Strategies for Solving the Kohn-Sham Equations

Computational Strategies for Solving the Kohn-Sham Equations 

Once an approximate XC functional has been selected, the need to solve the KS equations remains. Several strategies have emerged, which differ mainly in two ways the type of basis in which the KS orbitals are expanded and the manner in which the electrostatic component the KS equations is handled. The large number feasible DFT approaches is in sharp contrast to the more familiar HF formalism where the linear combination Gaussian-type orbitals (LCGTO) approach clearly dominates. The wide variation in DFT schemes is a direct result of the foct that exchange is no longer a nonlocal phenomenon as 

It should be first pointed out that the KS equations can be solved numerically in a basis-set-free approach. Such a scheme is used in NUMOL. Assuming the grids employed are sufficiently accurate, this approach has the distinct advantage of being limited only by the quality of the XC energy functional employed and is, therefore, ideally suited for benchmarking their performance on various systems. However, in the vast majority of DFT programs, the KS orbitals, are expressed as a linear combination of atom-centered basis functions. 

One may now ask why such a large number of LCAO approaches are feasible. Expanding the KS orbitals in an LCAO basis as in Eq. [12], the KS equations can be recast in matrix form. 

At this point, we are confronted with the same four-centered integrals 

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