Exact Generator and its High Order Expansions in Time-Convolutionless Generalized Master Equation: Applications to Spin-Boson Model and Exictation Energy Transfer?

The time-convolutionless (TCL) quantum master equation provides a powerful tool to simulate reduced dynamics of a quantum system coupled to a bath. The key quantity in the TCL master equation is the so-called kernel or generator, which describes effects of the bath degrees of freedom. Since the exact TCL generators are usually hard to calculate analytically, most applications of the TCL generalized master equation have relied on approximate generators using second and fourth order perturbative expansions. By using the hierarchical equation of motion (HEOM) and extended HEOM methods, we present a new approach to calculating the exact TCL generator and its high order perturbative expansions. The new approach is applied to the spin-boson model with different sets of parameters, to investigate the convergence of the high order expansions of the TCL generator. We also discuss circumstances where the exact TCL generator becomes singular for the spin-boson model, and a model of excitation energy transfer in the Fenna-Matthews-Olson complex.     

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.     

Second-order quantized Hamilton dynamics coupled to classical heat bath

Starting with a quantum Langevin equation describing in the Heisenberg representation a quantum system coupled to a quantum bath, the Markov approximation and, further, the closure approximation are applied to derive a semiclassical Langevin equation for the second-order quantized Hamilton dynamics (QHD) coupled to a classical bath. The expectation values of the system operators are decomposed into products of the first and second moments of the position and momentum operators that incorporate zero-point energy and moderate tunneling effects. The random force and friction as well as the system-bath coupling are decomposed to the lowest classical level. The resulting Langevin equation describing QHD-2 coupled to classical bath is analyzed and applied to free particle, harmonic oscillator, and the Morse potential representing the OH stretch of the SPC-flexible water model.     

Reduced density matrix hybrid approach: an efficient and accurate method for adiabatic and non-adiabatic quantum dynamics

We present a new approach to calculate real-time quantum dynamics in complex systems. The formalism is based on the partitioning of a system's environment into "core" and "reservoir" modes with the former to be treated quantum mechanically and the latter classically. The presented method only requires the calculation of the system's reduced density matrix averaged over the quantum core degrees of freedom which is then coupled to a classically evolved reservoir to treat the remaining modes. We demonstrate our approach by applying it to the spin-boson problem using the noninteracting blip approximation to treat the system and core, and Ehrenfest dynamics to treat the reservoir. The resulting hybrid methodology is accurate for both fast and slow baths, since it naturally reduces to its composite methods in their respective regimes of validity. In addition, our combined method is shown to yield good results in intermediate regimes where neither approximation alone is accurate and to perform equally well for both strong and weak system-bath coupling. Our approach therefore provides an accurate and efficient methodology for calculating quantum dynamics in complex systems.     

Semiclassical initial value representation study of internal conversion rates

Internal conversion is an inherently quantum mechanical process. To date, "on the fly" computation of internal conversion rates is limited to harmonic approximations, which would seem to be especially unsuitable, given that the typical transition to the ground electronic state occurs at energies which are far from the harmonic limit. It is thus of interest to study the applicability of the semiclassial initial value representation (SCIVR) approach which is in principle amenable to on the fly studies even with "many" degrees of freedom. In this paper we study the applicability of the Herman-Kluk (HK) SCIVR to a model system with two coupled and anharmonic degrees of freedom. We find that (a) the HK SCIVR is a good approximation to the exact quantum dynamics; (b) computation of the first order correction to the HK-SCIVR approximation corroborates the accuracy; (c) by studying a large parameter range, we find that the harmonic approximation is mostly unsatisfactory; and (d) for the specific model used, the coupling between the modes was found to be relatively unimportant. These results imply that the HK-SCIVR methodology is a good candidate for on the fly studies of internal conversion processes of "large" molecules.     

Benchmark calculations for dissipative dynamics of a system coupled to an anharmonic bath with the multiconfiguration time-dependent Hartree method

In this paper, we present benchmark results for dissipative dynamics of a harmonic oscillator coupled to an anharmonic bath of Morse oscillators. The microscopic Hamiltonian has been chosen so that the anharmonicity can be adjusted as a free parameter, and its effect can be isolated. This leads to a temperature dependent spectral density of the bath, which is studied for ohmic and lorentzian cases. Also, we compare numerically exact multiconfiguration time-dependent Hartree results with approximate solutions using continuous configuration time-dependent self-consistent field and local coherent state approximation.     

Finite temperature application of the corrected propagator method to reactive dynamics in a condensed-phase environment

The recently proposed mixed quantum-classical method is extended to applications at finite temperatures. The method is designed to treat complex systems consisting of a low-dimensional quantum part (the primary system) coupled to a dissipative bath described classically. The method is based on a formalism showing how to systematically correct the approximate zeroth-order evolution rule. The corrections are defined in terms of the total quantum Hamiltonian and are taken to the classical limit by introducing the frozen Gaussian approximation for the bath degrees of freedom. The evolution of the primary system is governed by the corrected propagator yielding the exact quantum dynamics. The method has been tested on a standard model system describing proton transfer in a condensed-phase environment: a symmetric double-well potential bilinearly coupled to a bath of harmonic oscillators. Flux correlation functions and thermal rate constants have been calculated at two different temperatures for a range of coupling strengths. The results have been compared to the fully quantum simulations of Topaler and Makri [J. Chem. Phys. 101, 7500 (1994)] with the real path integral method.     

