Exchange correlation functionals, local

In the Kohn-Sham Hamiltonian, the SVWN exchange-correlation functional was used. Equation 4.12 was applied to calculate the electron density of folate, dihydrofolate, and NADPH (reduced nicotinamide adenine dinucleotide phosphate) bound to the enzyme— dihydrofolate reductase. For each investigated molecule, the electron density was compared with that of the isolated molecule (i.e., with VcKt = 0). A very strong polarizing effect of the enzyme electric field was seen. The largest deformations of the bound molecule s electron density were localized. The calculations for folate and dihydrofolate helped to rationalize the role of some ionizable groups in the catalytic activity of this enzyme. The results are,... 

Proynov, E. I., A. Vela, and D. R. Salahub. 1994. Gradient-free exchange-correlation functional beyond the local-spin-density approximation. Phys. Rev. A 50, 2421. 

The problem is that the exchange correlation functional Exc is unknown. Approximate forms have to be used. The most well-known is the local density approximation (LDA) in which the expressions for a uniform electron gas are... 

To perform excited-state calculations, one has to approximate the exchange-correlation potential. Local self-interaction-free approximate exchange-correlation potentials have been proposed for this purpose [73]. We can try to construct these functionals as orbital-dependent functionals. There are different exchange-correlation functionals for the different excited states, and we suppose that the difference between the excited-state functionals can be adequately modeled through the occupation numbers (i.e., the electron configuration). Both the OPM and the KLI methods have been generalized for degenerate excited states [37,40]. 

Local exchange-correlation functionals such as generalized gradient approximations (GGA) are continuum approximations, which can, at best, average over the discontinuity. In regions where the HOMO and LUMO are significant, they provide an approximate average description [39—41] ... 

The wave functions are expended in a plane wave basis set, and the effective potential of ions is described by ultrasoft pseudo potential. The generalized gradient approximation (GGA)-PW91, and local gradient-corrected exchange-correlation functional (LDA)-CAPZ are used for the exchange-correlation functional. 

This approximation uses only the local density to define the approximate exchange-correlation functional, so it is called the local density approximation (LDA). The LDA gives us a way to completely define the Kohn-Sham equations, but it is crucial to remember that the results from these equations do not exactly solve the true Schrodinger equation because we are not using the true exchange-correlation functional. 

A critical feature of this quantity is that it is nonlocal, that is, a functional based on this quantity cannot be evaluated at one particular spatial location unless the electron density is known for all spatial locations. If you look back at the Kohn-Sham equations in Chapter 1, you can see that introducing this nonlocality into the exchange-correlation functional creates new numerical complications that are not present if a local functional is used. Functionals that include contributions from the exact exchange energy with a GGA functional are classified as hyper-GGAs. 

Three types of exchange/correlation functionals are presently in use (i) functionals based on the local spin density approximation, (ii) functionals based on the generalized gradient approximation, and (iii) functionals which employ the exact Hartree-Fock exchange as a component. The first of these are referred to as local density models, while the second two are collectively referred to as non-local models or alternatively as gradient-corrected models. 

Kohn-Sham Equations. The set of equations obtained by applying the Local Density Approximation to a general multi-electron system. An Exchange/Correlation Functional which depends on the electron density has replaced the Exchange Energy expression used in the Hartree-Fock Equations. The Kohn-Sham equations become the Roothaan-Hall Equations if this functional is set equal to the Hartree-Fock Exchange Energy expression. 

In order to calculate the band structure and the density of states (DOS) of periodic unit cells of a-rhombohedral boron (Fig. la) and of boron nanotubes (Fig. 3a), we applied the VASP package [27], an ab initio density functional code, using plane-waves basis sets and ultrasoft pseudopotentials. The electron-electron interaction was treated within the local density approximation (LDA) with the Geperley-Alder exchange-correlation functional [28]. The kinetic-energy cutoff used for the plane-wave expansion of... 

All electron calculations were carried out with the DFT program suite Turbomole (152,153). The clusters were treated as open-shell systems in the unrestricted Kohn-Sham framework. For the calculations we used the Becke-Perdew exchange-correlation functional dubbed BP86 (154,155) and the hybrid B3LYP functional (156,157). For BP86 we invoked the resolution-of-the-iden-tity (RI) approximation as implemented in Turbomole. For all atoms included in our models we employed Ahlrichs valence triple-C TZVP basis set with polarization functions on all atoms (158). If not noted otherwise, initial guess orbitals were obtained by extended Hiickel theory. Local spin analyses were performed with our local Turbomole version, where either Lowdin (131) or Mulliken (132) pseudo-projection operators were employed. Broken-symmetry determinants were obtained with our restrained optimization tool (136). Pictures of molecular structures were created with Pymol (159). 

---