1549380323-Statistical Mechanics Theory and Molecular Simulation

568 Quantum time-dependent statistical mechanics

0 0.25 0.5 0.75 1

t (ps)

0

5

10

15

20

K

vv

(t

) (Å

2 /ps

2 )

CMD

RPMD

0 0.035 0.07

t(K

-1

)

0

0.2

0.4

0.6

0.8

1

R

2 (

t) (Å

2 )

CMD-Exact

CMD-Recons.

RPMD-Exact

RPMD-Recons.

Fig. 14.9(a) Velocity autocorrelation functions forpara-hydrogen atT= 14 K for CMD
and RPMD simulations. (b) Exact imaginary-time mean-square displacements and imagi-
nary-time mean-square displacements reconstructed from the approximate CMD and RPMD
real-time correlation functions in part (a).

CMD. Interestingly, although Braams and Manolopoulos (2006) showed that RPMD
is a more accurate approach at very short times, CMD seems to givea slightly better
approximation to the true correlation function overall.

14.7 Problems

14.1. a. Derive eqns. (14.4.7) and (14.4.8).

b. Show that the Fourier transforms of correlation functions〈ˆx(0)ˆx(t)〉and

〈xˆ(t)ˆx(0)〉are related by

1

2 π

∫∞

−∞

dte−iωt〈xˆ(0)ˆx(t)〉= eβ ̄hω

1

2 π

∫∞

−∞

dte−iωt〈xˆ(t)ˆx(0)〉.

c. Show that the Fourier transform of〈ˆx(t)ˆx(0)〉is related to its classical

counterpart by

1

2 π

∫∞

−∞

dte−iωt

1

2

〈[ˆx(t),xˆ(0)]+〉=

β ̄hω

2

tanh(β ̄hω/2)

×

1

2 π

∫∞

−∞

dte−iωt〈x(t)x(0)〉cl.

14.2. a. Derive eqns. (14.6.6) and (14.6.12).