My PhD thesis - Condensed Matter Theory - Imperial College London

CHAPTER 2.
MECHANICS
THE SIMPLIFICATION OF MANY-ELECTRON QUANTUM
The Coulomb operator ĵ represents the interaction of each electron with the average
charge density; the unrestricted sum means that its own contribution to this charge
density is included. This would give rise to an unphysical self-interaction, were it
not for the fact that a corresponding term exists in ˆk, and the two cancel.
An important point about the non-local exchange operator ˆk may be highlighted
by making explicit the dependence of φ i on spin:
φ i (x) = φ i (r, σ) = ζ i (r)δ σσi . (2.21)
Substituting this expression into equation (2.20) gives
ˆkφ i (x) = ∑ j
φ j (x)δ σi σ j
∫ ζ
∗
j (r ′ )ζ i (r ′ )
|r − r ′ |
dr ′ (2.22)
showing that the exchange interaction only affects electrons with like spins. In
Hartree-Fock theory, electrons of like spin are kept apart, and this lowers the energy
expectation value. The exchange energy is the difference between the Hartree and
Hartree-Fock values.
Unfortunately, the theory does nothing to keep electrons of opposite spin apart.
Such electrons interact only via the average charge density appearing in the Coulomb
operator; there is no pairwise interaction to make it unfavourable for electrons of
opposite spin to come together. This means that the ground-state energy calculated
in Hartree-Fock theory is always higher than the true ground-state energy. The
correlation energy is defined as the difference between the exact energy of the system
and that calculated in Hartree-Fock theory.
2.4 Density-functional theory
The theorem underlying density-functional theory, due to Hohenberg and Kohn [35],
states that the ground state of an N-electron system is completely determined by
the one-electron density
N∑
∫
n(r) = |Ψ(x 1 , . . . , x N )| 2 δ(r − r i ) dx 1 · · · dx N . (2.23)
i=1
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