Abstract
Imaginary-time path-integral or ‘ring-polymer’ methods have been used to simulate quantum (Boltzmann) statistical properties since the 1980s. This article reviews the more recent extension of such methods to simulate quantum dynamics, summarising the chain of approximations that links practical path-integral methods, such as centroid molecular dynamics (CMD) and ring-polymer molecular dynamics (RPMD), to the exact quantum Kubo time-correlation function. We focus on single-surface Born–Oppenheimer dynamics, using the infrared spectrum of water as an illustrative example, but also survey other recent applications and practical techniques, as well as the limitations of current methods and their scope for future development.Graphic abstract
Highlights
Since the 1980s, imaginary-time path-integrals [1] or ‘ring-polymers’ have been exploited as a practical technique for simulating quantum Boltzmann statistical properties [2,3,4,5].1 This review addresses the more recent extension of such methods to calculate the dynamical properties of atomic nuclei in systems which are too large to treat by wave-function methods
After reviewing jagged and smooth imaginary-time Feynman paths and their application to thermal expectation values (Sect. 2), we describe the chain of theoretical approximations that links the exact quantum dynamics of imaginary-time Feynman paths (Sect. 3) through Matsubara dynamics (Sect. 4) to mean-field Matsubara dynamics and centroid molecular dynamics (CMD) (Sect. 5) and (T) ring-polymer molecular dynamics (RPMD)
This compactness ensures that the dynamics on F0(R0) gives a good approximation to the Matsubara dynamics of the polar centroid, and that the polar centroid is close to the cartesian centroid, allowing it to be used in place of the cartesian centroid when computing time-correlation functions
Summary
Since the 1980s, imaginary-time path-integrals [1] or ‘ring-polymers’ have been exploited as a practical technique for simulating quantum Boltzmann statistical properties [2,3,4,5] (and in some cases Bosonic statistical properties [5,6,7]). This review addresses the more recent extension of such methods to calculate the dynamical properties of atomic nuclei in systems which are too large to treat by wave-function methods. The aim is not to review all that has been done using pathintegral dynamics methods, since there are already several excellent reviews in the literature [8,9,10,11], nor is it to cover allied topics, such as real-time path-integral methods [12,13,14,15,16,17] or the application of static path-integral methods to infer dynamical properties (e.g., tunnelling splittings [18,19] or quantum instanton rates [20]). I hope to convince a perhaps sceptical reader that (imaginary-time) path-integral dynamics methods are approximations to the exact quantum dynamics, which are necessarily crude when compared with wavefunction methods, but are often good enough to capture at least the essential physics.
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