Orbital localization

Highest occupied molecular orbital Intermediate neglect of differential overlap Linear combination of atomic orbitals Local density approximation Local spin density functional theory Lowest unoccupied molecular orbital Many-body perturbation theory Modified INDO version 3 Modified neglect of diatomic overlap Molecular orbital Moller-Plesset... 

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. 

LORG (localized orbital-local origin) technique for removing dependence on the coordinate system when computing NMR chemical shifts LSDA (local spin-density approximation) approximation used in more approximate DFT methods for open-shell systems LSER (linear solvent energy relationships) method for computing solvation energy... 

Further examination of Table 1.10 reveals that the lowest unoccupied molecular orbital, i/ 4, is a pure p orbital, localized on carbon, since the coefficients are zero for all but the 2p orbital. The MO picture is in agreement with the usual qualitative hybridization picture for the methyl cation. 

We have extended the linear combination of Gaussian-type orbitals local-density functional approach to calculate the total energies and electronic structures of helical chain polymers[35]. This method was originally developed for molecular systems[36-40], and extended to two-dimensionally periodic sys-tems[41,42] and chain polymers[34j. The one-electron wavefunctions here are constructed from a linear combination of Bloch functions c>>, which are in turn constructed from a linear combination of nuclear-centered Gaussian-type orbitals Xylr) (in ihis case, products of Gaussians and the real solid spherical harmonics). The one-electron density matrix is given by... 

Finally, examine the highest-occupied molecular orbital (HOMO) of phenoxide anion. Is the HOMO the best electron-donor orbital Is the orbital localized primarily on oxygen or on carbon Is the observed product consistent with orbital control Explain your answers. 

When an electronegativity perturbation occurs, two molecular orbitals originally containing equal components of two atomic orbitals localize so that the lower one is primarily on the more electronegative atom, the upper one on the less electronegative atom. 

In several previous Sections (1.15, 1.19) we considered the interaction of lone-pair orbitals with orbitals localized in other regions of the molecules. In some cases an atom (ether oxygen or thioether sulfur, halogens) may carry several lone pairs. These lone pairs are generally described as localized orbitals pointing in tetrahedral directions on the atom to which they belong as in Fig. 47. The elee-... 

In Fig. 2 are depicted anion and neutral potential curves that are qualitatively illustrative of (lb,4d) the X II NH case mentioned earlier. In this anion, the HOMO is a non-bonding 2pn orbital localized almost entirely on the N atom. As such, its LCAO-MO coefficients are not strongly affected by motion of the N-H bond (because it is a non-bonding orbital). Moreover, the anion and neutral surfaces have nearly identical Re and a>e values, and similar De values, as a result of which these two sur ces are nearly parallel to one... 

Reduced nicotinamide-adenine dinucleotide (NADH) plays a vital role in the reduction of oxygen in the respiratory chain [139]. The biological activity of NADH and oxidized nicotinamideadenine dinucleotide (NAD ) is based on the ability of the nicotinamide group to undergo reversible oxidation-reduction reactions, where a hydride equivalent transfers between a pyridine nucleus in the coenzymes and a substrate (Scheme 29a). The prototype of the reaction is formulated by a simple process where a hydride equivalent transfers from an allylic position to an unsaturated bond (Scheme 29b). No bonds form between the n bonds where electrons delocalize or where the frontier orbitals localize. The simplified formula can be compared with the ene reaction of propene (Scheme 29c), where a bond forms between the n bonds. 

The UHF solution appears when the hopping integral t becomes small and leads to a spin density wave. The localization of the MOs leads to a and b atom centered orbitals, localized around odd and even labelled atoms respectively. 

Haberlen, O.D. and Rdsch, N. (1992) A scalar-relativistic extension of the linear combination of Gaussian-type orbitals local density functional method application to AuFl, AuCl and Au2. Chemical Physics Letters, 199, 491-496. 

Fournier, R., Andzelm, J., Salahub, D. R., 1989, Analytical Gradient of the Linear Combination of Gaussian-Type Orbitals-Local Spin Density Energy , J. Chem. Phys., 90, 6371. 

Remark 1. The concept for local stable and unstable manifolds becomes clear when one represents the stable and unstable manifolds of the hyperbolic fixed point (periodic orbit) locally. For details see (Wiggins, 1989) or (Wiggins, 1988). 

To illustrate these energy levels, Figure 2.1 shows formaldehyde as an example, with all the possible transitions. The n —> n transition deserves further attention upon excitation, an electron is removed from the oxygen atom and goes into the n orbital localized half on the carbon atom and half on the oxygen atom. The n-n excited state thus has a charge transfer character, as shown by an increase in the dipole moment of about 2 D with respect to the ground state dipole moment of C=0 (3 D). 

One of the first consequences of the above ideas was the development of the Orbital Local Plasma Approximation (OLPA) by Meltzer et al. [37-39]. The main ingredients in the OLPA consist in approximating the orbital weight factors by the orbital occupation numbers and adapting the Lindhard-Scharff Local Plasma Approximation (LPA) [10-12] to an orbital scheme whereby the orbital mean excitation energy was originally defined as [37,38]... 

E.s.r. measurements and X-ray studies show that the extra electron in the tricobalt compound occupies an antibonding orbital made up primarily of a combination of two cobalt 3d orbitals localized in the plane of the three cobalt atoms. Replacement of one Co by Fe again leads to a decrease in the metal-metal bond lengths [0.039 A (av)]. The effect is therefore not so marked as with the sulphur analogues and the effect of replacing sulphur by selenium in [C03-(CO)gS] is to decrease the metal-metal distances by 0.021 The relevant data for the three compounds are given below ... 

The electron density described by a core orbital space will of course strongly affect the nature of the active orbitals. The form of the inactive orbitals may be influenced by placing symmetry restrictions on them, or by invoking an initial orbital localization [11-15]. The localized orbitals that are not of interest for the VB description may then be placed in the core in the subsequent CASSCF or fully variational VB calculation and, if necessary, some or all of them may be frozen. 

At the RHF level of theory, which uses a wavefunction that is relatively straightforward to interpret, the subtle differences between the half- and full-arrow reaction schemes would remain well-hidden within the doubly-occupied, usually delocalized orbitals. While it can be argued that the application of an orbital localization procedure could produce a semblance of the SC description for the 1,3-dipolar cycloaddition of fulminic acid to ethyne, the double-occupancy restriction makes it impossible to obtain the analogue of a half-arrow SC mechanism using an RHF wavefunction. 

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