Electronic harmonic oscillator

In anisotropic materials, the electronic bonds may have different polarizabilities for different directions (you may think of different, orientation-dependent spring constants for the electronic harmonic oscillator). Remembering that only the E-vector of the light interacts with the electrons, we may use polarized light to test the polarizability of the material in different directions, lno is one of the most important electro-optic materials and we use it as an example. The common notations are shown in Figure 4.7. If the E-vector is in plane with the surface of the crystal, the wave is called a te wave. In this example, the te wave would experience the ordinary index na of LiNbOs (nG 2.20). If we rotate the polarization by 90°, the E-ve ctor will be vertical to the surface and the wave is called tm. In lno, it will experience the extraordinary index ne 2.29. Therefore these two differently polarized waves will propagate with different phase velocities v c/n. In the example of Figure 4.7, the te mode is faster than the tm mode. 

A more elegant derivation, known as the Drude model, approximates the molecule as an electron harmonically oscillating about a positively charged core, and it results in a similar equation ... 

Infrared (ir) transmission depends on the vibrational characteristics of the atoms rather than the electrons (see Infrared and Raman spectroscopy). For a diatomic harmonic oscillator, the vibrational frequency is described by... 

To give a simple classical model for frequency-dependent polarizabilities, let me return to Figure 17.1 and now consider the positive charge as a point nucleus and the negative sphere as an electron cloud. In the static case, the restoring force on the displaced nucleus is d)/ AtteQO ) which corresponds to a simple harmonic oscillator with force constant... 

The deep philosophical significance of the new theory lies precisely at this point, and consists in replacing a somewhat metaphysical concept of the harmonic oscillator (which could never be produced experimentally) by the new concept of a physical oscillator of the limit cycle type, with which we are dealing in the form of electron tube circuits and similar self-excited systems. 

Figure 10.6 Graph of the Boltzmann distribution function for the CO molecule in the ground electronic state for (a), the vibrational energy levels and (b), the rotational energy levels. Harmonic oscillator and rigid rotator approximations have been used in the calculations.

Instead of the quantity given by Eq. (15), the quantity given by Eq. (10) was treated as the activation energy of the process in the earlier papers on the quantum mechanical theory of electron transfer reactions. This difference between the results of the quantum mechanical theory of radiationless transitions and those obtained by the methods of nonequilibrium thermodynamics has also been noted in Ref. 9. The results of the quantum mechanical theory were obtained in the harmonic oscillator model, and Eqs. (9) and (10) are valid only if the vibrations of the oscillators are classical and their frequencies are unchanged in the course of the electron transition (i.e., (o k = w[). It might seem that, in this case, the energy of the transition and the free energy of the transition are equal to each other. However, we have to remember that for the solvent, the oscillators are the effective ones and the parameters of the system Hamiltonian related to the dielectric properties of the medium depend on the temperature. Therefore, the problem of the relationship between the results obtained by the two methods mentioned above deserves to be discussed. 

The reason that AG/ = Ef — ) is due to the fact that our system consists of a collection of harmonic oscillators with displaced surfaces between the initial and final electronic states. In this case, the vibrational partition functions are the same between the initial and final electronic states. 

The first term is the intrinsic electronic energy of the adsorbate eo is the energy required to take away an electron from the atom. The second term is the potential energy part of the ensemble of harmonic oscillators we do not need the kinetic part since we are interested in static properties only. The last term denotes the interaction of the adsorbate with the solvent the are the coupling constants. By a transformation of coordinates the last two terms can be combined into the same form that was used in Chapter 6 in the theory of electron-transfer reactions. 

In the general case R denotes a set of coordinates, and Ui(R) and Uf (R) are potential energy surfaces with a high dimension. However, the essential features can be understood from the simplest case, which is that of a diatomic molecule that loses one electron. Then Ui(R) is the potential energy curve for the ground state of the molecule, and Uf(R) that of the ion (see Fig. 19.2). If the ion is stable, which will be true for outer-sphere electron-transfer reactions, Uf(R) has a stable minimum, and its general shape will be similar to that of Ui(R). We can then apply the harmonic approximation to both states, so that the nuclear Hamiltonians Hi and Hf that correspond to Ui and Uf are sums of harmonic oscillator terms. To simplify the mathematics further, we make two additional assumptions ... 

In practice, the harmonic oscillator has limits. In the ideal case, the two atoms can approach and recede with no change in the attractive force and without any repulsive force between electron clouds. In reality, the two atoms will dissociate when far enough apart, and will be repulsed by van der Waal s forces as they come closer. The net effect is the varying attraction between the two in the bond. When using a quantum model, the energy levels would be evenly spaced, making the overtones forbidden. 

For one-electron electrochemical reactions, the harmonic oscillator ("Marcus ) model (21) yields the following predicted dependence of AG upon the electrode potential ... 

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