1549380323-Statistical Mechanics Theory and Molecular Simulation

Numerical evaluation 475

Q(N,V,T) =

∮

Dr 1 (τ)···DrN(τ)

×exp

{

−

1

̄h

∫β ̄h

0

dτ

∑N

i=1

1

2

mir ̇^2 i(τ) +U(r 1 (τ),...,rN(τ))

}

. (12.5.11)

Using the techniques from Section 12.3, eqn. (12.5.10) yields the following estimators
for the energy and pressure:

ǫP

(

{r(1),...,r(P)}

)

=

dNP

2 β

−

∑P

k=1

∑N

i=1

1

2

miω^2 P

(

r

(k)

i −r

(k+1)

i

) 2

+

1

P

∑P

k=1

U(r

(k)

1 ,...,r

(k)

N)

PP

(

{r(1),...,r(P)}

)

=

NP

βV

−

1

dV

∑P

k=1

∑N

i=1

[

miω^2 P

(

r(ik)−r(ik+1)

) 2

+

1

P

r(ik)·∇r(k)

i

U

]

,

(12.5.12)

where{r(1),...,r(P)}represents the full set ofNparticle paths. IfUhas an explicit
volume dependence, then an additional term

−

1

P

∑P

k=1

∂

∂V

U(r( 1 k),...,r(Nk),V)

must be added to eqn. (12.5.12) (Martynaet al., 1999). In eqn. (12.5.12), the quantum
kinetic energy terms grow linearly withP. From a numerical viewpoint, this is prob-
lematic, as these harmonic terms become quite stiff for systems withstrong quantum
effects and exhibit large, rapid fluctuations, making them difficult to converge. In the
remainder of this chapter, we will discuss numerical techniques forthe evaluation of
path integrals that explicitly address how to handle these stiff harmonic interactions.

12.6 Numerical evaluation of path integrals

Early numerical studies using path integrals to study condensed-phase problems fo-
cused on one or a few quantum particles in simple liquids (Spriket al., 1985; Sprik
et al., 1986; Cokeret al., 1987; Wallqvistet al., 1987; Martynaet al., 1993; Liu and
Berne, 1993). Somewhat later, path integrals were applied to study quantum effects in
bulk fluids such as water (Delbuonoet al., 1991; Chenet al., 2003; Fanourgakiset al.,
2006; Paesaniet al., 2007; Paesani and Voth, 2009; Morrone and Car, 2008) and aque-
ous solutions (Marxet al., 1999; Tuckermanet al., 2002), and in biological processes
such as enzyme catalysis (Hwanget al., 1991; Hwang and Warshel, 1996). Numerical
path integration is also central to studies of lattice gauge theories(Weingarten and
Petcher, 1981; Weingarten, 1982; Brown and Christ, 1988; Brownet al., 1991). The
application of path integrals to many types problems using increasingly sophisticated
models and computational algorithms is now becoming routine. In thissection, we will
discuss the use of molecular dynamics and Monte Carlo techniques toevaluate path in-
tegrals numerically, identifying several technical challenges that affect the construction
of the numerical algorithms and the formulation of the thermodynamic estimators.