2' . /\

/\

(97)

Since the vibrational angular momentum is directed along the axis of symmetry Z’, then ˆlx' = 0, ˆly' = 0. The last term is independent of the rotational quantum numbers and can be omitted in this context. The calculation of the energies of rotational states is reduced to replacing /\

J 2 → J (J + 1), Jˆz' → K, ˆlz' → ς l, where ς is some constant that is obtained by calculating the Coriolis interaction of vibrating atoms and the general rotation of the molecule. With these substitutions,

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the energy can be represented in the form EνJK = Bν J (J + 1) + (Aν − Bν )K 2 − 2ς Aν Kl, J = 0, 1, 2, . . . , l = ν, ν − 2, . . . − ν.

(98)

For this formula to work for molecules such as a flattened symmetric rotator, the constant Aν must be replaced with a constant Cν . An essential element of this formula is the last term, which is proportional to the product of the first powers of the quantities K and l. Although in this case the twofold degeneracy is preserved, it has a different character—for the same absolute values of the quantities K and l, the energies of the levels will not change if the signs of these quantities are simultaneously changed. For many rotational levels (see the analysis presented in [4]) this degeneracy can be lifted in higher orders of the perturbation theory.

2.5 Diatomic Molecule with Identical Nuclei The presence of identical nuclei in a molecule can significantly change the structure of the rotational spectra. This is due to the statistics of nuclear spins. The total wave function of a molecule must be symmetrized in a certain way with respect to the permutation of identical nuclei, namely, it must be symmetric with respect to this permutation if the identical nuclei have integer spins, and antisymmetric if the identical nuclei have half-integer spins. In the simplest case of a diatomic molecule, a permutation of identical nuclei is equivalent to an inversion of space. Since the total wave function of a molecule includes the electronic, vibrational, rotational and nuclear parts, the total symmetry of the wave function depends on the spatial parity of the electronic, vibrational and rotational functions and on the parity of the nuclear function with respect to the permutation of nuclei. In order to explain how this affects the structure of the rotational spectra, let us consider a simplified situation, assuming that the electronic and vibrational functions of a molecule in their ground states are symmetric. In this case, they can be excluded from consideration; the total symmetry of the molecule depends only on the symmetries of the rotational and nuclear parts of the wave function. Therefore, this part of the wave function can be represented in the following form ΦJMIMI (α, β, σ1 , σ2 ) = ψJM (α, β)ΞIMI (σ1 , σ2 ) [ JM iM = √ YJM (α, β) Cim1Iim2 χim1 (σ1 )χim2 (σ2 ), 2π m1 m2

(99)

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241

where ψJM (α, β) is the rotational function (72) of a diatomic molecule, ΞIMI (σ1 , σ2 ) is the total spin function of identical nuclei with spins i, characterized by the total spin I and its projection MI onto the quantization axis. The functions χim1 (σ1 ) and χim2 (σ2 ) are spin functions of identical nuclei depending on the spin variables of the nuclei σ1 and σ2 . The kernel permutation operator can be defined as follows: Pˆ i ΦJMIMI (α, β, σ1 , σ2 ) = ψJM (α + π, π − β)ΞIMI (σ2 , σ1 ) [ JM = (−1)J ψJM (α, β) Cim1Iim2 χim1 (σ2 )χim2 (σ1 ) m1 m2 = (−1)J +2i−I ψJM (α, β)

[

JMI Cim χ (σ1 )χim2 (σ2 ) 1 im2 im1

m1 m2 = (−1)J +2i−I ψJM (α, β)ΞIMI (σ1 , σ2 ) = (−1)J +2i−I ΦJMIMI (α, β, σ1 , σ2 ).

(100)

Two-fold application of the operator Pˆ i is the identical operation. Wave functions of correct symmetry are constructed as follows: 1 [ ΨJMIMI (α, β, σ1 , σ2 ) = √ ΦJMIMI (α, β, σ1 , σ2 ) 2 ] +(−1)2i Pˆ i ΦJMIMI (α, β, σ1 , σ2 ) ,

(101)

that is, the wave function is symmetric with respect to the nuclear permutation if the nuclear spins are integral, and antisymmetric if the nuclear spins are half-integral. Substituting here the result of the action of the permutation operator on the functions under consideration [formula 100], we obtain that ] 1[ 1 + (−1)2i (−1)J +2i−I ΦJMIMI (α, β, σ1 , σ2 ) 2 ] 1[ = 1 + (−1)J −I ΦJMIMI (α, β, σ1 , σ2 ) (102) 2

ΨJMIMI (α, β, σ1 , σ2 ) =

Thus, the requirement of a certain type of symmetry with respect to the permutation of identical nuclei leads to the fact that rotational levels with even J can occur only for an even total spin of nuclei I, and rotational levels with an odd J only for an odd value of I. As an illustration of this rule, it is convenient to consider the rotational spectra of the molecules of hydrogen, H2 , and deuterium D2 . The ground electronic states of these molecules are—1 Σg+ these states are symmetric with respect to coordinate

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V. K. Khersonsky and E. V. Orlenko

inversion. The fundamental vibrational states are always symmetric. That is, both molecules exactly correspond to the conditions adopted for determining the function (99). The nuclear spins of the proton ip and deuteron id have values ip = 1/2, id = 1. In the general case, the total number of possible states with respect to the nuclear spin is (2i + 1)2 of which (i + 1)(2i + 1) are symmetric, and i(2i + 1) are antisymmetric. Here we are talking about an antisymmetric or symmetric nuclear wave function corresponding to the total spin introduced in formula (100), that is, [

ΞIMI (σ2 , σ1 ) =

JMI Cim χ (σ2 )χim2 (σ1 ) 1 im2 im1

m1 +m2 =MI

In a hydrogen molecule, one antisymmetric state corresponds to the value of the total nuclear spin I = 0, M I = 0 (singlet state—para-hydrogen). Symmetric states in a hydrogen molecule correspond to the values of the total spin I = 1, M I = +1, 0, −1 (the triplet state is ortho-hydrogen). (1) A singlet state 00 Ξ00 (σ1 , σ2 ) =C 00 1 1 1 1 χ 1 1 (σ1 )χ 1 − 1 (σ2 ) + C 1 1 1 1 χ 1 − 1 (σ1 )χ 1 1 (σ2 ) − − 22 2 2 2 222 2 2 222 2 22 ) ( 1 = √ χ 1 1 (σ1 )χ 1 − 1 (σ2 ) − χ 1 − 1 (σ1 )χ 1 1 (σ2 ) , 2 22 2 2 2 2 22 Ξ00 (σ2 , σ1 ) = − Ξ00 (σ1 , σ2 ), (103)

A permutation (σ1 , σ2 ) → (σ2 , σ1 ) changes the sign of this wave function. This spin function is really antisymmetric. (2) A triplet states Ξ1MI (σ1 , σ2 ) =

[

I C 1M 1

m1 m2

2

1 χ 1 (σ1 )χ 1 m2 (σ2 ), m1 m2 2 m1 2 2

Ξ11 (σ1 , σ2 ) = C 11 1 1 1 1 χ 1 1 (σ1 )χ 1 1 (σ2 ) = χ 1 1 (σ1 )χ 1 1 (σ2 ), Ξ10 (σ1 , σ2 )

22 2222 22 10 = C 1 1 1 1 χ 1 1 (σ1 )χ 1 − 1 (σ2 ) − 2 2 222 2 22

22

22 10 C 1 1 1 1 χ 1 − 1 (σ1 )χ 1 1 (σ2 ) − 22 2 222 2 2

+ ) ( 1 = √ χ 1 1 (σ1 )χ 1 − 1 (σ2 ) + χ 1 − 1 (σ1 )χ 1 1 (σ2 ) , 2 22 2 2 2 2 22 Ξ1−1 (σ1 , σ2 ) = C 1−1 χ = χ (σ )χ (σ ) 1 1 1 1 (σ1 )χ 1 1 (σ2 ). 1 2 1 1 1 1 1−1 − − − 2

−

22

−

2

2

2

2

2

2

2

2

2

(104)

Elements of Theory of Angular Moments as Applied to Diatomic Molecules …

243

It is easy to see that all three triplet functions are symmetric with respect to permutation (σ1 , σ2 ) → (σ2 , σ1 ). Thus, in the singlet state, the rotational spectrum of the hydrogen molecule includes levels corresponding only to even values of J. On the contrary, in the triplet state, the rotational levels correspond only to odd values of J. Now consider a molecule of deuterium. Since the spin of the deuterium nucleus is equal to unity, there are nine states in nuclear spin, of which six are symmetric. Of these, one singlet (I = 0, M I = 0) and a quintet (I = 2, M I = +2, +1, 0, −1, − 2). The rotational energy spectrum of these states corresponds only to even values J = 0, 2, 4, … Three antisymmetric states correspond to quantities, I = 1, M I = +1, 0, −1, and their rotational energy spectrum of these states corresponds only to odd values J = 1, 3, 5, … (1) Symmetric states Ξ2MI (σ2 , σ1 ) =

[

2MI C1m χ (σ1 )χ1m2 (σ2 ), 1 1m2 1m1

m1 +m2 =MI 20 20 Ξ20 (σ2 , σ1 ) = C111−1 χ11 (σ1 )χ1−1 (σ2 ) + C1−111 χ1−1 (σ1 )χ11 (σ2 ) 20 χ10 (σ1 )χ10 (σ2 ) + C1010 1 = √ (χ11 (σ1 )χ1−1 (σ2 ) + χ1−1 (σ1 )χ11 (σ2 ) + 2χ10 (σ1 )χ10 (σ2 )), 6 21 21 Ξ21 (σ2 , σ1 ) = C1110 χ11 (σ1 )χ10 (σ2 ) + C1011 χ10 (σ1 )χ11 (σ2 ) 1 = √ (χ11 (σ1 )χ10 (σ2 ) + χ10 (σ1 )χ11 (σ2 )), 2 2−1 2−1 χ1−1 (σ1 )χ10 (σ2 ) + C101−1 χ10 (σ1 )χ1−1 (σ2 ) Ξ2−1 (σ2 , σ1 ) = C1−110 1 = √ (χ1−1 (σ1 )χ10 (σ2 ) + χ10 (σ1 )χ1−1 (σ2 )), 2 22 χ11 (σ1 )χ11 (σ2 ) = χ11 (σ1 )χ11 (σ2 ), Ξ22 (σ2 , σ1 ) = C1111 2−2 χ1−1 (σ1 )χ1−1 (σ2 ) = χ1−1 (σ1 )χ1−1 (σ2 ), Ξ2−2 (σ2 , σ1 ) =C1−11−1 00 00 χ11 (σ1 )χ1−1 (σ2 ) + C1−111 χ1−1 (σ1 )χ11 (σ2 ) Ξ00 (σ2 , σ1 ) = C111−1 00 χ10 (σ1 )χ10 (σ2 ) + C1010 1 = √ (χ11 (σ1 )χ1−1 (σ2 ) + χ1−1 (σ1 )χ11 (σ2 ) − χ10 (σ1 )χ10 (σ2 )). 3 (105)

(2) Antisymmetric states Ξ1MI (σ2 , σ1 ) =

[

1MI C1m χ (σ1 )χ1m2 (σ2 ), 1 1m2 1m1

m1 +m2 =MI 10 10 Ξ10 (σ2 , σ1 ) = C111−1 χ11 (σ1 )χ1−1 (σ2 ) + C1−111 χ1−1 (σ1 )χ11 (σ2 )

244

V. K. Khersonsky and E. V. Orlenko 10 + C1010 χ10 (σ1 )χ10 (σ2 ) 1 = √ (χ11 (σ1 )χ1−1 (σ2 ) − χ1−1 (σ1 )χ11 (σ2 )), 2 11 11 χ11 (σ1 )χ10 (σ2 ) + C1011 χ10 (σ1 )χ11 (σ2 ) Ξ11 (σ2 , σ1 ) = C1110 1 = √ (χ11 (σ1 )χ10 (σ2 ) − χ10 (σ1 )χ11 (σ2 )), 2 1−1 1−1 χ10 (σ1 )χ1−1 (σ2 ) Ξ1−1 (σ2 , σ1 ) = C1−110 χ1−1 (σ1 )χ10 (σ2 ) + C101−1 1 = √ (χ1−1 (σ1 )χ10 (σ2 ) − χ10 (σ1 )χ1−1 (σ2 )). (106) 2

Note that if the spins of identical nuclei are equal to zero, then the rotational spectrum of the molecule contains only levels corresponding to even J.

3 Diatomic Molecules with Electronic Angular Moments The model of a rigid rotator or rotator with centrifugal stretching and vibrationalrotational interactions in a molecule turns out to be insufficient when it is necessary to take into account relativistic corrections arising from the presence of a nonzero spin of the electron shell S. The spin interacts with the orbital motions of electrons and the rotation of molecules as a whole, and these interactions can affect the structure of the rotational spectrum. Among these corrections, the spin-orbit interaction plays a central role, determining, in particular, the structure of the rotational spectrum. The most important effects of this type are associated with spin-orbit interactions that arise in diatomic and linear molecules, where large gradients of the electric field along the axes of the molecules affect the orbital motions of electrons, naming their momenta and moments of motion. Although the total angular momentum of the electrons L is not an integral of motion in such an axially symmetric field, its projection onto the axis, Λ, is. Therefore, the electron shell of a diatomic or linear molecule can be characterized by a quantum number Λ. In turn, electrons moving along spiral trajectories along the axis of the molecule generate a magnetic field, the magnitude of which is greater, the greater the projection of the orbital momentum onto the axis of the molecule. The electron spin of the electron shell, S, interacts with this magnetic field. In addition to this, this spin-orbit interaction occurs in a rotating molecule, which is characterized by an angular momentum R. Therefore, it can also change the structure of the rotational spectrum of the molecule. Thus, this structure depends on how all these moments are related to each other. In part, similar effects can occur in molecules of the symmetric rotator type, although such molecules with a nonzero value in the ground state are encountered very rarely. In asymmetric polyatomic molecules, the orbital motions of electrons do not have axial symmetry and the total spin of the electron shell is the only factor that can affect the structure of the rotational spectrum if it is nonzero. Thus, the electronic

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structure of the molecular shell turns out to be the most interesting from the point of view of the application of methods of the theory of angular momentum in diatomic and linear molecules. We will discuss diatomic molecules here. It is clear that a polyatomic linear molecule has a rich vibrational spectrum, including many vibrational modes, while a diatomic molecule has only one vibrational mode. However, here we believe that the molecules under consideration are in the ground fully symmetric vibrational state and therefore the differences noted are not of fundamental importance. Our goal is to obtain expressions for the wave electron-rotational wave functions for molecules in which the electron shells have orbital and spin moments that are not equal to zero. Below we will discuss how these wave functions can be used to calculate the strengths of radiative transition lines, taking into account the electronic structure. Nuclear spins will not be covered in this section. They interact weakly with electron spins or nuclear motions. These interactions cause hyperfine splitting of rotational levels and are taken into account using perturbation theory.

3.1 Coordinates. General Integrals of Motion Let us denote the wave function of the molecule as ΨςJM πi (q, Q),, where q = {ri } and Q = {RA , RB } are the sets of coordinates of electrons and nuclei, which are defined below, and ς, J , M , πi there is a set of quantum numbers characterizing the quantum state of the molecule. This set includes the minimum set of quantities, which consists of quantum numbers corresponding to the general integrals of motion. There are the total angular momentum of the molecule, J (nuclear spins are not included), its projection onto the quantization axis in the LSC (Laboratory Coordinate System), M, (Z-axis) and πi is the parity of the wave function with respect to inversion of all spatial coordinates. No other angular momenta—total orbital momentum of electrons L, the total electron spin S, or the rotational moment of the molecule R, generally speaking, are not integrals of motion. However, in certain cases, some of them may be so in a fairly good approximation. A quantity ς is the set of all the other quantum numbers required to describe the state of a molecule. Let the origin of the laboratory LSC coordinate system (XYZ) be placed at the center of inertia of the molecule. In this coordinate system, the direction of the molecular axis is determined by the polar angle ϑ and azimuthal angle ϕ, and the distance between the nuclei is R. Let us define the molecular coordinate system (MSC) (X ' Y ' Z ' ) rigidly connected to the molecule, the origin of which coincides with the origin of the LSC, and the Z’ axis coincides in direction with the axis of the molecule. The Euler angles describing the rotations that align the LSC and MSC can be chosen as α = ϕ, β = ϑ. The angle γ corresponding to the rotation around the molecular axis (Z’ axis) is chosen by different authors as γ = 0 (see, for example, [4]), or γ = π/2 see, for example, [1]. In this set, we choose it equal to zero. However, this angle will be discussed in a slightly different context, which will now be explained.

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The coordinates of electrons in the LSC are determined by the {ri = (xi , yi , zi )} set or in spherical coordinates {ri = (r / i', ϑi ,(ϕi' )}.' Later )/ it will be convenient to ' r = x , y , z or in spherical coordinates define these coordinates in the MSC as i i i i / ' ( ' ' ' )/ ri = ri , ϑi , ϕi , where the polar angle ϑi' is measured from the molecule axis (Z' axis), and the azimuthal angle ϕi' measures rotation about the molecular axis. Since in the case of axial symmetry of the system of electrons, the origin of this angle is not fixed by any physical conditions, it is convenient to choose the azimuthal angle of one of the electrons as the origin, say, the first, ϕ1' = γ , and the azimuthal angles of all other electrons to count from this value. In other words, the azimuthal angles of electrons are replaced by ϕ1' = γ , ϕ2' → ϕ2' − γ , ϕ3' → ϕ3' − γ , . . .

(107)

As a result of this substitution, the potential energy of electron interaction turns out to be independent of the angle γ , which is a cyclic coordinate. Therefore, we can immediately say that the dependence of the electron wave function on this angle is factorized in the form of a phase factor ei/\γ , where /\ = 0, 1, 2, . . .. That is, you can write that Ψς JM πi (q, Q) = ei/\γ ϒς/\JM πi (q, Q),

(108)

where ϒς/\JM πi (q, Q) is some new function, and the set of quantum numbers contains the value /\ in an explicit form. The result of such factorization is nothing more than a consequence of the conservation of the magnitude L z’ of the projection of the total orbital angular momentum of electrons L onto the axis of the molecule in an axially symmetric field. The quantity /\ is an important characteristic of molecular terms— each quantity |/\| corresponds to a separate molecular term: /\ = 0 − Σ-therm, /\ = 1 − ∏-term, /\ = 2 − Δ-term, etc. Therefore, below we will indicate the value /\ in the set of quantum numbers, characterizing the state of the molecule, that is, we will write Ψς/\JM πi (q, Q),. The set of {ri , R(βα)} variables includes only spatial coordinates. In addition to these coordinates, there are N e spin variables of electrons that must be added to the number of variables of the Ψς/\JM πi ({ri }R(βα)) wave function. Such a function could be used to fully describe the system of electrons and nuclei in the most general case, taking into account the spins of electrons interacting with each other and with other types of motions (orbital and rotational). However, in order to find such a function, it is necessary to write relativistic equations or at least equations of the type of Breit’s equations. Such equations for many-electron systems are not only impossible to solve, but often difficult to formulate. Therefore, as is known, in the theory of molecules, the approximation is used, within the framework of which a nonrelativistic Hamiltonian Hˆ is considered, which naturally does not include electron spins. In such a nonrelativistic formulation, the wave function Ψς/\JM πi ({ri }R(βα)) satisfies four

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247

equations that do not depend on the particle spins, Hˆ Ψς/\JM πi ({ri }R) = Eς/\J πi Ψς/\JM πi , /\

J 2 Ψς/\JM πi ({ri }R) = J (J + 1)Ψς/\JM πi ({ri }R), /\

Jz Ψς/\JM πi ({ri }R) = M Ψς/\JM πi ({ri }R), ll i Ψς/\JM πi ({ri }R) = πi Ψς/\JM πi ({ri }R).

(109)

/\

Here, Eς/\J πi is the energy of the electronic-vibrational-rotational state, the operator Pi is the operator of the inversion of spatial coordinates, and parity πi is the eigenvalue of this operator. In addition to this set of equations, as follows from (108), the function Ψς/\JM πi ({ri }R) also satisfies the equation /\

Lz' Ψς/\JM πi ({ri }R) = /\Ψς/\JM πi ({ri }R),

(110)

/\

where Lz' the operator of the projection of the electrons orbital momentum on the Z’-axis, Lz' = −i ∂γ∂ . After the function is found, the electron spins are added semi phenomenologically by multiplying the obtained electronic functions by the corresponding electron spin functions. (With subsequent antisymmetrization of these products over the permutations of the spatial and spin electronic variables) We take into account the rules for adding the total electron spin with the orbital and rotational moments of the molecule. The solution of the system of Eq. (109) is a complex problem. For the solution of it the so-called adiabatic approximation or, what is the same, the Born-Oppenheimer approximation, which we briefly summarize here (see, for example [11, 19]), is used to explain some general considerations that are used to construct the wave function taking into account the electronic angular momenta. /\

3.2 Adiabatic Approximation The adiabatic approximation is based on the idea that, since nuclear motions are characterized by significantly slower speeds than electronic ones, a system of electrons can be described assuming that the nuclei are motionless, separated by a distance R. The Hamiltonian of a system of electrons and two nuclei, A and B, can be represented as the sum of the following parts Hˆ = Tˆ e + Tˆ n + U ({ri }, R)

(111)

where U ({ri }, R) is the potential energy of the entire system of charges, RA , RB are the radius vectors of the positions of nuclei A and B in the LSC, R = | RA − RB |, T e and T n are the operators of the kinetic energy of electrons and nuclei.

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Ne Ne h2 [ h2 [ 2 ∇r2i = − ∇ '' , Tˆ e = − 2me i=1 2me i=1 ri

h2 2 h2 2 h2 ∇RA − ∇RB = − ∇R2 , Tˆ n = − (112) 2MA 2MB 2μ Ne Ne Ne [ [ [ ZB e2 e2 ZA e2 ZA ZB e2 I I. + − + U ({ri }, R) = − I |ri − RA | i=1 |ri − RB | R r − rj I i=1 i/=j=1 i Here μ = MA MB /(MA + MB ) is a reduced mass. The formula for the operator of the kinetic energy Tˆ e of electrons takes into account that this operator is invariant with respect to rotations of the electronic coordinate system. Therefore, everywhere below, we can consider the set {r’i }. Consider an equation that describes only electron motions in the field of nuclei separated by a distance R (/ / ) (/ / ) Hˆ e Fς r'i , R = ες (R)Fς r'i , R ,

(113)

where Hˆ e is the electronic Hamiltonian, the (/ energy / ) eigenvalues of which are ες (R) and the corresponding eigenfunctions Fς r'i , R have a parametric dependence on the distance R, and the subscript ζ denotes the entire set of quantum numbers that determine the quantum state at a given distance. In the (/ absence / ) of external fields, the electronic Hamiltonian Hˆ e and wave functions Fς r'i , R do not depend on the orientation of the system, determined by the angles β, α, and depend only on the internuclear distance R. For the found solution of this equation, the eigenvalue of the energy ες (R) is called the electronic term of the quasi-molecule, described by the set of quantum numbers ζ. A molecule is stable if the function ες (R) has a minimum at a certain distance between the nuclei R = Re , which is called the equilibrium internuclear distance. (/ / ) The wave functions Fς r'i , R form a complete orthonormal system of functions, / Ve'

(/ / ) (/ / ) Fς∗' r'i , R Fς r'i , R dVe' = δςς ' ,

[ ς

(/ / ) (/ / ) (/ / / /) Fς r'i , R Fς∗ r''i , R = δ r'i − r''i

(114)

Integration is ll performed over the entire configuration space of the electron Ne 3 ' system: dVe' = i=1 d ri . Therefore, the exact wave function of the molecule Ψ ζ /\JMπ i ({ri },R) can be expanded in terms of these functions, (/ / ) [ (/ / ) Ψς r'i , R = χη (R)Fη r'i , R η

(115)

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249

where χη (R) are some functions of R, and q is some quantum number or a set of quantum numbers corresponding to (corresponding) to various states that form a given superposition. We substitute this expansion into the first of Eq. (109) and take into account that ] [ [ (/ / ) [ (/ ' / ) ˆ χη (R)Fη ri , R = Fη r'i , R Tˆ N χη (R) TN η

η

[

(/ / ) h2 ∂ Fη r'i , R 2μ ∂R η [ (/ / ) + χη (R)Tˆ N Fη r'i , R

−2

∇R χη (R)

(116)

η

Then we get the following, [

) (/ / )( Fη r'i , R Tˆ N + εη (R) − Eς J πi χη (R)

η

=2

[ η

∇R χη (R)

(/ / ) [ (/ / ) h2 ∂ χη (R)Tˆ N Fη r'i , R Fη r'i , R − 2μ ∂R η

(117)

(/ / ) Multiplying Eq. (117) by Fς∗ r'i , R and integrating over the configuration space of all electrons, we obtain the following equation for the function of nuclei: ( ) [ ℵˆ ςη (R)χη (R), Tˆ N + ες (R) − Eς χς (R) = η

h2 ℵˆ ςη (R) = μ

/ Ve'

/ −

(/ / ) ∂ (/ / ) Fη r'i , R dVe' ∇R Fς∗ r'i , R ∂R (/ / ) (/ / ) Fς∗ r'i , R Tˆ N Fη r'i , R dVe'

(118)

Ve'

where the operator ℵˆ ςη (R) takes into account the "mixing" of neighboring molecular terms in the process of motion of atoms with a changing distance parameter. Equation (118) is exact. The adiabatic approximation for Eq. (118) will take into account the above-mentioned fact that the ratio of the velocity of nuclei to the velocity of electrons is a small parameter. It means that it can be taken in the vicinity of equilibrium nuclear configurations, R ≈ Re , (Re , is the distance corresponding to the minimum electron energy of the quasi-molecule) where, basically, the nuclear wave function is localized, the electron wave function does not change, and is equal to its value at R = Re . In this case, the derivatives in integral (118) vanish and the operator can be replaced by zero. That is, under the assumptions made, the admixture of neighboring terms is considered negligible in the description of the characteristics of this term.

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Now Eq. (118) has the form ( ) [ ℵˆ ςη (R)χη (R) = 0, Tˆ N + ες (R) − Eς χς (R) = η

( ) Tˆ N + ες (R) χς (R) = Eς J π i χς (R).

(119)

Equation (119) is the equation of motion of nuclei in an electronic potential ες (R). In expansion (115) of the total wave function, only one term remains, with η = ς : (/ / ) (/ / ) Ψς r'i , R = χς0 (R)Fς r'i , R

(120)

The operator of the kinetic energy of nuclei can be represented as the sum of the operator of the kinetic energy of oscillations Tˆ osc (R) and the operator Tˆ rot (R, α, β) of the kinetic energy of rotation of nuclei: Tˆ n (R, α, β) = Tˆ osc (R) + Tˆ rot (R, α, β), ( ) h2 ∂ 2 ∂ ˆ R , Tosc (R) = − 2μR2 ∂R ∂R h2 ˆ 2 R Tˆ rot (R, α, β) = − 2μR2

(121)

ˆ 2 of rotation of kernels, the rotation components Here we denote the operator R of which are determined by formulas in Cartesian and cyclic bases as follows: ) ( ) ( ∂ 1 ∂2 1 ∂ ˆ2 = − . sin β + R sin β ∂β ∂β sin2 β ∂α 2 ) ( ∂ ∂ , Rˆ x = −i y − z ∂z ∂y ) ( ∂ ∂ −x , Rˆ y = −i z ∂x ∂z ) ( ∂ ∂ Rˆ z = −i x − y ∂y ∂x ) iα ( ∂ ∂ e + i cot β Rˆ +1 = − √ ∂α 2 ∂β ∂ Rˆ 0 = −i ∂α ) −iα ( ∂ ∂ ˆR−1 = − e√ − i cot β . ∂α 2 ∂β

(122)

The described adiabatic approximation, which is important for constructing the wave function of the system in slow collision with the formation of quasi-molecular

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states, when the total orbital and spin moments of the electron shell are equal to zero, and the total angular momentum of the molecule is equal to its rotational moment, J = R. In this case, we can approximately assume that electronic motions are not associated with rotational and rotational angular momentum R is the integral of motion. The eigenfunctions and eigenvalues corresponding to this integral of motion are determined from the equations: ˆ 2 ΘRN (α, β) = R(R + 1)ΘRN (α, β) R Rˆ Z ΘRN (α, β) = NΘRN (α, β).

(123)

Solutions to Eq. (123) are spherical harmonics ΘRN (α, β) = YRN (α, β). In this case, the variables describing nuclear motions in Eq. (123) can be separated, which corresponds to the representation of the wave function χς0 (R) in the form: χς0 (R) = χς0RN (R) = uςν (R)YRN (α, β)

(124)

where the set of electronic quantum numbers ζ is defined above, the quantum number ν is vibrational, R is the angular momentum, and N is the projection of the angular momentum onto the Z-axis of the LSC. The function describes the vibrational motion of nuclei with respect to the equilibrium position for the electronic term εζ (R). This function is a solution to the equation for oscillations, which is obtained from Eq. (119) by substituting a solution in the form (124) into it and using the first of Eqs. (123). Really, (

) Tˆ osc (R) + Tˆ rot (R, α, β) + ες (R) χς (R) = EςJ π i χς (R), ( ) Tˆ osc (R) + ες (R) uςν (R)YRN (α, β) + Tˆ rot (R, α, β)uςν (R)YRN (α, β)

= Eς J π i uςν (R)YRN (α, β), ( ) h2 R(R + 1)uςν (R) = 0, Tˆ osc (R) + ες (R) − Eς J π i uςν (R) + 2μR2 [ )] ( 2 ( ) h ˆ R(R + 1) − Eς J π i − ες (R) uςν (R) = 0, Tosc (R) + 2μR2 ) ( ) ( ) 1 ∂ 2μ ( 1 2 ∂uςν (R) R + Eς J π i − ες (R) − 2 R(R + 1) uςν (R) = 0. R2 ∂R ∂R h2 R (125) Below, as an example, we present one of the frequently used solutions of the vibrational equation. Here we only note that the quantity ) ( ΔςνR = EςνR − ες (R)

(126)

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V. K. Khersonsky and E. V. Orlenko

is the eigenvalue of Eq. (125). Therefore, the total energy of a molecule in a quantum state ς νR can be expressed as ) ( EςνR = ΔςνR + ες (R)

(127)

At small values of the vibrational and rotational quantum the quantity ) ( numbers, can be expanded in powers of the quantities R(R + 1) and ν + 21 and their products. Thus, the total energy of a molecular term can be represented as Eς vR

( ) 1 + Be h2 R(R + 1) ≈εςπ i (Re ) + hωe v + 2 ( ( ) ) 1 1 2 2 2 R(R + 1) + . . . + De h (R(R + 1)) − αe v + − xe hωe v + 2 2 (128)

where ωe , Be , xe , De , αe are some of the spectroscopic constants that are designated as is customary in the literature on molecular spectroscopy. The complete electronoscillation-rotation function of the molecule is (/ / ) (/ / ) ΨςνRN r'i , R = Fς r'i , Re uςν (R)ΘRN (α, β) (129) ΘRN (α, β) = YRN (α, β). In the further discussion of molecules with nonzero electronic angular momenta, we will be mainly interested in the electronic and rotational wave functions—it will be assumed that the molecule is in the ground totally symmetric vibrational state. In the case considered above, the corresponding types of movements correspond to El ≈ ες (R) , Erot = Be h2 R(R + 1).

(130)

In many cases, the assumptions made within the adiabatic approximation are not accurate enough. Therefore, we have to use other approximations. We note here only the following, so-called, variational adiabatic approximation, which turns out to be convenient in determining the wave functions of nuclear motion in diatomic molecules. Within the framework of this approximation, the eigenenergies and eigenfunctions are sought in the process of determining the minimum of the energy functional, which is represented by an integral calculated with trial functions. Omitting details (see [4]), we immediately present the equation for the nuclear function, which is obtained in this approximation [(

Tˆ N (R)

) ςς

] + ες (R) χς (R) = EςJ χς (R),

(131)

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253

( ) is the kinetic energy operator averaged on the electron state where Tˆ N (R) ςς (/ ' / ) Fς ri , Re , ( ) Tˆ N (R)

/

ςς

=

(/ / ) (/ / ) Fς∗ r'i , Re Tˆ N (R)Fς r'i , Re dVe .

(132)

Ve

Its action on some function of nuclear coordinates g(R) can be expressed as / ( ) (/ / ) ] [ (/ / ) ˆ TN (R) g(R) = Fς∗ r'i , Re Tˆ N (R) Fς r'i , Re g(R) dVe . ςς

(133)

Ve

If an electronic term characterized by a quantum number is m—multiply degenerate, these formulas are generalized. Let the electron wave function describing this state be represented by a superposition, which is considered as a test function in the variational adiabatic approximation. m (/ / ) [ (/ / ) Ψς r'i , R = Fςλ r'i , Re χςλ (R).

(134)

λ=1

Here we have to use two indices—ζ and λ, since the index λ lists (/ its own / func) tions of the degenerate state denoted by the index ζ. The functions Fςλ r'i , Re are orthonormalized by the conditions /

(/ ' / ) (/ / ) ∗ ri , Re Fςλ' r'i , Re dVe . = δλλ' Fςλ

(135)

Ve

and the functions χςλ (R) satisfy the following normalization conditions m / [

I I Iχςλ (R)I2 dVN = 1,

(136)

VN

λ=1

where the integration is performed over all nuclear variables. This condition follows directly from the condition for normalizing the total wave function (134) to unity. The equation for determining functions χςλ (R) has the form of a secular equation, m [( ) [ Tˆ N (R) λ' =1

( ) the values Tˆ N (R)

ςλςλ'

ςλςλ'

+ ες (R)δ

λλ'

] χςλ' (R) = 0,

(137)

are matrix elements, which are determined by their action

on some function g(R) of nuclear coordinates as

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V. K. Khersonsky and E. V. Orlenko

( ) Tˆ N (R)

/ ςλςλ

g(R) = '

(/ ' / ) (/ / ) ] [ ∗ ri , Re Tˆ N (R) Fςλ' r'i , Re g(R) dVe . Fςλ

(138)

Ve

Below we use this approximation when discussing the properties of the nuclear wave function.

3.3 Electronic Wave Function Symmetry (/ / ) (A) Heteronuclear molecules. The electron wave function Fς r'i , Re is an eigenfunction of the operators He and LZ ' and is defined as the solution of the following two equations /\

/\

(/ / ) (/ / ) He Fς r'i , R = ες (Re )Fς r'i , R , (/ / ) (/ / ) LZ ' Fς r'i , R = /\Fς r'i , R . /\

(139)

/\

As noted above, the second of these equations follows directly from the conservation of the projection of the total orbital angular momentum of the electrons on the symmetry axis (Z’axis) of the molecule, which corresponds to the axially symmetric distribution of electrons relative to the axis of the diatomic molecule. The eigenvalue /\ of the corresponding operator LZ ' takes integer values /\ = 0, 1, 2,…. Therefore, in the set of quantum numbers (/ η, / you ) can select this value and write the electron wave function in the form Fς/\ r'i , R instead of Eq. (115). In particular, an expression similar to formula Eq. (108) can be rewritten as /\

(/ / ) (/ / ) 1 Fς/\ r'i , R = √ ei/\γ fς/\i r'i , R . 2π

(140)

A multiplier √12π has been added to ensure correct normalization on the angle γ. The solution of Eq. (139) for a multi-electron system is a complex problem. Methods for solving this type of problem are discussed in quantum chemistry and will not interest us here. However, we will need to use some of the symmetry properties of electronic functions that can be understood without trying to get these functions explicitly. This can be done using the requirements for the electron wave function imposed on it by the symmetry of the electron system. Indeed, due to the axial symmetry of the system, the Hamiltonian He is invariant with respect to three transformations: an identical transformation E , two rotations—one Cω at an arbitrary angle ω, and the second C−ω at an angle −ω around the molecule axis, and reflection σν in any plane passing through the molecular axis. These three transformations form a group C∞ν . The characters of irreducible representations are given in Table 3. The requirements are imposed on the wave electronic function, which are that when this function is acted upon by operations from a group, it must transform /\

/\

/\

/\

/\

Elements of Theory of Angular Moments as Applied to Diatomic Molecules … Table 3 Characters of irreducible group representations

/\

/\

255 /\

C∞ν

E

2 Cω

σν

Σ+

1

1

1

Σ−

1

1

−1

∏

2

2cosω

0

Δ

2

2cos2ω

0

Φ

2

2cos3ω

0

…

…

…

…

according to the irreducible representations of this C∞ν group. The first two representations are one-dimensional and correspond to the case /\ = 0 (terms Σ + and Σ − ). This means that the character is equal to the matrix element of the transformation, that is, Σ+ :

(/ / ) (/ / ) E Fς/\=0 r'i , R = Fς/\=0 r'i , R , (/ / ) (/ / ) Cω Fς/\=0 r'i , R = Fς/\=0 r'i , R , (/ / ) (/ / ) σν Fς/\=0 r'i , R = Fς/\=0 r'i , R ; Σ− : (/ / ) (/ / ) E Fς/\=0 r'i , R = Fς/\=0 r'i , R , (/ / ) (/ / ) Cω Fς/\=0 r'i , R = Fς/\=0 r'i , R , (/ / ) (/ / ) σν Fς/\=0 r'i , R = −Fς/\=0 r'i , R

/\

/\

/\

(141)

/\

/\

/\

Here was taken into account that for /\ = 0 from (140) we have (/ / ) (/ / ) (/ / ) 1 1 σν Fς 0 r'i , R = √ σν fς 0 r'i , R = ± √ fς 0 r'i , R → Σ ± . 2π 2π /\

/\

(142)

The irreducible group C∞ν representations for terms /\ > 0 are two-dimensional. First, this means that the states with /\ > 0 are doubly degenerate. States with +|/\| and −|/\| have the same energy (they are split(/ by /relativistic interactions—see below). To ) understand how the wave functions Fς/\ r'i , R are transformed during operations from group C∞ν , it is necessary to find out how the basis of a two-dimensional irreducible representation is transformed. It is easy to see that the components x' and y' of some arbitrary vector ρ(x' , y' ) lying in the X' Y ’plane form such a basis. Indeed, let us express these components in cylindrical coordinates x' = ρ ' cos φ, __y' = ρ ' sin φ, where φ is some angle, and consider the cyclic components ) ρ' ρ' ( ' x±1 = ∓ √ x' ± iy' = ∓ √ e±iφ . 2 2

(143)

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Obviously, the same components considered in a coordinate system rotated by an angle ω around the Z' axis can be expressed as ρ' ' ' ' x±1 = Cω x±1 , (ω) = ∓ √ e±i(φ+ω) = e±iω x±1 2

(144)

( ' ) ( +iω )( ' ) ) ' 0 x+1 x+1 e x+1 (ω) = ⇒ = C ω ' ' ' x−1 x−1 x−1 0 e−iω (ω) ) ( +iω 0 e Cω = . 0 e−iω

(145)

or (

/\

/\

/\

The character χ (ω) of the irreducible representation Cω is determined as χ (ω) = SpCω , then /\

/\

χ (ω) = SpCω = Sp

(

e+iω 0 0 e−iω

)

= e+iω + e−iω = 2 cos ω.

(146)

It corresponds to the case /\ = 1 (i.e., for ∏-states). In order to construct a basis for an irreducible representation in the case /\ ' = > 1, one can proceed in the same way, forming from vector components x±1 / ) ) ( ( π π i/\φ ' 4π ρ 3 Y1±1 2 , φ constructions of the type of spherical functions YL/\ 2 , φ ∝ e , corresponding to the symmetry of molecular state, YL/\

(π 2

)

,φ =

/

2L + 1 (L + /\)!(L − /\)!× 4π ( ' )(L+/\)/2 ( ' )(L−/\)/2 [ x−1 x+1 1 ' ρ' ((L + /\)/2)!((L − /\)/2)! ρ (L + /\) > 0, (L − /\) > 0 (147) 1

2L/2

' Thus, jo vector components will have the form x±1 = ( π the) considered construction i/\φ ρYL/\ 2 , φ = /\/\ (φ) = /\/\ (0)e . Then the required transformation is

Cω /\/\ (φ) = /\/\ (φ + ω) = /\/\ (0)ei/\(φ+ω) = /\/\ (φ)ei/\ω , ⇒ Cω = ei/\ω ) ( i/\ω 0 e , Cω = 0 e−i/\ω /\

/\

χ (ω) = SpCω = 2 cos /\ω.

(148)

Elements of Theory of Angular Moments as Applied to Diatomic Molecules …

257

Since the basis of the considered two-dimensional irreducible representation transforms the operator Cω , the requirement imposed on the wave function (/ / with ) Fς/\ r'i , R by the symmetry of the system is that it must be transformed by the operator according to (148). Let’s go back to formula (107) and replace the angles as follows: /\

ϕ1' = γ ⇒ ϕ1' + ω = γ + ω, ϕ2' → ϕ2' − ϕ1' = ϕ2' − γ , ⇒ ϕ2' + ω − γ − ω = ϕ2' − γ ϕ3' → ϕ3' − γ ⇒ ϕ3' − γ , . . .

(149)

then the difference the angles ϕi' − ϕ1' , which are the arguments of the (/ ' / between ) function fς/\i ri , R , will not change, while the angle ϕ1' ϕ 1 also becomes equal to γ + ω. Therefore, the considered transformation takes the form (/ / ) (/ / ) (/ / ) 1 Cω Fς/\ r'i , R = √ ei/\(γ +ω) fς/\i r'i , R = ei/\ω Fς/\ r'i , R , 2π or (/ / ) ) ( i/\ω (/ / ) ) ( )( 0 Fς/\ (/r'i /, R ) Fς/\ (/r'i /, R ) e Cω = Fς −/\ r'i , R Fς −/\ r'i , R 0 e−i/\ω (/ ' / ) ) ( i/\ω r ,R e F =, −i/\ω ς/\ (/i ' / ) . Fς −/\ ri , R e /\

/\

(150)

(/ / ) Indeed, the functions Fς/\ r'i , R are transformed as basic functions of the irreducible representation /\/\ (φ) corresponding to the quantity |/\|. The action of the reflection operator σν in the plane (x ' z' ) on the basis functions is reduced to the change of variables /\

x' → x' , y' → y' , z → z '

(151)

then, (

) ( ) ( ' ) ( ' ) )( ' ' x+1 0 −1 −x−1 (φ) x+1 (−φ) x+1 (φ) (φ) = = = , ' ' ' ' x−1 x−1 −x+1 x−1 −1 0 (φ) (φ) (φ) (−φ) (152) ( ) 0 −1 σν = . −1 0 /\

σν

/\

Reflection in the plane containing the molecular axis changes the sign of the angle. For example, for the case |/\|=1, acting by the reflection operator σν on (143), we have /\

258

V. K. Khersonsky and E. V. Orlenko

) ρ' ρ' ( ' x±1 = ∓ √ x' ± iy' = ∓ √ e±iφ . 2 2 ( ' ) ) ( )( √ρ ' +iφ ) ( √ρ ' −iφ ) ( ' − 2e − 2e x+1 (−φ) x (φ) 0 −1 = = σν +1 . = ' ' ' ' x−1 x−1 −1 0 (−φ) (φ) + √ρ 2 e+iφ + √ρ 2 e−iφ (153) /\

The same can be seen in more complex constructions (147), σν YL/\

/ ) 1 2L + 1 , φ = L/2 (L + /\)!(L − /\)!× 2 2 4π [ 1

(π

(

' σν x+1

)(L+/\)/2 (

ρ'

' σν x−1

)(L−/\)/2

ρ'

((L + /\)/2)!((L − /\)/2)! (L + /\) > 0, (L − /\) > 0 / 2L + 1 1 = L/2 (L + /\)!(L − /\)!× 2 4π ( ' )(L+/\)/2 ( ' )(L−/\)/2 [ x+1 x−1 (−1)L

ρ' ((L + /\)/2)!((L − /\)/2)! ρ ' (L + /\) > 0, (L − /\) > 0 / 2L + 1 1 = L/2 (L + /\)!(L − /\)!× 2 4π ( ' )(L−/\)/2 ( ' )(L+/\)/2 [ x+1 x−1 (−1)L (−1)/\−L ρ' ((L + /\)/2)!((L − /\)/2)! ρ ' /\ ↔ −/\ (L + /\) > 0, (L − /\) > 0 / (−1)/\ 2L + 1 = L/2 (L + /\)!(L − /\)!× 2 4π ( ' )(L−/\)/2 ( ' )(L+/\)/2 [ x+1 x−1 1 /\ ↔ −/\ (L + /\) > 0, (L − /\) > 0

((L + /\)/2)!((L − /\)/2)!

