The spectrum of the atomic orbital overlap matrix and the locality of the virtual electronic density matrix

Abstract

It is well known that near-linear dependencies in the atomic orbital (AO) basis will impede electronic-structure calculations since the inverse AO overlap matrix will be ill-defined. However, small eigenvalues will also impact the locality of the virtual density matrix. The virtual density matrix is relevant for developing efficient local approaches in electronic-structure theory, and recent literature shows that orthogonal molecular orbitals (MOs) are not optimal for exploiting locality. As size of molecules treated is increasing and high-quality basis sets are used, the problem is becoming more pronounced and challenges similar to those for extended systems appear. Here it is shown that the spectrum of the AO overlap matrix puts severe restrictions on the locality of the virtual density matrix, and that locality cannot be recovered by excluding components corresponding to near-singular eigenvalues. The effect is seen even when eigenvalues are orders of magnitude from being near-singular, and occurs also for small basis sets such as cc-pVDZ depending on the molecular system. Non-orthogonal orbitals do not constitute a solution to the problem, but they may be more convenient since lack of locality in the density matrix can be shifted from MO coefficients to the inverse MO overlap matrix.