Near Equivalence of Intrinsic Atomic Orbitals and Quasiatomic Orbitals.

A direct relationship between the Intrinsic Atomic Orbitals (IAO) method (Knizia, G. J. Chem. Theory Comp. 2013, 9, 4834-4843) and earlier work on the same topic, quasiatomic minimal basis set orbitals (QUAMBO) (Lu, W. C.; Wang, C. Z.; Schmidt, M. W.; Bytautas, L.; Ho, K. M.; Ruedenberg J. Chem. Phys. 2004, 2629) and later modifications (quasiatomic orbitals, QUAO) is investigated. It will be demonstrated mathematically that IAOs are almost identical to the original formulation of QUAMBOs and span the same space as a later QUAO modification. The construction of QUAOs involves minimization of a functional that requires matrix diagonalization, or singular matrix decomposition, while the IAO method provides a direct solution by projections. As a byproduct of this proof, it will be shown that (a) under mild conditions a simpler projection yields identical IAOs and (b) an alternative proof is obtained that IAOs span the full space of molecular orbitals if they are linearly independent. Utilization of QUAMBOs as the defining basis set results in rock-solid numerical stability of Pipek-Mezey localization and Mulliken or Löwdin population analysis in very large systems. The charges do not depend on the basis set used, as already shown by Knizia for smaller systems. In this paper, more difficult cases of large semiperiodic systems with strong linear dependency are tested, and it is shown that QUAMBOs perform extremely well.