ρ'

ρ'

= (−1)/\ YL−/\

(π 2

) ,φ ,

(154) ⇒ σν YL/\

) ) (π , φ = YL/\ , −φ ∝ e−i/\φ . 2 2

(π

From (148) it follows that the electron wave function should change the signs of the azimuthal angles of the electrons during the reflection operation (recall that ϕ1' = γ ), (/ / ) (/ / ) σν Fς/\ r'i , R = σν Fς/\ ri , ϑi' , ϕi' , R

Elements of Theory of Angular Moments as Applied to Diatomic Molecules …

[ (/ / ) ] 1 = √ σν fς/\ ri , ϑi' , ϕi' − ϕ1' , R ei/\γ 2π (/ / ) 1 = √ fς/\ ri , ϑi' , ϕi' − ϕ1' , R e−i/\γ . 2π

259

(155)

On the other hand, when reflected in a plane containing the molecular axis, the MSC is transformed from right to left and vice versa. This means that the direction of the axis of the molecule changes, or, equivalently, the sign of the projection of the orbital momentum (/ / LZ) on the axis of the molecule, /\ → −/\. Therefore, the wave function Fς/\ r'i , R should change during reflection as follows: /\

(/ / ) (/ / ) (/ / ) 1 σν Fς/\ r'i , R = Fς −/\ r'i , R = √ fς −/\ ri , ϑi' , ϕi' − ϕ1' , R e−i/\γ . (156) 2π /\

Comparing (155) and (156), we get that (/ / ) (/ / ) fς/\ ri , ϑi' , ϕi' − ϕ1' , R = fς −/\ ri , ϑi' , ϕi' − ϕ1' , R , ⇒ (/ / ) (/ / ) σν fς/\ ri , ϑi' , ϕi' − ϕ1' , R = fς −/\ ri , ϑi' , ϕi' − ϕ1' , R . /\

(157)

We also note that the electron wave function has a similar symmetry with respect to complex conjugation [4], (/ ' / ) (/ / ) ∗ Fς/\ ri , R = Fς −/\ r'i , R , (/ / ) (/ / ) ∗ ri , ϑi' , ϕi' − ϕ1' , R = fς −/\ ri , ϑi' , ϕi' − ϕ1' , R . fς/\

(158)

Formally, it should be added that the identical transformation does not change the wave function for any /\, however, the characters of these transformations are equal to the dimensions of the representations, and that is, they are equal to 2 for, /\ > 0 as indicated in the table. We summarize the results obtained as follows: (/ / ) (/ / ) E Fς/\ r'i , R = Fς/\ r'i , R , Σ± : (/ / ) (/ / ) Cω Fς/\=0 r'i , R = Fς/\=0 r'i , R , (/ / ) (/ / ) (159) σν Fς/\=0 r'i , R = ±Fς/\=0 r'i , R ; /\

/\

/\

∏, Δ, Φ : (/ / ) (/ / ) Cω Fς/\ r'i , R = ei/\ω Fς/\ r'i , R , (/ / ) (/ / ) σν Fς/\ r'i , R = +Fς −/\ r'i , R ,

/\

/\

and also property (158).

260

V. K. Khersonsky and E. V. Orlenko

(B) Homonuclear molecules. So far, we have considered the symmetry properties of the electron wave function of heteronuclear molecules. If a molecule has identical nuclei, then the distribution of electron density in the shell of the molecule is symmetric with respect to the operation of reflection in the plane perpendicular to the axis of the molecule and dividing the internuclear distance in half (let’s call it the plane H). This operation is denoted as σh . The Hamiltonian of a homonuclear molecule also has symmetry with respect to the inversion of space Pi , and with respect to rotations by an angle π relative to any axis lying in the plane H and crossing the axis of symmetry of the molecule in its middle. We will refer to this operation as C’2 . The combination of reflection σh and rotation by an angle π around an axis of symmetry gives an inversion, that is, σh C2' = Pi . The combination of inversion Pi and rotation C2' gives a reflection σh , that is, i.e., Pi C2' = σh . All the listed elements, together with ˆ form a group, D∞h which can be represented as the the identity transformation, E, result of the group multiplication of the group C∞ν and the group CI of the inversion (containing two elements—Eˆ and Pi ), D∞h = C∞ν × CI . This means that each irreducible representation of the group C∞ν creates two irreducible representations of the group D∞h , which differ in the parities of the wave functions with respect to the coordinate inversion in the MSC. All electronic states are divided into even ones, denoted by the subscript g (from the German word gerade, that means even), and odd ones, by the subscript u (from the German word ungerade, that is odd). The functions of even states do not change when inverted, while functions of odd states change sign. The characters of the irreducible representations of the group D∞h are presented in Table 4. Further discussion of the symmetry of wave functions associated with inversion will be carried out below in the context of considering the properties of the total wave functions of diatomic molecules. /\

/\

/\

/\ /\

/\

/\

/\

/\

/\/\

/\

/\

Table 4 The characters of the irreducible representations of the group D∞h /\

/\

/\

/\

/\/\

Pi

Pi Cω

Pi σν = C2'

1

1

1

1

1

−1

−1

−1

/\

D∞ν

E

2 Cω

σν

Σg+

1

1

Σu+ Σg− Σu−

1

1

/\

/\

1

1

−1

1

1

−1

1

1

−1

−1

−1

1

∏g

2

2cosω

0

2

2cosω

0

∏u

2

2cosω

0

−2

−2cosω

0

Δg

2

2cos2ω

0

2

2cos2ω

0

Δu

2

2cos2ω

0

−2

−2cos2ω

0

Φg

2

2cos3ω

0

2

2cos3ω

0

…

…

…

…

…

…

…

Elements of Theory of Angular Moments as Applied to Diatomic Molecules …

261

3.4 The Angular Momentum of the Molecule. Accounting the Orbital Angular Momentum of Electrons Below, when defining wave functions, we need formulas expressing the projections of the total angular momentum of a molecule on the LSC and MSC axes. In those cases when the total spin moment of the electron shell S is equal to zero, the angular momentum of the molecule N is determined by the sum of the rotational moment R and the total orbital moment of the electrons L. When calculating this sum, the following should be taken into account. The rotational angular momentum of a diatomic molecule is directed perpendicular to the axis of the molecule and is determined by two rotational coordinates, α and β in the LSC. The angle γ is often assumed to be equal to π/2 [1] or zero [4]. We will assume γ = 0. The average value of the total orbital angular momentum L of electrons is a vector directed along the axis of the molecule, and its value is determined by the eigenvalue of the operator LZ ' = −i ∂γ∂ of projection onto the axis of the molecule (axis Z' MSC), the direction of which n(α, β) is determined by the angles in the LSC. Thus, in order to perform the addition required to calculate the moment N, it is necessary to project a vector L = LZ ' n(α, β) on the LCS axis. This is easily done by pointing the molecular axis along the Z ’axis. Then the projection of the vector L onto the line of nodes LP is equal to LP = LZ ' / sin β and onto the Z axis—LP = LZ ' / cos β. Next, you need to project the resulting value on the X and Y axes, multiplying it by and, respectively. As a result, we get that /\

/\

/\

/\

/\

/\

cos α ∂ cos α , LZ ' = −i Lˆ X = sin β ∂γ sin β sin α ∂ sin α , LZ ' = −i Lˆ Y = sin β ∂γ sin β 1 ∂ 1 LZ ' = −i . Lˆ Z = cos β cos β ∂γ /\

/\

(160)

/\

Let us now consider the Cartesian components of the total angular momentum operator N. As clearly follows from above said, the sum of the operators of the rotational R and the orbital momenta L, which forms the operator of the total angular momentum N, must take into account the following rotations: ˆ (1) rotation of RZ (α) around the Z-axis by an angle α (operator R), ˆ (2) rotation RY1 (β) around the new Y1 axis by the angle β (operator R), (3) rotation of RZ (γ ) around the new Z-axis by the angle γ (operator LZ ). /\

However, this is exactly the set of rotations, which forms the complete operator of rotation of the coordinate system [14]. Therefore, to obtain the Cartesian components of this operator, one can immediately use formulas [14], which we represent here in the form

262

V. K. Khersonsky and E. V. Orlenko

) ( ) 1 ( cos α ∂ ∂ ∂ NX = √ N−1 − N+1 = i sin α −i , + cotβ cos α ∂β ∂α sin β ∂γ 2 ' '' ' ' '' ' /\

/\

/\

LX

RX

) ( ) i ( sin α ∂ ∂ ∂ NY = √ N−1 + N+1 = i − cos α −i , (161) + cotβ sin α ∂β ∂α sin β ∂γ 2 ' '' ' ' '' '

/\

/\

/\

LY

RY

∂ NZ = N0 = −i . ∂α

/\

/\

Comparison of the terms proportional to the derivatives with respect to the angle γ shows that these are precisely the terms that contribute to the orbital angular momentum of the electrons LZ ' . On the other hand, the terms enclosed in brackets ˆ [see coincide with the expressions for the purely rotational angular momentum R formulas (122)], if the angle γ is chosen equal to a constant value (γ = 0). Thus, expressions (161) can be rewritten as /\

(

) ∂ ∂ cos α + cotβ cos α + LZ ' , NX = RX + LX = i sin α ∂β ∂α sin β ( ) ∂ ∂ sin α + cotβ sin α + LZ ' , NY = RY + LY = i − cos α ∂β ∂α sin β ∂ NZ = RZ = −i . ∂α /\

/\

/\

/\

/\

/\

/\

/\

/\

(162)

/\

/\

The square operator for the total angular momentum, N 2 , can be represented by the formula /\

/\

/\

/\

N 2 =NX2 + NY2 + NZ2 [ 2 ( 2 )] 2 ∂ ∂ ∂ 1 ∂ ' ' =− + cot β − 2i cos βL + − L . Z Z ∂β 2 ∂β ∂α sin2 β ∂α 2 /\

/\

(163)

The projections of the operator of the total angular momentum on the MSC axis, NX ' ,NY ' ,NZ ' can be obtained by transforming the projections of this operator on the LSC axis using the rotation matrix. The result of such a transformation for cyclic contravariant components is presented by the following formulas /\

/\

/\

'±1

/\

N

( ) '0 i ∂ 1 ∂ = √ ±i cot βLZ ' + i ∓ , __N = Lˆ Z ' , ∂β sin β ∂α 2 /\

/\

We use these formulas for γ = 0 to obtain the desired projections

(164)

Elements of Theory of Angular Moments as Applied to Diatomic Molecules … /\

i ∂ + cotβLZ ' , sin β ∂α ∂ =i , ∂β /\

NX ' = /\

NY ' /\

263

(165)

/\

NZ ' = LZ ' . /\

Note that [4] the operator N 2 is expressed in terms of these projections by the following formula /\

/\

/\

/\

/\

N 2 = NX2 ' + NY2' − NY ' cot β + NZ2'

(166)

3.5 Equation for the Nuclear Wave Function Since in the case when the total electron spin of a molecule is zero, the projection of the total orbital angular momentum of electrons onto the axis of the molecule is an integral of motion, we will explicitly indicate the value /\ in the set of quantum numbers characterizing the quantum state. The quantity /\ numbers the electronic terms of a diatomic molecule. Above, when describing the symmetry of the wave function, it was already noted that the states of /\ = 0 (Σ-terms) under symmetry transformations from a group C∞ν are transformed according to one-dimensional irreducible representations of this group. This means that the Σ-terms are nondegenerate. The states with /\ = 1, 2, 3,… (∏, Δ, Φ…) are transformed according to two-dimensional representations of the group C∞ν , that is, they are doubly degenerate. States characterized by quantities +|/\| and −|/\| correspond to the same energy (in the absence of an external field, the energy cannot depend on whether the molecule axis is directed along or against the projection of the orbital angular momentum on this axis). These features must be taken into account when formulating equations for the nuclear wave function. As noted in Sect. 3.2, the equation for the nuclear wave function (131), obtained in the variational adiabatic approximation, contains the operator of the kinetic energy nuclei averaged over a given electronic state. We will rewrite this equation here, highlighting the dependence on the quantum number. For Σ-terms this equation has the form [( ] ) ˆ + ες 0 (R) χς0 (R) = Eς 0 χς 0 (R), (167) TN (R) ς 0ς 0

For terms corresponding to /\ > 0, due to the presence of degeneracy, it is necessary to consider the following system of equations

264

V. K. Khersonsky and E. V. Orlenko

[( ) Tˆ N (R)

ς/\ς/\

] ( ) + ες |/\| (R) χς/\ (R) + Tˆ N (R)

ς/\ς −/\

= Eς |/\| χς/\ (R), [( ( ) ) χς/\ (R) + Tˆ N (R) Tˆ N (R) ς−/\ς/\

ς−/\ς −/\

χς −/\ (R)

] + ες |/\| (R) χς −/\ (R)

(168)

= Eς |/\| χς−/\ (R). As we have already noted, the energy of the electronic term in the case under consideration cannot depend on the sign of the /\ value. Therefore, ες/\ = ες −/\ = ( ) in these formulas ες |/\| and Eς/\ = Eς −/\ = Eς |/\| . The operators Tˆ N (R) ' ς/\ς/\

are determined by their action on the nuclear coordinate functions by the following expression /

( ) Tˆ N (R)

ς/\ς/\'

=

(/ ' / ) (/ / ) ∗ ri , Re Tˆ N (R)Fς/\' r'i , Re dVe . Fς/\

(169)

Ve

It is assumed that the electronic wave functions are normalized by condition Eq. (135). The reduction of Eqs. (168) and (169) to the forms that can be used to determine the nuclear wave function is considered in more detail (/ / in [4]. ) We define the electronic wave functions Fς/\ r'i , Re in MSC. Therefore, the ( )2 ˆ − Lˆ , which determines the square of the angular momentum of rotation R2 = N operator of the kinetic energy of rotation of nuclei, Tˆ rot , must be represented by the ˆ and orbital momentum operators of the projections of the total angular momentum N ˆ L of electrons on the axis of the MSC. This representation can be obtained if we take into account that the operator of the projection of the total angular momentum onto the axis of the molecule NZ ' is equal to the operator of the projection of the orbital angular momentum LZ ' onto the same axis (because the angular momentum of rotation R is perpendicular to this axis). Then we can write that /\

/\

/\

/\

( )2 ( ) ' ' ˆ − Lˆ = N ˆ 2 + L2X ' + L2Y ' + L2Z' − 2LZ ' NZ ' + 2 L'+1 N−1 R2 = N + L'−1 N+1 , /\

/\

/\

/\

/\

/\/\

/\/\

/\/\

/\

Taking into account NZ ' = LZ ' we can write ) ( ( )2 ' ' ˆ − Lˆ = N ˆ 2 + L2X ' + L2Y ' − L2Z' + 2 L'+1 N−1 . + L'−1 N+1 R2 = N /\

/\

/\

/\

/\

/\

/\/\

/\/\

(170)

Here we denote by L'μ and Nν' the cyclic components of the moment projection ˆ on the MSC axis. operators Lˆ and N Now the operator Tˆ N of the kinetic energy of nuclei can be represented in the form

Elements of Theory of Angular Moments as Applied to Diatomic Molecules …

( ) h2 2 h2 ∂ 2 ∂ ˆ R + R TN = − 2μR2 ∂R ∂R 2μR2 ( ) ∂ h2 ∂ R2 =− 2μR2 ∂R ∂R )] ( h2 [ ˆ 2 2 2 2 ' ' ' ' N + + L + L − L . + 2 L N + L N +1 −1 −1 +1 X' Y' Z' 2μR2

265

/\

/\

/\

/\

/\/\

/\/\

(171)

After substituting this operator into integral (169), it can be represented as a sum of three terms, (

Tˆ N (R)

) ς/\ς/\'

/\

/\

/\

= A ς/\/\' (R) + B ς/\/\' (R) + C ς/\/\' (R, α, β, γ ),

(172)

where h2 A ς/\/\' (R) = − 2μR2 /\

h2 B ς/\/\' (R) = 2μR2

/\

/

/ Ve

∗ Fς/\

( ) (/ ' / ) ∂ (/ / ) 2 ∂ R Fς/\' r'i , R dVe , ri , R ∂R ∂R

) (/ ' / )( 2 (/ / ) ∗ ri , R LX ' + L2Y ' Fς/\' r'i , R dVe , Fς/\ /\

/\

Ve /\

C ς/\/\' (R, α, β, γ ) / )) ( (/ / ) (/ ' / )( 2 h2 ∗ ' ' ' ˆ − L2Z' + 2 L'+1 N−1 = Fς/\' r'i , R dVe . + L N r , R N F ς/\ i −1 +1 2μR2 /\

/\/\

/\/\

Ve

(173) Following [5], we clarify the properties of these integrals. First of all, it can be argued that all off-diagonal elements with /\' = −/\ are equal to zero. To show this, we act by the symmetry operator Cω on the azimuthal angles of the electrons in the integrands of the integrals (173). This should not change the values of the integrals. As you can see, the action of the symmetry operator Cω on the operators in the integrands either does not change them, or multiplies these operators by e±iω . Indeed, the operator ( in the )first integral does not depend on electronic variables at all. The operator L2X ' + L2Y ' in the second integral is a real quantity that expresses the square of the projection of the orbital momentum vector onto the plane perpendicular to the axis of symmetry—this quantity in an axially symmetric field does not depend on the azimuthal angles of the radius vectors of electrons. The operators of the cyclic projections of the orbital momenta L'+1 and /\

/\

/\

/\

/\

/\

/\

L'−1 change under the action of the operator Cω [14],

266

V. K. Khersonsky and E. V. Orlenko /\/\

/\

/\/\

/\

Cω L'+1 = e+iω L'+1 , Cω L'−1 = e−iω L'−1 /\/\

/\/\

(174) /\

Cω L'0 = Cω L'Z ' = L'Z ' , /\

' ˆ 2 and N±1 do not depend on electronic variables. Therefore, in the and the operators N third integral, ( ) the action of the symmetry operator Cω does not change the operator 2 2 ˆ N − LZ' , while the term containing the operators L'+1 and L'−1 changes as /\

/\

/\

/\

( ) ' ' ' ' Cω L'+1 N−1 = e+iω L'+1 N−1 + e−iω L'−1 N+1 . + L'−1 N+1 /\

/\/\

/\/\

/\/\

/\/\

(175)

/\

We also recall that upon symmetry transformation Cω , the electronic wave functions acquire a phase factor ei/\ω , (/ / ) (/ / ) Cω F∗ς/\ r'i , R = e−i/\ω Fς/\ r'i , R , (/ / ) (/ / ) Cω Fς −/\ r'i , R = e−i/\ω Fς −/\ r'i , R , /\

/\

(176) /\

and the entire integral is multiplied by e−2i/\ω . Thus, if we transform the symmetry Cω of the electronic coordinates in the integrands of integrals (173), then these integrals will change as follows: /\

/\

A ς/\−/\ (R) → e−2i/\ω A ς/\−/\ (R), /\

/\

B ς/\−/\ (R) = e−2i/\ω B ς/\−/\ (R)

/\

C ς/\−/\ (R, α, β, γ ) → ) 2 / (/ ' / )( 2 (/ / ) ∗ −2i/\ω h ˆ − L2Z' Fς −/\ r'i , R dVe + e F r , R N ς/\ i 2 2μR Ve / ) (/ ' / )( ' (/ / ) h2 ∗ ' +e−i(2/\−1)ω 2 Fς −/\ r'i , R dVe + ri , R L+1 N−1 Fς/\ μR Ve / ) (/ ' / )( ' (/ / ) h2 ∗ ' +e−i(2/\+1)ω 2 ri , R L−1 N+1 Fς/\ Fς −/\ r'i , R dVe . μR Ve /\

(177)

/\/\

/\/\

It is easy to see, none of the additional phase factors turns into unity at /\ = 1, 2, 3… This contradicts the statement that the symmetry operation acting on the integrand does not change the value of the integral. Consequently, the considered matrix elements are equal to zero (at /\' = −/\). A formula similar to (176) for the case /\' = /\ can be represented as

Elements of Theory of Angular Moments as Applied to Diatomic Molecules …

267

/\

C ς/\/\ (R, α, β, γ ) → / ) (/ ' / )( 2 (/ ' / ) h2 ∗ 2 ˆ r , R N F − L ς/\ i Z' Fς/\ ri , R dVe + 2 2μR /\

Ve

h2 +eiω 2 μR

/

) (/ / ) (/ ' / )( ' ∗ ' Fς/\ r'i , R dVe + ri , R L+1 N−1 Fς/\ /\/\

Ve

h2 +e−iω 2 μR

/

(178)

) (/ / ) (/ ' / )( ' ∗ ' Fς/\ r'i , R dVe . ri , R L−1 N+1 Fς/\ /\/\

Ve

Using the same reasoning as in the case /\' = −/\, we can conclude that the second and third integrals in this expression should vanish. Therefore, the integral C ς/\/\ (R, α, β, γ ) can be redefined as follows: /\

/\

C ς/\/\ (R, α, β, γ ) → /\

C ς/\ (R, α, β, γ ) =

h2 2μR2

/

) (/ ' / )( 2 (/ / ) ∗ ˆ − L2Z' Fς/\ r'i , R dVe . ri , R N Fς/\ /\

(179)

Ve /\

Here we also omitted one of the indices in the notation C ς/\ (R, α, β, γ ), since only the diagonal terms are nonzero. Changing in the same way the notation for integrals A ς/\/\' (R) and B ς/\/\' (R) (173), we write that /\

/\

/\

A ς/\ (R) = −

h2 2μR2

h2 B ς/\ (R) = 2μR2

/\

/

/ Ve

( ) (/ ' / ) ∂ (/ / ) ∂ ∗ R2 Fς/\ r'i , R dVe , ri , R Fς/\ ∂R ∂R

) (/ / ) (/ ' / )( 2 ∗ ri , R LX ' + L2Y ' Fς/\ r'i , R dVe , Fς/\ /\

/\

(180)

Ve

After that, formula (172) takes the form ( ) Tˆ N (R)

/\

ς/\ς/\

/\

/\

= A ς/\ (R) + B ς/\ (R) + C ς/\ (R, α, β, γ ),

Let’s look at each of the three operators in this formula. These operators act on the nuclear function χς/\JM πi (R). Consequently, the result of the action of the first of them, by successive differentiation and using the relation of orthogonality of electronic wave functions (135), is reduced to the form

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V. K. Khersonsky and E. V. Orlenko

[ ( ) ∂ h2 (2) 2 ∂ R + Gς/\ Aς/\ (R)χς/\JM πi (R) = − (R) 2μR2 ∂R ∂R . ( )] ∂ 1 (1) +Gς/\ + χς/\JM πi (R), (R) ∂R R /\

(181)

where (1) Gς/\ (R) =

/ Ve

(2) Gς/\ (R) =

/

Ve

(/ ' / ) ∂ (/ / ) ∗ Fς/\ r'i , R dVe , ri , R Fς/\ ∂R (182)

(/ ' / ) ∂ 2 (/ ' / ) ∗ ri , R ri , R dVe Fς/\ F ς/\ ∂R2 /\

Using the properties of orthogonality and symmetry σν of the electronic wave (1) functions, it is easy to show that Gς/\ (R) = 0. Indeed, we differentiate expression (135) with respect to R. Then we obtain /

(/ / ) (/ ' / ) ∂ ∗ Fς/\ r'i , R dVe ri , R Fς/\ ∂R Ve [ ] / (/ / ) ∂ ∗ (/ ' / ) Fς/\ ri , R Fς/\ r'i , R dVe = 0, ⇒ + ∂R

(183)

Ve

(1) Gς/\ (R)

(1)∗ + Gς/\ (R) = 0. /\

Let us act on the integrand of the first of these integrals by the operation σν , which should not change the value of the integral. Considering that (/ / ) (/ / ) (/ ' / ) ∗ ri , R , σν Fς/\ r'i , R = Fς −/\ r'i , R = Fς/\ (/ ' / ) (/ ' / ) ∗ σν Fς/\ ri , R = Fς/\ ri , R , /\

/\

(184)

(1) we obtain that both integrals in (183) are equal to each other. Hence Gς/\ (R) = 0. The (2) second integral Gς/\ (R) can be transformed by the second differentiation of equality (183) with respect to R, which gives the following equality

Elements of Theory of Angular Moments as Applied to Diatomic Molecules …

/ Ve

(/ / ) ∂ ∗ (/ ' / ) ∂ r ,R F Fς /\ r'i , R dVe + ∂R ς /\ i ∂R / [

+ /

+ Ve

Ve

(/ / ) ∂ 2 (/ / ) Fς∗/\ r'i , R Fς /\ r'i , R dVe + ∂R2

∂2 ∂R

Ve

] (/ ' / ) (/ / ) ∗ F r , R Fς /\ r'i , R dVe i 2 ς /\

/

269

(/ / ) ∂ ∗ (/ ' / ) ∂ r ,R F Fς /\ r'i , R dVe = 0 ⇒ ∂R ς /\ i ∂R

(2) (2)∗ Gς /\ (R) + Gς /\ (R) + 2

I / I I ∂ (/ / )I2 I Fς /\ r' , R I dVe = 0. i I ∂R I

Ve

(185) (2)∗ If in the second integral, Gς/\ (R), we again use the transformation of electronic wave functions induced by the operator σν , then, as in the previous case, when (1) analyzing the integral Gς/\ (R), the use of transformations (184) shows that the second (2)∗ (2) integral in this equality is equal to the first, Gς/\ (R) = Gς/\ (R). Hence /\

(2) Gς/\ (R)

I / I I ∂ (/ ' / )I2 (2) I I =− I ∂R Fς/\ ri , R I dVe = Gς |/\| (R).

(186)

Ve

Substitution of the value /\ instead of |/\| in the subscripts of the function under consideration emphasizes that this function does not depend on the sign of /\. Thus, the operator A ς/\ (R) can be represented in the following form, /\

( ) h2 ∂ 2 ∂ R + Dς |/\| (R), 2μR2 ∂R ∂R I / I I ∂ (/ ' / )I2 h2 (2) I G (R) = − I Fς/\ ri , R II dVe . Dς |/\| (R) = − 2μR2 ς |/\| ∂R

/\

A ς/\ (R) = −

(187)

Ve

/\

Let the operator B ς/\ (R) defined by expression (180). The oper( us now consider ) 2 2 ator LX ' + LY ' included in this integral is real and positive. This operator is invariant /\

/\

/\

with respect to operation σν . Therefore, taking into account (184), we obtain that Gς(3) −/\ (R) /

/ =

) (/ / )( (/ / ) Fς∗−/\ r'i , R L2X ' + L2Y ' Fς −/\ r'i , R dVe /\

Ve

) (/ / ) (/ ' / )( 2 (3) ∗ ri , R LX ' + L2Y ' σν Fς/\ r'i , R dVe = Gς/\ σν Fς/\ (R), /\

/\

=

/\

Ve (3) ⇒ Gς/\ (R) = Gς(3) |/\| (R),

/\

/\

(188)

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V. K. Khersonsky and E. V. Orlenko

/\

/\

or B ς/\ (R) = B ς |/\| (R). That is, this operator does not depend on the sign of /\. The most important for determining the rotational wave function of nuclear motion ( ) is the third operator C ς/\ (R, α, β, γ ) in Tˆ N (R) (172) with taking into account /\

ς/\ς/\

(178) and (179), which can be easily calculated taking into account that the electron (/ / ) wave function Fς/\ r'i , R is an eigenfunction of the operator LZ ' (see formula 139). Therefore, /\

(/ / ) (/ / ) L2Z' Fς/\ r'i , R = /\2 Fς/\ r'i , R /\

(189)

For the operator of the square of the total angular momentum (166), we obtain (/ / ) (/ / ) N 2 Fς/\ r'i , R = R/\ (α, β)Fς/\ r'i , R [ 2 )] (190) ( 2 ∂ 1 ∂ ∂ ∂ 2 + − /\ + cot β − 2i/\ cos β R/\ (α, β) = − . ∂β 2 ∂β ∂α sin2 β ∂α 2 /\

/\

/\

This does not act on the arguments of the electronic wave functions (/ /operator ) Fς/\ r'i , R . Therefore, using the orthonormality of these functions, we obtain the following expression for the operator C ς/\ (R, α, β, γ ) /\

/\

/\

C ς/\ (R, α, β, γ ) ≡ C ς/\ (R, α, β) =

) h2 ( 2 R . β) − /\ (α, /\ 2μR2 /\

(191)

We have removed the angle γ from the list of variables for this operator. Summarizing, we can present the following expression for the average kinetic energy operator (172), ( ) Tˆ N (R)

ς/\ς/\

/\

/\

/\

= A ς/\ (R) + B ς/\ (R) + C ς/\ (R, α, β, γ ) ( ) ) h2 ( h2 ∂ 2 ∂ R + R/\ (α, β) − /\2 =− 2 2 2μR ∂R ∂R 2μR + Bς |/\| (R) + Dς |/\| (R). /\

(192)

This expression for the operator of the kinetic energy of nuclei can be used in the Schrödinger equation for the nuclear wave function. As shown earlier in this section, the off-diagonal elements of the averaged operator are zero. Therefore, the equations in system (168) are decoupled, and the nuclear wave function can be defined as the solution of one equation [see. formulas (167) and (168)],

Elements of Theory of Angular Moments as Applied to Diatomic Molecules …

[(

] + ες |/\| (R) χς/\ (R) = Eς |/\| χς/\ (R), ς/\ς/\ [ ( ) ] ) 2 h h2 ( ∂ 2 ∂ 2 − R + R + E (R) χς/\ (R) β) − /\ (α, /\ ς |/\| 2μR2 ∂R ∂R 2μR2 = Eς |/\| χς/\ (R), Eς |/\| (R) = ες |/\| (R) + Bς |/\| (R) + Dς |/\| (R). Tˆ N (R)

271

)

/\

(193)

Here Eς |/\| (R) is a potential energy with corrections. This equation can be used for arbitrary values of /\.

3.6 The Total Wave Function of a Molecule for the Case S = 0 Equation (193) allows the decoupling of variables. We will seek for a nuclear motion function in the form of a product χς/\NM (R) = uς |/\|ν (R)ΘNM /\ (α, β),

(194)

where uς |/\|ν (R) is the wave function of oscillating motion describing oscillations of the molecule, which electronic states are characterized by the quantum numbers ς |/\|. The oscillational quantum number ν denotes the number of the oscillation quantum state of the molecule. ΘNM /\ (α, β) is the wave function of the rotational motion of the molecule corresponding to the quantum numbers N, M, /\. Note that we do not yet indicate the parity of the state πi in the full set of quantum numbers. This quantity πi will be introduced below after discussing the symmetry of the total wave function. This set adds a vibrational quantum number ν. Substituting this product into Eq. (193) and then dividing each term by the same product gives the following equation ( ) h2 ∂ 2 ∂ R uς |/\|ν (R)ΘNM /\ (α, β) 2μR2 ∂R ∂R ) ( + Eς |/\| (R) − Eς |/\| uς |/\|ν (R)ΘNM /\ (α, β) ) h2 ( 2 uς |/\|ν (R)ΘNM /\ (α, β), R =− β) − /\ (α, /\ 2μR2 ( ) ] [ 1 h2 ∂ 2 ∂ R uς |/\|ν (R) + Eς |/\| − Eς |/\| (R) uς |/\|ν (R) 2μR2 ∂R ∂R ) 1 h2 ( 2 = N R ΘNM /\ (α, β) β) − /\ (α, /\ ΘM /\ (α, β) 2μR2

−

/\

/\

(195)

272

V. K. Khersonsky and E. V. Orlenko /\

Recall that the operator R/\ (α, β) is obtained by substituting the following quantities in the formulas for the square of the total angular momentum instead of the derivatives with respect to the angle γ ∂ → i/\, ∂γ ∂2 → −/\2 . ∂γ 2

(196)

That means, that the functions ΘNM /\ (α, β) are the eigenfunctions of the angular /\

/\

momentum quadrat operator N 2 and the operator of its projection NZ on the quantization axes Z in the LCS with the fixed projection /\ of this angular momentum on the axes Z’ of the MCS. In order for these functions to be eigenfunctions of the ∂ , that is, so that they satisfy the equation operator NZ = −i ∂α /\

/\

NZ ΘNM /\ (α, β) = M ΘNM /\ (α, β)

(197)

the dependence of the angle α should be factored by the factor eiM α , so then ΘNM /\ (α, β) ∝ eiM α

(198)

Taking into account (198), we can make the following replacements ∂ → iM , ∂α ∂2 → −M 2 , ∂α 2

(199)

/\

in the operator R/\ (α, β) )] ( 2 [ 2 1 ∂ ∂ ∂ ∂ 2 → + − /\ R/\ (α, β) = − + cot β − 2i/\ cos β ∂β 2 ∂β ∂α sin2 β ∂α 2 ) ( 2 ) 1 ( 2 ∂ ∂ R/\M (β) = − + M − 2/\M cos β + /\2 . + cot β 2 2 ∂β ∂β sin β (200) /\

/\

As follows from formulas (163) and [14], the eigenfunctions of this operator are the dMN /\ (β) Wigner functions, and the eigenvalues are the quantities N (N + 1), /\

R/\ (α, β)dMN /\ (β) = N (N + 1)dMN /\ (β). The total rotational function ΘNM /\ (α, β) has the following form

(201)

Elements of Theory of Angular Moments as Applied to Diatomic Molecules …

ΘNM /\ (α, β) = CN eiM α dMN /\ (β),

273

(202)

where CN —is a normalization factor. If we normalize this function by the condition /2π

/π dα

0

( ' ) N∗ N sin βd β ΘM = δN ' N δM ' M , ' /\ (α, β)ΘM /\ (α, β)

(203)

0

then taking into account the properties of Wigner D-function [14], we have / CN =

2N + 1 . 4π

(204)

Substitution of function (202) into Eq. (195) with allowance for (200) leads to the following equation for the function describing the vibrations of nuclei in a field represented by potential energy [

( ) ] h2 ∂ 2 ∂ R + Eς |/\| − Eς |/\| (R) uς |/\|ν (R) = const · uς |/\|ν (R), 2μR2 ∂R ∂R ) h2 ( 2 R ΘNM /\ (α, β) = constΘNM /\ (α, β), ⇒ β) − /\ (α, /\ 2μR2 ) h2 ( N (N + 1) − /\2 , ⇒ const = 2 2μR [ ( ) ] ] 1 ∂ N (N + 1) − /\2 2μ [ 2 ∂ R − + 2 Eς|/\| − Eς |/\| (R) uς |/\|ν (R) = 0. R2 ∂R ∂R R2 h (205) /\

The discrete spectrum of eigenstates of this equation arises in those cases when the potential energy curve Eς |/\| (R) has a minimum at a certain value of Re . The energies of vibrational-rotational states are determined in the process of calculating the eigenvalues of Eq. (205). We briefly illustrate it in the next section. Here we assume that the eigenfunctions uς |/\|ν (R) are defined and normalized by the condition /∞

uς∗ |/\|ν ' (R)uς |/\|ν (R)R2 dR = δν ' ν .

0

The total wave function of a molecule can be represented as the product / (/ ' / ) (/ / ) 2N + 1 Fς/\ r'i , R uς |/\|ν (R)ΘNM /\ (α, β) Ψςν/\NM ri , R = 4π / 2N + 1 (/ ' / ) = fς/\ ri , R uς |/\|ν (R)eiM α dMN /\ (β)ei/\γ 8π 2

(206)

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V. K. Khersonsky and E. V. Orlenko

/

2N + 1 (/ ' / ) N∗ = fς/\ ri , R uς |/\|ν (R)DM /\ (α, β, γ ) 8π 2 / (/ / ) 2N + 1 N∗ Fς/\ r'i , R uς |/\|ν (R)DM = /\ (α, β, 0). 4π

(207)

3.7 Oscillatory and Rotational Energy of the Molecule As noted above, the discussion of the vibrational spectra of molecules and related problems lies outside the context of this chapter and we will not discuss these problems in detail. However, it is convenient to determine the rotational energy from Eq. (205) together with the oscillatory energy, because the rotational energy obtained in this method contains correction terms corresponding to centrifugal perturbations. Therefore, we will briefly illustrate the solution of Eq. (205), modeling the potential energy curve by the Morse formula (see, for example, [20]—this solution is given below), [ ]2 Eς |/\| (R) = Eς |/\| (Re ) + D 1 − e−α(R−Re ) ,

(208)

where D—is a dissociation energy in a given electron state of molecule and α is some constant that determines the curvature of the potential function. Note that, in addition to the Morse formula, several other approximations are used to obtain analytical solutions to Eq. (205). Among them, for example, is the Kratzer potential, which gives sufficiently accurate solutions only for very small deviations from the equilibrium position, or potential [21–23]. Another approximation is based on the use of the Pöschl-Teller potential, which gives a good description of potential curves even at significant deviations from equilibrium, but the analytical solutions of Eq. (205) obtained with this potential are somewhat more complicated than using the Morse potential. Substitute the Morse formula into Eq. (205) and instead of the function uς |/\|ν (R) we will use the function uς |/\|ν (R) =

U (R) . R

(209)

Then Eq. (205) will have a form [ ∂ 2 U (R) N (N + 1) − /\2 + − ∂R2 R2 ] ] 2μ [ −2α(R−Re ) −α(R−Re ) U (R) = 0. + 2 Eς|/\| − Eς |/\| (R) − D − De + 2De h or

Elements of Theory of Angular Moments as Applied to Diatomic Molecules …

[ d 2 U (R) N (N + 1) − /\2 + − dR2 R2 ] [ ]2 ] [ 2μ U (R) = 0 + 2 Eς |/\| − Eς |/\| (Re ) − D 1 − e−α(R−Re ) h

275

(210)

Let’s make a substitution y = e−α(R−Re )

(211)

and represent the term that determines the dependence on N and /\ in the form of an expansion at R ≈ Re , then Eq. (210) is transformed as follows: d 2 U (y) dy2 1 d U (y) + + y dy h2 N (N R= 2μ

[ ] 2μ Eς |/\| − Eς |/\| (Re ) − D 2D R R2e U (R) = 0, − D − + h2 α 2 y2 y y2 R2 + 1) − /\2 , R2e (212)

We rewrite (212) in the following form d 2 U (y) 1 d U (y) + dy2 y dy [ ] 2μ Eς |/\| − Eς |/\| (Re ) − D − c0 2D − c1 − D − c2 U (y) = 0. + 2 2 + h a y2 y

(213)

Here the following coefficients is denoted ) ( 3 3 , c0 = R 1 − + αRe (αRe )2 ) ( 6 4 c1 = R , − αRe (αRe )2 ) ( 3 1 c2 = R − . + αRe (αRe )2 Making the following substitutions,

(214)

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V. K. Khersonsky and E. V. Orlenko

U (y) = e−z/2 z b/2 F(z), z = 2ηy, / 2μ η= (D + c2 ), h2 α 2 / ) 8μ ( b= D + c0 − Eξ ν|/\|N πi + Eξ ν|/\| (Re ) , 2 2 h α

(215)

we have the equation for the degenerate hypergeometric function ( ) d 2 F(z) b+1 dF(z) ν + − 1 + F(z) = 0. 2 dz z dz z μ 1 ν = 2 2 (2D − c1 ) − (b + 1). h α d 2

(216)

A general solution for this equation is F(z) = C1 M (−ν, b + 1, z) + C2 U (−ν, b + 1, z),

(217)

where M (−ν, b + 1, z) and U (−ν, b + 1, z) are Kummer’s (confluent hypergeometric) functions and the coefficients C1 and C2 are arbitrary coefficients, one of them is determined from the boundary conditions, and the second from the normalization condition. As the first boundary condition, the requirement of the finiteness of the solution when R → ∞ is used. In this case, z → 0 the function U (−ν, b + 1, z) also increases indefinitely, U (−ν, b + 1, z) ∝ z −b . Therefore, the coefficient C2 should be chosen equal to zero and the solution to the equation can be represented in the form U (y) = C1 e−z/2 z b/2 M (−ν, b + 1, z).

(218)

The second boundary condition should be that when R → 0, U (y) → 0. To satisfy this condition, we might require that the function M (−ν, b + 1, z) be a polynomial. In turn, this would require the parameter ν to take positive values or zero. The fulfillment of these boundary requirements is not precise. The fact is that the solution U (y) → 0 could vanish at R → 0, only if the Morse potential then turned to infinity. But, y(R)|R→0 → y(0) = eαRe ⇒ De2αRe < ∞,

(219)

for the majority of molecules, the value of 2αRe is so large that the considered boundary condition provides a sufficiently good accuracy in determining the eigenvalues and eigenfunctions. Therefore, fulfilling this boundary condition with the noted caveat, we assume that the parameter ν must take positive integer values or zero, ν = 0, 1, 2, … In turn, this means that

Elements of Theory of Angular Moments as Applied to Diatomic Molecules …

μ h2 a 2 d

1 (2D − c1 ) − (b + 1) = 0, 1, 2, . . . 2

277

(220)

Resolving this equation for Eς |/\|νN πi , one can obtain an expression for the total energy of the electronic-vibrational-rotational state, (D − c1 /2)2 Eς |/\|νN πi ≈ες |/\|πi (R) + D + c0 − D + c2 ( ) ( ) 1 α 2 h2 1 2 2αh(D − c1 /2) ν+ − ν+ . + √ √ 2 2μ 2 2μ D + c2

(221)

Usually, this expression is expanded in small parameters c1 /D and c2 /D to bring it to the form in which it is used in spectroscopy, ) ( ) ( 1 2 1 (π hcν˜ e )2 − ν+ Eς |/\|νN πi − ες |/\|πi (R) ≈ hcν˜ e ν + 2 4π 2 D 2 ] [ ] h2 [ 2 N (N + 1) − /\2 + De h2 N (N + 1) − /\2 + 2Ie ] [ −h3 π α(ν+ 21 ) N (N + 1) − /\2 . / 2D , νe = a μ

(222)

1 1 =− , 8μ3 ν˜ e2 re6 c2 16π 2 Dμ2 a2 re6 c2 ) ( 1 3˜νe 1 − 2 2 . αe = 2μre2 D are a re

De = −

In practical work with this formula, it is simplified by the fact that the terms proportional to /\2 (independent of N and v) are included in the electron energy ες |/\|πi (Re ). In what follows, we will be mainly interested in the rotational energy, E/\N the expressions for which it will be convenient to discuss in the form in which it is given in formula (220). Neglecting the terms corresponding to vibrational-rotational interactions, rotational energy can be represented in the form, E/\N = +

] [ ]2 h2 [ N (N + 1) − /\2 + De h2 N (N + 1) − /\2 . 2Ie

(223)

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V. K. Khersonsky and E. V. Orlenko

3.8 Symmetries of the Total Wave Function. Parity In Sect. 3.3, we considered the symmetry properties of the electron wave function with respect to transformations from the group C∞ν . These were transformations of electronic coordinates, defined in the MSC. The total wave function (207) also depends on nuclear coordinates. In what follows, we will only be interested in rotational spectra. Therefore, for the sake of simplicity, we will restrict our consideration to only the ground vibration state. This state is completely symmetrical. The following discussion will not deal with the vibrational part of the total wave function. As for the symmetry of the electron-rotational part of the wave function, here, in addition to the already considered operations of transforming electronic coordinates, it is important to discuss two types of transformations: molecular rotations characterized by coordinates α , β and coordinate inversion, Pi . The complete Hamiltonian of the molecule must be invariant under these transformations and, therefore, certain integrals of motion must correspond to each of these transformations. As we already know, the invariance of the Hamiltonian with respect to the rotations of the molecule (or the corresponding rotations of the coordinate system) corresponds to the conservation of the total angular momentum of the system N and its projection M onto the quantization axis (Z-axis of the LSC). Therefore, below in this section, we mainly concentrate on discussing the invariance of the Hamiltonian under the operation of space inversion, which consists in changing the signs of all coordinates. It was already noted above that parity is an integral of motion. Mathematically, this is expressed in the fact that the total wave functions of molecular states should be eigenfunctions of the inversion operator Pi and the parity πi (= ±1) must be the eigenvalue of this operator. The desired (/ / wave ) functions at /\ > 0 can be constructed from functions (207), Ψςν/\NM r'i , R , in the form of the following linear combinations, (/ / ) (/ / )) (/ / ) 1 ( Ψςν/\NM r'i , R + πi Pi Ψςν/\NM r'i , R , Ψςν/\NM πi r'i , R = N

(224)

where the factor (/ /1/N) is introduced for the correct normalization of the function Ψςν/\NM πi r'i , R . Indeed, these functions are eigenfunctions of the inversion operator, (/ / ) (/ / ) (/ / )) 1 ( Pi Ψςν/\NM r'i , R + πi Pi Pi Ψςν/\NM r'i , R , Pi Ψςν/\NM πi r'i , R = N ˆ Pi Pi = 1, (/ / ) (/ / ) (/ / )) 1 ( Pi Ψςν/\NM r'i , R + πi Ψςν/\NM r'i , R Pi Ψςν/\NM πi r'i , R = N (/ ' / ) (/ / )) 1 ( = πi πi Pi Ψςν/\NM ri , R + Ψςν/\NM r'i , R = πi Ψςν/\NM πi . N (225)

Elements of Theory of Angular Moments as Applied to Diatomic Molecules …

279

ˆ Thus, The inversion operator Pi possess the idempotency property: Pi Pi = 1. in order to construct functions of a certain parity, it is necessary to understand what mathematically represents (/ / ) the result of the operation of the inversion operator on a function Ψςν/\NM r'i , R . Nuclear coordinate inversion is equivalent to the following spherical coordinate transformation Pi {X , Y , Z} = {−X , −Y , −Z} ⇒ Pi {R, α, β} = {R, π + α, π − β}.

