1549380323-Statistical Mechanics Theory and Molecular Simulation

Approximations 567

0 5 10 15 20

-0.4

-0.2

0

0.2

0.4

0.6

0.8

K

xx

(t

)

RPMD

CMD

Exact

0 5 10 15 20

t

-0.1

-0.05

0

0.05

0.1

0.15

K

xx

(t

)

RPMD

CMD

Exact

b=1

b=8

Fig. 14.8Kubo-transformed position autocorrelation function for a quartic potential
U(x) =x^4 /4 at inverse temperaturesβ= 1 (top) andβ= 8 (bottom).

potential model of Silvera and Goldman (1978). Following Miller and Manolopoulos
(2005) and Honeet al.(2006), the system is simulated at a temperature ofT= 14
K, a density ofρ=0.0234 ̊A−^1 , andN= 256 molecules subject to periodic boundary
conditions. In addition, we takeP= 32 beads to discretize the path integral, and for
CMD, the adiabaticity parameter is taken to beγ^2 = 0.0444. Fig. 14.9(a) shows the
Kubo-transformed velocity autocorrelation functions for this system from CMD and
RPMD. The two methods appear to be in excellent agreement with each other. In fact,
if these velocity autocorrelation functions are used to compute the diffusion constant
using the Green-Kubo theory in eqn. (13.3.33), we obtainD= 0. 306 ̊A^2 /ps for CMD
andD= 0. 263 ̊A^2 /ps for RPMD, both of which are in reasonable agreement with the
experimental value of 0.4 ̊A^2 /ps (Miller and Manolopoulos, 2005). Interestingly, we see
that the correlation function of this condensed-phase system decays to zero in a short
time, something which is not uncommon in the condensed phase at finite tempera-
ture. In Fig. 14.9(b), we show the imaginary-time mean-square displacementsR^2 (τ)
computed directly from an imaginary-time path-integral calculationand estimated
from the CMD and RPMD approximate real-time correlation functions. Both approx-
imations miss the true imaginary-time data, particularly in the peak region around
τ=β ̄h/2. Theχ^2 error measure for both cases is 0.0089 for RPMD and 0.0056 for