Canonical Hartree-Fock equations

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.


Show that


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


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.


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