Dipole Oscillator Strength Sums

The set of dipole oscillator strengths fno, defined in Eq. (7.69), is often called the dipole oscillator strength distribution (DOSD). Summed over all excited states, bound as well as continuum states, they are related to several other molecular properties, as will be shown in the following. One defines two types of energy-weighted moments of the dipole oscillator strength distribution  

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. 

Exercise 7.2 Derive the Thomas-Reiche-Kuhn sum rule Eq. (7.83). 

Hint Start with the mixed representation of the oscillator strengths in Eq. (7.73). Use the fact that the set of excited states is complete, i.e. 

Comparing the definition of the components of the oscillator strength in the length representation, Ekj. (7.70), with the expression for a component of the frequency-dependent polarizability in Eq. (7.28) we can see that the polarizability can be written in terms of the oscillator strengths as 

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Recalling that the frequency-dependent polarizability is related to the (( fia A/3 ))w propagator, Eq. (7.26), we can express the even dipole oscillator strength sums also as derivatives of this polarization propagator, i.e. 

Alternatively, one can make use of the fact that the frequency dependence of the polarizability can be expressed in terms of dipole oscillator strength sums Eq. (7.86). This expansion, however, converges only for frequencies below the first excitation energy, i.e. tuv < minjE — Nevertheless, the expansion can be extended... 

It is not easy to calculate oscillator strengths from first principles except in some very simple cases. On the other hand, the oscillator strength distribution must fulfill certain sum rules, which in some cases help to unravel their character. Referring the (dipole) oscillator strength for the transition from the ground state with excitation energy n to state n as fn, a sum may be defined by... 

Instead of one resonance frequency per individual electron, Bethe recovered the spectrum of resonance frequencies for the atom, weighted by dipole oscillator strengths satisfying the sum rule... 

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.

In the local response model each electron density volume element is separately characterized by a two-parameter formula giving electric dipole oscillator strength as a function of frequency [12]. One of the two parameters is fixed by the oscillator strength sum rule, while the other is an effective mean excitation energy, taken to be the plasma energy huip by Andersson et al [9]. This model requires introduction of a low-density cutoff of the dipole response, because a... 

Jhanwar BL, Meath WJ (1982) Dipole oscillator strength distributions, sums, and dispersion energy coefficients for CO and C02. Chem Phys 67 185—199... 

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). 

The second, and more important kind is the giant dipole resonance intrinsic to the delocalised closed shell of a metallic cluster. Such resonances have received a great deal of attention [684]. They occur at energies typically around 2-3 eV for alkali atoms, and have all the features characteristic of collective resonances. In particular, they exhaust the oscillator strength sum rule, and dominate the spectrum locally. 

Oto state m for molecule A andy is the corresponding dipole oscillator strength averaged over degenerate final states. Similarly, the sum-over-states formula for the mean, frequency-dependent, polarizability can be written as... 

Table 8. The oscillator strength sum for Zn-Te dipoles in measured samples of ZnxCdyHgl-x-yTe...

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

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