Principles and Practices of Molecular Properties

Patrick Norman is Professor and Head of Theoretical Chemistry and Biology at KTH Royal Institute of Technology, Stockholm, Sweden. His research interests include response theory for non-resonant and resonant external fields in the UV/vis and X-ray regions. He is a co-author of the Dalton program. Kenneth Ruud is Professor of Theoretical Chemistry at the University of Tromsø – The Arctic University of Norway. His research interests include linear and nonlinear response theory for mixed electric and magnetic fields as well as vibrational and medium effects. He is a co-author of the Dalton program. Trond Saue is a directeur de recherché of the French National Center for Scientific Research (CNRS) working at Université Toulouse III-Paul Sabatier in France. His research focuses on relativistic methods in theoretical chemistry. He is a principal author of the DIRAC program.

• Preface xi1 Introduction 12 Quantum Mechanics 112.1 Fundamentals 112.1.1 Postulates of Quantum Mechanics 112.1.2 Lagrangian and Hamiltonian Formalisms 112.1.3 Wave Functions and Operators 182.2 Time Evolution of Wave Functions 222.3 Time Evolution of Expectation Values 252.4 Variational Principle 27Further Reading 293 Particles and Fields 313.1 Microscopic Maxwell’s Equations 323.1.1 General Considerations 323.1.2 The Stationary Case 343.1.3 The General Case 383.1.4 Electromagnetic Potentials and Gauge Freedom 393.1.5 Electromagnetic Waves and Polarization 413.1.6 Electrodynamics: Relativistic and Nonrelativistic Formulations 453.2 Particles in Electromagnetic Fields 483.2.1 The Classical Mechanical Hamiltonian 483.2.2 The Quantum-Mechanical Hamiltonian 523.3 Electric and Magnetic Multipoles 573.3.1 Multipolar Gauge 573.3.2 Multipole Expansions 593.3.3 The Electric Dipole Approximation and Beyond 633.3.4 Origin Dependence of Electric and Magnetic Multipoles 643.3.5 Electric Multipoles 653.3.5.1 General Versus Traceless Forms 653.3.5.2 What We Can Learn from Symmetry 683.3.6 Magnetic Multipoles 693.3.7 Electric Dipole Radiation 703.4 Macroscopic Maxwell’s Equations 723.4.1 Spatial Averaging 723.4.2 Polarization and Magnetization 733.4.3 Maxwell’s Equations in Matter 773.4.4 Constitutive Relations 793.5 Linear Media 813.5.1 Boundary Conditions 823.5.2 Polarization in Linear Media 863.5.3 Electromagnetic Waves in a Linear Medium 923.5.4 Frequency Dependence of the Permittivity 963.5.4.1 Kramers–Kronig Relations 973.5.4.2 Relaxation in the Debye Model 983.5.4.3 Resonances in the Lorentz Model 1013.5.4.4 Refraction and Absorption 1053.5.5 Rotational Averages 1073.5.6 A Note About Dimensions, Units, and Magnitudes 110Further Reading 1114 Symmetry 1134.1 Fundamentals 1134.1.1 Symmetry Operations and Groups 1134.1.2 Group Representation 1174.2 Time Symmetries 1204.3 Spatial Symmetries 1254.3.1 Spatial Inversion 1254.3.2 Rotations 127Further Reading 1345 Exact-State Response Theory 1355.1 Responses in Two-Level System 1355.2 Molecular Electric Properties 1455.3 Reference-State Parameterizations 1515.4 Equations of Motion 1565.4.1 Time Evolution of Projection Amplitudes 1575.4.2 Time Evolution of Rotation Amplitudes 1595.5 Response Functions 1635.5.1 First-Order Properties 1665.5.2 Second-Order Properties 1665.5.3 Third-Order Properties 1695.5.4 Fourth-Order Properties 1745.5.5 Higher-Order Properties 1795.6 Dispersion 1795.7 Oscillator Strength and Sum Rules 1835.8 Absorption 1855.9 Residue Analysis 1905.10 Relaxation 1945.10.1 Density Operator 1955.10.2 Liouville Equation 1965.10.3 Density Matrix from Perturbation Theory 2005.10.4 Linear Response Functions from the Density Matrix 2015.10.5 Nonlinear Response Functions from the Density Matrix 2045.10.6 Relaxation in Wave Function Theory 2045.10.7 Absorption Cross Section 2075.10.8 Einstein Coefficients 210Further Reading 2116 Electronic and Nuclear Contributions