Path integral molecular dynamics for exact quantum statistics of multi-electronic-state systems

Abstract

An exact approach to compute physical properties for general multi-electronic-state (MES) systems in thermal equilibrium is presented. The approach is extended from our recent progress on path integral molecular dynamics (PIMD), Liu et al. [J. Chem. Phys. 145, 024103 (2016)] and Zhang et al. [J. Chem. Phys. 147, 034109 (2017)], for quantum statistical mechanics when a single potential energy surface is involved. We first define an effective potential function that is numerically favorable for MES-PIMD and then derive corresponding estimators in MES-PIMD for evaluating various physical properties. Its application to several representative one-dimensional and multi-dimensional models demonstrates that MES-PIMD in principle offers a practical tool in either of the diabatic and adiabatic representations for studying exact quantum statistics of complex/large MES systems when the Born-Oppenheimer approximation, Condon approximation, and harmonic bath approximation are broken.

Accurate calculation of equilibrium reduced density matrix for the system-bath model: A multilayer multiconfiguration time-dependent Hartree approach and its comparison to a multi-electronic-state path integral molecular dynamics approach

“…The recently developed multi-electronic-state path integral molecular dynamics (MES-PIMD) approach [20] in principle offers a practical tool in either of the diabatic or adiabatic representations for studying exact quantum statistics of general MES systems when the Born-Oppenheimer approximation, Condon approximation, and harmonic bath approximation break down. The MES-PIMD approach employs a unified efficient thermostat scheme (the "middle" scheme) for PIMD (or MD) which applies to either stochastic or deterministic thermostats [53][54][55][56][57].…”

“…We have proposed three splitting schemes, namely the "diagonalization", "first-order expansion", and "hyperbolic function" methods, the details of which are described in Ref. [20].…”

“…(4.18) must be coupled to a thermostating method to ensure a proper canonical distribution. For many thermostats, the integration in one time step ∆t can be split into three parts [20,[53][54][55][56][57], the steps for updating coordinates, momenta, and thermostat, denoted as "R", "p", and "T", respectively. In this case the "equations of motion" may be expressed as A useful approach is to employ the forward Kolmogorov equation to express the evolution of the density distribution in the phase space [20,[53][54][55][56][57]…”

“…The purpose of this paper is to present our implementation of the ML-MCTDH theory to evaluate equilibrium reduced density matrix (ERDM) for the systembath model and compare the results to those obtained by the multi-electronic-state path integral molecular dynamics (MES-PIMD) approach [20]. The ML-MCTDH work is a generalization of the previous work on calculating energy eigenstates and other thermodynamic quantities [21][22][23][24][25].…”

“…In Section II we will summarize the ML-MCTDH theory, followed by a discussion on practical calculations of ERDM in Section III. Then we will provide in Section IV a brief introduction to the recently developed MES-PIMD approach [20]. In Section V we will present examples on the spin-boson model.…”