Charge and Energy Transfer Dynamics in Molecular Systems

Volkhard May studied physics at Humboldt University, Berlin, and received his Ph.D. in Theoretical Physics in 1981, and his Habilitation at the College of Education, Güstrow, in 1987. His research activities focused on the theory of transfer phenomena in molecular nanostructures. He is retired since 2019. Oliver Kühn received his PhD from Humboldt University, Berlin. Since 2008 he is a Professor of Theoretical Physics at the University of Rostock. His current research focusses on dynamics of condensed phase systems such as molecular aggregates and novel materials.

• Preface to the Fourth Edition xiiiPreface to the Third Edition xvPreface to the Second Edition xviiPreface to the First Edition xix1 Introduction 12 Electronic and Vibrational Molecular States 72.1 Introduction 72.2 Molecular Schrödinger Equation 92.3 Born–Oppenheimer Separation 112.3.1 Born–Oppenheimer Approximation 132.4 Electronic Structure Methods 152.4.1 The Hartree–Fock Equations 172.4.2 Density Functional Theory 192.5 Potential Energy Surfaces 212.5.1 Harmonic Approximation and Normal Mode Analysis 242.5.2 Operator Representation of the Normal Mode Hamiltonian 272.5.3 Construction of System–Bath Models 312.6 Adiabatic versus Diabatic Representation of the Molecular Hamiltonian 362.6.1 Adiabatic Picture 362.6.2 Diabatic Picture 372.6.3 Two-State Case 402.7 Condensed-phase Approaches 422.7.1 Dielectric Continuum Model 432.7.1.1 Medium Electrostatics 432.7.1.2 Reaction Field Model 472.7.2 Explicit Quantum-classical Solvent Model 492.8 Supplement 512.8.1 Franck–Condon Factors 512.8.2 The Two-level System 522.8.3 The Linear Molecular Chain and the Molecular Ring 55References 57Further Reading 573 Dynamics of Isolated and Open Quantum Systems 593.1 Introduction 603.2 Time-dependent Schrödinger Equation 663.2.1 Wave Packets 663.2.2 The Interaction Representation 693.2.3 Multidimensional Wave Packet Dynamics 713.3 The Golden Rule of Quantum Mechanics 753.3.1 Transition from a Single State into a Continuum 753.3.2 Transition Rate for a Thermal Ensemble 783.3.3 Green’s Function Approach 813.4 The Nonequilibrium Statistical Operator and the Density Matrix 843.4.1 The Density Operator 843.4.2 The Density Matrix 863.4.3 Equation of Motion for the Density Operator 883.4.4 Wigner Representation of the Density Operator 903.4.5 Dynamics of Coupled Multilevel Systems in a Heat Bath 933.5 The Reduced Density Operator and the Reduced Density Matrix 963.5.1 The Reduced Density Operator 963.5.2 Equation of Motion for the Reduced Density Operator 973.5.3 Mean-field Approximation 983.5.4 The Interaction Representation of the Reduced Density Operator 993.5.5 The Nakajima–Zwanzig Equation 1013.5.6 Second-order Equation of Motion for the Reduced Density Operator 1053.6 Quantum Master Equation 1073.6.1 Markov Approximation 1093.7 The Reservoir Correlation Function 1123.7.1 General Properties of C uv (t) 1123.7.2 Harmonic Oscillator Reservoir 1143.7.3 The Spectral Density 1163.7.4 Linear Response Theory for the Reservoir 1203.7.5 Classical Description of C uv (t) 1223.8 Reduced Density Matrix in Energy Representation 1233.8.1 The Quantum Master Equation in Energy Representation 1233.8.2 Multilevel