Chapter 4
Perturbation Theory for the Electron
Propagator
As in the familiar Rayleigh–Schrödinger perturbation theory (RSPT) for the N -
electron ground state discussed in Appendix A.1, the starting point for perturbation
theory is a division of the original hamiltonian (3.1) into two parts,
ˆ
H = ˆ
T + ˆ
V = ˆ
H 0 + ˆ
H I
(4.1)
where ˆ
H 0 , defining the zeroth-order part, is a one-particle hamiltonian associated
with non-interacting particles, and
ˆ
H I = ˆ
H − ˆ
H 0 = ˆ
W + ˆ
V
(4.2)
is the interaction part, comprising the bare electron repulsion, ˆ
V , and, possibly, a
remainder of the one-particle part of the hamiltonian,
ˆ
W = ˆ
T − ˆ
H 0 =
w rs c
†
r c s
(4.3)
In the so-called Møller–Plesset (MP) partitioning,
ˆ
H 0 =
r c
†
r c r
(4.4)
is the Hartree–Fock (HF) hamiltonian, based on the solutions of the HF equations,
t rs +
k
V rk[sk] n k = r δ rs
(4.5)
where n k = 1, 0 denotes the usual (HF) ground-state occupation numbers. Accordingly, the matrix elements of the one-particle interaction part ˆ
W are given by
w rs = t rs − r δ rs = −
k
V rk[sk] n k
(4.6)
© Springer Nature Switzerland AG 2018
J. Schirmer, Many-Body Methods for Atoms, Molecules and Clusters, Lecture
Notes in Chemistry 94, https://doi.org/10.1007/978-3-319-93602-4_4
45