Pseudospinors normalization

This pseudospinor is normalized on the metric G and is orthogonal to other pseudospinors on the same metric. The coefficients can be determined to minimize the density of the pseudospinor in the core region. For this reason, we will call the sum of core spinors the core tail. 

At this point, we have a valence wave function given in terms of pseudospinors that have little or no core contribution, and a set of valence operators that only operate on the valence space. We do however have explicit use of the valence projection operators, which are composed of infinite sums. What we would like to do is to write the Hamiltonian in terms of the normal Hamiltonian and a correction, which is termed a pseudopotential. This we can do by replacing with 1 - and extracting out the unprojected Hamiltonian,... 

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... 

The exchange terms can be partitioned in an analogous fashion. The final form of the pseudopotential for the normalized pseudospinors is... 

The second consequence is that the pseudopotential has a long-range part that comes from the renormalization term. Beyond the range of the core spinors, the valence electron density from the normalized pseudospinors is less than the Hartree-Fock density because of the normalization. The reduced density results in a reduced Coulomb repulsion from the valence electrons, and the reduction in the Coulomb repulsion is compensated for in the pseudopotential. The decrease in electron density in the outer... 

To ensure that there is no renormalization problem, this pseudospinor must itself be normalized,... 

In practice, it is not the function fy(r) but the function gy(r) that is used to determine the pseudospinor, by polynomial expansion in r. The coefficients are determined by equating the derivatives of the polynomial and the Hartree-Fock spinor at the matching point, rc, with the constraints that the pseudospinor has the minimum number of turning points and is normalized. Further details on the methodology for generating pseudopotentials is given later. 

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