Molecular Timescale Generalized Langevin Equation Research Articles

Molecular time scale generalized Langevin equation theory and polynomial maximum entropy imaging of spectral densities

Molecular time scale generalized Langevin equation (MTGLE) theory is discussed as an approach to condensed phase dynamics. A polynomial maximum entropy (MaxEnt) process for imaging required MTGLE spectral densities based on knowledge of the moments of the spectral density is introduced. The process is based on the use of interpolation polynomials which serve both to image the spectral density as well as provide a numerical procedure to compute the inverse Hessian matrix in a Newton-type minimization. A default model is added to allow for the inclusion of additional information in forming the image. The polynomial MaxEnt imaging process is found to be a fast, numerically stable, computational procedure which produces images comparable in quality to images obtained by other imaging processes. The polynomial MaxEnt imaging process is examined in the context of imaging MTGLE bath spectral densities with special emphasis on a coupled linear chain model. Standard harmonic oscillator, Hamiltonian bath models such as Ohmic-exponential and Ohmic-Gaussian are shown to possess regions of parameter space for which the MTGLE adiabatic frequency is imaginary. When the adiabatic frequency is zero, it is shown that imaging of the friction kernel is the best approach.

Maximum entropy imaging and quantum molecular timescale generalized langevin equation theory

Abstract A new maximum entropy (MaxEnt) imaging procedure is introduced to treat the inverse problem in quantum molecular timescale generalized Langevin equation (MTGLE) theory. In particular MaxEnt imaging is used to construct the MTGLE bath spectral density from knowledge of its moments.

A graphical approach to quantum molecular time scale generalized Langevin equation theory

A graphical approach for obtaining the algebraic and topological structure of time derivatives of time correlation functions is proposed in the context of quantum molecular timescale generalized Langevin equation (MTGLE) theory. A sequence of approximate MTGLE procedures is introduced using the graphical approach, and computations on a single-vibron Hamiltonian model are carried out using the procedures. In particular a new unlinked procedure for obtaining estimates of higher-order time derivatives is revealed by the graphical approach. The graphical approach is shown to be a new and powerful tool to be used with MTGLE theory.

Maximum entropy approach in molecular timescale generalized Langevin equation theory

The principle of maximum entropy is used to develop a procedure for obtaining the best fit to a spectral distribution within the context of molecular timescale generalized Langevin equation theory. Results are obtained for a single-vibron model and compared with previously computed results. Excellent agreement is demonstrated and the maximum entropy approach is shown to be a robust procedure.

Nonlinearly coupled vibrational modes and self-trapped states: Single-vibron-oscillator case with alpha -helix parameters.

Quantum molecular-time-scale generalized-Langevin-equation (MTGLE) theory is applied to calculate spectral linewidths and redshifts of a single vibron oscillator nonlinearly coupled by the Davydov Hamiltonian in the \ensuremath{\alpha}-helix parameter regime to a chain of linearly coupled molecules. Exact values can also be calculated for this problem. The MTGLE theory gives very accurate results for linewidths and less accurate but acceptable results for redshifts over a wide range of temperatures from 10 to 300 K. These results successfully extend previous calculations carried out in the l-alanine parameter regime to the \ensuremath{\alpha}-helix parameter regime with a 30-fold increase in frequency of the vibron oscillator and a 1000-fold increase in the square of the nonlinear coupling constant. In constrast to the l-alanine case, the vibron-oscillator frequency is more than 16 times greater than the largest frequency in the phonon manifold.

Time correlation functions for finite systems using second-quantized molecular time scale generalized Langevin equation theory: Effect of rotating wave approximation

Second-quantized molecular time scale generalized Langevin equation (MTGLE) theory is applied to the computation of time correlation functions for a finite system and shown to be convergent as a function of temperature and nonlinear coupling parameter. The system chosen is a simple nonlinear or quartic oscillator in which the rotating wave approximation has been made. The effect of this approximation in the context of an MTGLE approach to computing dipole spectra is explored. As a consequence of these computations, a new pathology of the MTGLE approach is uncovered; namely, coupling frequencies ω4cn can become negative. A procedure for dealing with this problem is demonstrated and shown to work successfully.

Electronic bath theory: Multi-orbital primary zone

Second-quantized molecular time scale generalized Langevin equation (MTGLE) theory is extended to include a multi-orbital primary zone for fermion systems. A generic one-electron Hamiltonian is treated using this approach and MTGLE expressions are obtained. The specific case of a semi-infinite tight-binding chain is addressed and the resulting response function is shown to behave as a damped oscillation with respect to time. An approximation suitable for computer implementation is presented and results are given for the semi-infinite chain.

