Shape-Consistent Pseudospinors and Pseudopotentials

The removal of the core projectors from the metric to the potential is only the first step in the removal of all references to the core spinors. The next step is the partitioning of the Fock operator into core and valence parts. 

Our goal is to write the pseudopotential in a form that looks like an all-electron operator plus a one-electron correction term. We would also like to ensure that the pseudopotential has no long-range terms. To do this we partition the nuclear attraction terms as well as the electron repulsion terms into a core and a valence part, 

The valence repulsion terms also need to be partitioned because these are expressed in terms of the spinors, not the pseudospinors. Expanding the spinors using (20.12) and (20.13), the direct potential is 

The first term is the potential for the pseudospinor the remainder, which includes all the core tail terms, goes into the pseudopotential. We can make a similar partitioning for the exchange potential. 

So far, we have not normalized the pseudospinors, but we must do so now because we are using the unit metric with the pseudopotential. As we noted at the end of the previous section, the diagonal elements of the overlap matrix, which are the norms of the pseudospinors, are greater than 1. From (20.40), the normalized pseudospinor is given by 

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