Dirac theory energy surfaces

We now need to know how the probability of finding an electron of specified energy varies across the permitted band. In the first and simplest version of the Electron Band Theory, electrons were assumed to move in a field of uniform positive potential (i.e. ion cores were neglected), and mutual electrostatic repulsion was ignored. Application of the Schrodinger equation and Fermi-Dirac statistics leads to the conclusion that a collection of N electrons at the absolute zero occupies the N/2 lowest levels, those at the maximum being said to be at the Fermi surface Ef. 

For studies of the Fermi surface in the lanthanide compounds, it is necessary to develop a reliable theoretical method in which hybridization of the 4f electrons with other electrons as well as the relativistic effect can be taken into account quantitatively. For that purpose, the relativistic APW method proposed by Loucks (1967) provides a good starting basis. Loucks derived his original method from the Dirac one-electron equation, which is a natural extension of Slater s non-relativistic APW method (Slater 1937). It proved to be a powerful method comparable to a relativistic KKR method (Onodera and Okazaki 1966, Takada 1966). Loucks method does not accocunt for the symmetrization of the wave functions by group theory, nor it is a self-consistent method. These shortcomings are serious limitations for calculations of the energy band structure in the lanthanide compounds. 

Yamagami and Hasegawa carried out a self-consistent calculation of the energy band structure by solving the Kohn-Sham-Dirac one-electron equation by the density-functional theory in a local-density approximation (LDA). This self-consistent, symmetrized relativistic APW approach was applied to many lanthanide compounds and proved to give quite accurate results for the Fermi surface. 

Based on an empirical correlation between adsorption enthalpies of single atoms on Au surfaces with their sublimation enthalpy (see Fig. 38), for Cn a value of A/is = 39( ( kJ moP (= A.% kcal moP ) results [136]. This value is significantly lower compared to a theoretical prediction based on solid-state theory using relativistic Dirac-Kohn-Sham calculations, which predicted that Cn is a semiconductor with a cohesive energy of about 110 kJ moP [138]. 

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