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. 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. For molecules with an open electron shell, in which some molecular orbitals are singly occupied, the electrons of alpha and beta spin must be localized separately. 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.
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:
⟨ L ^ ⟩ = ∑ i = 1 n ⟨ ϕ i ϕ i | L ^ | ϕ i ϕ i ⟩ {\displaystyle \langle {\hat {L}}\rangle =\sum _{i=1}^{n}\langle \phi _{i}\phi _{i}|{\hat {L}}|\phi _{i}\phi _{i}\rangle } , where L ^ {\displaystyle {\hat {L}}} is the localization operator and ϕ i {\displaystyle \phi _{i}} is a molecular spatial orbital. Many methodologies have been developed during the past decades, differing in the form of L ^ {\displaystyle {\hat {L}}} . 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.
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