to Molecular Properties 2136.1 Born–Oppenheimer Approximation 2136.2 Separation of Response Functions 2166.3 Molecular Vibrations and Normal Coordinates 2216.4 Perturbation Theory for Vibrational Wave Functions 2256.5 Zero-Point Vibrational Contributions to Properties 2276.5.1 First-Order Anharmonic Contributions 2276.5.2 Importance of Zero-Point Vibrational Corrections 2316.5.3 Temperature Effects 2346.6 Pure Vibrational Contributions to Properties 2356.6.1 Perturbation Theory Approach 2356.6.2 Pure Vibrational Effects from an Analysis of the Electric-Field Dependence of the Molecular Geometry 2386.7 Adiabatic Vibronic Theory for Electronic Excitation Processes 2446.7.1 Franck–Condon Integrals 2486.7.2 Vibronic Effects in a Diatomic System 2506.7.3 Linear Coupling Model 2526.7.4 Herzberg–Teller Corrections and Vibronically Induced Transitions 252Further Reading 2537 Approximate Electronic State Response Theory 2557.1 Reference State Parameterizations 2557.1.1 Single Determinant 2557.1.2 Configuration Interaction 2637.1.3 Multiconfiguration Self-Consistent Field 2667.1.4 Coupled Cluster 2687.2 Equations of Motion 2717.2.1 Ehrenfest Theorem 2717.2.2 Quasi-Energy Derivatives 2757.3 Response Functions 2767.3.1 Single Determinant Approaches 2767.3.2 Configuration Interaction 2817.3.3 Multiconfiguration Self-Consistent Field 2817.3.4 Matrix Structure in the SCF, CI, and MCSCF Approximations 2817.3.5 Coupled Cluster 2857.4 Residue Analysis 2887.5 Relaxation 291Further Reading 2938 Response Functions and Spectroscopies 2958.1 Nuclear Interactions 2968.1.1 Nuclear Charge Distribution 2968.1.2 Hyperfine Structure 3018.1.2.1 Nuclear Magnetic Dipole Moment 3018.1.2.2 Nuclear Electric Quadrupole Moment 3058.2 Zeeman Interaction and Electron Paramagnetic Resonance 3108.3 Polarizabilities 3178.3.1 Linear Polarizability 3178.3.1.1 Weak Intermolecular Forces 3218.3.2 Nonlinear Polarizabilities 3258.4 Magnetizability 3268.4.1 The Origin Dependence of the Magnetizability 3288.4.2 Magnetizabilities from Magnetically Induced Currents 3318.4.3 Isotropic Magnetizabilities and Pascal’s Rule 3328.5 Electronic Absorption and Emission Spectroscopies 3358.5.1 Visible and Ultraviolet Absorption 3388.5.2 Fluorescence Spectroscopy 3438.5.3 Phosphorescence 3448.5.4 Multiphoton Absorption 3478.5.4.1 Multiphoton Absorption Cross Sections 3488.5.4.2 Few-State Models for Two-Photon Absorption Cross Section 3508.5.4.3 General Multiphoton Absorption Processes 3518.5.5 X-ray Absorption 3548.5.5.1 Core-Excited States 3558.5.5.2 Field Polarization 3588.5.5.3 Static Exchange Approximation 3608.5.5.4 Complex or Damped Response Theory 3628.6 Birefringences and Dichroisms 3648.6.1 Natural Optical Activity 3668.6.2 Electronic Circular Dichroism 3728.6.3 Nonlinear Birefringences 3758.6.3.1 Magnetic Circular Dichroism 3768.6.3.2 Electric Field Gradient-Induced Birefringence 3798.7 Vibrational Spectroscopies 3818.7.1 Infrared Absorption 3818.7.1.1 Double-Harmonic Approximation 3818.7.1.2 Anharmonic Corrections 3838.7.2 Vibrational Circular Dichroism 3848.7.3 Raman Scattering 3888.7.3.1 Raman Scattering from a Classical Point of View 3888.7.3.2 Raman Scattering from a Quantum Mechanical Point of View 3928.7.4 Vibrational Raman Optical Activity 4028.8 Nuclear Magnetic Resonance 4088.8.1 The NMR Experiment 4088.8.2 NMR Parameters 413Further Reading 417AppendiciesA Abbreviations 419B Units 421C Second Quantization 423C.1 Creation and Annihilation Operators 423C.2 Fock Space 425C.3 The Number Operator 426C.4 The Electronic Hamiltonian on Second-Quantized Form 427C.5 Spin in Second Quantization 429D Fourier Transforms 431E Operator Algebra 435F Spin Matrix Algebra 439G Angular Momentum Algebra 441H Variational Perturbation Theory 445I Two-Level Atom 451I.1 Rabi Oscillations 452I.2 Time-Dependent Perturbation Theory 454I.3 The Quasi-energy Approach 455Index 457