Hartree-Fock theory variational conditions

Although we may keep the redundant parameters fixed (equal to zero) during the optimization of the Hartree-Fock state, we are also free to vary them so as to satisfy additional requirements on the solution - that is, requirements that do not follow from the variational conditions. In canonical Hartree-Fock theory (discussed in Section 10.3), the redundant rotations are used to generate a set of orbitals (the canonical orbitals) that diagonalize an effective one-electron Hamiltonian (the Fock operator). This use of the redundant parameters does not in any way affect the final electronic state but leads to a set of MOs with special properties. 

Under the Hartree-Fock (i.e., HF) approximation, the function of in variables for the solutions of the electronic Hamiltonian is reduced to n functions, which are referenced as molecular orbitals (MOs), each dependent on only three variables. Each MO describes the probability distribution of a single electron moving in the average field of all other electrons. Because of the requirements of the Pauli principle or antisymmetry with respect to the interchange of any two electrons, and indistinguishability of electrons, the HF theory is to approximate the many-electron wavefunction by an antisymmetrized product of one-electron wavefunctions and to determine these wavefunctions by a variational condition applied to the expected value of the Hamiltonian in the resulting one-electron equations,... 

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