Frozen core approximation, combination with

Full ab initio treatments for complex transition metal systems are difficult owing to the expense of accurately simulating all of the electronic states of the metal. Much of the chemistry that we are interested in, however, is localized around the valence band. The basis functions used to describe the core electronic states can thus be reduced in order to save on CPU time. The two approximations that are typically used to simplify the basis functions are the frozen core and the pseudopotential approximations. In the frozen core approximations, the electrons which reside in the core states are combined with the nuclei and frozen in the SCF. Only the valence states are optimized. The assumption here is that the chemistry predominantly takes place through interactions with the valence states. The pseudopotential approach is similar. 

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. 

While plane waves are a good representation of delocalized Kohn-Sham orbitals in metals, a huge number of them would be required in the expansion (O Eq. 7.67) to obtain a good approximation of atomic orbitals, in particular near the nucleus where they oscillate rapidly. Therefore, in order to reduce the size of the basis set, only the valence electrons are treated explicitly, while the core electrons (i.e., the inner shells) are taken into account implicitly through pseudopotentials combining their effect on the valence electrons with the nuclear Coulomb potential. This frozen core approximation is justified as typically only the valence electrons participate in chemical interactions. To minimize the number of basis functions the pseudopotentials are constructed in such a way as to produce nodeless atomic valence wavefunctions. Beyond a specified cutoff distance from the nucleus, Ecut the nodeless pseudo-wavefunctions are required to be identical to the reference all-electron wavefunctions. 

A third approach to include scalar relativistic effects in chemical shift calculations has been suggested, namely a combination of a first-order quasirelativistic approach with the frozen-core approximation. These authors have shown that the frozen-core approximation is valid if implemented properly. First results suggest good accuracy for this approach. ... 

When combined with a frozen core, this approximation is equivalent to pseudopotential or ab initio model potential methods, which are developed in chapter 20. These methods incorporate relativistic effects into a one-electron operator for heavy atoms, and the rest of the molecule is treated nonrelativistically. 

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