Numerical Methods in Quantum Mechanics - Dipartimento di Fisica

Chapter 6
Self-consistent Field
A way to solve a system of many electrons is to consider each electron under the
electrostatic field generated by all other electrons. The many-body problem is
thus reduced to the solution of single-electron Schrödinger equations under an
effective potential. The latter is generated by the charge distribution of all other
electrons in a self-consistent way. This idea is formalized in a rigorous way in
the Hartree-Fock method and in Density-Functional theory. In the following
we will see an historical approach of this kind: the Hartree method.
6.1 The Hartree Approximation
The idea of the Hartree method is to try to approximate the wave functions,
solution of the Schrödinger equation for N electrons, as a product of singleelectron
wave functions, called atomic orbitals. As we have seen, the best
possible approximation consists in applying the variational principle, by minimizing
the expectation value of the energy E = 〈ψ|H|ψ〉 for state |ψ〉.
The Hamiltonian of an atom having a nucleus with charge Z and N electrons
is
H = − ∑ ¯h 2
∇ 2 i − ∑ Zqe
2 + ∑ qe
2 (6.1)
2m
i e r
i i r
〈ij〉 ij
where the sum is over pairs of electrons i and j, i.e. each pair appears only
once. Alternatively:
∑ N−1 ∑ N∑
=
(6.2)
〈ij〉
i=1 j=i+1
It is convenient to introduce one-electron and two-electrons operators:
With such notation, the Hamiltonian is written as
f i ≡ − ¯h2
2m e
∇ 2 i − Zq2 e
r i
(6.3)
g ij ≡ q2 e
r ij
(6.4)
H = ∑ i
f i + ∑ 〈ij〉
g ij (6.5)
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