Second-quantized molecular time scale generalized Langevin equation theory and the quantum oscillator

The short-time Gaussian approximation to the molecular time scale generalized Langevin equation (MTGLE) friction kernel is introduced and used to compute the time dependence of a fluctuation time correlation function. The fluctuations are shown to have two time scales, namely, the dissipation time scale and a quantum time scale given by exp[−2πkTt/ℏ]. Absorption and emission spectral functions for a quantum oscillator coupled to a bath are derived with no approximations made. The second-quantized MTGLE approach is applied to the problem of a quantum oscillator coupled linearly to a bath of quantum oscillators. The method is shown to be consistent with previous work and to provide a systematic methodology to examine more general many-body boson problems.

Dynamics of liquid state chemical reactions: Vibrational energy relaxation of molecular iodine in liquid solution

The dynamics of vibrational energy relaxation of highly excited molecular iodine in three monatomic solvents is studied via stochastic classical trajectory simulations based on the molecular timescale generalized Langevin equation (MTGLE) of motion for liquid state chemical reactions [S. A. Adelman, J. Chem. Phys. 73, 3145 (1980)]. Also presented for comparison purposes are parallel studies based on a matrix Langevin equation of motion characterized by friction coefficients which depend on the instantaneous I2 internuclear separation R. The qualitative features of the energy relaxation may be interpreted as effects arising from modifications of the solute dynamics due to molecular timescale correlations between its motion and that of its solvation shells. Such dynamical solvent effects are realistically described by the MTGLE equation of motion but not by the Langevin equation. Thus, for example, the marked slowdown of the rate of I2 energy relaxtion in simple solvents when the I2 vibrational quantum number drops below a solvent-dependent critical value, earlier predicted by Nesbitt and Hynes, is predicted by MTGLE dynamics but not by Langevin dynamics. Finally, practical algorithms for numerically constructing the MTGLE and Langevin equations for specific solute–solvent systems are presented.

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 production of quasibound pairs of iodine atoms even if the static or mean I2 effective interaction is repulsive. The fundamental vibrational frequency of the quasibound pairs is governed by ω2e0(R) which determines the modification of the static effective interaction due to instantaneous elastic cage restoring forces exerted on the I2 molecule; (ii) largely determine the efficiency of cage breakout (and hence the nongeminate yield) which occurs during the initial stages of the photolysis process. This breakout efficiency depends on a competition between ω2e0(R) [large ω2e0(R) retards breakout] and ω2c1(R) [large ω2e0(R)promotes breakout]; (iii) determine, via the strength of solute-shell coupling as measured by ω2c1(R), the rate and qualitative character of the energy relaxation of highly vibrationally excited nascent I2 product. Moreover, the interplay between the efficiencies of the initial cage breakout step and the final vibrational stabilization step of the photolysis process determines the observable picosecond timescale transient electronic absorption spectrum. Because of this interplay, systems with very different underlying dynamics can exhibit similar transient absorption spectra (on the timescale of typical experiments). This is illustrated by the similarities in the absorption spectra of I2 in liquid CCl4 (high geminate yield and relatively inefficient vibrational stabilization) and I2 in liquid C2H6 (low geminate yield and relatively efficient vibrational stabilization). Finally, simplified methods to handle the complicating effects of nonlinear solvent response to very fast initial solute motion and of nonadiabatic motion involving several I2 electronic states are described.

Dynamics of liquid state chemical reactions: Modeling of R-dependent correlation functions

Methods for the simplified construction of those reagent configuration point r0-dependent static and dynamic solvent correlation functions required to implement the molecular time scale generalized Langevin equation (MTGLE) approach to problems in liquid state chemical reaction dynamics [S. A. Adelman, J. Chem. Phys. 73, 3145 (1980)] are presented. These methods permit one to bypass straightforward but laborious construction of the correlation functions via molecular dynamics simulations of the solvent motion about the solute for many solute configurations r0. The methods are based on two approximations: (i) The matrix r0-dependent friction kernel of the reagents β1(t;r0) is modeled via a matrix generalization of the conventional Gaussian approximation for scalar time correlation functions. This modeling permits one to reduce the problem of constructing β1(t;r0) to an equilibrium calculation which may be carried out by Monte Carlo as opposed to molecular dynamics techniques. (ii) The equilibrium r0-dependent solvent correlation functions required to construct the Gaussian approximation to β1(t;r0) are calculated using the superposition approximation. This permits one to construct β1 (t;r0), and hence the MTGLE chain models, from equilibrium solute–solvent and solvent–solvent pair correlation functions. For simple liquids, the pair correlation functions may be obtained as solutions to the Percus–Yevick integral equation. The simplified methods are applied to construct the MTGLE chain models for three Lennard-Jones systems designed to simulate infinitely dilute solutions of molecular iodine in: liquid carbon tetrachloride, dense gaseous ethane, and liquid ethane. The results are tested against molecular dynamics for the I2 in CCl4 model solution.

