Spectroscopy

The interaction of light with matter provides some of the most important tools for studying structure and dynamics on the microscopic scale. Atomic and molecular spectroscopy in the low pressure gas phase probes this interaction essentially on the single particle level and yields information about energy levels, state symmetries, and intramolecular potential surfaces. Understanding environmental effects in spectroscopy is important both as a fundamental problem in quantum statistical mechanics and as a prerequisite to the intelligent use of spectroscopic tools to probe and analyze molecular interactions and processes in condensed phases. Spectroscopic observables can be categorized in several ways. We can follow a temporal profile or a frequency resolved spectrum; we may distinguish between observables that reflect linear or nonlinear response to the probe beam; we can study different energy domains and different timescales and we can look at resonant and nonresonant response. This chapter discusses some concepts, issues, and methodologies that pertain to the effect of a condensed phase environment on these observables. For an in-depth look at these issues the reader may consult many texts that focus on particular spectroscopies. With focus on the optical response of molecular systems, effects of condensed phase environments can be broadly discussed within four categories: 1. Several important effects are equilibrium in nature, for example spectral shifts associated with solvent induced changes in solute energy levels are equilibrium properties of the solvent–solute system. Obviously, such observables may themselves be associated with dynamical phenomena, in the example of solvent shifts it is the dynamics of solvation that affects their dynamical evolution. Another class of equilibrium effects on radiation– matter interaction includes properties derived from symmetry rules. A solvent can affect a change in the equilibrium configuration of a chromophore solute and consequently the associated selection rules for a given optical transition. Some optical phenomena are sensitive to the symmetry of the environment, for example, surface versus bulk geometry. 2. The environment affects the properties of the radiation field; the simplest example is the appearance of the dielectric coefficient ε in the theory of radiation–matter interaction.

Chemical Reactions In Condensed Phases

Understanding chemical reactions in condensed phases is essentially the understanding of solvent effects on chemical processes. Such effects appear in many ways. Some stem from equilibrium properties, for example, solvation energies and free energy surfaces. Others result from dynamical phenomena: solvent effect on diffusion of reactants toward each other, dynamical cage effects, solvent-induced energy accumulation and relaxation, and suppression of dynamical change in molecular configuration by solvent induced friction. In attempting to sort out these different effects it is useful to note that a chemical reaction proceeds by two principal dynamical processes that appear in three stages. In the first and last stages the reactants are brought together and products are separated from each other. In the middle stage the assembled chemical system undergoes the structural/chemical change. In a condensed phase the first and last stages involve diffusion, sometimes (e.g. when the species involved are charged) in a force field. The middle stage often involves the crossing of a potential barrier. When the barrier is high the latter process is rate-determining. In unimolecular reactions the species that undergoes the chemical change is already assembled and only the barrier crossing process is relevant. On the other hand, in bi-molecular reactions with low barrier (of order kBT or less), the rate may be dominated by the diffusion process that brings the reactants together. It is therefore meaningful to discuss these two ingredients of chemical rate processes separately. Most of the discussion in this chapter is based on a classical mechanics description of chemical reactions. Such classical pictures are relevant to many condensed phase reactions at and above room temperature and, as we shall see, can be generalized when needed to take into account the discrete nature of molecular states. In some situations quantum effects dominate and need to be treated explicitly. This is the case, for example, when tunneling is a rate determining process. Another important class is nonadiabatic reactions, where the rate determining process is hopping (curve crossing) between two electronic states. Such reactions are discussed in Chapter 16.

Linear Response Theory

Equilibrium statistical mechanics is a first principle theory whose fundamental statements are general and independent of the details associated with individual systems. No such general theory exists for nonequilibrium systems and for this reason we often have to resort to ad hoc descriptions, often of phenomenological nature, as demonstrated by several examples in Chapters 7 and 8. Equilibrium statistical mechanics can however be extended to describe small deviations from equilibrium in a way that preserves its general nature. The result is Linear Response Theory, a statistical mechanical perturbative expansion about equilibrium. In a standard application we start with a system in thermal equilibrium and attempt to quantify its response to an applied (static- or time-dependent) perturbation. The latter is assumed small, allowing us to keep only linear terms in a perturbative expansion. This leads to a linear relationship between this perturbation and the resulting response. Let us make these statements more quantitative. Consider a system characterized by the Hamiltonian Ĥ0.

