Equivalent localized orbitals

In the case of two valence electrons there is hardly any difference between the localized orbital and the canonical valence orbital, except for the fact that the localization has separated the valence shell somewhat from the other shells. — In the case of four valence electrons, the sigma bonding and the sigma antibonding canonical orbitals yield two equivalent localized orbitals which resemble distorted atomic (2s) orbitals on each of the two atoms. They are precursors of what will be seen to be sigma lone pairs and are denoted by oC and ok . The absence of a bond can be ascribed to the nonbonded repulsion between these orbitals. This corresponds to the case of the unstable Be2 molecule. —... 

The first row represents an alternative set of equivalent localized orbitals which is as strongly localized as the one just discussed. 56) They extend essentially over three atoms. Whereas the Kekuld type localized orbitals are symmetric with respect to the plane bisecting a bond, the localized orbitals in the first row are symmetric with respect to a plane containing two opposite atoms. The negative lobe extends only over one atom, and the... 

Molecular orbitals are not unique. The same exact wave function could be expressed an infinite number of ways with different, but equivalent orbitals. Two commonly used sets of orbitals are localized orbitals and symmetry-adapted orbitals (also called canonical orbitals). Localized orbitals are sometimes used because they look very much like a chemist s qualitative models of molecular bonds, lone-pair electrons, core electrons, and the like. Symmetry-adapted orbitals are more commonly used because they allow the calculation to be executed much more quickly for high-symmetry molecules. Localized orbitals can give the fastest calculations for very large molecules without symmetry due to many long-distance interactions becoming negligible. 

The MO and VB methods provide altema ive but equivalent descriptions of the bonding in a molecule. A set of molecular orbitals can always be transformed into a corresponding set of more localized orbitals, and vice versa. For example, according to the MO de-... 

Finally, it must be mentioned that localized orbitals are not always simply related to symmetry. There are cases where the localized orbitals form neither a set of symmetry adapted orbitals, belonging to irreducible representations, nor a set of equivalent orbitals, permuting under symmetry operations, but a set of orbitals with little or no apparent relationship to the molecular symmetry group. This can occur, for example, when the symmetry is such that sev-... 

This leads to modifications of the localized it orbitals. In benzene, for example, a Kekule localization which mixes the a and ir orbitals to form double banana bonds is preferred over the other equivalent ir localizations discussed. 60) In naphthalene a Kekule type structure is found similar to the one presently discussed, but different in that the (jtE2) are hybridized with corresponding o-CC bonding orbitals to form banana bonds, whereas the (ttC2 ) remains a pure jt orbital. 61 > While this is of interest in the discussion of the whole molecule, it is clear that certain intrinsic properties of the ir-electrons are more readily recognized by the localization which has been discussed here. We hope to discuss elsewhere localized orbitals involving a bonds in organic molecules. 

Ellipsoid of inertia, 198-202,439-440 Emission of radiation spontaneous, 121-122 stimulated, 118, 120,122,135-139 Energy conversion factors, 468 Energy-localized orbitals, 69, 103-104 Equilibrium frequencies, 147, 262 Equivalent representations, 400 Ethane ... 

So now we have two equivalent bonding orbitals Vt and if2 with the same energy. Moreover, and 1 2 are localized orbitals ir is localized between Be and Ha and 2 between Be and H. They are 2c-2e bonds. 

Fig. 11.3 illustrates the relative momentum profile of the 15.76 eV state in a later experiment at =1200 eV, compared with the plane-wave impulse approximation with orbitals calculated by three different methods. The sensitivity of the reaction to the structure calculations is graphically illustrated. A single Slater-type orbital (4.38) with a variationally-determined exponent provides the worst agreement with experiment. The Hartree-Fock—Slater approximation (Herman and Skillman, 1963), in which exchange is represented by an equivalent-local potential, also disagrees. The Hartree—Fock orbital agrees within experimental error. 

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