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Notes: 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 increase in the degree of resonance overlap between ωl and the solvent frequency spectrum ρF(ω)∼β(ω). Moreover indirect energy flow pathways, i.e., those which involve solute vibration ↔ solute translation–rotation ↔ solvent energy transfer, are found to play a qualitatively essential role for many of the systems studied here. Finally a study of the temperature and density dependence of T1 for iodine in xenon in the experimentally accessible range is presented.

Notes: This paper gives a theoretical treatment of liquid-phase activated barrier crossing that is valid for chemical reactions which occur on typical (e.g., high activation barrier) potential-energy surfaces. This treatment is based on our general approach [S. A. Adelman, Adv. Chem. Phys. 53, 61 (1983)] to problems in liquid-phase chemical dynamics. We focus on the early-time regime [times short compared to the relaxation time of 〈F˜(t)F˜〉0, the fluctuating force autocorrelation function of the reaction coordinate] in which the solvent is nearly "frozen.'' This regime has been shown to be important for the determination of the rate constant in the molecular-dynamics simulations of model aqueous SN2 reactions due to Wilson and co-workers. Our treatment is based on a generalized Langevin equation of motion which naturally represents the physics of the early-time regime.In this regime the main features of the reaction dynamics are governed by the instantaneous potential WIP[y,F˜], which accounts for the cage confinement forces which dominate the liquid-phase effects at early times, rather than by the familiar potential of mean force. The instantaneous potential is derived from the t→0 limit of the equation of motion and its properties are developed for both symmetric and nonsymmetric reactions. The potential is then shown to account for both the early-time barrier recrossing processes found to determine the transmission coefficient κ in the SN2 simulations and the dependence of these processes on environmental fluctuations modeled by F˜. Making the parabolic approximation for the gas-phase part of WIP[y,F˜] yields the following result for the transmission coefficient: κ=ω−1PMFx+=ω−1PMFωMIP[1+ω−2 MIPaitch-thetaˆ(x+)]1/2≠ ω−1PMFω MIP[1+ (1)/(2) ω−2MIPaitch-thetaˆ(ωMIP)], where ωMIP and ωPMF are, respectively, the barrier frequencies of WIP[y,F˜=0] and of the potential of mean force, and where aitch-thetaˆ(x+)=∫∞0 exp(−x+t)aitch-theta(t)dt with aitch-theta(t)≡(kBT)−1〈F˜(t)2F〉0. This result for κ, which is equivalent to a result of Grote and Hynes, but which more naturally represents the physics of the early-time regime, permits a straightforward interpretation of the variation of the transmission coefficients for the model SN2 systems.

Notes: 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 construction of T1 is the formula T1=β−1(ωl) where β(ω)=∫∞0 β(t)cos ω dt and where β(t)=[kBT]−1 〈F˜(t)F˜〉0 is the friction kernel of the solute normal mode. This formula is valid if T1(very-much-greater-than)T2=vibrational phase relaxation time. The approximate formulas for T1 are worked out in detail for diatomic solutes. The approximations are tested for this diatomic solute case by comparing with molecular dynamics results.

Notes: MTGLE and Langevin equation stochastic classical trajectory studies of the mobile silver cation channeling and activated barrier crossing processes which underly superionic conduction in α-AgI are presented. To interpret non-Kramers effects on the channeling kinetics, a new description of the condensed phase effect on activated barrier crossing is developed. This picture emphasizes the competition between "whipback'' of the reaction coordinate, arising from instantaneous environmental restoring forces, and frequency-dependent dissipation of reaction coordinate energy due to environmental relaxation. The simulations show that while the roughest features of the channeling trajectories are determined by the cation friction coefficient the finer details are governed by short-time scale cation-cage correlations. Analogously, for α-AgI, the gross features of the kinetics, e.g., the rough extent of the breakdown of the transition state model, are described by a Kramers-like picture. The finer details of the kinetics, however, are governed by the competition between whipback, which makes realistic barrier crossing less efficient than Kramers model barrier crossing, and dissipation, which works in the opposite direction.

