Electronic transitions in intermediate coupling

As we have seen in Chapter 11, the energy levels of atoms and ions, depending on the relative role of various intra-atomic interactions, are classified with the quantum numbers of different coupling schemes (11.2)— (11.5) or their combinations. Therefore, when calculating electron transition quantities, the accuracy of the coupling scheme must be accounted for. The latter in some cases may be different for initial and final configurations. Then the selection rules for electronic transitions are also different. That is why in Part 6 we presented expressions for matrix elements of electric multipole (Ek) transitions for various coupling schemes. 

In various pure coupling schemes the intensities of spectral lines may differ significantly. Some lines, permitted in one coupling scheme, are forbidden in others. Comparison of such theoretical results with the relevant experimental data may serve as an additional criterion of the validity of the coupling scheme used. 

The wave function in intermediate coupling is a linear combination of the relevant quantities of pure coupling (see (11.10)). Line strength in intermediate coupling is defined by (24.4), whose submatrix element equals 

Here E(pJ) — E(fi J ) = AE 0, the operators in general are defined by (4.12) and (4.13), AE and submatrix elements are presented in atomic units. Then WEk is in s-1. Let us recall that the oscillator strengths are dimensionless quantities. The relevant expressions for magnetic dipole transitions are given by formulas (27.8) and (27.9), respectively. 

As was shown in Chapters 24 and 27, the use of intermediate coupling essentially changes the selection rules with respect to approximate quantum numbers. Let us discuss a few examples of transitions between configurations, whose energy levels for neutral or a-few-times-ionized atoms are calculated in intermediate coupling, but classified with the help of LS coupling. Thus, for 1-transition d1p — d8, instead of selection rules described by (24.23) for k = 1, we have 

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Superposition of configurations already allows two-electron transitions in the pure coupling scheme. However, to obtain more accurate values, intermediate coupling must be utilized. Unfortunately, this leads to very rapid increase of the order of energy matrices to be diagonalized. 

Three oxidations states are potentially available in a binuclear iron center. Enzymes with octahedral fi-o o bridged iron clusters can be isolated in each of the three states the diferric and diferrous states appear to be the functional terminal oxidation states for most of the enzymes, while the mixed valence state may be an important intermediate or transition state for some reactions (Que and True, 1991). In these enzymes the cluster participates primarily as a two-electron partner in the redox of substrates, perhaps using sequential one-electron steps. Without additional coupled redox steps the enzyme is in a new oxidation state after one turnover. In contrast only the diferric and mixed valence oxidation states have been found for 2Fe 2S clusters. The diferrous state may not be obtainable because of the high negative charge on [2Fe 2S(4RS)] versus -1 or 0 net charge for the diferrous octahedral (i.e., non-Fe S) clusters. The 2Fe 2S proteins either are one-electron donor/acceptors or serve as transient electron transfer intermediates. 

A total set of selection rules consists of the sum of all selection rules both exact and approximate . Transition is forbidden if at least one selection rule is violated. The transitions may be forbidden to a different extent -by one, two, three, etc. violated conditions. If an electronic transition is between complex configurations, then, as we shall see in the next section, there may be a large number of selection rules. However, the majority of them, especially with regard to the quantum numbers of intermediate momenta, are rather approximate, even when a specific pure coupling scheme is valid. This is explained by the presence of interaction between the momenta. 

In these formulas the energy difference AE is measured in atomic units, transition probabilities are obtained in seconds and the submatrix element in the cases of one or two shells of equivalent electrons for LS coupling may be taken from (27.3) or (27.4). When calculating using intermediate coupling one has to bear in mind that the appropriate wave functions are of the form (11.10). 

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