Exact quantum master equation via the calculus on path integrals

We investigate the calculation of approximate non-equilibrium quantum time correlation functions (TCFs) using two popular path-integral-based molecular dynamics methods, ring-polymer molecular dynamics (RPMD) and centroid molecular dynamics (CMD). It is shown that for the cases of a sudden vertical excitation and an initial momentum impulse, both RPMD and CMD yield non-equilibrium TCFs for linear operators that are exact for high temperatures, in the t = 0 limit, and for harmonic potentials; the subset of these conditions that are preserved for non-equilibrium TCFs of non-linear operators is also discussed. Furthermore, it is shown that for these non-equilibrium initial conditions, both methods retain the connection to Matsubara dynamics that has previously been established for equilibrium initial conditions. Comparison of non-equilibrium TCFs from RPMD and CMD to Matsubara dynamics at short times reveals the orders in time to which the methods agree. Specifically, for the position-autocorrelation function associated with sudden vertical excitation, RPMD and CMD agree with Matsubara dynamics up to O(t4) and O(t1), respectively; for the position-autocorrelation function associated with an initial momentum impulse, RPMD and CMD agree with Matsubara dynamics up to O(t5) and O(t2), respectively. Numerical tests using model potentials for a wide range of non-equilibrium initial conditions show that RPMD and CMD yield non-equilibrium TCFs with an accuracy that is comparable to that for equilibrium TCFs. RPMD is also used to investigate excited-state proton transfer in a system-bath model, and it is compared to numerically exact calculations performed using a recently developed version of the Liouville space hierarchical equation of motion approach; again, similar accuracy is observed for non-equilibrium and equilibrium initial conditions.

Section: B Excited-state Proton Transfermentioning

confidence: 99%

Non-equilibrium dynamics from RPMD and CMD

Welsch

Song

Shi

et al. 2016

“…Both unbiased and biased two-site systems are considered with ε 12 = 0 and 100 cm −1 . To be compared with the calculation of the hierarchy equation, [35][36][37][38] a quantum bath with the Debye spectral density is applied, given by…”

Section: Quantum Kinetic Expansion In a Two-site Systemmentioning

confidence: 99%

“…In this paper, all the exact quantities are obtained using the hierarchy equation method. [35][36][37][38] In Sec. IV, the RQKE rate kernels are used to predict the detailed population evolution and are calibrated with the exact result.…”

Section: Introductionmentioning

confidence: 99%

A continued fraction resummation form of bath relaxation effect in the spin-boson model

Gong

Tang

Mukamel

et al. 2015

Articles you may be interested inIn the spin-boson model, a continued fraction form is proposed to systematically resum high-order quantum kinetic expansion (QKE) rate kernels, accounting for the bath relaxation effect beyond the second-order perturbation. In particular, the analytical expression of the sixth-order QKE rate kernel is derived for resummation. With higher-order correction terms systematically extracted from higher-order rate kernels, the resummed quantum kinetic expansion approach in the continued fraction form extends the Pade approximation and can fully recover the exact quantum dynamics as the expansion order increases. C 2015 AIP Publishing LLC. [http://dx

“…For Gaussian bath, exact QDT can be formulated with path integral 1,2,3 or its differential version in terms of hierarchical equations of motion (HEOM). 4,5,6,7,8,9,10,11,12,13,14,15,16,17 However, exact approaches are numerically expensive in general. In this paper, we propose an approximate HEOM theory, which will be termed hereafter as hierarchical quantum master equation (HQME).…”

Section: Introductionmentioning

confidence: 99%

Hierarchical quantum master equation with semiclassical Drude dissipation

Tian

et al. 2009

Self Cite

We propose a nonperturbative quantum dissipation theory, in term of hierarchical quantum master equation. It may be used with a great degree of confidence to various dynamics systems in condensed phases. The theoretical development is rooted in an improved semiclassical treatment of Drude bath, beyond the conventional high temperature approximations. It leads to the new theory a simple modification but important improvement over the conventional stochastic Liouville equation theory, without extra numerical cost. Its broad range of validity and applicability is extensively demonstrated with two-level electron transfer model systems, where the new theory can be considered as the modified Zusman equation. We also present a criterion, which depends only on the system-bath coupling strength, characteristic bath memory time, and temperature, to estimate the performance of the hierarchical quantum master equation.