1549380323-Statistical Mechanics Theory and Molecular Simulation

The ideal fermion gas 427

E=−

(

∂

∂β

lnZ(ζ,V,T)

)

ζ,V

, (11.5.49)

from which it follows that

E=

3

2 β

V g

λ^3

∑∞

l=1

(−1)l+1ζl

l^5 /^2

. (11.5.50)

Comparing eqns. (11.5.48) and (11.5.50), we see thatEandPare related by

E=

3

2

PV ⇒ P=

2

3

E

V

. (11.5.51)

Note that the energy and the pressure atT= 0 are not zero. The zero-temperature
values of these quantities are

E=

3

5

〈N〉εF

P=

2

5

〈N〉

V

εF. (11.5.52)

These are referred to as thezero-pointenergy and pressure and are purely quantum
mechanical in nature. The fact that the pressure does not vanishatT= 0 is again
a consequence of the Pauli exclusion principle and the effective repulsive interaction
that also appeared in the low density, high-temperature limit.

11.5.4 Derivation of the local density approximation

In Section 11.5.3, we referred to the local density approximation todensity functional
theory. In this section, we will derive the local density approximation to the exact
exchange energy in density functional theory. The functional wewill obtain is still used
in many density functional calculations and serves as the basis for more sophisticated
approximation schemes. The exact exchange energy is a component of the electronic
structure method known as Hartree–Fock theory. It takes theform

Ex=−

e^2

4

∫

drdr′

|ρ 1 (r,r′)|^2

|r−r′|

, (11.5.53)

whereρ 1 (r,r′) is known as theone-particle density matrix:

ρ 1 (r,r′) =

∑

s,s′

∑

m

∑

n

〈fnm〉φnm(x)φ∗nm(x′). (11.5.54)

Thus, for this calculation, we need both the energy levels and the corresponding eigen-
functions of the quantum ideal gas. We will show that for an ideal gas of electrons,
the exchange energy is given exactly by

Ex=V Cxρ^4 /^3 , (11.5.55)