A derivation of the mixed quantum-classical Liouville equation from the influence functional formalism

We show that the mixed quantum-classical Liouville equation is equivalent to linearizing the forward-backward action in the influence functional. Derivations are provided in terms of either the diabatic or adiabatic basis sets. An application of the mixed quantum-classical Liouville equation for calculating the memory kernel of the generalized quantum master equation is also presented. The accuracy and computational feasibility of such an approach is demonstrated in the case of a two-level system nonlinearly coupled to an anharmonic bath.     

Mixed quantum-classical Redfield master equation

Redfield master equation is derived from mixed quantum-classical Liouville equation using product initial conditions. Simple two-level system example is given and comparison with Fermi golden rule is made.     

Nanoparticle dynamics: a multiscale analysis of the Liouville equation

A theory of nanoparticle dynamics based on scaling arguments and the Liouville equation is presented. We start with a delineation of the scales characterizing the behavior of the nanoparticle/host fluid system. Asymptotic expansions, multiple time and space scale techniques, the resulting coarse-grained dynamics of the probability densities of the Fokker-Planck-Chandrasekhar (FPC) type for the nanoparticle(s), and the hydrodynamic models of the host medium are obtained. Collections of nanoparticles are considered so that problems such as viral self-assembly and the transition from a particle suspension to a solid porous matrix can be addressed via a deductive approach that starts with the Liouville equation and a calibrated atomic force field, and yields a generalized FPC equation. Extensions allowing for the investigation of the rotation and deformation of the nanoparticles are considered in the context of the space-warping formalism. Thermodynamic forces and dissipative effects are accounted for. The notion of configuration-dependent drag coefficients and their implications for coagulation and consolidation are shown to be natural consequences of the analysis. All results are obtained via formal asymptotic expansions in mass, size, and other physical and kinetic parameter ratios.     

Adaptive resolution simulation of liquid para-hydrogen: testing the robustness of the quantum-classical adaptive coupling

Adaptive resolution simulations for classical systems are currently made within a reasonably consistent theoretical framework. Recently we have extended this approach to the quantum-classical coupling by mapping the quantum nature of an atom onto a classical polymer ring representation within the path integral approach [Poma & Delle Site, Phys. Rev. Lett., 2010, 104, 250201]. In this way the process of interfacing adaptively a quantum representation to a classical one corresponds to the problem of interfacing two regions with a different number of effective "classical" degrees of freedom; thus the classical formulation of the adaptive algorithm applies straightforwardly to the quantum-classical problem. In this work we show the robustness of such an approach for a liquid of para-hydrogen at low temperature. This system represents a highly challenging conceptual and technical test for the adaptive approach due to the extreme thermodynamical conditions where quantum effects play a central role.     

Implicit numerical schemes for the stochastic Liouville equation in Langevin form

We present and numerically test implicit as well as explicit numerical schemes for solving the Stochastic Liouville Equation in Langevin form. It is found that implicit schemes provide significant gain in robustness, for example, when nonsecular Hamiltonian terms cannot be ignored in electron and nuclear spin resonance. Implicit schemes open up several spectroscopic relaxation problems for direct interpretation using the Stochastic Liouville Equation. To illustrate the proposed numerical schemes, studies are presented for an electron paramagnetic resonance problem involving a coordinated copper complex and a fluorescence problem.     

Stochastic Liouville equation simulation of multidimensional vibrational line shapes of trialanine

The line shapes detected in coherent femtosecond vibrational spectroscopies contain direct signatures of peptide conformational fluctuations through their effect on vibrational frequencies and intermode couplings. These effects are simulated in trialanine using a Green's function solution of a stochastic Liouville equation constructed for four collective bath coordinates (two Ramachandran angles affecting the mode couplings and two diagonal energies). We find that fluctuations of the Ramachandran angles which hardly affect the linear absorption can be effectively probed by two-dimensional spectra. The signal generated at k(1)+k(2)-k(3) is particularly sensitive to such fluctuations.     