Some formal properties of the MTGLE representation of generalized Brownian motion theory

The molecular time scale generalized Langevin equation (MTGLE) representation of generalized Brownian motion theory [S. A. Adelman, Adv. Chem. Phys. 44, 143 (1980)] is reformulated to yield a more rigorous and general theory. Key results include: (i) The theory is extended so as to be valid for an arbitrary real scalar process ṙ0(t) whose dynamics are governed by an arbitrary Hermitian Liouville operator L and whose statistics are governed by an arbitrary inner product 〈 〉; (ii) a rigorous development of the equivalent harmonic chain representation of ṙ0(t) is presented; (iii) the dynamical and statistical properties of the equivalent chain are rigorously shown to be formally identical to the dynamical and statistical properties of physical harmonic chains obeying classical Boltzmann statistics; (iv) the concept of clamping of equivalent chain atoms is developed for both dynamics and statistics in a rigorous manner. The clamping concept plays a role in the MTGLE theory somewhat analogous to the role played by successive projection operations in the Mori theory; (v) the MTGLE random forces Ṙp(t) are shown to be nonstationary processes if the inner product is 〈 〉. A new inner product 〈 〉r0,r1,⋅⋅⋅rp−1 is developed which renders the random forces stationary; (vi) the normal modes of the equivalent chain are shown to be simply related to the eigenfunctions of L; (vii) the MTGLE fluctuation–dissipation theorems are shown to be direct consequences of the orthonormality of the eigenfunctions of L; (viii) the connection between the MTGLE and Mori representations is made.

Generalized Langevin theory for many-body problems in chemical dynamics: Gas-surface collisions, vibrational energy relaxation in solids, and recombination reactions in liquids

Model classical trajectory simulations of gas-solid inelastic collisions, vibrational energy relaxation of a diatom in a solid, and atomic recombination reactions in a liquid are presented. These simulations are first applications of the molecular timescale generalized Langevin equation (MTGLE) theory [S. A. Adelman, Adv, Chem. Phys. 44, 143 (1980)] for condensed phase energy transfer and chemical reaction processes. The main conclusions of these model studies are as follows: (i) The MTGLE theory often provides computationally practical methods for condensed phase chemical problems. These methods are applicable with equal ease to both solid and liquid state processes. (ii) The equivalent harmonic chain heatbath modeling for reducing many-body chemical problems to effective few-body problems often converges rapidly. This rapid convergence is found for both short (subpicosecond) timescale processes (gas–solid collisions) and long (nanosecond) timescale processes (vibrational energy relaxation). (iii) The simplest harmonic chain model (heatbath replaced by a single fictitious atom) is often found to yield a qualitatively correct description of heatbath influence on both solid and liquid state processes. (iv) The MTGLE parameters ωe0 (chemical system Einstein frequency) and ω2c1 (chemical system/heatbath coupling constant) are the quantities which often determine the gross magnitude of heatbath influence on condensed phase chemical processes. For both solid and liquid state processes ωe0 determines the effective vibrational frequencies (normal modes) of the chemical system and ω2c1 determines the gross efficiency of chemical system/heatbath energy transfer. For liquid state processes ωe0 also determines the frequency of oscillations in the solvent cage, and ω2c1 also determines rigidity of the solvent cage. (ω2c1=0 means the cage is perfectly rigid).

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+12(r0)}, p=0,1,2,⋅⋅⋅, which account for dynamical effects of solvent–solute interaction. These parameters may be generated as a function of r0 by performing molecular dynamics simulations of the solvent for a number of solute configurations r0. A generalized equivalent harmonic chain representation for solute dynamics is then developed. The force constants of the equivalent chain are the MTGLE parameters. Techniques for approximating the non-Markovian chain equations by Markovian model chain equations are presented. These Markovian equations are directly applicable to simulation of liquid state reactions.

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 vibrational frequencies of the chemical system are those which govern the molecular time scale response of the system. In the conventional Langevin representation, the apparent vibrational frequencies are those which govern the static reponse of the chemical system. The MTGLE is next rigorously transformed into an effective equation of motion for a fictitious nearest neighbor collinear harmonic chain. The parameters characterizing the chain are calculable from the velocity autocorrelation function of the chemical system. The rigorous chain representation of many-body dynamics: (i) provides a simple quasiphysical picture of energy transfer between the heat bath and chemical system during a chemical process; (ii) provides an equivalent Hamiltonian for the chemical system; (iii) permits one to construct a sequence of heat bath models whose response converges to that of the true heat bath in a rapid and systematic manner. The heat bath models are sets of collinear nearest neighbor harmonic chains composed of N=1,2,3,... fictitious atoms. Atom N in the chain is subject to time-local functional damping and is driven by a white noise stochastic force. The remaining atoms 1,2,...N-1 are undamped. The heat bath models (i) produce effective equations of motion for the chemical process which do not involve time nonlocal kernels and which may, hence, be readily solved by standard classical trajectory methods; (ii) allow one to rigorously generalize the Fokker–Planck theory of stochastic dynamics to non-Markovian systems. Finally, the treatment of selected chemical processes via the theory is briefly outlined.