Quantum Chemistry and Dynamics of Excited States

Name: Quantum Chemistry and Dynamics of Excited States | Methods and Applications
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Methods and Applications

Leticia González Roland Lindh(Herausgeber*in)

Wiley (Verlag)

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Erschienen am 10. November 2020

688 Seiten

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Inhalt

Cover
Title Page
Copyright
Contents
List of Contributors
Preface
Chapter 1 Motivation and Basic Concepts
1.1 Mission and Motivation
1.2 Atomic Units
1.3 The Molecular Hamiltonian
1.4 Dirac or Bra-Ket Notation
1.5 Index Definitions
1.6 Second Quantization Formalism
1.7 Born-Oppenheimer Approximation and Potential Energy Surfaces
1.8 Adiabatic Versus Diabatic Representations
1.9 Conical Intersections
1.10 Further Reading
1.11 Acknowledgments
Part I Quantum Chemistry
Chapter 2 Time-Dependent Density Functional Theory
2.1 Introduction
2.2 TDDFT Fundamentals
2.2.1 The Runge-Gross Theorems
2.2.2 The Time-Dependent Kohn-Sham Approach
2.2.3 Solutions of Time-Dependent Kohn-Sham Equations
2.2.3.1 Real-Time TDDFT
2.2.3.2 Linear-Response TDDFT
2.3 Linear-Response TDDFT in Action
2.3.1 Vertical Excitations and Energy Surfaces
2.3.1.1 Vertical Excitations: How Good are They?
2.3.1.2 Reconstructed Energy Surfaces: How Good are They?
2.3.2 Conical Intersections
2.3.3 Coupling Terms and Auxiliary Wave Functions
2.3.3.1 The Casida Ansatz
2.3.3.2 Time-Derivative Non-Adiabatic Couplings
2.3.4 Non-Adiabatic Dynamics
2.4 Excited States and Dynamics with TDDFT Variants and Beyond
2.5 Conclusions
Acknowledgments
References
Chapter 3 Multi-Configurational Density Functional Theory: Progress and Challenges
3.1 Introduction
3.2 Wave Function Theory
3.3 Kohn-Sham Density Functional Theory
3.3.1 Density Functional Approximations
3.3.2 Density Functional Theory for Excited States
3.3.2.1 Issues Within the Time-Dependent Density Functional Theory Ansatz
3.3.2.2 Self-Interaction Error
3.3.2.3 Degeneracies, Near-Degeneracies and the Symmetry Dilemma
3.4 Multi-Configurational Density Functional Theory
3.4.1 Semi-Empirical Multi-Configurational Density Functional Theory
3.4.2 Multi-Configurational Density Functional Theory Based the On-Top Pair Density
3.4.2.1 Density Matrices and the On-Top Pair Density
3.4.2.2 Energy Functional and Excited States with the On-Top Pair Density
3.4.3 Multi-Configurational Density Functional Theory Based on Range-Separation
3.4.3.1 Energy Functional and Excited States in Range-Separated Methods
3.4.3.2 The Range-Separation Parameter in Excited State Calculations
3.5 Illustrative Examples
3.5.1 Excited States of Organic Molecules
3.5.2 Excited States for a Transition Metal Complex
3.6 Outlook
Acknowledgments
References
Chapter 4 Equation-of-Motion Coupled-Cluster Models
4.1 Introduction
4.2 Theoretical Background
4.2.1 Coupled-Cluster Wave Function
4.2.2 The Equation-of-Motion Approach
4.2.3 Similarity-Transformed Hamiltonian
4.2.4 Davidson Diagonalization Algorithm
4.3 Excited States: EE-EOM-CC
4.3.1 EE-EOM-CCSD Model
4.3.2 EE-EOM-CCSDT Model
4.3.3 EE-EOM-CC Results
4.4 Ionized States: IP-EOM-CC
4.4.1 IP-EOM-CCSD Model
4.4.2 IP-EOM-CCSDT Model
4.4.3 IP-EOM-CC Results
4.5 Electron-Attached States: EA-EOM-CC
4.5.1 EA-EOM-CCSD Model
4.5.2 EA-EOM-CCSDT Model
4.5.3 EA-EOM-CC Results
4.6 Doubly-Ionized States: DIP-EOM-CC
4.6.1 DIP-EOM-CCSD Model
4.6.2 DIP-EOM-CCSDT Model
4.6.3 DIP-EOM-CC Results
