Perturbation degenerate orbitals

In quantum states with n > 1, quantum number / assumes different values for instance, for n = 3, / = 0, 1, 2. When / is equal to or greater than 1, several independent wave functions exist (21 + 1). An electron of the second level, sub-level p, can occupy three 2p orbitals of the same energy, described by three distinct wave functions. These orbitals, which, in the absence of external perturbation, rigorously have the same energy, are called atomic degenerate orbitals (ADs). 

In order to find the energy x, we use first order perturbation theory. The degenerate orbitals involved are the doubly occupied AO s of Cl- and I-, and the empty NBMO of ArCHa. The necessary matrix elements are found as before in terms of the CC1 and Cl resonance integrals fta, ft. Solving the resulting three-row secular equation, we find for the perturbed energies—... 

Figure 2.22. Schematic representation of the perturbation-induced shifts and splittings of the degenerate orbitals of 4 -electron perimeters. The indicated transitions correspond to the main contributions to the observed absorption bands the broken arrows symbolize the double excitation (adapted from Hdweler et al., 1989).

So far we have assumed that the overlap between the molecular orbitals of the two molecules is negligible in an excimer complex. At short distances, say rP... P < 300 pm, orbital overlap leads to further stabilization of an excimer. As can be seen from Figure 2.25, first-order perturbation of the degenerate orbitals (Section 4.3) due to... 

Perturbation theory gives reliable results only as long as the perturbation 8 of a given orbital is small compared with the difference between its energy sj and that of all other orbital energies sb sj-Si 8sjl To account for perturbations on degenerate orbitals, st = sj, we... 

Clearly, first-order perturbation theory for degenerate orbitals cannot be done as described above, because the choice of any linear combination of degenerate... 

We now return to first-order perturbation theory for degenerate orbitals (Section 4.3). As any linear combination of two degenerate orbitals i/q and xj/j is equally valid, we set up a trial wavefunction function xj/jc = akixl/i-F akj t//7 and we have to solve the secular Equation 4.18. The eigenvalues of the unperturbed system will be equal for all linear combinations, = = (0). 

Equation 4.18 First order perturbation of two degenerate orbitals... 

If we choose the linear combination of the degenerate orbitals such that one is symmetric and the other antisymmetric with respect to a symmetry element that is retained in the perturbed molecule, then the off-diagonal elements will vanish, hki = hik = 0, because integration of any function over the complete range of its variable x vanishes if the function is antisymmetric Jf(x)dx = 0 (Figure 4.11). A function is said to be antisymmetric (a) if f(—x) = —f(x). The product of two antisymmetric functions is symmetric (s) we write a a = s similarly, s s = s but a s = s a = a. 

These results are equal to the total first-order pair correlation energy, obtained in Exercise 5.3, for the dimer using localized orbitals. The total first-order pair correlation energy is identical to the second-order many-body perturbation result for the correlation energy (see Chapter 6). The above results are a reflection of the fact that many-body perturbation theory is invariant to unitary transformations of degenerate orbitals. 

Mixing of Degenerate Orbitals—First-Order Perturbations... 

This is a second-order perturbation because of the square in the numerator, and typically second-order orbital interactions are smaller than those between degenerate orbitals. In the situation where overlap is neglected, we have Equation 14.61. 

These rules follow directly from the quantum-mechanical theory of perturbations and the resolution of the secular equations for the orbital interaction problem. The (small) interaction between orbitals of significantly different energ is the familiar second order type interaction, where the interaction energy is small relative to the difference between EA and EB. The (large) interaction between orbitals of same energy is the familiar first order type interaction between degenerate or nearly degenerate levels. 

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