The Resonance Phenomenon

The concept of resonance played an important part in the discussion of the behavior of certain systems by the methods of classical mechanics. Very shortly after the discovery of the new quantum mechanics it was noticed by Heisenberg that a quantum-mechanical treatment analogous to the classical treatment of resonating system can be applied to many problems, and that the results of the quantum-mechanical discussion in these cases can be given a simple interpretation as corresponding to a quantum-mechanical resonance phenomenon. It is not required 

Resonance in Classical Mechanics.—A striking phenomenon is shown by a classical mechanical system consisting of two parts between which there is operative a small interaction, the two parts being capable of executing harmonic oscillations with the same or nearly the same frequency. It is observed that the total oscillational energy fluctuates back and forth 

It is illuminating to consider this system in greater detail. Let x-i and x2 be the coordinates for two oscillating particles each of mass m (such as the bobs of two pendulums restricted to small amplitudes, in order that their motion be harmonic), and let v0 be their oscillational frequency. We assume for the total potential energy of the system the expression 

These expressions correspond to pure harmonic oscillation of the two variables and y (Sec. la), each oscillating with constant amplitude, with the frequency / vl + X and y with the frequency Y2 — X, according to the equations 

1 These are the normal coordinates of the system, discussed in Section 37. 

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Curve 1 represents the total energy of the hydrogen molecule-ion as calculated by the first-order perturbation theory curve 2, the naive potential function obtained on neglecting the resonance phenomenon curve 3, the potential function for the antisymmetric eigenfunction, leading to elastic collision. 

The contour lines represent points of relative density 1.0, 0.9, 0.8,..0.1 for a hydrogen atom. This figure, with the added proton 1.06 A from the atom, gives the electron distribution the hydrogen molecule-ion would have (in the zeroth approximation) if the resonance phenomenon did not occur it is to be compared with figure 6 to show the effect of resonance. 

An attempt was made by Unsold (33) to evaluate to the second-order the interaction of a proton and a hydrogen atom. He found, neglecting the resonance phenomenon, that the second-order perturbation energy is given approximately by the expression... 

It will be seen that the second-order treatment leads to results which deviate more from the correct values than do those given by the first-order treatment alone. This is due in part to the fact that the second-order energy was derived without considerar-tion of the resonance phenomenon, and is probably in error for that reason. The third-order energy is also no doubt appreciable. It can be concluded from table 3 that the first-order perturbation calculation in problems of this type will usually lead to rather good results, and that in general the second-order term need not be evaluated. 

The interaction of two alkali metal atoms is to be expected to be similar to that of two hydrogen atoms, for the completed shells of the ions will produce forces similar to the van der Waals forces of a rare gas. The two valence electrons, combined symmetrically, will then be shared between the two ions, the resonance phenomenon producing a molecule-forming attractive force. This is, in fact, observed in band spectra. The normal state of the Na2 molecule, for example, has an energy of dissociation of 1 v.e. (44). The first two excited states are similar, as is to be expected they have dissociation energies of 1.25 and 0.6 v.e. respectively. 

A useful equivalent way of looking at the resonance phenomenon is in a coordinate system rotating with the angular frequency of Hi about the z axis which coincides with the direction of Ho, the applied static field 44)-In this rotating coordinate system the effective fields are Hi, which is now time-independent, and a field Ho — u/y, where w is the angular frequency... 

The concept of resonance was introduced into quantum mechanics by Heisenberg16 in connection with the discussion of the quantum states of the helium atom. He pointed out that a quantum-mechanical treatment somewhat analogous to the classical treatment of a system of resonating coupled harmonic oscillators can be applied to many systems. The resonance phenomenon of classical mechanics is observed, for example, for a system of two tuning forks with the same characteristic frequency of oscillation and attached to a common base, hich... 

Because the resonating system does not have a structure intermediate between those involved in the resonance, but instead a structure that is further changed by the resonance stabilization, I prefer not to use the word mesomerism, suggested by In gold in 1933 for the resonance phenomenon (C. K. In gold, /. Chem. Soc., 1933 1120). 

Here again, however, we have neglected the resonance phenomenon for the structure with electron 2 attached to nucleus A and electron 1 to nucleus B is just as stable as the equivalent structure assumed above, and in accordance with quantum-mechanical principles we must consider as a representation of the normal state of the system neither one structure nor the other, but rather a combination to which the two contribute equally that is, we must make the calculation in. such a way as to take into consideration the possibility of the exchange of places of the two electrons ... 

The resonance concept indicates that the actual molecular structure lies somewhere between these various approximations, but is not capable of objective representation. The idea can be applied to any molecule, organic or inorganic in which and electron pair bond is present. The term resonance hybrid denotes a molecule that has this property. Such molecules do not vibrate back and forth between two or more structures, nor are they isotopes or mixtures. The resonance phenomenon is rather an idealized expression of an actual molecule that cannot be accurately pictured by any graphic device. 

The unique spectrum when in the liquid crystalline state is due to the resonance phenomenon exhibited by the two terminal groups and the conjugated carbon chain between them, and the ability of electrons to be shared between the individual molecules of the liquid crystal. 

Metal nanoparticles have been used for many applications because of their unique characteristics, even before they were visualized as small particles of nano-meter order by using a transmission electron microscope [118]. For example, colored glasses, which gained in popularity in medieval times, contain nanoparticles of noble metals. These colors originate from the SPR of metal nanoparticles, which is the resonance phenomenon of surface electron density wave with incident light wave at the metal surface [119]. Since this resonance is sensitive to the dielectric constant of surrounding media, the phenomenon has... 

The case when the wave function is given as a standing wave on one side of the barrier requires a detailed treatment, since the resonance phenomenon may occur. Putting in (4.30a)... 

In this section, the resonance phenomenon that is so troublesome in these problems will be investigated. Additionally, a novel set of resonance-free equations of motion based on the isokinetic ensemble that can be used in MD simulations to avoid the resonance problem will be discussed [37]. 

To illustrate the resonance phenomenon, discussed above, consider a single coordinate q with momentum p with associated equations of motion [33] ... 

All these results show in detail the tuning mechanism. Finally, the question of whether the resonance phenomenon Is responsible for... 

On the other hand, the further splitting due to the integrals K, and Kp was not satisfactorily interpreted before the development of the quantum mechanics. It will be shown in Section 41 that we may describe it as resulting from the resonance phenomenon of the quantum mechanics. The zeroth-order wave function for the state with W = J, + K for example, is... 

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