The canonical Hartree-Fock equations are a set of eigenvalue equations, which when solved iteratively gives the eigenvalues of a modified form of the non-relativistic multi-electron Hamiltonian.
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Answer
Substituting eq75, eq76 and eq102 in and changing the dummy variables
and
to
and
respectively, we have
where .
With reference to eq84, is the matrix element of an
matrix, which is the product of three matrices. In matrix notation,
Since is Hermitian, we can always select a pair of matrices
and
such that
produces a diagonal matrix. Therefore,
and
To derive the canonical equations, we perform a unitary transformation of in eq96, i.e. by substituting eq88, eq92 and eq106 in eq96 to give
where
If we apply the functional variation method as per the previous article, we have
Since , we can remove n terms of
, where
, from
, resulting in
.
Therefore, eq107 becomes
To satisfy the above equation, we select a set of such that n terms in eq108 are zero, ensuring all other variable functions
and
are independent. Using the same logic described in the steps taken from eq20 to eq21, we have the canonical Hartree-Fock equations:
where eq110 is the complex conjugate of eq109.