1549380323-Statistical Mechanics Theory and Molecular Simulation

Approximations 565

x ̇k=

pk

m

p ̇k=−

m

β^2 P ̄h^2

[2xk−xk− 1 −xk+1]−

∂U

∂xk

. (14.6.25)

In this dynamics, no thermostats are used on any of the beads since all beads are
treated as dynamical variables. As discussed in Section 13.2.1, however, the use of
eqn. (14.6.23) assumes that the distribution exp[−βPHcl,P(x,p)] can be adequately
sampled, and as Section 12.6 makes clear, this requires some care. Thus, RPMD is
optimally implemented by performing a fully thermostatted path-integral molecular
dynamics calculation in staging or normal modes. From this trajectory, path configu-
rations are periodically transformed back to primitive variables and saved. From these
saved path configurations, independent RPMD trajectories are initiated, and these
trajectories are then used to compute the approximate Kubo-transformed correlation
function. The position autocorrelation function in RPMD is accuratefor short times
up toO(t^8 ) (Braams and Manolopoulos, 2006).

14.6.3 Self-consistent quality control of time correlation functions

The quality of CMD and RPMD correlation functions is often difficult to assess, and
therefore, it is important to have an internal consistency check for these predicted time
correlation functions. The measure we will propose allows the inherent accuracy of the
CMD or RPMD approximation to be evaluated for a given model withouthaving to
rely on experimental data as the final arbiter.
Recall that a CMD or RPMD simulation yields an approximation to the Kubo-
transformed time correlation function. Consider, for example, the velocity autocorre-
lation functionCvv(t) and its Fourier transformC ̃vv(ω). LetC ̃vv(est)(ω) denote a CMD
or RPMD approximation toC ̃vv(ω). Eqn. (14.6.16) allows us to reconstruct the asso-

ciated imaginary time correlation functionR ̄^2 (τ) fromC ̃vv(est)(ω). The approximation
R ̄^2 (τ) can then be compared directly to the numerically exact mean square displace-
ment functionR^2 (τ) computed from the same simulation (see eqn. (14.6.15)). P ́erez
et al.(2009) suggested a dimensionless quantitative descriptor for thequality of an
approach (CMD/RPMD) can be defined by

χ^2 =

1

β ̄h

∫β ̄h

0

dτ

[ ̄

R^2 (τ)−R^2 (τ)

R^2 (τ)

] 2

. (14.6.26)

If CMD or RPMD were able to generate exact quantum time correlation functions,
thenχ^2 would be exactly zero. Thus, the largerχ^2 , the poorer is the CMD or RPMD
approximation to the true correlation function.
As illustrative examples of the CMD and RPMD schemes and the error measure
in eqn. (14.6.26), we consider two one-dimensional systems with potentials given by
U(x) =x^2 /2 + 0. 1 x^3 + 0. 01 x^4 andU(x) =x^4 /4. These potentials are simulated at
inverse temperatures ofβ= 1 andβ= 8 withP = 8 andP = 32 beads, respec-
tively. For the CMD simulations, the adiabaticity parameter isγ^2 = 0.005. Figs. 14.7