AbstractA condition for the equivalence of the Löwdin orthogonalization method and various localization methods is derived, taking the matrix elements of the localizing operator into consideration. In the example of the π atomic orbitals of benzene, it is shown that the “ultralocalized” functions defined in the Anderson fashion with help of the Boys minimum‐fluctuation criterion do not fulfill this condition, in contrast with a Ruedenberg‐type localization.

As of today, there is certainly no doubt about the quantum character of the atomistic world, most straightforwardly calculated by using wave mechanics and Schrödinger's fundamental equation from 1926. Even though one century has passed, the paramount importance of the wave function, which determines everything down to the last detail, remains unchanged, and the wave function is most conveniently approximated by a combination of orbitals, one-electron wave functions for atoms, molecules, and also solids. And it is precisely this "orbital basis" that serves as a gateway to understanding the very interactions that cause atoms to condense into solids, just like for molecules. The analysis of quantum-chemical interactions and the nature of the chemical bonding between atoms in solids by use of orbitals will be our topic in this perspective, starting with the glorious past, going over to the current practice and, of course, the magnificent prospects for the future. As electronic structures for periodic solids are most often calculated using plane waves (instead of orbitals), for simple reasons of translational symmetry and Bloch's fundamental theorem, a unitary transformation to atomic or molecular orbitals is needed for final inspection, technically solved by the LOBSTER quantum-chemistry package. LOBSTER allows for the calculation of wave function-based atomic charges, various population analyses and periodic bonding indicators, first-principles bond orders, two- and multi-centre bonding analysis, fragment-molecular analysis, and a lot more. All those techniques are illustrated from three solid-state systems deriving from carbonate chemistry.

Per-Olov Löwdin (October 28, 1916 – October 6, 2000) was a Swedish physicist, professor at the University of Uppsala from 1960 to 1983, and in parallel at the University of Florida until 1993. A former graduate student under Ivar Waller, Löwdin formulated in 1950 the symmetric orthogonalization scheme for atomic and molecular orbital calculations, greatly simplifying the tight-binding method. This Per-Olov Löwdin (October 28, 1916 – October 6, 2000) was a Swedish physicist, professor at the University of Uppsala from 1960 to 1983, and in parallel at the University of Florida until 1993. A former graduate student under Ivar Waller, Löwdin formulated in 1950 the symmetric orthogonalization scheme for atomic and molecular orbital calculations, greatly simplifying the tight-binding method. This scheme is the basis of the zero-differential overlap (ZDO) approximation used in semiempirical theories. In 1956 he introduced the canonical orthogonalization scheme, which is optimal for eliminating approximate linear dependencies of a basis set. These orthogonalization procedures are widely used today in all modern quantum chemistry calculations. The famous 'Löwdin's pairing theorem' used in restricted open-shell Hartree–Fock (ROHF), unrestricted Hartree–Fock (UHF) and generalized valence bond (RES-GVB) theories is not his. According to himself, George G. Hall and King made the formal proposition after an informal suggestion by Löwdin. His Löwdin partitioning technique for quantum chemistry problems is best appreciated through the series of 14 papers on perturbation theory published between 1963 and 1971. He was also a very active teacher, starting the Summer Schools of Quantum Chemistry at Uppsala around 1958. In 1959 and 1960, Löwdin started the Quantum Theory Project at the University of Florida as a sister project to the Uppsala Quantum Chemistry Group. In 1964 he was joined by John C. Slater from MIT. The International Winter Institutes (held initially at Sanibel Island, and later at Gainesville) provided the initiation into quantum chemistry for hundreds of young Latin American scientists during the 1980s and 1990s. In 1960 he founded the Sanibel Symposium in conjunction with the Winter Institute, held every year since then. Löwdin was elected a member of the Royal Swedish Academy of Sciences in 1969, the American Philosophical Society in 1983, and was a member of the committee for the Nobel Prize in Physics from 1972 to 1984. He was the founder of the International Journal of Quantum Chemistry and of the series Advances in Quantum Chemistry. He was a…

A linear-scaling implementation of Hartree-Fock and Kohn-Sham self-consistent field (SCF) theories is presented and illustrated with applications to molecules consisting of more than 1000 atoms. The diagonalization bottleneck of traditional SCF methods is avoided by carrying out a minimization of the Roothaan-Hall (RH) energy function and solving the Newton equations using the preconditioned conjugate-gradient (PCG) method. For rapid PCG convergence, the Löwdin orthogonal atomic orbital basis is used. The resulting linear-scaling trust-region Roothaan-Hall (LS-TRRH) method works by the introdu

The orthogonality between sets of basis vectors, referred below to as pair-orthogonality, is examined from a variational point of view. Both how to constitute pair-orthogonal vectors and how to retain the pair-orthogonality conditions in the course of variation are presented. The relation of the vectors to canonically orthonormalized Löwdin vectors is noted.