10.2.6 Oxidation State Localized Orbitals

Oxidation State Localized Orbitals (OSLO) is a new localization scheme focused on molecular fragments for the purposes of oxidation state assignment.^{Aldossary:2021} The method has been developed to avoid some pitfalls encountered in the LOBA method where the PM orbitals can spread across many fragments without reaching the 60% threshold. OSLO starts by looking for fragments’ centers of charges, and localizes on radial spread (in the real space) to those centers. Then, it admits orbitals to its list of orbitals if they are above a slowly increasing threshold of a criterion called the fragment orbital localization index (FOLI). To understand this criterion, one needs to know Pipek’s delocalization measure. This is defined as

where the $N_A^i$ is the population of the $i$th orbital on center $A$ and the summation runs over all centers. When an orbital is entirely localized on a given center, its delocalization measure is $d_i=1$. If the orbital is perfectly delocalized among two centers $A$ and $B$ then $N_A=N_B=1/2$ and the delocalization measure is $d_i=2$, etc. The square helps make it less sensitive to the ratio of the population to each other compared to how many are related. In fact, minimizing the sum over all occupied orbitals is precisely what leads to Pipek-Mezey localization procedure.⁸⁹⁷ However, for our purposes, out of the localized orbitals generated from fragment $A$ with low delocalization measure, we are interested in those that are also highly localized on fragment $A$. Defining the fragment orbital localization index (FOLI),

it is easy to see $d_i^A$ tends to unity for fragment $A$ when orbital $i$ is perfectly localized on $A$ ($d_i=N_A=1$), tends to 2 when the localized orbital is perfectly delocalized over two fragments ($d_i=2,N_A=0.5$), and gradually increases as the orbital becomes more delocalized as well as less centered on fragment $A$. Thus, among the redundant set of $M{n}_{occ}$ localized orbitals, one selects all orbitals above the smallest FOLI by a threshold (0.01 by default) and assigns each orbital to the originator fragment. The orbitals which are then symmetrically orthogonalized, and projected out from the space of remaining unassigned occupied space. In case a set of orbitals is redundant due to a symmetry in the system or simple covalency, then the orbitals are split over all contributing fragments (originators). After the iterative process, each molecular fragment has associated a set of localized orbitals derived from the simplest orbital spread criterion, which in turns determines the fragment’s formal charge or oxidation state in a natural manner.

The new method expects fragments, otherwise it localizes on atomic centers instead. Although this method was developed for oxidation state, it produces a set of localized orbitals on fragments or atoms that can be used like any other localization method.

These orbitals do not suffer from the multiple minima problem like most localization procedure that do iterative rotations discussed in the ER section. This is mainly because the iterative process essentially takes the low-hanging fruit orbitals first before going to higher up ones that are more ambiguous. In very few examples, it was found that the orbitals above the threshold could demonstrate another solution, i.e., another Lewis picture. Therefore, the algorithm was augmented to look into the next set of orbitals, print them out, and see if they are similar enough to the current ones, by looking at the singular values of the overlap of the current set and the next set.

More precise control of OSLO goes under the $loco input section, with keywords that are introduced below. See the example for reference.