Atomic Structure Theory

230 7 Introduction to MBPT
From the Schrödinger equation
one easily shows
(H0 + V )Ψl = ElΨl,
HeffΨ (0)
l
(0)
= ElΨ l , (7.113)
where
Heff = PH0P + PVΩP (7.114)
is an effective Hamiltonian defined within the model space that gives the exact
energy. A perturbation expansion of Heff leads to a perturbation expansion
for the energy.
7.6.2 First-Order Perturbation Theory
The first-order approximation to Heff is PH0P + PVP. Substituting into
(7.113) leads immediately to the d-dimensional eigenvalue equation
〈Φl|H0 + V |Φk〉ck = E (1)
l cl. (7.115)
k
Note that the first-order energy above corresponds to the sum of the zerothand
first-order energy for systems with one valence electron. There is no well
defined counterpart of the zeroth-order energy in cases where the model space
consists of non-degenerate states.
In the magnesium example, we let the index k in (7.115) represent the
two-particle state vw(JM) and the index l represent the state xy(JM). The
eigenvalue equation then becomes
where
v≤w
x≤y
Vxy,vw = ηxyηvw
[(ɛv + ɛw)δvxδwy + Vxy,vw] cvw = Ecxy,
(−1)
L
J+L+jy+jv
jx jy J
TL(xyvw)
jw jv L
+(−1) L+jy+jv
jx jy J
TL(xywv) . (7.116)
jv jw L
In the relativistic case, TL(rskl) =XL(rskl)+BL(rskl)isthesumofCoulomb
(7.38) and Breit (7.92) interaction matrix elements. In first order, we are
led to a d-dimension two-particle CI calculation, essentially identical to that
described in Section 7.5.1. The basic orbitals here, however, are obtained in
the V (N−2) HF potential of the ionic core.
As an illustration, we consider the magnesiumlike ion P +3 . The diagonal
terms in the first-order Hamiltonian matrix are dominated by sums ɛv + ɛw