Notes: The theoretical treatment in Paper I [D. W. Miller and S. A. Adelman, J. Chem. Phys. 117, 2672, (2002), preceding paper] of the vibrational energy relaxation (VER) of low-frequency, large mass dihalogen solutes is extended to the VER of the high-frequency, small mass molecular hydrogen solutes H2 and D2 in a Lennard-Jones argon-like solvent. As in Paper I, values of the relaxation times T1 predicted by the theory are tested against molecular dynamics (MD) results and are found to be of semiquantitative accuracy. To start, it is noted that standard Lennard-Jones site–site potentials derived from macroscopic data can be very inaccurate in the steep repulsive slope region crucial for T1. Thus, the H–Ar Lennard-Jones diameter σUV is not taken from literature values but rather is chosen as σUV=1.39 Å, the value needed to make the theory reproduce the experimental H2/Ar gas phase VER rate constant. Next, by MD simulation it is shown that the vibrational coordinate fluctuating force autocorrelation function 〈F˜(t)F˜〉0 of Paper I decays roughly an order of magnitude more rapidly for the molecular hydrogen solutions than for the dihalogen solutions. This result implies a relatively slow decay for the molecular hydrogen friction kernels β(ω)=(kBT)−1∫0∞〈F˜(t)F˜〉0 cos ω tdt, yielding for the H2/Ar and D2/Ar systems at T=150 K physical millisecond values for T1=β−1(ωl) despite the high liquid phase vibrational frequencies ωl of H2 and D2. The rapid decay of 〈F˜(t)F˜〉0 is due to both the steepness of the repulsive slope of the H–Ar potential and the small masses of H and D. Thus, the small value chosen for σUV is needed to avoid unphysically long T1's. Next, an analytical treatment of the H2/D2 isotope effect on T1, based on the theory, is found to predict that the H2/Ar and D2/Ar T1's are close in value due to the compensating effects of lower ωl but slower decay of 〈F˜(t)F˜〉0 for D2/Ar, a result in qualitative agreement with experiments. Applying the theory to numerically study the isothermal ρ dependencies of the VER rate constant k(T,ρ)=T1−1 at 150 K reveals that for both H2/Ar and D2/Ar, as for the solutions of Paper I, k(T,ρ) can be factorized as in the isolated binary collision (IBC) model. Moreover, the molecular theory and IBC rate isotherms differ only slightly for both solutions, a result interpreted in terms of the form of the H–Ar pair correlation function. The theoretical and experimental rate isotherms at 150 K are then compared. Agreement is very good for the H2/Ar solution, but for the D2/Ar solution the theoretical rates are about four times too large. Finally, the isochoric T dependencies of k(T,ρ) in the range 200–1000 K are found for both solutions to conform to an Arrhenius rate law. © 2002 American Institute of Physics.

Notes: A generalized Langevin equation framework for the treatment of liquid state influence on solute dynamics in molecular solvents, which generalizes an earlier framework restricted to monatomic solvents [S. A. Adelman, Adv. Chem. Phys. 53, 61 (1983)], is developed. This framework permits one to realistically treat energy exchange between the solute atoms and the solvent vibrational (V) degrees of freedom. This energy exchange can qualitatively influence the rates of liquid state processes when the solute frequencies relevant to the process of interest (e.g., a solute normal mode frequency if one is interested in the vibrational energy relaxation of that mode) substantially overlap the V bands of the frequency spectrum describing local solvent density fluctuations. The main result of the present analysis is an infinite set of equivalent chain equations governing the dynamics of the solute configuration point in molecular solvents. These are equations of motion for the solute configuration point and for the coordinates of an infinite set of abstract chain "molecules''. Each molecule is composed of r+1 "atoms'' where r is the number of normal modes of a real solvent molecule. The nearest neighbor chain of fictitious molecules may alternatively be regarded as a set of r+1 nearest neighbor cross-linked "atomic'' chains. Atomic chain 1 executes low frequency"acoustical'' motions which simulate the influence of local solvent translational–rotational (TR) density fluctuations. Atomic chains 2,3,. . .,r+1 execute high frequency "optical'' motions which simulate the influence of normal mode V local solvent density fluctuations. The coupling of the optical and acoustical branches of the chain by the crosslinks is the chain formalism equivalent of the physical coupling, due to liquid state effects, of the temporal development of the TR and V modes of motion. The theory presented in this paper provides a conceptual foundation for the treatment of liquid state reactions occuring in molecular solvents by stochastic dynamics techniques. This conceptual foundation may be made the basis of practical simulation methods for treating reactions of diatomic solutes in molecular solvents and may be further developed to provide practical simulation methods for polyatomic solutes in molecular solvents.