Time Correlation Functions

In the previous chapter we have seen how spatial correlation functions express useful structural information about our system. This chapter focuses on time correlation functions that, as will be seen, convey important dynamical information. Time correlation functions will repeatedly appear in our future discussions of reduced descriptions of physical systems. A typical task is to derive dynamical equations for the time evolution of an interesting subsystem, in which only relevant information about the surrounding thermal environment (bath) is included. We will see that dynamic aspects of this relevant information usually enter via time correlation functions involving bath variables. Another type of reduction aims to derive equations for the evolution of macroscopic variables by averaging out microscopic information. This leads to kinetic equations that involve rates and transport coefficients, which are also expressed as time correlation functions of microscopic variables. Such functions are therefore instrumental in all discussions that relate macroscopic dynamics to microscopic equations of motion. It is important to keep in mind that dynamical properties are not exclusively relevant only to nonequilibrium system. One may naively think that dynamics is unimportant at equilibrium because in this state there is no evolution on the average. Indeed in such systems all times are equivalent, in analogy to the fact that in spatially homogeneous systems all positions are equivalent. On the other hand, just as in the previous chapter we analyzed equilibrium structures by examining correlations between particles located at different spatial points, also here we can gain dynamical information by looking at the correlations between events that occur at different temporal points. Time correlation functions are our main tools for conveying this information in stationary systems. These are systems at thermodynamic equilibrium or at steady state with steady fluxes present.

Introduction To Solids And Their Interfaces

The study of dynamics of molecular processes in condensed phases necessarily involves properties of the condensed environment that surrounds the system under consideration. This chapter provides some essential background on the properties of solids while the next chapter does the same for liquids. No attempt is made to provide a comprehensive discussion of these subjects. Rather, this chapter only aims to provide enough background as needed in later chapters in order to take into consideration two essential attributes of the solid environment: Its interaction with the molecular system of interest and the relevant timescales associated with this interaction. This would entail the need to have some familiarity with the relevant degrees of freedom, the nature of their interaction with a guest molecule, the corresponding densities of states or modes, and the associated characteristic timescales. Focusing on the solid crystal environment we thus need to have some understanding of its electronic and nuclear dynamics. The geometry of a crystal is defined with respect to a given lattice by picturing the crystal as made of periodically repeating unit cells. The atomic structure within the cell is a property of the particular structure (e.g. each cell can contain one or more molecules, or several atoms arranged within the cell volume in some given way), however, the cells themselves are assigned to lattice points that determine the periodicity. This periodicity is characterized by three lattice vectors, ai, i = 1, 2, 3, that determine the primitive lattice cell—a parallelepiped defined by these three vectors.

Electron Transfer and Transmission at Molecule–Metal and Molecule–Semiconductor Interfaces

This chapter continues our discussion of electron transfer processes, now focusing on the interface between molecular systems and solid conductors. Interest in such processes has recently surged within the emerging field of molecular electronics, itself part of a general multidisciplinary effort on nanotechnology. Notwithstanding new concepts, new experimental and theoretical methods, and new terminology, the start of this interest dates back to the early days of electrochemistry, marked by the famous experiments of Galvani and Volta in the late eighteenth century. The first part of this chapter discusses electron transfer in what might now be called “traditional” electrochemistry where the fundamental process is electron transfer between a molecule or a molecular ion and a metal electrode. The second part constitutes an introduction to molecular electronics, focusing on the problem of molecular conduction, which is essentially electron transfer (in this context better termed electron transmission) between two metal electrodes through a molecular layer or sometimes even a single molecule. In Chapter 16 we have focused on electron transfer processes of the following characteristics: (1) Two electronic states, one associated with the donor species, the other with the acceptor, are involved. (2) Energetics is determined by the electronic energies of the donor and acceptor states and by the electrostatic solvation of the initial and final charge distributions in their electronic and nuclear environments. (3) The energy barrier to the transfer process originates from the fact that electronic and nuclear motions occur on vastly different timescales. (4) Irreversibility is driven by nuclear relaxation about the initial and final electronic charge distributions. How will this change if one of the two electronic species is replaced by a metal? We can imagine an electron transfer process between a metal substrate and a molecule adsorbed on its surface, however the most common process of this kind takes place at the interface between a metal electrode and an electrolyte solution, where the molecular species is an ion residing in the electrolyte, near the metal surface. Electron transfer in this configuration is the fundamental process of electrochemistry.

The Quantum Mechanical Density Operator And Its Time Evolution: Quantum Dynamics Using The Quantum Liouville Equation

