1549380323-Statistical Mechanics Theory and Molecular Simulation

428 Quantum ideal gases

where

Cx=−

3

4

(

3

π

) 1 / 3

e^2. (11.5.56)

As we did for the kinetic energy, the volume factor in eqn. (11.5.55) can be written as
an integral:

Ex=

∫

drCxρ^4 /^3. (11.5.57)

The local density approximation consists in replacing the constant density in eqn.
(11.5.57) with the spatially varying densityρ(r) of a system of interacting electrons.
When this is done, we obtain the local density approximation to the exchange energy:

Ex=

∫

drCxρ^4 /^3 (r). (11.5.58)

The remainder of this section will be devoted to the derivation of eqn. (11.5.55).
Since we are interested in theT= 0 limit, we will make use the zero-temperature
occupation numbers in eqn. (11.5.33), and we will assume that the fermions are elec-
trons (spin-1/2) so that the spin degeneracy factor isg= 2. The first step in the
derivation is to determine the one-particle density matrix using the eigenvalues and
eigenfunctions in eqn. (11.2.15). Substituting these into eqn. (11.5.54) gives

ρ 1 (r,r′) =

1

V

∑

s,s′

∑

m

∑

n

χm(s)χm(s′)e^2 πin·(r−r

′)/L

θ(εF−εn)

=

1

V

∑

s,s′

∑

m

∑

n

δmsδms′e^2 πin·(r−r

′)/L

θ(εF−εn)

=

2

V

∑

n

e^2 πin·(r−r

′)/L

θ(εF−εn)

=

2

V

∫

dne^2 πin·(r−r

′)/L

θ(εF−εn), (11.5.59)

where in the last line, the summation has been replaced by integration, and the factor
of 2 comes from the summation over spin states. At this point, notice thatρ 1 (r,r′)
does not depend onrandr′separately but only on the relative vectors=r−r′.
Thus, we can write the last line of eqn. (11.5.59) as

ρ 1 (r) =

2

V

∫

dne^2 πn·s/Lθ(εF−εn). (11.5.60)

The integral overncan be performed by orienting thencoordinate system such that
the vectorslies along thenzaxis. Then, transforming to spherical polar coordinates
inn, we find thatρ 1 only depends on the magnitudes=|s|ofsand the angle between
sandn:

ρ 1 (s) =

2

V

∫∞

0

dn n^2 θ(εF−εn)

∫ 2 π

0

dφ

∫π

0

dθsinθe^2 πnscosθ/L. (11.5.61)