Professor Leticia González teaches at the Department of Chemistry at the University of Vienna, Austria. She is a theoretical chemist world-known for her work on molecular excited states and ultrafast dynamics simulations. Besides publishing over 250 papers and several reviews on excited states and dynamics, she has developed the SHARC program package to simulate non-adiabatic dynamics. Professor Roland Lindh currently teaches at Uppsala University, Sweden. He is a member of the editorial board of International Journal of Quantum Chemistry and the MOLCAS quantum chemistry program project. He co-authored the book "Multiconfigurational Quantum Chemistry" and is an expert on method development for multiconfigurational wave function theory.

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• List of Contributors xixPreface xxiii1 Motivation and Basic Concepts 1Sandra Gómez, Ignacio Fdez. Galván, Roland Lindh, and Leticia Gonzalez1.1 Mission and Motivation 11.2 Atomic Units 41.3 The Molecular Hamiltonian 51.4 Dirac or Bra-Ket Notation 61.5 Index Definitions 71.6 Second Quantization Formalism 71.7 Born–Oppenheimer Approximation and Potential Energy Surfaces 91.8 Adiabatic Versus Diabatic Representations 101.9 Conical Intersections 111.10 Further Reading 121.11 Acknowledgments 12Part I Quantum Chemistry 132 Time-Dependent Density Functional Theory 15Miquel Huix-Rotllant, Nicolas Ferre, and Mario Barbatti2.1 Introduction 152.2 TDDFT Fundamentals 162.2.1 The Runge–Gross Theorems 162.2.2 The Time-Dependent Kohn–Sham Approach 182.2.3 Solutions of Time-Dependent Kohn–Sham Equations 192.2.3.1 Real-Time TDDFT 192.2.3.2 Linear-Response TDDFT 202.3 Linear-Response TDDFT in Action 222.3.1 Vertical Excitations and Energy Surfaces 222.3.1.1 Vertical Excitations: How Good are They? 232.3.1.2 Reconstructed Energy Surfaces: How Good are They? 252.3.2 Conical Intersections 282.3.3 Coupling Terms and Auxiliary Wave Functions 302.3.3.1 The Casida Ansatz 302.3.3.2 Time-Derivative Non-Adiabatic Couplings 312.3.4 Non-Adiabatic Dynamics 322.4 Excited States and Dynamics with TDDFT Variants and Beyond 342.5 Conclusions 35Acknowledgments 36References 363 Multi-Configurational Density Functional Theory: Progress and Challenges 47Erik Donovan Hedegård3.1 Introduction 473.2 Wave Function Theory 503.3 Kohn–Sham Density Functional Theory 503.3.1 Density Functional Approximations 533.3.2 Density Functional Theory for Excited States 543.3.2.1 Issues Within the Time-Dependent Density Functional Theory Ansatz 553.3.2.2 Self-Interaction Error 553.3.2.3 Degeneracies, Near-Degeneracies and the Symmetry Dilemma 563.4 Multi-Configurational Density Functional Theory 573.4.1 Semi-Empirical Multi-Configurational Density Functional Theory 573.4.2 Multi-Configurational Density Functional Theory Based the On-Top Pair Density 583.4.2.1 Density Matrices and the On-Top Pair Density 593.4.2.2 Energy Functional and Excited States with the On-Top Pair Density 603.4.3 Multi-Configurational Density Functional Theory Based on Range-Separation 613.4.3.1 Energy Functional and Excited States in Range-Separated Methods 623.4.3.2 The Range-Separation Parameter in Excited State Calculations 623.5 Illustrative