15.3 Expectation value of the Hamiltonian with a given Slater determinant 501
is readily reduced to ∫
Φ∗Hˆ 0 Φdτ=N!
∫
ΦH∗Hˆ 0 AΦHdτ,where we have used eqs. (15.13) and (15.14). The next step is to replace the anti-symmetry
operator by its definition eq. (15.10) and to replaceHˆ 0 with the sum of one-body operators
∫
Φ∗Hˆ 0 Φdτ=N
∑
i= 1∑
P(−)P
∫
ΦH∗hˆiPΦHdτ. (15.15)The integral vanishes if two or more electrons are permuted in only one of the Hartree-
functionsΦHbecause the individual orbitals are orthogonal. We obtain then
∫
Φ∗Hˆ 0 Φdτ=N
∑
i= 1∫
ΦH∗hˆiΦHdτ. (15.16)Orthogonality allows us to further simplify the integral, and we arrive at the following expres-
sion for the expectation values of the sum of one-body Hamiltonians
∫
Φ∗Hˆ 0 Φdτ=N
∑
μ= 1∫
ψμ∗(ri)hˆiψμ(ri)dri. (15.17)The expectation value of the two-body Hamiltonian is obtained in a similar manner. We
have ∫
Φ∗HˆIΦdτ=N!
∫
Φ∗HAHˆIAΦHdτ, (15.18)which reduces to ∫
Φ∗HˆIΦdτ=N
∑
i≤j= 1∑
P(−)P
∫
ΦH∗1
ri j
PΦHdτ, (15.19)by following the same arguments as for the one-body Hamiltonian. Because of the dependence
on the inter-electronic distance 1 /ri j, permutations of two electrons no longer vanish, and we
get
∫
Φ∗HˆIΦdτ=
N
∑
i<j= 1∫
ΦH∗1
ri j
( 1 −Pi j)ΦHdτ. (15.20)wherePi jis the permutation operator that interchanges electronsiandj. Again we use the
assumption that the orbitals are orthogonal, and obtain
∫
Φ∗HˆIΦdτ=1
2
N
∑
μ= 1N
∑
ν= 1[∫
ψ∗μ(ri)ψ∗ν(rj)1
ri jψμ(ri)ψν(rj)dxidxj −∫
ψ∗μ(ri)ψν∗(rj)1
ri jψν(ri)ψμ(ri)dxidxj]
.
(15.21)
The first term is the so-called direct term or Hartree term, while the second is due to the
Pauli principle and is called the exchange term or the Fock term. The factor 1 / 2 is introduced
because we now run over all pairs twice.
Combining Eqs. (15.17) and (15.21) we obtain the functional
E[Φ] =
N
∑
μ= 1∫
ψμ∗(ri)hˆiψμ(ri)dri+1
2
N
∑
μ= 1N
∑
ν= 1[∫
ψμ∗(ri)ψν∗(rj)1
ri j
ψμ(ri)ψν(rj)dridrj−(15.22)−
∫
ψ∗μ(ri)ψν∗(rj)1
ri j
ψν(ri)ψμ(rj)dridrj