Quantum-classical approximation beyond Redfield theory

Fokker-Planck quantum master equation for mixed quantum-semiclassical dynamics

We revisit Caldeira-Leggett's quantum master equation representing mixed quantum-classical theory, but with limited applications. Proposed is a Fokker-Planck quantum master equation theory, with a generic bi-exponential correlation function description on semiclassical Brownian oscillators' environments. The new theory has caustic terms that bridge between the quantum description on primary systems and the semiclassical or quasi-classical description on environments. Various parametrization schemes, both analytical and numerical, for the generic bi-exponential environment bath correlation functions are proposed and scrutinized. The Fokker-Planck quantum master equation theory is of the same numerical cost as the original Caldeira-Leggett's approach but acquires a significantly broadened validity and accuracy range, as illustrated against the exact dynamics on model systems in quantum Brownian oscillators' environments, at moderately low temperatures.

Semiclassical extension of the Landau-Teller theory of collisional energy transfer

A semiclassical version of the quantum coupled-states approximation for the vibrational relaxation of diatomic molecules in collisions with monatomic bath gases is presented. It is based on the effective mass approximation and a recovery of the semiclassical Landau exponent from the classical Landau-Teller collision time. For an interaction with small anisotropy, the Landau exponent includes first order corrections with respect to the orientational dependence of the collision time and the effective mass. The relaxation N(2)(v=1)-->N(2)(v=0) in He is discussed as an example. Employing the available vibrationally elastic potential, the semiclassical approach describes the temperature dependence of the rate constant k(10)(T) over seven orders of magnitude across the temperature range of 70-3000 K in agreement with experimental data and quantum coupled-states calculations. For this system, the hierarchy of corrections to the Landau-Teller conventional treatment in the order of importance is the following: quantum effects in the energy release, dynamical contributions of the rotation of N(2) to the vibrational transition, and deviations of the interaction potential from a purely repulsive form. The described treatment provides significant simplifications over complete coupled-states calculations such that applications to more complex situations appear promising.

Statistical theory of nonadiabatic transitions

Based on results of the preceding paper, and assuming fast equilibration in phase space to the temperature of the surrounding media compared to the time scale of a reaction, we formulate a statistical theory of intramolecular nonadiabatic transitions. A classical mechanics description of phase space dynamics allows for an ab initio treatment of multidimensional reaction coordinates and easy combination with any standard molecular dynamics (MD) method. The presented approach has several features that distinguishes it from existing methodologies. First, the applicability limits of the approach are well defined. Second, the nonadiabatic transitions are treated dynamically, with full account of detailed balance, including zero-point energy, quantum coherence effects, arbitrarily long memory, and change of the free energy of the bath. Compared to popular trajectory surface hopping schemes, our MD-based algorithm is more efficient computationally, and does not use artificial ad hoc constructions like a "fewest switching" algorithm, and rescaling of velocities to conserve total energy. The enhanced capabilities of the new method are demonstrated considering a model of two coupled harmonic oscillators. We show that in the rate regime and at moderate friction the approach precisely reproduces the free-energy-gap law. It also predicts a general trend of the reaction dynamics in the low friction limit, and is valid beyond the rate regime.

Restoring detailed balance in the Landau–Teller probabilities for collision-induced vibrational transitions

The general quasi-classical treatment for collision-induced vibrational transitions in diatomic molecules, under near-adiabatic conditions, is used to derive quantum corrections for probabilities, calculated in the external field approximation originally used by Landau and Teller. The quantum corrections are expressed through the Landau-Teller classical collision time. The first-order correction to the classical exponent restores detailed balance for up- and down-transitions and does not depend on the properties of the bath except for its temperature. The limits of applicability of the first-order correction are discussed.

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.

Theory of solvent influence on reaction dynamics

A generalization of the recently published quantum-classical approximation [A. A. Neufeld, J. Chem. Phys., 119, 2488 (2003)] for the purposes of reaction dynamics in condensed phase is presented. The obtained kinetic equations treat a solvent influence in a nonphenomenological way, account for the change of the free energy of the surrounding media, allow for different solvent dynamics in each reaction channel, and constitute a powerful framework for an accurate modeling of solvent effects, including ultrafast processes. The key features of the approach are its differential form, which considerably facilitates practical applications, and well defined wide applicability limits. The developed methodology fully accounts for an arbitrary long memory of the canonical bath and covers solvent-induced processes from a subpicosecond time scale.

Non-Markovian theory of open systems in classical limit

A fully classical limit of the recently published quantum-classical approximation [A. A. Neufeld, J. Chem. Phys. 119, 2488 (2003)] is obtained and analyzed. The resulting kinetic equations are capable of describing the evolution of an open system on the entire time axis, including the short-time non-Markovian stage, and are valid beyond linear response regime. We have shown, that proceeding to the classical mechanics limit we restrict the class of allowed correlations between an open system and a canonical bath, so that the initial conditions and the relaxation operator has to be appropriately modified (projected). Disregard of the projection may lead to unphysical behavior, since mechanism of the decay of some correlations is essentially of quantum-mechanical nature, and is not correctly described by classical mechanics. The projection (quantum correction to the kinetics) is particularly important for the non-Markovian regime of relaxation towards canonical equilibrium. The conformity of the developed method to the conventional approaches is demonstrated using a model of Brownian motion (heavy particle in the bath of light ones), for which the obtained non-Markovian equations are reduced to the standard Fokker-Planck equation in phase space.