Redfield Equations 1263.8.2.1 Population Transfer: a = b, c = d 1273.8.2.2 Coherence Dephasing: a ≠ b, a = c, b = d 1293.8.2.3 Remaining Elements of R ab,cd 1293.8.3 The Secular Approximation 1303.8.4 State Expansion of the System–Reservoir Coupling 1313.8.4.1 Some Estimates 1323.9 Coordinate and Wigner Representation of the Reduced Density Matrix 1333.10 The Path Integral Representation of the Density Matrix 1353.11 Hierarchy Equations of Motion Approach 1403.12 Coherent to Dissipative Dynamics of a Two-level System 1433.12.1 Coherent Dynamics 1433.12.2 Dissipative Dynamics Using Eigenstates 1443.12.3 Dissipative Dynamics Using Zeroth-order States 1473.13 Trajectory-based Methods 1493.13.1 The Mean-field Approach 1493.13.2 The Surface Hopping Method 1523.14 Generalized Rate Equations: The Liouville Space Approach 1553.14.1 Projection Operator Technique 1563.14.2 Generalized Rate Equations 1573.14.3 Rate Equations 1593.14.4 The Memory Kernels 1593.14.5 Second-order Rate Expressions 1613.14.6 Fourth-order Rate Expressions 1643.14.6.1 Three-level System with Sequential Coupling 1653.15 Supplement 1683.15.1 Thermofield Dynamics 1683.15.2 Stochastic Schrödinger Equation 172References 175Further Reading 1764 Interaction of Molecular Systems with Radiation Fields 1774.1 Introduction 1784.2 Absorption of Light 1824.2.1 Linear Absorption Coefficient 1824.2.2 Dipole–Dipole Correlation Function 1844.3 Nonlinear Optical Response 1864.3.1 Nonlinear Polarization 1864.3.2 Nonlinear Response Functions 1894.3.3 Eigenstate Expansion of the Response Functions 1914.3.4 Cumulant Expansion of the Response Functions 1944.3.5 Rotating Wave Approximation 1974.3.6 Pump–Probe Spectroscopy 1984.3.7 Two-dimensional Spectroscopy 2024.4 Field Quantization and Spontaneous Emission of Light 206References 208Further Reading 2095 Vibrational Dynamics: Energy Redistribution, Relaxation, and Dephasing 2115.1 Introduction 2115.2 Intramolecular Vibrational Energy Redistribution 2155.2.1 Zeroth-order Basis and State Mixing 2155.2.2 Golden Rule and Beyond 2195.3 Intermolecular Vibrational Energy Relaxation 2235.3.1 The System–Reservoir Hamiltonian 2235.3.2 Instantaneous Normal Modes 2265.3.3 Generalized Langevin Equation 2285.3.4 Classical Force–Force Correlation Functions 2315.3.5 Dissipative Dynamics of a Harmonic Oscillator 2345.4 Polyatomic Molecules in Solution 2375.4.1 System–Reservoir Hamiltonian 2375.4.2 Higher Order Multiquantum Relaxation 2385.5 Quantum–Classical Approaches to Relaxation and Dephasing 243References 247Further Reading 2476 Intramolecular Electronic Transitions 2496.1 Introduction 2496.1.1 Optical Transitions 2506.1.2 Internal Conversion Processes 2556.2 The Optical Absorption Coefficient 2556.2.1 Golden Rule Formulation 2556.2.2 The Density of States 2586.2.3 Absorption Coefficient for Harmonic Potential Energy Surfaces 2606.2.4 Absorption Lineshape and Spectral Density 2636.2.5 Cumulant Expansion of the Absorption Coefficient 2646.2.6 Absorption Coefficient for Model Spectral Densities 2666.3 Absorption Coefficient and Dipole–Dipole Correlation Function 2696.3.1 Absorption Coefficient and Wave Packet Propagation 2696.3.2 Absorption Coefficient and Reduced Density Operator Propagation 2736.3.3 Mixed