Characterization and Generation of Local Occupied and Virtual Hartree–Fock Orbitals

Abstract

The scope of this review article is to discuss the locality of occupied and virtual orthogonal Hartree-Fock orbitals generated by localization function optimization. Locality is discussed from the stand that an orbital is local if it is confined to a small region in space. Focusing on locality measures that reflects the spatial extent of the bulk of an orbital and the thickness of orbital tails, we discuss, with numerical illustrations, how the locality may be reported for individual orbitals as well as for sets of orbitals. Traditional and more recent orbital localization functions are reviewed, and the locality measures are used to compare the locality of the orbitals generated by the different localization functions, both for occupied and virtual orbitals. Numerical illustrations are given also for large molecular systems and for cases where diffuse functions are included in the atomic orbital basis. In addition, we have included a discussion on the physical and mathematical limitations on orbital locality.

The divide–expand–consolidate coupled cluster scheme

“…These include the square of the second and fourth central moments where the localized orbitals are less system‐dependent than the Boys or Pipek–Mezey localized orbitals. The more advanced localization functions combined with a robust optimization strategy makes it possible to determine localized virtual orbitals, even for systems that are traditionally considered to have delocalized electronic structures . The localized virtual orbitals constitute an orthogonal and nonredundant set, and are very convenient for a straightforward application of standard CC implementations in local correlation methods.…”

“…The three most important features of the DEC scheme compared to other local correlation methods are the following: The sizes of the fragments are determined dynamically in a black‐box manner to ensure error control. The DEC scheme is combined with a massively parallel implementation, with three levels of parallelism. The virtual space is described using localized virtual orbitals …”

“…The three partitioning schemes yield independent ways of evaluating the correlation energy and may therefore be used to estimate the error associated with a DEC calculation. Virtual orbitals are usually less local than occupied orbitals and larger fragments occur in the virtual and Lagrangian partitioning scheme than in the occupied partitioning scheme, which is therefore generally preferred. However, the virtual partitioning scheme is necessary in DEC calculations of molecular gradients …”

“…The fragment spaces tend to be bigger for systems characterized by a delocalized electronic structure such as graphene than for systems containing only nonconjugated covalent bonds. In particular, for systems with a delocalized electronic structure it is important to use the most advanced orbital localization functions, such as the squared fourth moment localization function, since these localization functions are capable of giving localized sets of orbitals that are very little system‐dependent …”

Localized orbitals for optimal decomposition of molecular properties

“…To transform delocalized orbitals (either the canonical MOs obtained as a solutions of self‐consistent field Hartree‐Fock or Kohn‐Sham equations, or the Lowding natural orbitals obtained by diagonalization of the first‐order density matrix corresponding to a correlated wavefunction) into the localized ones, a number of procedures have been developed (see Ref. for a recent review). However, most of them date back to the times preceding the “mini revolution” of 1970s, when personal computers became widely spread among chemists, and therefore these procedures were rarely implemented as a handy computer codes, with probably the only exception of the Natural Bond Orbital (NBO) method implemented in NBO program .…”