(226)

The value of R has the meaning of the distance between the nuclei, R = |RA − RB |, which does not change during inversion. Let how the transformation of angles affects the wave functions (/ us / consider ) Fς/\ r'i , R and ΘNM /\ (α, β). The inversion of all coordinates is defined in the LCS. In order to check how the inversion of all coordinates affects electron wave function, it is necessary to see how the electronic coordinates of the wave function (defined in the MSC) change during inversion. For this, we use relations, which determine the transformation of coordinates, when the coordinate system is rotated by angles from the LCS to the MCS. These ratios can be represented as follows: ⎛

⎞ ⎡ ⎤⎛ ⎞ xi' cos α cos β sin α cos β − sin β xi ⎝ y' ⎠ = ⎣ − sin α ⎦⎝ yi ⎠. cos α 0 i zi' zi cos α sin β sin α sin β cos β

(227)

Substitute here the transformation of angles α → π + α, β → π − β and the transformation of coordinates (xi , yi , zi ) → (−xi , −yi , −zi ). Then we get that ⎞ ⎛ '⎞ ⎤⎛ −xi cos α cos β sin α cos β − sin β −xi ⎠ ⎝ ⎣ ⎦ ⎝ −yi = yi' ⎠. sin α − cos α 0 −zi zi' − cos α sin β − sin α sin β − cos β ⎡

(228)

Thus, the inversion of the electronic coordinates determined with respect to the LCS leads to the fact that the electronic coordinates determined ( ) with respect (to the ) MCS are transformed with the operation of reflection σν y' z ' in the plane y' z ' . Therefore, the action of the coordinate inversion operation Pi on the total wave function can be represented as (/ / ) Pi Ψςν/\NM r'i , R / (/ / ) 2N + 1 uς|/\|ν (R)ΘNM /\ (α + π, π − β)σν Fς/\ r'i , R = 4π / ( ) (/ / ) 2N + 1 N = (−1) uς |/\|ν (R)ΘNM /\ (α, β)σν y' z ' Fς/\ r'i , R , 4π

(229)

where we use property of the Wigner D-function. For further transformation of this expression, it is necessary to refer to the symmetry properties of the electron wave

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V. K. Khersonsky and E. V. Orlenko

function presented in Sect. 3.3. Recall that, according to the definition of the transformation of reflection in a plane passing through the axis of symmetry of the molecule (152)–(156), the plane (x’z’) is chosen as the plane of reflection. Transformation ' ' (227) to the ( ' 'corresponds ) ( 'plane ) of reflection (y z ). In order to go from operation ' σν x z to operation σν y z , we use the rules of group multiplication [4], ( ) ( ) σν y' z ' = Cπ σν x' z ' , Now, if /\ > 0, we can directly use properties (150) and (155), ( ) (/ / ) ( ) (/ / ) σν y' z ' Fς/\ r'i , R = Cπ σν x' z ' Fς/\ r'i , R (/ / ) (/ / ) (/ / ) = Cπ Fς −/\ r'i , R = e−i/\π Fς −/\ r'i , R = (−1)/\ Fς −/\ r'i , R .

(230)

Hence, (/ / ) Pi Ψςν/\NM r'i , R / ( ) (/ / ) 2N + 1 N uς |/\|ν (R)ΘNM /\ (α, β)σν y' z ' Fς/\ r'i , R = (−1) 4π / (/ / ) N +/\ 2N + 1 = (−1) uς |/\|ν (R)ΘNM −/\ (α, β)Fς −/\ r'i , R 4π (/ / ) = (−1)N +/\ Ψςν−/\NM r'i , R .

(231)

Thus, functions (224) have the form (/ / ) (/ / )) (/ / ) 1 ( Ψςν/\NM r'i , R + πi (−1)N +/\ Ψςν−/\NM r'i , R Ψςν/\NM πi r'i , R = N (/ / ) (/ / )) 1 ( = Ψςν/\NM r'i , R + πi/\N Ψςν−/\NM r'i , R , N πi/\N = πi (−1)N +/\ , ⇒ πi = (−1)N +/\ πi/\N . (232) Formula (232) shows that the parity of a state affects the wave function (/ as / follows. ) ' r wave function is the sum of the functions Ψ For, πi =(/1, the ςν/\NM i , R and / ) Ψςν−/\NM r'i , R if N + /\ is even and the difference if N + A is odd. When πi = −1, the sum or difference of these functions is realized if N + /\ is odd and even numbers, respectively. That is, each electronic term consists of two states described by wave functions (232) with πi/\N = 1 and πi/\N = −1. In the zero approximation, the energies of these states are the same, that is, these states are degenerate. Taking into account the interactions between the electronic state and rotation, the electronic terms with /\ > 0 split. This effect is called /\-doubling. However, /\ -doubling rapidly decreases with increasing /\ and is really significant only for terms with /\ = 1. If /\ = 0, then we have following

Elements of Theory of Angular Moments as Applied to Diatomic Molecules …

( ) (/ / ) ( ) (/ / ) σν y' z ' Fς 0 r'i , R = Cπ σν x' z ' Fς 0 r'i , R (/ / ) (/ / ) = πΣ Cπ Fς 0 r'i , R = πΣ Fς 0 r'i , R ,

281

(233)

± where π(/ = ±1 Σ / ) is a parity of the Σ terms. In this case, the wave function ' Ψςν0NM ri , R itself is an eigenfunction of the inversion operator,

Pi Ψςν0NM

(/ ' / ) (/ / ) ri , R = πΣ (−1)N Ψςν0NM r'i , R

(234)

then πi = πΣ (−1)N and (/ / ) (/ / ) Ψςν0NM πi r'i , R = Ψςν0NM r'i , R . (/ / ) The wave function Ψςν/\NM πi r'i , R can be presented in a form that will be convenient for calculating matrix elements for any /\ > 0. For this, we introduce the value κ/\Ni = (1 − δ/\0 )π/\Ni .

(235)

Then the wave function can be represented as (/ / ) Ψςν/\NM πi r'i , R =

N

√

( (/ / ) (/ / )) 1 Ψςν/\NM r'i , R + κ/\Ni Ψςν−/\NM r'i , R . 2 − δ/\0 (236)

In conclusion of this section, we note that if the molecule is homonuclear, additional symmetry appears with respect to the permutation of two nuclei with spins i. The form of the wave function corresponding to this type of symmetry was discussed above. The total wave function includes the nuclear function χIMI (σi1 , σi2 ), which depends on the spin variables of both nuclei, σi1 and σi2 , and describes states with the total nuclear spin I and its projection MI onto the quantization axis (Z-axis of the LSC). Although the interaction of nuclear spins i with rotation and the electron shell is extremely weak (it causes hyperfine splitting of rotational levels), the rules associated with the rearrangement of spins allow the existence of some and forbid the existence of other rotational levels. The total wave function has the form (/ / ) Ψςν/\N πi IFMF r'i , R, σi1 , σi2 ] [ FMF (/ / ) 1 [ (237) = CNMIMI Ψςν/\NM r'i , R χIMI (σi1 , σi2 ), √ 1 + (−1)N +I N 2 MMI

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where F is the quantum number of the total angular momentum taking into account the nuclear spin, M F is the quantum number of the projection of this angular momentum onto the quantization axis (Z-axis of the LSC), and χIMI (σi1 , σi2 ) =

i [

i [

IMI Cim χ (σi1 )χimi2 (σi2 ). i1 imi2 imi1

(238)

mi1 =−i mi2 =−i

If the total nuclear spin is an even number, the rotational spectrum contains states corresponding only to even numbers N. On the contrary, if the total spin is an odd number, the rotational states correspond to odd values of N.

3.9 Λ-Doubling It has been repeatedly noted above that the states corresponding to a certain /\ value turn out to be twofold degenerate in the (±) |/\| sign. The levels corresponding to positive and negative values have the same energy. The question arises as to whether these levels can be split by some kind of interaction. To answer this question, we recall that we are talking about a perturbation of electronic motion (perturbation of electronic terms with a certain /\). On the other hand, rotation is the only dynamic factor present in the operator of kinetic energy of nuclei. Therefore, we are talking about the interaction of rotational and electronic motions, and the element of the operator of the kinetic energy of nuclei responsible for this interaction is represented by the operator Dˆ ≡

1 N (α, β) · L({ri }). 2μR2 /\

/\

(239)

In order for the corresponding terms ±|/\| to be split, they must be connected by matrix elements of the interaction operator, Ithat is, matrix elements of the form I < ' < ˆ − /\NM > A and ς ' − /\N ' M ' ID|ς ˆ /\NM >. However, as shown in ς /\N ' M ' ID|ς Sect. 3.5 [see. transformation of the integral C ς/\−/\ (R, α, β, γ ) (176)], matrix I I < < elements ς ' /\IN −μ Lμ |ς − /\> and ς ' − /\IN −μ Lμ |ς /\> where μ = ±1 are equal to zero for reasons of symmetry. The operator L0 cannot connect the states characterized by the values I /\ and -/\. That is, matrix elements of the form < ' I < ˆ − /\> and ς ' − /\ID|ς ˆ /\> are equal to zero. Therefore, the only way ς /\ID|ς to connect these states Iis Ito use a transition throughI an intermediate term /\'' , for < >< ˆ /\NM >. example, ς ' − /\N ' M ' IDˆ Iς '' /\'' N '' M '' ς '' /\'' N '' M '' ID|ς Electronic terms with different values /\ are separated by large electronic energies (compared to the energies of rotational levels). Consequently, the transitions between them could be taken into account according to the perturbation theory, in which the small parameter is precisely the ratio of the energy of the rotational levels to the /\

/\

/\

/\

/\

/\

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283

difference in the energies of electronic terms with different /\ ones. To this it must be added that the operator L'' μ can change the value /\ only by one. Therefore, in order to take into account all the necessary transitions for binding states /\ → −/\, that is, /\

ˆ ˆ ˆ ˆ . . . ˆ − 1> ˆ . . . = √ N (2 − δ/\0 )(2 − δ/\' 0 ) I < (244) +κ/\' N ' i' ς ' − /\' N ' M ' πi' ILμ ({ri })|ς /\NM πi >+ < ' ' ' ' 'I +κ/\Ni ς /\ N M πi ILμ ({ri })|ς − /\NM πi >+ ] I < + κ/\Ni κ/\' N ' i' ς ' − /\' N ' M ' πi' ILμ ({ri })|ς − /\NM πi > . /\

/\

/\

/\

/\

/\

/\

Consider the first matrix element by substituting expressions (242), (140), and (243) into it. Then you can get that

Elements of Theory of Angular Moments as Applied to Diatomic Molecules …




'

π 0

285

√ (2N + 1)(2N ' + 1) = × 4π

'

N 1∗ DM ' /\' (α, β, 0)Dμν (α, β, 0)×

(/ / ) (/ /) (/ / ) N∗ Fς∗' /\' r'i R L' ν r'i Fς/\ r'i R dVe' Dμν (α, β, 0) sin βd βd α /\

Ve'

√

(2N + 1)(2N ' + 1) × 8π 2 1 / 2π / π [ N' 1∗ DM ' /\' (α, β, 0)Dμν (α, β, 0)×

=

ν=−1 0 [/ 2π

0

'

e−i/\ γ +i/\γ ×

/

0

(/ / ) ' (/ /) (/ / ) ] N∗ fς∗' /\' r'i R Lν r'i fς/\ r'i R dVe' Dμν (α, β, 0) sin βd βd αd γ = /\

Ve'

√

(2N + 1)(2N ' + 1) × 8π 2 1 / 2π / π / 2π [ [ N' 1∗ DM ' /\' (α, β, γ )Dμν (α, β, γ )×

=

ν=−1 0

/

0

(245)

0

(/ / ) (/ /) (/ / ) ] N∗ fς∗' /\' r'i R L' ν r'i fς/\ r'i R dVe' Dμν (α, β, γ ) sin βd γ d βd α /\

Ve'

√

1 [ < ' ' I ' (/ ' /) (2N + 1)(2N ' + 1) ς /\ IL ν ri |ς /\>× × = 8π 2 ν=−1 / 2π / π / 2π [ ' N 1∗ N∗ DM × ' /\' (α, β, γ )Dμν (α, β, γ )Dμν (α, β, γ ) sin βd γ d βd α 0 0 0 / 1 (2N + 1) N ' M ' [ N ' /\' < ' ' II ' (/ ' /) C = CN /\1ν ς /\ L ν ri |ς /\> NM 1μ (2N ' + 1) ν=−1 / I < (/ /) (2N + 1) N ' M ' N ' /\' CNM 1μ CN /\1(/\' −/\) ς ' /\' IL' /\' −/\ r'i |ς /\> = ' (2N + 1) < ' ' ' I (/ ' /) ς /\ N |IL ri ||ς /\N > N ' M ' CNM 1μ ; = √ (2N ' + 1) /\

/\

/\

/\

where the reduced matrix element is defined as / < ' ' ' I (/ ' /) < ' 'I ' (/ ' /) N ' /\' I ς /\ N |IL ri ||ς /\N > = (2N + 1)CNM 1(/\' −/\) ς /\ L /\' −/\ ri |ς /\>. (246) /\

/\

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Here we have introduced the matrix element of the orbital momentum operator which is calculated by integrating over the electronic coordinates,

' (/ /) Lν r'i , /\

I (/ /) ς ' /\' IL' /\' −/\ r'i |ς /\> =


/ < ' 'I ' (/ ' /) (2N + 1) N ' M ' N ' /\' I C ri |ς /\> C = ' −/\) ς /\ L /\' −/\ NM 1μ N /\1(/\ (2N ' + 1) < ' ' ' I (/ ' /) ς /\ N |IL ri ||ς /\N > N ' M ' CNM 1μ . = √ (2N ' + 1)


N ' M ' CNM 1μ . = √ (2N ' + 1)


∗ .


I < ' = (−1)N +N +1 ς ' /\' N ' M ' ILμ ({ri })|ς − /\NM >∗ .


= N ] [< I I (/ /) < (/ /) ' ' ς ' /\' IL/\' −/\ r'i |ς /\> + (−1)/\ +/\+1 πi πi' ς ' /\' IL' /\' −/\ r'i |ς /\i >∗ . '

'

'

/\

πi' |IL({ri })||ς

/\

/\

(255)

The selection rules for the quantum number /\ are as follows: /\' = /\, /\ ± 1. If we formally consider the case when /\' > 1 and /\ < −1, then in expression ' ' /\ (255) the coefficient CNN/\1/\ ' −/\ vanishes and in the matrix element only the last row

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˜ = −/\, then this contributes. However, if we introduce a new quantum number /\ result will again be reduced to (252), up to a phase factor (−1)N +/\ πi . In the case when, /\ = 0, then κ/\Ni = 0, we have
+ (1 − δ/\' 0 )(−1) πi' ς /\ L /\' ri |ς 0> . '

'

'

/\

πi' |IL({ri })||ς 0N πi >

/\

/\

The selection rule for the quantum number /\' is /\' = 0, ±1, N ' /= N. In the case 0 when N ' = N, coefficient CNN010 = 0 at /\' = 0. Therefore, in this case, the value /\' ' can take only two values, /\ = ±1. Finally, when /\' = 0, then
. ς ' 0N ' πi' |IL({ri })||ς 0N πi > = 2N + 1CNN01/\ ' ς 0 L0 /\

/\

(257)

In this expression, the value N' } N—it can take values N ' = N ± 1. ˆ (239) describing the interaction of rotation and elecReturning to the operator D tronic motion and its matrix elements, it should be noted that according to the general formulas of perturbation theory, the second-order correction to the state energy can be formally represented as EςNM π ' = Eς 1N + i

[

/\

ς ' /\' = 0, 2, . . . N ' M ' πi' [

= Eς 1N +

I2 I< 1 I I ' ' ' ' ' II I ς /\ N M πi D|ς 1NM πi >I Eς 1N − Eς ' /\' N ' I I2 I< ' ' ' ' ' I 1 I 1 I ς /\ N M π I I . })|ς > β) · L N 1NM π (α, ({r i i i I I Eς 1N − Eς ' /\' N ' 2μR2 /\

ς ' /\' = 0, 2, . . . N ' M ' πi'

/\

(258) Let’s calculate the matrix element of the scalar product I ' ' ' 'I 1 ς /\ N M πi 2μR2 N (α, β) · L({ri })|ς 1NM πi >. This matrix element can be represented in the form


= 2N + 1 I I [ II I >< '' < II (−1)N +N ς ' /\' N ' πi' IIN (α, β)IIς ' /\' N '' πi'' ς ' /\' N '' πi'' IL({ri })|ς /\N πi >.


I Here is a reduced matrix element ς ' /\' N ' πi' IIN (α, β)IIς ' /\' N '' πi'' that has the following form: /\

Elements of Theory of Angular Moments as Applied to Diatomic Molecules …

289

II III > I ς ' /\' N ' πi' IIN (α, β)IIς ' /\' N '' πi''


/\

/ 1 + πi πi' δς ' ς δ/\' /\ δN ' N N (N + 1)(2N + 1), __/\ /= 0, 2 − δ/\0 I < ' ' ' III I ς 0N πi IIN (α, β)I|ς 0N πi > / = δς ' ς δ/\' /\ δN ' N N (N + 1)(2N + 1), __/\ = 0.

=

(261)

/\

It follows from formulas (260), (261) that the matrix elements of the operators of the projections of the angular momentum on the LSC axis bind the states of the same NI = N' . This clearly follows from the first same parity πi πi' = 1 and with the < ' ' ' ' III I formulas in (259) for ς /\ N πi IIN (α, β)I|ς /\N πi >. As for the second formula, the angular momentum operator does not change the value of the angular momentum N ' = N ' . Therefore, πi πi' = πΣ2 (−1)N +N = (−1)2N = 1, as it should be for an axial vector. The quantities M and M ' are related by the relation M = M ' + μ. Diagonality in N is a consequence of the fact that the total angular momentum is a conserved quantity. We substitute in (259) the reduced matrix element of the operator (260) and take into account that this matrix element is diagonal over all four quantum numbers ς ' /\' N ' πi' . Then you can get that /\

I ς ' /\' N ' M ' πi' IN (α, β) · L({ri })|ς /\NM πi > / N (N + 1) < ' ' ' ' II ς /\ N πi | L({ri })||ς /\N πi > = δN ' N δM ' M N (2N + 1)


/ I N (N + 1) < ' ς 0N πi' |IL({ri })||ς 1N πi > = δN ' N δM ' M (2N + 1) [< I ] / < I ' ' δN N δM M = N (N + 1) ς ' 0IL−1 ({ri })|ς 1> + (−1)N πi ς ' 0IL+1 ({ri })|ς − 1> 2 ] [< I < I δN ' N δM ' M / N (N + 1) ς ' 0IL−1 ({ri })|ς 1> + (−1)N πi ς ' 0IL−1 ({ri })|ς 1>∗ . = 2 (265)


, qr = Eς 1N − Eς ' 0N ς'

291

q1 =

/\

qi =

[ ς'

(267)

)]2 [ (< I B2 Im ς ' 0IL−1 {ri }|ς 1> . − Eς ' 0N /\

Eς 1N

As follows from formula (266), each rotational level of the ∏-term splits into two sublevels[ of opposite parity. The corresponding sublevel πi = +1 is shifted by an ] amount q1 + q2 N (N + 1) = qr N (N [ + 1) at] even N, while the shift of the sublevel corresponding to parity πi = −1 is q1 − q2 N (N + 1) = qi N (N + 1) at the same N. If N is an odd number, the order of the energy displacements is reversed. As can be concluded from formula (266), the values of the energy displacements 2 are of the order of E∏B−EΣ . 3.9.1

Diatomic Molecules with Nonzero Electron Spin

In those cases when the total electron spin S of the electron shell is not equal to zero, the structure of the spectra significantly gets complicated. The spin S does not interact directly with the axially symmetric electric field of the electron charge distribution, since in the electromagnetic sense, spin is a purely magnetic moment. However, if /\ /= 0, then the movement of electrons generates a magnetic field directed along the axis of the molecule. The interaction of the spin magnetic moment with this field is called the spin-orbit interaction. Spin can also interact with a magnetic field that is generated by rotation. In general, the structure of the spectra depends on the sequence in which the four angular momenta are associated—the orbital momentum of the electrons L, the spin momentum of the electrons S, the angular momentum of rotation of nuclei R, and the vector directed along the axis of the molecule—to the total angular momentum of the molecule J. Hund first carried out the analysis of various communication schemes of angular momenta in molecules [24]. There are two main (that is, the most common in diatomic molecules) Hund’s schemes known as scheme (a) and scheme (b) and two additional schemes—schemes (c), (d)—are relatively rare—the first in heavy molecules and the second in Rydberg states of molecules, in which electrons are excited to high states and, on average, spend a lot of time away from nuclei. In this section, we briefly describe these schemes and indicate some conserved quantities in each of them. In the next section, when discussing the wave functions for each of the schemes, we will refine this description. Hund’s scheme (a). The connection diagram of vectors in case (a) is shown in Fig. 3a. In this case, a strong axially symmetric electric field causes the precession of the orbital angular momentum L relative to the axis of the molecule, as noted above. If the rotation of the molecule is slow, then the cone of this precession has time to

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V. K. Khersonsky and E. V. Orlenko

follow the axis adiabatically. The total spin S interacts with an axially symmetric magnetic field directed along the axis of the molecule. In the general case, this interaction is weaker than the interaction of the orbital angular momentum with the electric field. However, nevertheless, with a very slow rotation, it can be the cause of the precession of the spin S around the axis of the molecule. More often, however, the spin S is associated with the axis not by direct interaction, but by interaction with the projection Λ of the orbital angular momentum L on to MSC—spin-orbit interaction. Therefore, in states with Λ = 0 spin, as a rule, it is not associated with the axis—a situation described by Hund’s scheme Fig. 3b (see below). We remind that Λ—is the projection of the total orbital angular momentum of electrons on the axis of the molecule (in the MSC), and /\—is the projection of the total spin of the electrons on to molecular axis (in the MSC). The vectors Λ and /\ form the total vector Ω = Λ + /\ directed along the axis of the molecule. Further, the vector is associated with the rotational angular momentum R, forming the total angular momentum J. The conserved quantities in the strict sense are the total angular momentum J, its projection onto the quantization axis M, and the projection of the Fig. 3 Coupling of vectors L, S, and R in Hund’s schemes (a) and (b), (c), and (d)

Elements of Theory of Angular Moments as Applied to Diatomic Molecules …

293

total angular momentum onto the molecule axis Ω. However, the quantities Λ and /\ are also good enough quantum numbers. Hund’s scheme (b). This case (Fig. 3b) corresponds to fast rotation, when the total spin, being relatively weakly coupled with the axis of the magnetic field, does not have time to follow this axis, while the precession cone of the orbital angular momentum has time to rotate along with the axis. In this case, the vector Λ is associated with the rotational angular momentum R at the total angular momentum N. In turn, the angular momentum N is associated with the total spin S at the total angular momentum of the molecule J. Good quantum numbers are the total spin S, the total angular momentum J and its projection M on the Z-axis of the LCS. Hund’s scheme (c). In this case, the spin-orbit interaction is so large compared to the interaction with the axis that the orbital angular momentum L and the spin moment S are linked at the momentum J a , which is then linked to the axis with the projection Ω. Then the vector Ω is associated with the rotational angular momentum R at the total angular momentum of the molecule J. Good quantum numbers in this case are the quantum number of the total angular momentum J, its projection onto the Z-axis of the LSC, M, and its projection onto the molecular axis, Ω. /\ and Σ are not good quantum numbers. Hund’s scheme (d). (see Fig. 3d) This scheme describes a situation when the orbital angular momentum is practically not associated with the axis of the molecule, but is associated with the angular momentum R, forming the angular momentum N. The total electron spin of the molecule S is bound with the angular momentum N, forming the total angular momentum of the molecule J. In various problems of molecular spectroscopy, it is necessary to use the wave functions of diatomic molecules that correctly reflect the angular momentum coupling patterns. When discussing these functions in the next section, we will concentrate on communication schemes (a) and (b) and the intermediate communication schemes (ab) between them, which are of most interest in applications.

3.9.2

Wave Functions Taking Into Account the Electron Spin. Hund’s Scheme (a)

In order to analyze the structure of the spectra, taking into account various possible cases of coupling of angular momenta, it is necessary to consider how the definitions of the zero-order wave functions change when the electron spins are taken into account. We will call the spinless Hamiltonian the nonrelativistic Hamiltonian that was used to construct the wave functions (236). Interactions associated with spins are usually taken into account using the perturbation theory, in which the wave functions (236), extended so as to correctly take into account the total spin of the electron shell in accordance with a certain Hund scheme, are functions of the zero approximation. By themselves, the interactions under consideration can be of two types, these are

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V. K. Khersonsky and E. V. Orlenko

spin-orbit interaction and spin-spin interaction. The latter is nothing more than the interaction of the magnetic dipole moments of electrons. It is usually small and is not taken into account when constructing wave functions. The spin-orbit interaction is the interaction of the total magnetic moment associated with the total spin of the electron shell with the magnetic field induced by the orbital motion of electrons. Approximately in MSC it can be represented as Ne )( ) ( [ ' c r'i , r'j ˆli · sj ,

Vˆ SO =

/\

(268)

i> B speaking,) the function expansion in the Hund scheme (a) should be a good approximation, while in the opposite case Aς/\νS = C (a) 1 II1, − , , JM πi + C (a) 3 II1, , , JM πi , Ω= J Ω= J 2 2 2 2 2

(331)

2

where, to simplify the notation in the designation of the coefficients, we have omitted (a) (a) = CςνS/\ΩJM all inessential indices, CΩJ πi , and to designate the basis functions, we use the bra-c-ket designation, omitting the symbols ς, ν, S, since they denote constant values. We add for clarity a symbol Σ that, although not necessary, so how Σ = Ω − /\, but will be useful in the discussion below. Thus, the notation of basis functions in formula (331) means the following, ) (/ / |/\, Σ, Ω, JM πi > ≡ ΨςνS/\ΩJM πi r'i , {σi }R . Here we must clarify what we mean when we talk about oscillatory functions in the context of the problem under consideration. When determining the total wave functions in the Hund scheme (a), the vibrational functions were determined from Eq. (286), in which the potential energy was written taking into account the accepted assumption that the spin-orbit interaction significantly exceeds the molecular rotation energy and has a significant influence on the formation of an electronic term. That is why the potential energy turns out to be dependent on the quantum numbers /\ and Ω. However, in the context of the problem under consideration, the spin-orbit

Elements of Theory of Angular Moments as Applied to Diatomic Molecules …

315

interaction is considered comparable to the energies of the rotational levels—they are both considered together as perturbations. This means that the vibrational wave function must be determined within the framework of the nonrelativistic approach, as is done in Eq. (205), where the potential energy depends on |/\|, but not on, the quantum numbers /\ and Ω. Obviously, this is also a limiting statement. In any case, we will use the assumption that is usually made when solving this kind of problem, and in any case is accurate enough for the illustrative purposes that we have in mind. Assumptions that the potential curves of relatively close terms 2 ∏1/2 and 2 ∏3/2 have the same form. Then, to calculate the integrals of the products of vibrational functions included in the matrix < elements, one can use the orthonormality relation, which will lead to the result ν ' |ν> = δν ' ν . Therefore, in what follows, we do not mention vibrational functions at all in this section. With this clarification, the basic function is [see. formula (320)] ) 1 ( |/\, Σ, Ω, JM πi > = √ |/\, Σ, Ω, JM > + (−1)J +1/2 |−/\, −Σ, −Ω, JM > , 2 (332) where |/\, Σ, Ω, JM πi > =

√ S ) (2J + 1) [ S 1/2 (/ / DMS Σ (α, β, 0)Fς MS 1 r'i , {σ i }, R uς 1ν (R). 4π M =−S S

(333) Wave functions (332) should be normalized by the condition = 1,

(334)

whence, due to the conditions of orthonormality of the basis functions, it follows (a) that the coefficients C (a) 1 ≡ C (a) 1 and C 3 satisfy the equality Ω= J 2

2

J

2

J

I I I I I (a) I2 I (a) I2 IC I + IC I = 1. I 1JI I 3JI 2

(335)

2

In order to determine the splitting between the terms 2 ∏1/2 and 2 ∏3/2 , that is, to find the corrections to the electron energy due to rotation and spin–orbital interaction, it is necessary to use the perturbation theory, where the perturbation operator Vˆ is the sum of the operator of the kinetic energy of nuclear rotation Tˆ rot , [see. formula (282)], and the operator of spin-orbit interaction, Vˆ˜ SO ,

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V. K. Khersonsky and E. V. Orlenko

)2 ( ) h2 ( ˆ ˆ ˆ + Aς/\νS Lˆ · Sˆ J − L − S 2μR2 ) ( 2 [ h ˆ 2 − 2J '0 S '0 + Sˆ 2 − 2 J '0 − S '0 L'0 J = 2μR2 ) ( )2 ( (336) + L'0 − L'+1 L'−1 + L'−1 L'+1 ( ) [( ) ( ) ]] +2 J '+1 S '−1 + J '−1 S '+1 − 2 J '−1 − S '−1 L'+1 + J '+1 − S '+1 L'−1 ( ) −Aς/\νS L'+1 S '−1 − L'0 S '0 − L'−1 S '+1 .

Vˆ = Tˆ rot + Vˆ˜ SO =

/\/\

/\

/\/\

/\/\

/\

/\

/\/\

/\/\

/\/\

/\

/\/\

/\

/\

/\

/\

/\

/\

/\/\

It is important to emphasize here that all scalar products are expressed in terms of contravariant components of the operators angular moments, which are projections on the MSC axis. (a) According to standard perturbation theory, the coefficients CΩJM πi satisfy the system of homogeneous equations I I \ ) I I 1 1 1 1 (ε) I I ˆ 1, − , , JM πi IV I1, − , , JM πi − E C (a) 1 J 2 2 2 2 2 I I \) (/ I I 1 3 1 1 + 1, − , , JM πi IIVˆ II1, , , JM πi C (a) 3 = 0, J 2 2 2 2 2 I I (/ \) I I 1 1 1 3 1, , , JM πi IIVˆ II1, − , , JM πi C (a) 1 J 2 2 2 2 I I \ 2 ) (/ I I 1 3 1 3 + 1, , , JM πi IIVˆ II1, , , JM πi − E (ε) C (a) 3 = 0, J 2 2 2 2 2 (/

(337)

where E (ε) (ε = 1, 2) is the eigenvalue of the matrix representing this system of equations [in Hund’s scheme (a)]. More compact ( ) (a) Vτ τ − E (ε) C (a) 1 + Vτρ C 3 = 0, 2

Vρτ C (a) 1 J 2

J

2

J

) ( + Vρρ − E (ε) C (a) 3 = 0, 2

(338)

J

) ) ( ( where τ = 1, − 12 , 21 , JM πi ,ρ = 1, 21 , 23 , JM πi . The characteristic equation for the system of Eq. (338) has the form ( ) E (ε)2 − E (ε) Vτ τ − Vρρ + Vτ τ Vρρ − Vτρ Vρτ = 0

(339)

Solving this equation, we obtain corrections to the electron energy 2 ∏-term

Elements of Theory of Angular Moments as Applied to Diatomic Molecules …

( ) E (ε)2 − E (ε) Vτ τ + Vρρ + Vτ τ Vρρ − Vτρ Vρτ = 0 [ ] / ( )2 ) ) ( 1 ( (ε) Vτ τ − Vρρ ± Vτ τ + Vρρ − 4 Vτ τ Vρρ − Vτρ Vρτ = E1,2 2 ] [ / ( ) ( )2 1 Vτ τ − Vρρ ± Vτ τ − Vρρ + 4Vτρ Vρτ . = 2

317

(340)

Obviously, one of these eigenvalues should correspond to a state 2 ∏1/2 in the limit when the spin-orbit interaction significantly exceeds the rotational energy, and the second to a state in the same limit. We will see below that this is indeed the case. The coefficients are determined from one of the Eq. (338), for example, the first, (a) (a) C (a) 1 = ξτρ C 3 , 2

J

(a) ξτρ

2

J

(341)

Vτρ = (ε) , E − Vτ τ

and normalization condition (335). The joint solution of these two equations gives the expansion coefficients in the form C (a) 1 J 2

=/

τρ

1+

C (a) 3 = / 2

I (a) I Iξ I

J

(

(a) ξτρ

)2 , (342)

1 (

(a) 1 + ξτρ

)2 .

In order to finally find the required expansion, it is necessary to determine the matrix elements in Eq. (337) or (338). Consider those terms in operator (336) that contribute to the matrix elements of interest to us. You can immediately see that the terms that contain operators L'±1 (but not products L'±1 L'∓1 ) should be omitted, since they shift the value /\ by one, while we consider matrix elements diagonal in /\. The rest of the terms can be divided into two groups, united by operators that correspond to diagonal matrix elements, Vˆ diag , and non-diagonal matrix elements, Vˆ ndiag , /\

/\/\

) ( ( )2 h2 [ ˆ 2 '0 '0 ˆ 2 − 2 J '0 − S '0 L'0 + L'0 J − 2J Vˆ diag = S + S 2μR2 )] ( − L'+1 L'−1 + L'−1 L'+1 + Aς/\νS L'0 S '0 , ) h2 ( '+1 '−1 '−1 '+1 . S + J J S Vˆ ndiag = μR2 /\/\

/\/\

/\/\

/\/\

/\

/\

/\

/\

/\/\

(343)

/\/\

Since the functions |/\, Σ, Ω, JM πi > are eigenfunctions of the operators ˆJ2 , Sˆ 2 , J0' = JZ ' , S0' = SZ ' , L' 0 = LZ ' , then /\

/\

/\

/\

/\

/\

318

V. K. Khersonsky and E. V. Orlenko = J (J + 1), = S(S + 1), /\

/\

= Ω, /\

= Σ, /\

/\

(344)

= /\,

therefore, the diagonal matrix element can be represented as =

h2 {J (J + 1) 2μR2

(345) −2ΩΣ + S(S + 1) − 2Ω/\ + 2/\Σ + /\2 ( ] ) + + Aς/\νS /\Σ, /\/\

/\/\

The last term is the matrix element of the operator of nonconserved quantity. In other words, functions |/\, Σ, Ω, JM πi > are not eigenfunctions of this operator. But since it does not change the values, /\, Σ, and Ω, we can consider its diagonal matrix element as some quantity that characterizes the unsplit ∏-term and include it in the electron energy, dropping it in expression (345). Hence, I I \ [ ] I I 1 1 h2 1 1 1 1 I I ˆ J (J + 1) + − Aς/\νS , = 1, − , , JM πi IV I1, − , , JM πi = 2 2 2 2 2μR2 4 2 I I \ [ ] / I I 1 3 h2 7 1 1 3 I I ˆ J (J + 1) − + Aς/\νS , = 1, , , JM πi IV I1, , , JM πi = 2 2 2 2 2μR2 4 2 (346) /

Vτ τ Vρρ

Let us now consider the non-diagonal matrix elements of the operator Vˆ ndiag . Wave functions |/\, Σ, Ω, JM πi > in the general sense should behave like spin functions under the action of the operators of spin and total angular momentum. We have already used this property in formula (305) to determine the result of the action of the covariant component of the spin operator on the electron wave function, in which the spin variables were determined with respect to the LSC. In the case under consideration, the contravariant components of the spin operators S '±1 , and the total angular momentum J '±1 are projections on the MSC axis and act on the Wigner S J∗ D-functions, DM (α, β, 0) and DM (α, β, 0), which determine the dependences of SΣ ) (/ ' /Ω the wave function ΨςνS/\ΩJM πi ri , {σ i }, R [see. Formula (332) and (333)] from the angles of turns α, β. Therefore, to determine the non-diagonal matrix elements, it is necessary to use formulas, according to which for μ = ±1 we have /\

/\

Elements of Theory of Angular Moments as Applied to Diatomic Molecules …

/ SΣ+μ S S S 'μ DM (α, β, 0) = − S(S + 1)CSΣ1μ DM (α, β, 0) S Σ+μ SΣ / S(S + 1) − Σ(Σ + μ) S =μ DMS Σ+μ (α, β, 0). 2

319

/\

(347)

For contravariant components of the total angular momentum operator, such a formula should take into account the fact that this operator acts on the complex J∗ conjugate function DM Ω (α, β, 0). Therefore, ]∗ [ −μ ]∗ [ J J∗ 'μ∗ J DM Ω (α, β, 0) = − J ' DM J 'μ DM Ω (α, β, 0) Ω (α, β, 0) = J / J Ω−μ J∗ = J (J + 1)CJ Ω1−μ DM Ω−μ (α, β, 0) / J (J + 1) − Ω(Ω − μ) J ∗ DM Ω−μ (α, β, 0). = 2 /\

/\

/\

(348)

Let us calculate the off-diagonal matrix elements in Eq. (337) using these relations. First of all, we note that the off-diagonal matrix element of the operator can be transformed to the following form

( ) h2

= 2 μR h2 ( + = μR2 ) . /\/\

/\/\

/\

/\

/\

/\

(349) Here we emphasize that the projection operator of the total angular momentum J (μ = ±1) connects states that differ in magnitude Ω, and the total spin projection operator S '±1 (μ = ±1) connects states that differ in magnitude Σ. We calculate the matrix element of the operator in this form using the basis (332) (for generality, we consider the matrix element completely off-diagonal in Ω and Σ), /\

'±1

/\

I /\, Σ, Ω' , JM πi IJ 'μ |/\, Σ, Ω, JM πi > I 1 (< /\, Σ, Ω' , JM πi IJ 'μ |/\, Σ, Ω, JM πi > = 2 I < +(−1)J +1/2 πi −/\, −Σ, −Ω' , JM πi IJ 'μ |/\, Σ, Ω, JM πi > I < +(−1)J +1/2 πi /\, Σ, Ω' , JM πi IJ 'μ |−/\, −Σ, −Ω, JM πi > I < +(−1)2J +1 πi −/\, −Σ, −Ω' , JM πi IJ 'μ |−/\, −Σ, −Ω, JM πi >,


/ / [ ] 2J + 1 [ 2π π J S∗ S 'μ J ∗ = DM Ω' (α, β, 0)DM ' (α, β, 0) J DM Ω (α, β, 0) DMS Σ (α, β, 0) sin βd βd α Σ S 4π 0 0 /\

/\

MS

/ 2π / π 2J + 1 / J Ω−μ J J∗ S∗ = J (J + 1)CJ Ω1−μ DM Ω' (α, β, 0)DM Ω−μ (α, β, 0)DMS Σ (α, β, 0)× 4π 0 0 [ S∗ S DM Σ ' (α, β, 0)DM Σ (α, β, 0) sin βd βd α MS

S

S

δΣ ' Σ

/ 2π / π 2J + 1 / J Ω−μ J J∗ S∗ = J (J + 1)CJ Ω1−μ DM Ω' (α, β, 0)DM Ω−μ (α, β, 0)DMS Σ (α, β, 0) sin βd βd α δΣ ' Σ 4π 0 0 / J Ω−μ = J (J + 1)CJ Ω1−μ δΩ' Ω−μ δΣ ' Σ

δΩ' Ω−μ

(352)

The fourth integral in (350), as is easy to see, differs only in the substitution /\ → −/\, Σ → −Σ, Ω → −Ω. Therefore, we can immediately conclude that its value will be equal to expression (352), in which it is necessary to make a replacement Ω → −Ω. Therefore, the matrix element (350) is equal to

Elements of Theory of Angular Moments as Applied to Diatomic Molecules …

I /\, Σ, Ω' , JM πi IJ 'μ |/\, Σ, Ω, JM πi > [ ] / J Ω−μ J −Ω−μ = J (J + 1) CJ Ω1−μ δΩ' Ω−μ + (−1)2J +1 CJ −Ω1−μ δ−Ω' Ω−μ δΣ ' Σ [ ] / J Ω−μ J Ω+μ = J (J + 1) CJ Ω1−μ δΩ' Ω−μ − CJ Ω1μ δΩ' Ω+μ δΣ ' Σ .
[ ] / SΣ+μ SΣ−μ = S(S + 1) CSΣ1μ δΣ ' Σ+μ − CSΣ1−μ δΣ ' Σ−μ δΩ' Ω .


0, then the term 2 ∏3/2 lies above the term 2 ∏1/2 —this is the case of a regular fine structure, and vice versa, if Aς/\νS < 0—this is the case of an irregular fine structure. Substitution of matrix elements (346) and (357) into expression (360) gives the (a) , parameter ξτρ (a) ξτρ =−

2κJ , λ−2±X

(362)

(a) that determines the coefficients C (a) 1 and C 3 (342). With large λ, 2 (a) ξτρ =−

J

2

J

2κJ 1 ) . ( λ − 2 1 ± 1 ± 2κJ 2

(363)

λ−2

The to the solution E1(ε) . In this case, at |λ|>> 1, I κ I sign in (257) corresponds I upper I (a) (a) Iξ I ≈ I J I > 1. corresponds to the second solution (E2(ε) ). In this case, at |λ|>> 1, Iξτρ κJ

(a) Therefore, C (a) 1 → 1,C 3 → 0, that is, the wave function is defined as the basis 2

J

2

J

function of the term 2 ∏1/2 . In the case of an irregular fine structure (Aς/\νS < 0), the solution E2(ε) should be interpreted as the energy of a term that asymptotically passes into 2 ∏1/2 , and to 2 ∏3/2 . In this the solution E1(ε) corresponds to a term that passes asymptotically I I I I I I case, we have for the term 2 ∏1/2 at |λ|>> 1, Iξ (a) I ≈ I λ I >> 1. Therefore, C (a) → τρ

1,C (a) 3 → 0. At the same J 2 (a) C 1 → 0,C (a) 3 → 1. J J 2 2

time, for the ∏3/2 2

κJ

1

J

2 I I I (a) I I ≈ I κJ I in the basis obtained for the Hund scheme (b). With the same remarks that were made regarding the use of the basis in the Hund scheme (a), we can represent the desired wave function in the form of an expansion I I \ \ I I 1 1 1 1 (b) I I |JM πi > = C 1 I1, J − , , JM πi + C 1 I1, J + , , JM πi , J − ,J J + ,J 2 2 2 2 2 2 (b)

(364)

where, to simplify the notation in the designation of the coefficients, we have omitted (b) (b) = Cςν/\NSJM all insignificant indices, CNJ πi and to denote the basis functions we again use the bra-c-ket notation, omitting the symbols ς, ν. Thus, the notation of the basis functions in formula (364) means ) (/ / |/\, N , S, JM πi > ≡ Ψςν/\NSJM πi r'i , {σi }R = / (365) / 1 |/\, N , S, JM > + (1 − δ/\0 )(−1)N +/\ πi |−/\, N , S, JM > , = √ N 2 − δ/\0 where √

N S (2J + 1) [ [ JM N∗ |/\, N , S, JM > = CNMN SMS DM (α, β, 0)FςSMS /\ N /\ 4π MN =−N MS =−S ) (/ / (366) × r'i , {σ i }, R .

