Localized molecular 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:

${\displaystyle ⟨\hat{L}⟩=\sum _{i=1}^n⟨{\varphi }_i{\varphi }_i|\hat{L}|{\varphi }_i{\varphi }_i⟩}$,

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

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.

Foster–Boys localization

The Foster–Boys localization or Boys localization method was developed by Samuel Francis Boys and Jamie M. Foster in 1960. It minimizes the spatial extent of the orbitals by minimizing ${\displaystyle ⟨\hat{L}⟩}$, where ${\displaystyle \hat{L}=|{\overrightarrow{r}}_1-{\overrightarrow{r}}_2{|}^2}$. This turns out to be equivalent to the easier task of maximizing ${\displaystyle \sum _i^n[⟨{\varphi }_i|\overrightarrow{r}|{\varphi }_i⟩{]}^2}$. In one dimension, the Foster-Boys (FB) objective function can also be written as

${\displaystyle ⟨{\hat{L}}_{\text{FB}}⟩=\sum _i⟨{\varphi }_i|(\hat{x}-⟨i|\hat{x}|i⟩{)}^2|{\varphi }_i⟩}$.

Fourth moment procedure

The fourth moment (FM) procedure was developed by Ida-Marie Høyvik, Branislav Jansik and Poul Jørgensen in 2012. It is analogous to Foster–Boys scheme, however the orbital fourth moment is used instead of the orbital second moment. The objective function to be minimized is

${\displaystyle ⟨{\hat{L}}_{\text{FM}}⟩=\sum _i⟨{\varphi }_i|(\hat{x}-⟨{\varphi }_i|\hat{x}|{\varphi }_i⟩{)}^4|{\varphi }_i⟩}$.

The fourth moment method produces more localized virtual orbitals than Foster–Boys method, since it implies a larger penalty on the delocalized tails. For graphene (a delocalized system), the fourth moment method produces more localized occupied orbitals than Foster–Boys and Pipek–Mezey schemes.

Edmiston–Ruedenberg localization

Edmiston–Ruedenberg localization was developed by Clyde Edmiston and Klaus Ruedenberg in 1963. It maximizes the electronic self-repulsion energy by maximizing ${\displaystyle ⟨{\hat{L}}_{\text{ER}}⟩}$, where ${\displaystyle \hat{L}=|{\overrightarrow{r}}_1-{\overrightarrow{r}}_2{|}^{-1}}$.

Pipek–Mezey localization

Pipek–Mezey localization was developed by János Pipek and Paul G. Mezey in 1989. It takes a slightly different approach, maximizing the sum of orbital-dependent partial charges on the nuclei:

${\displaystyle ⟨\hat{L}{⟩}_{\text{PM}}=\sum _A^{\text{atoms}}\sum _i^{\text{orbitals}}|q_i^A{|}^2}$.

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 (see intrinsic bond orbitals)", Bader charges, or "fuzzy atom" charges. 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. 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.

Informal usage in organic chemistry

Organic chemistry is often discussed in terms of localized molecular orbitals in a qualitative and informal sense. Historically, much of classical organic chemistry was built on the older valence bond / orbital hybridization models of bonding. To account for phenomena like aromaticity, this simple model of bonding was 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 across the molecule and do not admit a ready correspondence with the "bonds" and "lone pairs", as conceived by the practicing chemist, the most common approach is to instead consider the interaction between filled and unfilled localized molecular orbitals that do have a rough correspondence to σ bonds, π bonds, lone pairs, and their virtual counterparts. These orbitals and typically given the notation σ (sigma bonding), π (pi bonding), n (occupied nonbonding orbital, "lone pair"), p (unoccupied (p-type) nonbonding orbital, "empty p orbital"; less commonly, the symbols a for "acceptor" or "atomic" orbital and n* for unoccupied nonbonding orbital are also used), π* (pi antibonding), and σ* (sigma antibonding). (Woodward and Hoffmann introduced the symbol 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 < π* < σ* when ranked by increasing energy.

Orbital interactions in elementary steps of reactions and in ground state stereoelectronic interactions are often expressed in terms of these symbols. For example, the S_N2 reaction of hydroxide ion on methyl bromide can be expressed as a n_O → σ*_C–Br process, while the stabilization of alkenes by hyperconjugation by neighboring C–H bonds on alkyl groups is symbolically expressed as a σ_C–H → π*_C=C interaction.

The localized molecular orbitals that organic chemists often depict can be thought of as qualitative renderings of orbitals generated by the computational methods described above. However, they do not map onto any single approach, nor are they used consistently. For instance, the lone pairs of water are usually treated as two equivalent sp^x hybrid orbitals, while the corresponding "nonbonding" orbitals of carbenes are generally treated as a filled σ(out) orbital and an unfilled pure p orbital, even though the lone pairs of water could be described analogously by filled σ (out) and p orbitals (for further discussion, see the article on lone pair and the discussion above on sigma-pi and equivalent-orbital models). In other words, the type of localized orbital invoked depends on context and considerations of convenience and utility.