Is a Bit over Okay? - Overcomplete Sets of Localized Molecular Orbitals

Quantum chemical bases are mostly linearly independent, whether orthonormal or not. However, in certain applications, using an overcomplete set of functions, not forming a basis in the strict sense offers distinct advantages. Localized molecular orbitals (LMO) projected from atomic orbitals (PAO) are a primary example. Use of redundant LMOs is facilitated by the mathematical theory of frames. Along with a systematic presentation of the necessary background, it is shown that the trace of one-electron operators can be obtained by a formula analogous to the orthonormal case. For instance, the electronic dipole moment can be expressed as a sum of frame LMO centroids multiplied by their fractional populations as weights. An advantage of frame LMOs is demonstrated when following reaction coordinates. When a system passes through a high-symmetry point between lower-symmetry arrangements, orthogonal LMOs often exhibit abrupt changes. This occurs when a specific set of localized orbitals does not represent the molecular point group of the high-symmetry structure, resulting in equivalent sets of orthogonal LMOs that may induce discontinuities in LMO based quantities. It is shown that PAO related frame LMOs transition smoothly in these scenarios, without any extra algorithmic effort.