1549380323-Statistical Mechanics Theory and Molecular Simulation

494 The Feynman path integral

whereεn(X) are the eigenvalues that result from the solution of the

Schr ̈odinger equation

[

−

̄h^2

2 m

∂^2

∂x^2

+U(x,X)

]

ψn(x;X) =εn(X)ψn(x;X)

for the light degree of freedom at a fixed valueXof the heavy degree

of freedom. This approximation is known as thepath-integral Born–

Oppenheimer approximation(Cao and Berne, 1993). The eigenvalues

εn(X) are the Born-Oppenheimer surfaces.

b. Under what conditions can the sum overnin the above expression be

approximated by a single term involving only the ground-state surface

ε 0 (X)?

12.9. a. Consider making the following transformation in eqn. (12.3.14):

r=

1

2

(x+x′)/ 2 , s=x−x′.

Show that the ensemble average ofAˆ(ˆp) can be written as

〈Aˆ〉=

1

Q(L,T)

∫

dpdr a(p)ρW(r,p),

whereρW(r,p), known as theWigner distribution functionafter Eugene

P. Wigner (1902–1995), is defined by

ρW(r,p) =

∫

dseips/ ̄h

〈

r−

s

2

∣

∣

∣e−β

Hˆ∣∣

∣r+

s

2

〉

.

b. CalculateρW(r,p) for a harmonic oscillator of massmand frequency

ωand show that in the classical limit,ρW(r,p) becomes the classical

canonical distribution.

12.10. Show that the transition path ensemble of Section 7.7 can be formulated as

a kind of path integral when the limitn→∞and ∆t→0 is taken. Give an

explicit functional integral expression for partition function in eqn. (7.7.5).

12.11. Write down a complete set of path-integral molecular dynamics equations of

motion (using Nos ́e-Hoover chain thermostats of lengthM) for the numerical

evaluation of an imaginary-time path integral for a system ofN quantum

particles inddimensions obeying Boltzmann statistics at temperatureT.