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## Self-Interaction Effects

As well as accounting for the difference in the kinetic energy between the fully interacting system and the non-interacting Kohn-Sham scheme, exchange and correlation also cancels the self-interaction effect that originates from the Hartree term. The only case in which self-interaction can be fully defined is for a system containing exactly one electron. In this case let one of the spin densities be zero, say , then,

 (2.27)

In this instance and are the exact kinetic and potential energies of the the system, and as there are no electron-electron interactions . Since the probability of finding another electron in the system is zero, the pair-correlation function must also be zero at all points in space, which leads to the result,
 (2.28)

and therefore the exchange-correlation energy term directly cancels the Hartree energy of the system. More specifically, since condition (2.29) arises from the non-zero form of the exchange sum rule, then,
 (2.29)

so the self-energy of the electron is cancelled by exchange. Similarly the exchange and correlation potentials are,
 (2.30) (2.31)

where the constants and arise because the exchange-correlation potential can only be defined up to a constant [37]. Relations (2.31) and (2.32) show that the electron moves in the bare external potential . For most approximate functionals the above exact relations are not satisfied, which results in a self-interaction error in the computed total energies,
 (2.32)

This equation can be used as a measure of the degree of self-interaction exhibited by a given exchange-correlation approximation. Nearly all of the conventional functionals used in DFT possess self-interaction errors i.e. for a one-electron system.

Self-interaction corrected (SIC) functionals have been devised in the past such as that of Perdew and Zunger [38]. When applied to atoms the error in the total exchange-correlation energies is greatly reduced [39,40], and the highest occupied orbitals of isolated atoms are in better agreement with experimental ionisation energies [38], which in theory should be identical if the arbitrary constant in is set to zero. Problems are however encountered when applying the Perdew-Zunger SIC prescription to solids since the energy functional is not invariant under a unitary transformation of the occupied orbitals, also, a localised set of basis functions must be used since the SIC energy is zero when Bloch functions are employed. Nevertheless, examples of the method have been demonstrated in solids using localised orbitals, and the most notable result is that the electronic band-gap of wide-gap insulators [41,42] and transition-metal monoxides [43,44] is significantly improved.

Next: Exchange-Correlation Approximations Up: Fundamentals Previous: The Pair-Correlation Function   Contents
Dr S J Clark
2003-05-04