Self-interaction-correction functional

The linear photoresponse of metal clusters was successfully calculated for spherical [158-160, 163] as well as for spheroidal clusters [164] within the jellium model [188] using the LDA. The results are improved considerably by the use of self-interaction corrected functionals. In the context of response calculations, self-interaction effects occur at three different levels First of all, the static KS orbitals, which enter the response function, have a self-interaction error if calculated within LDA. This is because the LDA xc potential of finite systems shows an exponential rather than the correct — 1/r behaviour in the asymptotic region. As a consequence, the valence electrons of finite systems are too weakly bound and the effective (ground-state) potential does not support high-lying unoccupied states. Apart from the response function Xs, the xc kernel /xc[ o] no matter which approximation is used for it, also has a self-interaction error. This is because /ic[no] is evaluated at the unperturbed ground-state density no(r), and this density exhibits self-interaction errors if the KS orbitals were calculated in LDA. Finally the ALDA form of /,c itself carries another self-interaction error. 

All applications quoted so far were for the linear response. Very few investigations have dealt with the higher-order response described in Sect. 5.2. The frequency-dependent third-order hyperpolarizabilities of rare-gas atoms were calculated by Senatore and Subbaswamy [86] within the ALDA the calculated values turned out to bee too large by a factor of two, further indicating the need for self-interaction corrected functionals in the calculation of response properties. The effect of adsorbates on second-harmonic generation at simple metal surfaces was invested by Kuchler and Rebentrost [205, 206]. Most recently, the second-order harmonic generation in bulk insulators was calculated within the ALDA [207]. 

The self-interaction corrected functional (MLSDSIC) proposed for calculating exchange energies for excited states is given as... 

Perdew J P and Zunger A 1981 Self-interaction correction to density-functional approximations for many-electron systems Phys. Rev. B 23 5048... 

Svane A and Gunnarsson Q 1990 Transition-metal oxides in the self-interaction-corrected density-functional formalism Phys. Rev. Lett. 65 1148... 

Perdew J P and A Zunger 1981. Self-Interaction Correction to Density-Functional Approximations for Many-Electron Systems. Physical Review B23 5048-5079. 

Encl[p] is the non-classical contribution to the electron-electron interaction containing all the effects of self-interaction correction, exchange and Coulomb correlation described previously. It will come as no surprise that finding explicit expressions for the yet unknown functionals, i. e. T[p] and Encl[p], represents the major challenge in density functional theory and a large fraction of this book will be devoted to that problem. 

Of course, this self-correction error is not limited to one electron systems, where it can be identified most easily, but applies to all systems. Perdew and Zunger, 1981, suggested a self-interaction corrected (SIC) form of approximate functionals in which they explicitly enforced equation (6-34) by substracting out the unphysical self-interaction terms. Without going into any detail, we just note that the resulting one-electron equations for the SIC orbitals are problematic. Unlike the regular Kohn-Sham scheme, the SIC-KS equations do not share the same potential for all orbitals. Rather, the potential is orbital dependent which introduces a lot of practical complications. As a consequence, there are hardly any implementations of the Perdew-Zunger scheme for self-interaction correction. 

Csonka, G. I., Johnson, B. G., 1998, Inclusion of Exact Exchange for Self-Interaction Corrected H, Density Functional Potential Energy Surface , Theor. Chem. Acc., 99, 158. 

Johnson, B. G., Gonzales, C. A., Gill, P. M. W., Pople, J. A., 1994, A Density Functional Study of the Simplest Hydrogen Abstraction Reaction. Effect of Self-Interaction Correction , Chem. Phys. Lett., 221, 100. 

Heaton, R.A., Harrison, J.G. and Lin, C.C. (1983) Self-interaction correction for density-functional theory of electronic energy bands of solids, Phys. Rev., B28, 5992-6007. 

Successful density functional approximations such as the PW91 GGA or the self-interaction correction (SIC) [57] to LSD recover [19] LSD values for the on-top hole density and cusp. The weighted density approximation (WDA) [41,42], which recovers the LSD exchange hole density but not the LSD correlation hole density [19] in the limit u -> 0, needs improvement in this respect. 

DPT schemes, which allow to calculate the electron affinities of atoms are based on the exact [59,60] and generalized (local) [61,62] exchange self-interaction-corrected (SIC) density functionals, treating the correlation separately in some approximation. Having better asymptotic behavior than GGA s, like in the improved SIC-LSD methods, one should obtain more... 

A number of different methods have been proposed to introduce a self-interaction correction into the Kohn-Sham formalism (Perdew and Zunger 1981 KUmmel and Perdew 2003 Grafenstein, Kraka, and Cremer 2004). This correction is particularly useful in situations with odd numbers of electrons distributed over more than one atom, e.g., in transition-state structures (Patchkovskii and Ziegler 2002). Unfortunately, the correction introduces an additional level of self-consistency into the KS SCF process because it depends on the KS orbitals, and it tends to be difficult and time-consuming to converge the relevant equations. However, future developments in non-local correlation functionals may be able to correct for self-interaction error in a more efficient manner. 

M. A. Whitehead and S. Manoli, Phys. Rev. A, 38,630 (1988). Generalized-Exchange-Local-Spin-Density-Functional Theory Self-Interaction Correction. 

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. 

Vydrov OA, Scuseria GE (2005) Ionization potentials and electron affinities in the Perdew-Zunger self-interaction corrected density-functional theory, J Chem Phys, 122 184107... 

Garza J, Nichols JA, Dixon DA (2000) The optimized effective potential and the self-interaction correction in density functional theory Application to molecules, J Chem Phys, 112 7880-7890... 

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