About this title: This text provides a uniform and consistent approach to diversified problems encountered in the study of dynamical processes in condensed phase molecular systems. Given the broad interdisciplinary aspect of this subject, the book focuses on three themes: coverage of needed background material, <br>in-depth introduction of methodologies, and analysis of several key applications. The uniform approach and common language used in all discussions help to develop general understanding and insight on condensed phases chemical dynamics. The applications discussed are among the most fundamental<br>processes that underlie physical, chemical and biological phenomena in complex systems.<br><br>The first part of the book starts with a general review of basic mathematical and physical methods (Chapter 1) and a few introductory chapters on quantum dynamics (Chapter 2), interaction of radiation and matter (Chapter 3) and basic properties of solids (chapter 4) and liquids (Chapter 5). In the<br>second part the text embarks on a broad coverage of the main methodological approaches. The central role of classical and quantum time correlation functions is emphasized in Chapter 6. The presentation of dynamical phenomena in complex systems as stochastic processes is discussed in Chapters 7 and<br>8. The basic theory of quantum relaxation phenomena is developed in Chapter 9, and carried on in Chapter 10 which introduces the density operator, its quantum evolution in Liouville space, and the concept of reduced equation of motions. The methodological part concludes with a discussion of linear<br>response theory in Chapter 11, and of the spin-boson model in chapter 12. Thethird part of the book applies the methodologies introduced earlier to several fundamental processes that underlie much of the dynamical behaviour of condensed phase molecular systems. Vibrational relaxation and<br>vibrational energy transfer (Chapter 13), Barrier crossing and diffusion controlled reactions (Chapter 14), solvation dynamics (Chapter 15), electron transfer in bulk solvents (Chapter 16) and at electrodes/electrolyte and metal/molecule/metal junctions (Chapter 17), and several processes pertaining<br>to molecular spectroscopy in condensed phases (Chapter 18) are the main subjects discussed in this part.
Table Of Contents
PART I BACKGROUND
1 Review of some mathematical and physical subjects
1.1 Mathematical background
1.1.1 Random variables and probability distributions
1.1.2 Constrained extrema
1.1.3 Vector and fields
1.1.4 Continuity equation for the flow of conserved entities
1.1.5 Delta functions
1.1.6 Complex integration
1.1.7 Laplace transform
1.1.8 The Schwarz inequality
1.2 Classical mechanics
1.2.1 Classical equations of motion
1.2.2 Phase space, the classical distribution function, and the Liouville equation
1.3 Quantum mechanics
1.4 Thermodynamics and statistical mechanics
1.4.1 Thermodynamics
1.4.2 Statistical mechanics
1.4.3 Quantum distributions
1.4.4 Coarse graining
1.5 Physical observables as random variables
1.5.1 Origin of randomness in physical systems
1.5.2 Joint probabilities, conditional probabilities, and reduced descriptions
1.5.3 Random functions
1.5.4 Correlations
1.5.5 Diffusion
1.6 Electrostatics
1.6.1 Fundamental equations of electrostatics
1.6.2 Electrostatics in continuous dielectric media
1.6.3 Screening by mobile charges
2 Quantum dynamics using the time-dependent Schr?inger equation
2.1 Formal solutions
2.2 An example: The two-level system
2.3 Time-dependent Hamiltonians
2.4 A two-level system in a time-dependent field
2.5 A digression on nuclear potential surfaces
2.6 Expressing the time evolution in terms of the Green's operator
2.7 Representations
2.7.1 The Schr?inger and Heisenberg representations
2.7.2 The interaction representation
2.7.3 Time-dependent perturbation theory
2.8 Quantum dynamics of the free particles
2.8.1 Free particle eigenfunctions