Here, the vibrational wave function is omitted for reasons that were presented at the beginning of the section. In order to determine the splitting between the levels 2 ∏3/2 and 2 ∏1/2 in this case, it is necessary to use the perturbation theory, again considering as the perturbation operator Vˆ as the sum of the operator of the kinetic energy of nuclear rotation Trot , and the operator of spin-orbit interaction, V˜ SO . However, in this case, the operator of 2 ˆ 2. the kinetic energy of nuclei is expressed in the form Trot = h 2 N /\

/\

/\

2μR

324

V. K. Khersonsky and E. V. Orlenko

As for the spin-orbit interaction, it can no longer be used in the form in which it appears in expression (336), since the quantities Ω and Σ are not good quantum numbers. Therefore, to take into account the spin-orbit interaction, one has to consider operators of the type V˜ SO = A˜ ς/\νS n · Sˆ that are constructed as a result of double averaging—first over the electronic state, and then over the rotation of the molecule (see, for example, the discussion in [11, 25]). The calculation of the matrix elements and the subsequent solution of a secular equation of the type (337) makes it possible to find corrections to the energy of terms and the expansion coefficients of wave functions in a basis corresponding to the Hund scheme (b). Nevertheless, the averaging process itself introduces some specificity and such an averaged operator can differ significantly from those used in calculations in the Hund scheme (a). The problem under consideration can be solved in another way. Instead of calculating the matrix elements in the bond circuit (b), you can get these elements by expanding them into the matrix elements that were obtained in the bond circuit (a). In this approach, to calculate the position of the terms and expansion coefficients (408), the same operator of spin-orbit interaction is used that was used to calculate in the Hund scheme (a). We will discuss this hike in some detail here. The sought-for matrix elements in the binding scheme (b) can be represented as follows: /\


of the operator Vˆ are calculated in the communication scheme (a). When writing this formula, we used relations (335), (336). Substituting matrix elements here, Vτ τ , Vρρ and Vτρ , as well as the Clebsch–Gordan coefficients

Elements of Theory of Angular Moments as Applied to Diatomic Molecules …

/

2J + 3 , 4J / 3 J 2J − 1 2 C 1 11 =− , J+ 1 4(J + 1) 2 22 / 1 J 2J − 1 2 , C 1 1 1 = J− 1 − 4J 2 2 2 / 1 J 2J + 3 C 21 1 1 = . J+ 1 − 4(J + 1) 2 2 2 C

J

3 2

1 11 J− 1 2 22

325

=

(368)

it is easy to calculate all matrix elements (262), diagonal for N ' = N = J ± 1, and off-diagonal for N = J ± 1, N ' = J ∓ 1. We will demonstrate this calculation for only one off-diagonal matrix element—all the others are calculated in the same way. Consider the matrix element for N = J − 1, N ' = J + 1

√ 4J (J + 1) × = 2J + 1 [ I I / \ 1 1 I I J J 1 1 1 1 C 2 1 1 1 C 2 1 1 1 1, − , , JM πi IIVˆ II1, − , , JM πi J+ 1 − J− 1 − 2 2 2 2 2 2 2 2 2 2 +C

3 J 2

1 J 2

1 J 2

3 J 2

1 1 1C 1 1 1 J+ 1 − J− 1 − 2 2 2 2 2 2

+C

Vτ τ

I I \ I I 1 3 1 1 I I ˆ 1, , , JM πi IV I1, − , , JM πi 2 2 2 2

/

1 1 1C 1 11 J+ 1 − J− 1 2 2 2 2 22

Vρτ

I I \ I I 1 3 1 1 I I ˆ 1, − , , JM πi IV I1, , , JM πi 2 2 2 2

/

(369)

Vτρ

⎫ ⎪ I I ⎪ \ ⎬ 3 3 I I J J 3 3 1 1 2 2 I I ˆ +C 1 1 1 C 1 1 1 1, , , JM πi IV I1, , , JM πi ⎪ J+ 1 J− 1 2 2 2 2 ⎪ 2 22 2 22 ⎭ Vρρ √ [ [ ] B 4J (J + 1) κJ B Aς/\νS BκJ = J (J + 1) + − −√ √ 2J + 1 8 4 J (J + 1) 2 J (J + 1) [ ]] B κJ 7B Aς/\νS −√ J (J + 1) − + 8 4 J (J + 1) 2 Aς/\νS =− κJ . 2J + 1 /

The matrix element for the reverse transition is equal to the same expression, i.e.,

326

V. K. Khersonsky and E. V. Orlenko

= = = −

(370)

Aς/\νS κJ . 2J + 1

Diagonal matrix elements can be calculated similarly. We do not repeat the calculations, but we present the results. )( [( 1 J+ J− 2 Aς/\νS + , 2J + 1 )( [( 1 = B J + J+ 2 Aς/\νS − . 2J + 1

= B

1 2

3 2

)

] −1

)

] −1 (371)

These matrix elements can be used to determine the energies of multiplet states and coefficients in the expansion of the wave function in a basis built from functions corresponding to the Hund scheme (b). Of course, the energies of the levels in the multiplet should not depend on which of the Hund schemes the expansion is performed in. Let us check whether these energies calculated with the matrix elements (370) and (371) are equal to the values (359) obtained in the Hund scheme (a). In this case, the secular equation for the energy corrections can be represented as I [( I B J − 1 )(J + 1 ) − 1] + I 2 2 I A I − 2Jς /\νS +1 κJ

Aς /\νS 2J +1

− E (ε)

A

− 2Jς /\νS κJ [( )( ) +1 ] 1 B J + 2 J + 23 − 1 −

Aς /\νS 2J +1

I I I Ihfill (ε) −E I

= 0.

(372) The solutions of this quadratic equation have the form ⎛

E (1)

E (2)

⎞ ( 2 λ − λ + ⎠ = B κJ2 + = B⎝κJ2 + 4 ⎛ ⎞ /( )2 ( 2 1 λ J+ = B⎝κJ2 − − λ + ⎠ = B κJ2 − 2 4 /(

1 J+ 2

)2

) 1 X , 2 ) 1 X , 2

(373)

where we again use the notation (360). These solutions are exactly equal to solutions (359). Let us now determine the expansion coefficients of the wave function (364). A relation of the form (341) relates these coefficients,

Elements of Theory of Angular Moments as Applied to Diatomic Molecules …

C (b) 1

J − ,J 2

327

(b) (b) = ξτρ C 1 , J + ,J 2

κJ Aς/\νS (b) ξτρ =− )( ) ] [( 1 2J + 1 E (ε) − B J − J + 21 − 1 − 2 2λκJ =− . 2 (2J + 1) ± (2J + 1)X − 2λ Now the coefficients C (b) 1 similar to formulas (342),

J − ,J 2

and C (b) 1

J + ,J 2

Aς/\νS 2J +1

can be calculated by formulas that are

I (b) I I I Iξ I I (b) I τρ IC I I J − 1 ,J I = / ( )2 , 2 (b) 1 + ξτρ I I I (b) I 1 IC I 1 I= / I ( ) . J + ,J 2

(374)

(b) 1 + ξτρ

(375)

2

In the case when the spin-orbit interaction disappears (λ → 0), we obtain the following, X → 2J + 1, ⇒ (( (1) E →B J+ (( (2) E →B J−

)( ) ) 3 1 J+ − 1 = B[N (N + 1) − 1], 2 2 )( ) ) 1 1 J+ − 1 = B[N (N + 1) − 1], 2 2

(376)

That is, the difference between the two levels disappears and the energy is char(b) → 0 and one of the acterized only by the quantum number N. In this case ξτρ coefficients disappears—in our definition of these coefficients CN(b),J , this is the first of them. The second coefficient becomes equal to one, C (b) 1 = CN(b),J and J + ,J 2

I \ I 1 I |JM πi > → I1, N , , JM πi . 2 In other words, we arrive at a pure Hund scheme (b). Figure 6 demonstrates the transition from the level structure in the Hund scheme (a) corresponding to the regular fine structure (Aς/\νS < 0) to the level structure in the Hund scheme (b) and then again to the level structure in the Hund scheme (a), but corresponding to the irregular fine structure (Aς/\νS < 0).

328

V. K. Khersonsky and E. V. Orlenko

Fig. 6 Transition from the level structure in the Hund scheme a corresponding to the regular fine structure (Aς /\νS < 0) to the level structure in the Hund scheme b and then again to the level structure in the Hund scheme (a), but corresponding to the irregular fine structure (Aς /\νS < 0). The figure is borrowed from the monograph [1]

4 Application of Theory In this section, we will consider methods for calculating the probabilities of radiative transitions between energy levels of rotators of various types. Since in this chapter we are mainly interested in the problems of molecular spectroscopy, we will consider mainly electrical dipole transitions. We will also discuss methods for calculating the probabilities of radiative transitions between the rotational levels of diatomic molecules in various electronic configurations. We will continue to be interested exclusively in electric dipole transitions, since it is the type of radiative transitions that is of primary interest in the study of molecular spectra. Some special cases involving other types of radiative transitions will be dealt with separately if necessary.

4.1 Forces of Radiation Transition Lines in the Spectra of Rotators The quantum state of the rotator is characterized by a certain set of quantum numbers ς corresponding to the integrals rotator movement (see Sect. 2). In the absence of external fields, such a set always contains the quantum number of the total angular momentum of the rotator, J, the quantum number of the projection M of this moment onto the quantization axis (the Z' axis of the LSC), and the parity of the state, πi , with

Elements of Theory of Angular Moments as Applied to Diatomic Molecules …

329

respect to the inversion of coordinates. Therefore, we will write this set of quantum numbers as ς JM πi . The letter ς now denotes the set of other quantum numbers that may be important in describing the quantum state of the rotator. The spontaneous probability (for a detailed derivation, see Chap 2), the transition (ς JM πi ) → (ς ' J ' M ' πi ' ) is determined by the Einstein coefficient A

ς ' J ' M ' πi' ς JM πi

(E1) =

/ dw(sp) fi (E1, ϑ, ϕ) Ω

dΩ

dΩ

(377)

(sp)

where wfi (E1, ϑ, ϕ) is the differential spontaneous probability of an electric dipole transition, determined by formula (119), Aς ' J ' M ' πi' ς JM πi (E1) =

I2 32π 3 ( νς ' J ' M ' πi' ς JM πi )3 II< ' ' ' ' II ˆ I I ς J M πi DM ' −M |ς JM πi >I , 3h (378)

and the quantities Dˆ μ (μ = 0, ±1) (are the cyclic components (projections) of the dipole moment operator on the LSC axis. Here we do not consider the internal structure of the rotator—the components Dˆ ν' of the rotator’s dipole moment are considered to be their given projections on the MSC axis. These quantities are the quantum-mechanical averages of the dipole moment of the distribution of electrons in the considered electronic-vibrational states. Since the projections of the dipole moment are the components of an irreducible tensor of the first rank, the connections between the projections Dˆ μ and Dˆ ν' are determined by the relations Dˆ μ (α, β, γ ) =

[

1∗ ˆ ν' . Dμν (α, β, γ )D

(379)

ν=0,±1

Using the Wigner–Eckart theorem, probability in (377) can be represented in the form, Aς ' J ' M ' πi' ς JM πi (E1) =

32π 3 ( νς ' J ' M ' πi' ς JM πi 3h

I2 I< I I ˆ M ' −M ||ς JM πi >II ( )3 I ς ' J ' M ' πi' I|D )2 (380) J 'M ' C . JM 1M ' −M 2J ' + 1

In the absence of external fields, when all directions in space are equally probable, the states of the rotator are degenerate in the quantum number M, and the transition frequencies do not depend on M, M ' . In this case, we will be interested in the probability of a transition Aς ' J ' M ' πi' ς JM πi from a given level M to all levels M ' allowed by the selection rules and induced by all three components of the dipole moment

330

V. K. Khersonsky and E. V. Orlenko

Aς ' J ' ς J =

[

Aς ' J ' M ' ς JM =

μM '

32π 3h

3(

I2 I< I I ' ' II ˆ ) )2 > | D||ς J ς J I [( ' ' I 3 νς ' J ' ς J JM = C ' −M JM 1M 2J ' + 1 μM '

I2 I< I I ' ' II ˆ 32π 3 ( νς ' J ' M ' ς JM )3 I ς J |D||ς J >I = . 3h 2J ' + 1

(2J ' +1)(2J +1)

(381)

The magnitude of the square of the modulus of the reduced matrix element is called the line strength and is denoted as I2 I ( ) II< ˆ J >II . S ς ' J ' ς J = I ς ' J ' I|D||ς

(382)

Taking this definition into account, the discussed transition probability can be represented as A

ς 'J 'ς J

( ) 32π 3 ( νς ' J ' M ' ς JM )3 S ς ' J ' ςJ = . 3h 2J ' + 1

(383)

4.2 Matrix Elements for Rotational Transitions of a Symmetric Rotator Calculation of the matrix elements of rotational dipole electrical transitions in molecules of this type can be easily performed using wave functions (108) and is reduced to calculating the integral of three Wigner D-functions. Due to the symmetry of the rotator, the vector of the dipole moment is directed along the axis of symmetry, that is, the only component of this vector that is not equal to zero is its projection onto the Z' axis of the MSC, which is equal to itself. In other words, one term remains in formula (379), which corresponds to ν = 0. Therefore, I Jf Mf Kf IDˆ μ (α, β, γ )|Ji Mi Ki > / ( ) (2Ji + 1) 2Jf + 1 Mi −Mf −Ki +Kf =i × 8π 2 / 2π / π / 2π J Ji ∗ 1∗ DMf f Kf (α, β, γ )Dμ0 (α, β, γ )DM (α, β, γ ) sin βd γ d βd α i Ki 0 0 0 / (2Ji + 1) Jf Kf Jf Mf Mi −Mf ( )C =i C . 2Jf + 1 Ji Ki 10 Ji Mi 1μ
III II 1 Jf Mf < IIDˆ μ I|Ji Ki >, J K C = iMi −Mf /( f f J M 1μ ) i i 2Jf + 1


= (2Ji + 1)CJifKif10 δKf Ki .


1 Jf Mf Kf πνf IDˆ μ IJi Mi Ki πνi = /( )( )× 2 1 + πνi δKi 0 1 + πνf δKf 0 I I [< < ˆ μ |Ji Mi Ki > + πνf Jf Mf − Kf IDˆ μ |Ji Mi Ki > Jf Mf Kf I D ] I I < < +πνi Jf Mf Kf IDˆ μ |Ji Mi − Ki > + πνi πνf Jf Mf − Kf IDˆ μ |Ji Mi − Ki > .


iMi −Mf Jf Mf Jf Mf Kf πνf IDˆ μ IJi Mi Ki πνi = /( ) CJi Mi 1μ × 2Jf + 1 ⎧ √ ⎨ [ (2Ji + 1) J K J −K /( D' 0 CJifKif10 δKf Ki + πνf CJifKi 10f δ−Kf Ki )( ) ⎩2 1 + π δ 1+π δ


. The first and fourth terms can be combined as follows:


can be written in the form √ III II (2Ji + 1) ' Jf Kf πνf IIDˆ μ I|Ji Ki πνi > = /( )( ) D 0× 2 1 + πνi δKi 0 1 + πνf δKf 0 ] [[ ] JK [ ] JK 1 + (−1)Ji +Jf +1 πνi πνf CJifKif10 δKf Ki + πνi 1 + (−1)Ji +Jf +1 πνi πνf CJifKif10 δKf Ki δKi 0. √ [ ] JK ( ) (2Ji + 1) Ji +Jf +1 ' = /( πνi πνf CJifKif10 δKf Ki 1 + δKi 0. )( ) D 0 1 + (−1) 2 1 + πνi δKi 0 1 + πνf δKf 0 ]/ ( [ ) / 1 + (−1)Ji +Jf +1 πνi πνf 1 + πνi δKi 0 J K ' ) C f f δK f K i . ( = D 0 (2Ji + 1) 2 2 1 + πνf δKf 0 Ji Ki 10 (392)


= D Z −Ji , if _Jf = Ji − 1.







Kf =−Jf Ki =−Ji

= iMi −Mf /

2π

/

π

/ ( ) (2Ji + 1) 2Jf + 1 [ 8π 2

/

2π

Dˆ ν'

ν=0,±1

Jf [

Ji [

J ∗

gτff Kf gτJiiKi iKf −Ki ×

Kf =−Jf Ki =−Ji

(396)

Ji ∗ 1∗ DMf f Kf (α, β, γ )Dμν (α, β, γ )DM (α, β, γ ) sin βd γ d βd α i Ki 0 0 0 ⎧ ⎫ Jf Ji ⎨ ⎬ Mi −Mf [ [ [ / i J ∗ J K J M 2Ji + 1 gτff Kf gτJiiKi CJifKif1ν . Dˆ ν' iν =/ CJifMif1μ ⎩ ⎭ 2Jf + 1 J

ν=0,±1

Kf =−Jf Ki =−Ji

The expression in curly braces is the reduced matrix element for the transition Ji Mi τi → Jf Mf τf , [ III II ( ) ˆ μ I|Ji τi > = ˆ ν' T ν Ji τi Jf τf , Jf τf IID D


= Jf τf IID Dˆ ν' T ν Ji τi Jf τf = Dˆ · T ν=0,±1

( ) ( ) ( ) = Dˆ X' ' TX ' Ji τi Jf τf + Dˆ Y' ' TY ' Ji τi Jf τf + Dˆ Z' ' TZ ' Ji τi Jf τf , where

Elements of Theory of Angular Moments as Applied to Diatomic Molecules …

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( ) ( ) ( )] 1 [ TX ' Ji τi Jf τf = √ T −1 Ji τi Jf τf − T 1 Ji τi Jf τf 2 Ji ( ) [ i / J ∗ J K +1 J ∗ J K −1 = − √ 2Ji + 1 gτJiiKi gτifKi +1 CJifKii11 + gτff Ki −1 CJifKii1−1 , 2 Ki =−Ji ( ) ( ) ( )] i [ TY ' Ji τi Jf τf = √ T −1 Ji τi Jf τf + T 1 Ji τi Jf τf 2 J i ( ) [ 1 / J ∗ J K +1 J ∗ J K −1 gτJiiKi gτifKi +1 CJifKii11 − gτff Ki −1 CJifKii1−1 , = √ 2Ji + 1 2 Ki =−Ji

(398)

Ji [ ( ) ( ) / J ∗ J K TZ ' Ji τi Jf τf = T 0 Ji τi Jf τf = 2Ji + 1 gτJiiKi gτifKi +1 CJifKii10 . Ki =−Ji

The quantities, Dˆ X' ' , Dˆ Y' ' , Dˆ Z' ' are the Cartesian components of the rotator dipole moment vector Dˆ (the projection of the dipole moment on the X ' , Y ' , Z ' axes of the MSC). If it is preferable to have an expression for the matrix element that would explicitly contain the dependences on the parities πνi , πνf of the initial and final states, and, then the calculations can be carried out with wave functions, which are expansions in Wang functions. In this case, the reduced matrix element can again be represented in the form (292), [ III II < ( ) ˆ ν' T ν Ji τi πνi Jf τf πνf , Jf τf πνf IIDˆ I|Ji τi πνi > = D ν=0,±1

where J ∗

Jf Ji [ [ / wτff Kf wτJiiKi ( ) ν /( T Ji τi πνi Jf τf πνf = i 2Ji + 1 )( )× 1 + πνf δKf 0 1 + πνi δKi 0 Kf =0 Ki =0 2 [ ] J K J −K J K J −K CJifKif1ν + πνf CJifKi 1νf + πνi CJif−Kf i 1ν + πνi πνf CJif−Kif1ν . ν

(399) Having performed simple but cumbersome algebraic transformations, which include changing the signs of quantities Ki , Kf , using the symmetry properties of the J Clebsch–Gordan coefficients and coefficients wτJ K , wτJ√ −K = πν wτ K , and taking into J J J account the relationship between )wτ K and gτ K , wτ K = 2 − δK0 gτJK , it can(be shown) ( ν that quantity T Ji τi πνi Jf τf πνf (399) is exactly equal to quantity T ν Ji τi Jf τf (399). We will not do this here, but we will transform the resulting expression to a more convenient form that allows us to directly see the selection rules associated with the parities of states. Let us define the Cartesian X' -component of the vector

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using its cyclic contravariant components (399). Using the symmetry properties of the Clebsch–Gordan coefficients, this component can be represented as ( ) ( ) ( )] 1 [ TX ' Ji τi πνi Jf τf πνf = √ T −1 Ji τi πνi Jf τf πνf − T +1 Ji τi πνi Jf τf πνf 2 ] [ −i / = √ 2Ji + 1 1 + πνf πνi (−1)Ji +Jf +1 × 2 J

f Ji [ [

Kf =0 Ki =0

[ JK ( J −K )] Jf Kf Jf −Kf f f f f WKi Kf CJ K 11 + CJ K 1−1 + πνf CJ K 11 + CJ K 1−1 i i i i i i i i

] [ −i / = √ 2Ji + 1 1 + πνf πνi (−1)Ji +Jf +1 × 2 Ji [ [ ]] [ Jf Ki +1 Jf Ki −1 Jf −Ki +1 Jf Ki −1 WKi Ki+1 CJ K 11 + WKi Ki−1 CJ K 11 + πνf WKi −Ki−1 CJ K 11 + WKi 1−Ki CJ K 1−1 , i i

Ki =0

i i

i i

i i

Jf ∗

wτ K wτJi K i i f f WKi Kf ≡ WKi Kf Ji τi πνi Jf τf πνf = /( . )( ) 2 1 + πνf δKf 0 1 + πνi δKi 0 (

)

(400) When obtaining the last equality in formula (400), we used the triangle rule for the Clebsch–Gordan coefficients, which eliminates the sum over K ’. However, when performing this operation, it must be remembered that the value Kf takes positive ] [ J values or zero in the interval Kf ∈ 0, Jf . In other words, the coefficients wτff Kf are determined only at 0 ≤ Kf ≤ Jf . This means that the values WKi Kf for which Kf take negative values vanish. For example, the value WKi −Ki−1 ≡ WKi −(Ki+1) is zero for any Ki in the interval Ki ∈ [0, Ji ]. The value WKi 1−Ki = 0 at Ki > 1. The value WKi 1−Ki = 0 at Ki = 0. Taking these rules into account, formula (130) can be simplified, ( ) TX ' Ji τi πνi Jf τf πνf ] [ −i / = √ 2Ji + 1 1 + πνf πνi (−1)Ji +Jf +1 × 2 [ [ ] J1 J 0 W01 1 + πνf (−1)Ji +Jf +1 CJif011 + W10 CJif111 ] Ji [ ] [ Jf Ki +1 Jf Ki −1 WKi Ki+1 CJi Ki 11 + WKi Ki−1 CJi Ki 1−1 . +

(401)

Ki =1

Similarly, you can) get expressions for the) Cartesian components of the vector ( ( T ,T' Ji τi πνi Jf τf πνf and TZ ' Ji τi πνi Jf τf πνf ,

Elements of Theory of Angular Moments as Applied to Diatomic Molecules …

337

( ) ( ) ( )] −i [ TY ' Ji τi πνi Jf τf πνf = √ T −1 Ji τi πνi Jf τf πνf + T +1 Ji τi πνi Jf τf πνf 2 ] [ 1 / = √ 2Ji + 1 1 + πνf πνi (−1)Ji +Jf × 2 Jf Ji [ ( )] [[ J K J K J −K J −Kf WKi Kf CJifKif11 − CJifKif1−1 + πνf CJifKi 11f − CJifKi 1−1 Kf =0 Ki =0

] [ 1 / = √ 2Ji + 1 1 + πνf πνi (−1)Ji +Jf × 2 [ [ ] J1 J 0 W01 1 + πνf (−1)Ji +Jf +1 CJif011 + W10 CJif111 ] Ji [ ] [ Jf Ki +1 Jf Ki −1 WKi Ki+1 CJi Ki 11 + WKi Ki−1 CJi Ki 1−1 , + Ki =1

( ) ( ) TZ ' Ji τi πνi Jf τf πνf = T 0 Ji τi πνi Jf τf πνf = =

/

Jf Ji [ ] [ ][ [ J K J −K 2Ji + 1 1 + πνf πνi (−1)Ji +Jf +1 WKi Kf CJifKif10 + πνf CJifKi 10f Kf =0 Ki =0

[ ] Ji [ / [ ] [ ] J 0 J K f f i = 2Ji + 1 1 + πνf πνi (−1)Ji +Jf +1 W01 1 + πνf CJi 010 + WKi Ki CJi Ki 10 . Ki =1

(402) To illustrate the use of the obtained ) formulas, we calculate the matrix elements ( for the transition (Ji τi ) = 22 → Jf τf = 11. We will again use the Hamiltonian representation of the asymmetric Hamiltonian rotator (see (32)). The coefficients gτJK for the states and in this representation were calculated by using (57) and (58).

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V. K. Khersonsky and E. V. Orlenko

/ ⎧ 1+κ 1 3 ⎪ ⎪ / , Ki = −2, − ⎪ ⎪ √ ⎪ 2 2 ⎪ 2 + 6 − (3 − κ) κ 2 + 3 ⎪ 2κ ⎪ ⎪ ⎪ ⎪ ⎪ 0, Ki = −1, ⎪ ⎪ ⎪ √ ⎪ ⎪ ⎨ 1 3 − κ − 2 κ2 + 3 2 , Ki = 0, g2K = −2 / √ i ⎪ 2 + 6 − (3 − κ) κ 2 + 3 ⎪ 2κ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ i = +1, ⎪ 0, K/ ⎪ (403) ⎪ ⎪ ⎪ 1 3 1+κ ⎪ ⎪ / − , Ki = +2, ⎪ ⎪ √ ⎩ 2 2 2κ 2 + 6 − (3 − κ) κ 2 + 3 ⎧ 1 ⎪ ⎪ + √ , Kf = −1, ⎪ ⎪ ⎪ 2 ⎨ 1 = 0, 0, K = g1K f f ⎪ ⎪ ⎪ 1 ⎪ ⎪ ⎩ − √ , Kf = +1. 2 ( ) Let us calculate the components T ν Ji τi Jf τf (397) using these coefficients and the numerical values of the Clebsch–Gordan coefficients for the considered transition, ) ( T +1 Ji = 2, τi = 2, Jf = 1, τf = 1 ≡ T +1 (2, 2, 1, 1) ⎛ ⎞ √ √ 2κ + κ 2 + 3 1−1 2 1 2 1 11 ⎠ ⎝ , = i 5 g2−2 g1−1 C2−211 +g20 g11 C2011 = −i / √ √ √ = 1/2·5 2 2κ 2 + 6 − (3 − κ) κ 2 + 3 = 3/5 ⎛ ⎞ √ 1−1 ⎠ 2 1 11 2 1 T −1 (2, 2, 1, 1) = i 5⎝g22 g11 C221−1 +g20 g1−1 C201−1 √ = 3/5

√ = 1/2·5

√ 2κ + κ 2 + 3 , = −i / √ 2 2κ 2 + 6 − (3 − κ) κ 2 + 3 T 0 (2, 2, 1, 1) = 0

(404) Since the components T +1 and T −1 are equal to each other, and T 0 = 0, then only one Cartesian component of the vector T is nonzero, (

)

√

κ2 + 3 . T' Ji = 2, τi = 2, πνi , Jf = 1, τf = 1, πνf = √ / √ 2 2κ 2 + 6 − (3 − κ) κ 2 + 3 (405) 2κ +

Elements of Theory of Angular Moments as Applied to Diatomic Molecules …

339

Therefore, the reduced matrix element of the transition Ji = 2, τi = 2 → Jf = 1, τf = 1, is ( ) III II Jf = 2τf = 2IIDˆ μ I|Ji = 1τi = 1> = Dˆ · T √ ( ) 2κ + κ 2 + 3 ' ' ˆ ˆ . = DY ' TY ' Ji τi Jf τf = DY ' √ / √ 2 2κ 2 + 6 − (3 − κ) κ 2 + 3


= Dˆ 0 2Ji + 1CJi 010 .

(407)

The selection rules follow from the properties of the Clebsch–Gordan coefficient in this formula, which is nonzero only in that Jf + Ji + 1 the case when the sum is equal to an even number, that is, Jf = Ji ± 1. If a linear molecular rotator has identical nuclei, the symmetry associated with their rearrangement must be taken into account when calculating the reduced matrix element, since it affects the selection rules for rotational transitions. As an illustration, consider a diatomic rotator with identical nuclei. It should be noted right away that the electric dipole moment of such a system is zero. Therefore, the rotational levels of such a rotator cannot be related by an electric dipole transition. But they can be related by transitions of higher multipolarities, the first of which is the electric quadrupole transition. The probability of a rotational electric quadrupole transition can be calculated using the following general formula from Chap. 2,

340

V. K. Khersonsky and E. V. Orlenko sp−e wnf Lf Sni Li S

( ) Jp + 1 2 = ( ) [( ) ] h Jp 2Jp + 1 (2Ji + 1) 2Jp − 1 !! 2 2Jp +1 I

I2 III II I I< × 2Jp +1 I nf Lf SJf IIQ2 I|ni Li SJi >I , c ωfi

(408)

/\

putting in it Jp = 2, S = 0 and L = J, I2 ωfi5 II< III II ) 2 3 sp−e ( IIQ2 I|ni Li SJi >II , → n wJf Ji kJp = 2; E2 = L SJ I f f f h 2 · 5(2Ji + 1)[3!!]2 c5 5I ωfi I< III II II2 1 = I Jf IIQ2 I|Ji >I 15h(2Ji + 1) c5 (409) /\

/\

/\

where Q2 is the operator of the quadrupole moment of the electric charge distribution in the electron shell of the molecule. Due to the axial symmetry of the considered molecule, only one of the five / independent components of the quadrupole moment /\

' = 23 Dzz (this value is the quantum-mechanical average tensor is nonzero, Q20 of the dipole moment of the distribution of electrons in the considered electronicvibrational state). Therefore, the transformation formula for the components of the quadrupole moment tensor upon passing from the LSC to the MSC has the form

/\

' Q2μ =

/ =

2 [

2∗ ' 2∗ ' ˆ 2ν ˆ 20 Dμν = Dμ0 (α, β, 0)Q (α, β, 0)Q

ν=−2

(410)

3 2π Yˆ 2μ (α, β)Dzz' . 5

To calculate the matrix element of this operator, it is necessary to use rotationalnuclear wave functions (150), I Jf Mf If MIf IQ2μ |Ji Mi Ii MIi > I ][ ]< 1[ = 1 + (−1)Ji +Ii 1 + (−1)Jf +If Jf Mf If MIf IQ2μ |Ji Mi Ii MIi > 2 ][ ] 1[ = 1 + (−1)Ji +Ii 1 + (−1)Jf +If /2 I < 3 × 2π Dzz' Jf Mf If MIf IYˆ 2μ (β, α)|Ji Mi Ii MIi >. 5


= Jf Mf IYˆ 2μ (β, α)|Ji Mi > If MIf I Ii MIi >


= 1 + (−1) 2 2 /\

(414)

Substitution of this result into (409) completes the calculation of the probability of the rotational quadrupole transition, ) sp−e ( wJf Ji kJp = 2; E2 = I I2 / I ωfi5 II 1 [ ] 3(2Ji + 1) ' Jf 0 1 I Ji +Ii C δ δ 1 + D = (−1) I I I I J J ±2 f i i i zz J 020 i I 15h(2Ji + 1) c5 I 2 2

(415)

] '2 ( Jf 0 )2 1 ωfi [ Ji +Ii 1 + Dzz CJi 020 δJi Ji ±2 . (−1) 10h c5 5

=

If there is an Ii even number, the rotational states of the molecule are characterized by only even values Ji . Accordingly, the considered quantum transitions occur between rotational states with even values of J. In addition, vice versa, in cases where the total nuclear spin is an odd number, rotational transitions are allowed only between levels characterized by odd values of the rotational quantum number.

4.5 Line Strengths of Electrical Dipole Transitions Between Rotational Levels of Diatomic Molecules In this section, we will consider methods for calculating the probabilities of radiative transitions between the rotational levels of diatomic molecules in various electronic configurations. We will be interested exclusively in dipole electrical transitions, since it is this type of radiative transitions that is of primary interest in the study of molecular

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V. K. Khersonsky and E. V. Orlenko

spectra. In Sect. 4.1, we presented general formulas for the probabilities and line strengths of electric dipole rotational transitions in a rigid rotator [see formulas 111– 113]. The formulas that interest us in this section differ from these formulas in the sense that to calculate the matrix elements included in them. It is necessary to use the total wave functions of the molecular states of a diatomic molecule discussed above in the Sect. 3, and take into account that the operator of the dipole moment of the electron system is dˆ = e

Ne [

ri =

Ne [

i=1

dˆ i ,

(416)

i=1

where dˆ i = eri and all radius vectors of electron positions are measured from the center of mass. In order to find the value of the dipole moment of the molecule in a given electronic state, we have to integrate the operator of the dipole moment (416) with the density of the electronic distribution, which is given by the square of the modulus of the electronic wave functions and whose coordinates are determined in the MSC. In turn, in order to make this integration possible, it is necessary either to transform the coordinates of the electronic wave functions to the LSC, or to transform the coordinates included in the definition of the dipole moment to the MSC. The latter turns out to be a simpler procedure. For the cyclic components of the dipole moment operator (irreducible tensor operator of rank 1), we have dˆ 1μ ({ri }) = =

[

Ne ( ) Ne ( ) [ [ [ 1∗ dˆ i = dˆ i' Dμν (α, β, 0) i=1

μ

ν=0,±1

1∗ ' Dμν (α, β, 0)dˆ 1μ ({ri }).

i=1

ν

(417)

ν=0,±1

There is a dipole moment dˆ i = er'i associated with one electron, and is determined in the MSC. We will characterize the quantum state of a molecule by a set of quantum numbers ς JM πi . The probability of a spontaneous radiative transition between states is characterized by sets of quantum numbers ςi Ji Mi πii and ςf Jf Mf πif can be expressed in terms of the reduced matrix element of the dipole transition see formulas (378)–(381). We will write this probability as I< I I2 32π 3 ( νςf Jf Mf πif ςi Ji Mi πii )3 I ςf Jf Mf πif Id1Mf −Mi |ςi Ji Mi πii >I Aςf Jf Mf ςi Ji Mi = 3h 2Jf + 1 I I I2 I< (418) I I I II I )2 )3 I ςf Jf πif Idˆ 1 I|ςi Ji πii >I ( Jf Mf 4 ( CJi Mi 1Mf −Mi . = 3 ωςf Jf Mf πif ςi Ji Mi πii 3h 2Jf + 1

Elements of Theory of Angular Moments as Applied to Diatomic Molecules …

343

III II < Here the quantity ςf Jf πif IIdˆ 1 I|ςi Ji πii > is the reduced matrix element of the operator of the dipole moment of the system of electrons. As in Sects. 4.1 and 4.4, we will be interested in the total transition probability, which is obtained by summing the probability (381) over all values of μ = Mf − Mi and Mf for a given Mi . We have already obtained a formula for this probability in the reviewed section Aςf Jf πif ςi Ji πii =

[

Aςf Jf Mf πfi ςi Ji Mi πii =

μ,Mf

)3 4 ( ωςf Jf Mf πif ςi Ji Mi πii 3h3

I2 I< III II I I I ςf Jf πif IIdˆ 1 I|ςi Ji πii >I 2Ji + 1 ) ) ( 3 (ν 32π ςf Jf Mf πif ςi Ji Mi πii 3 S ςf Jf πif ςi Ji πii = . 3h 2Ji + 1

(419)

Below we will be interested in the question of how to calculate the strength of the transition line. If this value is found, then the calculation of the actual transition probability is not difficult. Therefore, we will consider methods for calculating the reduced matrix element of the dipole moment operator (417). Let us first consider the matrix elements of the radiative transition between the rotational levels of a molecule whose total electron spin, S, is zero. In this case, the total angular momentum of the molecule (excluding nuclear spins) is its moment N, that is, J = N, and the wave functions are determined by formula (236) of Sect. 3.8, which we will write here in the form (/ / ) Ψςν/\JM πi r'i , R =

N

√

( (/ / ) (/ / )) 1 Ψςν/\JM r'i , R + κ/\Ni Ψςν−/\JM r'i , R . 2 − δ/\0 (420)

The matrix element of the dipole transition ςi Ji Mi πii → ςf Jf Mf πif can be represented as
=

1 /( )( )× N 2 2 − δ0/\f 2 − δ0/\i

I ςf νf /\f Jf Mf Id1μi |ςi νi /\i Ji Mi > I (f ) < +κ/\Ji ςf νf − /\f Jf Mf Id1μi |ςi νi /\i Ji Mi > I (i) < +κ/\J ςf νf /\f Jf Mf Id1μi |ςi νi − /\i Ji Mi > i ] I (f ) (i) < Id1μ |ςi νi − /\i Ji Mi > . ς ν − /\ J M + κ/\Ji κ/\J f f f f f i i /
[ / ∞ / 2π / π / (/ / ) 1∗ (/ /) Ψς∗f νf /\f Jf Mf r'i R Dμν = (α, β, 0)d ' 1ν r'i


/( ) 2Jf + 1 (2Ji + 1) × = D0 (ς /\ν) 4π / 2π / π J∗ J 1∗ DM (α, β, 0)DM (α, β, 0)Dμ0 (α, β, 0) sin βd βd α i /\ f /\ 0 0 / (2Ji + 1) Jf /\ Jf Mf )C = D0 (ς /\ν) ( C . 2Jf + 1 Ji /\10 Ji Mi 1μ


= D0 (ς /\ν) ( 2Jf + 1 Ji −/\10 Ji Mi 1μ / (2Ji + 1) Jf /\ Jf Mf Ji −Jf +1 )C = (−1) D0 (ς /\ν) ( C . 2Jf + 1 Ji /\10 Ji Mi 1μ (428)
ς ν/\J π f if I < J M I |ς > / ς ν/\Jf Mf πf d1μ ν/\Ji Mi πi = CJifMif1μ , 2Jf + 1

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V. K. Khersonsky and E. V. Orlenko

where the result is presented in the form of the Wigner–Eckart theorem, and the reduced matrix element is / III II 1 − πif πii J /\ D0 (ς /\ν) (2Ji + 1)CJif/\10 . ς ν/\Jf πif IIdˆ 1 I|ς ν/\Ji πii > = 2


of eigenfunctions [form ] Ψn0 (r1 , ..., r N , ξ1 ...ξ N ) , is normalized by the condition Ψn0 I Ψm0 = δnm . This set ] I >< I ˆ Here we use the Dirac possesses the completeness property n IΨn0 Ψn0 I = 1. I 0 > I 0 > −i En t symbols for description wave vectors IΨn (t) = IΨn e h in Schrodinger representation. If the electron system is under an electromagnetic field, or other timedependent fields, influenced by the electron system, we consider this interaction of system with the fields as a perturbation in the case, when this interaction I Ithe electron Iˆ I IV (t)I = Hˆ 0 + Vˆ (t) |Ψ(t)> i ∂t

(2)

which is written in Dirac’s notations. I > I > The zero-approximation wavevector IΨi0 of the initial state |Ψ(t)>|t=0 = IΨi0 I > I > is an eigenvector of the Hamiltonian of the unperturbed system: H 0 IΨi0 = E i0 IΨi0 , where we have rewritten the Schrödinger equation of the unperturbed system in Dirac’s symbols. We search for the wavevector |Ψ(t)> which is the solution to the Schrödinger equation, by the method of successive approximations, as it is typically solved in conventional perturbation I 0 theory. IFor> this, we will search |Ψ(t)> as an > IΨ (t) = IΨ 0 e−i Ehn t by using the completeness expansion on the basis vectors n >< n I ] I property: |Ψ(t)> = n IΨn0 (t) Ψn0 (t)I Ψ(t)>. Substituting this expansion in the timedependent Schrödinger equation, we obtain /\

( )[I I >< h ∂ [ II 0 >< 0 II IΨ 0 (t) Ψ 0 (t)I Ψ(t)>, Ψn (t) Ψn (t) Ψ(t)> = Hˆ 0 + Vˆ (t) n n i ∂t n n ) ( [I >∂< 0 I h [ i 0 II 0 >< 0 II 0 I I − Ψn (t) Ψ (t) Ψ(t)> = − E n Ψn (t) Ψn (t) Ψ(t)> + i h ∂t n n n )I [( I >< = E n0 + Vˆ (t) IΨn0 (t) Ψn0 (t)I Ψ(t)>,

−

n

[I [ I I I >< > < IΨ 0 (t) ∂ Ψ 0 (t)I Ψ(t)> = − i Vˆ (t)IΨn0 (t) Ψn0 (t)I Ψ(t)>, ⇒ n n ∂t h n n ∂ < 0 II i [ < 0 II ˆ II 0 >< 0 II Ψ (t) Ψ(t)> = − Ψk (t) V (t) Ψn (t) Ψn (t) Ψ(t)>. ∂t k h n I I > I > I > > |Ψ(t)> = IΨi0 (t) + IΨ (1) (t) + IΨ (2) (t) + ... + IΨ (n) (t) .

(3)

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E. V. Orlenko and V. K. Khersonsky

For the first-order correction: ∂ < 0 II (1) > i [ < 0 II ˆ II 0 > < 0 II 0 > Ψk (t) V (t) Ψn (t) Ψn (t) Ψi (t) , Ψk (t) Ψ (t) = − ∂t h n I I > ∂ < 0 II (1) > i< Ψk (t) Ψ (t) = − Ψk0 (t)IVˆ (t)IΨi0 (t) , ∂t h (t < 0 ( ' )I ( ' )I 0 ( ' )> ' < 0 I (1) > i Ψk t IVˆ t IΨi t dt = Ψk (t)I Ψ (t) = − h

δni

0

=−

i h

(t

I ( )I > ' Ψk0 IVˆ t ' IΨi0 eiωki t dt ' , (k /= i )


Ψi0 IVˆ t ' IΨi0 dt ' .


-state to the |i>-state. Then, the first-order correction to the wave vector has the form [ i 0I > I (1) > IΨ (t) = − i e− h Ek t IΨk0 h k

(t

I ( )I > ' Ψk0 IVˆ t ' IΨi0 eiωki t dt ' , (k /= i )


tν > tν−1 > ... > t2 > t1 > 0.

(8)

The last equation can be transformed to a more symmetric form by introducing the time ordering operator (Chronological operator) τˆ : ( ) ( ) I (ν) > i 1 1 ν [ ' [ ' [ ' II 0 > IΨ Ψn ν exp − E n ν t × = ... ν! ih h n n n ν

(t ×τˆ

(t ...

0

ν−1

1

I I > ( ) < I { Ψn0ν I Vˆ (tν )I Ψn0ν−1 exp i (ωn ν n ν−1 tν ...

(9)

0

I > < I ( ) I ... Ψn01 I Vˆ (t1 )I Ψi0 exp i ωn 1 i t1 }dt1 ...dtν .

This expression defines the corrections to the< waveI vector to the ν-th order of 0 I perturbation. The expansion >< 0 CIn = Ψn (t) Ψ(t)> in the expansion of ] II coefficient 0 wave function |Ψ(t)> = I n Ψ>n (t) Ψn (t)I Ψ(t)> determines the probability to find a quantum system in the IΨn0 (t) state at the time moment t. In I other > words, the coefIΨ 0 (t) to the final state ficient corresponding to the transition from the initial state i I \ I 0 IΨ f (t) can be written in an invariant form, taking into account the representation of the wave vector as I I > I > I > > |Ψ(t)> = IΨi0 (t) + IΨ (1) (t) + IΨ (2) (t) + ... + IΨ (ν) (t) + ... :

(10)

Emission and Absorption of Photons in Quantum Transitions …

355

(11) where we introduced the perturbation operator in the interaction representation i ˆ0 i ˆ0 Wˆ (t) = e h H t Vˆ (t)e− h H t

(12)

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E. V. Orlenko and V. K. Khersonsky

I I2 The probability of transition at the time t is given by P f i (t) = IC f (t)I when a transition occurs between states of a discrete spectrum, and it is given by d P f i (t) = I I IC f (t)I2 dν f when the transition occurs in the range dν f of the continuous spectrum.

