Core-valence separation definitions

This classification of the one-electron orbitals is displayed in Figure 10.1. While, due to this definition, it is possible to have completely unoccupied core-valence orbitals or (completely) occupied valence orbitals in the model space, they could be eliminated always by a proper re-definition of the many-electron vacuum state 0). The distinction of the valence states into core-valence and valence orbitals has the great advantage that we need not to deal explicitly with the particular choice of the vacuum in the derivation of perturbation expansions. As we shall see below, namely, the projection upon the vacuum appears rather frequently in the derivations for open-shell structures and can be carried out formally, if all the core and core-valence orbitals (i.e. all the orbitals up to the Fermi level) are taken into account separately from the rest of the orbital functions. 

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. 

There is much precedent but no particular justification for omitting the core orbitals from the molecular calculation. To determine the consequences of (6) separating the core orbitals and electrons, we divide the set of atomic orbitals into two classes, Xp, core orbitals with energies Ep, and valence orbitals for which we retain the symbol . Unlike the Crrti, the coefficients of the core orbitals are not free for variation to minimize the energy but are determined by requiring that arbitrary admixture of the core orbitals in the valence molecular orbitals do not change the energy of the latter. The final matrix equation (6) is of the order of the number of valence orbitals, but the definitions of the S and H matrix elements are modified ... 

The most interesting aspect of the electroific structure of Ceo is what happens to the electrons in the p orbitals, since the electrons in sp orbitals form the a-bonds that correspond to low-energy states, well separated from the corresponding antibonding states. We will concentrate then on what happens to these 60 pz states, one for each C atom. The simplest approximation is to imagine that these states are subject to a spherical effective potential, as implied by the overall shape of the cluster. This effective potential would be the result of the ionic cores and the valence electrons in the a-manifold. A spherical potential would produce eigenstates of definite angular momentum (see Appendix B), with the sequence of states shown in Fig. 13.5. However, there is also icosahedral symmetry introduced by the atomic... 

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