Local-density approximation closed-shell

A time-dependent generalisation of the RKS-equation (3.25) has been suggested by Parpia and Johnson [49]. While a rigorous foundation of this approach is not available to date, this method has been successfully applied to the photoionisation of Hg [50] and Xe [49] as well as the evaluation of the polarisabilities of heavy closed-shell atoms [51] (using a direct time-dependent extension of the local density approximation for [ ]). 

For multi-electron systems, it is not feasible, except possibly in the case of helium, to solve the exact atom-laser problem in 3 -dimensional space, where n is the number of electrons. One might consider using time-dependent Hartree Fock (TDHF) or the time-dependent local density approximation to represent the state of the system. These approaches lead to at least njl coupled equations in 3-dimensional space which is much more attractive computationally. For example, in TDHF the wave function for a closed shell system can be approximated by a single Slater determinant of time dependent orbitals,... 

Table 14 Fano parameters for different ionization resonances for some closed-shell atoms. Except for the calculations marked EXX, the Kohn-Sham single-particle eigenvalues have in all cases been shifted to match the ionization potential. Exact KS represents results with the exact Kohn-Sham potential, GGA those with a gradient-corrected density functional, EXX exact-exchange without correlation, EXX- -LDA the same but with the inclusion of correlation effects with a local-density approximation, and Exp. experimental results. The results are from ref. 91... 

The different densities for different spins" case (analogous to DODS in HF theory) for which the two spatial densities must be retained and used separately. This method will generate the closed-shell case if the system is, indeed, actually spin-paired. This is often called the Local Spin Density Approximation (LSD) for obvious reasons. 

In the LDA, the total density is considered to be the sum of a and p spin densities. This assumption is satisfied in closed-shell systems, but not in open-shell systems. If we extend the LDA to the latter case, we arrive to the so-called local spin density approximation (LSDA). The exchange energy in this LSDA is ... 

In this review we will give an overview of the properties (asymptotics, shell-structure, bond midpoint peaks) of exact Kohn-Sham potentials in atomic and molecular systems. Reproduction of these properties is a much more severe test for approximate density functionals than the reproduction of global quantities such as energies. Moreover, as the local properties of the exchange-correlation potential such as the atomic shell structure and the molecular bond midpoint peaks are closely related to the behavior of the exchange-correlation hole in these shell and bond midpoint regions, one might be able to construct... 

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