The starting point of the classical description of motion is the Newton equations that yield a phase space trajectory (rN (t), pN (t)) for a given initial condition (rN (0), pN (0)). Alternatively one may describe classical motion in the framework of the Liouville equation (Section (1.2.2)) that describes the time evolution of the phase space probability density f (rN , pN ; t). For a closed system fully described in terms of a well specified initial condition, the two descriptions are completely equivalent. Probabilistic treatment becomes essential in reduced descriptions that focus on parts of an overall system, as was demonstrated in Sections 5.1–5.3 for equilibrium systems, and in Chapters 7 and 8 that focus on the time evolution of classical systems that interact with their thermal environments. This chapter deals with the analogous quantum mechanical problem. Within the limitations imposed by its nature as expressed, for example, by Heisenbergtype uncertainty principles, the Schrödinger equation is deterministic. Obviously it describes a deterministic evolution of the quantum mechanical wavefunction. The analog of the phase space probability density f (rN , pN ; t) is now the quantum mechanical density operator (often referred to as the “density matrix”), whose time evolution is determined by the quantum Liouville equation. Again, when the system is fully described in terms of a well specified initial wavefunction, the two descriptions are equivalent. The density operator formalism can, however, be carried over to situations where the initial state of the system is not well characterized and/or a reduced description of part of the overall system is desired. Such situations are considered later in this chapter.

Stochastic Equations Of Motion

We have already observed that the full phase space description of a system of N particles (taking all 6N coordinates and velocities into account) requires the solution of the deterministic Newton (or Schrödinger) equations of motion, while the time evolution of a small subsystem is stochastic in nature. Focusing on the latter, we would like to derive or construct appropriate equations of motion that will describe this stochastic motion. This chapter discusses some methodologies used for this purpose, focusing on classical mechanics as the underlying dynamical theory. In Chapter 10 we will address similar issues in quantum mechanics. The time evolution of stochastic processes can be described in two ways: 1. Time evolution in probability space. In this approach we seek an equation (or equations) for the time evolution of relevant probability distributions. In the most general case we deal with an infinite hierarchy of functions, P(zntn; zn−1tn−1; . . . ; z1t1) as discussed in Section 7.4.1, but simpler cases exist, for example, for Markov processes the evolution of a single function, P(z, t; z0t0), fully characterizes the stochastic dynamics. Note that the stochastic variable z stands in general for all the variables that determine the state of our system. 2. Time evolution in variable space. In this approach we seek an equation of motion that describes the evolution of the stochastic variable z(t) itself (or equations of motion for several such variables). Such equations of motions will yield stochastic trajectories z(t) that are realizations of the stochastic process under study. The stochastic nature of these equations is expressed by the fact that for any initial condition z0 at t = t0 they yield infinitely many such realizations in the same way that measurements of z(t) in the laboratory will yield different such realizations. Two routes can be taken to obtain such stochastic equations of motions, of either kind: 1. Derive such equations from first principles. In this approach, we start with the deterministic equations of motion for the entire system, and derive equations of motion for the subsystem of interest. The stochastic nature of the latter stems from the fact that the state of the complementary system, “the rest of the world,” is not known precisely, and is given only in probabilistic terms.

Vibrational Energy Relaxation

An impurity molecule located as a solute in a condensed solvent, a solid matrix or a liquid, when put in an excited vibrational state will loose its excess energy due to its interaction with the surrounding solvent molecules. Vibrational energy accumulation is a precursor to all thermal chemical reactions. Its release by vibrational relaxation following a reactive barrier crossing or optically induced reaction defines the formation of a product state. The direct observation of this process by, for example, infrared emission or more often laser induced fluorescence teaches us about its characteristic timescales and their energetic (i.e. couplings and frequencies) origin. These issues are discussed in this chapter. Before turning to our main task, which is constructing and analyzing a model for vibrational relaxation in condensed phases, we make some general observations about this process. In particular we will contrast condensed phase relaxation with its gas phase counterpart and will comment on the different relaxation pathways taken by diatomic and polyatomic molecules. First, vibrational relaxation takes place also in low density gases. Collisions involving the vibrationally excited molecule may result in transfer of the excess vibrational energy to rotational and translational degrees of freedom of the overall system. Analysis based on collision theory, with the intermolecular interaction potential as input, then leads to the cross-section for inelastic collisions in which vibrational and translational/rotational energies are exchanged. If C∗ is the concentration of vibrationally excited molecules and ρ is the overall gas density, the relaxation rate coefficient kgas is defined from the bimolecular rate law When comparing this relaxation to its condensed phase counterpart one should note a technical difference between the ways relaxation rates are defined in the two phases.

An Overview Of Quantum Electrodynamics And Matter–Radiation Field Interaction

Many dynamical processes of interest are either initiated or probed by light, and their understanding requires some knowledge of this subject. This chapter is included in order to make this text self contained by providing an overview of subjects that are used in various applications later in the text. In particular, it aims to supplement the elementary view of radiation–matter interaction as a time-dependent perturbation in the Hamiltonian, by describing some aspects of the quantum nature of the radiation field. This is done on two levels: The main body of this chapter is an essentially qualitative overview that ends with a treatment of spontaneous emission as an example. The Appendix gives some more details on the mathematical structure of the theory. In elementary treatments of the interaction of atoms and molecules with light, the radiation field is taken as a classical phenomenon.