Hybrid orbitals orthogonal sets

In contrast to the four tetrahedrally oriented elliptic orbits of the Sommer-feld model, the new theory leads to only three, mutually orthogonal orbitals, at variance with the known structure of methane. A further new theory that developed to overcome this problem is known as the theory of orbital hybridization. In order to simulate the carbon atom s basicity of four an additional orbital is clearly required. The only possible candidate is the 2s orbital, but because it lies at a much lower energy and has no angular momentum to match, it cannot possibly mix with the eigenfunctions on an equal footing. The precise manoeuvre to overcome this dilemma is never fully disclosed and appears to rely on the process of chemical resonance, invented by Pauling to address this, and other, problems. With resonance, it is assumed that, linear combinations of an s and three p eigenfunctions produce a set of hybrid orbitals with the required tetrahedral properties. 

There are also many examples of linear M—O—M units, and in these more delocalized n bonding occurs. The [ClsMOMCls]"- ions provide straightforward examples. It may be assumed that by using a pair of sp hybrid orbitals on the oxygen atom a linear pair of M—O—M a bonds is formed. There are then two orthogonal, occupied pn orbitals on the oxygen atom that can interact with suitable metal dn orbitals. Now, however, there is a total of four such dn orbitals and two sets of n interactions (16-11) (one in the xz plane and the other just like it in the yz plane)... 

Each set can be shown to be orthonormalized, provided that the s- and p-functions are normalized and orthogonal. Fig. 10.5 shows a contour plot of a tetrahedral sp hybrid. The strong directional character of this and other hybrid orbitals enhances the overlap with neighboring orbitals, thus contributing to stronger bonds. 

These are acceptable for many purposes, but the complex number VI (= — 1) makes them difficult to visualize. Given a set of solutions to the wave equation, which we find inconvenient, it is always permissible to transform these into an equal number of new functions obtained by taking linear combinations of the original ones, provided that orthogonality is upheld, that is, the overlap integral f dr i j) is always zero. (This procedure is utilized in other familiar situations, for example, in the construction of hybrid orbitals, and of MOs as linear combinations of AOs see Molecular Orbital Theory.) Eliminating i from equations (1-5) by equations (6-9),... 

Natural bond orbital (NBO) analysis The NBO analysis transforms the canonical delocalized Hartree-Fock (HF) MOs and non-orthogonal atomic orbitals (AOs) into the sets of localized natural atomic orbitals (NAOs), hybrid orbitals (NHOs), and bond orbital (NBOs). Each of these localized basis sets is complete, orthonormal, and describes the wavefunction with the minimal amount of filled orbitals in the most rapidly convergent fashion. Filled NBOs describe the hypothetical, strictly localized Lewis structure. NPA charge assignments based on NBO analysis correlate well with empirical charge measures. ... 

The cage system is treated quantum mechanically. In the original version of the model all valence electrons were included and to allow a natural definition of the cage, orthogonalized atomic hybrid orbitals were used as a basis set [215]. This allows to avoid problems with the saturation of dangling bonds since all hybrids on the same atom may belong to the cage with a wave function obtained by solution of a closed-shell secular equation. 

Like the molecular orbitals, the hybrid orbitals should comprise a mutually orthogonal set. For example, the two sp hybrid orbitals just used are orthogonal ... 

This proof relied on the atomic orbitals used to create the hybrid orbitals being themselves normalized and orthogonal. Other combinations, such as the three sp or the four sp hybrid orbitals, are also orthogonal sets. 

FIGURE 13.26 Operation of the symmetry classes of T on the sp orbitals. The a, b, c, and ti labels are used only to keep track of the individual hybrid orbitals. The nrnnber of hybrid orbitals that do not move when a symmetry operation occurs is listed in the final coliunn. This set of mrmbers is the reducible representation F of the sp orbitals. The great orthogonality theorem is used to reduce F into its irreducible representation labels. 

A second condition, which does not apply generally to orbitals but which does apply to different atomic orbitals on the same atom is that they do not overlap with each other. The correct terminology for orbitals that have zero overlap is that they are orthogonal. We have seen that the overlap of two orbitals is found by integrating over space the product 951952- Since our s and p orbitals, and also the resulting hybrids, are on the same atom, we require for any pair in the s, p set and also for any pair in the hybrid set that Equation A1.2 be satisfied ... 

After the CASSCF calculation with the above choice of orbitals, in order to perform an efficient VB analysis, it is better in this case to resort to an overcomplete non-orthogonal hybrid set. The five active orbitals, in fact, can be split into ten hybrids, in term of which the VB transcription of the wavefunction turns out to be the simplest and the most compact. Such kinds of overcomplete basis sets are commonly used in constructing the so called non-paired spatial orbital structures (NPSO, see for example [35]), but it should be remarked that their use is restricted to gradient methods of wavefunction optimization, such as steepest descent, because other methods, which need to invert the hessian matrix (like Newton--Raphson) clearly have problems with singularities. 

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