A simple model for the treatment of imaginary frequencies in chemical reaction rates and molecular liquids

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: Appendix A: Connection To Condon Approximationmentioning

confidence: 99%

Non-equilibrium dynamics from RPMD and CMD

Welsch

Song

Shi

et al. 2016

“…The LSC-IVR retains the Boltzmann quantum statistics inside a Wigner transform, 26 is exact in the zero-time, harmonic and high-temperature limits, and has been developed into a practical method by several authors. [19][20][21][22][23] However, it has a serious drawback: the classical dynamics does not in general preserve the quantum Boltzmann distribution, and thus the quality of the statistics deteriorates over time.…”

Section: Introductionmentioning

confidence: 99%

“…für Physikalische Chemie, ETH Zürich, CH-8093 Zürich, Switzerland b) Corresponding author: sca10@cam.ac.uk approximation), 3,[15][16][17][18][19][20][21][22][23][24][25][26][27] in which the quantum Liouvillian is expanded as a power series in 2 , then truncated at 0 . Miller 15,16 and later Shi and Geva 17 showed that this approximation is equivalent to linearizing the displacement between forward and backward Feynman paths in the exact quantum time-propagation, which removes the coherences, thus making the dynamics classical.…”

Section: Introductionmentioning

confidence: 99%

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

Hele

Willatt

Muolo

et al. 2015

113

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.

“…[26][27][28]54 In this approach, Feynman paths in the forward and backward real-time propagators are assumed to cancel out unless they are very close together, such that the difference in their actions can be expanded to linear order in position. This approximation has the effect of removing all real-time Feynman paths, except the forwardbackward pairs that lie along classical trajectories joining the start and end points, which reduces the timecorrelation function to its classical Wigner approximation.…”

Section: Introductionmentioning

confidence: 99%

“…* Corresponding author: sca10@cam.ac.uk most famous instance of this being the Wigner-Eyring formula. 13 A more systematic approach is to use semiclassical theories [14][15][16][17][18][19][20][21][22][23][24][25][26][27][28] (some of which we discuss below), or the quantum instanton approach. 29,30 However, there is one approach 20,[31][32][33][34][35][36][37][38][39][40][41][42][43][44][45][46] that is especially relevant to this article, which is to note that classical TST takes the form of a free-energy combined with a flux factor, and then to exploit the fact that it is relatively easy to compute a quantum free energy, using 'ring-polymer' pathintegration.…”

Section: Introductionmentioning

confidence: 99%

Derivation of a true (t → 0+) quantum transition-state theory. I. Uniqueness and equivalence to ring-polymer molecular dynamics transition-state-theory

Hele¹,

Althorpe²

2013

112

215

Surprisingly, there exists a quantum flux-side time-correlation function which has a non-zero t → 0+ limit, and thus yields a rigorous quantum generalization of classical transition-state theory (TST). In this Part I of two articles, we introduce the new time-correlation function, and derive its t → 0+ limit. The new ingredient is a generalized Kubo transform which allows the flux and side dividing surfaces to be the same function of path-integral space. Choosing this common dividing surface to be a single point gives a t → 0+ limit which is identical to an expression introduced on heuristic grounds by Wigner in 1932, but which does not give positive-definite quantum statistics, causing it to fail while still in the shallow-tunnelling regime. Choosing the dividing surface to be invariant to imaginary-time translation gives, uniquely, a t → 0+ limit that gives the correct positivedefinite quantum statistics at all temperatures, and which is identical to ring-polymer molecular dynamics (RPMD) TST. We find that the RPMD-TST rate is not a strict upper bound to the exact quantum rate, but a good approximation to one if real-time coherence effects are small. Part II will show that the RPMD-TST rate is equal to the exact quantum rate in the absence of recrossing.