Density functionals Slater-Kohn-Sham-type methods

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. 

The density functional (DF) method has been successful and quite useful in correlating experimental results when model densities are used in the calculations. In fact, the equations characteristic of the DF method can be derived from a variational approach as Kohn and Sham showed some time ago. In this approach, when model densities are introduced, it is not always possible to relate such densities to corresponding wave functions this is the N-representability problem. Fortunately, for any normalized well behaved density there exists a Slater single determinant this type of density is then N-representable. The problem of approximately N-representable density functional density matrices has been recently discussed by Soirat et al. [118], In spite... 

The most uniformly successful family of methods begins with the simplest possible n-electron wavefunction satisfying the Pauli antisymmetry principle - a Slater determinant [2] of one-electron functions % r.to) called spinorbitals. Each spinorbital is a product of a molecular orbital xpt(r) and a spinfunction a(to) or P(co). The V /.(r) are found by the self-consistent-field (SCF) procedure introduced [3] into quantum chemistry by Hartree. The Hartree-Fock (HF) [4] and Kohn-Sham density functional (KS) [5,6] theories are both of this type, as are their many simplified variants [7-16],... 

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. 

We have anployed the parametrized DFTB method of Porezag et al. [33,34]. The approximate DFTB method is based on the density-functional theory of Hohenberg and Kohn in the formulation of Kohn and Sham [43,44]. In this method, the single particle wave functions l (r) of the Kohn-Sham equations are expanded in a set of atomic-like basis functions < > , with m being a compound index that describes the atom on which the function is centered, the angular dependence of the function, as well as its radial dependence. These functions are obtained from self-consistent density functional calculations on the isolated atoms employing a large set of Slater-type basis functions. The effective Kohn-Sham potential Feff(r) is approximated as a simple superposition of the potentials of the neutral atoms... 

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