Pipek-Mezey localization

The Pipek-Mezey localization scheme corresponds to maximizing the sum of the Mulliken atomic charges. The contribution from the hh MO to atom A is given as (eq. (9.5))... 

Although the localization by energy criteria (Edmiston-Ruedenberg) may be considered more fundamental than one based on distance (Boys) or atomic charge (Pipek-Mezey), the difference in computational effort means that the Boys or Pipek-Mezey procedures are often used in practice, especially since there is normally little difference in the shape of the final LMOs. 

Figure 33 1H-NMR chemical shifts in HX (X = F, Cl, Br, I) from scalar-relativistic (SR) and relativistic (SR+SO) EC-PP DFT calculations for common gauge (CG) and individual gauge for Pipek-Mezey (PM) localized orbitals (LO) in comparison to experimental gas-phase values.

A recently often used practical method is that of proposed by Pipek and Mezey [26], Their intrinsic localization is based on a special mathematical measure of localization. It uses no external criteria to. define a priori orbitals. The method is similar to the Edmiston-Ruedenberg s localization method in the a-n separation of the orbitals while it works as economically as the Boys procedure. For the application of their localization algorithm, the knowledge of atomic overlap integrals is sufficient. This feature allows the adoption of their algorithm for both ab initio and semiempirical methods. The implementation of die procedure in existing program systems is easy, and this property makes the Pipek-Mezey s method very attractive for practical use. 

In the local MP2 method [131] the occupied canonical MOs of the preceding HF calculation are localized using the Pipek-Mezey [37] localization procedure and keeping the localized orbitals orthonormal. Even though the localization is formally an 0 N ) step, the localization time is negligible for aU the local MP2 calculations. 

The basic strategy adopted for normal octet and hypercoordinate molecules XYn was first to carry out a standard closed-shell RHF calculation and then to localize the orbitals according to the population or overlap criterion introduced by Pipek and Mezey [12]. In all cases, it was straightforward to identify localized molecular orbitals (LMOs) associated with particular X—Y bonds. 

The same advantageous algorithmic complexity characterizes the so-called population localization method of Pipek and Mezey [21], where the functional of the form... 

The NBO procedure is not the only localization technique for transforming delocalized MOs into the intuitive Lewis structure description. Foster and Boys, Edmiston and Ruedenberg," and Pipek and Mezey reported alternative localization procedures that provide additional bridges between MO and VB theories. From the organic chemist s point of view, these approaches are conceptually similar to NBO and, for the sake of brevity, will not be discussed here. 

Pipek, J., Mezey, P. G. (1989). A fast intrinsic localization procedure applicable for ab initio and semi-empirical linear combination of atomic orbital wave functions. Journal of Chemical Physics, 90,4916-4926. 

J. Pipek and P. G. Mezey, /. Chem. Phys., 90, 4916 (1989). A Fast Intrinsic Localization Procedure Applicable lor Ab initio and Semiempirical LCAO Wavefunctions. 

The a posteriori localization of the subspace of the occupied orbitals is a relatively standard procedure, which can be achieved following a large variety of localization criteria (for a succinct overview, see Ref. [53]). In the context of correlation energy calculations, i.e., in various local correlation approaches , the most widely used localization methods are based either on the criterion of Foster and Boys [24] or that proposed by Pipek and Mezey [54]. For reasons which become clearer below, in the present work we will use the Foster-Boys localization criterion, which can be expressed in various equivalent forms [26]. In its the most suggestive formulation, the Foster-Boys localization procedure consists in the maximization of the squared distance between the centroids of the orbitals ... 

It is also possible to localize the MOs by localization procedures. We have shifted the emphasis from previous localization procedures by Boys and Ruedenberg to a new localization procedure by Pipek and Mezey. It maximizes the population localization and is suitable for ab initio and semiempirical methods. 

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