Pipek-Mezey method

There is little experience with the von Niessen method, but for most molecules the remaining three schemes tend to give very similar LMOs. The main exception is systems containing both a- and vr-bonds, such as ethylene. The Pipek-Mezey procedure preserves the cr/yr-separation, while the Edmiston-Ruedenberg and Boys schemes produce bent banana bonds. Similarly, for planar molecules which contain lone pairs (like water), the Pipek-Mezey method produces one in-plane cr-type lone pair and one out-of-plane yr-type lone pair, while the Edmiston-Ruedenberg and Boys schemes produce two equivalent rabbit ear lone pairs. 

Still another way to calculate LMOs is the Pipek-Mezey method, which maximizes a certain sum that is related to the Mulliken gross populations of the orbitals [J. Pipek and P. G. Mezey, J. Chem. Phys., 90,4916 (1989)]. Unlike the Edmiston-Ruedenberg and the Boys LMOs, the Pipek-Mezey LMOs for a double bond consist of one a and one tt MO. (See Section 15.9.)... 

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

The population method due to Pipek and Mezey [37] maximizes a certain sum of gross atomic Mulliken populations. The latter are not realistic when the LCAO basis includes diffuse atomic orbitals as is necessary for crystalhne sohds. 

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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