Notes: A liquid state analog of the gas phase harmonic oscillator-rigid top model is developed. This analog, which we call the harmonic mean field model, permits one to compute the equilibrium and dynamic properties of real, i.e., vibrating, molecular solvents from the structure and dynamics of the corresponding rigid solvents. The harmonic mean field model is based on: (i) A mean field harmonic model for the solvent vibrational (V) force field. (ii) A rigid solvent model treatment of translational–rotational (TR) fluctuations. (iii) Complete neglect of explicit coupling between V and TR fluctuations. (Implicit coupling is included in the vibrational force field.) The model is developed for statics via a sequence of physically motivated approximations to the exact canonical ensemble phase space distribution function of the solvent, fCA[S]. This yields a model distribution function f(0)CA[S] =f(0)CA[pyy]fCA[pww; v0], where f(0)CA[pyy] is an effective harmonic vibrational phase space distribution function which describes mean field harmonic V fluctuations and where f(0)CA[pww; v0] is the rigid solvent canonical ensemble distribution function. The nonequilibrium version of the model is defined as the solvent dynamics generated by a model Liouville operator L(0). This is defined via the model equilibrium Liouville equation L(0)f(0)CA [S]=0. Explicit results for equilibrium averages and time correlation functions of molecular solvents are obtained. The frequency spectra of the time correlation functions contain a low frequency "acoustic'' branch arising from solvent TR motions and high frequency "optical'' branches arising from collective solvent V motions. A detailed analysis of the frequency spectra of autocorrelation functions of diatomic solvents is presented.

Notes: A molecular theory of liquid phase vibrational energy relaxation (VER) [S. A. Adelman et al., Adv. Chem. Phys. 84, 73 (1993)] is applied to study the temperature T and density ρ dependencies of the VER rate constant k(T,ρ)=T1−1, where T1 is the energy relaxation time, of model Lennard-Jones systems that roughly simulate solutions of high-mass, low-frequency dihalogen solutes in rare gas solvents; specifically the I2/Xe, I2/Ar, and ICI/Xe solutions. For selected states of these systems, the theory's assumptions are tested against molecular dynamics (MD) results. The theory is based on the expression T1=β−1(ωl), where ωl and β(ω) are, respectively, the solute's liquid phase vibrational frequency and vibrational coordinate friction kernel. The friction kernel is evaluated as a cosine transform of the fluctuating force autocorrelation function of the solute vibrational coordinate, conditional that this coordinate is fixed at equilibrium. Additionally, the early-time decay of the force autocorrelation function is approximated by a Gaussian function which is exact to order t2. This Gaussian approximation permits evaluation of T1 in terms of integrals over equilibrium solute–solvent pair correlation functions. The pair correlation function formulas yield T1's in semiquantitative agreement with those found by MD evaluations of the Gaussian approximation, but with three orders of magnitude less computational effort. For the isothermal ρ dependencies of k(T,ρ), the theory predicts for all systems that the Gaussian decay time τ is nearly independent of ρ. This in turn implies that k(T,ρ) factorizes into a liquid phase structural contribution and a gas phase dynamical contribution, yielding a first-principles form for k(T,ρ) similar to that postulated by the isolated binary collision model. Also, the theory predicts both "classical" superlinear rate isotherms, and "nonclassical" sublinear isotherms similar to those recently observed by Troe and co-workers for azulene relaxation in supercritical fluids. The isochoric T dependencies of k(T,ρ) are studied in the range 300 to 1000 K. For none of the solutions are the rate isochores found to accurately conform to either Arrhenius or Landau–Teller kinetics. © 2002 American Institute of Physics.