Transport properties of quantum-classical systems

Correlation function expressions for calculating transport coefficients for quantum-classical systems are derived. The results are obtained by starting with quantum transport coefficient expressions and replacing the quantum time evolution with quantum-classical Liouville evolution, while retaining the full quantum equilibrium structure through the spectral density function. The method provides a variety of routes for simulating transport coefficients of mixed quantum-classical systems, composed of a quantum subsystem and a classical bath, by selecting different but equivalent time evolution schemes of any operator or the spectral density. The structure of the spectral density is examined for a single harmonic oscillator where exact analytical results can be obtained. The utility of the formulation is illustrated by considering the rate constant of an activated quantum transfer process that can be described by a many-body bath reaction coordinate.     

Self‐energy fields solutions to the generalized reduced Liouville equation in the perturbative approach

A direct approach for the generalized reduced Liouville equation of motion decoupling problem associated with open quantum systems within the superoperator formalism is presented. The procedure is based on inversion of the perturbation series for the energy representation of the density operator so as to obtain one for the proper self‐energy fields that emerge as a consequence of the analytic character of the associated spectral representation. Thus, the perturbation series that arises from the iteration of the energy‐dependent matrix elements hierarchy involved in the statistical operator allows, upon further expansion of the inverse of such series, to get formally exact expressions for the corrections to all orders of the self‐energy fields. The lower order corrections of these fields are discussed in terms of resonant and nonresonant contributions. The present approach provides matrix equations that show the close relation between the environment effects represented by the self‐energy fields and the relaxation kernel that drives the system–reservoir interaction. © 2000 John Wiley & Sons, Inc. Int J Quant Chem 79: 280–290, 2000     

An adaptive stepsize method for the chemical Langevin equation

Mathematical and computational modeling are key tools in analyzing important biological processes in cells and living organisms. In particular, stochastic models are essential to accurately describe the cellular dynamics, when the assumption of the thermodynamic limit can no longer be applied. However, stochastic models are computationally much more challenging than the traditional deterministic models. Moreover, many biochemical systems arising in applications have multiple time-scales, which lead to mathematical stiffness. In this paper we investigate the numerical solution of a stochastic continuous model of well-stirred biochemical systems, the chemical Langevin equation. The chemical Langevin equation is a stochastic differential equation with multiplicative, non-commutative noise. We propose an adaptive stepsize algorithm for approximating the solution of models of biochemical systems in the Langevin regime, with small noise, based on estimates of the local error. The underlying numerical method is the Milstein scheme. The proposed adaptive method is tested on several examples arising in applications and it is shown to have improved efficiency and accuracy compared to the existing fixed stepsize schemes.     

Simultaneous integration of mixed quantum-classical systems by density matrix evolution equations using interaction representation and adaptive time step integrator

A density matrix evolution method [H. J. C. Berendsen and J. Mavri, J. Phys. Chem., 97, 13464 (1993)] to simulate the dynamics of quantum systems embedded in a classical environment is applied to study the inelastic collisions of a classical particle with a five-level quantum harmonic oscillator. We improved the numerical performance by rewriting the Liouville–von Neumann equation in the interaction representation and so eliminated the frequencies of the unperturbed oscillator. Furthermore, replacement of the fixed time step fourth-order Runge–Kutta integrator with an adaptive step size control fourth-order Runge–Kutta resulted in significantly lower computational effort at the same desired accuracy. © 1996 by John Wiley & Sons, Inc.     

Stochastic Liouville equation treatment of the electron paramagnetic resonance line shape of an S-state ion in solution

The current approaches used for the analysis of electron paramagnetic resonance spectra of Gd3+ complexes suffer from a number of drawbacks. Even the elaborate model of [Rast et al., J. Chem. Phys. 113, 8724 (2000)] where the electron spin relaxation is explained by the modulation of the zero-field splitting (ZFS), by molecular tumbling (the so called static contribution), and deformations (transient contribution), is only readily applicable within the validity range of the Redfield theory [Advances in Magnetic Resonance, edited by J.-S. Waugh (Academic, New York, 1965), Vol. 1, p. 1], that is, when the ZFS is small compared to the Zeeman energy and the rotational and vibrational modulations are fast compared to the relaxation time. Spin labels (nitroxides and transition metal complexes) have been studied for years in systems that violate these conditions. The theoretical framework commonly used in such studies is the stochastic Liouville equation (SLE). The authors shall show how the physical model of Rast et al. can be cast into the SLE formalism, paying special attention to the specific problems introduced by the [Uhlenbeck and Ornstein, Phys. Rev. 36, 823 (1930)] process used to model the transient ZFS. The resulting equations are very general and valid for arbitrary correlation times, magnetic field strength, electron spin S, or symmetry. The authors demonstrate the equivalence of the SLE approach with the Redfield approximation for two well-known Gd3+ complexes.     

