Resonance energy functions

A, B and C being the Coulomb energy functions, a, 0, and y the resonance energy functions over the distances XY, YZ and XZ. Hence, if these functions are known (a usual approximation is to simply subtract 10 % of the total energies for the Coulomb interaction), the saddle energy may be calculated for certain distances as being the activation energy of the reaction... 

Each of the two first VB stmctures contributes 40% to the wave function, and each of the remaining three contributes 6%. The stability of benzene in the SCVB picture is due to resonance between these VB structures. It is furthermore straightforward to calculate the resonance energy by comparing the full SCVB energy with that ealeulated from a VB wave function omitting certain spin coupling functions. 

The secular equation has been solved for Q — W, with assuming values between 450 and 750, and the ratios of the coefficients Kj and kii of the wave functions for structures I and II in the wave function of the actual molecule, 1ft = Kqi/q + KnxfilIt have been evaluated. The resonance energy, E, is the difference between W and Q (Q includes the strain energy), and since it can easily be shown that in this case... 

The energy of the bond for this function too is essentially resonance energy. Dickinson2 introduced an additional term, dependent on two additional parameters, in order to take polarization into account. He wrote for the (not yet normalized) variation function... 

By equating to zero the differential of this with respect to r, we find the equilibrium distance as a function of x and the ratio of force constants C2/C1. For Q/Ci = 3, r2 = 1.34, and f = 1.54, this function is that given by Equation 1. The value 3 for the ratio of force constants is somewhat larger than that indicated by Badger s rule, which is about 2.3 it is probable that this increase is needed to compensate for the neglect of resonance energy in the assumed potential function. 

Jovin, T. and Arndt-Jovin, D. (1989). FRET microscopy Digital imaging of fluorescence resonance energy transfer. Application in cell biology. In Cell Structure and Function by Microspectrofluorometry (Kohen, E., JG, H. and Ploem, J., eds.). Academic Press, London, pp. 99-117. 

Equations such as this were normally solved by graphing before the days in which a calculator removed the need for such tedious techniques. Using numerical techniques, the roots can be found to be x = — 2.55, —1.15, —0.618, 1.20, and 1.62. The three lowest energy states are populated with six electrons (nitrogen is presumed to contribute two electrons to the bonding). Therefore, the resonance energy is 6a + 7.00/3 — (6a + 8.64/3) = - 1.64/3. After the constants ax. .. as are evaluated, the wave functions can be shown to be... 

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