4.7 Doubly Electron-Attached States: DEA-EOM-CC
4.7.1 DEA-EOM-CCSD Model
4.7.2 DEA-EOM-CCSDT Model
4.7.3 DEA-EOM-CC Results
4.8 Size-Extensivity Issue in the EOM-CC Theory
4.9 Final Remarks
References
Chapter 5 The Algebraic-Diagrammatic Construction Scheme for the Polarization Propagator
5.1 Original Derivation via Green's Functions
5.2 The Intermediate State Representation
5.3 Calculation of Excited State Properties and Analysis
5.3.1 Excited State Properties
5.3.2 Excited-State Wave Function and Density Analyses
5.4 Properties and Limitations of ADC
5.5 Variants of EE-ADC
5.5.1 Extended ADC(2)
5.5.2 Unrestricted EE-ADC Schemes
5.5.3 Spin-Flip EE-ADC Schemes
5.5.4 Spin-Opposite-Scaled ADC Schemes
5.5.5 Core-Valence Separated (CVS) EE-ADC
5.6 Describing Molecular Photochemistry with ADC Methods
5.6.1 Potential Energy Surfaces
5.6.2 Environment Models within ADC
5.7 Brief Summary and Perspective
Bibliography
Chapter 6 Foundation of Multi-Configurational Quantum Chemistry
6.1 Scaling Problem in FCI, CAS and RAS Wave Functions
6.2 Factorization and Coupling of Slater Determinants
6.2.1 Slater Condon Rules
6.3 Configuration State Functions
6.3.1 The Unitary Group Approach (UGA)
6.3.1.1 Analogy between CSFs and Spherical Harmonics
6.3.1.2 Gel'fand-Tsetlin Basis
6.3.1.3 Paldus and Weyl Tables
6.3.1.4 The Step-Vector
6.3.2 The Graphical Unitary Group Approach (GUGA)
6.3.3 Evaluation of Non-Vanishing Hamiltonian Matrix Elements
6.3.3.1 One-Body Coupling Coefficients
6.3.3.2 Two-Body Matrix Elements
6.4 Configuration Interaction Eigenvalue Problem
6.4.1 Iterative Methods
6.4.1.1 Lanczos Algorithm
6.4.1.2 Davidson Algorithm
6.4.2 Direct-CI Algorithm
6.5 The CASSCF Method
6.5.1 The MCSCF Parameterization
6.5.2 The MCSCF Gradient and Hessian
6.5.3 One-Step and Two-Step Procedures
6.5.4 Augmented Hessian Method
6.5.5 Matrix form of the First and Second Derivatives in MCSCF
6.5.6 Quadratically Converging Method with Optimal Convergence
6.5.7 Orbital-CI Coupling Terms
6.5.8 Super-CI for the Orbital Optimization
6.5.9 Redundancy of Active Orbital Rotations
6.6 Restricted and Generalized Active Space Wave Functions
6.6.1 GUGA Applied to CAS, RAS and GAS Wave Functions
6.6.2 Redundancies in GASSCF Orbital Rotations
6.6.3 MCSCF Molecular Orbitals
6.6.4 GASSCF Applied to the Gd2 Molecule
6.7 Excited States
6.7.1 Multi-State CI Solver
6.7.2 State-Specific and State-Averaged MCSCF
6.8 Stochastic Multiconfigurational Approaches
6.8.1 FCIQMC Working Equation
6.8.2 Multi-Wave Function Approach for Excited States
6.8.3 Sampling Reduced Density Matrices
Bibliography
Chapter 7 The Density Matrix Renormalization Group for Strong Correlation in Ground and Excited States
7.1 Introduction
7.2 DMRG Theory
7.2.1 Renormalization Group Formulation
7.2.2 Matrix Product States and Matrix Product Operators
7.2.2.0 Singular Value Decomposition
7.2.2.0 Left- and Right-Normalization
7.2.2.0 Overlaps
7.2.2.0 Matrix Product Operators
7.2.2.0 Expectation Values
7.2.3 MPS-MPO Formulation of DMRG
7.2.3.0 Sweeping in the MPS-MPO Formulation
7.2.3.0 One-Site and Two-Site DMRG
7.2.4 Connection between the Renormalization Group and the MPS-MPO Formulation of DMRG
7.2.5 Developments to Enhance DMRG Convergence and Performance
7.3 DMRG and Orbital Entanglement
7.4 DMRG in Practice
7.4.1 Calculating Excited States with DMRG
7.4.2 Factors Affecting the DMRG Convergence and Accuracy
7.4.2.0 Initial Guess
7.4.2.0 Orbital Choice and Ordering
7.4.2.0 Number of Renormalized States
