S. A. Adelman

Generalized Langevin equation approach for atom/solid‐surface scattering: General formulation for classical scattering off harmonic solids

A general theoretical framework for introducing many‐body or lattice effects into gas/solid scattering is presented. The theory is presently restricted to classical scattering off harmonic lattices but is otherwise completely general. It is nonperturbative and valid for arbitrary lattice temperature. The theory is based on a formulation of lattice dynamics suggested by and related to the Kubo–Mori theory of generalized Brownian motion. This formulation leads to a generalized Langevin equation (GLE) in which only the coordinates of the gas atom and the n∼1–6 surface atoms directly struck by the gas atom appear explicitly. The remainder of the lattice, which functions as a harmonic heat bath, affects the collision through a friction kernel and a Gaussian random force appearing in the GLE. The GLE can be solved in terms of a tractable number of (n+1) ‐particle gas–surface trajectories using approximate stochastic techniques. Stochastic solution yields thermally averaged temporal gas particle probability distribution functions (pdf). From the long time limit of these pdf’s all temperature dependent gas–surface cross sections can be found. In the limit of zero friction, the theory gives a convenient method for calculating atom–oscillator thermally averaged cross sections which circumvents laborious Monte Carlo classical trajectory sampling and which can be generalized to treat other gas phase collision problems.

Fokker–Planck equations for simple non‐Markovian systems

Exact generalized Fokker–Planck equations are derived from the linear Mori–Kubo generalized Langevin equation for the case of Gaussian but non‐Markovian noise. Fokker–Planck equations which generate the momentum and phase space probability distribution functions (pdf’s) for free Brownian particles and the phase space pdf for Brownian oscillators are presented. Also given is the generalized diffusion equation for the free Brownian particle pdf in the zero inertia limit. The generalized Fokker–Planck equations are similar in structure to the corresponding phenomenological equations. They, however, involve time‐dependent friction and frequency functions rather than phenomenological constants. Explicit results for the frequency and friction functions are given for the Debye solid model. These functions enter as simple multiplicative factors rather than as retarded kernels. Further the phase space Fokker–Planck equations contain an extra diffusive term, a mixed phase space second partial derivative, not occurr...

Generalized Langevin theory for many‐body problems in chemical dynamics: General formulation and the equivalent harmonic chain representation

A comprehensive theoretical framework for classical trajectory simulations of many‐body chemical processes is presented. This framework generalizes previous theoretical methods designed to treat the many‐body problems arising in gas molecule collisions off perfect harmonic solids [S. A. Adelman and J. D. Doll, J. Chem. Phys. 64, 2374 (1976); S. A. Adelman and B. J. Garrison, J. Chem. Phys. 65, 3571 (1976)]. The present version of the theory is not restricted to the harmonic systems and thus provides a formal framework for treating liquid state as well as solid state chemical phenomena. Basic to the theory is the molecular time scale generalized Langevin equation (MTGLE), a formally exact representation of the dynamics of a chemical system coupled to an arbitrary heat bath in an arbitrary manner. The MTGLE is equivalent to but distinct from the conventional [Mori–Kubo] generalized Langevin representation. It, however, is more natural for chemical dynamics simulation work because in the MTGLE the apparent v...

Generalized Langevin theory for gas/solid processes: Dynamical solid models

A new class of smooth and structured solid models is developed from the generalized Langevin theory of gas/solid processes [S. A. Adelman and J. D. Doll, J. Chem. Phys. 64, 2375 (1976)], and numerical results for scattering off the simplest of these model solids are presented. The models, which may be refined to arbitrary precision, allow one to treat the many‐body or lattice effect in gas/solid dynamics in a qualitatively correct but computationally simple manner. Scattering calculations based on the models may be carried out using standard classical trajectory methodology; the many‐body dynamics modifies the usual classical equations of motion through noise terms and auxiliary variables. Collisional studies based on the simplest of the new models reveal the importance of many‐body dynamics on energy transfer and trapping thresholds. The percentage of energy transfer due to many‐body effects is found to be a rapidly increasing function of solid Debye temperature ΘD; at ΘD≳225°K the many‐body contribution...

The effective direct correlation function: An approach to the theory of liquid solutions

An approach to the theory of liquid solutions based on the concept of the effective solute direct interaction is presented. The Ornstein–Zernike (OZ) matrix equation for an n‐component mixture containing p solute components and q=n−p solvent components is rigorously transformed to an effective p‐dimensional OZ equation explicitly involving just the solute components. Solvent effects enter the solute OZ equation through a pxp matrix of modified or effective solute direct correlation functions. The osmotic thermodynamic properties of the solution are related to the effective direct correlation function by formulas identical in structure to those which relate the corresponding thermodynamic properties of a pure fluid to the direct correlation function of the fluid. An analysis of the small wave vector k limits of several effective direct correlation functions important in the description of polar–ionic mixtures is given. The effective ion–ion direct correlation function for small k is proportional to a Coulo...