Examples 643.5.1 Excited States of Organic Molecules 643.5.2 Excited States for a Transition Metal Complex 653.6 Outlook 66Acknowledgments 67References 674 Equation-of-Motion Coupled-Cluster Models 77Monika Musiał4.1 Introduction 774.2 Theoretical Background 794.2.1 Coupled-ClusterWave Function 794.2.2 The Equation-of-Motion Approach 804.2.3 Similarity-Transformed Hamiltonian 814.2.4 Davidson Diagonalization Algorithm 824.3 Excited States: EE-EOM-CC 844.3.1 EE-EOM-CCSD Model 844.3.2 EE-EOM-CCSDT Model 864.3.3 EE-EOM-CC Results 874.4 Ionized States: IP-EOM-CC 894.4.1 IP-EOM-CCSD Model 894.4.2 IP-EOM-CCSDT Model 894.4.3 IP-EOM-CC Results 904.5 Electron-Attached States: EA-EOM-CC 914.5.1 EA-EOM-CCSD Model 924.5.2 EA-EOM-CCSDT Model 924.5.3 EA-EOM-CC Results 924.6 Doubly-Ionized States: DIP-EOM-CC 944.6.1 DIP-EOM-CCSD Model 954.6.2 DIP-EOM-CCSDT Model 954.6.3 DIP-EOM-CC Results 964.7 Doubly Electron-Attached States: DEA-EOM-CC 974.7.1 DEA-EOM-CCSD Model 984.7.2 DEA-EOM-CCSDT Model 984.7.3 DEA-EOM-CC Results 984.8 Size-Extensivity Issue in the EOM-CC Theory 1004.9 Final Remarks 102References 1035 The Algebraic-Diagrammatic Construction Scheme for the Polarization Propagator 109Andreas Dreuw5.1 Original Derivation via Green’s Functions 1105.2 The Intermediate State Representation 1125.3 Calculation of Excited State Properties and Analysis 1145.3.1 Excited State Properties 1145.3.2 Excited-State Wave Function and Density Analyses 1165.4 Properties and Limitations of ADC 1175.5 Variants of EE-ADC 1195.5.1 Extended ADC(2) 1195.5.2 Unrestricted EE-ADC Schemes 1205.5.3 Spin-Flip EE-ADC Schemes 1215.5.4 Spin-Opposite-Scaled ADC Schemes 1225.5.5 Core-Valence Separated (CVS) EE-ADC 1235.6 Describing Molecular Photochemistry with ADC Methods 1255.6.1 Potential Energy Surfaces 1255.6.2 Environment Models within ADC 1265.7 Brief Summary and Perspective 126Bibliography 1276 Foundation of Multi-Configurational Quantum Chemistry 133Giovanni Li Manni, Kai Guther, Dongxia Ma, and Werner Dobrautz6.1 Scaling Problem in FCI, CAS and RASWave Functions 1366.2 Factorization and Coupling of Slater Determinants 1386.2.1 Slater Condon Rules 1406.3 Configuration State Functions 1416.3.1 The Unitary Group Approach (UGA) 1426.3.1.1 Analogy between CSFs and Spherical Harmonics 1436.3.1.2 Gel’fand-Tsetlin Basis 1436.3.1.3 Paldus andWeyl Tables 1456.3.1.4 The Step-Vector 1486.3.2 The Graphical Unitary Group Approach (GUGA) 1486.3.3 Evaluation of Non-Vanishing Hamiltonian Matrix Elements 1536.3.3.1 One-Body Coupling Coefficients 1546.3.3.2 Two-Body Matrix Elements 1576.4 Configuration Interaction Eigenvalue Problem 1586.4.1 Iterative Methods 1596.4.1.1 Lanczos Algorithm 1596.4.1.2 Davidson Algorithm 1606.4.2 Direct-CI Algorithm 1626.5 The CASSCF Method 1656.5.1 The MCSCF Parameterization 1676.5.2 The MCSCF Gradient and Hessian 1696.5.3 One-Step and Two-Step Procedures 1706.5.4 Augmented Hessian Method 1716.5.5 Matrix form of the First and Second Derivatives in MCSCF 1716.5.6 Quadratically Converging Method with Optimal Convergence 1756.5.7 Orbital-CI Coupling Terms 1786.5.8 Super-CI for the Orbital Optimization 1796.5.9 Redundancy of Active Orbital Rotations 1816.6 Restricted and