The Generalized Philips-Kleinman Pseudopotential

What we would like to do is to remove all explicit reference to core spinors and to core basis functions from the Hamiltonian, incorporating their effect into some kind of local or nonlocal potential. The foundation for all such approaches is the Philips-Kleinman (1959) procedure, which was generalized by Weeks and Rice (1968). It starts from the all-electron self-consistent field (SCF) equation, which we will assume here is converged. 

is a valence spinor, we may insert the valence projection operator as defined by (20.8) into (20.17) to obtain a projected equation that is satisfied for any valence spinor. 

The sum on the right is restricted to the valence spinors because the terms involving the core spinors vanish. Now that we have introduced the projection operators, the equation is satisfied by the valence pseudospinors (t) of (20.12) as well. This is the SCF equation we would get from the projected frozen-core Hamiltonian. 

If there is only one valence spinor, and we substitute the pseudospinor for the spinor (20.20) simplifies to 

To write this equation in terms of the normal operator plus a correction, we move the projector from the right-hand side and rearrange, 

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The term in the square brackets is the generalized Philips-Kleinman pseudopotential, yGPK... 

Since the core spinors are eigenfunctions of /, the generalized Philips-Kleinman pseudopotential can be written... 

We would also like to partition the Fock operator into a valence and a core part this we will do later. In the rest of this section, we explore the development of the generalized Philips-Kleinman pseudopotential for many valence electrons and many valence spinors. 

The generalized Philips-Kleinman pseudopotential depends on the eigenvalue of the spinor v, unlike the frozen-core pseudopotential, which depends only on the core spinors. The appearance of the term in the pseudopotential came about because we transferred a term from right-hand side of (20.21). This means that we have changed the metric, which has implications for orthogonality that we pursue later. The new operator makes the core spinors degenerate with the valence spinor ... 

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