1549380323-Statistical Mechanics Theory and Molecular Simulation

564 Quantum time-dependent statistical mechanics

definition holds for operatorsAˆandBˆthat are functions of momentum only. As dis-
cussed by Hernandezet al.(1995), eqn. (14.6.21) can be generalized for operators that
are functions of both position and momentum using a procedure known as “Weyl op-
erator ordering”(Weyl, 1927; Hilleryet al., 1984), which we alluded to in Section 9.2.5
(see eqn. (9.2.51)). CMD is exact in the classical limit and in the limit of a purely har-
monic potential. Away from this limit, position autocorrelation functions are accurate
up toO( ̄h^3 ) (Martyna, 1996) for short times up toO(t^6 ) (Braams and Manolopoulos,
2006).

14.6.2 Ring-polymer molecular dynamics

The method known as ring-polymer molecular dynamics (RPMD), originally intro-
duced by Craig and Manolopoulos (2004), is motivated by the primitivepath-integral
algorithm of eqn. (12.6.4). Craig and Manolopoulos posited that the primitive equa-
tions of motion could be used to extract approximate real-time information. Indeed,
like CMD, the dynamics generated by eqns. (12.6.4) possess the correct harmonic and
classical limits.
The principal features that distinguish RPMD from CMD are threefold. First, the
RPMD fictitious masses are chosen such that each imaginary time sliceor bead has the
physical massm. Second, RPMD uses the full chain to approximate time correlation
functions. Thus, a quantum observableAˆ(ˆx) is assumed to evolve in time according to

AP(t) =

1

P

∑P

k=1

a(xi(t)). (14.6.22)

RPMD approximates the Kubo-transformed time correlation functionKAB(t) as

KAB(t)≈

1

(2π ̄h)PQP(N,V,T)

∫

dPxdPp AP(0)BP(t) e−βPHcl,P(x,p), (14.6.23)

whereβP=β/P(RPMD simulations are typically carried out atP times the actual
temperature) and

Hcl,P(x,p) =

∑P

k=1

p^2 k

2 m

+

m

2 βP^2 ̄h^2

∑P

k=1

(xk−xk+1)^2 +

∑P

k=1

U(xk) (14.6.24)

withxP+1=x 1. Note that the harmonic bead-coupling and potential energy terms
are taken to beP times larger than their counterparts in eqn. (12.6.3). We adopt
this convention for consistency with Craig and Manolopoulos (2004); it amounts to
nothing more than a rescaling of the temperature fromTtoPT. For operators linear
in position or momentum, the CMD and RPMD representations of observables are the
same; however, they generally differ for functions that are nonlinear in these variables.
The third difference is that RPMD is purely Newtonian. The equations of motion
are easily derived from eqn. (14.6.24):