Atomic Structure Theory

4.4 Atoms with One or Two Valence Electrons 121
Ev = Ecore + ɛv +(VHF − U) vv . (4.73)
If we let U be the Hartree-Fock potential of the core, then the valence orbital
is just the V N−1
HF orbital discussed in the previous section. As we found
previously, ɛv is the difference between the energy of the atom and ion. This
rule will, of course, be modified when we consider corrections from higherorder
perturbation theory. For atoms with one valence electron, the secondquantization
approach leads easily to results obtained previously by evaluating
matrix elements using Slater determinants.
Two valence electrons:
Now, let us turn to atoms having two valence electrons. As an aid to evaluating
the energy for such atoms, we make use of the identities
〈0c|awav : a †
i a†
j alak : a † va † w|0c〉 =(δivδjw − δjvδiw) ×
(δkvδlw − δlvδkw) , (4.74)
〈0c|awav : a †
i aj : a † va † w|0c〉 = δivδjv + δiwδjw . (4.75)
From these identities, we find for the lowest-order energy,
and for the first-order energy,
E (0)
vw = 〈vw|H0|vw〉 = E0 + ɛv + ɛw , (4.76)
E (1)
vw = 〈vw|V |vw〉
= V0 +(VHF − U) vv +(VHF − U) ww + gvwvw − gvwwv . (4.77)
Combining, we find to first order
Evw = Ecore +ɛv +ɛw +(VHF − U) vv +(VHF − U) ww +gvwvw −gvwwv . (4.78)
For the purpose of illustration, we assume that U = VHF in (4.78), and we
measure energies relative to the closed core. We then have E (0)
vw = ɛv + ɛw and
E (1)
vw = gvwvw − gwvvw. As in the case of helium, the degenerate states v and
w can be combined to form eigenstates of L2 , Lz, S2 and Sz. The expression
for E (1) in an LS basis is found from (4.50) to be:
E (1)
vw,LS = η2
k
(−1) L+k+lv+lw
+(−1) S+k+lv+lw
lv lw L
lv lw k
lv lw L
lw lv k
Xk(vwvw)
Xk(vwwv)
. (4.79)
Here, η =1/ √ 2 for the case of identical particles (nv = nw and lv = lw), and
η = 1 otherwise. For the identical-particle case, the sum L + S must be an
even integer.