Kohn-Sham equations methodology

For direct Af-electron variational methods, the computational effort increases so rapidly with increasing N that alternative simplified methods must be used for calculations of the electronic structure of large molecules and solids. Especially for calculations of the electronic energy levels of solids (energy-band structure), the methodology of choice is that of independent-electron models, usually in the framework of density functional theory [189, 321, 90], When restricted to local potentials, as in the local-density approximation (LDA), this is a valid variational theory for any A-electron system. It can readily be applied to heavy atoms by relativistic or semirelativistic modification of the kinetic energy operator in the orbital Kohn-Sham equations [229, 384],... 

The methodology focuses, as many density-functional schemes do, on the key role of the electron density. The Schrodinger equation is then solved self-consistently in the Kohn-Sham scheme.86 Initial approaches dealt with a jellium-adatom system, which would at first sight seem rather unchemical, lacking microscopic detail. But there is much physics in such an effective medium theory, and with time the atomic details at the surface have come to be modeled with greater accuracy. 

The details of implementation of scalar relativity in GTOFF were presented in [41] and reviewed in [75], so we summarize the essential assumptions and methodological features here. First, all practical DFT implementations of relativistic corrections of which we are aware assume the validity (either explicitly or implicitly) of an underlying Dirac-Kohn-Sham four-component equation. We do also. The Hamiltonian is therefore a relativistic free particle Hamiltonian augmented by the usual non-relativistic potentials... 

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