1549380323-Statistical Mechanics Theory and Molecular Simulation

566 Quantum time-dependent statistical mechanics

0 5 10 15 20

-1

-0.5

0

0.5

1

1.5

K

xx

(t

)

RPMD

CMD

Exact

0 5 10 15 20

t

-0.1

0

0.1

0.2

K

xx

(t

)

RPMD

CMD

Exact

b=1

b=8

Fig. 14.7 Kubo-transformed position autocorrelation function for amildly anharmonic po-
tentialU(x) =x^2 /2+0. 1 x^3 +0. 01 x^4 at inverse temperaturesβ= 1 (top) andβ= 8 (bottom).

and 14.8 show the Kubo-transformed time correlation functionsKxx(t) for these two
problems, respectively, comparing CMD and RPMD to the exact correlation func-
tions, which are available for these one-dimensional examples via numerical matrix
multiplication (Thirumalaiet al., 1983). Since the first potential is very close to har-
monic, we expect CMD and RPMD to perform well compared to the exact correlation
functions, which, as Fig. 14.8 shows, they do. For the strongly anharmonic potential
U(x) =x^4 /4, both methods are poor approximations to the exact correlationfunc-
tion. We notice, however, that the results improve at the higher temperature (lowerβ)
for the mildly anharmonic potential, which is expected, as the higher temperature is
closer to the classical limit. This trend is consistent with a study by Wittet al.(2009),
who found significant deviations of vibrational spectra from the correct results at low
temperatures. In particular, in a subsequent study by Ivanovet al.(2010), CMD was
shown to produce severe artificial redshifts in high-frequency regions of vibrational
spectra (however, see Paesani and Voth (2010)). For the quartic potential, the results
are actually worse at high temperature, indicating that at low temperature (highβ),
the quartic potential is closer to the harmonic limit for which CMD and RPMD are
exact.
The second example is a more realistic one of fluidpara-hydrogen, described by a