Generalising localisation schemes of orthogonal orbitals to the localisation of non-orthogonal orbitals

ABSTRACT

When a set of orthogonal orbitals is localised using orthogonal transformations, the orbital metric is conserved as the unit matrix during the localisation. In this paper we describe how non-orthogonal orbitals may be localised by requiring that the orbital metric is conserved. In particular, we demonstrate how exponential mappings may be used to parametrise orbital transformations such that the orbital metric is conserved. Using this parametrisation, the localisation of non-orthogonal orbitals becomes a generalisation of the localisation for orthogonal orbitals, where the conservation of a unit metric is replaced by the conservation of a non-unit metric. As a result, standard orbital localisation functions and optimisation algorithms that have been used for localising orthogonal orbitals may also be used to localise non-orthogonal orbitals. Numerical illustrations show that the thickness of the orbital tails may be reduced when a set of non-orthogonal orbitals is localised compared to the orthogonal counterpart, and further that the physical interpretation of occupied orbitals for representing chemical bonds is preserved.