Quantum–Classical Computation of the Absorption Coefficient 2756.4 The Emission Spectrum 2806.5 Optical Preparation of an Excited Electronic State 2816.5.1 Wave Function Formulation 2816.5.1.1 Case of Short Pulse Duration 2846.5.1.2 Case of Long Pulse Duration 2846.5.2 Density Matrix Formulation 2846.6 Internal Conversion Dynamics 2866.6.1 The Internal Conversion Rate 2876.6.2 Ultrafast Internal Conversion 2886.7 Supplement 2906.7.1 Absorption Coefficient for Displaced Harmonic Oscillators 290References 294Further Reading 2947 Electron Transfer 2957.1 Classification of Electron Transfer Reactions 2957.2 Theoretical Models for Electron Transfer Systems 3057.2.1 The Electron Transfer Hamiltonian 3057.2.2 The Electron–Vibrational Hamiltonian of a Donor–Acceptor Complex 3107.2.2.1 The Spin-Boson Model 3127.2.2.2 Two Independent Sets of Vibrational Coordinates 3137.2.3 Electron–Vibrational State Representation of the Hamiltonian 3147.3 Regimes of Electron Transfer 3157.3.1 Landau–Zener Theory of Electron Transfer 3197.4 Nonadiabatic Electron Transfer in a Donor–Acceptor Complex 3237.4.1 High-temperature Case 3237.4.2 High-temperature Case: Two Independent Sets of Vibrational Coordinates 3277.4.3 Low-temperature Case: Nuclear Tunneling 3307.4.4 The Mixed Quantum–Classical Case 3337.4.5 Description of the Mixed Quantum–Classical Case by a Spectral Density 3357.5 Bridge-Mediated Electron Transfer 3367.5.1 The Superexchange Mechanism 3387.5.2 Electron Transfer Through Arbitrary Large Bridges 3407.5.2.1 Case of Small Intrabridge Transfer Integrals 3407.5.2.2 Case of Large Intrabridge Transfer Integrals 3417.6 Nonequilibrium Quantum Statistical Description of Electron Transfer 3437.6.1 Unified Description of Electron Transfer in a Donor–Bridge–Acceptor System 3447.6.2 Transition to the Adiabatic Electron Transfer 3477.7 Heterogeneous Electron Transfer 3477.7.1 Nonadiabatic Charge Injection into the Solid State Described in a Single-Electron Model 3487.7.1.1 Low-temperature Case 3517.7.1.2 High-temperature Case 3527.7.1.3 HET-induced Lifetime 3527.7.2 Ultrafast Photoinduced HET from a Molecule into a Semiconductor. A Case Study 3547.7.3 Nonadiabatic Electron Transfer from the Solid State into the Molecule 3557.8 Charge Transmission Through Single Molecules 3567.8.1 Inelastic Charge Transmission 3597.8.1.1 An Example 3607.8.2 Elastic Charge Transmission 3617.8.2.1 An Example 3647.8.2.2 Inclusion of Vibrational Levels 3657.9 Photoinduced Ultrafast Electron Transfer 3677.9.1 Quantum Master Equation for Electron Transfer Reactions 3727.9.2 Rate Expressions 3777.10 Supplement 3787.10.1 Landau–Zener Transition Amplitude 3787.10.2 The Multimode Marcus Formula 3797.10.3 Second-order Electron Transfer Rate 3807.10.4 Fourth-order Donor–Acceptor Transition Rate 3827.10.5 Rate of Elastic Charge Transmission Through a Single Molecule 385References 387Further Reading 3888 Proton Transfer 3898.1 Introduction 3898.2 Proton Transfer Hamiltonian 3958.2.1 Hydrogen Bonds 3958.2.2 Reaction Surface Hamiltonian for Intramolecular Proton Transfer 3998.2.3 Tunneling Splittings 4008.2.4 The Proton Transfer Hamiltonian in the Condensed Phase 4048.2.4.1 Adiabatic Representation 4058.2.4.2 Diabatic Representation 4068.3 Adiabatic Proton