7.4.3 Post-DMRG Methods for Dynamic Correlation and Environment Effects
7.4.4 Analytical Energy Gradients and Non-Adiabatic Coupling Matrix Elements
7.4.5 Tensor Network States
7.5 Applications in Quantum Chemistry
7.6 Conclusions
Acknowledgment
References
Chapter 8 Excited-State Calculations with Quantum Monte Carlo
8.1 Introduction
8.2 Variational Monte Carlo
8.3 Diffusion Monte Carlo
8.4 Wave Functions and their Optimization
8.4.1 Stochastic Reconfiguration Method
8.4.2 Linear Method
8.5 Excited States
8.5.1 Energy-Based Methods
8.5.2 Time-Dependent Linear-Response VMC
8.5.3 Variance-Based Methods
8.6 Applications to Excited States of Molecular Systems
8.7 Alternatives to Diffusion Monte Carlo
Bibliography
Chapter 9 Multi-Reference Configuration Interaction
9.1 Introduction
9.2 Basics
9.2.1 Configuration Interaction and the Variational Principle
9.2.2 The Size-Extensivity Problem of Truncated CI
9.2.3 Multi-Reference Configuration Spaces
9.2.4 Many-Electron Basis Functions: Determinants and CSFs
9.2.5 Workflow
9.3 Types of MRCI
9.3.1 Uncontracted and Contracted MRCI
9.3.2 MRCI with Extensivity Corrections
9.3.3 Types of Selection Schemes
9.3.4 Construction of Orbitals
9.4 Popular Implementations
9.5 Conclusions
References
Chapter 10 Multi-Configurational Reference Perturbation Theory with a CASSCF Reference Function
10.1 Rayleigh-Schrödinger Perturbation Theory
10.1.1 The Single-State Theory
10.1.1.1 The Conventional Projectional Derivation
10.1.1.2 The Bi-Variational Approach
10.1.2 Convergence Properties and Intruder States
10.1.2.1 Real and Imaginary Shift Techniques
10.2 Møller-Plesset Perturbation Theory
10.2.1 The Reference Function
10.2.2 The Partitioning of the Hamiltonian
10.2.3 The First-Order Interacting Space and Second-Order Energy Correction
10.3 State-Specific Multi-Configurational Reference Perturbation Methods
10.3.1 The Generation of the Reference Hamiltonian
10.3.2 CAS-MP2 Theory
10.3.3 CASPT2 Theory
10.3.3.1 The Partitioning of the Hamiltonian
10.3.3.2 The First-Order Interacting Space
10.3.3.3 Other Active Space References
10.3.3.4 Benchmark Results
10.3.3.5 IPEA Shift
10.3.4 MRMP2 Theory
10.3.4.1 The Partitioning of the Hamiltonian
10.3.4.2 The First-Order Interacting Space
10.3.5 NEVPT2 Theory
10.3.5.1 The Partitioning of the Hamiltonian
10.3.5.2 The First-Order Interacting Space
10.3.6 Performance Improvements
10.4 Quasi-Degenerate Perturbation Theory
10.5 Multi-State Multi-Configurational Reference Perturbation Methods
10.5.1 Multi-State CASPT2 Theory
10.5.2 Extended MS-CASPT2 Theory
10.6 Summary and Outlook
Acknowledgments
References
Appendix
Part II Nuclear Dynamics
Chapter 11 Exact Quantum Dynamics (Wave Packets) in Reduced Dimensionality
11.1 Introduction
11.2 Fundamentals of Molecular Quantum Dynamics
11.2.1 Wave Packet Dynamics
11.2.2 Time-Propagator Schemes
11.2.3 Excited State Wave Packet Dynamics
11.2.4 Surfaces and Coupling Elements in Reactive Coordinates
11.3 Choice of Dynamical Coordinates and Hamiltonian in Reduced Dimensionality
11.3.1 Manual Selection by Chemical Intuition
11.3.2 The G-Matrix Formalism
11.3.2.1 General Setup
11.3.2.2 Practical Computation of the G-Matrix Elements
11.3.2.3 Photorelaxation of Uracil in Linear Reactive Coordinates
11.3.3 Automatic Generation of Linear Coordinates
11.3.3.1 IRC Based Approach
11.3.3.2 Trajectory-Based Approach
11.3.3.3 Comparison of Both Techniques for Linear Subspaces
11.3.4 Automatic Generation of Non-Linear Coordinates
11.4 Summary and Further Remarks
References
