426 Quantum ideal gases
E=
∫
D(V)drE
V
=CK
∫
D(V)drρ^5 /^3 =V CKρ^5 /^3. (11.5.46)In one of the early theories of the electronic structure of multielectron atoms, the
Thomas–Fermi theory, eqns. (11.5.45) and (11.5.46) were used toderive an expres-
sion for the electron kinetic energy. In a fermion ideal gas, the densityρis constant,
whereas in an interacting many-electron system, the densityρvaries in space and is,
therefore, a functionρ(r). A key assumption in the Thomas–Fermi theory is that in
a multielectron atom, the spatial variation inρ(r) is mild enough that the kinetic en-
ergy can be approximated by replacing the constantρin eqn. (11.5.45) withρ(r) and
then performing a spatial integration over both sides. The result isan approximate
kinetic-energy functional given by
T[ρ] =CK∫
drρ^5 /^3 (r). (11.5.47)Since the functional in eqn. (11.5.47) depends on the functionρ(r), it is known as
adensity functional. In 1964, Hohenberg and Kohn proved that the total energy of
a quantum multielectron system can be expressed as a unique functionalE[ρ] of the
densityρ(r) and that the minimum of this functional over the set of all densitiesρ(r)
derivable from the set of all ground-state wave functions leads tothe ground-state
density of the particular system under consideration. The implication is that knowl-
edge of the ground-state densityρ 0 (r) uniquely defines the quantum Hamiltonian of
the system. This theorem has led to the development of the moderntheory of elec-
tronic structure known asdensity functional theory, which has become one of the most
widely used electronic structure methods. The Hohenberg–Kohn theorem amounts to
an existence proof, since the exact form of the functionalE[ρ] is unknown. The kinetic
energy functional in eqn. (11.5.47) is only an approximation to the exact kinetic-energy
functional known as alocal density approximationbecause the integrand of the func-
tional depends only on one spatial pointr. Eqn. (11.5.47) is no longer used for actual
applications because it, together with the rest of Thomas–Fermi theory, is unable to
describe chemical bonding. In fact, Thomas–Fermi theory and itsvariants have been
largely supplanted by the orbital-dependent version of density functional theory intro-
duced by Kohn and Sham (1965); however, important recent workis showing that a
nonlocal version of orbital-free density functional theory that builds on the Thomas–
Fermi model can also yield accurate results in some cases (Huang and Carter, 2010).
In Section 11.5.4 below, we will use our solution to the fermion ideal gasto derive
another approximation commonly used in density functional theory, which is still used
within the Kohn–Sham theory for certain classes of systems.
The pressure atT= 0 can now be obtained straightforwardly. We first recognize
that the pressure is given by the sum in eqn. (11.5.5):
PV
kT=
V g
λ^3∑∞
l=1(−1)l+1ζl
l^5 /^2= lnZ(ζ,V,T). (11.5.48)Recall, however, the total energy can be obtained as a thermodynamic derivative of
the partition function via