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.
Substituting eq59 in and using
i) eq65
ii) the Hermitian property of
iii) the Hermitian property of twice
yields
Substitute eq62 and eq63 in the above equation,
Substituting and
in eq85 and simplifying gives
where .
Given that any expectation value of a physical property of an electron is invariant to a unitary transformation of the electron’s wavefunction 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.