2.8.2 Free particle density of states
2.8.3 Time evolution of a one-dimensional free particle wavepacket
2.8.4 The quantum mechanical flux
2.9 Quantum dynamics of the harmonic oscillator
2.9.1 Elementary considerations
2.9.2 The raising/lowering operators formalism
2.9.3 The Heisenberg equations of motion
2.9.4 The shifted harmonic oscillator
2.9.5 Harmonic oscillator at thermal equilibrium
2.10 Tunneling
2.10.1 Tunneling through a square barrier
2.10.2 Some observations
2A Some operator identities
3 An Overview of Quantum Electrodynamics and Matter?adiation Field Interaction
3.1 Introduction
3.2 The quantum radiation field
3.2.1 Classical electrodynamics
3.2.2 Quantum electrodynamics
3.2.3 Spontaneous emission
3A The radiation field and its interaction with matter
4 Introduction to solids and their interfaces
4.1 Lattice periodicity
4.2 Lattice vibrations
4.2.1 Normal modes of harmonic systems
4.2.2 Simple harmonic crystal in one dimension
4.2.3 Density of modes
4.2.4 Phonons in higher dimensions and the heat capacity of solids
4.3 Electronic structure of solids
4.3.1 The free electron theory of metals: Energetics
4.3.2 The free electron theory of metals: Motion
4.3.3 Electronic structure of periodic solids: Bloch theory
4.3.4 The one-dimensional tight binding model
4.3.5 The nearly free particle model
4.3.6 Intermediate summary: Free electrons versus noninteracting electrons in a periodic potential
4.3.7 Further dynamical implications of the electronic band structure of solids
4.3.8 Semiconductors
4.4 The work function
4.5 Surface potential and screening
4.5.1 General considerations
4.5.2 The Thomas?ermi theory of screening by metallic electrons
4.5.3 Semiconductor interfaces
4.5.4 Interfacial potential distributions
5 Introduction to liquids
5.1 Statistical mechanics of classical liquids
5.2 Time and ensemble average
5.3 Reduced configurational distribution functions
5.4 Observable implications of the pair correlation function
5.4.1 X-ray scattering
5.4.2 The average energy
5.4.3 Pressure
5.5 The potential of mean force and the reversible work theorem
5.6 The virial expansion?he second virial coefficient
Part II METHODS
6 Time correlation functions
6.1 Stationary systems
6.2 Simple examples
6.2.1 The diffusion coefficient
6.2.2 Golden rule rates
6.2.3 Optical absorption lineshapes
6.3 Classical time correlation functions
6.4 Quantum time correlation functions
6.5 Harmonic reservoir
6.5.1 Classical bath
6.5.2 The spectral density
6.5.3 Quantum bath
6.5.4 Why are harmonic baths models useful?
7 Introduction to stochastic processes
7.1 The nature of stochastic processes
7.2 Stochastic modeling of physical processes
7.3 The random walk problem
7.3.1 Time evolution
7.3.2 Moments
7.3.3 The probability distribution
7.4 Some concepts from the general theory of stochastic processes
7.4.1 Distributions and correlation functions
7.4.2 Markovian stochastic processes
7.4.3 Gaussian stochastic processes
7.4.4 A digression on cumulant expansions
7.5 Harmonic analysis
7.5.1 The power spectrum
7.5.2 The Wiener?hintchine theorem
7.5.3 Application to absorption
7.5.4 The power spectrum of a randomly modulated harmonic oscillator
7A Moments of the Gaussian distribution
7B Proof of Eqs (7.64) and (7.65)
7C Cumulant expansions
7D Proof of the Wiener?hintchine theorem
8 Stochastic equations of motion
8.1 Introduction
8.2 The Langevin equation
8.2.1 General considerations
8.2.2 The high friction limit
8.2.3 Harmonic analysis of the Langevin equation
8.2.4 The absorption lineshape of a harmonic oscillator
8.2.5 Derivation of the Langevin equation from a microscopic model
8.2.6 The generalized Langevin equation
8.3 Master equations
8.3.1 The random walk problem revisited
8.3.2 Chemical kinetics
8.3.3 The relaxation of a system of harmonic oscillators
8.4 The Fokker?lanck equation
8.4.1 A simple example
8.4.2 The probability flux
8.4.3 Derivation of the Fokker?lanck equation from the Chapman?olmogorov equation
