Degenerate wavefunctions

The 7T-electron MOs Of and Of are degenerate, in that they correspond to the same tt-electron orbital energy (a— ft). As is the case with any degenerate wavefunctions, any combination of Of and Of will be equally valid. It is convenient to combine Of and Of in such a way as to obtain real MOs, this can be done by ... 

Since the expression for Ejis symmetrical with respect to the exchange of quantum numbers j and k, i.e. Ej k = Ekj. In other words, i// and ifkj are degenerate wavefunctions. 

When the unperturbed energy levels are n-fold degenerate (e.g., n valence-state unperturbed atomic C wavefunctions for the n C atoms in an alkane molecule C H2n + 2), then caution must be used to avoid divisions by zero in equations such as Eq. (3.13.8). The "fix" is to choose the degenerate wavefunctions more closely, adding a suffix i for the zth member of an n-fold degenerate set ... 

Because an SCF calculation does not allow for correlation, the wavefunctions of the configurations are not eigenfunctions of the Hamiltonian operator H. We set up a Cl wavefunction (Equation 4.35) considering only the two degenerate wavefunctions, xP = C X -2Jr C2X2 - 1, and we have to solve the 2 x 2 secular problem (Equation 4.36). 

Section 4.2, any normalized linear combination of degenerate wavefunctions is an equally valid wavefunction. The linear combinations of the NBMOs with the smallest possible amount of local overlap for 3 and 17 are shown in Figure 5.14. Those of 3 are separated entirely in space to the starred and unstarred sets of atoms. Hence 3 is a disjoint biradical. The same cannot be achieved by any linear combination of the NBMOs of 17 it is classified as a nondisjoint biradical. 

The same answer is obtained from equation 4.8, irrespective of wliich of Xi " Xa is used to start with. For the doubly degenerate wavefunction however, using equation 4.8 with xi gives... 

Therclore, i// (c) and lJiy e) are a set of two orthonormal orbitals belonging to the e representation. In fact, there is considerable choice as to the form of the degenerate wavefunctions. Any linear combination of t td i// (c) are also valid wave-functions. 4.10shows the form of the functions of equations 4.19 and 4.27. Orbitals that look like these will occur again and again throughout the book. 

Degenerate and nondegenerate perturbation theories share the feature that the first-order correction to the energy is the expectation value of the perturbation operator using the unperturbed wavefunction. Often we can evaluate or estimate this expectation value by inspection. The special problem for degenerate wavefunctions or orbitals is that there is an infinite number of ways to express the unperturbed wavefunctions, and only one of these ways is proper for finding The mathematical procedure for finding the proper set of functions tends to complicate and obscure the situation. However, it is often easy to deduce the proper unperturbed wavefunctions by inspection. 

Assume a particle is confined to a cubical box. For what set of three quantum numbers will there first appear degenerate wavefunctions For what sets of different quantum numbers will there first appear degenerate wavefunctions ... 

FIGURE 11.17 The energy level diagram for a hydrogen atom, showing the n and quantum numbers for the levels. The quantized energy levels are labeled. Degenerate wavefunctions are... 

The restriction m = n comes from the derivation of equation 12.21. The integration in the numerator is over the complete space of the system. The requirement that this is nondegenerate perturbation theory also eliminates the possibility that two energies E ° and E ° might be equal due to degenerate wavefunctions. (The extension of perturbation theory to degenerate wavefunctions will not be discussed here.)... 

As shown in Chapter 4, there is always a choice to be made in writing degenerate wavefunctions. Recall that two new functions may be generated by taking a linear combination of the old. An alternative pair of wavefunctions is shown... 

The existence of several degenerate wavefunctions distinguished only by the relative orientation of the angular spin momentum. Defined by the total angular spin momentum 5, the multiplicity of a state is given by 25 -f- 1. 

As for the wavefunctions of the G3/2 representation, there are two sets of 4-fold degenerate wavefunctions corresponding to the spatial wavefunctions of T2 and E. The wavefunctions that correspond to the spatial wavefunctions are as follows ... 

The same procedure as in the non-degenerate case is now continued to obtain the first-order correction to the energy. Equation 4-28 is now multiplied by the complex conjugate of one the unperturbed degenerate wavefunctions, m[Pg.78]

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