Localized orbitals Edmiston-Ruedenberg

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. 

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). 

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... 

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. 

In CH4, the equivalent C—H orbitals are determined by symmetry. In other molecules with less symmetry, e.g. HjO, CjHg, etc., t may be determined so as to minimize the magnitude of the exchange terms - in Eq. (144), or by a similar criterion. Localized orbitals have been studied extensively by Pople, Linnett and Hall >. Very recently, Ruedenberg and Edmiston have developed a procedure for calculating them based on the "minimum exchange criterion. ... 

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. 

It is obvious that the problem is the delocalized nature of the MOs. Fortunately it has been known, based on the work of Lennard-Jones [192] and worked out mainly by Edmiston and Ruedenberg [193] and also by Foster and Boys [194], that it is possible to transform the (occupied) MOs to localized orbitals (LMOs) which span the same space, and which are orthogonal to each other, but which are mainly localized in rather restricted areas of space. 

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. 

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. 

Edmiston, C., Ruedenberg, K. (1963). Localized Atomic and Molecular Orbitals. Reviews of Modem Physics, 35, 451-465. 

In the early 1960s, Edmiston and Ruedenberg placed the localization of molecular orbitals on a somewhat more objective foundation by transforming to that basis in which interorbital exchange is a minimum. Lipscomb and coworkers found that when applied to diborane this approach indeed leads to localized three-centre bonds for the B-H-B bridge. Lipscomb recalls how the localization of molecular orbitals... 

The method described above is essentially that due to Edmiston and Ruedenberg (1963). Another commonly used localization criterion (Boys and Foster, 1960) is that of maximum separation of the centroids of the transformed orbitals, and other methods have also been successfully applied (see e.g. Gilbert, 1964 Magnasco and Perico, 1967 McWeeny and Del Re, 1968). 

---