1.1.2

Time-Dependent Representations in Quantum Mechanics

Now, a few words about time-dependent representations in Quantum Mechanics. In quantum mechanics, there are three representations of operators and wave vectors, which dependent on time and are related to each other by unitary transformations, they are: the Schrödinger representation, the Heisenberg representation, and the Interaction representation. It is known that in the Schrödinger representation all operators of physical quantities Lˆ = Lˆ Sch do not depend on time, the time dependence, which determines the time dependence of the average value of a physical ˆ is in the wave function Ψ(t). The time evolution of quantity L(t) = , the wave function is described by the Schrödinger equation: ∂t∂ |Ψ(t)> = − hi Hˆ |Ψ(t)>. Hˆ When implementing a unitary transformation, of the form Uˆ = ei h t , Uˆ † = † ˆ ˆ H H e−i h t = e−i h t = Uˆ −1 , the time dependence passes from the wave function to the operators of physical quantities. Now all the operators corresponding to physHˆ Hˆ ical quantities are explicitly time-dependent Lˆ H (t) = ei h t Lˆ Sch e−i h t , and the wave Hˆ functions ψ H = Uˆ Ψ(t) = ei h t Ψ Sch (t) are independent of time. The reverse is also ˆ ˆ H H Hˆ true: Lˆ Sch = e−i h t Lˆ H (t)ei h t , Ψ Sch (t) = e−i h t ψ H . The third temporal representation is an interaction representation. It is useful in the case when the total Hamiltonian can be written as Hˆ = Hˆ 0 + Vˆ (t) a sum of the Hamiltonian of the unperturbed system and additional interaction of the system with Hˆ 0 an external object (field or particles). The unitary transformation Uˆ = ei h t implemented to the wave functions and operators given in Schrödinger representation transHˆ 0 Hˆ 0 lates them to the Interaction representation: Lˆ int (t) = ei h t Lˆ Sch e−i h t , Φint (t) = Hˆ 0 Uˆ Ψ(t) = ei h t Ψ Sch (t). The reverse translation is also true. In this representation a Schrödinger equation has the form: ∂ ∂ ∂ Hˆ 0 Hˆ 0 Hˆ 0 Φint (t) = ei h t Ψ Sch (t) = i Φint (t) + ei h t Ψ Sch (t) = ∂t ∂t h ∂t ) Hˆ 0 i i Hˆ 0 t ( ˆ 0 Φint (t) − e h H + Vˆ (t) Ψ Sch (t) = =i h h i Hˆ 0 i Hˆ 0 Hˆ 0 Φint (t) − Hˆ 0 ei h t Ψ(t) − ei h t Vˆ (t)Ψ Sch (t) = =i h h h Φint (t) i Hˆ 0 i Hˆ 0 Hˆ 0 = − ei h t Vˆ (t)e−i h t ei h t Ψ Sch (t) ≡ − Wˆ int (t)Φint (t), ⇒ h h ∂ i Φint (t) = − Wˆ int (t)Φint (t) . ∂t h

(13)

Emission and Absorption of Photons in Quantum Transitions …

1.1.3

357

The Probabilities of Quantum Transitions

Consider some quantum system that is in the initial state i and can be described by a set of quantum numbers ηi . Although this set must be detailed in each specific case, here we note that throughout below we will consider isolated quantum systems like atoms and molecules. In the most general sense, such a system is characterized by the total angular momentum J (or F) and its projection M J (or M F ) onto the quantization axis (the Z axis of the laboratory coordinate system) and other quantum numbers (set γ), which may include, for example, the principal quantum number if we are talking about atoms. On the other hand, it may be a number denoting the electronic term of a molecule or the characteristics of its vibrational states. In addition, the set {γ} may contain quantum numbers corresponding to other angular momenta, which together form the total angular momentum. For example, if the total angular momentum of an atom, taking into account the hyperfine structure, is F, and its projection M F , that is η = γ F M F , then the set γ should include the nuclear spin I and the angular momentum J (the total angular momentum of the electron shell). In this case, we will write that η = γ J I F M F , always implying that γ there is a collection of other quantum numbers. First, consider the case when the states of a quantum system are nondegenerate. Suppose that the system is in an electromagnetic field in volume V. Electromagnetic field can be considered as a set of plane waves, and in the quantum description—a set of photons n kλ with different momenta p = hk (k is the wave vector), being in polarization states λ (λ = 1,2) and having frequencies ν = ck/2π = ω/2π where ω is an angular frequency. The number of photons of each type will be denoted as n kλ . This description corresponds to the three-dimensional transverse calibration of the electromagnetic field, at which the vector potential satisfies the conditions di vA = 0, A0 = eΦ = 0. WeIwill use \ notation |ηi > for the state of the system, and for the state of the system I (i) |i> = In kλ ηi and the electromagnetic field. Under the influence of photons of this field, described by the interaction of electrons with photons Vˆeph , the electronic I \ I (f) system goes over to the state f (or | f > = In kλ η f ). In this state it can be described by a set of quantum numbers η f of the material matter, and the electromagnetic field, characterized by a set of number photons with the same wave vectors and in the same (f) polarization states, n kλ . As a result of the transition, one or more photons are emitted or absorbed, in some polarization states with different pulses and frequencies. Differences in momenta and polarization states for photons initiating the transition and for photons emitted/absorbed as a result of the transition determine the angular and polarization characteristics of the emitted or absorbed radiation. They are described by (e) differential transition probabilities, dw (a) f i (for absorption) and dw f i (for emission). We will pay considerable attention to the discussion of these quantities. The differences in frequencies between initiating and emitted/absorbed photons are described by the profile of the emission/absorption spectral line, which is formed due to finite lifetimes of quantum states. To take these effects into account, it would be necessary

358

E. V. Orlenko and V. K. Khersonsky

to consider the initial and finalI states of a( quantum )system not as stationary, that > is |ηi > ∝ exp(−i E i t/h), and Iη f ∝ exp −i E f t/h , but as decaying with some relaxation constant, γi and γ f , or a probability of decay per unit time (within one second) of a( given state, that)determine line width, |ηi > ∝ exp(−i E i t/h − γi t) and I > Iη f ∝ exp −i E f t/h − γ f t . Since the theory of line widths is not the purpose of this book, in general we will not discuss in detail the issues related to the profile of spectral lines. In many cases, although not always, differential transition probabilities and profiles can be discussed separately and we will try to follow this whenever possible. As is well-known [1–3] (and will be discussed below), the interaction of electrons with an electromagnetic field can be considered by methods of perturbation theory, in which the smallness parameter is the fine structure constant. According to the general formulas of the theory of time-dependent perturbations, the probability P f i (t) of transition from state |i> to state | f > under the influence of a perturbation that acts in the time interval [0,t] can be represented in the form I2 I I I P f i (t) = I< f |Wˆ (1) (t)|i> + < f |Wˆ (2) (t)|i> + ...I .

(14)

Here are i < f |Wˆ (1) (t)|i> = −

h

(t

( ) < f |Vˆ t ' |i>dt ' ,

0

' ( )2 [ ( t (t ( ') ( ) i (2) < f |Vˆ t |k> dt '' dt ' , < f |Wˆ (t)|i> = − h

k/=i 0

0

' '' ( )3 [ [ ( t (t (t ( ') ( '' ) ( ) i (3) < f |Vˆ t |l> it will be fair to write † ˆ nˆ kλ |n kλ > = bˆkλ bkλ |n kλ > = n kλ |n kλ >.

(21)

360

E. V. Orlenko and V. K. Khersonsky

In what follows, these states |n kλ > with a certain number of energy quanta in a given mode (photons) will be called Fock states. They are characterized by the fact that the number of photons in them is precisely defined, while the phase in them is absolutely undefined. This combination of operators, up to a factor, the dimension of energy, determines the operator of the energy of the electromagnetic field, written for one mode: ( ) ( ( ) [ 1 1 † ˆ 2 2 0 ˆ ˆ ˆ ˆ E + H dV = hωk bkλ bkλ + H = 8π 2 kλ V ( ) [ 1 = hωk nˆ kλ + . (22) 2 kλ Thus, there is a set of eigenvalues {n} of the indicated operator and the corresponding set of states {|n>} corresponding, as a solution of the Sturm–Liouville problem, to the condition of orthogonality = δmn and completeness ] ˆ n |n>kλ -vector of state |n> as an infinite-dimensional one-column matrix, and, accordingly, its conjugate kλ = bˆkλ |n kλ >: ( ) † ˆ ˆ † nˆ kλ |φ> = bˆkλ − 1 bˆkλ |n kλ > = bkλ bkλ |n kλ > = bˆkλ bˆkλ † ˆ = bˆkλ bˆkλ bkλ |n kλ > − bˆkλ |n kλ > =

(23)

= n kλ bˆkλ |n kλ > − bˆkλ |n kλ > = (n kλ − 1)|φ>. [ ] † = δk' k δλ' λ . The We used here the commutation rules for operators bˆk' λ' bˆkλ resulting Eq. (8) means that the state vector |φ> = bˆkλ |n kλ > is an eigenvector † ˆ of the operator nˆ kλ = bˆkλ bkλ and corresponds to the eigenvalue (n − 1), that is, up to a factor φ, corresponds to the state described by the vector |n kλ − 1>, namely |φ> = φ|n kλ − 1>. Then we can write the matrix element of the operator bˆkλ as = φ, and the operator itself can be considered the operator of decreasing the eigenvalue by 1: bˆkλ |n kλ > = φ|n kλ − 1>. In a similar way, † on the state |n>kλ we can show that the result of the action of the operator bˆkλ † ˆ is the vector bkλ |n kλ > = |η> = η|n kλ + 1>, the matrix element of which will be, † = = η that is, we are dealing with an operator of raising an eigenvalue by 1. Taking into account the Hermitian conjugacy of the lowering and increasing operators with respect to each other, we have the obvious equality:

Emission and Absorption of Photons in Quantum Transitions … † † = = φ ∗

361

(24)

then from (6), taking into account the properties of orthogonality and completeness of eigenstates, it obviously follows that † ˆ n kλ =kλ = [ † = = m kλ † |n − 1>kλkλ kλ = |φ|2 = . We here mean transitions within a quantum electronic system, for example, transitions between the rotational states of a molecule, which are accompanied by the emission or absorption of two-photons, therefore they are called two-photon transitions. Next, we will discuss one-quantum transitions.

362

1.1.4

E. V. Orlenko and V. K. Khersonsky

One-Quantum Transitions

The probabilities Eq. (14) of emission and absorption transitions I I \ in one-quantum \ I (i) I (f) (one-photon transitions) between states |i> ≡ In kλ ηi and | f > ≡ In kλ η f can be represented as I2 I I I (t I I ( ) i ' (e) iωk t 'I (e) I ˆ < f |V π p |i>e dt I = P f i (t) = I− I I h 0 I t I2 I( I ( ) i (ωk +ω f i )t ' ' I 1I I , π e = 2 II M (e) dt p fi I h I I 0

I I2 I I (t I I ( ) i ' (a) −iω t ' (a) k I < f |Vˆ π p |i>e P f i (t) = I− dt II = I h I 0 I t I2 I( I I ( ) 1 II ' (a) i (ω f i −ωk )t 'I dt I , = 2 I M f i πp e h I I

(28)

0

where we singled out the time factors in the wave functions of the initial and final stationary states and introduced the designations for the transition frequency ω f i = E η f −E ηi h

and transition amplitude ( ) M (e) f i πp = ( ) M (a) f i πp

/

/ =

2π h V ωk 2π h V ωk

/

n (i) kλ

( ( ) ∗ ˆjη f ηi (r) · ekλ e−ik·r d V , +1 V

/

( ( ) (i ) ∗ ˆjη f ηi (r) · ekλ eik·r d V . n kλ

(29)

V

Determining these values, we took into account Eq. (26) and the fact that the vector potential of a plane electromagnetic wave in a three-dimensional transverse gauge has the form Eq. (18) and the matrix element of the electron current density is (see [1–3]) ( ) ( ) ˆjη f ηi (r) = ecΨη† r j αΨ ˆ ηi r j . f

(30)

Emission and Absorption of Photons in Quantum Transitions …

363

Assuming that the matrix elements are independent or weakly dependent on time, formulas Eq. (28) can be written in the form, I2 1 II (e) ( ) (e) I π f , t) (ω I , IM p k fi fi h2 I2 1 II (a) ( ) (a) I P (a) f i (t) = 2 IM f i π p f f i (ωk , t)I , h P (e) f i (t) =

(31)

where ( ) I t I2 I I2 2 ω f i ±ωk I( I I I i (ω f i ±ωk )t sin t I I 2 − 1I ' Ie ) I = ( = II ei (ω f i ±ωk )t dt ' II = I ( )2 I I ω ±ω ω ± ω fi k fi k I I 0 2 ( ) 2 ω f i ±ωk sin t ) ( 2 lim ( → δ ω f i ± ωk . ) 2 t→∞ ω ±ω t f i2 k (e ) f f ia (ωk , t)

(32)

As you can see, the obtained probabilities have resonances at ω f i = ±ωk for the emission and absorption. ( ) ) (e) ( The differential probabilities dW (a) f i π p (for absorption) and dW f i π p (for emission), ( ) of quantum ( ) transitions per unit time under the influence of perturbations Vˆ (a) π p and Vˆ (e) π p , are defined as, ( ) ) ( ) (e) ( dW (e) f i π p = T f i π p ρ E ph d E ph , ( ) ) ( ) (a) ( dW (a) f i π p = T f i π p ρ E ph d E ph .

(33)

Let us determine the included ( quantities ) ) in these formulas. (a) ( π π The quantities T (e) and T p p are called the probabilities of emission and fi fi absorption calculated per unit time and are determined by the ratios ] 1 (e) = lim P f i (t) , t→∞ t ] [ ) 1 (a) (a) ( T f i π p = lim P f i (t) . t→∞ t ( ) T (e) f i πp

[

(34)

It is easy to calculate the limits included in these definitions. For example, for the first of them, we get,

364

E. V. Orlenko and V. K. Khersonsky

⎡

( )⎤ 2 ω f i +ωk t sin 2 1 (e) 1 ⎢1 ⎥ lim f (ωk , t) = 2 lim ⎣ ( )2 ⎦ = t→∞ t f i h t→∞ t ω f i +ωk ]

[

(

=

ω f i + ωk π δ 2 h 2

2

) =

(35)

) 2π ( δ E ηi − E η f − hωk . h

The obtained connection between the energies of the initial E ηi and final E η f states and the energy of the emitted photon E ph = hωk expresses the energy conservation law, E ηi = E η f +hωk . Similarly, for the formula that corresponds to the absorption of ) ( a photon, we get 2π δ E ηi − E η f + hωk that again expresses the law of conservation h of energy, E η f = E ηi + hωk . Now the probabilities of transitions calculated per unit of time can be written in the form, ( ) 2π II (e) ( )II2 ( ) T (e) IM f i π p I δ E ηi − E η f − hωk , f i πp = h ) 2π II (a) ( )II2 ( ) (a) ( T f i πp = I M f i π p I δ E ηi − E η f + hωk . h

(36)

Taking into account these relations, the differential probabilities can be represented in the form of the “Fermi golden rule”, ( ) 2π II (e) ( )II2 ( ) ( ) dW (e) IM f i π p I δ E ηi − E η f − hωk ρ E f d E f , f i πp = h (37) ) ( ) ) 2π II (a) ( )II2 ( (a) ( dW f i π p = IM f i π p I δ E ηi − E η f + hωk ρ E f d E f . h ) ( ) ( The quantity ρ E f d E f = ρ E ph d E ph is the number of final( states ) (f ) of the energy of the electromagnetic field E ph , the density of which is ρ E ph , within the interval d E ph . This value is introduced in order to take into account the following circumstance. Let the energy of a photon emitted during the transition of an atomic system to a discrete state be E ph = hωk . The number of degrees of freedom of the electromagnetic field determines the number of final states of a system including an atomic system and an electromagnetic field. Because of the atomic system is in a strictly defined state corresponding to energy E η f , and the electromagnetic field contains many (or infinitely many) final states with energy equal to or close to energy E ph . Indeed, there is d N ph = V

p 2ph dp ph dΩ (2π h)3

=V

E 2ph dp ph dΩ c2 (2π h)3

(38)

states of photons with energies near E ph [= hωk and momenta ] p ph = E ph /c falling into the interval of absolute values of p ph , p ph + dp ph , lying in a solid angle dΩ = sin ϑdϑdϕ and having a certain polarization ekλ . Therefore, the question of a quantum transition to a definite final state loses its meaning. One can only consider

Emission and Absorption of Photons in Quantum Transitions …

365

the probability of transition to one of the final states presented in the interval d N ph . dp Since d Ephph = 1c , the density of the number of final states per unit energy interval is ( ) d N ph ω2 dΩ ρ E ph = =V k . d E ph h(2π c)3

(39)

If, when discussing the angular and polarization characteristics of emitted and absorbed photons, one is not interested in the frequency composition of these photons, then it is convenient to use the differential transition probabilities integrated over the energies of the final states. The differential probabilities introduced in this way have the form

(40)

366

E. V. Orlenko and V. K. Khersonsky (

(a)

dw f i (k, λ) = Ef

2π (a) ( ) dW f i π p = h

( I I ( ) ( ) I (a) ( )I2 IM f i π p I δ E ηi − E η f + hωk ρ E ph d E f = Ef

I I2 I ( ( I/ ( ) ) ω2 dΩ I 2π h / (i ) ( I 2π ∗ ik·r I I δ E η − E η + hωk V k ˆjη η (r) · e n d V dE f = = e f i i f kλ I I kλ h h(2π c)3 I V ωk I Ef

V

I2 I I ( I( ( ) ) ω2 dΩ I 2π h (i ) II (ˆ 2π ∗ ik·r n kλ I d V II δ E ηi − E η f + hωk V k 3 d E f = = jη f ηi (r) · ekλ e h V ωk h(2π c) I I Ef

V

I2 I I ( I( ( ( ) ) hω dΩ I (i ) II k ∗ eik·r d V I δ E − E = n kλ I dE f = jˆη f ηi (r) · ekλ ηi η f + hωk I 2π h2 (c)3 I I Ef

V

(i ) = n kλ

I I2 I( ( I ) I I ∗ ik·r I I dΩ, ˆjη η (r) · e e d V f i kλ I I 3 2 2π h (c) I I hω f i

V

(a) dw f i (k, λ)

dΩ

I I2 I( ( I ) I ∗ ik·r I . ˆjη η (r) · e e d V f i kλ I I

(i ) ω f i II = n kλ 2π hc3 II V

(41) Since the squares of the moduli in formulas Eqs. (40) and (41) represent the same quantity, we can write that dw (e) f i (k, λ) dw (a) f i (k, λ)

=

) ( n (ikλ) + 1 n (i) kλ

,

(42)

and the probability dw (e) f i (k, λ) can be represented as (sp)

(ind) dw (e) f i (k, λ) = dw f i (k, λ) + dw f i (k, λ) (sp)

(sp)

= n (ikλ) dw f i (k, λ) + dw f i (k, λ), I2 I I I( ) I ω f i II (ˆ (sp) ∗ ik·r jη f ηi (r) · ekλ e d V II dΩ dw f i = I 3 2π hc I I

(43)

V

(sp)

The last quantity, dw f i , is the probability of spontaneous emission, which is initiated by the interaction of the quantum system with the vacuum states of the is the electromagnetic field, in the absence of external photons. The quantity dw (ind) fi probability of induced radiation, which is initiated by external photons—the emitted photons have the same characteristics as the initiating photons.

Emission and Absorption of Photons in Quantum Transitions …

367

The number n (ikλ) of photons propagating in the k direction, and having polarization ekλ and frequency ωk , can be related to the spectral intensity Iωkλ of radiation of frequency ωk , polarization ekλ and propagating in the k direction, 3 n (i) kλ = (2π )

c2 Iωkλ . ω2 hω

(44)

Then, taking into account Eqs. (41)–(43) we obtain that (ind) dw (a) = f i (k, λ) = dw f i

(

2π c ω

)3

1 (sp) Iωkλ dw f i . hc

(45)

The differential probabilities of the angular distribution of photons are obtained by summing the obtained formulas Eqs. (40) and (41) in two possible directions of polarization dw (e) f i (ϑ, ϕ) dΩ

=

[ dw (e) f i (k, λ) dΩ

λ=1,2

dw (a) f i (ϑ, ϕ) dΩ

=

[ dw (a) f i (k, λ) dΩ

λ=1,2

, (46) .

The total transition probabilities are determined by integrating the differential probabilities Eqs. (40) and (41) over the angles and summing over the polarizations Eq. (46), that is, w (e) fi =

[ ( dw (e) f i (k, λ) λ=1,2 Ω

w (a) fi

=

dΩ

[ ( dw (a) f i (k, λ)

λ=1,2 Ω

dΩ

dΩ, (47) dΩ.

If the incident radiation is isotropic and not polarized, that is, Iωkλ =

c 1 Iω = ρω , 2 4π

(48)

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E. V. Orlenko and V. K. Khersonsky

where ρω is the spectral density of radiation, then integration Eq. (30) over angles and summation over polarizations gives the relation for the total probabilities 2 3 c (sp) π c I = w ρω ω f i hω3 hω3

(sp) 4π

(ind) w (a) = wfi fi = wfi

3 2

(49)

Similar formulas can be obtained when the initial and final states are degenerate system can be found with equal and have statistical weights gi , g f . If anI atomic > probability in any of the gi initial states Iηin , then the total transition probabilities (taking into account degeneracy) are determined by summing over all sublevels of the final state, f n , and averaging over the sublevels of the initial state i n , that is Wfi =

1 [ [ w fm in , gi i ∈|η > n i f m ∈|η f >

(50)

where W f i , w fm in are any of the three probabilities—absorption, induced or spontaneous emission. Formulas for differential probabilities are generalized in the same way in the case of degenerate states. The Eq. (49) can be rewritten as (sp) π

(ind) g f Wi(a) = gi W f i f = gi W f i

2 3

c ρω . hω3

(51)

If we separate out the dependence on the external radiation field in the probabilities of absorption and induced radiation in the form Wi(a) f = Bi f ρω ,

(52)

W (ind) = B f i ρω , fi (sp)

then three quantities Bi f , B f i , and A f i = W f i form a set independent of the external field and determined only by the properties of the atomic system. These quantities are called Einstein’s coefficients. Using relation Eq. (51), the relationship between these coefficients can be represented as g f Bi f = gi B f i = gi A f i

π 2 c3 . hω3

(53)

Emission and Absorption of Photons in Quantum Transitions …

369

1.2 Dipole Transitions The basis for further discussion of one-quantum transitions is formula Eq. (46) for the differential probability of a spontaneous transition. Since, as discussed in Sect. 1.1.4, the differential probabilities of absorption and induced emission are related by conditions Eq. (45), and the total probabilities by conditions Eq. (49), knowledge of the angular and polarization characteristics of spontaneous emission allows one to determine the same properties for stimulated emission and absorption. In order to calculate the volume integral in formula Eq. (43), we use the fact that in a large number of applied problems requiring the calculation of the probabilities of quantum transitions. A quantity (k:r) on a linear scale comparable to the dimensions of a molecular system turns out to be a small parameter (the so called long-wave approximation) if to be interested in radiation wavelengths lying in the optical or ultraviolet regions of the spectrum. Indeed, if E is the energy of the atom (on the order of the energy of the first excited state with the principal quantum number n = 2), then the wavelength corresponding to this energy is λ ∼ hc/E. But, since at a nucleus charge Z = 1, E ∼ e2 /a0 it follows that k·r ∼

e2 1 e2 ∼ =α= = − η f I ekλ · Hˆ 0 rˆ |ηi > = h ] < I( ) ) e [< II ˆ 0† ( η f H ekλ · rˆ |ηi > − η f I ekλ · rˆ Hˆ 0 |ηi > = =− h < I( < I( ] ) ) e[ = − E η f η f I ekλ · rˆ |ηi > − η f I ekλ · rˆ |ηi >E ηi = h( ) < I( ) ) e E η f − E ηi < I( η f I ekλ · rˆ |ηi > = −ω f i η f I ekλ · eˆr |ηi > = =− ) ( h < I ˆ i > = −ω f i ekλ · d f i = −ω f i d f i · ekλ . = −ω f i ekλ · η f Id|η

(64)

Here we have introduced the dipole moment operator dˆ = eˆr and the matrix element d f i of this operator, < I ˆ i >. d f i = η f Id|η

(65)

The described approximation is called a dipole. The differential probability of spontaneous emission in the dipole approximation is (sp)

dw f i (El, k, λ) =

I ω3f i I Id f i · e I2 dΩ kλ 3 2π hc

(66)

The identification E1 was introduced in order to distinguish in the future the probabilities calculated in the electric dipole approximation from other types of probabilities. Formula Eq. (66) contains all the information about the angular distribution of radiation at a certain polarization of photons. Summation over the polarizations leads to the following angular distribution, (sp)

dw f i (El, ϑ, ϕ) =

I ω3f i I Id f i × nk I2 dΩ, 3 2π hc

(67)

where nk is the unit vector in the direction of the wave vector k. This formula fully corresponds to the formula for the intensity of dipole radiation in classical electrodynamics [4, 5]. Consider some simple special cases of formulas Eqs. (66) and (67). First, let the system be characterized by the orbital angular momentum L, its projection onto a

Emission and Absorption of Photons in Quantum Transitions …

373

certain quantization axis and the principal quantum number n. In other words, we consider the simplest model of a hydrogen-like atom in the nonrelativistic approximation. To simplify the indexing of the initial and final states, we denote the wave function of the initial and final states as |ηi > = |n i , L i , m i >, I > I > Iη f = In f , L f , m f .

(68)

In this case, the matrix element of the dipole moment can be represented as I < < I ˆ i > = n f , L f , m f Idˆ 1 |n i , L i , m i > d f i = η f Id|η

(69)

where the index 1 was introduced to emphasize that the dipole moment operator is an irreducible tensor of the first rank 1. Taking into account that the dipole moment operator is proportional to the radius vector r, that is dˆ 1 = eˆr = erˆ nr , and using the Wigner–Eckart theorem, formula Eq. (66) can be rewritten as I I2 I I ω3f i II[ < I ν I |n > d n L m L m = (e ) I kλ f f f 1ν i i i I dΩ = I 2π hc3 I ν I I2 I I e2 ω3f i II[ < I ν I |n > r n n = L m L m (e ) I kλ f f f 1ν i i i I dΩ. I 2π hc3 I ν

(sp) dw f i (El, k, λ)

(70)

ν Here the quantities ekλ are the cyclical contravariant components of the vector ekλ , defined in the laboratory coordinate system (LSC). The wave functions of a hydrogen-like system are the products of radial functions Rn L (r ) and spherical functions Y Lm (ϑ, ϕ), that is,

|n Lm> = Rn L (r )Y Lm (ϑ, ϕ).

(71)

Since the angular variables are completely decoupled from the radial variable in both the operator and the wave function, the matrix element in Eq. (70) is easily calculated using the Wigner–Eckart theorem [4] I I I < < < e n f L f m f Ir n1ν |n i L i m i > = e n f L f Ir |n i L i > L f m f In1ν |L i m i > = I d0 n Lf L mf < =√ L f I|nr1 ||L i > = C L i fm i ;1ν RnifL i ;1 2L ' + 1 / 2L i + 1 n f L f L f m f L 0 = d0 C C f , R 2L f + 1 ni L i ;1 L i m i ;1ν L i 0;10

(72)

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E. V. Orlenko and V. K. Khersonsky

where we used the formula for the reduced matrix element of the unit vector operator: < I √ L 0 L f I|nr1 ||L i > = 2L i + 1C L i f010 [6]. The quantity d0 = ea0 is introduced to mainn L

f tain the correct dimension of the matrix element. The radial matrix element RnifL i ;1 can be considered as dimensionless, since the dimension is taken into account by the value d0 . This matrix element is a special case of the more general radial matrix n Lf , which we will consider later in this chapter and which we will define element RnifL i ;J as

n Lf RnifL i ;J

(∞ =

Rni L i (r )Rn f L f (r )r J +2 dr.

(73)

0

If the radial wave functions are exact solutions of the Schrödinger equation for one electron in the Coulomb field of the nucleus (hydrogen atom), then the integral n Lf RnifL i ;1 is calculated in terms of hypergeometric functions (see, for example, [7]) and is equal to n f L f =L i −1

Rn L ;1 i i

(∞ Rn i L i (r )Rn f L f =L i −1 (r )r 3 dr =

= 0

/( ( ) L +1 ( )n +n −2L i −2 ) n f + L i − 1 !(n i + L i )! 4n i n f i ni − n f i f (−1)n f −L i ( ) × = ( )n i +n f 4(2L i − 1)! (n i − L i − 1)! n f − L i ! ni − n f ⎧ ( ) ( )2 ( )⎫ ⎬ ⎨ 4n i n f 4n i n f ni − n f F −n ir − 2 − n f r , −2L i , − ( . − F −n ir − n f r , 2L i , − ( ) ) 2 2 ⎭ ⎩ ni + n f n −n n −n i

f

i

f

(74) Here n ir = n i − L i − 1, n f r = n f − L i . A complete analysis of this expression is outside the context of this chapter. Therefore, we will only give a few useful formulas that allow us to calculate the radial matrix element in some simple cases, ( ( ( (

n1 R10;1 n1 R20;1 n2 R21;1 n0 R21;1

)2 )2 )2 )2

= = = =

28 n 7 (n − 1)2n−5

, (n + 1)2n+5 ) ( 217 n 7 n 2 − 1 (n − 2)2n−6

(n + 2)2n+6 ( ) 219 n 9 n 2 − 1 (n − 2)2n−7 3(n + 2)2n+7 ( ) 215 n 9 n 2 − 1 (n − 2)2n−6

L−1 L Rnn L−1;1 = Rnn L;1

3(n + 2)2n+6 3 / = n n2 − L 2. 2

, , ,

(75)

Emission and Absorption of Photons in Quantum Transitions …

375

In models of hydrogen-like atoms, the radial wave functions can be represented by more complex dependences on the radius, and the radial integrals must be calculated taking these features into account. Let us return to the formula matrix element Eq. (72) for the matrix element of the dipole transition. According to the properties of the Clebsch–Gordan coefficients, the coefficients are nonzero only if the sum L i + L f + 1 = 2g,

(76)

where g is an integer. In this case, /

L 0

2L f + 1g! ) ( (g − L i )! g − L f !(g − 1)! / ) ( (2g − 2L i )! 2g − 2L f !(2g − 2)! , (2g + 1)!

C L i f010 = (−1)g−L f

(77)

Requirement Eq. (76) is nothing more than a consequence of the parity conservation law—the states connected by a dipole transition must have the opposite parity. In addition to this, the moments L i and L f must satisfy the triangle rules, |L i − 1| ≤ L f ≤ |L i + 1|. Therefore, the probabilities of dipole transitions differ from zero only in those cases when L f = L i ± 1. Using again the rules of addition L mf , we can of moments, expressed in terms of the Clebsch–Gordan coefficient C L i fm i ;1ν conclude that the projections of the moments in the initial and final states satisfy the rule m f = m i + ν. The listed rules for changing the moments L and their projections m are called selection rules. Substituting the matrix element Eq. (72) into formula Eq. (70), we obtain the differential probability of the dipole transition in the form (sp)

)2 I I ω3f i d02 2L + 1 ( n f L f L f 0 )2 ( L f 0 I m −m i I2 C R C I(ekλ ) f I dΩ = L L 010 m 1m −m n L ;1 ' 3 i i i f i i i 2π hc 2L + 1 ( )2 ( ) nf Lf Lf0 2L + 1 R C n i L i ;1 L i 010 Dλ El; L i m i → L f m f dΩ, 2L ' + 1 )2 I ( I ) Lf0 I m −m i I2 L f m f = C L m 1m −m I(ekλ ) f I

dw f i (El, k, λ) = =

e2 ω3f i ω3f i d02 2π hc3 2π hc3

( Dλ El; L i m i →

i

i

f

i

(78) The resulting formula describes the angular distribution of spontaneous emission depending on the type of polarization and quantum numbers L i → L f and m i → mf. If all directions in space are equivalent, then the atomic system can be in any of m i - states with equal probability. Therefore, the transition probability n i L i m i → n f L f m f can be obtained by summing over the m f final states and averaging over the initial m i states,

376

E. V. Orlenko and V. K. Khersonsky (sp)

dw f i (El, ϕk , ϑk ) =

[ 1 (sp) dw f i (El, k, λ) = 2L i + 1 m ,m f

=

ω3f i d02 2πhc3

(

1 n Lf L 0 C L i f010 RnifL i ;1 2L f + 1

i

)2 [

(

)

(79)

Dλ El; L i m i → L f m f dΩ.

m f ,m i

Considering that [ m f ,m i

[ Lf0 ( ) [ ' L 0 Dλ El; L i m i → L f m f = C L i m i 1ν C L i fm i 1ν ' = (ekλ )ν∗ (ekλ )ν m f ,m i

νν '

2L f +1 δνν ' 3

2L f + 1 2L f + 1 [ ' , = (ekλ )ν∗ (ekλ )ν = 3 3 ν=−1,0,1

(80)

=1

we obtain 2L f + 1 ( n f L f L f 0 )2 1 Rni L i ;1 C L i 010 dΩ = 2π hc3 2L f + 1 3 ω3f i d02 ( n f L f L f 0 )2 C dΩ. R = 6πhc3 ni L i ;1 L i 010 (sp)

dw f i (El, ϕk , ϑk ) =

ω3f i d02

(81)

Thus, in this case, the dependence on the polarization disappears, and the dependence on the angle ϑk enters only through a solid angle dΩ. In other words, if all directions in space are equal, the radiation of an atomic system is isotropic (all solid angles are equivalent) and unpolarized. If we integrate this formula over the angles and sum over the polarizations, we get the total probability of spontaneous emission (sp)

w f i (El) =

4ω3f i d02 ( 3π hc3

n L

L 0

f C L i f010 RnifL i ;1

)2

.

(82)

Let us consider the angular distributions and polarizations of dipole radiation in some special cases. This analysis can be performed as follows. We will set some simple types of polarizations (that is, set different directions of the two vectors that determine the polarization states, ek1 and ek2 ) and determine the angular distributions of the radiation probabilities corresponding to these polarizations. We note right away that for m i − m f = 0 I2 ( ) ( L m )2 I Dλ El; L i m i → L f m f = C L i fm i i10 I(ekλ )0 I .

(83)

only the z-component of the vector ekλ contributes to the transition. This means that if we choose one of the vectors (for example, ek1 ) coinciding with the vector e0 of the

Emission and Absorption of Photons in Quantum Transitions …

377

cyclic basis (ek1 )0 = 1, then we have the case when a photon in a given polarization state, λ = 1, is characterized by a polarization vector directed to the Z axis and ( ) ( L m )2 Dl El; L i m i → L f m f = C L i fm i i10 .

(84)

This component of the radiation (corresponding to the transition with Δm = 0) is called the π-component. So the photons the corresponding Δm = 0 ones are linearly polarized along the Z axis if the polarization is determined in the cyclic basis on which the laboratory coordinate system (LSC) is built, that is, ekλ = (ekλ )+1 e+1 + (ekλ )0 e0 + (ekλ )−1 e−1 . For transitions with Δm = ±1, the quantity (ekλ )Δm does not vanish if we choose ek1 = e±1 and ek2 = e∓1 . This choice corresponds to the fact that the vectors and describe the right/left circular and left/right circular polarization, respectively, when viewed along the Z axis—both vectors rotate in the (XY) plane— this rotation is characterized by an angle ϕk ∈ [0, 2π ], and the wave vector is directed along the axis Z, (e0 ) and ϑk = 0 ( ) ( L m ±1 )2 Dl El; L i m i → L f m i ± 1 = C L i fm i i1±1 , ( ) ( L m ∓1 )2 D2 El; L i m i → L f m i ∓ 1 = C L i fm i i1∓1 .

(85)

The radiation components corresponding to transitions with Δm = ±1 are called the σ-components. Let us define vectors ekλ in the Cartesian basis with the Z axis coinciding with the direction of the wave vector k. Cartesian unit vectors in this basis are spherical unit vectors ek (directed along the k-axis Z k ), eϑk (polar unit vector, directed along longitude) and eϕk (azimuthal unit vector, directed along latitude). Let us direct the vectors ekλ along the polar ek1 = eϑk and ek2 = eϕk azimuthal units. As in the previous case, we will consider transitions with Δm = 0 and Δm = ±1, which corresponds to the projection of the vectors ekλ ( )0 ( )±1 (ek1 )0 = eϑk , (ek1 )±1 = eϑk , ( )0 ( )±1 (ek2 )0 = eϕk , (ek2 )±1 = eϕk .

(86)

To calculate these components, we use formulas [4], which connect spherical and cyclic unit vectors. We need the following two ratios, 1 1 eϑk = −e+1 √ cos ϑk e−i ϕk − e0 sin ϑk + e−1 √ cos ϑk e+i ϕk , 2 2 i +iϕk i −i ϕk eϕk = e+1 √ e + e−1 √ e . 2 2

(87)

These relations show that for transitions without changing the quantum number m,

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E. V. Orlenko and V. K. Khersonsky

( )0 (ek1 )0 = eϑk = − sin ϑk , ( )0 (ek2 )0 = eϕk = 0,

(88)

whence it follows that I2 ( L m )2 ( ) ( L m )2 I D1 El; L i m i → L f m i = C L i fm i i10 I(ek1 )0 I = C L i fm i i10 (sin ϑk )2 , I2 ( ) ( L m )2 I D2 El; L i m i → L f m i = C L i fm i i10 I(ek2 )0 I = 0,

(89)

that is, the probability of radiation polarized linearly in the polar direction varies as (sin ϑk )2 , whereas the probability of radiation polarized in the azimuthal direction is zero. For transitions with a change in the quantum number m, we obtain that ( )±1 1 = ∓ √ cos ϑk e∓iϕk (ek1 )±1 = eϑk 2 ) ( i ±1 = √ e∓i ϕk , (ek2 )±1 = eϕk 2

(90)

( ) 1 ( L f m i ±1 )2 C D1 El; L i m i → L f m i ± 1 = (cos ϑk )2 , 2 L i m i 1±1 I2 ( ) ( L m )2 I 1 D2 El; L i m i → L f m i ± 1 = C L i fm i i10 I(ek2 )±1 I = . 2

(91)

whence it follows that

That is, in this case, the probability of radiation polarized in the azimuthal direction is not zero, but does not depend on the polar angle. In a similar way, one can obtain formulas that determine the angular distributions of the radiation probability for photons with a certain helicity. Let us define vectors ekλ in the helicoid basis, ek1 = ε+1 and ek2 = ε−1 , where ε±1 , ε0 are helicoid orts. We will use the relations [6] that connect the spiral unit vectors with the cyclic ones—we need two of these relations sin ϑk 1 + cos ϑk −iϕk 1 − cos ϑk iϕk e e , + e0 √ + e−1 2 2 2 1 − cos ϑk −iϕk 1 + cos ϑk iϕk sin ϑk e e . = e+1 − e0 √ + e−1 2 2 2

ε+1 = e+1 ε−1

(92)

Emission and Absorption of Photons in Quantum Transitions …

379

Using these relations, one can see that sin ϑk (ek1 )0 = (ε+1 )0 = √ , 2 1 ± cos ϑk ∓iϕk e , (ek1 )±1 = (ε+1 )±1 = 2 sin ϑk (ek2 )0 = (ε−1 )0 = − √ , 2 1 ∓ cos ϑk ∓i ϕk e . (ek2 )±1 = (ε−1 )±1 = 2

(93)

The corresponding angular distributions have the form, ( D1 El; L i m i → ( D1 El; L i m i → ( D2 El; L i m i → ( D2 El; L i m i →

⎧ ϑk ⎪ , ⎨ cos2 ) ( ) L f m i ±1 2 2 L f m i ± 1 = C L i m i 1±1 ⎪ ⎩ sin2 ϑk , 2 ) 1 ( L f m i )2 2 C sin ϑk , L f mi = 2 L i m i 10 ) ( L m )2 L f m i = C L i fm i i10 sin2 ϑk , ⎧ ϑk ⎪ , ⎨ sin2 ) ( ) L f m i ±1 2 2 . L f m i ± 1 = C L i m i 1±1 ⎪ ⎩ cos2 ϑk , 2

(94)

Let us return to Eq. (72), which represents the matrix element of the dipole transition in the LM-representation. In applications, it is often necessary to calculate the matrix elements of dipole transitions with wave functions in representations when the orbital angular momentum L is related to the electron spin S at the total angular momentum of the electron J, J = L + S. In this case, we are talking about the transition. This situation has a place, for example, when calculating the probabilities of dipole between the >levels of a fine structure I >transitions I |ηi > = |n i , L i , Si , Ji , Mi > → Iη f = In f , L f , S f , J f , M f . In this case, in approximation, which is used here, the wave function corresponding to the total angular momentum of the electron is the product of the Schrödinger radial wave function Rn L (r ) and the spherical spinor Ω LJ M (ϑ, ϕ) [compare with Eq. (70)], |n L S J M> = Rn L (r )Ω LJ M (ϑ, ϕ) = Rn L (r )

[ mσ

JM C Lm Sσ Y Lm (ϑ, ϕ)χ Sσ (σ S ).

(95)

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E. V. Orlenko and V. K. Khersonsky

(Recall that, in accordance with the generally accepted notation, we do not list the spin variable σ S in the argument list of the spherical spinor). The dipole moment operator does not affect the spin variable. Therefore, the transition matrix element can be written as I < e n f L f S f J f M f Ir n1ν |n i L i Si Ji Mi > = I I < < = e n f L f Ir |n i L i > L f J f M f In1ν |L i Ji Mi >δ S f Si = [ I < n Lf J M i I δ S f Si C L ff m ff ;Si σ C LJii M = d0 RnifL i ;1 m i ;Si σ L f m f (nr )1ν |L i m i > = (96) m i ,m f ,σ

[ I d0 n Lf J M Lfmf < i I =/ δ S f Si C L ff m ff ;Si σ C LJii M RnifL i ;1 m i ;Si σ C L i m i ;1ν L f |nr1 ||L i >. 2L f + 1 m i ,m f ,σ We use the formula for summing the product of three Clebsch–Gordan coefficients [ m i ,m f ,σ

J M

L m

f f i C L ff m ff ;Si σ C LJii M m i ;Si σ C L i m i ;1ν =

[ ] / ( ) J Mf L i Si Ji . = (−1) Si +Ji +L f +1 (2Ji + 1) 2L f + 1 C Ji fMi 1ν Jf 1 L f

(97)

Then we get the following expression for the matrix element I e n f L f S f J f M f Ir n1ν |n i L i Si Ji Mi > = [ ] / L i Si Ji < II Jf Mf nf Lf Si +Ji +L f +1 L f |nr1 ||L i > = = (−1) d0 Rni L i ;1 δ S f Si 2Ji + 1C Ji Mi 1ν Jf 1 L f [ ] / L i Si Ji nf Lf Lf0 J Mf Si +Ji +L f +1 C Ji fMi 1ν = (−1) d0 (2Ji + 1)(2L i + 1)Rni L i ;1 δ S f Si C L i 010 , Jf 1 L f (98)
I δ S f Si dΩ = I 3 I 2π hc I ν ]2 ) [ ( n L L f 0 2 L i Si Ji f f + 1)(2L + 1) R C (2J i i n i L i ;1 L i 010 Jf 1 L f 2π hc3 ( ) δ S f Si Dλ El; Ji Mi → J f M f dΩ, =

d02 ω3f i

Emission and Absorption of Photons in Quantum Transitions …

I2 )2 I ( ) ( Jf Mf I I Dλ El; Ji Mi → J f M f = C J M 1M −M I(ekλ ) M f −Mi I . i i f i

381

(99)

The resulting expression retains the selection rules for the quantum number of the orbital angular momentum L, but also adds new rules for the total angular momentum of the electron J and its projection M. As follows from the properties of the Clebsch– Gordan coefficients and 6j-symbols, these rules are J f = Ji , Ji ± 1, M f = Mi , Mi ± 1.

(100)

All the angular and polarization characteristics considered above are retained if they are replaced L i → Ji , m i → Mi , L f → Jf ,m f → Mf .

