Self interaction correction electron density with

This term does not exactly vanish for a one electron system since it contains the spurious interaction of the density with itself. Hence, for equation (6-32) to be correct, we must demand that J[p] exactly equals minus Exc[p] such that the wrong self-interaction is cancelled... 

Finally, in careful comparative studies of the molecular electron densities generated by HF, correlated ab initio, and pure, self-interaction corrected, and hybrid DFT calculations, Cremer et al have made a very interesting observation [72, 73]. They found that the pure DFT generated densities differed from those obtained with accurate ab initio methods in a particular way, and that both hybrid, and self-interaction corrected DFT methods, yielded densities closer to the correct ones. Based on this observation, they suggested that mixing in of exact exchange in hybrid functionals serves as a proxy for the self-interaction correction. 

Another disadvantage of the LDA is that the Hartree Coulomb potential includes interactions of each electron with itself, and the spurious term is not cancelled exactly by the LDA self-exchange energy, in contrast to the HF method (see A1.3I. where the self-interaction is cancelled exactly. Perdew and Zunger proposed methods to evaluate the self-interaction correction (SIC) for any energy density functional [40]. However, full SIC calculations for solids are extremely complicated (see, for example [41. 42 and 43]). As an alternative to the very expensive GW calculations, Pollmann et al have developed a pseudopotential built with self-interaction and relaxation corrections (SIRC) [44]. 

During the completion of the present review we became aware of all-electron frozen-core (R ls -4d °) QR-DFT (quasirelativistic density-functional theory) calculations for selected lanthanide diatomics by Wang and Schwarz (1995). Results obtained with the Amsterdam program (Snijders and Baerends 1978, Snijders et al. 1979), gradient-corrected exchange (Becke 1988) and self-interaction corrected (Stoll et al. 1978, 1980) correlation (Vosko et al. 1980) energy funetionals are summarized in table 17. The agreement with experimental data is quite satisfactory. 

In this paper a method [11], which allows for an a priori BSSE removal at the SCF level, is for the first time applied to interaction densities studies. This computational protocol which has been called SCF-MI (Self-Consistent Field for Molecular Interactions) to highlight its relationship to the standard Roothaan equations and its special usefulness in the evaluation of molecular interactions, has recently been successfully used [11-13] for evaluating Eint in a number of intermolecular complexes. Comparison of standard SCF interaction densities with those obtained from the SCF-MI approach should shed light on the effects of BSSE removal. Such effects may then be compared with those deriving from the introduction of Coulomb correlation corrections. To this aim, we adopt a variational perturbative valence bond (VB) approach that uses orbitals derived from the SCF-MI step and thus maintains a BSSE-free picture. Finally, no bias should be introduced in our study by the particular approach chosen to analyze the observed charge density rearrangements. Therefore, not a model but a theory which is firmly rooted in Quantum Mechanics, applied directly to the electron density p and giving quantitative answers, is to be adopted. Bader s Quantum Theory of Atoms in Molecules (QTAM) [14, 15] meets nicely all these requirements. Such a theory has also been recently applied to molecular crystals as a valid tool to rationalize and quantitatively detect crystal field effects on the molecular densities [16-18]. 

With respect to correlation functionals, corrections to the correlation energy density following Eq. (8.29) include B88, P86, and PW91 (which uses a different expression than Eq. (8.27) for the LDA correlation energy density and contains no empirical parameters). Another popular GGA correlation functional, LYP, does not correct the LDA expression but instead computes the correlation energy in toto. It contains four empirical parameters fit to the helium atom. Of all of the correlation functionals discussed, it is the only one tliat provides an exact cancellation of the self-interaction error in one-electron systems. 

We applied the generating functional approach to the periodic Anderson model. Calculation of the electron GFs gdd, 9ds, 9sd and gss reduces to calculation of only the d-electron GF. For this, an exact matrix equation was derived with the variational derivatives. Iterations with respect to the effective matrix element Aij(to) allow to construct a perturbation theory near the atomic limit. Along with the self-energy, the terminal part of the GF Q is very important. The first order correction for it describes the interaction of d-electrons with spin fluctuations. In the paramagnetic phase this term contains a logarithmic singularity near the Fermi-level and thus produces a Kondo-like resonance peak in the d-electron density of states. The spin susceptibility of d-electrons... 

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