Kuhn sum rule

In this contribution, we have shown that the Bethe sum rule, like the Thomas-Reiche-Kuhn sum rule, is satisfied exactly in the random phase approximation for a complete basis. Thus, in calculations that are related to the generalized oscillator strengths of a system, the Bethe sum rule may be used as an indicator of completeness of the basis set, much as the Thomas-Reiche-Kuhn... 

Figure 2 Dipole oscillator strength distribution in gaseous water [29, curve. A], in liquid water [31, curve, B] and in gaseous cyclohexane [32, curve, C]. Data in liquid water are obtained from an analysis of UV-reflectance and that in cyclohexane, from synchrotron-UV absorption. The Thomas-Kuhn sum rule is satisfied approximately in each case.

The usefulness of the oscillator strengths stems in part from the fact that they satisfy several sum rules. The Thomas-Reiche-Kuhn sum rule is given by1... 

It should be noted that in the RPA, the dipole oscillator strengths calculated in dipole velocity, dipole length, or mixed representation and all sum rules would be identical, and the TRK sum rule, Eq. (13), would be fulfilled exactly, that is, be equal to the number of electrons if the computational basis were complete [30,34,35]. Comparison of the oscillator strengths calculated in the different formulations thus gives a measure of the completeness of the computational basis in addition to the fulfillment of the Thomas-Reiche-Kuhn sum rule (vide infra). 

Table 5.5 The number of electrons, the Thomas-Reiche-Kuhn sum rule in the length representation So, and the mean excitation energy, /q, for the nucleobases...

Equation (4.12) implies an equivalence similar to (4.14) for true operators and eigenfunctions which may be used to derive the Thomas-Reiche-Kuhn sum rule. Equation (4.14), however, does not produce a sum rule for transition moments as shown in Appendix D. A general study of sum rules for effective operators will be presented elsewhere [79]. 

Equation (4.14) provides the equivalence between the dipole length and dipole velocity transition moments for a system of n identical particles of mass m with state-independent effective operator definition A. To see that this equivalence does not produce a sum rule, consider first the usual derivation of the Thomas-Reiche-Kuhn sum rule for the true operators. Left- and right-multiplying equation (4.12) by ( l and ), respectively, the z component yields... 

This is called the Thomas-Reiche-Kuhn sum rule for a one-electron spectrum. If Z electrons are involved, then the sum is simply Z. 

For k = Q one obtains the Thomas-Reiche-Kuhn sum rule for the oscillator strength /, which states that the zeroth moment is proportional to the number N e ) of electrons. 

We will now show the Thomas-Reiche-Kuhn sum rule ... 

Several dipole oscillator strength sums are related to other molecular properties by so-called dipole oscillator strength sum rules. The best known is the Thomas Reiche-Kuhn sum rule that relates the S(0) sum to the number of electrons N of the system, i.e. 

These results express well known very general quantum mechanical constraints for example, equation (56) is a generalized Condon sum rule, and equation (57) is the Thomas-Reiche-Kuhn sum rule within mixed length-velocity formalism. 

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