Kohn-Sham equations numerical basis sets

The application of density functional theory to isolated, organic molecules is still in relative infancy compared with the use of Hartree-Fock methods. There continues to be a steady stream of publications designed to assess the performance of the various approaches to DFT. As we have discussed there is a plethora of ways in which density functional theory can be implemented with different functional forms for the basis set (Gaussians, Slater type orbitals, or numerical), different expressions for the exchange and correlation contributions within the local density approximation, different expressions for the gradient corrections and different ways to solve the Kohn-Sham equations to achieve self-consistency. This contrasts with the situation for Hartree-Fock calculations, wlrich mostly use one of a series of tried and tested Gaussian basis sets and where there is a substantial body of literature to help choose the most appropriate method for incorporating post-Hartree-Fock methods, should that be desired. 

Once the functional has been chosen, one needs to solve the Kohn-Sham equations, typically using some sort of basis set, although numerical approaches have seen some use [17]. Here we will focus on the use of Gaussian basis sets which are used in a number of software packages (Gaussian [18], NWCHEM [19],... 

These expressions can be numerically implemented for a set of coefficients for the initial atomic orbitals in the system, as well as for other basis functions (e.g., of hydrogenic, Gaussian, or Slater type). An alternative method for computational implementation is to self-consistently solve the equations from the Hohenbeig-Kohn-Sham density functional theory, properly modified in order to include the extension of the spin characterization, wherefrom the molecular orbitals corresponding to the electronic distribution and of spin may directly result, hence, retaining only the HOMO and LUMO orbitals in the electronic frozen-core approximation with the help of which one can calculate and represent the contours of the frontier functions in any of the above (a) to (d) variants. 

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