Generalized Active Space Wave Functions 1826.6.1 GUGA Applied to CAS, RAS and GAS Wave Functions 1846.6.2 Redundancies in GASSCF Orbital Rotations 1866.6.3 MCSCF Molecular Orbitals 1876.6.4 GASSCF Applied to the Gd2 Molecule 1886.7 Excited States 1896.7.1 Multi-State CI Solver 1906.7.2 State-Specific and State-Averaged MCSCF 1916.8 Stochastic Multiconfigurational Approaches 1916.8.1 FCIQMC Working Equation 1926.8.2 Multi-Wave Function Approach for Excited States 1966.8.3 Sampling Reduced Density Matrices 196Bibliography 1987 The Density Matrix Renormalization Group for Strong Correlation in Ground and Excited States 205Leon Freitag and Markus Reiher7.1 Introduction 2057.2 DMRG Theory 2077.2.1 Renormalization Group Formulation 2077.2.2 Matrix Product States and Matrix Product Operators 2107.2.3 MPS-MPO Formulation of DMRG 2147.2.4 Connection between the Renormalization Group and the MPS-MPO Formulation of DMRG 2177.2.5 Developments to Enhance DMRG Convergence and Performance 2187.3 DMRG and Orbital Entanglement 2187.4 DMRG in Practice 2207.4.1 Calculating Excited States with DMRG 2207.4.2 Factors Affecting the DMRG Convergence and Accuracy 2207.4.3 Post-DMRG Methods for Dynamic Correlation and Environment Effects 2217.4.4 Analytical Energy Gradients and Non-Adiabatic Coupling Matrix Elements 2227.4.5 Tensor Network States 2247.5 Applications in Quantum Chemistry 2257.6 Conclusions 230Acknowledgment 231References 2318 Excited-State Calculations with Quantum Monte Carlo 247Jonas Feldt and Claudia Filippi8.1 Introduction 2478.2 Variational Monte Carlo 2498.3 Diffusion Monte Carlo 2528.4 Wave Functions and their Optimization 2568.4.1 Stochastic Reconfiguration Method 2588.4.2 Linear Method 2598.5 Excited States 2618.5.1 Energy-Based Methods 2618.5.2 Time-Dependent Linear-Response VMC 2638.5.3 Variance-Based Methods 2648.6 Applications to Excited States of Molecular Systems 2658.7 Alternatives to Diffusion Monte Carlo 269Bibliography 2709 Multi-Reference Configuration Interaction 277Felix Plasser and Hans Lischka9.1 Introduction 2779.2 Basics 2789.2.1 Configuration Interaction and the Variational Principle 2789.2.2 The Size-Extensivity Problem of Truncated CI 2809.2.3 Multi-Reference Configuration Spaces 2829.2.4 Many-Electron Basis Functions: Determinants and CSFs 2869.2.5 Workflow 2879.3 Types of MRCI 2899.3.1 Uncontracted and Contracted MRCI 2899.3.2 MRCI with Extensivity Corrections 2919.3.3 Types of Selection Schemes 2939.3.4 Construction of Orbitals 2939.4 Popular Implementations 2949.5 Conclusions 295References 29510 Multi-Configurational Reference Perturbation Theory with a CASSCF Reference Function 299Roland Lindh and Ignacio Fdez. Galván10.1 Rayleigh–Schrödinger Perturbation Theory 30010.1.1 The Single-State Theory 30010.1.1.1 The Conventional Projectional Derivation 30010.1.1.2 The Bi-Variational Approach 30410.1.2 Convergence Properties and Intruder States 30810.1.2.1 Real and Imaginary Shift Techniques 31010.2 Møller–Plesset Perturbation Theory 31310.2.1 The Reference Function 31410.2.2 The Partitioning of the Hamiltonian 31510.2.3 The First-Order Interacting Space and Second-Order Energy Correction 31610.3 State-Specific Multi-Configurational