The Thomas-Reiche-Kuhn sum rule

From the previous discussion, if an electron is in state j, the total oscillator strength available for transitions to all other levels must add up to the classical value for one oscillator, i.e. 

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. 

The sum rule can also be derived by elementary quantum mechanics from the definition of the dipole transition probability and its relation to the / value. 

The simplest way is from Heisenberg s equation for the time evolution 

A good example occurs in the alkali spectra the integrated oscillator strength evaluated over the optical range is approximately 1 for all alkalis. This proves that the other electrons (i.e. those in the core) hardly participate in the visible spectrum. If, however, observations are extended to very high excitation energies, one eventually breaks into deeper electronic shells (see chapter 7), and so a separate sum rule will apply for inner-shell excitation. For instance, we may expect an oscillator strength of 10 for excitation from a d, subshell. 

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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... 

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... 

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. 

A second method that exists for normalization of relative absorption curves is dependent on the Thomas-Reiche-Kuhn (TRK) sum rule (see Section II). The zeroth sum rule simply states that the total oscillator strength for photoabsorption is equal to the number of electrons in the atom or molecule. In cases where the electronic shells are well separated... 

The Thomas Reiche Kuhn f-sum rule shows that the summation of the oscillator strengths of all transitions in a chromophore is equal to the number of electrons. In this way, we could envisage an electron count in the same way NMR and electron paramagnetic resonance (EPR)... 

Exercise 7.3 Derive the Thomas-Reiche-Kuhn and 5(2) sum rules from Eq. (7.90). 

The Thomas-Reiche-Kuhn (TRK) sum rule, which can be written in terms of the dipole oscillator strengths as... 

Equation (3) is known as the Thomas-Kuhn-Reiche (TKR) sum rule. 

As noted earlier, in the complete basis set limit of fhe RPA fhe Thomas-Reiche-Kuhn (TRK) sum rule should be equal fo fhe number of elecfrons Ne and fhe mean excifafion energies calculated from oscillafor sfrengfhs in fhe lengfh, mixed, or velocify representation should have the same values. The fulfillment of Eq. (13) and the agreement between the mean excitation energies in different representations thus provide figures of merif for fhe basis... 

Equation (3) is known as the Thomas-Kuhn-Reiche (TKR) sum rule. The oscillator-strength distribution d//dii is proportional to the cross section a for the absorption of a photon of energy E, the so-called photoabsorption cross section. Note that the excitation energy is equivalent to the photon energy. Explicitly, one may write ... 

The terms in this sum rule tend to cancel since the emission f-values fj are negative fractions. This is known as the Thomas-Kuhn-Reiche sum rule. In more complex atoms useful approximate sum rules can be obtained by replacing the value unity on the right side of equations (4.32) and (4.33) by z, where z is the number of equivalent electrons in the valence shell of the atom. This approximation breaks down if there is any appreciable interaction between different electron configurations. 

These moments are related to many physical properties. The Thomas-Kuhn-Reiche sum rule says that 5 (0) equals the number of electrons in the molecule. Other sum rules [36] relate (2), S(l) and (-1) to ground state expectation values. The mean static dipole polarizability is (()) = 2)/jw. The Cauchy expansion... 

Using the Thomas-Kuhn-Reiche sum rule for the denominator [11] one has,... 

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