Chapter 12 Multi-Configuration Time-Dependent Hartree Methods: From Quantum to Semiclassical and Quantum-Classical
12.1 Introduction
12.2 Time-Dependent Variational Principle and MCTDH
12.2.1 Variational Principle and Tangent Space Projections
12.2.2 MCTDH: Variational Multi-Configurational Wave Functions
12.2.2.1 MCTDH Wave Function Ansatz
12.2.2.2 MCTDH Equations of Motion
12.2.3 ML-MCTDH: Hierarchical Representations
12.3 Gaussian-Based MCTDH
12.3.1 G-MCTDH and vMCG
12.3.1.1 G-MCTDH Wave Function Ansatz
12.3.1.2 G-MCTDH Equations of Motion
12.3.1.3 vMCG Equations of Motion
12.3.2 2L-GMCTDH
12.3.2.1 Wave Function Ansatz
12.3.2.2 Equations of Motion
12.4 Quantum-Classical Multi-Configurational Approaches
12.4.1 Quantum-Classical Limit of G-MCTDH
12.4.2 Quantum-Classical Scheme with Finite-Width Wave Packets
12.4.3 Related Approaches
12.5 How to use MCTDH & Co
12.6 Synopsis and Application to Donor-Acceptor Complex
12.6.1 Hamiltonian, Spectral Densities, and Potential Surfaces
12.6.2 Ultrafast Coherent Charge Transfer Dynamics
12.6.3 Comparison of Methods
12.7 Conclusions and Outlook
Acknowledgments
References
Chapter 13 Gaussian Wave Packets and the DD-vMCG Approach
13.1 Historical Background
13.2 Basic Theory
13.2.1 Gaussian Wave Packets
13.2.2 General Equations of Motion
13.2.2.1 Coefficients and Parameters
13.2.2.2 CX-Formalism
13.2.2.3 Nuclear and Electronic Degrees of Freedom
13.2.3 Variational Multi-Configurational Gaussian Approach
13.3 Example Calculations
13.4 Tunneling Dynamics: Salicylaldimine
13.5 Non-Adiabatic Dynamics: The Butatriene Cation
13.6 Direct Non-Adiabatic Dynamics: Formamide
13.7 Summary
13.8 Practical Implementation
Acknowledgments
References
Chapter 14 Full and Ab Initio Multiple Spawning
14.1 Introduction
14.2 Time-Dependent Molecular Schrödinger Equation in a Gaussian Basis
14.2.1 Central Equations of Motion
14.2.2 Dynamics of the Trajectory Basis Functions
14.3 Full Multiple Spawning
14.3.1 Full Multiple Spawning Equations
14.3.2 Spawning Algorithm
14.4 Extending Full Multiple Spawning
14.4.1 External Field in Full Multiple Spawning
14.4.2 Spin-Orbit Coupling in Full Multiple Spawning
14.5 Ab Initio Multiple Spawning
14.5.1 From Full- to Ab Initio Multiple Spawning
14.5.2 Testing the Approximations of Ab Initio Multiple Spawning
14.5.3 On-the-Fly Ab Initio Multiple Spawning
14.5.4 Ab Initio Multiple Spawning versus Trajectory Surface Hopping
14.6 Dissecting an Ab Initio Multiple Spawning Dynamics
14.6.1 The Different Steps of an Ab Initio Multiple Spawning Dynamics
14.6.2 Example of Ab Initio Multiple Spawning Dynamics - the Photo-Chemistry of Cyclohexadiene
14.7 In Silico Photo-Chemistry with Ab Initio Multiple Spawning
14.8 Summary
References
Chapter 15 Ehrenfest Methods for Electron and Nuclear Dynamics
15.1 Introduction
15.2 Theory of the (Simple) Ehrenfest Method
15.2.1 Wave Function Ansatz
15.2.2 Equations of Motion
15.3 Theory of the Multi-Configurational Ehrenfest Method
15.3.1 Wave Function Ansatz
15.3.2 Equations of Motion
15.3.3 Computational Aspects
15.4 Applications
15.4.1 Coupled Electron and Nuclear Dynamics Upon Sudden Ionization
15.4.2 Ultrafast Scattering as a Probe of Nuclear Dynamics
15.5 Conclusion
References
Chapter 16 Surface Hopping Molecular Dynamics
16.1 Introduction
16.2 Basics of Surface Hopping
16.2.1 Advantages and Disadvantages
16.2.2 General Algorithm
16.3 Surface Hopping Ingredients
16.3.1 Nuclear Motion
16.3.2 Wave Function Propagation
16.3.3 Decoherence