8.4.4 Derivation of the Smoluchowski equation from the Langevin equation: The overdamped limit
8.4.5 Derivation of the Fokker?lanck equation from the Langevin equation
8.4.6 The multidimensional Fokker?lanck equation
8.5 Passage time distributions and the mean first passage time
8A Obtaining the Fokker?lanck equation from the Chapman?olmogorov equation
8B Obtaining the Smoluchowski equation from the overdamped Langevin equation
8C Derivation of the Fokker?lanck equation from the Langevin equation
9 Introduction to quantum relaxation processes
9.1 A simple quantum-mechanical model for relaxation
9.2 The origin of irreversibility
9.2.1 Irreversibility reflects restricted observation
9.2.2 Relaxation in isolated molecules
9.2.3 Spontaneous emission
9.2.4 Preparation of the initial state
9.3 The effect of relaxation on absorption lineshapes
9.4 Relaxation of a quantum harmonic oscillator
9.5 Quantum mechanics of steady states
9.5.1 Quantum description of steady-state processes
9.5.2 Steady-state absorption
9.5.3 Resonance tunneling
9A Using projection operators
9B Evaluation of the absorption lineshape for the model of Figs 9.2 and 9.3
9C Resonance tunneling in three dimensions
10 Quantum mechanical density operator
10.1 The density operator and the quantum Liouville equation
10.1.1 The density matrix for a pure system
10.1.2 Statistical mixtures
10.1.3 Representations
10.1.4 Coherences
10.1.5 Thermodynamic equilibrium
10.2 An example: The time evolution of a two-level system in the density matrix formalism
10.3 Reduced descriptions
10.3.1 General considerations
10.3.2 A simple example the quantum mechanical basis for macroscopic rate equations
10.4 Time evolution equations for reduced density operators: The quantum master equation
10.4.1 Using projection operators
10.4.2 The Nakajima?wanzig equation
10.4.3 Derivation of the quantum master equation using the thermal projector
10.4.4 The quantum master equation in the interaction representation
10,4.5 The quantum master equation in the Schr?inger representation
10.4.6 A pause for reflection
10.4.7 System-states representation
10.4.8 The Markovian limit the Redfield equation
10.4.9 Implications of the Redfield equation
10.4.10 Some general issues
10.5 The two-level system revisited
10.5.1 The two-level system in a thermal environment
10.5.2 The optically driven two-level system in a thermal environment the Bloch equations
10A Analogy of a coupled 2-level system to a spin 2system in a magnetic field
11 Linear response theory
11.1 Classical linear response theory
11.1.1 Static response
11.1.2 Relaxation
11.1.3 Dynamic response
11.2 Quantum linear response theory
11.2.1 Static quantum response
11.2.2 Dynamic quantum response
11.2.3 Causality and the Kramers?ronig relations
11.2.4 Examples: mobility, conductivity, and diffusion
11A The Kubo identity
12 The Spin?oson Model
12.1 Introduction
12.2 The model
12.3 The polaron transformation
12.3.1 The Born Oppenheimer picture
12.4 Golden-rule transition rates
12.4.1 The decay of an initially prepared level
12.4.2 The thermally averaged rate
12.4.3 Evaluation of rates
12.5 Transition between molecular electronic states
12.5.1 The optical absorption lineshape
12.5.2 Electronic relaxation of excited molecules
12.5.3 The weak coupling limit and the energy gap law
12.5.4 The thermal activation/potential-crossing limit
12.5.5 Spin?attice relaxation
12.6 Beyond the golden rule
Part III APPLICATIONS
13 Vibrational energy relaxation
13.1 General observations
13.2 Construction of a model Hamiltonian
13.3 The vibrational relaxation rate
13.4 Evaluation of vibrational relaxation rates
13.4.1 The bilinear interaction model
13.4.2 Nonlinear interaction models
13.4.3 The independent binary collision (IBC) model
13.5 Multi-phonon theory of vibrational relaxation
13.6 Effect of supporting modes
13.7 Numerical simulations of vibrational relaxation
