Equivalent bond orbitals

The character table for D3h tells us that the nitrogen atom orbitals involved in bonding are s, p py. We now use the projection operator technique to find the linear combinations of oxygen ligand orbitals i,J that combine with s,px,py, 

But how can we have generated three basis functions for a doubly degenerate representation The answer is that eqs. (6), (7), and (8) are not LI. So we look for two linear combinations that are LI and will overlap with the nitrogen atom orbitals px and 

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It is probable that eight equivalent bond orbitals can be constructed when four d orbitals are available, as in [.Mo(CW)8] = = and [W(C iV)8] = =. The arrangement of these in space has not yet been determined either experimentally or theoretically. 

It is found on analysis of the problem that when only two d orbitals are available for combination with the s and p orbitals six equivalent bond orbital- of strength 2.923 (nearly as great as the maximum 3 for the best spd hybrid) can be formed, and that these six orbitals have their bond directions toward the corners of a regular octahedron. We accordingly conclude that complexes such as [CoCNHs ], [PdClc]—, and [PtCle] — should be octahedral in configuration. This conclusion is of course identical with the postulate made by Werner to account for isomerism in complexes with different substituent groups,1 and verified also by the x-ray examination of Co(NH8)el3, (NHOaPdCU, (NH4)2 PtCl , and other crystals (see Fig. 5-1). 

This description of the triple bond represents it as an axially symmetric a bond together with two perpendicular n bonds. This is appropriate for spectroscopy and must be used if we are discussing excited N2 or N2+. But for N2 in its ground state, another description in terms of three equivalent bonding orbitals can be obtained by applying the trigonal transformation to (24) and (25). Thus if we write... 

The final molecule of this series is methane, the tetrahedral structure of which follows if a fourth unit positive charge is removed from the nucleus in the ammonia lone-pair direction. There are now four equivalent bonding orbitals, which may be represented approximately as linear combinations of carbon s-p hybrid and hydrogen Is functions. The transformation from molecular orbitals into equivalent orbitals or vice versa is exactly the same as for the neon atom. 

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. 

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. 

The rows and columns of the four-by-four Hamiltonian matrix correspond to Bloch sums of equivalent bond orbitals of the form... 

The equivalent bent bonds description of the triple bond is obtained using the same arguments used before for the double bond. Three equivalent bond orbitals Bl,B2,B3 are obtained by the linear mixing ofcr. nx, ny ... 

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. 

For H2O, the Lewis electron-dot formula suggests the localized MOs to be a pair of equivalent bonding orbitals 6(OHj) and (OHj), an inner-shell Ij oxygen orbital i (0), and two lone-pair equivalent MOs /i(0) and 2(0) on oxygen. 

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 carbon-carbon double bond, such as that in ethylene, may be described as follows. From 2 , and two of the 2p orbitals, which we may take to be yp2p, and p2pyy we form three equivalent bond orbitals, which will have their maximum values in the xy plane and will be... 

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