Core-valence separation theories

Having reviewed the theoretical background to the core-valence separation, we now turn to the practical implementation of the theory. Starting from equations (31)— (34) we note that the valence pseudo-orbitals are eigenfunctions of an equation which can be written as... 

Mukherjee/91/ initially proved LCT for incomplete model spaces having n-hole n—particle determinants, showing also at the same time the validity of the core—valence separation. The corresponding open-shell perturbation theory of Brandow/20/ for such cases leads to unlinked terms and a breakdown of the core-valence separation, which used IN for O. Mukherjee emphasized that it is essential to have a valence-universal wave operator O within a Fock space formulation/91/ such that it also correlates the subduced valence sectors. Later on,... 

ANO basis set used gives a separation in good agreement with, but smaller than, the value deduced from a combination of theory and experiment. From the convergence of the result with expansion of the ANO basis set, it is estimated that the valence limit is about 9.05 + O.lkcal/mole. The remaining discrepancy with experiment is probably mostly due to core-valence correlation effects. However, as the valence correlation treatment is nearly exact, finer effects such as Bom-Oppenheimer breakdown and relativity must also be considered. While FCI calculations have shown that a very high level of correlation treatment is required for an accurate estimate of the CV contribution to the separation, theoretical calculations indicate that CV correlation will increase the separation by at most 0.35 kcal/mole (see later discussion). Therefore, it is now possible to achieve an accuracy of considerably better than one kcal/mole in the singlet-triplet separation in methylene. 

For the HgH system numerical wavefunctions were obtained for Hg using both relativistic (Desclaux programme87 was used) and non-relativistic hamiltonians. The orbitals were separated into three groups an inner core (Is up to 3d), an outer core (4s—4/), and the valence orbitals (5s—6s, 6p). The latter two sets were then fitted by Slater-type basis functions. This definition of two core regions enabled them to hold the inner set constant ( frozen core ) whilst making corrections to the outer set, at the end of the calculation, to allow some degree of core polarizability. The correction to the outer core was done approximately via first-order perturbation theory, and the authors concluded that in this case core distortion effects were negligible. 

His early works on adiabatic separation of valence electrons from fast moving core electrons in atoms attracted much attention a few years later, when the time came for the theory of pseudopotentials. These papers are still being referred to, more than forty years after their publication. 

Multiple line core spectra are produced also if the atom has an open valence shell, provided that the crystalline environment has not wiped out the J, L, S, M quantization of that shell the core vacancy is variously coupled to the open shell to yield a set of final states. For example, if the open shell has the one-electron orbital quantum numbers n and l and total spin S, a core s vacancy will be observed in two final states having spins (S +1/2) and (S — 1/2), with the latter spin state lying higher in energy. According to Condon-Slater-Racah theory, the energy separation is... 

In describing non-planar and/or saturated systems, the o-tc separation cannot be maintained and a merely 7c-electron theory cannot be justified. In most cases the treatment of inner shells is still separable from that of the valence shell to a certain degree of approximation. This leads to the so-called all-valence-electron methods. One may either construct a pseudopotential accounting for the effect of the inner-shell electrons on the valence shell, or simply consider a model where each nucleus is replaced by a core having a positive charge of ... 

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