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We extend the selected columns of the density matrix (SCDM) methodology [J.Chem. Theory Comput. 2015, 11, 1463--1469]---a non-iterative procedure forgenerating localized occupied orbitals for condensed-phase systems---to theconstruction of local molecular orbitals (LMOs) in systems described usingnon-orthogonal atomic orbital (AO) basis sets. In particular, we introducethree different variants of SCDM (referred to as SCDM-M, SCDM-L, and SCDM-G)that can be used in conjunction with the standard AO basis sets. The SCDM-M andSCDM-L variants are based on the Mulliken and L{\"o}wdin representations of thedensity matrix, and are tantamount to selecting a well-conditioned set ofprojected atomic orbitals (PAOs) and projected (symmetrically-) orthogonalizedatomic orbitals (POAOs), respectively, as proto-LMOs. The SCDM-G variantleverages a real-space (grid) representation of the wavefunction to select aset of well-conditioned proto-LMOs. A detailed comparative analysis revealsthat the LMOs generated by these three SCDM variants are robust, comparable inorbital locality to those produced with the iterative Boys or Pipek-Mezey (PM)localization schemes, and are agnostic towards any single orbital localitymetric. Although all three SCDM variants are based on the density matrix, wefind that the character of the generated LMOs can differ significantly betweenSCDM-M, SCDM-L, and SCDM-G. In this regard, only the grid-based SCDM-Gprocedure (like PM) generates LMOs that qualitatively preserve$\sigma\text{-}\pi$ symmetry and are well-aligned with chemical intuition.While the direct and standalone use of SCDM-generated LMOs should suffice formost applications, our findings also suggest that the use of these orbitals asan unbiased and cost-effective (initial) guess also has the potential toimprove the convergence of iterative orbital localization schemes.

Selected Columns of the Density Matrix in an Atomic Orbital Basis I: An  Intrinsic and Non-Iterative Orbital Localization Scheme for the Occupied  Space