562 Quantum time-dependent statistical mechanics
transform (see Appendix D). Unfortunately, the reverse process, transforming from
imaginary time back to real time, requires an inverse Laplace transform, which is an
extremely ill-posed problem numerically (see, for example, the discussion by Epstein
and Schotland (2008)). This is the primary reason that the analyticcontinuation of
imaginary time data to real-time data is such an immense challenge (Krilovet al.,
2001).
Before we discuss approximation schemes for quantum time correlation functions,
we need to point out that quantum effects in condensed-phase systems are some-
times squelched due to pronounced decoherence effects. In this case, off-diagonal ele-
ments of the density matrix exp(−βHˆ) tend to be small for large|x−x′|, and con-
sequently, the sums over forward and backward real-time paths are not appreciably
different. This means that there is considerable cancellation between these two sums,
a fact that forms the basis of a class of approximation schemes known assemiclassical
methods. These include the Herman-Kluk propagator (1984), thelinearized semiclas-
sical initial value representation (Miller, 2005), the linearized Feynman–Kleinert path-
integral method (Poulsenet al., 2005; Honeet al., 2008), and the forward–backward
approach (Nakayama and Makri, 2005), to name just a few. Although fascinating and
potentially very powerful, semiclassical approaches also carry a relatively high compu-
tational overhead, and we will not discuss them further here. Rather, we will focus on
two increasingly popular approximation schemes for quantum time correlation func-
tions that are based on the use of the imaginary-time path integral.Although these
schemes are somewhatad hoc, they have the advantage of being computationally in-
expensive and straightforward to implement. They must, however, be used with care
because there is no rigorous basis for this class of methods; because of this, we will
also introduce a procedure for checking the accuracy of their results.
14.6.1 Centroid molecular dynamics
In 1993, J. Cao and G. A. Voth introduced the centroid molecular dynamics (CMD)
method as an approximate technique for computing real-time quantum correlation
functions. The primary object in this approach is the path centroiddefined in eqn.
(12.6.21). Toward the end of Section 12.6.1, we briefly discussed thecentroid potential
of mean force (Feynman and Kleinert, 1986); the CMD approach is rooted in this
concept and in an idea put forth by Gillan (1987) for obtaining approximate quantum
rate constants from the centroid density along a reaction coordinate.
CMD is based on the notion that the time evolution of the centroid on this potential
of mean force surface can be used to garner approximate quantum dynamical properties
of a system. In CMD, the centroid for a single particle in one dimension, denoted here
asxc, is postulated to evolve in time according to the following equations ofmotion:
x ̇c=pc
m
, p ̇c=−dU 0 (xc)
dxc
≡F 0 (xc) (14.6.17)(Cao and Voth, 1994a; Cao and Voth, 1996), wheremis the physical mass,pcis a
momentum conjugate toxc, andU 0 (xc) is the centroid potential of mean force given
by