(101)

In those cases when it is necessary to calculate I the transitions >between theI levels > In i L i Si Ji Ii Fi M F and final Iη f ≡ |η > ≡ of the hyperfine structure, the initial i i I > In f L f S f J f I f F f M F states are constructed taking into account the connection of f the total angular momentum of the electron J with the nuclear spin I to the total angular momentum of the atomic system F, that is, |n L S J I F M F > = Rn L (r ) = Rn L (r )

[

[

C JFMMIFη Ω LJ M (ϑ, ϕ)χ I η (σ I ) =

Mη JM C JFMMIFη C Lm Sσ Y Lm (ϑ, ϕ)χ Sσ (σ S )χ I η (σ I ),

(102)

Mη mσ

where χ I η (σ I ) is the spin function of the nucleus and the quantity σ I is the spin variable of the nucleus. The dipole transition operator does not act on the spin variable of the nucleus. Now the matrix element of the dipole transition n i L i Si Ji Ii Fi M Fi → n f L f S f J f I f F f M F f can be represented in the form,

382

E. V. Orlenko and V. K. Khersonsky

I I > < e n f L f S f J f I f F f M F f Ir n1ν In i L i Si Ji Ii Fi M Fi = I I I < < > = e n f L f Ir |n i L i > L f S f J f I f F f M F f I(nr )1ν I L i Si Ji Ii Fi M Fi = I < n Lf = d0 RnifL i ;1 δ S f Si δ Ii I f L f m f I(nr )1ν |L i m i >× [ F f MF Fi M F J M i C Ji Mi Iii η C J f M f If f η C L ff m ff ;Si σ C LJii M m i ;Si σ = m i ,m f ,σ Mi M f η

< I d0 n Lf δ S f Si δ Ii I f L f I|nr1 ||L i >× RnifL i ;1 2L f + 1 [ F f MF Fi M F J M Lfmf i C Ji Mi Iii η C J f M f If f η C L ff m ff ;Si σ C LJii M m i ;Si σ C L i m i ;1ν =

=/

m i ,m f ,σ Mi M f η

/ < I n Lf δ S f Si δ Ii I f L f I|nr1 ||L i >× = (−1) Si +Ji +L f +1 2Ji + 1d0 RnifL i ;1 [ ] [ F f MF Fi M F L i Si Ji J Mf C Ji fMi 1ν C Ji Mi Iii η C J f M f If f η = Jf 1 L f Mi M f η / ( ) n Lf × = (−1) Si +Ii +Ji +J f +L f +Fi d0 (2Ji + 1) 2J f + 1 (2Fi + 1)RnifL i ;1 [ ][ ] F f MF f L i Si Ji Ji Ii Fi < II L f |nr1 ||L i >C Fi M F 1ν = δ S f Si δ Ii I f i Jf 1 L f Ff 1 Jf / ( ) n Lf × = (−1) Si +Ii +Ji +J f +L f +Fi (2Ji + 1) 2J f + 1 (2Fi + 1)(2L i + 1)d0 RnifL i ;1 [ ][ ] F f MF f L i Si Ji Ji Ii Fi L 0 C Fi M F 1ν . δ S f Si δ Ii I f C L i f010 i Jf 1 L f Ff 1 Jf (103)

When calculating this matrix element, we twice used the summation of the products of the three Clebsch–Gordan coefficients Eq. (97). Substitution of this matrix element in the formula for differential probability leads to the following result, ω3f i e2 (sp) dw f i (El; k, λ) = δS S δ I I × 2π hc3 f i i f I I2 I[ I I < >II I ν I I (ekλ ) n f L f S f J f I f F f M F f r (nr )1ν n i L i Si Ji Ii Fi M Fi I dΩ = I I ν I

)2 ( d02 ω3f i ( ) n Lf L 0 C L i f010 δ S f Si δ Ii I f × = (2Ji + 1) 2J f + 1 (2Fi + 1)(2L i + 1) RnifL i ;1 3 2π hc [ ]2 [ ]2 ( ) L i Si Ji Ji Ii Fi Dλ El; Fi M Fi → F f M F f dΩ, Jf 1 L f Ff 1 Jf I2 )2 I ( ) ( F f MF f I M F f −M Fi I Dλ El; Fi M Fi → F f M F f = C Fi M F 1M ) I(e I . kλ F −M F i

f

i

(104)

Emission and Absorption of Photons in Quantum Transitions …

383

This result introduces new selection rules in addition to the existing rules for the orbital angular momentum L and the total angular momentum of an electron J. These new rules also follow the properties of the Clebsch–Gordan coefficients and 6j-symbols. They can be summed up as follows, F f = Fi , Fi ± 1, M F f = M Fi , M Fi ± 1.

(105)

All the angular and polarization characteristics considered above are retained if they are replaced L i → Fi , m i → M Fi , L f → F f , m f → MF f .

(106)

The dipole approximation plays a very important role in applications, since in many cases it turns out to be sufficient for calculating the transition probabilities.

2 Coherent States The description of the processes of interaction of a molecular system with a field in the dipole approximation is carried out using the Hamiltonian, which takes into account as the contribution of the free electromagnetic field, above in the ) ( which is described ] † ˆ bkλ + 21 and the contribution Sect. 1 using (22) having the form Hˆ 0 = kλ hωk bˆkλ of interaction. This term takes into account the actual interaction of the field with the electronic system and was described above in relations Eqs. (18)–(20). A detailed calculation of the matrix elements Eq. (29) from the electronic states of the molecular system Eqs. (59) and (60), taking into account the averaging over the initial and final states of the molecular system, (see, for example, expression Eq. (72)), leads us to the Hamiltonian of the electromagnetic field, taking into account the effect of the electronic system on the photonic one: Hˆ = Hˆ 0 + Vˆ = ⎧ ⎫ / ( ) [ ( ) ]⎬ 2 [⎨ I < ( ) 2π hc 1 † † ˆ − η f Ie t ' + bˆkλ t ' = αˆ · ekλ · ik · r|ηi > bˆkλ bkλ + hωk bˆkλ , ⎩ ⎭ 2 V ω k kλ ( ) ] [ ( ) [[ ( ' )] 1 † † ˆ ' ˆ ˆ ˆ ˆ hωk bkλ bkλ + + Δkλ bkλ t + bkλ t , H= 2 kλ / / 2π hc2 < II 2L i + 1 n f L f L f m f L 0 Δkλ = −ei R C C f ,. η f αˆ · ekλ · k · r|ηi > = d0 V ωk 2L f + 1 ni L i ;1 L i m i ;1ν L i 0;10 (107)

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E. V. Orlenko and V. K. Khersonsky

For the description of most optical fields interacting with the matter (electronic system), the most acceptable basis is the basis of coherent states. In contrast to the Fock states considered above, with a certain number of photons in a given mode and a completely indefinite phase, it is convenient to use coherent states in cases where the phase of an electromagnetic wave itself and the phase difference play an important role for the physical phenomena associated with interference under consideration. Such states are relevant for the description of intense fields formed in laser systems when radiation interacts in atomic (molecular) gas cells with a “working” body, which is a non-molecular (molecular) gas, or a solid, if a solid-state laser (maser) based on spin transitions is considered. Coherent states are states with an indefinite number of photons, but at the same time have a definite phase, in contrast to states with a definite number of photons, but a completely random phase. The product of the uncertainties of the amplitude and phase of coherent fields is the minimum allowed by the uncertainty principle. In this sense, these quantum states are the closest to the states described in the framework of classical electrodynamics.

2.1 Hamiltonian of Coherent States Let us consider in detail the Hamiltonian describing one of the modes of the electromagnetic field, taking into account the interaction with the electronic system, in the following form: ( ) [ ] 1 † ˆ † + Δkλ bˆkλ + bˆkλ Hˆ kλ = hωk bˆkλ bkλ + 2

(108)

) / ( † ωk Pˆkλ Qˆ kλ + iω , bˆkλ = where the creation and annihilation operators bˆkλ = 2h k ) / ( ωk Pˆkλ Qˆ kλ − iω were introduced in Sect. 1 Eq. (20) for oscillatory states of a 2h k free electromagnetic field. To diagonalize the Hamiltonian, we introduce the following transformation of these operators: † † = Bkλ + u∗, bˆkλ

bˆkλ = Bkλ + u.

(109)

† where u is some real function, and operators Bkλ , Bkλ are new creation and annihilation operators, respectively, acting on a new vacuum function, also to be determined. † , bˆkλ : They satisfy the same commutation relations as the operators bˆkλ

Emission and Absorption of Photons in Quantum Transitions …

[

] † Bk' λ' Bkλ = δk' k δλ' λ .

385

(110)

Substituting Eq. (109) into Hamiltonian Eq. (108), we have ) (( ) 1 † + u ∗ (Bkλ + u) + Hˆ kλ = hωk Bkλ 2 [ ] † + Bkλ + 2Reu = + Δkλ Bkλ ( ) ) ( 1 † = hωk Bkλ Bkλ + + hωk |u|2 + 2ReuΔkλ 2 ( ) † + Bkλ + Bkλ (Δkλ + uhωk ).

(111)

( To obtain ) the diagonal form of the Hamiltonian, we choose the function Δkλ − hω = u , which will lead to the equality of the third term Eq. (109), and then k we obtain Hamiltonian Eq. (108) in the new representation: ( ) ( ) 1 † + hωk u 2 + 2uΔkλ = Bkλ + Hˆ kλ = hωk Bkλ 2 ( ( ) ) Δkλ 2 1 Δkλ † + hωk − = hωk Bkλ Bkλ + −2 Δkλ 2 hωk hωk ( ) 1 (Δkλ )2 † . = hωk Bkλ Bkλ + − 2 hωk

(112)

The energy eigenvalues, in this case, will be equal ( εkλ = hωk

1 αkλ + 2

) −

(Δkλ )2 . hωk

(113)

ˆ Let us now find a unitary transformation of the form Dˆ = ed , (where dˆ is † a Hermitian operator, dˆ = dˆ ) corresponding to the transformation of operators Eq. (109): ˆ

ˆ

ˆ d = bˆ + B = e−d be

Δkλ . hωk

(114)

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E. V. Orlenko and V. K. Khersonsky

where we will use the creation bˆ † and annihilation bˆ operators, omitting the indices (kλ) of the corresponding mode, since further consideration concerns each independent vibration mode. We use the expansion for the operator exponent [ ] [ [ ]] [ [ [ ]]] Δkλ ˆ ˆ Dˆ = e− D be = bˆ + bˆ Dˆ + bˆ bˆ Dˆ + bˆ bˆ bˆ Dˆ + ... (115) hωk

B = bˆ +

) ( Δkλ The operator dˆ will be sought in the form: Dˆ = c bˆ † − bˆ , then we find c = − hω , k ˆ that is, we have a special case of the unitary transformation of the type D(α) = (

α∗bˆ † −α bˆ

)

Δkλ e with a real α = − hω . This transformation transforms the “old” vacuum k function |n kλ > = |0>, corresponding to the ground state of the oscillator, into the “new” vacuum function, which corresponds to the coherent state:

ˆ |νkλ = 0> = |α> = D(α)|n kλ = 0> = e

(

α∗bˆ † −α bˆ (

Xˆ Yˆ

)

|n kλ = 0>.

(116)

) [ ] Xˆ +Yˆ + 21 Xˆ Yˆ

, which is a particBearing in mind the well-known identity e e = e ular case of The Baker–Campbell–Hausdorff formula, we obtain for the operators Xˆ = α ∗ bˆ † and Yˆ = −α bˆ the following equality: eα

∗ ˆ†

eα

∗ ˆ†

b

ˆ

e−αb = e

b −α bˆ

=e

[ ] ˆ |α|2 bˆ † bˆ α ∗ bˆ † −α b− 2

2 − |α|2

eα

∗ ˆ†

b

= eα

2

ˆ |α| b −α b+ 2

∗ ˆ†

= eα

2

b −α bˆ + |α|2

∗ ˆ†

e

,

ˆ

e−αb .

(117)

Thus, a coherent state can be obtained from the ground state of the oscillator using a unitary transformation: |α|2

b −α bˆ

∗ ˆ†

∗ ˆ†

ˆ

|α|2

|n kλ = 0> = e− 2 eα b e−αb |n kλ = 0> = e− 2 eα √ ∞ ∞ )ν 2 [ (α ∗ )ν ( 2 [ (α ∗ )ν ν! † − |α|2 − |α|2 ˆ |ν> = b |0> = e =e ν! ν! ν=0 ν=0

eα

= e−

|α|2 2

∞ [ ν=0

∗ ˆ†

b

|n kλ = 0> = (118)

∗ ν

(α ) √ |ν> = |α>kλ ν!

ˆ Here we took into account the equality Eq. (118) of Sect. 1, b|0> ≡ 0. These quantum superposition states of the different numbers of photons in the given mode we call coherent states.

Emission and Absorption of Photons in Quantum Transitions …

387

2.2 Determination of Coherent States Let us consider in more detail the states, which are eigenstates of the annihilation ˆ and which minimize the uncertainty relation [8]. They were first considoperator b, ered by E. Schrödinger in 1926 and are widely used in quantum optics. We denoted these states by |α>, then ˆ b|α> = α|α>

(119)

Let us find the expansion of the state |α> in the complete set of states |n> , then |α> =

∞ [

|n>

(120)

n=0

Multiplying the resulting expansion on the left by = α = = =

∞ [ n=0

|n> =

∞ [

αn |n> √ . n! n=0

(123)

The “zeroth” expansion coefficient is determined from the condition of normalization of the state |α>|:

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E. V. Orlenko and V. K. Khersonsky

=

∞ [

∞ I >< I [ =

n ' =0 ∞ [ 2

n=0 ∗n '

αn α < √ n ' |n> √ = n'! n!

= ||

n=0 n ' =0 ∞ [

= ||2

(124)

|α|2n 2 = ||2 e|α| = 1, n!

n=0

|| = e−|α|

2

/2

.

Finally, we find an expression for the coherent state |α> in the form of an expansion |α> =

∞ [

∞ [ αn αn 2 |n> √ = || = e−|α| /2 |n> √ . n! n! n=0 n=0

(125)

Let us now determine the average number of energy quanta in the state |α>: ˆ = = e−|α|

2

' ∞ ∞ [ [ |α|2n α ∗n < I ˆ αn 2 = n √ n ' Ibˆ † b|n> √ = e−|α| n! n'! n! n=0 n=0

n ' =0 2 −|α|2

= |α| e

∞ [ n=1

∞

[ |α|2k |α|2(n−1) 2 2 2 = |α|2 e−|α| e|α| = |α|2 . ≡ |α|2 e−|α| k! (n − 1)! k=1

(126)

Thus, coherent states |α> can be written as |α> = e

−/2

∞ [

αn |n> √ n! n=0

(127)

and the probability of finding the oscillator in the state |n> will have the form of the Poisson distribution: Wn (α) = e−

n n!

(128)

This means that the coherent state of a given mode is a state with a variable number of quanta (photons), the number of which obeys the Poisson distribution. This state most closely corresponds to the classical representation of light as an electromagnetic wave with a given frequency and phase. As already shown in Eq. (118), a coherent state can be obtained from the ground state of an oscillator using a unitary transformation: |α> = e−

|α|2 2

eα

∗ ˆ†

b

ˆ

e−αb |0> = e− 2 eα

∗ ˆ†

b

ˆ

e−αb |0>

(129)

Emission and Absorption of Photons in Quantum Transitions …

389

∗ ˆ† ˆ ˆ The unitary operator D(α) = e− 2 eα b e−αb , α ∗ = α,, which transfers from the vacuum state |0> to the coherent state |α>, has the following properties:

ˆ ˆ (1) D(α) Dˆ † (α) = Dˆ † (α) D(α) = 1, ˆ (2) Dˆ † (α) = D(−α), ˆ ˆ (3) D(β) D(α) = e−

βα ∗ −β ∗ α 2

ˆ + α), ⇒ D(β

(130)

∗ ∗ ˆ ˆ ˆ ˆ D(β) D(α) = e−βα −β α D(α) D(β).

It is not difficult to obtain other commutation relations: ] [ ˆ ˆ = α D(α), bˆ D(α) [ ] ˆ ˆ bˆ † D(α) = α ∗ D(α),

(131)

ˆ Dˆ † (α)bˆ D(α) = bˆ + α, ˆ Dˆ † (α)bˆ † D(α) = bˆ † + α ∗ . ˆ We will show below that the D(α) operators are offset operators.

2.3 Orthogonality, Normalization and Completeness of Coherent States Let us now consider a question related to the normalization and orthogonality of coherent states. Let there be two coherent states |α> and |β>, that differ from each other in the average number of energy quanta, and , respectively. Let us find the scalar product of the wave vectors of these coherent states = e−

+ 2

' ∞ ∞ ∗n n [ |α|2 +|β|2 [ α βn β α ∗n < = √ n ' |n> √ = e− 2 '! n! n n! n=0 n=0

n ' =0

n ' =0

) ( ) ( |α|2 +|β|2 |α|2 + |β|2 − 2α ∗ β (β − α)2 ∗ = e− 2 eα β = exp − = exp − , 2 2 ) ( ||2 = exp −|β − α|2 .

(132)

Whence it can be seen that the coherent states are not orthogonal, the normalization condition Eq. (124) is fulfilled in this case, that is, = 1 . Meanwhile, they form a system of state vectors with completeness. Let’s show it. To do this, consider the integral:

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E. V. Orlenko and V. K. Khersonsky

( |α> into integral Eq. (133), we find (

( [ ' ∞ < I α n α ∗n 2 |α> n ' Ie−|α| d 2 α = ' n! n ! n=0 2

n ' =0

< I (∞ (2π [ ∞ ] [ |n> n ' I ' ' −ρ 2 e ρdρ dϕ ρ n+n ei (ϕ−ϕ ) = = √ √ n! n ' ! n=0 0

n ' =0

= 2π

∞ [ |n> < >2 Δx 2 (t) = xˆ 2 (t) − x(t) ˆ = (< > < > ) 2 = xˆ 2 − xˆ cos2 (ωt)+

(< > < > ) sin2 (ωt) sin(2ωt) (< > < >< > ih ) 2 , pˆ xˆ − pˆ xˆ + + pˆ 2 − pˆ + ω2 ω 2 / ( h ˆ ˆ†) xˆ = b+b , 2ω / ( h ˆ ˆ†) pˆ = −i ω b−b , 2ω ] [(/ \ / \/ \ ) I/ \ / \ I 2I I 2 1 h 2 †ˆ † ˆ I I ˆ ˆ ˆ ˆ b b − b b + + I b − b I cos(2ωt − ϕ) , Δx (t) = ω 2 ] ) I/ \ / \ I [(/ \ / \/ \ 2I I 2 1 2 †ˆ † ˆ I I ˆ ˆ ˆ ˆ Δp (t) = hω − I b − b I cos(2ωt − ϕ) . b b − b b + 2

(139)

Then we have (( /

\ / \/ \ 1 )2 †ˆ ˆ b b − bˆ † bˆ + Δx (t) · Δp (t) = h − 2 ] (/ \ / \ )(/ \ / \ ) 2 2 bˆ 2 − bˆ cos2 (2ωt − ϕ) . − bˆ †2 − bˆ † 2

2

2

(140)

In order for the product Eq. (140) to take the minimum value at any moment of time (ΔxΔp = h/2), the following condition must be met: / \ / \2 / \ / \2 bˆ 2 = bˆ , bˆ †2 = bˆ †

(141)

Which will naturally entail the satisfaction of the following equality: /

\ / \/ \ bˆ † bˆ = bˆ † bˆ .

(142)

Suppose that there is a complete orthonormal set of states {Ψν } over which averaging is carried out, then condition Eq. (33) will be rewritten as: / \ [ ˆ i> = ˆ ν > = is orthogonal to all other ˆ i > /= 0, that is, there must be an states Ψν , in this case, the state itself |α> = b|Ψ

Emission and Absorption of Photons in Quantum Transitions …

393

ˆ i > = α|Ψi >, where α is in the general eigenstate of the operator bˆ such that: |α> = b|Ψ case a complex number. As shown in the previous section, such eigenstates of the annihilation operator do exist, they constitute a complete set of continuous states, but these states are not orthogonal. So the minimization of the uncertainty relation for the indicated coherent states is carried out only asymptotically and the inequality always holds: ΔxΔp = h/2.

(144)

2.5 Coordinate Representation of the Hamiltonian and Coherent States Here we will use expressions (5) for the creation-annihilation operators, and then we can write in the coordinate representation: † bˆkλ

/ =

) ) ( / ( ˆkλ ωk ˆ ω Pˆkλ P k , bˆkλ = → Q kλ + Qˆ kλ − 2h i ωk 2h i ωk

ˆ Qˆ kλ → x, ˆ Pˆkλ → p, / / / ( ) 2 p ˆ 2 2h ∂ ω † bˆ − bˆ = pˆ = − . = −i 2h i ω ωh ω ∂x

(145)

The “new” vacuum( function, ) obtained using the unitary transformation Eqs. √ α∗bˆ † −α bˆ ˆ (113)–(116), D(α) , α ∗ = α = = − Δkλ can be written as =e hωk

Ψα (x) = e α=−

( ) α bˆ † −bˆ

Δkλ , hωk

ψ0 (x) = e

√ 2h

−

ω

( α ∂∂x

ψ0 (x) = ψ0 x −

/

2h α ω

)

(146) (

/

Ψα (x) = ψ0 x +

where we denote X 0 =

/

2 Δ hω ω

2h α ω

=

)

/

2h ω

=

/

2 Δ , hω ω

that is, the new function is the ˆ wave( function of)an oscillator [ with a ]displaced center. ( Hamiltonian ) [Eq. (109) ] Hkλ = † † 1 1 † † hωk bˆkλ bˆkλ + 2 + Δkλ bˆkλ + bˆkλ → Hˆ = hω bˆ bˆ + 2 + Δ bˆ + bˆ written in the representation of generalized coordinates x with accounting Eq. (136) will have the form written for one mode, to which a term proportional to the generalized

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E. V. Orlenko and V. K. Khersonsky

coordinate is added: / ) ( [ ] 1( ) 2ω 1 † 2 2 2 †ˆ ˆ ˆ ˆ ˆ +Δ b +b = pˆ + ω xˆ + Δ H = hω b b + xˆ = 2 2 h ) ( / 1 2Δ 2ω 1 xˆ = = pˆ 2 + ω2 xˆ 2 + 2 2 2 ω h ( / )2 ( ) 1 2 1 2 Δ2 Δ 2 = pˆ + ω xˆ + . − 2 2 ω hω hω

(147)

Thus, we are really dealing with a Hamiltonian describing an oscillator with a center of inertia displaced in the space of generalized coordinates. The eigenfunction corresponding to the ground state of such a “biased” oscillator is the obtained function Eq. (146).

2.6 Exact Solution of the Time-Dependent Problem of the Interaction of Radiation with Matter Let us consider radiation in the model of quantum oscillators, which interacts with a material system, as a result of which dipole electric radiation is added to the photonic system due to transitions in atoms or molecules of the medium. The Hamiltonian of such a system will be written as ( ) ] [ 1 † ˆ † + Δkλ (t) bˆkλ + bˆkλ → Hˆ kλ (t) = hωk bˆkλ bkλ + 2 ) ( [ ] 1 − Δ(t) bˆ † + bˆ , Hˆ (t) = hω bˆ † bˆ + 2

(148)

where Δ(t) is a time-dependent function. Thus it is necessary to solve the timedependent Schrödinger equation: ih

∂Ψ(t) = Hˆ (t)Ψ(t). ∂t

(149)

We search for a solution in the form: Ψ(t) = C(t)eα(t)b eβ(t)b eγ (t)b b Ψ(−∞), †

†

(150)

where the wave function of the state Ψ(−∞) of the system before the start of interaction is chosen as the initial state, and α(t), β(t) and γ (t) are the sought functions of time. Substituting Eq. (150) into Eq. (149), we have the following relation:

Emission and Absorption of Photons in Quantum Transitions …

395

[ ˆ† ˆ ˆ† ˆ α(t)bˆ † β(t)bˆ γ (t)bˆ † bˆ ˙ ih C(t)e e e + C(t)α(t) ˙ bˆ † eα(t)b eβ(t)b eγ (t)b b + ˆ† ˆ β(t)bˆ eγ (t)bˆ† bˆ + ˙ be +C(t)eα(t)b β(t)

] ˆ† ˆ ˆ† ˆ + C(t)eα(t)b eβ(t)b eγ (t)b b γ˙ (t)bˆ † bˆ Ψ(−∞) = ) [ ( ] 1 ˆ† ˆ ˆ† ˆ † †ˆ ˆ ˆ ˆ − Δ(t)(b + b) C(t)eα(t)b eβ(t)b eγ (t)b b Ψ(−∞) = hω b b + 2

(151)

Taking into account the rules of commutation, † ˆ = −αeαbˆ† , ⇒ eαbˆ† bˆ = (bˆ − α)eαbˆ† , [eαb , b]

ˆ

ˆ

ˆ

ˆ

[eβ b , bˆ † ] = βeβ b , ⇒ eβ b bˆ † = (bˆ † + β)eβ b ,

(152)

we come to the following equation: ih

[˙ C(t) ˆ† ˆ ˆ† ˆ α(t)bˆ † β(t)bˆ γ (t)bˆ † bˆ C(t)eα(t)b eβ(t)b eγ (t)b b + bˆ † α(t)C(t)e ˙ e e + C(t) ˆ†

ˆ

ˆ† ˆ

˙ +β(t)( bˆ − α)C(t)eα(t)b eβ(t)b eγ (t)b b +

] ˆ† ˆ ˆ† ˆ + (bˆ † + β)(bˆ − α)γ˙ (t)C(t)eα(t)b eβ(t)b eγ (t)b b Ψ(−∞) = [˙ C(t) ˙ = ih bˆ − α) + + α(t) ˙ bˆ † + β(t)( C(t) ] ˆ† ˆ ˆ† ˆ + (bˆ † + β)(bˆ − α)γ˙ (t) C(t)eα(t)b eβ(t)b eγ (t)b b Ψ(−∞) = ) [ ( ] 1 ˆ C(t)eα(t)bˆ† eβ(t)bˆ eγ (t)bˆ† bˆ Ψ(−∞), − Δ(t)(bˆ † + b) = hω bˆ † bˆ + 2

(153)

whence we obtain an equation [ ˙ + C(t)α(t) ˙ ih C(t) ˙ bˆ † + β(t)C(t)( bˆ − α(t)) ] + C(t)(bˆ † + β)(bˆ − α(t))γ˙ (t) Ψ(−∞) = [ ] ˆ C(t)Ψ(−∞), = hω(bˆ † bˆ + 21 ) − Δ(t)(bˆ † + b) that allows us to write a system of equations for determining the coefficients:

(154)

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E. V. Orlenko and V. K. Khersonsky

(155)

or ˙ ( ) C(t) ω ˙ − β(t)i ω = −i , − α(t) β(t) C(t) 2 iΔ(t) α(t) ˙ + i ωα(t) = , h iΔ(t) ˙ − i ωβ(t) = , β(t) h γ˙ (t) = −i ω.

(156)

Thus, we have a general form of the solution for all the desired coefficients that completely determine the wave function at any time: ⎧ ⎨

1 C(t) = e−i ωt/2 exp − 2 ⎩ h α(t) =

(t

ie−iωt h

iei ωt β(t) = h

(t

⎛ dt ' ⎝Δ(t ' )e−iωt

−∞

(t '

'

−∞

'

Δ(t ' )ei ωt dt ' ,

(157)

−∞

(t

⎞⎫ ⎬ ( ) '' dt '' Δ(t '' )ei ωt ⎠ ⎭

'

Δ(t ' )e−iωt dt ' ,

−∞

γ (t) = −i ωt. The initial conditions can be taken as: α(−∞) = 0, β(−∞) = 0, C(−∞) = 1, Ψ(−∞) = |0>.

(158)

Let us consider the “evolution” of a wave packet that simulates a laser pulse in a medium (for example, in a system of molecules in resonance with the field) interacting with an electromagnetic field: |t|

Δ(t) = Δ0 e− τ ,

(159)

Emission and Absorption of Photons in Quantum Transitions …

397

Then, using the obtained general relations for the coefficients C(t), α(t), β(t), γ (t), Eq. (157), we obtain their explicit time dependence: ie−i ωt α(t) = h

=

ie−iωt h

→t→∞

β(t) =

(t

h

−∞

( Δ0

I 'I I I

t ' ie−i ωt Δ0 Δ0 e− τ ei ωt dt ' =

1

⎩

t' ' e τ eiωt dt '

−∞

t e− τ eiωt

1 τ + iω

⎧ 0 ⎨(

1 + 1 − 1 τ − iω τ − iω

) =

ie−iωt h

⎧ ⎪ ⎨

(t +

'

t ' e− τ eiωt dt '

⎫ ⎬ ⎭

0

t 2 e− τ ei ωt Δ0 ( )2τ − 1 ⎪ ⎩ 1 + ω2 τ − iω τ

= ⎫ ⎪ ⎬ ⎪ ⎭

→

2 ie−i ωt , Δ0 ( )2τ h 1 2 + ω τ

ieiωt Δ0 h

(t

e−

I 'I It I ' τ e−i ωt dt '

2 iei ωt Δ0 ( )2τ , h 1 2 + ω τ

=

−∞

γ (t) = −i ωt,

⎧ ⎛ ⎞⎫ ' ( ) ⎪ I '' I I 'I ⎪ (t ⎨ 1 (t ⎬ It I It I ' '' ⎜ ⎟ dt '' Δ0 e− τ ei ωt ⎠ = dt ' ⎝e− τ e−i ωt C(t) = e−i ωt/2 exp − 2 ⎪ ⎪ ⎩ h ⎭ −∞ −∞ ⎫ ⎧ ⎤2 ⎡ ⎪ ⎪ ⎪ ⎪ 2 ⎬ ⎨ Δ2 ⎥ ⎢ τ −i ωt/2 0 exp − 2 ⎣ ( )2 =e ⎦ . ⎪ ⎪ 2h 1 ⎪ ⎭ ⎩ + ω2 ⎪ τ

(160) Thus, the wave function, taking into account the initial conditions Eq. (158), will evolve in time as follows Ψ(t) = C(t)eα(t)b eβ(t)b eγ (t)b b |0> = †

= C(t)e

α(t)b†

†

|0> = C(t)

∞ [ α ν (t) ν=0

ν!

b†ν |0> =

∞ ∞ [ [ α ν (t) √ α ν (t) = C(t) ν!|ν> = C(t) √ |ν>, ν! ν! ν=0 ν=0

e

β(t)b γ (t)b† b

e

|0> = e

β(t)b

∞ [ γ ν (t) ν=0

ν!

ˆ nˆ ν |0> = eβ(t)b · 1|0> = |0>.

(161)

It is easy to show, that a normalization condition by Eq. (160) is totally fulfilled:

398

E. V. Orlenko and V. K. Khersonsky

= |C(t)|2

∞ [ |α(t)|2ν

ν!

ν=0

= |C(t)|2 e|α(t)| = 1, 2

|C(t)|2 = e−|α(t)| , ⇒ |C(t)| = e− 2 |α(t)| ⎧ [ ]2 ⎫ 2 ⎨ Δ2 ⎬ 2 1 exp − 02 ( )2 τ = e− 2 |α(t)| 1 ⎩ 2h ⎭ 2 +ω τ ( ) ( )2 2 2 Δ0 τ 2 (( ) )2 = |α(t)| . h 1 2 + ω2 τ 2

2

1

(162)

Thus, the energy flux of the electromagnetic field in the laser system will be defined as < j> = hωc ˆ = hωc|C(t)|2 = hωc|C(t)|2 |α(t)|2

∞ [ |α(t)|2κ κ=0

= |C(t)|2

κ!

=hωc|α(t)|2 |C(t)|2 e|α(t)| ,

∞ [ |α(t)|2ν ν=0

ν!

∞ [ |α(t)|2ν = ν ν! ν=0 2

(163)

= |C(t)|2 e|α(t)| = 1. 2

With respect to the normalization condition. Then the energy flux is < j> = hωc|α(t)|2 =

)2 ( τ 4ωcΔ20 . h 1 + (τ ω)2

In general, the exact solution found in the form Eq. (150) completely determines the time profile of the electromagnetic field signal in the resonating medium. In this case, along the way, it can be noted that the shape of the laser pulse time profile formed under the specified Eq. (158) conditions does not change, the packet in the specified environment does not spread over time.

2.7 Heisenberg Representation Let us consider the behavior of an oscillatory system under the action of a timedependent force from the point of view of the Heisenberg representation. The tranˆ Sˆ † (t) = sition to it in the general case is carried out using the unitary operator S(t), Sˆ −1 (t) called the S-matrix [3], then for an arbitrary operator of the physical quantity Aˆ and the wave function we have:

Emission and Absorption of Photons in Quantum Transitions …

ˆ ˆ = Sˆ † (t) Aˆ S(t), A(t) ˆ ψ(t) = S(t)ψ(0)

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(164)

ˆ and A(t), ˆ Then we have an equation for S(t) respectively: ˆ ∂ S(t) ˆ = Hˆ (t) S(t), ∂t ˆ S(0) =1

ih

ˆ ∂ A(t) ˆ = −[ Hˆ (t) A(t)], ∂t ˆ A(0) = 1,

ih

(165)

ˆ Hˆ (t) = Sˆ † (t) Hˆ S(t), In the general case, these equations are complex nonlinear operator equations, exact solutions of which exist in exceptional cases. It is in the case of an oscillator that these equations admit an exact solution, and the Heisenberg representation is the most convenient here. Let the oscillator with a constant frequency be acted upon by an external force f (t) that arbitrarily depends on time. Then for the operator ˆ ˆ = Sˆ † (t)bˆ S(t), from Eq. (165), we obtain: b(t) ˆ˙ = hωb(t) ˆ − ihb(t)

/

h f (t), 2ω

(166)

ˆ ˆ b(0) = b. Solving this equation, we find ˆ = (bˆ + u(t))e−iωt , b(t) (t i ' ei ωt f (t ' )dt ' . u(t) = √ 2hω

(167)

0

This shows that if the state at the initial time t = 0 was coherent, |ψ(0) = |α>, then it will remain coherent, up to a phase factor: I > |Ψ(t)> = eiξ(t) I(α + u(t))e−iωt .

(168)

Thus, as follows from Eq. (168), taking into account the force leads to an additional transformation of operators (see also Eq. (110)): † † = Bkλ + u∗, bˆkλ

bˆkλ = Bkλ + u.

(169)

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which defines the simplest canonical transformation (see Eqs. (110)–(113)). These equalities are a special case of J. von Neumann’s theorem, which asserts unitary equivalence under the canonical transformation of operators, that is, there exists a ˆ such that unitary operator Uˆ ,(Uˆ † Uˆ = Uˆ Uˆ † = 1) Bˆ = Uˆ † bˆUˆ , Bˆ † = Uˆ † bˆ † Uˆ .

(170)

Let us consider transformation Eqs. (110) and (170) in more detail, taking into account the time dependence. Using the operators bˆ and bˆ † , we construct a Hilbert space H whose basis is formed by the states n bˆ † |n> = √ |0>, n! ˆ = 0. b|0>

(171)

The operators Bˆ and Bˆ[† satisfy ] the same commutation relations (see Eq. (134)) as the operators bˆ and bˆ † , Bˆ Bˆ † = 1, thus one can construct a similar Hilbert space H˜ with basis n Bˆ † II ˜ \ |n> ˜ = √ I0 , n! I \ I Bˆ I0˜ = 0.

(172)

ˆ the transition from the Due to the unitary equivalence of the operators bˆ and B, space H to the space H˜ is carried out by the unitary transformation Eqs. (116) and (131): ˆ |n> ˜ = D(α)|n>, |n> = Dˆ † (α)|n>, ˜ I \ I˜ I0 = |α>.

(173)

˜ | in states | |m> coincide with Therefore, the expansion coefficients of the state |n> ˆ the matrix elements of the operator D(α) ˆ

˜ = = Dmn (α), 2 / I \ |α| I 0I0˜ = e− 2 . ˜ : Let us now form a generating function for the quantities

(174)

Emission and Absorption of Photons in Quantum Transitions …

˜ F(β, γ) =

401

[ βm γn ˜ √ . √ m! n! m,n

(175)

Summing over m, n we get I \ ˆ ˆ†I ˜ F(β, γ ) =

(15)

ˆj2 |λm> = λ|λm>. The set of eigenvectors {|λm>} is orthogonal and complete since Eq. (15) corresponds to the properties of the Sturm–Liouville problem: >

< ˆjx = ˆjx† , ⇒ λm' Iˆjx |λm> = = < ' I I ' >< ' I Iλm λm Iˆjx |λm> + =

m' =mmin m max Σ

= m2 + = m2 +

m' =mmin m max Σ

mmin

I I >< +

I I (19) >< =

m' =mmin

I I >II2 I III2 I III2 >II2 I I I = const|λ(m + 1)>, ˆj − |λm> = const|λ(m − 1)>.

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Proof: ]|χ > = ˆj + |λm>, [ ] ˆjz ˆjx = iˆjy ⇒ ˆjz ˆjx = ˆjx ˆjz + iˆjy [ ] ˆjy ˆjz = iˆjx ⇒ −iˆjx + ˆjy ˆjz = ˆjz ˆjy . ˆjz ˆjx − ˆjx ˆjz = iˆjy , ⇒ ˆjz ˆjx = iˆjy + ˆjx ˆjz , ˆjy ˆjz − ˆjz ˆjy = iˆjx ⇒ ˆjy ˆjz − iˆjx = ˆjz ˆjy , ( ) ˆjz |χ > = ˆjz ˆj + |λm> = ˆjz ˆjx + iˆjy |λm> = ˆjz ˆjx |λm> + iˆjz ˆjy |λm> = ( ) ( ) = ˆjx ˆjz + iˆjy |λm> + ˆjx + iˆjy ˆjz |λm> ) ( ) ( = mˆjx + iˆjy |λm> + ˆjx + iˆjy m |λm> = ( ) = m(ˆjx + iˆjy ) + ˆjx + iˆjy |λm> = (m + 1)ˆj + |λm> = (m + 1)|χ >,

(20)

|χ > = const|λ(m + 1)> = ˆj+ |λm>. ]|η> = ˆj− |λm>

( ) ˆjz |η> = ˆjz ˆj − |λm> = ˆjz ˆjx − iˆjy |λm> ) ( ) ( = ˆjx ˆjz + iˆjy |λm> − ˆjx + iˆjy ˆjz |λm> = ( ) ( ) ( ( )) = mˆjx + iˆjy |λm> − ˆjx + iˆjy m |λm> = mˆjx + iˆjy − ˆjx + iˆjy m |λm> = )) ( ( = m(ˆjx − iˆjy ) − ˆjx − iˆjy |λm> = (m − 1)ˆj− |λm> = (m − 1)|η>, |η> = ˆj− |λm> = const|λ(m − 1)>. Consequently, the operators ˆj − , ˆj + introduced by us can be considered as operators of decreasing and increasing by 1 the projection of the angular momentum on the z-axis, respectively.

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Theorem 1: The eigenvalue of the operator of the square of the angular momentum ˆj 2 is the number j(j + 1), where j is the maximal eigenvalue of the angular momentum projection onto the z-axis. Proof: ]mmax = j ˆj + |λmmax > = ˆj + |λj> = 0, ˆj2 |λmmax > = λ|λmmax >, ˆj 2 = ˆj − · ˆj + + ˆjz2 + ˆjz

(21)

ˆj 2 |λj> = (ˆj − · ˆj + + ˆjz2 + ˆjz )|λj> = (j2 + j)|λj> = j(j + 1)|λj>. Theorem 2: The minimum eigenvalue of the operator of the angular momentum projection on the z-axis is mmin = −j. Proof:

(22)

The second solution contradicts Lemma1. Thus, we come to the fundamental statement that the projection of the angular momentum on the quantization axis can change from the minimum value −j to the maximum value +j, running all possible values through one: −j, −j + 1, −j + 2, … j−2, j−1, j. The latter means that in nature there are only two types of angular momenta that satisfy this condition: integer, 0, 1, 2, … etc. and half-integers, 1/2, 3/2, … Matrix elements of the operators of projections on the x- and y-axes in the j z representation

Magnetic Ordering in a System of Identical Particles …

417

[ ] ˆj + ˆj − = 2ˆjz , ˆj − · ˆj + = ˆj + · ˆj − − 2ˆjz = ˆj 2 − ˆjz2 − ˆjz . ˆj + · ˆj − = ˆj 2 − ˆjz2 + ˆjz ; ˆjz |λm> = m|λm> = = = j(j + 1) − m(m − 1), I < < 'I − λm Iˆj |λm> = const λm' I|λ(m − 1)> = constδm' (m−1) I I > > = j(j + 1) − m(m − 1)|λm>, / ˆj− |λm> = j(j + 1) − m(m − 1)|λ(m − 1)>. Then, ( ) ˆjx = 1 ˆj + + ˆj − , 2 / = j(j + 1) − (m − 2)(m − 1) = 0 1/ = j(j + 1) − m(m − 1) = , 2 1/ = j(j + 1) − m(m − 1), 2 ( ) ˆjy = 1 ˆj + − ˆj − , 2i 1/ = j(j + 1) − m(m − 1) = −. 2i

(24)

Using the general expressions Eq. (24) obtained for the matrix elements of the operators of the projections of the angular momentum on the x-, y-, z-axes, we write out the matrices of these operators on the Cartesian basis in an explicit form.

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j = 1/2 ( ( 1) ) ) 1 01 1/2 0 0 2 ˆ = , , jx = sˆx = 1 0 0 −1/2 2 10 2 ( ( −i ) ) ˆjy = 0i 2 = 1 0 −i . 0 2 i 0 2

ˆjz = sˆz =

(

= = 1/ = j(j + 1) − m(m − 1), 2 = − = 1/ = j(j + 1) − m(m − 1), 2i = = 1/ 1 = 1/2(1/2 + 1) − 1/2(−1/2) = , 2 2 = − = 1/ −i = . 1/2(1/2 + 1) − 1/2(1/2 − 1) = 2i 2 (ii)

(25)

j=1 1/ = j(j + 1) − m(m − 1) = , 2 = = √ / 2 1 1 = = 1(1 + 1) − 1(1 − 1) = = √ , 2 2 2 = = 1 1/ 1(1 + 1) − 0(0 − 1) = √ , = 2 2 = − = 1/ j(j + 1) − m(m − 1), = 2i m = 1, 0, −1. ⎞ ⎛ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ 0 √1 0 010 0 −i 0 10 0 2 ⎟ ⎜ 1 1 1 ˆjz = ⎝ 0 0 0 ⎠, ˆjx = ⎜ √ 0 √1 ⎟ = √ ⎝ 1 0 1 ⎠, ˆjy = √ ⎝ i 0 −i ⎠. ⎝ 2 2⎠ 2 2 010 0 i 0 0 0 −1 0 √1 0 2

(26)

Magnetic Ordering in a System of Identical Particles …

(iii)

419

j = 3/2 ⎛

3/2 ⎜ 0 ˆjz = ⎜ ⎝ 0 0

⎞ 0 0 0 1/2 0 0 ⎟ ⎟, 0 −1/2 0 ⎠ 0 0 −3/2

1/ = j(j + 1) − m(m − 1) = , 2 √ / 1 3 = 3/2 · 5/2 − 1/2 · 3/2 = 2 2 = , 1/ = 15/4 + 1/4 = = 1, 2 √ 1/ 3 = = 15/4 − 3/2(1/2) = , 2 2 √ ⎞ ⎛ 3/2 0 0 √0 ⎜ 1 √0 ⎟ ⎟, ˆjx = ⎜ 3/2 0 ⎝ 0 3/2 ⎠ 1 √0 3/2 0 0 0 . (27) √ ⎞ ⎛ 3/2 0 0 0 −i √ ⎟ ⎜ 0 −i ⎟, ˆjy = ⎜ i 3/2 √0 ⎠ ⎝ 0 i √0 −i 3/2 0 0 0 i 3/2 / 1 = j(j + 1) − m(m − 1) = −, 2i √ / 1 3 = = −, 15/4 − 3/4 = −i 2i 2 1/ = 15/4 + 1/4 = −i = −, 2i √ / 1 3 = 15/4 − 3/4 = −i 2 2i = −.

