Computational self-interaction errors

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. 

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. 

Abstract This chapter discusses descriptions of core-ionized and core-excited states by density functional theory (DFT) and by time-dependent density functional theory (TDDFT). The core orbitals are analyzed by evaluating core-excitation energies computed by DFT and TDDFT their orbital energies are found to contain significantly larger self-interaction errors in comparison with those of valence orbitals. The analysis justifies the inclusion of Hartree-Fock exchange (HFx), capable of reducing self-interactions, and motivates construction of hybrid functional with appropriate HFx portions for core and valence orbitals. The determination of the HFx portions based on a first-principle approach is also explored and numerically assessed. 

From the above discussion it follows that the calculation of the Coulomb repulsion energy represents the most demanding computational task in Equation (5). The introduction of the variational approximation of the Coulomb potential reduces the formal scaling of this term to x M. Here M is the number of auxiliary functions which is usually two to three times N. The variational approximation of the Coulomb potential is based on the minimization of the following self-interaction error ... 

The biggest problem in DPT is the choice of the functional approximation. In many cases, computationally cheap (meta-)GGAs (e.g.,BLYP, PBE,orTPSS) can be recommended. Such functionals should not be used when the self-interaction error (e.g., in charged open shell systems) plays a role. Then, hybrid functionals are required which also have less tendency for over-polarization. The currently highest level of approximation in DPT is represented by double-hybrid functionals (e.g., B2PLYP) that also perform very well for non-covalent interactions. 

Another way to deal with the Coulomb self-interaction error is to use a hybrid functional that combines exact Hartree-Fock exchange with standard LDA/GGA. Recently, the hybrid HSE functional has been reported to describe successfully the localization of a single 4f electron in Cc203 (Da Silva et al., 2007). Even though the hybrid functional approach in some cases exhibits better results than the DFT -F U approach, DFT -F U can stiU compete well in terms of computational cost. Therefore, all the reported results in this chapter were obtained using DFT -F U calculations. 

The evaluation to the desired numerical accuracy of the density functional total energy has been a major obstacle to such calculations for many years. Part of the difficulty can be related to truncation errors in the orbital representation, or equivalently to basis set limitations, in variational calculations. Another part of the difficulty can be related to inaccuracies in the solution of Poisson s equation. The problem of maximizing the computational accuracy of the Coulomb self-interaction term in the context of least-squares-fitted auxiliary densities has been addressed in [39]. A third part of the difficulty may arise from the numerical integration, which is unavoidable in calculating the exchange and correlation contributions to the total energy in the density functional framework. 

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