Self-interaction errors

We see that the self-interaction error, J[p] + Exc[p], is in all cases in the order of 10 3 Eh or a few hundredths of an eV. In addition, the data in Table 6-2 reiterate some of the facts that we noted before. B3LYP, BP86 and BPW91 yield total energies below the exact result of -0.5 Eh, in an apparent contradiction to the variational principle (see discussion in Sec-... 

Table 13-8. Self-interaction error components for Coulomb and exchange energies (Ej + Ex) as well as for the correlation energy (Ec), and the resulting sum for the H atom, the H2 molecule, and the H3 transition structure [kcal/mol]. Data taken from Csonka and Johnson, 1998.

Perdew, J. P., Ernzerhof, M., 1998, Driving Out the Self-Interaction Error in Electron Density Functioruil Theory. Recent Progress and New Directions, Dobson, J. F., Vignale, G., Das, M. P. (eds.), Plenum Press, New York. 

Zhang, Y., Yang, W., 1998, A Challenge for Density Functionals Self-Interaction Error Increases for Systems With a Noninteger Number of Electrons , J. Chem. Phys., 109, 2604. 

Self-Interaction Error, Strongly Correlated Electron Systems, and DFT + U... 

Dispersion interactions are, roughly speaking, associated with interacting electrons that are well separated spatially. DFT also has a systematic difficulty that results from an unphysical interaction of an electron with itself. To understand the origin of the self-interaction error, it is useful to look at the Kohn-Sham equations. In the KS formulation, energy is calculated by solving a series of one-electron equations of the form... 

The fact that self-interaction errors are canceled exactly in HF calculations suggests that a judicious combination of an HF-like approach for localized states with DFT for everything else may be a viable approach for strongly correlated electron materials. This idea is the motivation for a group of methods known as DFT+U. The usual application of this method introduces a correction to the DFT energy that corrects for electron self-interaction by introducing a single numerical parameter, U — J, where U and J involve different aspects of self-interaction. The numerical tools needed to use DFT+U are now fairly widely implemented in plane-wave DFT codes. 

The simplest way to gain a better appreciation for tlie hole function is to consider the case of a one-electron system. Obviously, the Lh.s. of Eq. (8.6) must be zero in that case. However, just as obviously, the first term on the r.h.s. of Eq. (8.6) is not zero, since p must be greater than or equal to zero throughout space. In die one-electron case, it should be clear that h is simply the negative of the density, but in die many-electron case, the exact form of the hole function can rarely be established. Besides die self-interaction error, hole functions in many-electron systems account for exchange and correlation energy as well. 

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. 

Finally, it seems clear that routes to further improve DFT must be associated with better defining hole functions in arbitrary systems. In particular, the current generation of functionals has reached a point where finding efficient algorithms for correction of the classical self-interaction error are likely to have the largest qualitative (and quantitative) impact. 

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. 

However, this account does raise a number of issues, concerning transfer-ability (or universality), calibration, the role of exchange functionals in pure DFT, and the self-interaction error. 

The discrepancies between the calculated values of —eHOMO and experimental ionization potentials are frequently attributed to self-interaction error. Indeed, applying the Perdew-Zunger technique4 to correct this error of the LDA-, GGA-(PBE), and meta-GGA (TPSS) functionals improves the numerical values of —8HOmo-98 Interestingly, these studies showed that the Perdew-Zunger correction does not improve ionization potentials and electron affinities if calculated as energy differences (ASCF). 

GGA functionals afford self-interaction errors that can affect computed... 

HF exchange partially eliminate self-interaction errors and tend to be beneficial for calculations of CD spectra and OR, and to some extent for ROA, too. In response-theory computations, hybrids with a large fraction of HF exchange (50%) are often employed, also to address other common deficiencies listed here. 

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. 

I. Ciofini, C. Adamo, and H. Chermette (2005) Effect of self-interaction error in the evaluation of the bond length alternation in trans-polyacetylene using density-functional theory. J. Chem. Phys. 123, p. 121102... 

The procedure proposed by PZ (following earlier work by Lindgren ) involves a straight subtraction of the self-interaction error, orbital by orbital, to yield a self-interaction corrected exchange-correlation energy. 

From Table 1 we see the deficiencies of the LDA method, since it underestimates both total and exchange energies. Such discrepancies are increased for small confinement radii. As we discussed in the introduction, the main difference between LDA and LDA-SIC is the self-interaction error, thus this spurious contribution is more important when the electrons are localized within small regions, as it has been pointed out before [21]. 

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