Excited Determinants and Excitation Operators

In terms of the spatial orbitals the Hartree-Fock energy is given as 

In the Roothaan-Hartree-Fock approach (Roothaan, 1951) the molecular orbitals are expanded in atomic orbitals, xn, Eq. (9.4). Solving the Hartree-Fock equations for this ansatz then corresponds to finding the molecular orbital coefficients. 

The variational condition for the Hartree-Fock energy, Eq. (9.13), is then similarly 

Additional eigenfunctions of the Hartree-Fock Hamiltonian can be generated by including virtual orbitals in the determinants instead of some or all of the occupied orbitals. They are normally classified according to their relation to the Hartree-Fock wavefunction, also called the SCF determinant, A determinant in which one 

It is often convenient to express this in terms of general excitation operators which act on the Hartree-Fock determinant. The excitation level is indicated by the subscript i and /x refers to a particular operator of this general excitation level. The whole set of excitation operators of level i is often collected in a column vector denoted by hi- Alternatively, one often expresses excitation operators of a particular level in terms of the single excitation or orbital rotation operators where the subscript ai then refers to the involved virtual and occupied orbitals. 

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