Equivalent bond orbital models

E. Honegger, E. Heilbronner, in Theoretical Models of Chemical Bonding, Vol. 3, Z. B. Maksic, Ed., Springer Verlag, Berlin- Heidelberg, 1991, pp. 100-151. The Equivalent Bond Orbital Model and the Interpretation of PE Spectra. 

Following the work of Robin and Day, and Allen and Hush, the first rigorous solution to the vibronic Schrddinger equation for a mixed valence system was formulated by Piepho, Krausz, and Schatz. This model, based on the valence bond approach, is known as the PKS model. It is a model that has been modified to include orbital vibronic constants. A frilly equivalent molecular orbital model developed by Piepho is easier to conceptualize, however. The Piepho model also was the first to take advantage of the concept of orbital vibronic constants developed by Bersuker for Jahn-Teller systems (Figure 4). The Jahn-Teller theory is the basis for the PKS model. 

A sin e bonding model for a C v symmetry was used. Two equivalent bonding orbitals on the central iodine, generated from the s,Px and Py atomic orbitals, overlap pure p orbitals of the terminal atoms. The remaining orbitals of the central iodine as well as all other atomic orbitals were considered as being filled and non-bonding. 

The extension of independent electron treatments—e.g. of the type proposed by Huckel for n systems —to sigma systems, and in particular to hydrocarbons, has a long and well-known history. The early treatments used an orthonormal basis of atomic or bond orbitals with parametrized coulomb energies and interaction matrix elements restricted to nearest neighbours only. The most attractive approximation of this kind, proposed by Hall and Lennard-Jones in 1951, is the equivalent bond orbital (EBO) model, which has been used extensively since, with variations due mainly to Lorquet, Brailsford and Ford, Herndon, Murrell and Schmidt, and Gimarc . The conceptual consequences of such a treatment, in particular the phenomenon of -conjugation in saturated hydrocarbons, have been discussed in detail by Dewar ... 

The basis functions of such an EBO model are orthonormal two-centre equivalent bond orbitals (EBO) which we designate by = Their self-energies and interaction crossterms are defined as follows ... 

The important conclusion to be drawn from these observations is that the assumption of equivalent bond orbitals 2 as used in an EBO treatment can be validated by the properties of LMOs Aj derived from ab initio calculations. The mathematical formalism for handling LMOs according to equations, 32 and 33 is exactly the same as that of an EBO model, which implies that this Hiickel-type treatment is capable of yielding excellent approximations to the SCF orbital energies Sj and the corresponding CMOs (pj of saturated hydrocarbons. 

The bent-bond model can be expressed in orbital terms by assuming that the two components of the double bond are formed from sp3 hybrids on the carbon atoms (Figure 3.19) That this model and the ct-tt model are alternative and approximate, but equivalent, descriptions of the same total electron density distribution can be shown by converting one into the other by taking linear combinations of the orbitals, as shown in Figure 3.20. But neither form of the orbital model can predict the observed deviations from the ideal angles of 109° and 120°. 

Pauling150 has provided an alternative, valence bond treatment of the quadruple bond involving spd hybrid orbitals and four equivalent bent bonds. His model also explains the experimental facts described above and provides a good estimate of the bond length. 

This function is zero for all points equidistant front the two nuclei (that is, it has a nodal plane). The orbital y>j is large in the region between the nuclei (where lsA and 1% overlap and the electrostatic potential is low) and is generally referred to as a bonding orbital. Similarly, y>t, which keeps its electron away from the internuclear region, is antibonding. The two functions and yj2 are analogous to the symmetric and antisymmetric orbitals for the one-dimensional model. A similar transformation can be applied and two equivalent orbitals constructed. These are... 

There was much affinity between Coulson and Barriol, not only because of the many subjects they shared, but also because of their similar way of proceeding and thinking. They both conceded a high value, in many respects, to the determination of dipole moments. Both worked on methane (CH4) and more particularly on the dipole moment of the C-H group, for which Coulson gave a direction when Barriol s simple model could not. [25] It is highly interesting to compare the way how the two authors express themselves to show that experience or physical and chemical evidence had to correct the false inferences or deductions that square in no way with reality the description of the carbon electronic structure fails to account for four equivalent bonds. We have to admit that the C-orbitals that are... 

M—has also been reported for olefins and acetylenes ir-bonded to rhodium and to platinum (6, 21, 46, 87). In the case of rhodium, iy(i°3Rh—is between 10 and 16 Hz for a 7r-bonded olefin (see Table XXVII), while for the cr-bonded carbon in [(C5H5)Rh(ff-C3Hs)-(w-CsHb)], 7( ° Rh—is 26 Hz. It was suggested the bonding of the olefin results from a 60% contribution from a dsp -vnet X orbital and sp -carbon orbital 21). For the olefins and acetylenes w-bonded to platinum 7( Pt—is between 18 and 195 Hz (see Table XXIX) compared to the range of 360 to 1000 Hz reported for carbon cr-bonded to platinum. It was found that 7( Pt— C) is less for a 7r-bonded acetylene than for a rr-bonded ethylene. This was considered as evidence for the Chatt-Dewar-Duncanson molecular orbital model 39, 63) of TT-bonding (XIV), rather than the formally equivalent valence-bond treatment, (XV) and (XVI) 46). However, no allowance appears to have been made for the effect on the hybridization at the carbon of the pseudo-... 

It is possible in principle to calculate all of these modes from the theory of the electronic structure, which is equivalent to the calculation of all the force constants. Indeed we will see that this is possible in practice for the simple metals by using pseudopotential theory. In covalent solids, even within the Bond Orbital Approximation, this proves extremely difficult because of the need to rotate and to optimize the hybrids, and it has not been attempted. The other alternative is to make a model of the interactions, which reduces the number of parameters. The most direct approach of this kind is to reduce the force constants to as few as possible by symmetry, and then to include only interactions with as many sets of neighbors as one has data to fit- for example, interactions with nearest and next-nearest neighbors. This is the Born-von Karman expansion, and it has somewhat surprisingly proved to be very poorly convergent. This simply means that in all systems there arc rather long-ranged forces. 

Most clusters, however, caimot be described adequately in terms of two-center, two-electron bonds because the coimectivity of the vertices exceeds the number of valence orbitals that are available for bonding. Early efforts to rationalize such systems, such as Lipscomb s styx approach and Kettle s Topological Equivalent Orbital Method, are described by Mingos and Johnston. In these more difi cult cases, the simple valence-bond picture is inappropriate examples are deltahedral clusters composed of B-H vertices or conical M(CO)3 fragments, both of which usually have only three orbitals available for skeletal bonding. Theoretical models for describing these systems will be discussed in the next section. 

Although the extended valence bond PKS model probably can always be shown to be equivalent to a molecular orbital model for mixed valency systems involving complex bridging ligands such as found in the Creutz-Taube ion, these systems are best treated using a MO approach. A qualitative description is now given of the MO ideas developed by Piepho. ... 

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