The Slater-Condon rule for a one-electron operator is an expression of the expectation value of the one-electron operator involving Slater determinants.

Let’s consider the expectation value of the one-electron Hamiltonian operator: , where and is given by eq66.

###### Question

Show that the product of and is Hermitian.

###### Answer

or .

With reference to eq64 and the fact that , we have . Using eq65, we have , i.e. the product of and is Hermitian. This implies that , where is also Hermitian.

Substitute eq59 in and using

i) eq65

ii) the Hermitian property of

iii) the Hermitian property of twice

we have

Substitute eq62 and eq63 in the above equation,

Substitute and in eq85 and simplifying,

where .

Given that any expectation value of a physical property of an electron is invariant to a unitary transformation of the electron’s wavefunction that is expressed as a Slater determinant, we substitute eq74 and eq75 in eq86 to give

where we have changed the dummy variable from to .

Substitute eq82 in eq87,

Eq88 is used in the derivation of the Hartree-Fock equations.