Transfer 4078.4 Nonadiabatic Proton Transfer 4108.5 The Intermediate Regime: From Quantum to Quantum–Classical Hybrid Methods 4128.5.1 Multidimensional Wave Packet Dynamics 4138.5.2 Surface Hopping 4158.6 Proton-coupled Electron Transfer 417References 419Further Reading 4199 Excitation Energy Transfer 4219.1 Introduction 4219.2 The Aggregate Hamiltonian 4279.2.1 The Intermolecular Coulomb Interaction 4309.2.1.1 Dipole–Dipole Coupling 4329.2.2 The Two-level Model 4339.2.2.1 Classification of the Coulomb Interactions 4339.2.3 Single and Double Excitations of the Aggregate 4369.2.3.1 The Ground State Matrix Element 4389.2.3.2 The Single Excited State Matrix Elements 4389.2.3.3 The Double Excited State Matrix Elements 4399.2.3.4 Off-Diagonal Matrix Elements and Coupling to the Radiation Field 4409.2.3.5 Neglect of Intermolecular Electrostatic Coupling 4419.2.4 Introduction of Delocalized Exciton States 4419.2.4.1 The Molecular Heterodimer 4439.2.4.2 The Finite Molecular Chain and the Molecular Ring 4439.3 Exciton–Vibrational Interaction 4449.3.1 Exclusive Coupling to Intramolecular Vibrations 4459.3.2 Coupling to Aggregate Normal Mode Vibrations 4489.3.3 Differentiating Between Intramolecular and Reservoir Normal Mode Vibrations 4499.3.4 Exciton–Vibrational Hamiltonian and Excitonic Potential Energy Surfaces 4499.4 Regimes of Excitation Energy Transfer 4509.4.1 Quantum Statistical Approaches to Excitation Energy Transfer 4529.5 Transfer Dynamics in the Case of Weak Excitonic Coupling: Förster Theory 4539.5.1 The Transfer Rate 4549.5.2 The Förster Rate 4569.5.3 Nonequilibrium Quantum Statistical Description of Förster Transfer 4589.5.3.1 Case of Common Vibrational Coordinates 4629.5.3.2 Case of Vibrational Modulation of the Excitonic Coupling 4649.6 Transfer Dynamics in the Case of Strong Excitonic Coupling 4659.6.1 Rate Equations for Exciton Dynamics 4659.6.2 Density Matrix Equations for Exciton Dynamics 4669.6.3 Site Representation 4689.6.4 Excitation Energy Transfer Among Different Aggregates 4719.6.5 Exciton Transfer in the Case of Strong Exciton–Vibrational Coupling 4729.6.6 Nonperturbative and Non-Markovian Exciton Dynamics 4759.7 Optical Properties of Aggregates 4779.7.1 Case of No Exciton–Vibrational Coupling 4799.7.1.1 Static Disorder 4819.7.2 Inclusion of Exciton–Vibrational Coupling 4849.7.2.1 The n-Particle Expansion 4849.7.2.2 Weak Exciton–Vibrational Coupling 4879.7.2.3 Strong Exciton–Vibrational Coupling 4889.8 Excitation Energy Transfer Including Charge-transfer States 4909.8.1 Excitation Energy Transfer Via Two-electron Exchange 4909.8.2 Charge-transfer Excitons and Charge Separation 4939.9 Exciton–Exciton Annihilation 4969.9.1 Three-level Description of the Molecules in the Aggregate 4989.9.2 The Rate of Exciton–Exciton Annihilation 4999.10 Supplement 5009.10.1 Second Quantization Notation of the Aggregate Hamiltonian 5009.10.2 Photon-mediated Long-range Excitation Energy Transfer 5019.10.2.1 Preparatory Considerations for the Rate Computation 5039.10.2.2 Photon Correlation Functions 5059.10.2.3 The Rate of Photon-mediated Excitation Energy Transfer 5069.10.2.4 Some Estimates 5089.10.3 Fourth-order Rate of Two-electron-transfer-assisted EET 509References 513Further Reading 514Index 515