Oscillator strengths sum rules

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... 

Im. Z i(E) can be considered as a product of an ionic excitation density of states and an energy-dependent coupling constant. In model calculations one can independently vary the shape and the band with of the denstiy of states and the strength of the coupling constant. In the present case we can only vary these parameters indirectly by changing the atomic number Z. Since the self-energy involves the polarizability of the ionic system there must be an oscillator-strength sum rule such that... 

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. 

There is an oscillator strength sum rule that is instructive but frequently misunderstood. The sum of the /-values for all transitions that involve orbital promotions of the same electron sum to 1. If, as in the Ca 4s2 1S ground... 

An important consequence of the KK relations is the possibility of formulating numerous sum rules for the optical constants (Smith 1985). In particular, the oscillator strength sum rule is essential for the quantitative analysis of the optical data ... 

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. 

Oscillator strength sum rules. Sum rules for oscillator strengths are of considerable importance for checking the internal consistency of sets of calculations or measurements. It can be shown that if i is the lowest level in the energy-level system of a one-electron atom, then the absorption f-values obey the sum-rule (Problem 4.4)... 

The Bethe Sum Rule and Basis Set Selection in the Calculation of Generalized Oscillator Strengths... 

Thus the Bethe sum rule is fulfilled exactly in the RPA at all values of the momentum transferred, provided that a complete basis set is used. Therefore, as in the case of the TRK sum rule when optical transition properties (q = 0) are considered, we expect that the BSR sum rule will be useful in evaluating basis set completeness when generalized oscillator strength distributions are calculated, for example for use in calculating stopping cross sections. It should be noted [12] that the completeness of the computational basis set is dependent on q, and thus care needs be taken to evaluate the BSR at various values of q. 

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... 

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.

The vertical ionization potentials of the (lt2) and (2ai) ionic states are indicated in Fig. 2, while that of the (lai) ionic state is far away from the range in Fig. 2. Kameta et al. examined their photoabsorption cross sections c in terms of the TKR sum rule for the oscillator-strength distribution dfjdE, Eq. (3), following the conversion of a to dfjdE,... 

In principle, we already have in our disposal the SRPA formalism for description of the collective motion in space of collective variables. Indeed, Eqs. (11), (12), (18), and (19) deliver one-body operators and strength matrices we need for the separable expansion of the two-body interaction. The number K of the collective variables qk(t) and pk(t) and separable terms depends on how precisely we want to describe the collecive motion (see discussion in Section 4). For K = 1, SRPA converges to the sum rule approach with a one collective mode [6]. For K > 1, we have a system of K coupled oscillators and SRPA is reduced to the local RPA [6,24] suitable for a rough description of several modes and or main gross-structure efects. However, SRPA is still not ready to describe the Landau fragmentation. For this aim, we should consider the detailed Iph space. This will be done in the next subsection. 

Using the sum rule (1.52) with A=B= 1, we see that (7.29) equals nuclear wave function is normalized. Hence the nuclear wave functions are eliminated, and the oscillator strength involves only the bracketed electronic integral in (7.6). Substitution of ( eil eil ei) nt0 (3-49) and division by (7.28) then gives as the oscillator strength of an electronic absorption transition... 

The generalized oscillator strength defined by (II.4) has a number of important properties that have been listed by Inokuti.5 Of great importance practically are the sum rules... 

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