Evaluation of determinantal wavefunctions

From the known values of the Clebsch-Gordan coefficients, determinantal wavefunctions with selected angular momenta can easily be evaluated by adding to the parent function the new non-equivalent electron orbital and taking into account the Clebsch-Gordan coefficient ... 

For many of the commonly used renormalized methods, such as 2ph-TDA, NR2, and ADC(3), the operator space spans the h, p, 2hp, and 2ph subspaces [7,22]. Reference states are built from Hartree-Fock determinantal wavefunctions plus perturbative corrections. The resulting expressions for various blocks of the superoperator Hamiltonian matrix may be evaluated through a given order in the fluctuation potential. 

The calculation of expectation values of operators over the wavefunction, expanded in terms of these determinants, involves the expansion of each determinant in terms of the N expansion terms followed by the spatial coordinate and spin integrations. This procedure is simplified when the spatial orbitals are chosen to be orthonormal. This results in the set of Slater Condon rules for the evaluation of one- and two-electron operators. A particularly compact representation of the algebra associated with the manipulation of determinantal expansions is the method of second quantization or the occupation number representation . This is discussed in detail in several textbooks and review articles - - , to which the reader is referred for more detail. An especially entertaining presentation of second quantization is given by Mattuck . The usefulness of this approach is that it allows quite general algebraic manipulations to be performed on operator expressions. These formal manipulations are more cumbersome to perform in the wavefunction approach. It should be stressed, however, that these approaches are equivalent in content, if not in style, and lead to identical results and computational procedures. 

Fortunatly, the first four quintet states are each the lowest state of a particular symmetry, with single-determinantal wavefunction thus no formal and computational difficulties should have been expected. The same is true for triplets and septets. Only the B and B ll quintets are the succeeding states of II symmetry, the latter being also only the selected component of the parent configuration. The same objections as in the case of VO apply here and indeed, technical difficulties prohibited final evaluation of the first B state and the discrepancy of the latter with the experiment appeared to be larger than for the remaining states. 

The Hartree-Fock energy is the expectation value of the total electronic hamiltonian evaluated over the determinantal wavefunction (42). This is just the sum of the zero order and first-order energy coefficients. Explicitly, they take the form... 

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