Atomic Structure Theory

3.1 Two-Electron Systems 75
independent-particle approximation, the energy can be expressed in terms of
the radial wave function as
E1s1s = 〈Ψ1s1s|h0(r1)+h0(r2)+ 1
r12
|Ψ1s1s〉 . (3.25)
The expectation values of the single-particle operators h0(r1) andh0(r2) are
identical. The first term in (3.25) can be reduced to
〈Ψ1s1s|h0(r1)|Ψ1s1s〉 =
∞
0
dr
− 1
2 P1s(r) d2P1s dr
2 − Z
r P 2
1s(r)
. (3.26)
Integrating by parts, and making use of the previously derived expression for
the Coulomb interaction in (3.18), we obtain
∞ dP1s
2 E1s1s = dr − 2
dr
Z
r P 2 1s(r)+v0(1s, r)P 2
1s(r) . (3.27)
0
The requirement that the two-particle wave function be normalized,
〈Ψ1s1s|Ψ1s1s〉 = 1, leads to the constraint on the single electron orbital
N1s =
∞
0
P1s(r) 2 dr =1. (3.28)
We now invoke the variational principle to determine the radial wave functions.
We require that the energy be stationary with respect to variations of
the radial function subject to the normalization constraint. Introducing the
Lagrange multiplier λ, the variational principle may be written
δ(E1s1s − λN1s) =0. (3.29)
We designate the variation in the function P1s(r) byδP1s(r), and we require
δP1s(0) = δP1s(∞) = 0. Further, we note the identity
δ dP1s
dr
With the aid of (3.30) we obtain
δ(E1s1s − λN1s) =2
∞
0
= d
dr δP1s. (3.30)
− d2P1s − 2Z
dr2 r P1s(r)
+2v0(1s, r)P1s(r) − λP1s(r) δP1s(r) . (3.31)
Requiring that this expression vanish for arbitrary variations δP1s(r), satisfying
the boundary conditions, leads to the Hartree-Fock equation