Three-orbital mixing problem

The stability of this well-established bonding pattern can be understood from simple MO arguments. The B-H-B arrangement in diborane is another three-orbital mixing problem, like allyl. In fact, the mixing diagram we developed for allyl translates directly to the... 

The results of the three-orbital mixing problem can be summarized as follows ... 

Use the methods of secular determinants to derive the MOs and energies for the cyclic three-orbital mixing problem. Let the three orbitals be simple s orbitals. As with Hiickel theory, let the overlap integrals equal zero. The answer should be the... 

Cyclopropane is very well treated by a combination of group orbitals and the perturbation theory analysis described in this chapter and Chapter 1. First, however, we must consider a new aspect of orbital mixing, the general problem of combining three, equivalent orbitals in a cyclic array. We now have two choices on how to proceed in solving this problem. 

Why should this orbital mixing be stabilizing The evident problem is that this is a three-t tcixon system. One electron must occupy the antibonding orbital, and this is certainly destabilizing. Nevertheless, the overall system is still stabilized because two electrons are able to occupy the low-energy bonding orbital. 

Because the mapping approach treats electronic and nuclear dynamics on the same dynamical footing, its classical limit can be employed to study the phase-space properties of a nonadiabatic system. With this end in mind, we adopt a onemode two-state spin-boson system (Model IVa), which is mapped on a classical system with two degrees of freedom (DoF). Studying various Poincare surfaces of section, a detailed phase-space analysis of the problem is given, showing that the model exhibits mixed classical dynamics [123]. Furthermore, a number of periodic orbits (i.e., solutions of the classical equation of motion that return to their initial conditions) of the nonadiabatic system are identified and discussed [125]. It is shown that these vibronic periodic orbits can be used to analyze the nonadiabatic quantum dynamics [126]. Finally, a three-mode model of nonadiabatic photoisomerization (Model III) is employed to demonstrate the applicability of the concept of vibronic periodic orbits to multidimensional dynamics [127]. 

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 only bvo electrons in this simple cation but we need to mix the 7t bond (Jt and rt" orbitals) with the empty p orbital to give the MOs. One MO will be bonding all round the ring and this is the only one that matters to the structure. The others may have given you problems. We can mix the p-AO with the Jt-MO as they have the same s>Tnmetry in a three-membered ring but we cannot mix p with 7t. So our three MOs are t + p, ti — p, and n. ... 

It is found (Problem 8.6) that, for each Xi, the global contribution of the 2p orbitals is approximately three times that of the 2s. That is, the mixing of 2s and 2p C orbitals involved in the construction of equivalent and almost localized functions is approximately equivalent to the consideration of sp hybrid orbitals. Thus, ignoring residual delocalizations, the functions Xi can be represented as in Fig. 8.9. 

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