Wave, resonant energy

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... 

Finally, quantum mechanical trapping at the resonance energy can be verified using a time-delay analysis on the quantum S-matrix. In Fig. 8, the average time delay for the J = 0 partial wave of the F + HD — HF + D reaction, defined using Eq. (22), is plotted versus collision energy. A clear... 

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... 

Resonance Energies, Widths, and Wave Functions Using a Lanczos Method in Real Arithmetic. 

Resonance energy transfer between the aromatic amino acids proceeds by very weak coupling between the donor and acceptor.151,52) Very weak coupling implies that the interaction between the donor and acceptor wave functions is small enough so as not to perturb measurably the individual molecular spectra. This transfer process, which is distinct from the trivial process of absorption of an emitted photon, involves radiationless deexcitation of an excited-state donor molecule with concomitant excitation of a ground-... 

As a consequence of the quantum-mechanical selection rules for resonance energy transfer involving the spin wave functions of the donor and acceptor,... 

It is possible to make a successful comparison of theory with experiment for the resonance energy modified according to the Mulliken and Parr prescription[60], but there are still many assumptions that must be made that have uncertain consequences. A better approach is to attempt calculations that match more closely what experiment gives directly. This still requires making calculations on what is a nonexistent molecule, but the unreality pertains only to geometry, not to restricted wave functions. 

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