Two-orbital mixing problem

There is another way to solve the two-orbital mixing problem, which does not involve all the calculus shown above. It is a general approach that results from the variational theorem, and can be applied to ab initio, semi-empirical, or Hiickel theory. 

In the two-orbital mixing problem, we showed that when molecular orbitals are defined as linear combinations of atomic orbitals and are put into the Schrbdinger equation, followed by differentiation to minimize E, a series of simultaneous equations in the c s and E results. When mixing two orbitals, only two energies result along with two molecular orbitals. It is not much of a stretch to realize that there will be as many molecular orbitals with distinct energies as the number of atomic orbitals (or basis functions) we use to create the molecular orbitals. Moreover, there will be the same number of secular equations as the number of starting atomic orbitals or basis functions ( ), and hence the secular determinant will be n by n. 

It is actually easier to write the secular determinant first, and then create the secular equations (although we saw in the "Two-Orbital Mixing Problem" that they are derived in the opposite order). All secular determinants have the following form ... 

Putting this all in the context of only two adjacent carbon p orbitals gives the Huckel secular determinant for ethylene shown below. Compare this to the secular determinant for the "Two-Orbital Mixing Problem" given in Section 14.2.2. 

We can perform all the analysis in a manner similar to how the two-orbital mixing problem was addressed (Section 14.2.2). Alternatively, we can use some of the logic we have developed to this point in the chapter, and some of the rules of HMOT, to derive the answer. We choose the latter method here. 

The most simple and well known equivalence holds for the special case of two electrons in two orbitals, described by GVB(pp) and CASSCF(2,2) wave functions for covalent bonds [41]. When the bond has a mixed ionic-covalent character, the CASSCF(2,2) description is able to account approximately for this effect due to the presence of an extra ionic configuration. This problem is exactly replicated in a larger scale when... 

Now let the bands interact. The bands repel each other as they did in the H problem of Exercise 6.3. They mix in such a way that the lower band (valence band) will have dominant H character and the higher band (conduction band) dominant Li character. At k= ir/d, there is no interaction by symmetry. Unlike in the regular H problem these two orbitals are no longer degenerate and their linear combinations are not... 

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. 

The main drawback of the chister-m-chister methods is that the embedding operators are derived from a wavefunction that does not reflect the proper periodicity of the crystal a two-dimensionally infinite wavefiinction/density with a proper band structure would be preferable. Indeed, Rosch and co-workers pointed out recently a series of problems with such chister-m-chister embedding approaches. These include the lack of marked improvement of the results over finite clusters of the same size, problems with the orbital space partitioning such that charge conservation is violated, spurious mixing of virtual orbitals into the density matrix [170], the inlierent delocalized nature of metallic orbitals [171], etc. 

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]. 

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