Time Scale Generalized Langevin Equation Research Articles

Quantum generalized Langevin equation: Explicit inclusion of nonlinear system dynamics

We derive a quantum nonlinear generalized Langevin equation (GLE) which includes system anharmonic and nonlinear effects explicitly through either gas phase potentials or potentials of mean force. The GLE is applicable to a broad class of nonlinear Hamiltonians with time reversal invariance being the principal restriction. The constraint of linear coupling of the system to the bath is removed. Molecular time scale generalized Langevin equation theory (MTGLE) emerges as the limit case when the nonlinearities are removed explicitly from the system. Specifically, the usual harmonic approach to the dynamics of the MTGLE primary zone (or system in a system/bath partitioning) is replaced by a more general approach which allows for anharmonic and nonlinear effects. Appropriate statistical averages are developed which permit averaging over the bath and a reduction of the number of degrees of freedom to those present in the system. The final form of the quantum nonlinear GLE with attendant statistical relations is...

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.

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.

Molecular time scale generalized Langevin equation theory for coupled spin systems

The MTGLE theory is formulated for spin systems using the exact low temperature limit solution of the one dimensional ferromagnetic Heisenberg model as a guide. The final equations are solved to reproduce the exact time correlation function and GLE response function for the spin 1/2 XY model in the high temperature limit [Sur et al., Phys. Rev. B 12, 3845 (1975)]. The advantage of the method is that it can be applied to complex systems with relative ease and in the case of spin systems provides an avenue for investigating effects of anisotropic dipolar spin interactions. One such application to exchange narrowing of the ESR line shape is discussed.

Exact time correlation function for a nonlinearly coupled vibrational system

The time correlation function\(\dot \chi\)(t)=Re , which is related to the dipole spectrum and is the main focus of quantum molecular time scale generalized Langevin equation theory, is calculated for the Hamiltonian system in which a single oscillator is coupled by a nonlinear Davydov term to a chain of oscillators comprising a phonon heat bath. An exact expression for\(\dot \chi\)(t) is obtained. At long times we find that the time correlation function decays as a small power law atT=0K, but switches to exponential decay at higher temperature. This is a new result and bears on the long-standing issue of the existence of long-time tails.

A theory of electron baths: One-electron system dynamics

The second-quantized, many-electron, atomic, and molecular Hamiltonian is partitioned both by the identity or labeling of the spin orbitals and by the dynamics of the spin orbitals into a system coupled to a bath. The electron bath is treated by a molecular time scale generalized Langevin equation approach designed to include one-electron dynamics in the system dynamics. The bath is formulated as an equivalent chain of spin orbitals through the introduction of equivalent-chain annihilation and creation operators. Both the dynamics and the quantum grand canonical statistical properties of the electron bath are examined. Two versions for the statistical properties of the bath are pursued. Using a weak bath assumption, a bath statistical average is defined which allows one to achieve a reduced dynamics description of the electron system which is coupled to the electron bath. In a strong bath assumption effective Hamiltonians are obtained which reproduce the dynamics of the bath and which lead to the same results as found in the weak bath assumption. The effective (but exact) Hamiltonian is found to be a one-electron Hamiltonian. A reduced dynamics equation of motion for the system population matrix is derived and found to agree with a previous version. This equation of motion is useful for studying electron transfer in the system when coupled to an electron bath.

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.

Second-quantized molecular time scale generalized Langevin equation theory: Fermion equivalent chain

A second-quantized version of molecular time scale generalized Langevin equation theory is developed in an equivalent chain format for quantum fermion systems. The approach allows for nonlinear Hamiltonians and strong coupling to the bath. A bath average is defined which permits reduced dynamics prescriptions to be developed for the case of one-electron Hamiltonians. The bath average is shown to be defined such that perturbations of a primary zone damp away at long time.

Second-quantized molecular time scale generalized Langevin equation theory: Boson equivalent chain

A second-quantized version of molecular time scale generalized Langevin equation theory is developed in an equivalent chain format for quantum boson systems. The approach allows for nonlinear Hamiltonians and strong coupling to the bath. A bath average is defined which permits reduced dynamics prescriptions to be developed. The bath average is shown to be consistent with the notion that perturbations of a primary zone should damp away at long time.

Second-quantized molecular time scale generalized Langevin equation theory: Coupled oscillator model

A second-quantized, coupled oscillator model is presented which explicitly displays the structure of a second-quantized MTGLE theory. The Adelman ansatz [J. Chem Phys. 75, 5837 (1981)] for a quantum MTGLE response function is shown to generate the correct response function for the model. This result paves the way for the development of a general second-quantized MTGLE theory.

The molecular time scale generalized Langevin equation approach to problems in condensed-phase chemical reaction dynamics

The molecular time scale generalized Langevin equation approach to problems in condensed-phase chemical reaction dynamics

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: 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.