Localized molecular orbitals
Localized molecular orbitals are molecular orbitals which are concentrated in a limited spatial region of a molecule, for example a specific bond or a lone pair on a specific atom. They can be used to relate molecular orbital calculations to simple bonding theories, and also to speed up post-Hartree–Fock electronic structure calculations by taking advantage of the local nature of electron correlation. Localized orbitals in systems with periodic boundary conditions are known as Wannier functions.
Standard ab initio quantum chemistry methods lead to delocalized orbitals that, in general, extend over an entire molecule and have the symmetry of the molecule. Localized orbitals may then be found as linear combinations of the delocalized orbitals, given by an appropriate unitary transformation.
In the water molecule for example, ab initio calculations show bonding character primarily in two molecular orbitals, each with electron density equally distributed among the two O-H bonds. The localized orbital corresponding to one O-H bond is the sum of these two delocalized orbitals, and the localized orbital for the other O-H bond is their difference; as per Valence bond theory.
For multiple bonds and lone pairs, different localization procedures give different orbitals. The Boys and Edmiston-Ruedenberg localization methods mix these orbitals to give equivalent bent bonds in ethylene and rabbit ear lone pairs in water, while the Pipek-Mezey method preserves their respective σ and π symmetry.
Equivalence of localized and delocalized orbital descriptions
For molecules with a closed electron shell, in which each molecular orbital is doubly occupied, the localized and delocalized orbital descriptions are in fact equivalent and represent the same physical state. It might seem, again using the example of water, that placing two electrons in the first bond and two other electrons in the second bond is not the same as having four electrons free to move over both bonds. However, in quantum mechanics all electrons are identical and cannot be distinguished as same or other. The total wavefunction must have a form which satisfies the Pauli exclusion principle such as a Slater determinant (or linear combination of Slater determinants), and it can be shown  that if two electrons are exchanged, such a function is unchanged by any unitary transformation of the doubly occupied orbitals.
Localized molecular orbitals (LMO) are obtained by unitary transformation upon a set of canonical molecular orbitals (CMO). The transformation usually involves the optimization (either minimization or maximization) of the expectation value of a specific operator. The generic form of the localization potential is:
where is the localization operator and is a molecular spatial orbital. Many methodologies have been developed during the past decades, differing in the form of .
The optimization of the objective function is usually performed using pairwise Jacobi rotations. However, this approach is prone to saddle point convergence (if it even converges), and thus other approaches have also been developed, from simple conjugate gradient methods with exact line searches, to Newton-Raphson and trust-region methods.
The Foster-Boys (also known as Boys) localization method minimizes the spatial extent of the orbitals by minimizing , where . This turns out to be equivalent to the easier task of maximizing .
The fourth moment (FM) procedure is analogous to Foster-Boys, however the orbital fourth moment is used instead of the orbital second moment. The objective function to be minimized is where is a penalty exponent. The fourth moment method yields more local sets of orbitals than the Foster-Boys method, because it implies a larger penalty on delocal tails.
Edmiston-Ruedenberg localization maximizes the electronic self-repulsion energy by maximizing , where .
Pipek-Mezey localization takes a slightly different approach, maximizing the sum of orbital-dependent partial charges on the nuclei:
Pipek and Mezey originally used Mulliken charges, which are mathematically ill defined. Recently, Pipek-Mezey style schemes based on a variety of mathematically well-defined partial charge estimates have been discussed. Some notable choices are Voronoi charges, Becke charges, Hirshfeld or Stockholder charges, intrinsic atomic orbital charges, Bader charges, or "fuzzy atom" charges. Rather surprisingly, despite of the wide variation in the (total) partial charges reproduced by the different estimates, analysis of the resulting Pipek-Mezey orbitals has shown that the localized orbitals are rather insensitive to the partial charge estimation scheme used in the localization process. However, due to the ill-defined mathematical nature of Mulliken charges (and Löwdin charges, which have also been used in some works), as better alternatives are nowadays available it is advisable to use them in favor of the original version.
The most important quality of the Pipek-Mezey scheme is that it preserves σ-π separation in planar systems, which sets it apart from the Foster-Boys and Edmiston-Ruedenberg schemes that mix σ and π bonds. This property holds independent of the partial charge estimate used.
While the usual formulation of the Pipek-Mezey method invokes an iterative procedure to localize the orbitals, a non-iterative method has also been recently suggested.
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