Reference Perturbation Methods 32010.3.1 The Generation of the Reference Hamiltonian 32110.3.2 CAS-MP2 Theory 32210.3.3 CASPT2 Theory 32310.3.3.1 The Partitioning of the Hamiltonian 32410.3.3.2 The First-Order Interacting Space 32510.3.3.3 Other Active Space References 32810.3.3.4 Benchmark Results 32910.3.3.5 IPEA Shift 33010.3.4 MRMP2 Theory 33110.3.4.1 The Partitioning of the Hamiltonian 33110.3.4.2 The First-Order Interacting Space 33210.3.5 NEVPT2 Theory 33310.3.5.1 The Partitioning of the Hamiltonian 33310.3.5.2 The First-Order Interacting Space 33510.3.6 Performance Improvements 33610.4 Quasi-Degenerate Perturbation Theory 33810.5 Multi-State Multi-Configurational Reference Perturbation Methods 34110.5.1 Multi-State CASPT2 Theory 34110.5.2 Extended MS-CASPT2 Theory 34210.6 Summary and Outlook 343Acknowledgments 345References 345Appendix 350Part II Nuclear Dynamics 35511 Exact Quantum Dynamics (Wave Packets) in Reduced Dimensionality 357Sebastian Reiter, Daniel Keefer, and Regina de Vivie-Riedle11.1 Introduction 35711.2 Fundamentals of Molecular Quantum Dynamics 35811.2.1 Wave Packet Dynamics 35811.2.2 Time-Propagator Schemes 36011.2.3 Excited State Wave Packet Dynamics 36211.2.4 Surfaces and Coupling Elements in Reactive Coordinates 36211.3 Choice of Dynamical Coordinates and Hamiltonian in Reduced Dimensionality 36411.3.1 Manual Selection by Chemical Intuition 36411.3.2 The G-Matrix Formalism 36511.3.2.1 General Setup 36611.3.2.2 Practical Computation of the G-Matrix Elements 36711.3.2.3 Photorelaxation of Uracil in Linear Reactive Coordinates 36711.3.3 Automatic Generation of Linear Coordinates 36911.3.3.1 IRC Based Approach 36911.3.3.2 Trajectory-Based Approach 37111.3.3.3 Comparison of Both Techniques for Linear Subspaces 37211.3.4 Automatic Generation of Non-Linear Coordinates 37411.4 Summary and Further Remarks 378References 37912 Multi-Configuration Time-Dependent Hartree Methods: From Quantum to Semiclassical and Quantum-Classical 383M. Bonfanti, G. A. Worth, and I. Burghardt12.1 Introduction 38312.2 Time-Dependent Variational Principle and MCTDH 38512.2.1 Variational Principle and Tangent Space Projections 38512.2.2 MCTDH: Variational Multi-Configurational Wave Functions 38612.2.2.1 MCTDH Wave Function Ansatz 38612.2.2.2 MCTDH Equations of Motion 38812.2.3 ML-MCTDH: Hierarchical Representations 38912.3 Gaussian-Based MCTDH 39012.3.1 G-MCTDH and vMCG 39012.3.1.1 G-MCTDH Wave Function Ansatz 39112.3.1.2 G-MCTDH Equations of Motion 39212.3.1.3 vMCG Equations of Motion 39312.3.2 2L-GMCTDH 39412.3.2.1 Wave Function Ansatz 39412.3.2.2 Equations of Motion 39512.4 Quantum-Classical Multi-Configurational Approaches 39612.4.1 Quantum-Classical Limit of G-MCTDH 39612.4.2 Quantum-Classical Scheme with Finite-Width Wave Packets 39812.4.3 Related Approaches 39912.5 How to use MCTDH & Co 39912.6 Synopsis and Application to Donor–Acceptor Complex 40012.6.1 Hamiltonian, Spectral Densities, and Potential Surfaces 40012.6.2 Ultrafast Coherent Charge Transfer Dynamics 40212.6.3 Comparison of Methods 40312.7 Conclusions and Outlook 405Acknowledgments 406References 40613 Gaussian Wave Packets and the DD-vMCG Approach 413Graham A. Worth and Benjamin