Self-interaction-free functionals

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]. 

It is interesting to note that early x-fiinctionals preceded most GGA s[60,61]. These x-functionals (sometimes referred to as Meta-GGA s [62]), seem to be quite promising, as demonstrated by recent constructions [63-65] based upon fits to chemical data. They have also some important properties. In particular, the PBE fimctional, as well as other GGAs, is not self-interaction free, that is the correlation energy does not vanish for a one-electron density. In this sense, nonlocal functionals, using the kinetic energy density, are Self Interaction Correlation (SIC) free by construction. 

The energy functional should be self-interaction-free, i.e. the exchange energy for a one-electron system, such as the hydrogen atom, should exactly cancel the Coulomb energy, and the correlation energy should be zero. Although these seem like obvious requirements, none of the conamon functionals have this property. When the density becomes constant, the uniform electron gas result should be recovered. While this surely is a valid mathematical requirement, and important for applications in solid-state physics, it may not be as important for chemical applications, as molecular densities are relatively poorly described by uniform electron gas methods. 

It is easy to show that although the SIC-LSD energy functional in Eq. (20) appears to be a functional of all the one-electron orbitals, it can in fact be rewritten as a functional of the total (spin) density alone, as discussed by Svane (1995). The difference with respect to the LSD energy functional lies solely in the exchange-correlation functional, which is now defined to be self-interaction free. Since, however, it is rather impractical to evaluate this SIC exchange-correlation functional, one has to resort to the orbital-dependent minimization of Eq. (20). 

This condition then ensures that the functionals are self-interaction free and that each o-spin electron excludes another o-spin electron from its immediate vicinity. From a computational perspective, when the nonsymmetric form of the one-matrix is used, then the nonlinear equations (Equation 1.115) are decoupled in this case, the equations can be solved using Newton s method in one dimension. But when the... 

This transfer is driven by the external electric field and opposed by the chemical hardness of each H2 unit. Since self-interaction-free approaches (like the HF method) increase the chemical hardness of an H2 unit in comparison with semilocal density functionals, they reduce the charge transfer and the related linear and nonlinear responses. 

Eq. (125) is not valid for all the conventional XC functional and also for the EXX method, which is one-electron self-interaction free but not many-electron self-interaction free. ... 

Because the ionization threshold plays a crucial role in most strong field phenomena, Koopman s theorem, which relates the energy level of the KS HOMO to the ionization energy, should be satisfied. As standard functionals fail to satisfy Koopman s theorem due to their poor potentials (see Figure 3), this suggests the use of self-interaction free methods such as OEPI I"" or LDA-SIC. 

In these equations, (24)-(26), orthonormal orbits are denoted by indices Vs. Equation (26) means that the orbiting electron interacting with itself, that is self-interaction, exists. This is unphysical. In order to remove this unphysical term, the SIC is taken into account by the following procedure. The SIC for the LDA in the density functional method has been treated for free atoms and insulators [16], and found an important role in determining the energy levels of electrons. However, no established formula is known to take into account the SIC for semiconductors and metals. As a way of trial, in the present calculation, the atomic SIC potential is introduced for each angular momentum in a way similar to the SIC potential for atoms [17] as follows ... 

The decomposition of the irreducible part of the self-energy wave-function correction term is depicted in Fig. 2. The divergent terms are these with zero and one interaction in the binding potential present, below referred to as zero-potential term and one-potential term , respectively. The charge divergences cancel between both terms. In addition, a mass counter term dm has to be subtracted to obtain proper mass renormalization similar to the case of the free self energy [47] (for our schemes see also [44]). The zero- and one-potential... 

---