Semi-Core States

Three LMTO envelopes were used with the tail energies -0.01 Ry, -1 Ry and -2.3 Ry. In the first two of them, s,p,d orbitals were included and in the last one only. s and p were used. It was necessary to treat the Ti 3p and 3-s states in the semi-core state, i.e. to do a so called 2-panel calculation. The basis set for the second panel consisted of 3-s, 3p, 3d orbitals on the Ti sites and 3-s, 3p orbitals on the Si sites. The same quality k-mesh was used in all calculations to ensure maximum cancellation of numerical errors and to obtain accurate energy differences. 

It is sometimes possible to treat these states as core states, but often, especially when alloying with another element, hybridization broadens the valence band and the semi-core states and creates a continuous band, and in this case the semi-core states have to be treated as valence states. 

Figure 4.1. The density of states in hep La. The semi-core states can be seen as two peaks in the density of states located around -1.20 Ry. The valence band can be seen around the Fermi level.

Realistic SIC-LDA results for molecules or solids can only be obtained on the basis of some localization prescription for quasi-degenerate states [72]. Such a scheme essentially consists of using the localized linear combinations of for the evaluation of the SIC functional and has to be applied to all core and semi-core states. It is obvious that such a prescription becomes rather difficult to handle in more complicated molecules involving several types of atoms. One thus has to realize that none of the presently available implicit functionals can compete with the standard LDA or GGA for covalently bond molecules. 

The early LSD calculations of HJ do very poorly, predicting a Z " ground state with a bond length of 2.65 A and a vibrational frequency of 230 cm A With the benefit of hindsight this can probably be attributed to the same technical problem mentioned above for Scj (essentially the inability to handle situations in which the semi-core 3s or 3p electrons extend beyond the muffin-tin radius) which prevented examination of the true ground-state configuration in the region of the minimum. 

All electron LCGTO-LSD(VWN) calculations using a compact basis set yield, for a state, = 2.41 A, co = 380cm and = 2.7 eV (relative to sphericalized atoms). Relativity would likely decrease the distance by 0.05 A or so. A model-potential calculation, including the semi-core 4s and 4p electrons explicitly in the valence shell, changed the above values to 2.42 A, 330 cm" and 3.0 eV. While these results are preliminary, they do represent the most trustworthy theoretical estimates available. 

The transition metal ions, Cu , Ag and Au", all have a d ( 5) electronic state-configuration, with = 3,4 and 5, respectively. The RCEP used here were generated from Dirac-Fock (DF) all electron (AE) relativistic atomic orbitals, and therefore implicitly include the indirect relativistic effects of the core electron on the valence electrons, which in these metal ion systems are the major radial scaling effect. In these RCEP the s p subshells are included in the valence orbital space together with the d, ( + l)s and ( + l)p atomic orbitals and all must be adequately represented by basis functions. The need for such semi-core or semi-valence electrons to be treated explicitly together with the traditional valence orbitals for the heavier elements has been adequately documented The gaussian function basis set on each metal atom consists of the published 4 P3 distribution which is double-zeta each in the sp and n + l)sp orbital space, and triple-zeta for the nd electrons. 

Ab initio calculations for lanthanide solids were performed from the early days of band theory (Dimmock and Freeman, 1964). These pioneering calculations established that physical properties of the lanthanides could be described with the f-states being inert and treated as core states. For example, the crystal structures of the early lanthanides could be determined without consideration of the 4f-states (Duthie and Pettifor, 1977). Also the magnetic structures of the late lanthanides could be evaluated that way (Nordstrom and Mavromaras, 2000). Of course, one needed to postulate the number of s, p, and d valence electrons that is three in the case of a trivalent lanthanide solid or two in the case of a divalent lanthanide solid. Even this valence could be calculated in a semi-phenomenological way without taking the 4f-electrons explicitly into accoimt (Delin et al., 1997). 

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