1549380323-Statistical Mechanics Theory and Molecular Simulation

Problems 213

4.7. A classical system ofNnoninteracting diatomic molecules enclosed in a cubic

box of lengthLand volumeV =L^3 is held at a fixed temperatureT. The

Hamiltonian for a single molecule is

h(r 1 ,r 2 ,p 1 ,p 2 ) =

p^21

2 m 1

+

p^22

2 m 2

+ǫ|r 12 −r 0 |,

wherer 12 =|r 1 −r 2 |is the distance between the atoms in the diatomic.

a. Calculate the canonical partition function.

b. Calculate the Helmholtz free energy.

c. Calculate the total internal energy.

d. Calculate the heat capacity.

e. Calculate the mean-square molecular bond length

〈

|r 1 −r 2 |^2

〉

.

4.8. Write a program to integrate the Nos ́e–Hoover chain equations for a harmonic

oscillator with massm= 1, frequencyω= 1, and temperaturekT= 1 using

the integrator of Section 4.11. Verify that the correct momentumand position

distributions are obtained by comparison with the analytical results

f(p) =

1

√

2 πmkT

e−p

(^2) / 2 mkT
, f(x) =

√

mω^2

2 πkT

e−mω

(^2) x (^2) / 2 kT
.
∗4.9. Consider a system ofNparticles subject to a single holonomic constraint
σ(r 1 ,...,rN)≡σ(r) = 0.
Recall that the equations of motion derived using Gauss’s principle ofleast
constraint are
r ̇i=
pi
mi
p ̇i=Fi−

[∑

jFj·∇jσ/mj+

∑

∑ j,k∇j∇kσ··pjpk/(mjmk)

j(∇jσ)

(^2) /mj

]

∇iσ.

Show using the techniques of Section 4.9 that these equations of motion gen-

erate the partition function

Ω =

∫

dNpdNrZ(r)δ(H(r,p)−E)δ(σ(r))δ( ̇σ(r,p)),

where

Z(r) =

∑N

i=1

1

mi

(

∂σ

∂ri

) 2

.

This result was first derived by Ryckaert and Ciccotti (1983).