A quantum-classical treatment of non-adiabatic transitions

We have carried out molecular dynamics simulations of non-adiabatic processes with the help of a newly formulated potentially exact quantum-classical approach derived from a method proposed earlier [J. Chem. Phys. 118 (2003) 5302]. In this method, time-dependent Schroedinger equation is solved by representing Ψ on a moving Gauss–Hermite DVR grid, the motion of grid-centre being handled classically, but self consistently with the quantum evolution of the wavefunction. Electronic transitions are allowed anywhere in the configuration space among any number of coupled states. We have tested the method on three model problems proposed by J.C. Tully [J. Chem. Phys. 93 (1990) 1061]. These models are relevant to a wide range of gas-phase and condensed-phase phenomena occurring even at low energies. Excellent agreement of computed transition probabilities with corresponding quantum mechanical (DVR/FFT) results even in the deep quantum regime and its cost-efficiency (computational) are encouraging.     

Anharmonic nuclear dynamics in the mixed quantum-classical limit

This study employs mixed quantum-classical dynamics (MQCD) formalism to evaluate the linear electronic dipole moment time correlation function (DMTCF) in which a Morse oscillator serves to model the associated vibrations in a mixed quantum-classical (MQC) environment. While the main purpose of this work is to study the applicability of MQCD formalism to anharmonic systems in condensed phase, approximate schemes to physically evaluate the mathematically divergent integrals have been developed in order to deal with the essential singularities that arise while evaluating the Morse oscillator canonical partition function and the DMTCF in MQC systems in the classical limit. The motivation for numerically and analytically evaluating these divergent integrals is that a partition function of any system should lead to a finite value at any temperature and therefore this divergence is unphysical. Additionally, since a partition function is to signify the number of accessible states to the system at hand, divergent results are not physically acceptable. As such, straightforward approximate analytic expressions, at different levels of rigor, for both the classical Morse oscillator partition function and the DMTCF in MQC systems are derived, for the first time. Calculations of Morse oscillator partition function values using different approaches at various temperatures for CO, HCl, and I(2) molecules, showing good results, are presented to test the expressions derived herein. It is found that this divergence, due to singularity, diminishes upon lowering the temperature and only arises at high temperatures. The gradual diminishing of the singularity upon lowering the temperature is sensible since the Morse potential fits the parabolic potential at low temperatures. Model calculations and discussion of the DMTCF and linear absorption spectra in MQC systems using the molecular constants of CO molecule are provided. The linear absorption lineshape is derived by two methods, one of which is asymptotic expansion.     

A quantum-classical bracket that satisfies the Jacobi identity

A quantum-classical bracket is proposed and is shown to satisfy the Jacobi identity, in contrast to previous definitions that obey this property only up to higher order terms in the Planck constant variant Planck's over 2pi. The Jacobi identity is required of a true Lie bracket and ensures that the Lie bracket of constants of motion is also a constant of motion. An explicit calculation of the Jacobi identity highlights the difference between the proposed and traditional definitions. A further example illustrates that the proposed bracket generates a more consistent quantum-classical dynamics than the traditional bracket. The traditional quantum-classical dynamics in the Henon-Heiles system diverges due to higher order variant Planck's over 2pi terms. The divergence is eliminated with the proposed bracket.     

A quantum-classical approach to the photoabsorption spectrum of pyrazine

We have used the time-dependent discrete variable representation (TDDVR) method to simulate the photoabsorption spectrum of pyrazine. The time-dependent molecular dynamics of pyrazine after excitation to the S2 electronic state is considered as a benchmark to investigate the S2 absorption spectrum. We have carried out the dynamics on a basic four-mode model of pyrazine with the inclusion of five major modes as well as the rest of the vibrational modes as bath modes. Investigations reveal the effect of bath modes such as energy and population transfer from the subsystem to the bath. Calculated results demonstrate excellent agreement with traditional quantum-mechanical findings during the entire propagation and converge to the exact quantum results when enough gridpoints are used. It appears that TDDVR, as a numerical quantum dynamics methodology, is a good compromise between accuracy and speed.