Ring-polymer molecular dynamics rate-theory in the deep-tunneling regime: Connection with semiclassical instanton theory

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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%

Section: Introductionmentioning

confidence: 99%

Non-equilibrium dynamics from RPMD and CMD

Welsch

Song

Shi

et al. 2016

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“…The choice of global coordinates is not unique. We will find it convenient to use the normal modes of a free ringpolymer, 36,47 namely the linear combinations of q l that diagonalize T N (p, q) of Eq. (16).…”

Section: Definitionmentioning

confidence: 99%

Boltzmann-conserving classical dynamics in quantum time-correlation functions: “Matsubara dynamics”

Hele

Willatt

Muolo

et al. 2015

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109

225

We show that a single change in the derivation of the linearized semiclassical-initial value representation (LSC-IVR or 'classical Wigner approximation') results in a classical dynamics which conserves the quantum Boltzmann distribution. We rederive the (standard) LSC-IVR approach by writing the (exact) quantum time-correlation function in terms of the normal modes of a free ring-polymer (i.e. a discrete imaginary-time Feynman path), taking the limit that the number of polymer beads N → ∞, such that the lowest normal-mode frequencies take their 'Matsubara' values. The change we propose is to truncate the quantum Liouvillian, not explicitly in powers of 2 at 0 (which gives back the standard LSC-IVR approximation), but in the normalmode derivatives corresponding to the lowest Matsubara frequencies. The resulting 'Matsubara' dynamics is inherently classical (since all terms O( 2 ) disappear from the Matsubara Liouvillian in the limit N → ∞), and conserves the quantum Boltzmann distribution because the Matsubara Hamiltonian is symmetric with respect to imaginary-time translation. Numerical tests show that the Matsubara approximation to the quantum timecorrelation function converges with respect to the number of modes, and gives better agreement than LSC-IVR with the exact quantum result. Matsubara dynamics is too computationally expensive to be applied to complex systems, but its further approximation may lead to practical methods.

“…50 It is a major goal to formulate a nonadiabatic theory which is able to compute the rate constant over a wide range of coupling strengths, , for complex systems. It is obvious that in the adiabatic limit, where the Born-Oppenheimer approximation is valid, such a theory should reduce to something equivalent to classical rate theory, classical TST or ring-polymer TST, 51,52 which is known to be exact in the absence of recrossing. 53,54 In this limit, nonadiabatic dynamical approaches 40,41 can be applied to compute the small amount of dividing-surface recrossing due to nonadiabatic effects.…”

Section: Introductionmentioning

confidence: 99%

Non-oscillatory flux correlation functions for efficient nonadiabatic rate theory

Richardson

Thoss

2014

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Quantum mechanical canonical rate theory: A new approach based on the reactive flux and numerical analytic continuation methods J. Chem. Phys. 112, 2605 (2000); 10.1063/1.480834Erratum: "A unified framework for quantum activated rate processes. II. The nonadiabatic limit" [J. Chem. Phys. 106, 1769 There is currently much interest in the development of improved trajectory-based methods for the simulation of nonadiabatic processes in complex systems. An important goal for such methods is the accurate calculation of the rate constant over a wide range of electronic coupling strengths and it is often the nonadiabatic, weak-coupling limit, which being far from the Born-Oppenheimer regime, provides the greatest challenge to current methods. We show that in this limit there is an inherent sign problem impeding further development which originates from the use of the usual quantum flux correlation functions, which can be very oscillatory at short times. From linear response theory, we derive a modified flux correlation function for the calculation of nonadiabatic reaction rates, which still rigorously gives the correct result in the long-time limit regardless of electronic coupling strength, but unlike the usual formalism is not oscillatory in the weak-coupling regime. In particular, a trajectory simulation of the modified correlation function is naturally initialized in a region localized about the crossing of the potential energy surfaces. In the weak-coupling limit, a simple link can be found between the dynamics initialized from this transition-state region and an generalized quantum goldenrule transition-state theory, which is equivalent to Marcus theory in the classical harmonic limit. This new correlation function formalism thus provides a platform on which a wide variety of dynamical simulation methods can be built aiding the development of accurate nonadiabatic rate theories applicable to complex systems. © 2014 AIP Publishing LLC. [http://dx