Resonance condition formula

Here Psa is the transfer probability. Esa represents the resonance condition (in practise the spectral overlap of the emission of S and the relevant absorption of A) and occurs in both formulas. The quantity gsA comprises the optical strengths of the relevant transitions and a distance-dependence of the type rsA n=6,8, etc.). The quantity /sa, however, is proportional to the wave function overlap of S and A and comprises an exponential distance-dependence. 

In traditional discussions based on eq. (20), the resonance conditions have been expressed in terms of the magnetostriction coefficients. However, Du Tremolet de Lacheisserie (1995) has found that the elastic coefficients of the substrate (S 1 and S 2s) have entered his formulas of the resonance frequency. He showed that in the SMFMR method only the bY,2 and ba2 coefficients can be determined. 

Equation 5.5 clearly shows that the resonance condition is satisfied in the energy transfer. It also indicates that energy conservation is satisfied at the zeroth order of a, i.e E, + 2 = const., and that energy transfer requires a period, 2jt/(2(0i - CO2) = 2n/A, which is much longer than the frequencies of the two modes. This formula can easily be extended to a system with higher order coupling and more than three oscillators. 

This formula is applicable for metal colloids within the size range of 4-40 nm. Using this formula it is straightforward to determine the resonance conditions for a single cluster, thus the plasmon resonance energy. For a small 2 a resonance is obtained at ... 

This formula was first derived in ref. 6 when calculating the kinetics of donor luminescence decay in the presence of the randomly, i.e. chaotically, located acceptors under the condition n N and on the assumption of the resonance exchange mechanism of energy transfer. Similar equations were later used for the analysis of experimental data on the kinetics of electron tunneling reactions obtained under conditions of the chaotic distribution of the reagents and at n < N. As a rule, only the first term of the exponent in eqn. (23) has been taken into account, which is equivalent to employing the previously mentioned (see Sect. 2.1) stepwise approximation of the function 0(R,t) = exp[- 1V(jR)(]. In this case, one obtains... 

When 2-benzopyrylium cations have a substituent with a fairly mobile hydrogen atom (a-alkyl group or heteroatom group in any other position of the cation), deprotonation of such a substituent occurs even under mild conditions by an acid-base interaction as the primary step (Section III,A). Although deprotonation in both cases leads to compounds whose structures can be depicted by two resonance formulas, either with charge separation (betaines) or without (anhydrobases), on discussing products of C-deprotonation, the term (and the corresponding formula ) anhydro-base is more often used, whereas products of O-deprotonation are called betaines. ... 

If for any given compound two or more structures can be written differing only in the distribution of electronSj the properties of that compound will not be those to be expected of any of the formulas but rather they will be those to be expected of a hybrid of them all. An ion or a molecule in which resonance can occur will always be more stable than one-in which it cannot. It is obvious that when the principle is stated in such general terms no attempt has been made to explain the phenomenon, but rather the conditions have been described for recognizing its occurrence. Furthermore, since each covalent bond may have "a certain amount of partial ionic character, different electronic structures may actually be written for every chemical compound. The problem with which we are con-... 

Whilst no difference can be detected between the chemical properties of the two forms, the physical characteristios indicate that benzene does not actually exist as (6) or (7) but as a resonance hybrid of the two, a condition described as a mesomeric state. The energy of the mesomeric state is less than that of either of the conventional formulae (6) or (7), the difference being known as the resonance energy. 

---