Lasorne13.1 Historical Background 41313.2 Basic Theory 41513.2.1 Gaussian Wave Packets 41513.2.2 General Equations of Motion 41813.2.2.1 Coefficients and Parameters 41813.2.2.2 CX-Formalism 41913.2.2.3 Nuclear and Electronic Degrees of Freedom 42013.2.3 Variational Multi-Configurational Gaussian Approach 42213.3 Example Calculations 42413.4 Tunneling Dynamics: Salicylaldimine 42513.5 Non-Adiabatic Dynamics: The Butatriene Cation 42613.6 Direct Non-Adiabatic Dynamics: Formamide 42813.7 Summary 43113.8 Practical Implementation 431Acknowledgments 431References 43114 Full and Ab Initio Multiple Spawning 435Basile F. E. Curchod14.1 Introduction 43514.2 Time-Dependent Molecular Schrödinger Equation in a Gaussian Basis 43614.2.1 Central Equations of Motion 43614.2.2 Dynamics of the Trajectory Basis Functions 43914.3 Full Multiple Spawning 44014.3.1 Full Multiple Spawning Equations 44014.3.2 Spawning Algorithm 44214.4 Extending Full Multiple Spawning 44314.4.1 External Field in Full Multiple Spawning 44414.4.2 Spin-Orbit Coupling in Full Multiple Spawning 44514.5 Ab Initio Multiple Spawning 44714.5.1 From Full- to Ab Initio Multiple Spawning 44714.5.2 Testing the Approximations of Ab Initio Multiple Spawning 44914.5.3 On-the-Fly Ab Initio Multiple Spawning 45014.5.4 Ab Initio Multiple Spawning versus Trajectory Surface Hopping 45114.6 Dissecting an Ab Initio Multiple Spawning Dynamics 45414.6.1 The Different Steps of an Ab Initio Multiple Spawning Dynamics 45414.6.2 Example of Ab Initio Multiple Spawning Dynamics – the Photo-Chemistry of Cyclohexadiene 45514.7 In Silico Photo-Chemistry with Ab Initio Multiple Spawning 45914.8 Summary 462References 46315 Ehrenfest Methods for Electron and Nuclear Dynamics 469Adam Kirrander and Morgane Vacher15.1 Introduction 46915.2 Theory of the (Simple) Ehrenfest Method 47015.2.1 Wave Function Ansatz 47115.2.2 Equations of Motion 47215.3 Theory of the Multi-Configurational Ehrenfest Method 47415.3.1 Wave Function Ansatz 47415.3.2 Equations of Motion 47615.3.3 Computational Aspects 47915.4 Applications 48015.4.1 Coupled Electron and Nuclear Dynamics Upon Sudden Ionization 48115.4.2 Ultrafast Scattering as a Probe of Nuclear Dynamics 48515.5 Conclusion 490References 49116 Surface Hopping Molecular Dynamics 499Sebastian Mai, Philipp Marquetand, and Leticia Gonzalez16.1 Introduction 49916.2 Basics of Surface Hopping 50016.2.1 Advantages and Disadvantages 50016.2.2 General Algorithm 50116.3 Surface Hopping Ingredients 50316.3.1 Nuclear Motion 50316.3.2 Wave Function Propagation 50416.3.3 Decoherence 50516.3.4 Surface Hopping Algorithm 50716.3.5 Kinetic Energy Adjustment and Frustrated Hops 50916.3.6 Coupling Terms and Representations 51116.4 Practical Remarks 51316.4.1 Choice of the Electronic Structure Method 51316.4.2 Initial Conditions 51616.4.3 Example Application and Trajectory Analysis 51816.5 Popular Implementations 52116.6 Conclusion and Outlook 522Acknowledgments 522References 52217 Exact Factorization of the Electron–Nuclear Wave Function: Theory and Applications 531Federica Agostini and E. K. U. Gross17.1 Introduction 53117.2 The Time-Dependent Molecular Problem in the Exact-Factorization Formulation 53317.2.1 Wave