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: , where is the localization operator and is a molecular spatial orbital. 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. 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 The Foster-Boys (also known as Boys) localization method minimizes the spatial extent of the orbitals by minimizing , where . This turns out to be equivalent to the easier task of maximizing . In one dimension, the Foster-Boys (FB) objective function can also be written as . Fourth moment The fourth moment (FM) procedure 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 . 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 Edmiston-Ruedenberg localization maximizes the electronic self-repulsion energy by maximizing , where . Pipek-Mezey Pipek-Mezey localization 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. Some notable choices are Voronoi charges, Becke charges, Hirshfeld or Stockholder charges, intrinsic atomic orbital charges, Bader charges, or "fuzzy atom" charges. Rather surprisingly, despite of 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