Localized molecular orbitals
Localized molecular orbitals are molecular orbitals which are concentrated in a limited spatial region of a molecule, such as a specific bond or 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.
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 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.
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.
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 [1] that if two electrons are exchanged, such a function is unchanged by any unitary transformation of the doubly occupied orbitals.
[2][3] This applies to radical species such as nitric oxide and dioxygen.
Again, in this case the localized and delocalized orbital descriptions are equivalent and represent the same physical state.
The transformation usually involves the optimization (either minimization or maximization) of the expectation value of a specific operator.
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.
[5] 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,[6] to Newton-Raphson[7] and trust-region methods.
In one dimension, the Foster-Boys (FB) objective function can also be written as
For graphene (a delocalized system), the fourth moment method produces more localized occupied orbitals than Foster-Boys and Pipek-Mezey schemes.
Pipek-Mezey localization[13] 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.
[17] Rather surprisingly, despite 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.
[14] However, due to the ill-defined mathematical nature of Mulliken charges (and Löwdin charges, which have also been used in some works[18]), 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.
[19] Organic chemistry is often discussed in terms of localized molecular orbitals in a qualitative and informal sense.
To account for phenomena like aromaticity, this simple model of bonding is supplemented by semi-quantitative results from Hückel molecular orbital theory.
However, the understanding of stereoelectronic effects requires the analysis of interactions between donor and acceptor orbitals between two molecules or different regions within the same molecule, and molecular orbitals must be considered.
Because proper (symmetry-adapted) molecular orbitals are fully delocalized and do not admit a ready correspondence with the "bonds" of the molecule, as visualized by the practicing chemist, the most common approach is to instead consider the interaction between filled and unfilled localized molecular orbitals that correspond to σ bonds, π bonds, lone pairs, and their unoccupied counterparts.
(Woodward and Hoffmann use ω for nonbonding orbitals in general, occupied or unoccupied.)
When comparing localized molecular orbitals derived from the same atomic orbitals, these classes generally follow the order σ < π < n < p (n*) < π* < σ* when ranked by increasing energy.
In other words, the type of localized orbital invoked depends on context and considerations of convenience and utility.
