1549380323-Statistical Mechanics Theory and Molecular Simulation

404 Quantum ensembles

Q(β) =

∑∞

n=0

e−βEn=

∑∞

n=0

e−β(n+1/2) ̄hω. (10.4.17)

Recalling that the sum of a geometric series is given by

∑∞

n=0

rn=

1

1 −r

, (10.4.18)

where 0< r <1, the partition function becomes

Q(β) = e−β ̄hω/^2

∑∞

n=0

e−nβhω ̄ = e−β ̄hω/^2

∑

n=0

(

e−βhω ̄

)n

=

e−β ̄hω/^2

1 −e−β ̄hω

. (10.4.19)

From the partition function, various thermodynamic quantities canbe determined.
First, the free energy is given by

A=−

1

β

lnQ(β) =

̄hω

2

+

1

β

ln

(

1 −e−β ̄hω

)

, (10.4.20)

while the total energy is

E=−

∂

∂β

lnQ(β) =

̄hω

2

+

̄hωe−β ̄hω

1 −e−β ̄hω

=

(

1

2

+〈n〉

)

̄hω. (10.4.21)

Thus, even if〈n〉= 0, there is still a finite amount of energy, ̄hω/2 in the system. This
residual energy is known as thezero-point energy. Next, from the average energy, the
heat capacity can be determined

C

k

=

(β ̄hω)^2 e−β ̄hω

(1−e−β ̄hω)^2

. (10.4.22)

Note that as ̄h→0,C/k→1 in agreement with the classical result. Finally, the
entropy is given by

S=klnQ(β) +

E

T

=−kln

(

1 −e−β ̄hω

)

+

̄hω

T

e−β ̄hω

1 −e−β ̄hω

, (10.4.23)

which is consistent with the third law of thermodynamics, asS→0 asT→0.
The expressions we have derived for the thermodynamic observables are often used
to estimate thermodynamic quantities of molecular systems under the assumption
that the system can be approximately decomposed into a set of uncoupled harmonic
oscillators corresponding to the normal modes of Section 1.7. By summing the expres-
sions in eqns. (10.4.20), (10.4.22), or (10.4.23) over a set of frequencies generated in
a normal-mode calculation, estimates of the quantum thermodynamic properties such
as free energy, heat capacity, and entropy, can be easily obtained.
As a concluding remark, we note that the formulation of the quantum equilibrium
ensembles in terms of the eigenvalues and eigenvectors ofHˆsuggests that the computa-
tional problems inherent in many-body quantum mechanics have notbeen alleviated.