Quantifying the effects of the self-interaction error in DFT: When do the delocalized states appear?

The self-interaction error in density-functional theory leads to artificial stabilization of delocalized states, most evident in systems with an odd number of electrons. Clear examples are dissociations of carbocation radicals that often give delocalized states at long distances and large errors in computed binding energies. On the other hand, many mixed-valence transition-metal dimers known to exhibit valence trapping are correctly predicted to be localized. To understand the effects of the self-interaction error on these different systems, energy differences between delocalized and localized states are calculated with B3LYP. In the dissociation of radicals into symmetric fragments at infinite distance, this energy difference equals the error of the density-functional treatment. The energy difference decreases with increasing size of the system, from $55kcal∕mol$ in $H2+$ to $15kcal∕mol$ for $C12H26+$⁠. Solvent corrections stabilize the localized state and result in smaller errors. Most reactions are asymmetric and this decreases the effect of the self-interaction error. In many systems, delocalization will not occur if the cost to move the electron from one fragment to the other is $70–80kcal∕mol$ $(3.0–3.5eV)$⁠. This estimate refers to a situation where the distance between the fragments is infinite. The limit decreases with decreasing fragment distance. B3LYP calculations on the ferromagnetic state of a Mn(III,IV) dimer predict that the correct localized state is $22kcal∕mol$ more stable than the incorrect delocalized state. At short metal–metal distances the effect of the self-interaction error is predicted to be small. However, as the distance between the two manganese centers is increased to $7Å$⁠, the dimer starts to delocalize and the energy artificially decreases. In the dissociation limit, the error is $10kcal∕mol$⁠. This is interpreted as an artifact originating from the self-interaction error. Delocalization is not encountered in many systems due to relatively short metal–metal distances and asymmetric ligand environments. However, some charge-transfer complexes cannot be properly calculated and delocalized states may become a problem in large models of enzyme systems with multiple transition-metal complexes.