2.4 Addition of Angular Momenta in Quantum Mechanics. The Wigner–Eckart Theorem Consider two rotations. There can be two rotations in which the particle participates. For example, an electron in an atom has orbital rotation and proper rotation (spin),

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or, in another case, we have two electrons in an atom in certain orbital states l1 and l2 , forming a complete orbital state L. In any case, let us have two rotations j1 and j2 , where we do not care about the physical nature of these rotations, but we will be interested in the connection of these rotations with the formation of the total angular momentum J. The operator of the total angular momentum Jˆ = ex Jx + ey Jy + ez Jz connects with the operators j1 = ex j1x + ey j1y + ez j1z and j2 = ex j2x + ey j2y + ez j2z as follows: /\

/\

/\

/\

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/\

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/\

/\

/\

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Jˆ = ex Jx + ey Jy + ez Jz = ) ) ( ) ( ( = j1 + j2 = ex j1x + j2x + ey j1y + j2y + ez j1z + j2z . /\

/\

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(28)

And /\

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Jˆ · Jˆ = J 2 = Jx2 + Jy2 + Jz2 = ( ) ) ( ( ) = j1 + j2 · j1 + j2 = j12 + j22 + 2 j1 · j2 . /\

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(29)

It is easy to prove that [

] [ ] J 2 j12 = J 2 j22 = 0, [ ] J 2 Jz = 0, ] [ ] [ Jz j1z = Jz j2z = 0, [ ] J 2 j1z = −2ij1y j2x + 2ij1x j2y /= 0, [ ] J 2 j2z = −2ij2y j1x + 2ij2x j1y , /= 0. /\/\

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(30)

And [

] [ ] J 2 j12 = J 2 j22 = 0, [ ] [ ] j12 j1z = j22 j2z = 0, ] [ ] [ Jz j1z = Jz j2z = 0, [ ] J 2 j1z = −2ij1y j2x + 2ij1x j2y /= 0, [ ] J 2 j2z = −2ij2y j1x + 2ij2x j1y , /= 0. /\/\

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(31)

That means we have two different orthogonal and complete basis. The first one is the basis of the following eigenstates:

Magnetic Ordering in a System of Identical Particles …

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> {|JMj1 j2 >}, = J (J + 1)|JMj1 j2 >,

(32)

/\

Jz |JMj1 j2 > = M |JMj1 j2 >,

/\

j12 |JMj1 j2 > = j1 (j1 + 1), /\

j22 |JMj1 j2 > = j2 (j2 + 1). The second is {|j1 j2 m1 m2 > = |j1 m1 >|j2 m2 >}, I > = j1 (j1 + 1)|j1 m1 j2 m2 >,

(33)

/\

j22 |j1 m1 j2 m2 > = j2 (j2 + 1)|j1 m1 j2 m2 >, /\

j1z |j1 m1 j2 m2 > = m1 |j1 m1 j2 m2 >,

/\

j2z |j1 m1 j2 m2 > = m2 |j1 m1 j2 m2 >,

/\

Jz |j1 m1 j2 m2 > = M |j1 m1 j2 m2 >. m1 + m2 = M .

These bases are related by the unitary transformation: j1

|JMj1 j2 > =

j2 Σ

|j m1 j2 m2 >. CjJM 1 m1 j2 m2 1

(34)

m1 =−j1 m2 =−j2 . m1 +m2 =M

Here, the coefficients CjJM are the Clebsch–Gordan coefficients. This is the 1 m1 j2 m2 Wigner–Eckart theorem. These coefficients could be calculated by the exact algebraic method in the following way: Acting by the following operator on the wave vector (34) with respect to notations (23), we have ) ( J 2 = Jx2 + Jy2 + Jz2 == j12 + j22 + 2 j1 · j2 , ⇒ ( ) J 2 − j12 − j22 = 2 j1 · j2 = j1+ j2− + j1− j2+ + 2j1z j2z , /\

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(35)

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( ) J 2 − j12 − j22 |JMj1 j2 > = /\

/\

/\

j1

j2 Σ

[ [ + − − + C JM j1 m1 j2 m2 2j1z j2z + j1 j2 + j1 j2 |j1 m1 j2 m2 >, /\/\

/\/\

/\/\

m1 =−j1 m2 =−j2 . m1 +m2 =M j1

(J (J + 1) − j1 (j1 + 1) − j2 (j2 + 1))

j2 Σ

C JM j1 m1 j2 m2 |j1 m1 j2 m2 > =

m1 =−j1 m2 =−j2 . m1 +m2 =M j1

=

j2 Σ

C JM j1 m1 j2 m2 {2m1 m2 |j1 m1 j2 m2 >+

m1 =−j1 m2 =−j2 . m1 +m2 =M

/ / + j2 (j2 + 1) − (m2 + 1)m2 j1 (j1 + 1) − m1 (m1 − 1)|j1 (m1 − 1)j2 (m2 + 1)>+ [ / / + j1 (j1 + 1) − (m1 + 1)m1 j2 (j2 + 1) − m2 (m2 − 1)|j1 (m1 + 1)j2 (m2 − 1)> , (36) from which we have a system of equations determined by the coefficients CjJM 1 m1 j2 m2 = (J (J + 1) − j1 (j1 + 1) − j2 (j2 + 1) − 2m1 m2 )CjJM 1 m1 j2 m2 / / = CjJM j2 (j2 + 1) − (m2 − 1)m2 j1 (j1 + 1) − m1 (m1 + 1)+ 1 m1 +1j2 m2 −1 / / JM +Cj1 m1 −1j2 m2 +1 j1 (j1 + 1) − (m1 − 1)m1 j2 (j2 + 1) − m2 (m2 + 1), j1 j2

Σ

(

CjJM 1 m1 j2 m2

)2

(37)

= 1.

m1 =−j1 m2 =−j2 . m1 +m2 =M

As an example, we consider the Clebsch–Gordan coefficients for the addition of two spins s = 1 with a formation of the two different states |J = 2, M = 0, j1 = 1, j2 = 1 > and |J = 0, M = 0, j1 = 1, j2 = 1 > : 20 |J = 2, M = 0j1 = 1, j2 = 1> = C111−1 |j1 = 1, m1 = 1, j2 = 1, m2 = −1>+ 20 20 |j1 = 1, m1 = −1, j2 = 1, m2 = 1> + C1010 |j1 = 1, m1 = 0, j2 = 1, m2 = 0>, +C1−111 00 |J = 0, M = 0j1 = 1, j2 = 1> = C111−1 |j1 = 1, m1 = 1, j2 = 1, m2 = −1>+ 00 00 |j1 = 1, m1 = −1, j2 = 1, m2 = 1> + C1010 |j1 = 1, m1 = 0, j2 = 1, m2 = 0>. +C1−111

(38)

Magnetic Ordering in a System of Identical Particles …

423

For the |J = 2, M = 0j1 = 1, j2 = 1> state, we have the system of the following equations determined by the Clebsch–Gordan coefficients: I 20 I j1 = 1, m1 = 1, j2 = 1, m2 = −1>+ |J = 2, M = 0j1 = 1, j2 = 1> = C111−1 I 20 + C1−111 I j1 = 1, m1 = −1, j2 = 1, m2 = 1> I 20 I j1 = 1, m1 = 0, j2 = 1, m2 = 0>, + C1010 \ [ 20 I I j1 = 1, m1 = 1, j2 = 1, m2 = −1 + (2(2 + 1) − 1(1 + 1) − 1(1 + 1)) C111−1 \ I 20 I j1 = 1, m1 = −1, j2 = 1, m2 = 1 + C1−111 I 20 I j1 = 1, m1 = 0, j2 = 1, m2 = 0>} = + C1010 [[ [ I 20 I j1 = 1, m1 = 1, j2 = 1, m2 = −1>+ j1− j2+ + j1+ j2− + 2j1z j2z C111−1 I 20 I j1 = 1, m1 = −1, j2 = 1, m2 = 1> + C1−111 I 20 I j1 = 1, m1 = 0, j2 = 1, m2 = 0>} = + C1010 I / [ 20 / I = C111−1 1(1 + 1) − 0 · 1 1(1 + 1) − 1 · 0I j1 = 1, m1 = 0j2 = 1m2 = 0> /\/\

/\/\

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20 +C1−111 · 0+ \] / / 20 +C1010 1(1 + 1) − 0 · 1 1(1 + 1) − 1 · 0 |j1 = 1, m1 = −1, j2 = 1, m2 = +1> + / / [ 20 · 0 + 1(1 + 1) − 0 · 1 1(1 + 1) − 1 · 0 + C111−1 I 20 I j1 = 1, m1 = 0j2 = 1m2 = 0>+ C1−111 ] / / 20 1(1 + 1) − 0 · 1 1(1 + 1) − 1 · 0 |j1 = 1, m1 = 1, j2 = 1, m2 = −1> + +C1010 \ I 20 I j1 = 1, m1 = 1, j2 = 1, m2 = −1 + +{2(−1) · 1 C111−1 \ I 20 I j1 = 1, m1 = −1, j2 = 1, m2 = 1 +2 · 1 · (−1) C1−111 I 20 I +0 · C1010 j1 = 1, m1 = 0, j2 = 1, m2 = 0>}, ⎧ 20 (2(2 + 1) ⎪ ⎪ / − 1(1 + 1) − 1(1 / + 1))C111−1 ⎪ 20 20 ⎪ ⎪ 1(1 + 1) − 0 · 1 1(1 + 1) − 1 · 0 + 2(−1) · 1C111−1 , = C1010 ⎪ ⎪ 20 ⎪ ⎪ − 1(1 + 1) − 1(1 + 1))C ⎨ (2(2 + 1) 1−111 / / 20 20 1(1 + 1) − 0 · 1 1(1 + 1) − 1 · 0 + 2 · 1 · (−1)C1−111 , = C1010 ⎪ 20 ⎪ ⎪ = + 1) − 1(1 + 1) − 1(1 + 1))C (2(2 ⎪ 1010 / / ⎪ ⎪ 20 ⎪ = 1(1 + 1) − 0 · 1 1(1 + 1) − 1 · 0C1−111 ⎪ ⎪ / / ⎩ 20 +C111−1 1(1 + 1) − 0 · 1 1(1 + 1) − 1 · 0. (39)

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⎧ 20 20 ⎨ 2111−1 = C1010 20 20 , 2 = C1010 ⎩ 1−111 20 20 20 C1010 = C1−111 + C111−1 ( 20 )2 ( 20 )2 ( 20 )2 C111−1 + C1−111 + C1010 = 1, 1 20 20 C111−1 = √ = C1−111 , 6 / 2 2 20 C1010 = √ = . 3 6

(40)

1 |J = 2, M = 0j1 = 1, j2 = 1> = √ |j1 = 1, m1 = 1, j2 = 1, m2 = −1>+ 6 2 1 + √ |j1 = 1, m1 = −1, j2 = 1, m2 = 1> + √ |j1 = 1, m1 = 0, j2 = 1, m2 = 0> = 6 6 1 = √ {|j1 = 1, m1 = 1, j2 = 1, m2 = −1> + |j1 = 1, m1 = −1, j2 = 1, m2 = 1>+ 6 +2 |j1 = 1, m1 = 0, j2 = 1, m2 = 0>}. (41) For the state 00 |J = 0, M = 0j1 = 1, j2 = 1> = C111−1 |j1 = 1, m1 = 1, j2 = 1, m2 = −1>+ 00 |j1 = 1, m1 = −1, j2 = 1, m2 = 1> +C1−111 00 |j1 = 1, m1 = 0, j2 = 1, m2 = 0>, +C1010

we have

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I 00 I j1 = 1, m1 = 1, j2 = 1, m2 = −1>+ |J = 0, M = 0j1 = 1, j2 = 1> = C111−1 I 00 I j1 = 1, m1 = −1, j2 = 1, m2 = 1> + C1−111 I 00 I j1 = 1, m1 = 0, j2 = 1, m2 = 0>, + C1010 \ [ 00 I I j1 = 1, m1 = 1, j2 = 1, m2 = −1 + (−1(1 + 1) − 1(1 + 1)) C111−1 \ I 00 I j1 = 1, m1 = −1, j2 = 1, m2 = 1 + C1−111 I 00 I + C1010 j1 = 1, m1 = 0, j2 = 1, m2 = 0>} = [[ [ I 00 I j1 = 1, m1 = 1, j2 = 1, m2 = −1>+ j1− j2+ + j1+ j2− + 2j1z j2z C111−1 I 00 I j1 = 1, m1 = −1, j2 = 1, m2 = 1> + C1−111 I 00 I j1 = 1, m1 = 0, j2 = 1, m2 = 0>} = + C1010 I / [ 00 / I = C111−1 1(1 + 1) − 0 · 1 1(1 + 1) − 1 · 0I j1 = 1, m1 = 0j2 = 1, m2 = 0> /\/\

/\/\

/\/\

00 +C1−111 · 0+ / / 00 +C1010 1(1 + 1) − 0 · 1 1(1 + 1) − 1 · 0 \] |j1 = 1, m1 = −1, j2 = 1, m2 = +1> + / / [ 00 · 0 + 1(1 + 1) − 0 · 1 1(1 + 1) − 1 · 0 + C111−1 I 00 I j1 = 1, m1 = 0j2 = 1, m2 = 0>+ C1−111 / / 00 +C1010 1(1 + 1) − 0 · 1 1(1 + 1) − 1 · 0 ] |j1 = 1, m1 = 1, j2 = 1, m2 = −1> + \ I 00 I j1 = 1, m1 = 1, j2 = 1, m2 = −1 + +{2(−1) · 1 C111−1 \ I 00 I j1 = 1, m1 = −1, j2 = 1, m2 = 1 +2 · 1 · (−1) C1−111 I 00 I +0 · C1010 j1 = 1, m1 = 0, j2 = 1, m2 = 0>}, ⎧ 00 (−1(1 +/1) − 1(1 + 1))C111−1 ⎪ ⎪ / ⎪ 00 00 ⎪ ⎪ = C1010 1(1 + 1) − 0 · 1 1(1 + 1) − 1 · 0 + 2(−1) · 1C111−1 , ⎪ ⎪ 00 ⎪ ⎪ + 1) − 1(1 + 1))C (−1(1 ⎨ 1−111 / / 00 00 1(1 + 1) − 0 · 1 1(1 + 1) − 1 · 0 + 2 · 1 · (−1)C1−111 , = C1010 ⎪ 00 ⎪ ⎪ (−1(1 = + 1) − 1(1 + 1))C ⎪ 1010 / / ⎪ ⎪ 00 ⎪ = 1(1 / + 1) − 0 · 1 1(1 / + 1) − 1 · 0C1−111 ⎪ ⎪ ⎩ 00 +C111−1 1(1 + 1) − 0 · 1 1(1 + 1) − 1 · 0.

(42)

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⎧ 00 00 = C1010 ⎨ −C111−1 00 00 , −C1−111 = C1010 ⎩ 00 00 00 −2C1010 = C1−111 + C111−1 ( 00 )2 ( 00 )2 ( 00 )2 C111−1 + C1−111 + C1010 = 1, 1 00 00 00 C111−1 = C1−111 = −C1010 =√ . 3

(43)

Finally, we have for the state |J = 0, M = 0j1 = 1, j2 = 1> 1 |J = 0, M = 0j1 = 1, j2 = 1> = √ {|j1 = 1, m1 = 1, j2 = 1, m2 = −1> + 3 (44) +|j1 = 1, m1 = −1, j2 = 1, m2 = 1> − |j1 = 1, m1 = 0, j2 = 1, m2 = 0>}.

3 Spin Hamiltonian of Identical Particles with Arbitrary Spin 3.1 Interaction in a System of Two Identical Particles with Spin Consider a Hamiltonian describing pair interaction in the coordinate representation of two nonrelativistic identical particles. We represented the complete Hamiltonian as the sum of the Hamiltonians of two independent particles Hˆ 0 (r1 , r2 ) = hˆ I (r1 ) + hˆ II (r2 ) and the interaction operator Vˆ (r1 , r2 ): Hˆ (r1 , r2 ) = Hˆ 0 (r1 , r2 ) + Vˆ (r1 , r2 ).

(45)

In the nonrelativistic case, the total Hamiltonian Eq. (45), in the absence of spin– orbit interaction, is invariant with respect to the permutation of particles described by the operator Pˆ 1,2 and also commutes with the operator Sˆ of the total spin of particles included in the system: [

] Hˆ (r1 , r2 )Pˆ 1,2 = 0, [ ] Hˆ (r1 , r2 )Sˆ = 0.

(46)

This means that in the stationary case the total energy E of the system, the magnitude of the total spin S, its projection Sz , and the parity g of the permutation are conserved. These quantities are simultaneously precisely measurable and define the complete set of parameters that characterize the system. The eigenfunctions of the

Magnetic Ordering in a System of Identical Particles …

427

Hamiltonians hˆ I (r1 ) and hˆ II (r2 ), describing the motions of particles 1 and 2, corresponding to states I k, II l with energies εI and εII , respectively, are the products of the coordinate and spin parts of the first and second particles, respectively: hˆ I (r1 )ψI (r1 , ξ1 ) = εI ψI (r1 , ξ1 ), hˆ II (→r2 )ψII (r2 , ξ2 ) = εIl ψII (r2 , ξ2 ), ψI (r1 , ξ1 ) = φI (r1 )χ1 (ξ1 ),

(47)

ψII (r2 , ξ2 ) = φII (r2 )χ2 (ξ2 ). The solution of the stationary Schrödinger equation for two identical particles, in the absence of interaction, Hˆ 0 (r1 , r2 )ψI0,II (r1 , ξ1 , r2 , ξ2 ) = EI0,II ψI0,II (r1 , ξ1 , r2 , ξ2 ), in accordance with Eq. (46), is (2 s + 1) times degenerate in the value of the total spin S, where s is the value of the spin of the particle. The total wave function ψI0,II (r1 , ξ1 , r2 , ξ2 ), taking into account the above-indicated degeneracy, corresponding to the energy value EI0,II = εI + εII , is an eigenfunction of this Hamiltonian: Hˆ 0 (r1 , r2 )ψI0,II (r1 , ξ1 , r2 , ξ2 ) = (εI + εII )ψI0,II (r1 , ξ1 , r2 , ξ2 ),

(48)

and meets the properties of symmetry with respect to the permutation operation. Pˆ 1,2 ψI0,II (r1 , ξ1 , r2 , ξ2 ) = (−1)2s ψI0,II (r1 , ξ1 , r2 , ξ2 ).

(49)

Here, the parity of the complete permutation g = (−1)2s . The operation of a total permutation of indices 1 and 2 includes a permutation of the spatial coordinates of particles 1 and 2 and the values of the projections of the spins on the Z-axis of individual particles: Pˆ 1,2 = Pˆ r1 r2 Pˆ sz1 ,sz2 .

(50)

In the nonrelativistic case, a two-particle function can always be represented as a simple product of its coordinate and spin parts: ψI0,II (r1 , ξ1 , r2 , ξ2 ) = ΦI ,II (r1 , r2 ) · X(ξ1 , ξ2 ).

(51)

The symmetry properties of each part of the wave function with respect to the permutation operation are determined as follows: (1) The symmetry of the spin part of a system of two identical particles is determined by the symmetry of the Clebsch–Gordan coefficients [2]

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X(ξ1 , ξ2 ) = |S, Sz > =

Σ

CsSS1 sz1z s2 s2z |s1 , s1z >|s2 , s2z >,

s1z s2z

where CssSS1zz ss2z = (−1)2s−S CssSS2zz ss1z , and Pˆ sz1 ,sz2 X(ξ1 , ξ2 ) = (−1)2s−S X(ξ1 , ξ2 ).

(52)

Here, S is the total spin of the couple of particles. (2) The symmetry of the coordinate part of the two-particle function Eq. (51) is determined as the result of the action of the permutation operator Pˆ r1 r2 on the function Eq. (51), taking into account Eqs. (49) and (52): Pˆ 1,2 ψI ,II (r1 , ξ1 , r2 , ξ2 ) = Pˆ r1 ,r2 ΦI ,II (r1 , r2 )Pˆ sz1 ,sz2 X(ξ1 , ξ2 ) = = (−1)2s ψI ,II (r1 , ξ1 , r2 , ξ2 ), Pˆ r1 ,r2 ΦI ,II (r1 , r2 )(−1)2s−S X(ξ1 , ξ2 ) = (−1)2s ψI ,II (r1 , ξ1 , r2 , ξ2 ),

(53)

Pˆ r1 ,r2 ΦI ,II (r1 , r2 ) = (−1)S ΦI ,II (r1 , r2 ). Taking into account the interaction in the first order of the stationary perturbation theory gives the first correction to energy Eq. (48) in the following form: I I < > E (1) = ψI0,II (r1 , ξ1 , r2 , ξ2 )IVˆ (r1 , r2 )IψI0,II (r1 , ξ1 , r2 , ξ2 ) = I I < > = ΦI ,II (r1 , r2 )IVˆ (r1 , r2 )IΦI ,II (r1 , r2 ) = K ± A, K = ,

(54)

A = , where K is the “direct” Coulomb contribution and A is the exchange contribution. This type of solution is due to the symmetry of the coordinate part of the wave function (see, for example, [31]) which has the following form: ( ) ΦI ,II (r1 , r2 ) = N φI (r1 )φII (r2 ) + (−1)S φI (r2 )φII (r1 ) .

(55)

Thus, the correction to the energy due to the interaction in the coordinate representation is determined according to Eq. (55) by the value of the total spin: E (1) = K + (−1)S A.

(56)

It should be noted here that if we are dealing with a system of identical spin-less particles (s = 0), then in expressions (45)–(48), the spin part should be omitted (spin

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function χ1 (ξ1 ) = χ2 (ξ2 ) = X(ξ1 ξ2 ) = 1). Nevertheless, spin-less particles belong to the bosonic class (s = 0—is the integer number), then the condition Eq. (49) must be satisfied with the factor g = (−1)2s = 1, which means the symmetry of the total wave function of the system of two spin-less particles. The degeneracy of the two-particle state with the energy EI0,II = εI + εII is equal to (2 s + 1) = 1, i.e., this state is non-degenerated. That is why the energy correction Eq. (53) has only one solution corresponding to the symmetric form of the wave function E (1) = K + A, that is in an agreement with the general condition Eq. (56) for the total spin S = 0.

3.2 Spin Hamiltonian of a System of Identical Particles We showed above (see Eqs. 49, 50, 52, and 53) that the symmetries of the coordinate and spin parts of the wave function are uniquely related. This means that one can introduce operators acting in the spin space and directly acting on the spin variables of the wave function, in such a way that the eigenvalues of these operators will be numbers or combinations of numbers that are directly related to the symmetry of the coordinate part. Let us set ourselves the goal of constructing a Hamiltonian in the spin representation for a system of identical particles, which describes the interaction defined in the coordinate representation, and its eigenvalue will be a solution in the form Eq. (56) [32–35]. For this, it is necessary to obtain an explicit form of the parity operator of the permutation of the coordinate part of the wave function, denote it by Rˆ s1 ·s2 , which, however, acts on the spin variables of the wave function in the spin representation. The eigenvalue of this operator must be the parity value /\r = (−1)S of the permutation of the coordinate part of the wave function Eqs. (53), (55): Rˆ s1 ·s2 X(ξ1 , ξ2 ) = (−1)S X(ξ1 , ξ2 ), ⇔ Rˆ s1 ·s2 |S, Sz ; s1s2>, Rˆ s1 ·s2 |S, Sz ; s1s2> = (−1)S |S, Sz ; s1s2>, ⇒ Σ Σ CsSS1 sz1z s2 s2z |s1 , s1z >|s2 , s2z > = (−1)S CsSS1 sz1z s2 s2z |s1 , s1z >|s2 , s2z >. Rˆ s1 ·s2 s1z s2z

(57)

s1z s2z

We will look for the parity operator Rˆ s1 ·s2 in the form of a series expansion in powers of the scalar products of the spin operators of interacting particles: ( )2s ( )2s−1 ( ) + ... + c1 sˆ1 · sˆ2 + c0 . Rˆ s1 ·s2 = c2s sˆ1 · sˆ2 + c2s−1 sˆ1 · sˆ2

(58)

Substituting (41) into (40), and taking into account ) 1( ) 1( 2 S − s12 − s22 = S 2 − 2s2 , 2 2 /\

sˆ1 · sˆ2 =

/\

/\

/\

/\

(59)

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we have a system of algebraic equations that uniquely determines the sought coefficients in expansion Eq. (58): ⎧ ( ) ) ( S(S + 1) − 2s(s + 1) 2s−1 S(S + 1) − 2s(s + 1) 2s ⎪ ⎪ c + c + ... ⎪ 2s−1 2s ⎪ ⎪ 2 2 ⎪ ⎪ I ) ( ⎪ ⎪ I S(S + 1) − 2s(s + 1) ⎪ ⎪ ⎪ + c0 II = (−1)0 , +c 1 ⎪ ⎪ 2 ⎪ S=0 ⎪ ⎪ ( ( ) ) ⎪ ⎪ ⎪ S(S + 1) − 2s(s + 1) 2s−1 S(S + 1) − 2s(s + 1) 2s ⎪ ⎪ c2s + c2s−1 + ... ⎪ ⎪ 2 2 ⎪ ⎪ I ⎪ ( ) ⎪ I ⎪ S(S + 1) − 2s(s + 1) ⎪ ⎪ + c0 II = (−1)1 , +c1 ⎪ ⎪ ⎪ 2 ⎪ S=1 ⎪ ⎪ ⎨ ... (60) ( ( ) ) ⎪ S(S + 1) − 2s(s + 1) 2s S(S + 1) − 2s(s + 1) 2s−1 ⎪ ⎪ ⎪ c2s + c2s−1 + ... ⎪ ⎪ 2 2 ⎪ ⎪ I ( ) ⎪ ⎪ I ⎪ S(S + 1) − 2s(s + 1) ⎪ ⎪ +c1 + c0 II = (−1)S , ⎪ ⎪ 2 ⎪ S ⎪ ⎪ ⎪ ⎪ ... ⎪ ⎪ ⎪ ⎪ ) ) ( ( ⎪ ⎪ S(S + 1) − 2s(s + 1) 2s S(S + 1) − 2s(s + 1) 2s−1 ⎪ ⎪ + c2s−1 + ... ⎪ c2s ⎪ ⎪ 2 2 ⎪ ⎪ I ( ) ⎪ ⎪ I S(S + 1) − 2s(s + 1) ⎪ ⎪ ⎩ +c1 + c0 II = (−1)2s . 2 S=2s Thus, in the first approximation of perturbation theory, Hamiltonian Eq. (45) can be replaced by an effective Hamiltonian acting in the spin space, for which the commutation relations Eq. (45) remain valid. Hˆ = E 0 + K + ARˆ s1 ·s2 ,

(61)

where the parity operator Rˆ s1 ·s2 is uniquely determined by Eqs. (58) and (60). Generalizing to the case of a system of particles with an arbitrary spin s, and dropping the constant corresponding to the energy of the state E 0 of non-interacting particles, we obtain a Hamiltonian that takes into account the pair interaction in the form: Σ Σ Kkl + Akl Rˆ sk ,sl . (62) Hˆ int = k Iexp − A Rˆ S ·S IIS1,2,3,4 × 1,2 3,4 I I T I ( )I ( (1) ) II II < > A(1) ˆ A I I g5,6,7,8 S5,6,7,8 Iexp − RS5,6 ·S7,8 IIIS5,6,7,8,2 ...× f T T ( (1) ) I < A gN −3,N −2,N −1,N SN −3,N −2,N −1,N I f T I ( )I (1) I II > Iexp − A Rˆ S IISN −3,N −2,N −1,N , N −3,N −2 ·SN −1,N I I T

=

2 2 Σ Σ

...

2 Σ

(92)

where gklmn are degeneration factors of the spin projections onto an arbitrary quantization axis mentioned above. A matrix element of the operator exponent can be calculated in the following way: I ( )I II II > A(0) ˆ RsN −1 ·sN IIISN −1,N = SN −1,N IIIexp − T I I IΣ )η I ( (0) II II ∞ 1 < > A − SN −1,N III Rˆ sN −1 ·sN IIISN −1,N = T I η=0 η! I I I I∞ I ( (0) )η II IIΣ < > A 1 ηSN −1,N II I I = SN −1,N I − (−1) I SN −1,N = η! T I η=0 I I ( )I II II > < A(0) = SN −1,N IIIexp − (−1)SN −1,N IIISN −1,N = T ( (0) ) SN −1,N A . = exp −(−1) T


II > < II A(0) ˆ A(0) ˆ I I I I I I g1,2 S1,2 Iexp − Rs1 ·s2 I S1,2 · g3,4 S3,4 Iexp − Rs3 ·s4 IIIS3,4 = T T I ( )I (1) Σ ( A(1) ) II II < > A = Rˆ S1,2 ·S3,4 IIIS1,2,3,4 , ⇒ g1,2,3,4 S1,2,3,4 IIIexp − T T S1,2,3,4 , (94) ( ( ) ) A(0) A(0) · g3,4 exp −(−1)S3,4 = g1,2 exp −(−1)S1,2 T T ( (1) ) Σ ( A(1) ) A = f g1,2,3,4 exp −(−1)S1,2,3,4 . T T S ,
+ v|3/2, −3/2>, |χ− > = −v|3/2, 3/2> + u|3/2, −3/2>, I > Iχ1/2 = |3/2, 1/2>, I > Iχ−1/2 = |3/2, −1/2>,

(118)

u2 + v2 = 1. we diagonalize Hamiltonian Eq. (117) and set the parameters u = cos φ,v = sin φ, where tan φ = v/u, find the eigenvalues of energy corresponding to states Eq. (118) ( ) < >) < > ηA 1 < 0 > 1 ( Q2 + 9 + Q30 (u2 − v2 ) + Q33 2uv = E+ = Emin = − 6 2 5 ( ) < ( < > >) < > 1 ηA 1 0 Q2 + 9 + Q30 cos(2φ) + Q33 sin(2φ) , =− 6 2 5 ( ) < >) < > ηA 1 < 0 > 1 ( Q2 − 9 + Q30 (u2 − v2 ) − Q33 2uv = E− = Emax = − 6 2 5 ( ) < 0 >) < > ηA 1 < 0 > 1 ( Q2 − 9 + Q3 cos(2φ) − Q33 sin(2φ) , − 6 2 5 ( ) < >) ηA 1 < 0 > 1 ( Q2 − − Q30 , E1/2 = 2 6 5 ( ) (119) < >) ηA 1 < 0 > 1 ( Q2 + − Q30 . E−1/2 = 2 6 5 We chose the parameters u = sinϕ and v = cosϕ of the Bogoliubov transformations such that the state |χ + > corresponds to the normal sublevel with the minimum energy, which holds under the condition

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(( ) ) 1< > 3 5< > − ηA + Q30 · sin(2φ) − Q33 cos(2φ) = 0. 5 9 9

(120)

Bearing in mind relations Eqs. (114) and (115), we find the values of the multipole moments averaged over the above states
< > 41 Q30 = 5 (sz )3 − 4 < 2 > 15 < 0> Q2 = 3 (sz ) − 4 < 3 > 1 (/( + )3 \ /( − )3 \) s + s Q3 = 2

(121)

which take the following values for the state |χ + > =

) 3 3( 2 u − v2 = cos 2φ, 2 2

( < 0> < > 15 ) = 3 u2 + v2 = 3, Q2 = 3 (sz )2 − 4

(122)

> < > 41 ) 3 3( Q30 = 5 (sz )3 − = u2 − v2 = cos 2φ, 4 2 2 ) ( ( )3 ( )3 < 3 > uv + = Q3 = 2 = 3uv = 3 sin 2φ,
is the average value of the spin < projection > 0 Q Q is the average quadrupole moment, is the average octupole moment, and 2 3 < 3> Q3 is the average value of the octupole transition parameter. Substituting Eqs. (122) into Eq. (119), we find the expressions for the energies of the respective states 3ηA 4 ηA = 4

E+ = Emin = − E− = Emax E± 21 =

ηA 4

(123)

which vary according to the Bogoliubov transformation (the rotation transformation with the tangent of the rotation angle defined as tanϕ = v/u).

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Taking into account Eq. (123), condition Eq. (120) of the extreme of the state |χ + > with the minimum energy E is transformed to E+' =

η A cos(2φ) sin(2φ) = 0 5

(124)

from which the following possible solutions arise, which we will analyze for two fundamentally different cases of the values of the exchange integral, A > 0 and A < 0: Case I. A > 0 (1) The first solution of Eq. (124), cos(2φ1 ) = 0, ⇒ φ1 = π4 . This state is characterized by the average spin projection = 23 cos 2φ1 = 0, by the average < > < > = 3, by the average octupole moment quadrupole moment Q20 = 3 (sz )2 − 15 4 < 0> < 3 > 41 Q3 = 5 (sz ) − 4 = 23 cos 2φ1 = 0, and by the average value of the < > octupole transition parameter Q33 = 3 sin 2φ1 = 3. Thus, the first solution corresponds to a nematic state. To clarify the stability of the said state, we check the sign of the second derivative of E + in Eq. (123) Iwith respect to the angular parameter. '' A cos 4φ Iφ1 = − 2η A < 0. Thus, the above nematic state is unstable. : E+ = 2η 5 5 Following the method [41], the excitation spectrum in the nematic phase has the form 3η ε1,2 (k) = √ × 2 2 / (( ) ( ))[ (( ) ( )) 11 5 2 11 5 2 − A − (A − A(k)) 18 4 9 18 4 9 / (( ) ( ) ( ))] 9 3 11 63 2 1 + + = − A(k) − 2 8 2 18 16 9 ) ( Σ η(A − A(k)) ηA ik·bν = e , = √ 1 − Re √ 2 2 2 2 ν /( ) ] [( ) 2 2 9η ε3 (k) = (A − A(k)) (A − A(k)) − A(k)△1 = 4 9 9 ( ) Σ η(A − A(k)) η ik·bν = = A 1 − Re e , 6 6 ν ) ( ) ( ) ( 1 11 103 2 9 − + · = 0. △1 = − 8 2 18 16 9

(125)

( ) Here, we used the pre-factors from Eq. (113): the parameter c3 = 29 in front of ( 11 ) the bicubic term in Eq. (113), the parameter c2 = 18 in front of the biquadratic

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( ) term, and the parameter c1 = − 98 – in front of the bilinear (Heisenberg) term. Here, k is the magnon wave vector, bv is the lattice vector in the direction toward the vth nearest neighbor, A(k) is the Fourier transform of the exchange integral, and the gap parameter is △1 = 0, which corresponds to the state of the phase transition to a magnetic phase. At the corner point of the Brillouin zone, i.e., at k = 0, where A(k) = –A, we have / 135 9η / c2 9η A c3 (c1 − + c3 ) = A c3 △2 ; ε3 (k) = 4 2 16 4 135 2 c2 + c3 ) = △2 = (c1 − 2 16 9

(126)

( )2 and the condition c3 △2 = 29 > 0 ensures the stability of the nematic phase with respect to the phase transition to the antiferromagnetic state. This condition is always satisfied for any sign of the exchange integral A. In other words, the phase transition from the nematic phase to the antiferromagnetic state is prohibited. (2) The second solution sin 2φ2 = 0, ⇒ φ2 = 0, then • the average spin projection value is = 23 cos 2φ2 = 23 , < > • the average quadrupole moment is Q20 = 3, < 0> 3 • the average octupole moment is Q3 = 2 cos 2φ2 = 23 , • the average value of the octupole transition parameter is
Q33 = 3 sin(2φ2 ) = 0.

Thus, the second solution corresponds to a ferromagnetic state. STABILITY of THE SOLUTION. • We check the sign of the second derivative of E+ with respect to the angular I '' Iφ2 = 2η A > 0. Thus, the ferromagnetic state is A cos 4φ parameter: E+ = 2η 5 5 stable. The Excitation Spectrum in the Ferromagnetic Phase (A>0) In the nearest-neighbor approximation in the case of the isotropic magnet, three modes merge into a single gapless Goldstone magnon mode:

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Fig. 3 Schematic diagram of the possible phase transitions for the 3/2-spin system

(( ) ( ) ( )) 9 3 11 63 2 3η + + = ε1 (k) = (A − A(k)) − 4 8 2 18 16 9 η(A − A(k)) = , 2 ) ( ( ) 11 5 2 3η − + A△1 = ε2 (k) = (A − A(k)) 2 18 4 9 η = (A − A(k)), 2 ] [( ) 2 9η ε3 (k) = (A − A(k)) − A(k)△1 = 2 9 ( ) Σ η η(A − A(k)) ik·bν = A 1 − Re e , = 2 2 ν ) ( ) ( ( ) 1 11 103 9 2 − + = 0. △1 = − · 8 2 18 16 9

(127)

A gap parameter △1 = 0 corresponds to the state of the phase transition from a ferromagnetic phase to a nematic state. At the corner point of the Brillouin zone, i.e., at k = 0, where A(k) = –A, the condi( )2 tion c3 △2 = 29 > 0 ensures the stability of the ferromagnetic phase with respect to the phase transition to an antiferromagnetic phase. A transition to an antinematic phase is forbidden since under weakening of correlations, i.e., at A → 0, the system passes to a paramagnetic disordered phase. The total angular momentum remains maximal and spin waves with the Goldstone, the gapless spectrum can propagate on the background of ferromagnetic ordering (See Fig. 3). Case II, A Q2 = 3, the average quadrupole moment, < 0 >1 Q2 2 = 3, < 0> 3 Q3 1 = cos 2φ1 = 0, 2 the average octupole moment, < 0> 3 Q3 2 = − cos 2φ1 = 0, 2 the value of the octupole transition < 3 > average Q3 1 = 3 sin(2φ1 ) = 3, < 3> Q3 2 = −3 sin(2φ1 ) = −3. 1 =

parameter,

Thus, the first solution corresponds to the antinematic state with the inclusion of two sublattices. STABILITY of THE SOLUTION. We check the sign of the derivative of I second '' Iφ = − 2η A > 0 Thus, A cos 4φ E+ with respect to the angular parameter: E+ = 2η 1 5 5 the antinematic state is stable. The Excitation Spectrum in the Antinematic Phase (A < 0) should consist of three modes, but the first two modes disappear and only one Goldstone gapless mode remains. 3η / ε1,2 (k) = √ · (A + A(k))× 2 2 /(( ) ( ))[ (( ) ( )) 11 11 2 11 11 2 − A − 18 4 9 18 4 9 / (( ) ( ) ( ))] 9 3 11 191 2 1 − + = 0, − A(k) − 2 8 2 18 16 9 /( ) 2 9η ε3 (k) = (A − A(k))× 4 9 /[( ) ( ( ) ( )) ] 2 9 1 11 119 2 + △1 = (A − A(k)) − A(k) − − 9 8 2 18 16 9 ) ( Σ η(A − A(k)) η ik·bν e = . = √ A 1 − Re √ 2 2 ν

(128)

This mode is the same as in the case of the ferromagnetic phase considered above. The stability of the antinematic phase requires the fulfillment of two conditions: A△1 > 0 and A△2 < 0. The first condition is violated in principle, as △1 = 0, which corresponds to a phase transition to the antiferromagnetic state. The second condition does hold, since A△2 = 29 A < 0 and vanishes only under weakening of correlations,

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455

i.e., when the exchange integral vanishes. Thus, the antinematic (stable) state is in neutral equilibrium with respect to the transition to the antiferromagnetic state in the parameter AΔ1 = 0 and in absolutely stable equilibrium with respect to the transition to the forbidden ferromagnetic state in accordance with the parameter AΔ2 < 0 (See Fig. 3). (2) The second solution is given by the equation sin(2φ 2 ) = 0 yielding the parameter φ2 = 0, which determines the following parameters of two sublattices: •

•

• •

3 3 cos 2φ2 = , 2 2 the average spin projection value, 3 3 2 = − cos 2φ2 = − , 2 2 < 0> Q2 = 3, the average quadrupole moment, < 0 >1 Q2 2 = 3, < 0> 3 3 Q3 1 = cos 2φ2 = , 2 2 the average octupole moment, < 0> 3 3 Q3 2 = − cos 2φ2 = − , 2 2 the value of the octupole transition parameter, < 3average > Q3 1 = 3 sin(2φ2 ) = 0, < 3> Q3 2 = −3 sin(2φ2 ) = 0. 1 =

STABILITY of THE SOLUTION we check the sign of the derivative of I second '' Iφ2 = 2η A < 0. E+ with respect to the angular parameter: E+ = 2η A cos 4φ 5 5 Thus, the antiferromagnetic state is unstable. The excitation spectrum of the antiferromagnetic phase (A < 0) should consist of three modes. However, the first two modes disappear, leaving only one gapless Goldstone mode. c1 = - 9/8,_ c2 = 11/18, c3 = 2/9 ) ( ) ( )) ( ( / ) 5 11 191 2 3η ( 2 9 + − = 0, A − A2 (k) − − ε1 (k) = 4 8 2 18 16 9 ) ( )) ( ( ) ( / ) 3 11 9 147 2 3η ( 2 − − − = 0, ε2 (k) = A − A2 (k) 2 2 18 8 16 9 /( ) ( ( ) ( ) ( )) 2 2 2 1 11 9 119 2 2 9η 2 ε3 (k) = − − − A (k) − A = 4 9 2 18 8 16 9 ( )/ 9η 2 η/ 2 = A2 (k) − A2 = A (k) − A2 . 4 9 2

(129)

The condition for the stability of the antiferromagnetic state A△1 < 0,A△2 < 0 is satisfied only halfway, since A△1 = 0 which corresponds to a phase transition to the antinematic phase and A△2 < 0, which indicates the stability of the system with respect to the transition to the nematic state.

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Thus, the antiferromagnetic (unstable) state corresponds to a neutral equilibrium with respect to a transition to the antinematic state, according to the parameter A△1 = 0, and is absolutely stable with respect to a transition to the nematic state, which is forbidden, in accordance with the parameter A△2 < 0. (See Fig. 3.) Author Contribution Ernst Ising Contribution to the Theory of Ferromagnetism 1925 This excerpt of the Hamburg dissertation (1924) was first published in «Zeitschrift für Physik», vol. XXXI, 1925 (received on 9 December 1924).

Technical Note The underlined letters H, J and M represent the same letters of the original text in Old English letterface.

Appendix 1: Variational Heisenberg Model for the Spin-1/2 System A system of identical particles with spins 1/2 is described by the HDV Hamiltonian Eq. (64) Hˆ int1/2 = −

Σ

Jkl sˆk · sˆl .

(A1)

kb

i,a

where the i, j, k,… indices represent occupied orbitals, a, b, c,… represent virtual orbitals, and Cia , Ci j,ab , Ci jk,abc , . . . are CI coefficients. The computational cost of FCI scales factorially in both the number of electrons n and the number of basis functions N. One could truncate at a given excitation level, which leads to limited configuration interaction (or just CI for short): this approach suffers from size extensity errors that may rival actual reaction energies, and hence has become obsolete. Alternatively, one could apply a perturbation theory expression (many-body perturbation theory), which in practice only works well if the gap between occupied and unoccupied orbitals is sufficiently large. In coupled cluster theory (see Shavitt and Bartlett for a monograph [17]), the size extensivity problem of CI is eliminated by rewriting the trial wave function as an exponential ansatz: ) ( ( ) ψ FC I = exp T ψ0 = exp T 1 + T 2 + T 3 + · · · + T n ψ0 /\

/\

/\

/\

/\

(8)

If one does not truncate this expression, it merely represents a clumsier way of doing FCI. But if one truncates the cluster operator at a given excitation level, it can be shown (the linked-cluster theorem [18–20]) that the result is rigorously size-extensive. Consider for example a dimer A…B of two-electron systems A and B at infinite distance. CISD is an exact solution for each monomer on its own, but the dimer wave function at an infinite distance will be an antisymmetrized product of the monomer wave functions, which thus includes the cross-term ψ(A) ψ(B). It is clear that the latter entails quadruple excitations with respect to the reference. (This specific kind of quadruple excitations made up of simultaneous and independent double excitations is known as disconnected quadruples in coupled cluster lingo.)

