Projection operators valence spinor

In Eq. (1), the projection operators ljm> Dirac spinors, refers to the so-called residual RECP term, where L and J are taken as one larger than the largest angular momentum quantum numbers of the core electrons. These RECPs are calculated in numerical form, but are re-expressed as expansions in Gaussian-type orbitals (GTOs) to facilitate their use in molecular electronic structure codes that employ GTO basis sets for representing the valence electrons. 

This is not as bad as it sounds. If we ensure that the spinors are orthonormal on the valence projection operator,... 

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,... 

If 1 , 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 the core orthogonality is retained, there is no necessity to insert the projection operators around the core Fock operator in (20.5), but we must still ensure that core spinors are kept out of the valence space. In an atom it is easy to maintain the orthogonality, but in a molecule the basis functions on another center expand into a linear combination of functions on the frozen-core center, including core spinors. 

The equation that is used to ensure that the core spinors are projected out is the Huzinaga-Cantu equation (Huzinaga and Cantu 1971, Huzinaga et al. 1973), which we will now derive. Instead of writing the Fock equations for the valence spinors in terms of Lagrange multipliers and imposing the criterion that the overlaps must be zero at convergence, we can introduce projection operators into the Fock operator ... 

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