Edmiston-Ruedenberg localization

The Edmiston-Ruedenberg localization scheme uses the inverse of the distance between two electrons as the operator, and maximizes the expectation value. 

For such localizations to be effective in the present context, those orbital symmetry constraints that would prevent maximal localization in larger molecules must be abandoned. For instance, in the NCCN molecule, Ciy symmetry can be preserved during the localization process, but not left-right mirror symmetry. We have used Raffenetti s (55) version of the Edmiston-Ruedenberg localization method 54). 

The Edmiston-Ruedenberg localization procedure /58/ consists of an intrashell transformation of selfrepulsion energy until the sum... 

The first order electric moments obtained for bond and lone pair electron densities using Edmiston-Ruedenberg localization method are summarized in Table 4 and the geometry of the studied systems are presented in Table 3. The origin of (r) is taken at the corresponding heavy atom nucleus. The values... 

Given these encouraging preliminary results, further investigation of the effect of SIC on reaction barriers is warranted. For more complicated reactions, it is likely that the initial scheme used here will need to be elaborated, either by going to better localization methods (e.g. Edmiston-Ruedenberg localization [79]), or by incorporating the SIC term in the self-consistent procedure, which is more expensive. 

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. 

As to the localization of occupied orbitals the conventional procedures (Edmiston-Ruedenberg (Edmiston etal., 1963), Boys (Boys, 1966) might not be the most suitable because they do not restrict the magnitude of the off-diagonal Fock matrix elements. Regarding the localization of virtual orbitals, they cannot be localized uniquely into... 

These two criteria appear to be generalizations of the localization criteria named after Edmiston-Ruedenberg and Boys respectively [1, 2, 3]. Nevertheless the pair functions ipp, constructed by either of these criteria, are generally symmetry-adapted (in spite of having localized features as well) unlike for the one-electron case there is no strict alternative symmetry-adapted vs. localized [5],... 

Interestingly, and probably due to a very exciting connection between the Fermi-hole and the localized orbitals [28], various localization methods result in rather similar localized orbitals, except for the description of double bonds by a o- and 7r-orbital-pair or two equivalent r (banana) bonds. Boys localization gives r orbitals, while the Edmiston-Ruedenberg and the popula-... 

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. 

As with the smaller compounds, reliable computational descriptions of methyl phenyl sulfoxide excited states are not available. Ground state computations are easily accessible for molecules of this size. At the RHF/6-31G(d,p) level, the HOMO is 7t with regard to the SO bond but delocalized throughout the whole n-system. The next two descending orbitals are localized on the phenyl and SO, respectively. (The sulfur lone pair is the HOMO-2 when the valence bond orbitals are approximated by the Edmiston-Ruedenberg method.) While the LUMO is extensively delocalized, the LUMO-fl is entirely localized on the phenyl ring. 

Calculation of the Edmiston-Ruedenberg energy-localized MOs is very time consuming. Boys proposed a method to find localized MOs that is computationally much faster than the Edmiston-Ruedenberg method and that gives similar results in most cases see D. A. Kleier, J. Chem. Phys., 61,3905 (1974). 

Calculation of the Edmiston-Ruedenberg energy-localized MOs is very time consuming. Boys (and Foster) proposed a method to find localized MOs that is computationally much faster than the Edmiston-Ruedenberg method and that gives similar results in most cases see D. A. Kleier, J. Chem. Phys., 61, 3905 (1974). The Boys method defines the LMOs as those that maximize the sum of the squares of the distances between the centroids of charge of all pairs of occupied LMOs. The centroid of charge of orbital is defined as the point at (xc,yc>Zc), where Xc = ff>i x tf>i), yc = (I y I ). Zc = 4>i z (t>i) Tjj is, the distance between the centroids of LMOs i and j, the Boys LMOs maximize 2j>, 2, rfj. 

Numerical LMOs of this work are determined by the natural localized-molecular-orbital (NLMO) method A. E. Reed and F. Weinhold, J. Chem. Phys. 83 (1985), 1736. The LMOs determined by other methods (e.g., C. Edmiston and K. Ruedenberg, Rev. Mod. Phys. 34 [1963], 457 and J. M. Foster and S. F. Boys, Rev. Mod. Phys. 32 [1960], 300) are rather similar, and could be taken as equivalent for present purposes. 

Edmiston, C., Ruedenberg, K. J. Chem. Phys. 43, S 97 (1965). These localized orbitals are based on the canonical orbitals given in Ref. 34. Whenever possible, the BMO orbitals are plotted otherwise the contours are those of the SAO orbitals. 

A localization procedure similar to that of Edmiston and Ruedenberg has been proposed by Suthers and Linnett" for FSGO wave functions. The two electron... 

If the wave function that one considers, is a single Slater determinant , the spin orbitals (p, from which , is constructed, are not uniquely determined, but rather there is an infinity of equivalent sets of qo, related by unitary transformations. To some extent one can make the qo, unique if one requires either that they are canonical (diagonalize the Fock operator) and are symmetry-adapted, or localized (e.g. according to the criteria of Edmiston and Ruedenberg or Foster and Boys [1-3]). The localized spin orbitals have some advantages both for the chemical interpretation and for the computation of correlation corrections. 

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