Theory of vibrational energy relaxation in liquids: Construction of the generalized Langevin equation for solute vibrational dynamics in monatomic solvents

Algorithms which permit the explicit, albeit approximate, construction of a physically realistic generalized Langevin equation of motion for the energy relaxation dynamics of a specified solute normal mode coordinate y in a monatomic solvent are developed. These algorithms permit the construction, from equilibrium solute–solvent pair correlation functions, of the liquid state frequency ωl of the normal mode and of the Gaussian model approximation to the autocorrelation function 〈F(t)F〉0 of the fluctuating generalized force exerted by the solvent on the normal mode. From these quantities one may compute, from equilibrium solute–solvent pair correlation functions, the vibrational energy relaxation time T1 of the solute normal mode and also related quantities which permit one to assess the relative importance of direct [y coordinate→solvent] and indirect [y coordinate→solute translation–rotational coordinates→solvent] energy flow pathways in solute vibrational energy relaxation. The basis of the constructi...

Dynamics of liquid state chemical reactions: Photodissociation dynamics and geminate recombination of molecular iodine in liquid solution

The dynamics of photodissociation, cage breakout, and subsequent recombination of molecular iodine in model Lennard‐Jones solvents designed to simulate liquid carbon tetrachloride and ethane is studied via classical stochastic trajectory simulations based on the molecular timescale generalized Langevin equation (MTGLE) of motion for liquid state chemical reaction dynamics [S. A. Adelman, J. Chem. Phys. 73, 3145 (1980)]. Also presented for comparison purposes are parallel trajectory studies based on a matrix Langevin equation characterized by friction coefficients which depend on the I2 internuclear separation R. These studies show that dynamical solvent effects arising from molecular timescale correlated solute/solvation shell motions play a basic role in the I2 photodynamics. The nature and magnitude of these dynamical solvent effects are governed by the MTGLE force constants, especially, the basic force constants ω2e0(R) and ω2c1(R). Dynamical solvent effects in particular: (i) can lead to the productio...

Macroscopic model for solvated ion dynamics

A macroscopic treatment of solvated ion dynamics is developed and applied to calculate the limiting (zero concentration) conductance of cations in several aprotic solvents. The theory is based on a coupled set of electrostatic and hydrodynamic equations for the density, flow, and polarization fields induced in the polar solvent by a moving ion. These equations, which are derived by the Mori projection technique, include crucial local solvent structure (ion solvation) effects through solvent compressibility, and local constitutive parameters. If solvent structure is suppressed, the equations reduce to those derived previously by Onsager and Hubbard [J. B. Hubbard and L. Onsager, J. Chem. Phys. 67, 4850 (1977)]. The macroscopic equations are approximately decoupled into electrostatic and hydrodynamic parts. The decoupled equations are solved assuming a step density, viscosity, and dielectric constant model for the local solvent structure and dynamics. This yields analytic expressions for the viscous, ζV, an...

Generalized Langevin theory for many‐body problems in chemical dynamics: Reactions in liquids

A general theoretical framework for classical trajectory simulations of chemical reactions in liquids is presented. This framework is a development of the molecular timescale generalized Langevin equation (MTGLE) theory [S. A. Adelman, Adv. Chem. Phys. 44, 143 (1980)] for condensed phase chemical reaction dynamics. This generalization permits one to treat solute configuration (r0) dependent generalized damping forces in a computationally straightforward manner. Thus, for example, dynamical effects of common caging of reagents are realistically accounted for in the present theory. The theory is based on the following method. The nonequilibrium solvent density induced by small displacements Δr0(t) of the solute (chemical system) from a configuration point r0 is computed by linear response theory. The time dependent reaction force that the solvent exerts on the solute is then computed from the nonequilibrium solvent density. This leads to a set of solute configuration dependent MTGLE parameters {ωep2(r0), ωp...

Theory of vibrational energy relaxation in liquids: Diatomic solutes in monatomic solvents

A numerical study of the solute, solvent, and energy flow pathway dependence of the vibrational energy relaxation (VER) time T1 of a harmonic solute vibrational mode is presented for the prototype case of diatomic solutes in monatomic solvents. This study is based on formulas for T1 developed in the preceding paper, especially the relationship T1=β−1(ωl) where β(ω) is the frequency‐dependent friction kernel of the solute normal mode and where ωl is its liquid state frequency. These formulas permit evaluation of T1 and its energy flow pathway dependence from equilibrium solute–solvent pair correlation functions. Applications are made to VER of ground electronic state molecular iodine and bromine in the fluids xenon and argon and in model Lennard‐Jones solvents designed to simulate ethane and carbon tetrachloride. Satisfactory agreement between the present treatment and experimental and computer simulation results for 15 thermodynamic states is found. The VER rates (∼T−11) were found to increase with increa...