My PhD thesis - Condensed Matter Theory - Imperial College London

CHAPTER 2.
MECHANICS
THE SIMPLIFICATION OF MANY-ELECTRON QUANTUM
the optimum solution satisfies the condition:
(
δ
δφ ∗ i (x) 〈Ψ|Ĥ|Ψ〉 − ∑ j
λ j 〈φ j |φ j 〉
)
= 0. (2.8)
The normalisation constraints on the individual φ i are incorporated into this variational
equation through the Lagrange multipliers, λ j . It should be noted that Ĥ is
now the electronic Hamiltonian:
Ĥ({r i }) = − 1 ∑
∇ 2 r
2
i
+ 1 ∑ 1
2 |r i − r j | + ∑ i
i
i≠j
v ext (r i ) (2.9)
where the interaction of each electron with the nuclei has been condensed to the
external potential
v ext (r) = − ∑ α
Z α
|d α − r| . (2.10)
Carrying out the functional differentiation indicated in equation (2.8) gives an effective
Schrödinger equation for each one-electron wave function:
(
− 1 2 ∇2 r + ∑ ∫ )
|φj (r ′ )| 2
|r − r ′ | dr′ + v ext (r) φ i (r) = λ i φ i (r). (2.11)
j≠i
The Lagrange multipliers, λ i , are the equivalent of energy eigenvalues for these effective
Schrödinger equations. However, the total energy cannot be obtained simply
by summing them:
E[Ψ] = 〈Ψ|Ĥ|Ψ〉 ≠ ∑ i
λ i . (2.12)
The use of a wave function of the form described in equation (2.3) is called the
Hartree approximation [33, 34], the most obvious disadvantage of which is that the
wave function is not antisymmetric. The construction of a wave function which is
explicitly antisymmetric leads to Hartree-Fock theory.
2.3 Hartree-Fock theory
A many-electron wave function which is antisymmetric under interchange of electrons
may be constructed from a set of orthonormal one-electron orbitals in the form
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