Localized orbitals Pipek-Mezey

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. 

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. 

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 ... 

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