16.3.4 Surface Hopping Algorithm
16.3.5 Kinetic Energy Adjustment and Frustrated Hops
16.3.6 Coupling Terms and Representations
16.4 Practical Remarks
16.4.1 Choice of the Electronic Structure Method
16.4.2 Initial Conditions
16.4.3 Example Application and Trajectory Analysis
16.5 Popular Implementations
16.6 Conclusion and Outlook
Acknowledgments
References
Chapter 17 Exact Factorization of the Electron-Nuclear Wave Function: Theory and Applications
17.1 Introduction
17.2 The Time-Dependent Molecular Problem in the Exact-Factorization Formulation
17.2.1 Wave Function Ansatz
17.2.2 Equations of Motion
17.3 The Born-Oppenheimer Framework and the Exact Factorization
17.3.1 One-Dimensional Case: Time-Dependent Potential Energy Surface
17.3.2 Two-Dimensional Case: Time-Dependent Potential Energy Surface and Time-Dependent Vector Potential
17.4 Trajectory-Based Solution of the Exact-Factorization Equations
17.4.1 CT-MQC: The Approximations
17.4.2 CT-MQC: Photo-Induced Ring Opening in Oxirane
17.4.3 CT-MQC: The Algorithm
17.5 The Molecular Berry Phase
17.6 Conclusions
Acknowledgments
References
Chapter 18 Bohmian Approaches to Non-Adiabatic Molecular Dynamics
18.1 Introduction
18.2 A Practical Overview of Bohmian Mechanics
18.2.1 The Postulates
18.2.2 Computation of Bohmian Trajectories
18.2.2.1 Trajectories from the Schrödinger Equation
18.2.2.2 Trajectories from the Hamilton-Jacobi Equation
18.2.2.3 Trajectories from a Complex Action
18.2.3 Computation of Expectation Values
18.3 The Born-Huang Picture of Molecular Dynamics
18.3.1 The Molecular Schrödinger Equation in Position Space
18.3.2 Schrödinger Equation in the Born-Huang Basis
18.3.2.1 The Born-Oppenheimer Approximation: The Adiabatic Case
18.3.2.2 Non-Adiabatic Dynamics
18.4 BH-Based Approaches
18.4.1 The Non-Adiabatic Bohmian Dynamics Equations (NABDY)
18.4.2 Implementation in Molecular Dynamics: The Adiabatic Case
18.4.3 The Approximate Quantum Potential Approach
18.5 Non-BH Approaches
18.5.1 The Conditional Wave Function Approach
18.5.1.1 Hermitian Conditional Wave Function Approach
18.5.2 The Interacting Conditional Wave Function Approach
18.5.3 Time-Dependent Quantum Monte Carlo
18.6 Conclusions
References
Chapter 19 Semiclassical Molecular Dynamics for Spectroscopic Calculations
19.1 Introduction
19.2 From Feynman's Path Integral to van Vleck's Semiclassical Propagator
19.3 The Semiclassical Initial Value Representation and the Heller-Herman-Kluk-Kay Formulation
19.4 A Derivation of the Heller-Herman-Kluk-Kay Propagator
19.5 The Time-Averaging Filter
19.6 The Multiple Coherent States SCIVR
19.7 The "Divide-and-Conquer" SCIVR
19.8 Mixed SCIVR Dynamics: Towards Semiclassical Spectroscopy in Condensed Phase
19.9 Semiclassical Spectroscopy Workflow
19.10 A Taste of Semiclassical Spectroscopy
19.11 Summary and Conclusions
Acknowledgments
Bibliography
Chapter 20 Path-Integral Approaches to Non-Adiabatic Dynamics
20.1 Introduction
20.2 Semiclassical Theory
20.2.1 Mapping Approach
20.2.2 Linearized Semiclassical Dynamics
20.3 Non-Equilibrium Dynamics
20.3.1 Spin-Boson Systems
20.3.2 Non-Equilibrium Correlation Functions
20.4 Non-Adiabatic Path-Integral Theory
20.4.1 Mean-Field Path-Integral Sampling
20.4.2 Non-Adiabatic Ring-Polymer Molecular Dynamics
20.4.3 Alleviation of the Negative Sign
20.4.4 Practical Implementation of Monte Carlo Sampling
20.5 Equilibrium Correlation Functions
20.6 Conclusions
Acknowledgments
References
Index
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