Energetics of Pseudopotentials

We have so far discussed the theory and characteristics of pseudospinors and the corresponding one-particle pseudopotentials. The next stage is to examine the use of these pseudopotentials in a valence-only Hamiltonian and the properties of the energy and the SCF equations derived from this energy. The desired form of the pseudopotential Hamiltonian is one in which only the one-electron operator is modified. 

The valence energy using the frozen-core Hamiltonian can be written as 

The direct and exchange integrals are given in Mulliken notation by 

Obviously, this difference is not necessarily zero. It would be fairly straightforward to add the condition that this difference must be zero to the equations to determine the pseudospinors. Reproducing the valence energy is, however, only one consideration in obtaining a pseudopotential. More important is the ability of the pseudopotential to reproduce excitation energies, dissociation energies, and potential energy surfaces. 

The SCF equations derived for the pseudopotential from the valenee energy are exactly equivalent to the original SCF equations, provided that the spinor spaee is restricted to noncore spinors. In the atom, it is possible to exclude eore spinors from the calculation, but in molecules, basis functions on one atom can provide a eore-like component on the atom where the pseudopotential resides. In addition, we need to know what happens if the basis set contains a reasonable representation of a core spinor. We must therefore examine what happens when a core spinor is mixed in with a valence pseudospinor. 

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