Surface-hopping dynamics and decoherence with quantum equilibrium structure

In open quantum systems, decoherence occurs through interaction of a quantum subsystem with its environment. The computation of expectation values requires a knowledge of the quantum dynamics of operators and sampling from initial states of the density matrix describing the subsystem and bath. We consider situations where the quantum evolution can be approximated by quantum-classical Liouville dynamics and examine the circumstances under which the evolution can be reduced to surface-hopping dynamics, where the evolution consists of trajectory segments exclusively evolving on single adiabatic surfaces, with probabilistic hops between these surfaces. The justification for the reduction depends on the validity of a Markovian approximation on a bath averaged memory kernel that accounts for quantum coherence in the system. We show that such a reduction is often possible when initial sampling is from either the quantum or classical bath initial distributions. If the average is taken only over the quantum dispersion that broadens the classical distribution, then such a reduction is not always possible.     

Does ℏ play a role in multidimensional spectroscopy? Reduced hierarchy equations of motion approach to molecular vibrations

To investigate the role of quantum effects in vibrational spectroscopies, we have carried out numerically exact calculations of linear and nonlinear response functions for an anharmonic potential system nonlinearly coupled to a harmonic oscillator bath. Although one cannot carry out the quantum calculations of the response functions with full molecular dynamics (MD) simulations for a realistic system which consists of many molecules, it is possible to grasp the essence of the quantum effects on the vibrational spectra by employing a model Hamiltonian that describes an intra- or intermolecular vibrational motion in a condensed phase. The present model fully includes vibrational relaxation, while the stochastic model often used to simulate infrared spectra does not. We have employed the reduced quantum hierarchy equations of motion approach in the Wigner space representation to deal with nonperturbative, non-Markovian, and nonsecular system-bath interactions. Taking the classical limit of the hierarchy equations of motion, we have obtained the classical equations of motion that describe the classical dynamics under the same physical conditions as in the quantum case. By comparing the classical and quantum mechanically calculated linear and multidimensional spectra, we found that the profiles of spectra for a fast modulation case were similar, but different for a slow modulation case. In both the classical and quantum cases, we identified the resonant oscillation peak in the spectra, but the quantum peak shifted to the red compared with the classical one if the potential is anharmonic. The prominent quantum effect is the 1-2 transition peak, which appears only in the quantum mechanically calculated spectra as a result of anharmonicity in the potential or nonlinearity of the system-bath coupling. While the contribution of the 1-2 transition is negligible in the fast modulation case, it becomes important in the slow modulation case as long as the amplitude of the frequency fluctuation is small. Thus, we observed a distinct difference between the classical and quantum mechanically calculated multidimensional spectra in the slow modulation case where spectral diffusion plays a role. This fact indicates that one may not reproduce the experimentally obtained multidimensional spectrum for high-frequency vibrational modes based on classical molecular dynamics simulations if the modulation that arises from surrounding molecules is weak and slow. A practical way to overcome the difference between the classical and quantum simulations was discussed.     

Proton transfer reactions in model condensed-phase environments: Accurate quantum dynamics using the multilayer multiconfiguration time-dependent Hartree approach

The recently proposed multilayer multiconfiguration time-dependent Hartree (ML-MCTDH) approach to evaluating reactive quantum dynamics is applied to two model condensed-phase proton transfer reactions. The models consist of a one-dimensional double-well "system" that is bilinearly coupled to a "bath" of harmonic oscillators parameterized to represent a condensed-phase environment. Numerically exact quantum-mechanical flux correlation functions and thermal rate constants are obtained for a broad range of temperatures and system-bath coupling strengths, thus demonstrating the efficacy of the ML-MCTDH approach. Particular attention is focused on the regime where low temperatures are combined with weak system-bath coupling. Under such conditions it is found that long propagation times are often required and that quantum coherence effects may prevent a rigorous determination of the rate constant.     

Single molecule photon emission statistics in the slow modulation limit

A framework for calculating photon emission statistics for single chromophores perturbed by slow environmental fluctuations is introduced. When internal chromophore dynamics are significantly faster than time scales for environmental modulation it becomes possible to invoke a type of adiabatic approximation, allowing for straightforward calculation of photon counting moments including explicitly quantum effects. Unlike previous exact treatments, the present methodology involves calculation of dynamics reflecting only the modulation characteristics of the environment and quantum dynamics of an isolated chromophore separately, i.e., the complicated intermingling of chromophore quantum dynamics and the environmental modulation are suppressed via the adiabatic approximation. This leads to significant conceptual and computational simplifications. Within its regime of applicability, the present approximation reproduces exact calculations quantitatively. We demonstrate this accuracy explicitly for the case of a two-level chromophore modulated by a number of different stochastic models.