Basis Set Convergence and Extrapolation of Connected Triple Excitation …

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In contrast, the CCSD (coupled cluster with all singles and doubles [21]) wavefunction through exp(T2 ) = T2 + T2 2 /2 + T2 3 /6 + … naturally includes disconnected quadruple excitations of this type. To borrow a metaphor from Janesko [22], while a 42-electron full CI wavefunction for benzene is like an impossibly unwieldy “marriage between 42 people…[i]n real life, 42 people in a room don’t need to behave like they’re all married to each other! ‘Manners’, simple guidelines for behavior, suffice for most interactions in everyday life.” One might see CCSD as one such set of manners for couples (electron pairs) how to behave with each other on a dance floor. The computational cost of CCSD in a given basis set scales asymptotically as O(n2 N4 ), which represents a reduction from exponential to polynomial cost scaling. CCSD captures the breaking of a single bond quite well: however, for multiple bonds, the next higher term is required. From a somewhat naïve point of view, one might understand this as follows [23]: if one has a low-lying singly excited determinant, then it will be important in the wavefunction, but so will double excitations from it, i.e., connected triple excitations with respect to the reference determinant. If, on the other hand, one has a low-lying doubly excited determinant, you will have a substantial term of doubles out of doubles, i.e., connected quadruples. Unfortunately, CCSDT [24, 25] and CCSDTQ [26] have computational costs that scale asymptotically as O(n3 N5 ) and asymptotically as O(n4 N6 ), respectively. A felicitous compromise between accuracy and computational cost is represented by the CCSD(T) method [27, 28], where CCSD is augmented by a quasiperturbative estimate of the contribution of connected triples, T3 , with an asymptotic cost scaling O(n3 N4 ) rather than O(n3 N5 ) (Fig. 1). Empirically [23, 14, 29], and heavily relying on the arbitrary-order coupled cluster code developed by Kállay and coworkers [30–32], we have found that CCSD(T) represents a felicitous error compensation between neglect of higher-order triples (which are generally antibonding) and of connected quadruple excitations (which are universally bonding) (Fig. 2): Stanton [33] gives a theoretical rationale for this in terms of perturbation theory with Löwdin partitioning [34] starting from CCSD as the zero order reference; see also Kállay and Gauss [31]. Suffice to say that CCSD(T) has become the “gold standard” (T.H. Dunning [35]) of wavefunction ab initio theory.

1.3 Broader Context of the Problem Aside from some niche and emerging methods, computational quantum chemistry today is synonymous with two primary approaches. In wavefunction theory (WFT), highly accurate approximate solutions to the Schrödinger equation can be systematically refined to accuracy levels competing with the best experiments—or indeed surpassing them. The price one pays for this is their very steep CPU time scaling with the size of the system (N): as discussed above, for a truly exact solution within

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Fig. 1 Box-and-whiskers plot of the total atomization energy contributions in the W4-17 dataset[14] of 200 small molecules. Outer fences encompass 95% of the set, inner fences 80% and boxes 50%. Vertical lines span from population minimum to maximum

Fig. 2 Box-and-whiskers plot of the total atomization energy contributions of higher-order corrections in the W4-17 dataset [14]. Outer fences encompass 95% of the set, inner fences 80%, boxes 50%. Vertical lines span from population minimum to maximum

Basis Set Convergence and Extrapolation of Connected Triple Excitation …

473

the given basis set, like full CI, this would be factorial, while for the “gold standard” CCSD(T) method, it is ‘merely’ O(N7 ). In contrast, density functional theory (DFT) has comparatively gentle system size scaling, O(N3 ) or gentler—at the expense of introducing an unknown, and perhaps unknowable, exchange–correlation (XC) functional. Over the years, DFT has established itself as the ‘bread and butter tool’ of the molecular modeling community. But, while exchange–correlation functionals have come a long way, they still have to go quite a distance before they become competitive in accuracy with high-level WFT approaches. Recent developments in localized orbital approaches, such as the DLPNOCCSD(T) method of Neese and coworkers [36, 37], PNO-LCCSD(T) by Werner and coworkers [38], and LNO-CCSD(T) by Nágy and Kállay [39], appear to have revived interest in WFT methods even among computational chemists with modest computational resources. For sufficiently large systems, these methods boast nearlinear scaling, at the expense of introducing numerous cutoffs and thresholds that arguably introduce an empiricism of precision (instead of the empiricism of accuracy inherent in DFT methods). Another approach for large systems that have been gaining ground is to use △machine learning [40] to correct inexpensive calculated values to near-WFT quality. All such approaches require a substantial amount of high-accuracy WFT data for small molecules as a ‘training set’. The advantages in using accurate WFT data as a ‘primary standard’1 instead of experimental data are manifold. First of all, one is not restricted to the parts of the chemical space for which experimental data of the required accuracy is available. Second, data do not need to be isolated from experimental ‘confounding factors’ to bring them ‘on the same page’ with the simulation. All practical WFT approaches today rely on finite basis sets. This means that establishing basis set convergence becomes an essential aspect of any WFT study. Of all the ground-state properties of an atom or molecule, the most fundamental is the total energy. However, as we have seen above, absolute energies for even just first-row atoms are 2–3 orders of magnitude larger than any reaction energy of chemical interest (cf. Eq. (3)). Consider that for just a bare neon atom, Z = 10, we are already talking about more than 80,400 kcal/mol! These numbers only become more staggering as we add more atoms: clearly, attempting to reproduce total energies to within, say, 1 kcal/mol is a Sisyphean exercise. The next step down would be total atomization energies (TAEs), i.e., the energy required to separate a neutral molecule into its constituent atoms. (For charged molecules, this can be combined with ionization potentials and electron affinities.) The computed total atomization energy, in combination with atomic heats of formation in the gas phase, can be directly related to the gas phase heat of formation. (Thanks to the Active Thermochemical Tables project [41, 42, 43], a consistent set of mixed experimental–theoretical values based on a global thermochemical network is available.) 1

The term originates in quantitative analytical chemistry and is used here by analogy.

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1.4 Decomposition of the Total Atomization Energy The total atomization energy of a molecule Am Bn … is defined as the energy required to dissociate the molecule into separate atoms in their electronic ground states, all in the gas phase: TAE[Am Bn . . . ] = m E[A] + n E[B] + · · · − E[Am Bn . . . ]

(9)

Hence it also corresponds to the sum of all the bond energies in the molecule. It is the thermochemical cognate of the gas phase heat of formation: ◦

◦

◦

△H f [Am Bn . . . ] = m△H f [A(g)] + n△H f [B(g)] + · · · − TAE[Am Bn . . . ] (10) As discussed in detail in, e.g., Refs. [14, 44, 45], the electronic components of the total energy (and hence also of the TAE) of a first- or second-row molecule can be decomposed into the following components (see Table 1 for some representative molecules from the 200-molecule W4-17 dataset [14]): • The Hartree–Fock SCF component • The valence CCSD (coupled cluster with all singles and doubles [21]) correlation energy component • The valence (T) connected triples [28, 46] component, which corresponds to the CCSD(T)—CCSD difference • valence post-CCSD(T) correlation effects (discussed in detail in Refs. [23, 47]) • the contribution of inner-shell correlation (discussed and reviewed in detail in Ref. [48]) • scalar relativistic corrections [49] • spin–orbit coupling (which for light closed-shell species effectively amounts to the fine structures of the constituent atoms [50]) • diagonal Born–Oppenheimer corrections (DBOC) [51] All computational protocols for high-accuracy computational thermochemistry include all these terms in one fashion or another—be it Weizmann-4 (W4) and W4F12 from our own group [23, 44, 52], HEAT (High-accuracy Extrapolated Ab initio Thermochemistry) developed by a multinational team centered around Stanton [45, 53,54, 55], or the Feller-Peterson-Dixon (FPD) approach [56–62]. As expected, and as seen in Table 1, Hartree–Fock and valence CCSD correlation are the two dominant contributions: inner-shell correlation lies two orders of magnitude below that, as most of it cancels between the molecule and its separated atoms. (For a detailed discussion of its basis set convergence, see Ref. [48] and references therein.) Valence connected triple excitations, (T), are an order of magnitude less important than valence CCSD, yet still outweigh all remaining components by 1–2 orders of magnitude. The latter is true even for O3 and N2 O4 where nondynamical correlation effects [63] drive post-CCSD(T) contributions into the kcal/mol range— it was actually shown a decade and a half ago [52] that the relative importance of

Basis Set Convergence and Extrapolation of Connected Triple Excitation …

475

(T) in the CCSD(T) TAE is a good predictor for the thermochemical importance of post-CCSD(T) correlation effects.2 That leaves us with HF, valence CCSD, and valence (T) as the “big three”. Now for small molecules, the HF contribution is amenable to numerically exact calculation [65–68], or to calculations with very large finite basis sets, to the point that they become de-facto exact [69–71]). The valence CCSD contribution with its slow basis set convergence continues to be the focus of much research involving basis set extrapolation [52, 72–76] (vide infra) and explicitly correlated approaches [44, 77–79]. Thus (T) is left as the remaining major contributor, on which we will focus in the present chapter.

1.5 Gaussian Basis Sets Orbital-based electron correlation methods in practice require a finite basis set, as otherwise, the Löwdin expansion cannot be finite. In computational solid-state physics, plane waves form a very natural basis set, where only an energy cutoff needs to be specified. Such basis sets by construction assume a periodic (as in: unit cell-based) system, and hence are in practice unsuitable for molecular calculations. The latter are almost universally carried out in basis sets that are a linear combination of functions that at least resemble atomic orbitals. In the DFT world, both numerical orbitals (e.g., in FHI-AIMS [80, 81]) and Slater-type orbitals (e.g., in ADF [82, 83]) are used in some codes, but by far the most commonly used are Gaussian type orbitals (GTOs), i.e., products of spherical harmonics Y(θ,F) with a Gaussian function of r: A Y(θ,F).exp(−ζr2 ). The main reason for their near-universal adoption is computational convenience, i.e., the Gaussian product theorem, which dramatically speeds up the evaluation of the four-center two-electron integrals that occur in WFT calculations. Gaussian basis sets have been extensively reviewed, e.g., by Davidson and Feller [84], Shavitt [85], Peterson [86], Hill [87], Jensen [88], and most recently by Nagy and Jensen [89], who also cover other basis set types. For correlated wavefunction calculations, the correlation consistent polarized n-tuple zeta family [86, 90] in its many variants has become something of a de-facto standard. Atomic natural orbitals [91–93] have recently been revisited [94, 95]; polarization consistent basis sets [96, 97] have seen some adoption in DFT, while the widely used def2 Karlsruhe basis sets [98] offer something of a compromise between the demands of WFT and DFT, for a large swathe of the periodic table. Very recently, the nZaPa sets of Petersson [72, 99] offer a numerically somewhat better-behaved alternative to the correlation consistent family.

2

We also note in passing that as one goes further down the periodic table, relativistic effects will eventually come to rival the major contributors [64].

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Recently, explicitly correlated methods [77, 78] in which terms explicitly dependent on interelectronic distances, such as F12 geminals [100] are added to the basis set, have become a powerful addition to the WFT toolbox due to the greatly accelerated basis set convergence in MP2 and CCSD. Unfortunately for the subject at hand, triple excitations do not benefit from F12, as was shown in great detail by Köhn [101, 102].

1.6 Basis Set Extrapolation The convergence of the correlation energy is quite slow, but asymptotically systematic. For the MP2 (second order Møller-Plesset [103] perturbation theory) correlation energy of helium-like atoms, Schwartz [104, 105] showed that the “partial wave increment” of angular momentum l—that is, the total contribution to the correlation energy of all basis functions with angular momentum l—will be of the form: E(2) (l) = A(l + 1/2)−4 + B(l + 1/2)−6 + O(l−8 )

(11)

Hill [106] generalized this result to variational energies as (CI) E (l) (l) = A(l + 1/2)−4 + C(l + 1/2)−6 + O(l−6 )

(12)

In an analytical tour de force, Kutzelnigg and Morgan [107] generalized this work to arbitrary pair correlation energies in an atom: they found that for same-spin correlation energies, expansion starts at (l + 1/2)−6 rather than (l + 1/2)−4 . In order to estimate residual basis set incompleteness for a basis set that saturates partial waves through angular momentum L, we need to sum contributions from (L = l + 1) through infinity. This can be done analytically with the help of the polygamma [108] function ψ(n) (x). Replacing the latter by their asymptotic series expansions, we finally obtain the leading term (2) E (2) (l) = E ∞ + A.L −3 + O(L −5 )

(13)

From a second perspective, the principal expansion [109], Bunge [110] and Carroll, Silverstone, and Metzger (CSM) [111] independently found that the contribution of a single atomic orbital with quantum numbers n, l, m is essentially independent of the angular quantum number l and the magnetic quantum number m, and depends on the principal quantum number n as δE n,l,m = −A(n − 1/2)−6

(14)

For a given n, can run from 0 to n − 1, and m in turn from –l to l. Thus, there are [n−1 2 l=0 (2l + 1) = n essentially equal contributions. The contribution for each principal quantum number n thus acquires an ∝ n−4 leading dependence, and as above,

Basis Set Convergence and Extrapolation of Connected Triple Excitation …

477

summation leads to a leading ∝ n−3 dependence of the overall basis set incompleteness. Indeed, this “principal expansion” structure is exhibited by both of the major types of basis set sequences for correlated WFT calculations: the correlation consistent basis sets [86] (through clustering by atomic correlation energy contributions) and the atomic natural orbital basis sets [91–93] (through clustering by natural orbital occupation number). If the correlation energy (or a contribution to it) converges proportionally to L −β , then the basis set limit is trivially obtained from energies with two successive L as ( E = E[L] + (E[L] − E[L − 1])/ (

) L β ) −1 L −1

(15)

That an expression for pair correlation energies might also apply to the complete atomic correlation energy perhaps seems at least plausible. (Petersson [112] considered separate extrapolations of pair energies, using an L-shift as an adjustable parameter, vide infra.) That the correlation energy of a molecule, however, would behave similarly to a spherical atom requires more of a ‘leap of faith’: fortunately, during initial explorations for thermochemistry [113, 114] we found that this is essentially the case. This implies, incidentally, that the molecular correlation energy behaves largely like ‘atoms in molecules’. A very simple formula due to Halkier et al. [74] is widely used: E(L) = E ∞ + A/L 3 or hence E ∞ = E(L) + [E(L) − E(L − 1)]/[(

L 3 ) − 1] L −1 (16)

It was soon discovered empirically that better results could be obtained for smaller basis sets, and for specific components of the correlation energy, if the exponents were used as adjustable parameters: this, as well as the Petersson approach [72, 115] of using L-shifts in the same way, is actually equivalent to Schwenke’s [73] two-point linear extrapolation, as discussed in detail here [116]. (Extrapolation of the total energy for smaller basis sets leads to the misleading conclusion that overall convergence is exponential [117], which works well enough for HF and DFT energies but causes serious underestimates of the WFT correlation energy.) While it has been understood for about two decades (e.g., Helgaker and coworkers [118]) that (T) converges faster with the basis set than CCSD, this is one aspect where the Karton-Martin W4-11 and W4-17 thermochemical benchmarks [14, 29] might benefit from more accurate calibration. As a by-product thereof, we will present revised extrapolations of the (T) correlation energy for common basis set pairs. Two-point basis set extrapolations, inspired by the Schwartz-Kutzelnigg partial wave expansion [105, 106, 107] and the Klopper-Helgaker principal expansion [109], take the form: E(L) = E ∞ + A.(L + a)−α

(17)

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J. M. L. Martin

From which follows that E ∞ = E(L) +

E(L) − E(L − 1) ( L+a )α −1 L+a−1

(18)

where L is the largest angular momentum in the basis set used for calculating the total energy E(L), and the exponent α and the L-shift a are extrapolation parameters specific to the level of theory. Typically, one or both of α and a are frozen: Halkier et al. [74] set a = 0 and α = 3, the present author [113, 114] originally favored fixed a = 0 or a = 1/2 and fitted α, while the Petersson group [72] favor fixed α = 3 and fitted a. (Klopper [119], building on the landmark analytical work of Kutzelnigg and Morgan [107], advocated separate L −3 and L −5 extrapolation of singlet- and triplet-coupled CCSD pair correlation energies, respectively, which is the approach we adopted in W4 theory [52].) Schwenke [73] instead proposed to simply consider a two-point linear extrapolation of the form: E ∞ = E(L) + A L−1,L [E(L) − E(L − 1)]

(19)

where AL is a linear coefficient specific to the basis set pair and the level of theory.3 As discussed in Ref. [120], the Schwenke form is mathematically equivalent to Eq. (2) if the following relationships apply ( ) 1 log 1 + A L−1,L B ( L ) E L = E ∞ + α if α = L log L−1 E L = E∞ +

D if a = ( (L + a)3 1+

1 1

)1/3

A L−1,L

−1

(20) +1−L

(21)

Or, conversely A L−1,L = (

1

) L+a α L−1+a

−1

(22)

Ranasinghe and Petersson (RP) [72] determined Schwenke coefficients for (T) and their nZaPa basis set family[72] (n = 2–7, the largest basis set topping out at k functions) by fitting to a fairly large set of total energies of small first-and secondrow species. The reference data for the MP2, CCSD-MP2, and (T) components were obtained by least-square fitting of each component individually to expressions of the form E ∞ + A(L + a)−3 analogous to the CBS pair extrapolation by Petersson 3

He recommends eschewing nonlinear 3-point formulas, as they are not size-consistent. We note in passing that Schwenke also presents separate extrapolation coefficients for the Klopper-style [119] singlet-coupled and triplet-coupled CCSD correlation energy components.

Basis Set Convergence and Extrapolation of Connected Triple Excitation …

479

and coworkers [112], using E ∞ and a as fit parameters for each system separately to nZaPa (n = 4,5,6,7). Next, they fitted two-point Schwenke coefficients AL for {L-1,L} pairs for each component: specifically, for the (T) component, they found A2,3 = 0.466, A3,4 = 0.600, A4,5 = 0.849, A5,6 = 1.164, and A6,7 = 1.580. They noted that these coefficients are reproduced fairly well by the extrapolation formula: [( E(L) = E ∞ + A.

2 L− 3

)−3

( ) ] 2 −5 7 L− − 8 3

(23)

which yields A2,3 = 0.446, A3,4 = 0.604, A4,5 = 0.891, A5,6 = 1.199, and A6,7 = 1.517. For comparison, a simple L −3 extrapolation would yield, respectively, 0.421, 0.730, 1.049, 1.374, and 1.701. In this chapter, we will consider the basis set convergence of (T) for the W4-17 atomization energies in detail, and, as a by-product, obtain extrapolation parameters for a number of basis sets where they were hitherto unavailable. We will show that, leaving aside the behavior of contributions to the absolute correlation energy, TAE[(T)] actually converges reasonably rapidly with the basis set and can be obtained to 0.01 kcal/mol accuracy using no more than quintuple-zeta basis sets. We will also present evidence that radial flexibility of the basis set is more important for (T) than ‘piling on’ higher angular momenta.

2 Computational Methods All electronic structure calculations with basis set sequences requiring at most i functions were carried out using the MOLPRO 2020.2 electronic structure package [121] running on the ChemFarm HPC facility of the Faculty of Chemistry at the Weizmann Institute of Science. The 7ZaPa basis set of RP [72] requires k functions, which exceed the supported maximum angular momentum of MOLPRO, and hence Gaussian 16 rev. C.01 was employed for these [122]. Reference geometries of the W4-17 dataset were used ‘as is’ from the supporting information of Ref. [14] For open-shell cases, the restricted open-shell CCSD(T) definition of Watts et al. [28] was used throughout. Basis sets were used from the internal library of MOLPRO, except for the nZaPa, which were downloaded from https://www.basissetexchang e.org, version 2 of the Basis Set Exchange [123]. In the correlation consistent [86, 90] basis set sequences, we used cc-pVnZ on hydrogen [90, 124], aug-cc-pVnZ on first-row elements [125], and aug-cc-pV(n + d)Z on second-row elements [126, 127]. (The addition of a high-exponent d function is required for second-row elements in high oxidation states to ensure the proper description of the 3d orbital [127–129], which acts as a back-bonding recipient [130] from O and F.) This combo is indicated below as haVnZ + d. We also considered correlation consistent core-valence basis sets [131, 132] aug-cc-pCVnZ (or ACVnZ for short); in part, total valence energies from Ref. [48] were recycled for this purpose.

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In terms of basis sets for explicit correlation, we considered the cc-pVnZ-F12 basis sets [133, 134] as well as their augmented counterparts, [135] both in conventional CCSD(T) and in CCSD(T)-F12b [136] contexts. In the latter, the auxiliary basis sets employed were MOLPRO’s defaults for exchange [137], RI-MP2 [138], and complementary auxiliary basis sets [139], with the recommended geminal exponents for each basis set. For the W4-08 subset of W4-17, we were able to carry out CCSD(T)/7ZaPa calculations, except for NCCN (dicyanogen) which diverged for numerical reasons. CCSD(T)/haV6Z + d calculations proved possible for all of W4-17 except for benzene (near-linear dependence in the basis set) and n-pentane (lack of scratch storage); all of W4-17 could be treated with the remaining basis sets. We will use the {n-1,n} notation for extrapolation throughout: for example, ccpV{5,6}Z denotes basis set extrapolation from cc-pV5Z and cc-pV6Z basis sets.

3 Results and Discussion 3.1 Convergence in a Model System: Neon Atom How does (T) really converge for large basis sets, and can we make any sense of the somewhat erratic behavior for small basis sets? Thanks to the work of Barnes, Petersson, and coworkers [99, 115], we have nZaPa basis sets available for neon up to n = 10 (!). In addition, from Feller, Peterson, and Crawford [140], we have cc-pVnZ basis sets (and data) for the same system up to n = 10. All relevant data are given in Table 2. We were able to reproduce the data from Refs. [115, 140] (and hence also to extract individual components); In addition, we truncated the 10ZaPa and cc-pV10Z basis sets at successive angular momenta L = 2–9, leading to the results labeled “partial wave” in Table 2. Barnes et al. [115] already pointed out that E 4 (T) will be dominated by the E 4aab and E 4bba mixed-spin components, rather than the same-spin E 4aaa and E 4bbb terms. But how does this look in practical terms? In the upper pane of Table 2, one can see the E 4aaa /E 4aab ratio for the neon atom as a function of maximum angular momentum L ≤ 9 for both the cc-pVnZ and nZaP basis set sequences. One observation we can make is how stable the E 4 (T)aaa /E 4 (T)aab ratio remains as a function of the basis set, holding steady at about 0.054 except for the smallest cc-pVnZ and nZaP basis sets. Another is that the E 4(T)aaa /E 2aa ratio for sufficiently large n stabilizes at 0.0045, and the E 4(T)aab /E 2ab ratio at 0.0131. However, for small n these ratios are much smaller, 0.001 for cc-pVDZ and 0.004 for 2ZaP. In contrast, the partial-wave series yield ratios that are close to the limiting values even for L = 2, i.e., cc-pV10Z-L max = 2 and 10ZaPa-L max = 2. A comparison of the (T) correlation energies (Table 2, lower pane) also shows that for L = 2 and L = 3, the partial wave expansions recover a dramatically larger part of the (T) limit than the principal expansions. In general, the basis set convergence

4.12

4.39

4.46

4.48

4.49

4.50

4 4.50

5 4.51

6 4.50

7 4.50

8 4.50

9 13.08

13.08

13.08

13.07

12.95

12.35

4.50

4.50

4.50

4.50

4.50

4.43

3.94

−0.313327 −0.313981

−0.005575 −0.306758

−0.006098 −0.310965

−0.006290 −0.312767

−0.006379 −0.313642

−0.006425 −0.314089

−0.006449

4 −0.006291

5 −0.006394

6 −0.006433

7 −0.006450

8 −0.006458

9

−0.006455

−0.006448

−0.006430

−0.006391

−0.006288

−0.006001

−0.004912

−0.006421 −0.314241

−0.006377 −0.313793

−0.006295 −0.312919

−0.006099 −0.311117

−0.005538 −0.306912

−0.004245 −0.294119

−0.006446

13.07

13.08

13.07

13.04

12.92

12.39

10.74

4.03

nZaP

−0.314103

−0.313404

−0.312242

−0.310129

−0.305850

−0.295736

−0.268254

−0.195672

EcorrCCSD EcorrCCSD

13.08

13.10

13.13

13.19

13.27

13.46

12.93

Partial Wave

1000E 4 (T)aab /E 2ab

−0.001044 −0.259024

“Partial wave” corresponds to cc-pV10Z or 10ZaP truncated at angular momentum L

−0.312111

−0.309906

−0.305489

−0.294682

−0.266347

−0.004259 −0.293961

−0.189017

−0.000981 −0.258883

3 −0.006004

E(T)

4.50

4.49

4.48

4.45

4.38

4.14

3.33

1.19

Partial Wave nZaP

1000E 4 (T)aaa /E 2aa

EcorrCCSD EcorrCCSD E(T)

13.09

13.11

13.14

13.20

13.28

10.71

4.00

VnZ

2 −0.004915

E(T)

3.26

3 4.43

E(T)

12.94

1.08

2 3.94

13.47

Partial Wave

1000E 4 (T)aab/E 2ab

L Partial Wave VnZ

1000E 4 (T)aaa /E 2aa

Table 2 Basis set convergence of the (T) correlation energy (a.u.) for neon atom

0.0541

0.0541

0.0543

0.0545

0.0551

0.0559

0.0558

Partial nZaP Wave

E 4aaa /E 4aab

0.0541

0.0541

0.0542 0.0541 0.0542

0.0544 0.0541 0.0544

0.0547 0.0542 0.0546

0.0552 0.0545 0.0551

0.0560 0.0550 0.0559

0.0541 0.0559 0.0550

0.0509 0.0558 0.0555

Partial Wave VnZ

E 4aaa /E 4aab

Basis Set Convergence and Extrapolation of Connected Triple Excitation … 481

482

J. M. L. Martin

of the (T) energy is somewhat erratic for the cc-pVnZ and nZaP basis sets for small n, and much smoother for the partial wave expansions. This is less pronouncedly the case for the CCSD correlation energies, which may indicate that radial flexibility is more significant for the (T) term than for the MP2 or CCSD correlation energy.

3.2 Convergence and Basis Set Extrapolation for W4-17 Let us now turn to the W4-17 dataset (largest molecule: benzene) and its subset [141] W4-08 (largest molecules: B2 H6 and C2 H6 ). Root mean square deviations (RMSDs) and fitted extrapolation parameters are presented in Table 3. Full source data are made available as an Excel workbook in the Electronic Supporting Information. The reference data for W4-08 were obtained by {6,7}ZaPa extrapolation using RP’s optimized A6,7 = 1.580. However, the extrapolation covers just 0.047 kcal/mol RMSD from the raw 7ZaPa numbers, and is reasonably insensitive to the precise value of A6,7 : substituting 1.701 (which corresponds to simple L−3 extrapolation) changes the values by just 0.004 kcal/mol RMS. This means our reference data are not sensitive to fine details of the extrapolation—at least not to any energetic resolution we can realistically hope to achieve. Indeed, all three {h,i} extrapolation options—V{5,6}Z + d, A’V{5,6}Z + d, and {5,6}ZaPa—have RMSDs below 0.01 kcal/mol from the reference—regardless of whether one uses extrapolation coefficients from Schwenke and RP, or those optimized in the present work against the {6,7}ZaPa reference data. Using the simple L −3 extrapolation causes somewhat larger errors, especially for V{5,6}Z + d. The bottom line, however: it is possible to achieve 0.01 kcal/mol accuracy in the (T) term with ‘just’ spdfgh and spdfghi (i.e., L = {5,6}) basis sets. What about dialing down both basis sets one step? Here 0.01 kcal/mol is possible with both {4,5}ZaPa and A’V{Q,5}Z + d, and 0.02 kcal/mol with V{Q,5}Z + d, provided either the A{4,5} from RP and Schwenke, or the presently optimized version, are used: simple L −3 extrapolation triples or quadruples the error, bringing it in the range of the next basis set pair down with optimized exponents. At any rate, considering that reaching 0.1 kcal/mol RMSD in valence CCSD/{5,6} calculations will be the practical limit [79] for the CCSD term, it seems quite justified to limit the expensive and memory-hungry (T) calculation to {4,5} basis sets. For the still more economical {3,4} pair, 0.05 kcal/mol RMSD or better is achievable; L −3 is basically equivalent to the optimum here, as can be seen from the α exponents corresponding to our various optimized A{3,4}. For the Weigend-Ahlrichs [98] def2-TZVPP and def2-QZVPP basis sets, α = 3.155 is optimum, but the error from using just L −3 is quite tolerable. The same goes for {3,4}ZaPa.4

4

As a by-product, we can obtain the extrapolation exponents for MP2 and CCSD, which for the def2-{T,Q}ZVPP pair are αMP2 = 2.612 and αCCSD = 3.017, the latter nearly identical to 2.970 from Neese and Valeev [94], and both functionally equivalent to the simple L–3 extrapolation.

0.147

0.076

0.047

0.302

0.141

0.072

0.731

0.305

0.135

{5,6}ZaPa

{6,7}ZaPa

7ZaPa − Lmax = {2,3}

7ZaPa − Lmax = {3,4}

7ZaPa − Lmax = {4,5}

haV{D,T}Z + d

haV{T,Q}Z + d

haV{Q ,5}Z + d

0.034 0.012

0.707

0.260

0.117

0.063

0.984

0.422

0.188

0.097

ACV{D,T}Z

ACV{T,Q}Z

ACV{Q,5}Z

ACV{5,6}Z

V{D,T}Z + d

V{T,Q}Z + d

V{Q,5}Z + d

V{5,6}Z + d

0.029

0.060

0.046

0.246

0.007

0.020

0.054

0.304

0.242

0.073

haV{5,6}Z + d 0.040

0.006

0.013g

0.040

0.190

REF

0.008

0.013

0.009

0.040g

0.199

0.004

0.018

0.043

0.044g

0.025

0.037

0.062

0.048

0.287

{4,5}ZaPa

0.040

0.231

0.329

0.123

0.812

{3,4}ZaPa

Petersson, Schwenke, or Hillb

{2,3}ZaPa

Simple L−3 W4-08 Extrapolated

Pseudo-Marchetti-Wernera

Largest basis set

RMSD for W4-08

0.006

0.015

0.046

0.246

0.004

0.008

0.032

0.150

0.004

0.009

0.039

0.188

0.009

0.016

0.078

0.000

0.005

0.010

0.030

0.207

Fitted to W4-08

1.062

0.800

0.746

0.423

1.155

0.815

0.666

0.324

1.180

0.794

0.708

0.385

1.226

0.873

0.636

1.580

1.077

0.803

0.676

0.372

A{L-1,L}

Optimized Schwenke coefficientc

0.029 0.007 0.004 N/A

−0.165 −0.170 −0.770 −0.913 −0.368

3.639

3.633

2.957

2.991

3.421

3.587

3.186

3.473

3.367

3.654

3.062

3.155

2.673

2.654

2.331

3.181

3.601

3.626

3.156

0.130 0.034 0.006 0.003

−0.331 −0.201 −0.730 −0.672 0.050 −0.960

−0.777

(continued)

0.010

0.017

0.062

0.270

REF

−0.596

0.007

0.010

0.059

−0.070 −0.799

0.176

−0.120

0.546

0.450

0.700

0.204

a

α 3.219

Fitted to W4-08

Equiv L- shifte

Equiv exponentd

W4-17

Table 3 RMSD (kcal/mol) of connected triples contributions to the total atomization energy from best reference data (indicated as REF) for the W4-17 dataset and its W4-08 subset, as well as fitted extrapolation exponents (and equivalent Schwenke coefficients and Petersson shifts)

Basis Set Convergence and Extrapolation of Connected Triple Excitation … 483

1.010

0.411

0.687

0.299

0.161

0.679

0.316

0.169

0.503

0.227

0.121

def2-{TZVPP,QZVP}f

V{D,T}Z-F12 orb

V{T,Q}Z-F12 orb

V{Q,5}Z-F12 orb

V{D,T}Z-F12 F12b

V{T,Q}Z-F12 F12b

V{Q,5}Z-F12 F12b

aV{D,T}Z-F12 orb

aV{T,Q}Z-F12 orb

aV{Q,5}Z-F12 orb

Petersson, Schwenke, or Hillb

0.060

0.238

N/A

N/A

N/A

N/A

0.051

0.206

N/A

N/A

N/A

N/A

N/A

W4-08 Extrapolated

Simple L−3

0.024

0.024

0.105

0.035

0.038

0.119

0.026

0.031

0.107

0.051

0.238

Fitted to W4-08

1.092

0.818

0.668

1.119

0.861

0.708

1.050

0.819

0.672

0.680

0.424

A{L-1,L}

Optimized Schwenke coefficientc

2.913

2.775

2.256

2.861

2.679

2.172

2.998

2.774

2.247

3.144

0.134

0.280

0.804

0.217

0.414

0.931

0.002

0.281

0.050

0.045

0.113

0.039

0.036

0.101

0.071

−0.158 0.817

0.240

Fitted to W4-08

W4-17

0.009

a

α 2.989

Equiv L- shifte

Equiv exponentd

Eq. (25) b With extrapolation coefficients taken from RP [70] for nZaPa, Schwenke [71] for AVnZ, and Ref. [141, 142] for VnZ-F12 c E(CBS) = E(L) + A {L − 1,L} [E(L) − E(L − 1)] d E(CBS) = E(L) + [E(L) − E(L − 1)]/[(L/L − )α − 1] e E(CBS) = E(L) + [E(L) − E(L − 1)]/[((L + a)/(L + a − 1))3 − 1] f def2-{TZVPP,QZVP} extrapolation coefficient for valence CCSD, obtained in the same fashion, is A 3,4 = 0.715, or α = 3.041, or a = 0.046. For valence MP2, A3,4 = 0.896, or α = 2.605, or a = 0.524. g 3-point E + A.L −3 + B.L −5 extrapolation: haV{D,T,Q}Z 0.055 kcal/mol; haV{T,Q,5}Z 0.062 kcal/mol; haV{Q,5,6}Z 0.009 kcal/mol. ∞

a

Pseudo-Marchetti-Wernera

Largest basis set

RMSD for W4-08

def2-{SVP,TZVPP}

Table 3 (continued)

484 J. M. L. Martin

Basis Set Convergence and Extrapolation of Connected Triple Excitation …

485

In response to a suggestion by Prof. John F. Stanton (U. of Florida), we attempted a 3-point extrapolation using E∞ + AL −3 + BL −5 . With the help of the Mathematica computer algebra software, the closed-form solutions for L = {2,3,4}, {3,4,5}, and {4,5,6}, respectively, are found to be: E ∞ {2, 3, 4} = E[Q] + (673E[Q] − 729E[T] + 56E[D])/607

(24)

E ∞ {3, 4, 5} = E[5] + (14197E[5] − 16384E[Q] + 2187E[T])/7678

(25)

E ∞ {4, 5, 6} = E[6] + (12809E[6] − 15625E[5] + 2816E[Q])/4687

(26)

However, as seen in footnote (g) of Table 3, these offer no advantage over two-point extrapolation with fitted exponents. Finally, dropping down to the least expensive {2,3} pair entails 0.2 kcal/mol RMSD or worse, except when using the cc-pV{D,T}Z-F12 basis sets for explicitly correlated calculations in an orbital context. Interestingly, the optimum A{2,3} for the A’V{D,T}Z+d pair corresponds to α = 3.155, not too far from the α = 3.22 obtained from a small training set in W1 theory [75] (which yields essentially the same RMSD = 0.19 kcal/mol as the optimum). What about (T) in CCSD(T)-F12b calculations? First of all (vide supra), while F12 geminals [100] and explicitly correlated methods [77, 78] more generally, greatly accelerate basis set convergence of the CCSD correlation energy, Köhn [101, 102] showed in great detail that they do not benefit (T) in any way. Second, we find here that for (T), the V{Q,5}Z-F12 basis set pair does not seem to offer any advantages over V{T,Q}Z-F12, neither for CCSD(T)-F12b nor for orbital-only CCSD(T). Third, (T) corrections from CCSD(T)-F12b actually seem to be inferior in quality to those obtained from similar basis sets with ordinary CCSD(T) (which will have more modest scratch storage requirements). Fourth, an error below 0.05 kcal/mol RMS is quite achievable using the {T,Q} pair, and 0.1 kcal/mol using the {D,T} pair. In fact, the numbers make a good case for treating the (T) contribution in a separate non-F12 calculation, thus greatly reducing memory and scratch storage requirements for the remaining CCSD-F12b/VnZ-F12 calculation. We shall now compare performance for the whole W4-17 dataset. Based on its small RMSD = 0.004 kcal/mol from the reference for W4-08, we can use A’V{5,6}Z + d as the ‘secondary standard’. Clearly, A’V{Q,5}Z + d at RMSD = 0.01 kcal/mol is the basis set pair of choice for high-accuracy work, but V{T,Q}Z-F12 can easily meet a ±0.05 kcal/mol target. Considering the fairly constant E 4T /E 2 ratios, the mind wonders if these cannot be exploited to yield a parameter-free estimate for (T) at the basis set limit according to the equation: E (T) [CBS] ≈ E (T) [basis] × E corr,CCSD [CBS]/E corr,CCSD [basis]

(27)

486

J. M. L. Martin

This is actually a generalization of Marchetti-Werner scaling [142] which we previously considered in Ref. [134]. We can substitute here CCSD/{5,6}ZaPha for Ecorr CCSD. Then we find from Eq. (1) E(T) for nZaPha (n = 3–6) with RMSD = 0.123, 0.062, 0.037, and 0.025 kcal/mol, respectively—markedly better than the raw results but clearly inferior to extrapolation. While the 10ZaP basis set is only available for neon, we could trivially truncate 7ZaPa for other elements to generate ‘partial wave’ basis sets 7ZaPa-L max = lor any l< 7. Statistics for the W4-08 dataset with such basis sets can be found in Table 2. It can clearly be seen there that for small l, 7ZaPa-Lmax = l basis sets are markedly superior to lZaPa basis sets in terms of (T) recovery: for the first few l, 7ZaPa-Lmax = l has an RMSD comparable to (l + 1)ZaPa. In fact, we can recover (T) to better than 0.1 kcal/mol by 7ZaPa-L max = {2,3} extrapolation, and to better than 0.02 kcal/mol by 7ZaPa-L max = {3,4} extrapolation. While such basis sets are unwieldy for practical use, the results clearly indicate that radial saturation of the available angular momenta is more beneficial for recovering (T) than adding more angular momenta. This situation is very different from the long-standing received wisdom for the singles and doubles correlation energy, as reflected in the ‘principal expansion’ structure of both atomic natural orbital [91, 143] and correlation consistent [86, 90] basis sets.

4 Conclusions We can state with confidence that achieving basis set convergence for the valence triple excitations energy is much easier than for the valence singles and doubles correlation energy. In fact, it is quite possible to reach convergence on the order of 0.01 kcal/mol with just QZ and 5Z basis sets, and on the order of 0.05 kcal/mol with just TZ and QZ basis sets. It appears that radial flexibility in the basis set is more important here than adding angular momenta: apparently, replacing nZaPa basis sets with truncations of 7ZaPa at L = n gains about one angular momentum for small values of n. As already noted numerous times by Helgaker et al. [118], by RP, and by the present author and coworkers, the basis set convergence of the (T) component is fundamentally different from the CCSD correlation energies, and since partitioning between SCF, valence CCSD, and valence (T) energy components is trivially obtained,5 there is no reason not to extrapolate these components separately. Moreover, since basis set convergence for the CCSD component may be drastically speeded up through F12 methods, this allows one to keep basis sets down to at most QZ size.

5

Unlike the partitioning of the CCSD correlation energy in singlet-coupled and triplet-coupled pairs, which is not uniquely defined for open-shell cases.

Basis Set Convergence and Extrapolation of Connected Triple Excitation …

487

5 Future Outlook Quantum mechanical simulation is clearly here to stay. But where will it go from here? Advances continue to be made in the area of density functional theory. For molecules, empirical double-hybrid density functionals ([144] and references therein) offer a fairly low-cost approach (especially if the RI-MP2[145, 146] approximation is made) that approaches the accuracy of composite wavefunction methods. Double hybrids for solids have recently been implemented, [147] but intrinsically cannot be applied to conductors or semiconductors because of a denominator singularity in the GLPT2 part. Range-separated hybrids [148] and tuned range separated hybrids [149–151] are alternatives. For wavefunction calculations, the combination of localized orbital methods [36, 37, 38, 39] (which make size scaling of the calculation almost linear) and explicitly correlated approaches [44, 77, 78, 79] (which drastically speed up basis set convergence) has recently emerged as a powerful alternative [38, 152–154]. Thus, a rigorous, purely WFT-based alternative to DFT exists for large molecules; as the (T) correction does not benefit from explicit geminal correlation, the observations in the present article on the basis set convergence of (T) will be relevant. A caveat should be voiced, however, about possible errors from localized approaches in extended systems with significant near-degeneracy correlation (a.k.a., nondynamical correlation, static correlation): in Ref. [155] we have investigated this issue for the structures and transition states of polypyrrols, and concluded that (a) PNO-LCCSD(T) and DLPNO-CCSD(T) are vulnerable to static correlation; (b) the LNO-CCSD(T) approach [39] as implemented in MRCC [32] is much more resilient. For small molecules, canonical CCSD(T) combined with CCSD(F12*) [156] and with general post-CCSD(T) approaches such as CCSDT(Q) [31, 157, 158] as implemented in the MRCC [32] and CFOUR [159] program systems, offer a pathway to sub-kcal/mol accuracy, as we have shown at length in the W4-F12 paper [44]. But the impact of accurate calculations on smaller systems goes further. We have already mentioned their usefulness as “primary standards” for the parametrization of lower-cost empirical methods (as this has already happened, indirectly, via the large and chemically diverse GMTKN55 [160] and MGCDB84 [161] training sets which were mostly compiled from earlier benchmark calculations). But in addition, △machine learning [40, 162] offers a very attractive alternative that allows improving a low-cost DFT calculation through machine learning on the difference between low-cost and high-accuracy calculated values for a sufficiently large training set. Especially when the training is ad hoc to the problem at hand, this can be a very valuable alternative, for example for the acceleration of molecular dynamics on long time scales. Finally, such △-ML values (or those from a lower-cost empirical DFT functional or composite WFT method) could be used as “secondary standards” (in the analytical

488

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chemistry sense of the word) for training classical force fields, be they conventional, polarizable [163], or reactive [164]. Acknowledgements This research was funded in part by the Israel Science Foundation (Grant 1969/20) and by the Minerva Foundation (grant 2020/05). Supporting Information Microsoft Excel workbook with the relevant energetics is available at https://doi.org/10.34933/wis.000243.

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Author Biography Jan M. L. Martin FRSC (a.k.a., Gershom Martin) was born in Belgium in 1964. In 1987, he obtained his Licentiaat (integrated B.Sc./M.Sc. degree) in chemistry in 1987 from the University of Antwerp, then his Ph.D. (as an NFWO fellow) from the same institution in 1991, both summa cum laude. In 1992–3 he did postdoctoral research in the USA on Fulbright-Hays and NATO fellowships, first at NASA Ames Research Center with Drs. Timothy J. Lee and Peter R. Taylor, then following Dr. Taylor to his new position at San Diego Supercomputer Center. In 1994 he obtained his Aggregaat voor het Hoger Onderwijs (Habilitation) from U. of Antwerp. From 1995 until 1999 he was a senior research fellow with tenure of the NFWO, Belgium’s National Science Foundation. Following a 1996 Golda Meir fellowship with Prof. Chava Lifshitz z'' l at the Hebrew University of Jerusalem, he joined the faculty of the Weizmann Institute of Science and was promoted to full professor in 2005, succeeding Wolf Prize laureate Meir Lahav as the Baroness Thatcher Professor of Chemistry. In 2007–8 he spent a sabbatical year with Prof. George C. Schatz at Northwestern University. From 2010 until his return to Weizmann in 2012, he was a Distinguished University Research Professor at the Center for Advanced Scientific Computing and Modeling (CASCAM) at the University of North Texas. Dr. Martin is a Foreign Member of the Royal Academy of Sciences, Literature, and Arts of Belgium, a Fellow of the Royal Society of Chemistry, and a IUPAC Fellow. He was awarded the 2017 Israel Chemical Society Prize for Excellence, as well as the 2004 Dirac Medal of WATOC. He is married to a professional musician and they have one daughter. His avocations include linguistics, history, music theory, and creative writing.