Atomic Structure Theory

7
Introduction to MBPT
In this chapter, we take a step beyond the independent-particle approximation
and study the effects of electron correlation in atoms. One of the simplest and
most direct methods for treating correlation is many-body perturbation theory
(MBPT). In this chapter, we consider first-order MBPT corrections to manybody
wave functions and second-order corrections to the energies, where the
terms first-order and second-order refer to powers of the interaction potential.
Additionally, we give some results from third-order MBPT.
We retain the notation of Chap. 4 and write the many-electron Hamiltonian
H = H0 + VI in normally-ordered form,
H0 =
i
ɛi a †
i ai , (7.1)
VI = V0 + V1 + V2 , (7.2)
V0 =
1
2
a
VHF
− U , (7.3)
aa
V1 =
ij
V2 = 1
2
(∆V ) ij :a †
i aj : , (7.4)
ijkl
gijkl :a †
i a†
j alak : , (7.5)
where (∆V )ij =(VHF − U)ij with (VHF)
ij =
b (gibjb − gibbj) . The normal
ordering here is with respect to a suitably chosen closed-shell reference state
|Oc〉 = a † aa †
b ···a† n|0〉. If we are considering correlation corrections to a closedshell
atom, then the reference state is chosen to be the ground-state of the
atom. Similarly, if we are treating correlation corrections to states in atoms
with one or two electrons beyond a closed-shell ion, the reference state is
chosen to be the ionic ground-state.
We let Ψ be an exact eigenstate of the many-body Hamiltonian H and
let E be the corresponding eigenvalue. We decompose Ψ into an unperturbed
wave function Ψ0 satisfying