Function Ansatz 53317.2.2 Equations of Motion 53517.3 The Born–Oppenheimer Framework and the Exact Factorization 53617.3.1 One-Dimensional Case: Time-Dependent Potential Energy Surface 53817.3.2 Two-Dimensional Case: Time-Dependent Potential Energy Surface and Time-Dependent Vector Potential 54217.4 Trajectory-Based Solution of the Exact-Factorization Equations 54517.4.1 CT-MQC: The Approximations 54617.4.2 CT-MQC: Photo-Induced Ring Opening in Oxirane 54917.4.3 CT-MQC: The Algorithm 55117.5 The Molecular Berry Phase 55317.6 Conclusions 556Acknowledgments 556References 55618 Bohmian Approaches to Non-Adiabatic Molecular Dynamics 563Guillermo Albareda and Ivano Tavernelli18.1 Introduction 56318.2 A Practical Overview of Bohmian Mechanics 56518.2.1 The Postulates 56518.2.2 Computation of Bohmian Trajectories 56618.2.2.1 Trajectories from the Schrödinger Equation 56618.2.2.2 Trajectories from the Hamilton–Jacobi Equation 56718.2.2.3 Trajectories from a Complex Action 56818.2.3 Computation of Expectation Values 56918.3 The Born–Huang Picture of Molecular Dynamics 56918.3.1 The Molecular Schrödinger Equation in Position Space 56918.3.2 Schrödinger Equation in the Born–Huang Basis 57018.3.2.1 The Born–Oppenheimer Approximation: The Adiabatic Case 57118.3.2.2 Non-Adiabatic Dynamics 57218.4 BH-Based Approaches 57318.4.1 The Non-Adiabatic Bohmian Dynamics Equations (NABDY) 57318.4.2 Implementation in Molecular Dynamics: The Adiabatic Case 57518.4.3 The Approximate Quantum Potential Approach 57718.5 Non-BH Approaches 57918.5.1 The ConditionalWave Function Approach 57918.5.1.1 Hermitian ConditionalWave Function Approach 58118.5.2 The Interacting ConditionalWave Function Approach 58218.5.3 Time-Dependent Quantum Monte Carlo 58518.6 Conclusions 588References 58919 Semiclassical Molecular Dynamics for Spectroscopic Calculations 595Riccardo Conte and Michele Ceotto19.1 Introduction 59519.2 From Feynman’s Path Integral to van Vleck’s Semiclassical Propagator 59819.3 The Semiclassical Initial Value Representation and the Heller–Herman–Kluk–Kay Formulation 60119.4 A Derivation of the Heller–Herman–Kluk–Kay Propagator 60319.5 The Time-Averaging Filter 60419.6 The Multiple Coherent States SCIVR 60619.7 The “Divide-and-Conquer” SCIVR 61019.8 Mixed SCIVR Dynamics: Towards Semiclassical Spectroscopy in Condensed Phase 61519.9 Semiclassical Spectroscopy Workflow 61819.10 A Taste of Semiclassical Spectroscopy 61919.11 Summary and Conclusions 622Acknowledgments 624Bibliography 62420 Path-Integral Approaches to Non-Adiabatic Dynamics 629Maximilian A. C. Saller, Johan E. Runeson, and Jeremy O. Richardson20.1 Introduction 62920.2 Semiclassical Theory 63120.2.1 Mapping Approach 63120.2.2 Linearized Semiclassical Dynamics 63220.3 Non-Equilibrium Dynamics 63320.3.1 Spin-Boson Systems 63420.3.2 Non-Equilibrium Correlation Functions 63620.4 Non-Adiabatic Path-Integral Theory 64020.4.1 Mean-Field Path-Integral Sampling 64020.4.2 Non-Adiabatic Ring-Polymer Molecular Dynamics 64120.4.3 Alleviation of the Negative Sign 64420.4.4 Practical Implementation of Monte Carlo Sampling 64420.5 Equilibrium Correlation Functions 64620.6 Conclusions 648Acknowledgments 649References 649Index 655