13.8 Concluding remarks
14 Chemical reactions in condensed phases
14.1 Introduction
14.2 Unimolecular reactions
14.3 Transition state theory
14.3.1 Foundations of TST
14.3.2 Transition state rate of escape from a one-dimensional well
14.3.3 Transition rate for a multidimensional system
14.3.4 Some observations
14.3.5 TST for nonadiabatic transitions
14.3.6 TST with tunneling
14.4 Dynamical effects in barrier crossing?he Kramers model
14.4.1 Escape from a one-dimensional well
14.4.2 The overdamped case
14.4.3 Moderate-to-large damping
14.4.4 The low damping limit
14.5 Observations and extensions
14.5.1 Implications and shortcomings of the Kramers theory
14.5.2 Non-Markovian effects
14.5.3 The normal mode representation
14.6 Some experimental observations
14.7 Numerical simulation of barrier crossing
14.8 Diffusion-controlled reactions
14A Solution of Eqs (14.62) and (14.63)
14B Derivation of the energy Smoluchowski equation
15 Solvation dynamics
15.1 Dielectric solvation
15.2 Solvation in a continuum dielectric environment
15.2.1 General observations
15.2.2 Dielectric relaxation and the Debye model
15.3 Linear response theory of solvation
15.4 More aspects of solvation dynamics
15.5 Quantum solvation
16 Electron transfer processes
16.1 Introduction
16.2 A primitive model
16.3 Continuum dielectric theory of electron transfer processes
16.3.1 The problem
16.3.2 Equilibrium electrostatics
16.3.3 Transition assisted by dielectric fluctuations
16.3.4 Thermodynamics with restrictions
16.3.5 Dielectric fluctuations
16.3.6 Energetics of electron transfer between two ionic centers
16.3.7 The electron transfer rate
16.4 A molecular theory of the nonadiabatic electron transfer rate
16.5 Comparison with experimental results
16.6 Solvent-controlled electron transfer dynamics
16.7 A general expression for the dielectric reorganization energy
16.8 The Marcus parabolas
16.9 Harmonic field representation of dielectric response
16.10 The nonadiabatic coupling
16.11 The distance dependence of electron transfer rates
16.12 Bridge-mediated long-range electron transfer
16.13 Electron tranport by hopping
16.14 Proton transfer
16A Derivation of the Mulliken?ush formula
17 Electron transfer and transmission at molecule?etal and molecule?emiconductor interfaces
17.1 Electrochemical electron transfer
17.1.1 Introduction
17.1.2 The electrochemical measurement
17.1.3 The electron transfer process
17.1.4 The nuclear reorganization
17.1.5 Dependence on the electrode potential: Tafel plots
17.1.6 Electron transfer at the semiconductor?lectrolyte interface
17.2 Molecular conduction
17.2.1 Electronic structure models of molecular conduction
17.2.2 Conduction of a molecular junction
17.2.3 The bias potential
17.2.4 The one-level bridge model
17.2.5 A bridge with several independent levels
17.2.6 Experimental statistics
17.2.7 The tight-binding bridge model
18 Spectroscopy
18.1 Introduction
18.2 Molecular spectroscopy in the dressed-state picture
18.3 Resonance Raman scattering
18.4 Resonance energy transfer
18.5 Thermal relaxation and dephasing
18.5.1 The Bloch equations
18.5.2 Relaxation of a prepared state
18.5.3 Dephasing (decoherence)
18.5.4 The absorption lineshape
18.5.5 Homogeneous and inhomogeneous broadening
18.5.6 Motional narrowing
18.5.7 Thermal effects in resonance Raman scattering
18.5.8 A case study: Resonance Raman scattering and fluorescence from Azulene in a Naphtalene matrix
18.6 Probing inhomogeneous bands
18.6.1 Hole burning spectroscopy
18.6.2 Photon echoes
18.6.3 Single molecule spectroscopy
18.7 Optical response functions
18.7.1 The Hamiltonian
18.7.2 Response functions at the single molecule level
18.7.3 Many body response theory
18.7.4 Independent particles
18.7.5 Linear response
18.7.6 Linear response theory of propagation and absorption
18A Steady-state solution of Eqs (18.58): the Raman scattering flux
Index
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