Wave function valence pseudospinor

Once the projection operators have been introduced we may remove the requirement that the valence spinors should be orthogonal to the core spinors From the properties of determinants, we know that we ean always add a linear combination of the core spinors to the valence spinors without ehanging the total wave function. The resulting spinor we term a pseudospinor,... 

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,... 

The inclusion of virtual spinors in the pseudospinors also represents a departure from the strict frozen-core approximation. Mixing in virtual spinors means that the valence wave function is no longer equivalent to the all-electron version, and that the valence energy is no longer the same. However, we can always project out the appropriate linear combinations of the virtual spinors, as well as the core spinors, to remove their effect, and replace the core projector with a projector that includes these virtual spinors. 

However, if the valence spinors are modified to create pseudospinors—and this applies to both model potentials and pseudopotentials— the expectation of the valence wave function over the property operator is no longer the same as in the all-electron or the frozen-core case. For external fields, where the bulk of the contribution comes from the valence region of space, the deviation of the pseudospinor property integrals from their unmodified counterparts will be small. However, for nuclear fields, the approximation to the core part of the spinors will have serious consequences for the property. Pseudospinors used with pseudopotentials are designed to have vanishing amplitude at the nucleus, and therefore the property integrals will be much smaller than they should be. This means that properties such as NMR shielding constants calculated for the pseudopotential center will inevitably be erroneous. 

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