Localized bond orbital

Higuchi, J., J. Chem. Phys. 27, 825, (ii) Semi-localized bond orbital treatment of the allyl radical. Extension of VB. 

Localized bonding orbitals are then constructed from a linear combination of the orbital on each of the paired atoms. To do this we use the principle of maximum overlap, which states... 

Each B-H-B three-center two-electron bridge bond corresponds to a filled three-center localized bonding orbital requiring the hydrogen orbital and one hybrid orbital from each boron atom. 

The Forster-Coulson-Mofitt orbitals23 24, on the other hand, are considered to represent localized (bond) orbitals that help to visualize the bent bond character of the CC bonds of 1 and to explain its ring strain. They seem to be less suited to analyse substituent-ring interactions or the conjugati ve properties of 1. 

Hybridization. The simplest way to think about bonding in [1.1.1 Jpropellane is to characterize the state of hybridization at the two types of carbon atoms and then combine the hybrids pairwise into localized bond orbitals. In ordinary molecules, these have electron occupancy close to two, and the corresponding antibonds are nearly unoccupied. Bond delocalization can then be introduced by considering interactions of the occupied localized bond orbitals with the unoccupied localized antibond orbitals. 

The extension of the chain allows us to visualize the formation of bands. Figure 15 is for metal p. The complex band is the same as that of the metal but the fine structure of the levels is different and there is now a low level. In detail, everything takes place as if this level corresponded to an orbital localized on ethylene and on the first atom. The diagram of the levels of the complex is very like that of a metal having one electron less, as if indeed the first atom were blocked off. There is, thus, the appearance of a localized bond orbital, but this does not exclude considerable energetic contributions from the metallic mass as relatively unperturbed orbitals are very numerous. 

Localized Bond Orbitals A Pictorial Bridge between Molecular Orbital and Valence Bond Wave Functions... 

The standard MO wave function involves canonical MOs (CMOs), which are permitted to delocalize over the entire molecule. However, it is well known (8,9) that an MO wave function based on CMOs can be transformed to another MO wave function that is based on localized MOs (LMOs), known also as localized bond orbitals (LBOs) (10). This transformation is called unitary transformation, and as such, it changes the representation of the orbitals without affecting the total energy or the total MO wave function. This equivalence is expressed in Equation 3.64 ... 

FIGURE 3.4 Transformation of the valence orbitals of BeH2, from canonical MOs (left-hand side) to localized bond orbitals (right-hand side). This transformation leaves the polyelectronic Hartree-Fock function unchanged. 

As discussed in Chapter 1, the development of PES showed that the spectra could be simply interpreted if one assumed that electrons occupy delocalized molecular orbitals (25,26). In contrast, VB theory, which uses localized bond orbitals (LBOs), seems completely useless for interpretation of PES. Additionally, since VB theory describes equivalent electron pairs that occupy LBOs, the experimental PES results seem to be in discord with this theory. An iconic example of this failure of VB theory is the PES of methane that displays two different ionization peaks. These peaks correspond to the a and t2 MOs, but not to the four equivalent C—H LBOs in Pauling s hybridization theory. 

LBO Localized bond orbital. For a given molecule, the LBOs can be obtained from the canonical MOs (CMOs) by a unitary transformation that does not change either the total energy or the total wave function. (See Chapter 3.)... 

It can be easily checked that the hybrids given in either eq. (3.4.35) or eq. (3.4.36) are sp3 hybrids i.e., their hybridization index (population ratio of p orbitals and s orbital) is 3. The four sp3 hybrids of C are directed toward the corners of a tetrahedron and suitable for forming four localized bonding orbitals with four hydrogen Is orbitals. Such a bonding picture for CH4 is familiar to all chemistry students. 

Models of this type are present in the literature. The simplest ones are based on the use of local orbitals. It is the local self-consistent field (LSCF) approach [216,231, 265,266]. In it the chemical bonds between QM and MM regions are represented by strictly local bond orbitals (SLBOs). The BOs can be obtained by the a posteriori localization procedures known in the literature. The localized orbitals thus obtained have some degree of delocalization, i.e. they have non-zero contributions of the AOs centered on the atoms not incident to a given bond (or a lone pair) ascribed to this particular BO. These contributions are the so-called tails of the localized orbitals and neglecting them yields the strictly local BOs (SLBOs) which are used in the LSCF scheme. The QM part of the system is described by a set of delocalized MOs while the boundary is modeled by the frozen SLBOs. 

The effect of relativity on the bonding in this molecule is so large that even the qualitative features of the bonding cannot be correctly described by non-relativistic theory which (i) fails to predict any 5d-6s hybridization in the localized bonding orbital (ii) seriously underestimates 5d-6s hybridization in one of the nonbonding orbitals (NBOs) (iii) predicts incorrectly that one of the orbitals with mj 1/2 is entirely tt in character and has pure spin and (iv) erroneously predicts that the bond (in AuH) is formed solely from the interaction between the gold 6s and the hydrogen Is atomic orbitals because the non-relativistic molecular orbital (MO) wavefunction constructed from these two atomic orbitals predicts the AuH molecule to be unbound by 0.19 eV. 

Kem and co-workers have applied a previously developed method for analysing internal rotation in terms of localized bond orbital functions " to MeOH, using initially an MBS of STO orbitals for the SCF calculations and then extending these with a DZ basis. The essential result was that the dominant contribution to the barrier is due to overlap (exclusion-principle) interaction between closed-shell, localized bonds. The main effect of the lone-pair electrons on the barrier is partially to shield the oxygen nuclear charge from the bonding electrons. 

The local self-consistent field (LSCF) or fragment SCF method has been developed for treating large systems [105,134-139], in which the bonds at the QM/MM junction ( frontier bonds ) are described by strictly localized bond orbitals. These frozen localized bond orbitals are taken from calculations on small models, and remain unchanged in the QM/MM calculation. The LSCF method has been applied at the semiempirical level [134-137], and some developments for ab initio calculations have been made [139]. Gao et al. have developed a similar Generalized Hybrid Orbital method for semiempirical QM/MM calculations, in which the semiempirical parameters of atoms at the junction are modified to enhance the transferability of the localized bond orbitals [140]. Recent developments for ab initio QM/MM calculations include the method of Phillip and Friesner [141], who use Boys-localized orbitals in ab initio Hartree-Fock QM/MM calculations. These orbitals are again taken from calculations on small model systems, and kept frozen in QM/MM calculations. 

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