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

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

Substituting in eq85,

Substitute in the above equation and simplifying, we have

where and .

and are known as the Coulomb integral and the exchange integral respectively.

###### Question

Show that .

###### Answer

Since , we can add terms of , where to , resulting in , i.e.

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, eq75, eq76 and eq77 in of eq89 to give

where we have changed the dummy variables from and to and respectively.

Substituting eq82 in the above equation and simplifying

Similarly, substituting eq83 in the above equation and simplifying

Substituting eq74, eq75, eq76 and eq77 in of eq89 and repeating the above logic, we have

Substitute eq90 and eq91 back in eq89

Eq92 is used in the derivation of the Hartree-Fock equations.