Transformations between determinant and CSF bases

We now have two different and useful sets of orthonormal basis functions for the Fock space at our disposal - Slater determinants and CSFs. Neither set of functions is superior to the other in all respects - the CSFs exhibit the correct spin symmetry and lead to more compact expansions of the wave function but are more complicated than the Slater determinants, which in many situations are far easier to manipulate than the CSFs. Clearly, there will be situations where we would like to transform our representation of the wave function from one basis to the other. Such transformations are discussed in the present subsection. 

For this purpose, we label the Slater determinants m, i, j), where j specifies the type of orbital configuration as characterized by the number of unpaired electrons, i counts the orbital configurations of each type and m specifies the determinants in each configuration. Similarly, the CSFs are labelled n,i, 7), where i and j are the same indices as in the determinant basis and n counts the CSFs in each orbital configuration. Slater determinants and CSFs belonging to different configurations are orthogonal  

We note that the overlap depends on the type of orbital configuration j (as specified by the number of unpaired electrons) and also on the particular determinant m and CSF n that are chosen within this configuration type. However, the overlaps are the same for all orbital configurations with the same number of unpaired electrons see (2.6.10). 

We first assume that the wave function has been expanded in CSFs  

To transform to the determinant basis, we invoke the resolution of the identity and obtain 

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