Umbilical point

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.

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. Similarly, molecular orbital calculations show two nonbonding valence-shell orbitals: a roughly sp² hybrid orbital in the plane of the molecule and a pure p orbital perpendicular to this plane. The roughly tetrahedral sp³ hybrids of valence bond theory for the lone pairs can be compared to the sum and the difference of these nonbonding orbitals.

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.

Computation methods

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:

$\langle {\hat {L}}\rangle =\sum _{i=1}^{n}\langle \phi _{i}\phi _{i}|{\hat {L}}|\phi _{i}\phi _{i}\rangle$,

where ${\hat {L}}$ is the localization operator and $\phi _{i}$ is a molecular spatial orbital. Many methodologies have been developed during the past decades, differing in the form of ${\hat {L}}$.

Boys

The Boys (also known as Foster-Boys) localization method minimizes the spatial extent of the orbitals by minimizing $\langle {\hat {L}}\rangle$, where ${\hat {L}}=|{\vec {r}}_{1}-{\vec {r}}_{2}|^{2}$. This turns out to be equivalent to the easier task of maximizing $\sum _{i>j}^{n}[\langle \phi _{i}|{\vec {r}}|\phi _{i}\rangle -\langle \phi _{j}|{\vec {r}}|\phi _{j}\rangle ]^{2}$.

Edmiston-Ruedenberg

Edmiston-Ruedenberg localization maximizes the electronic self-repulsion energy by maximizing $\langle {\hat {L}}\rangle$, where ${\hat {L}}=|{\vec {r}}_{1}-{\vec {r}}_{2}|^{-1}$.

Pipek-Mezey

Pipek-Mezey localization takes a slightly different approach, maximizing the sum of orbital-dependent partial charges on the nuclei:

   $\langle {\hat {L}}\rangle _{\textrm {PM}}=\sum _{A}^{\textrm {atoms}}\sum _{i}^{\textrm {orbitals}}|q_{i}^{A}|^{2}$.

Pipek and Mezey originally used Mulliken charges, but also Löwdin charges have been used. Schemes equivalent to Pipek-Mezey with Bader or Becke charges have also been suggested.

Comparison

These three methods typically give very similar results, the main difference being that the Pipek-Mezey method does not mix $\sigma$ bonds and $\pi$ bonds.