476 The Feynman path integral
12.6.1 Path-integral molecular dynamics
We begin our discussion with the molecular dynamics approach. It must be mentioned
at the outset that molecular dynamics is used hereonlyas a means of sampling the
quantum canonical distribution. No quantum dynamical propertiescan be generated
using the techniques in this subsection. In Chapter 14, we will revisitthe quantum
dynamics problem and see howapproximatedynamical quantities can be generated
within a path-integral molecular dynamics framework.
Let us start by considering, once again, a single particle moving in a one-dimensional
potentialU(ˆx). In eqn. (12.3.9), we introduced the notion of the discrete partition
functionQP(L,T), which is explicitly defined to be
QP(L,T) =
(
mP
2 πβ ̄h^2)P/ 2 ∫
D(L)dx 1 ···dxP×exp{
−
1
̄h∑P
k=1[
mP
2 β ̄h(xk+1−xk)^2 +β ̄h
PU(xk)]}∣∣
∣
∣
∣
xP+1=x 1. (12.6.1)
Eqn. (12.6.1) can be manipulated to resemble the classical canonicalpartition function
of a cyclic polymer chain moving in a classical potentialU(x)/P by recasting the
prefactor as a set of Gaussian integrals over variables we will callp 1 ,...,pP so that
they resemble momenta conjugate tox 1 ,...,xP:
QP(L,T) =
∫
dp 1 ···dpP∫
D(L)dx 1 ···dxP×exp{
−β∑P
k=1[
p^2 k
2 m′+
1
2
mωP^2 (xk+1−xk)^2 +1
P
U(xk)]}∣∣
∣
∣
∣
xP+1=x 1. (12.6.2)
In the exponential of eqn. (12.6.2), we have replaced the prefactor of 1/ ̄hwith aβ
prefactor. We have also introduced a frequencyωP =
√
P/(β ̄h), which we call the
chain frequency, since it is the frequency of the harmonic nearest-neighbor coupling
of our cyclic chain. Finally, the parameterm′appearing in the Gaussian integrals
is formally given bym′=mP/(2π ̄h)^2. However, since the prefactor does not affect
any equilibrium averages, including those used to calculate thermodynamic estima-
tors, we are free to choosem′as we like. The resemblance of the partition function
in eqn. (12.6.2) to that of a classical cyclic polymer chain ofP points led Chandler
and Wolynes (1981) to coin the term “classical isomorphism” and to exploit the iso-
morphism between the classical and approximate (sincePis finite) quantum partition
functions. The classical isomorphism is illustrated in Fig. 12.10. The figure depicts
a cyclic polymer havingP = 8 with a harmonic nearest-neighbor coupling constant
k=mω^2 P. Because the cyclic polymer resembles a necklace, itsP points are often
referred to as “beads.” According to the classical isomorphism, wecan treat the cyclic
polymer using all the techniques we have developed thus far for classical systems to ob-
tain approximate quantum properties, and the latter can be systematically improved
simply by increasingP.