The Harmonic Oscillator Model

In this chapter we shall review the motion of atoms and molecules at surfaces. First we discuss how atoms vibrate about their equilibrium surface sites. Then, the elementary surface processes during the collisions of gas atoms and molecules with surfaces are described. We then discuss several elementary gas-surface interactions adsorption, surface diffusion, and desorption. 

2 SURFACE ATOM VIBRATIONS 4.2.1 The Harmonic Oscillator Model 

In studying the motion of surface atoms about their equilibrium positions, it is frequently useful to relate it to the motion of model systems. Perhaps the most useful 

The potential-energy change dV of the system as the particle is displaced by dx against the force exerted by the spring is 

Using a boundary condition in which the mass, in its equilibrium position, has zero displacement (x = 0), we integrate Eq. 4.3 to obtain, for the potential energy V, 

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Figure 1.13 Plot of potential energy, V(r), against bond length, r, for the harmonic oscillator model for vibration is the equilibrium bond length. A few energy levels (for v = 0, 1, 2, 3 and 28) and the corresponding wave functions are shown A and B are the classical turning points on the wave function for w = 28...

Rationalize nonzero zeio-point energies by reference to the harmonic oscillator model once again, and its energy ... 

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 aromaticity of 1,2,4-triazoles has been investigated and quantified using the harmonic oscillator model of aromaticity (HOMA) index, where a value of 1 is assigned to a molecule that is fully aromatic, 0 for a nonaromatic molecule, and a negative value for a molecule that is antiaromatic the data obtained were compared to other small-molecule heteroaromatics. It was determined that different tautomers of substituted and unsubstitued 1,2,4-triazoles have individual HOMA indices <2000JST(524)151>. 

Studies on the statistical deviation from an ideal bond order support the relatively high aromaticity of 1,2,5-thiadiazole (Table 7). The harmonic oscillator model of aromaticity (HOMA) value for 1,2,5-thiadiazole has not yet been reported. 

In addition to the above prescriptions, many other quantities such as solution phase ionization potentials (IPs) [15], nuclear magnetic resonance (NMR) chemical shifts and IR absorption frequencies [16-18], charge decompositions [19], lowest unoccupied molecular orbital (LUMO) energies [20-23], IPs [24], redox potentials [25], high-performance liquid chromatography (HPLC) [26], solid-state syntheses [27], Ke values [28], isoelectrophilic windows [29], and the harmonic oscillator models of the aromaticity (HOMA) index [30], have been proposed in the literature to understand the electrophilic and nucleophilic characteristics of chemical systems. 

It is seen that the "electrochemical estimates of values of AG diverge from the straight line predicted from the harmonic oscillator model to a similar, albeit slightly smaller, extent than the experimental values. Admittedly, there is no particular justification for assuming that the reduction half reactions obey the harmonic oscillator model. However, it turns out that the estimates of AG are relatively insensitive to... 

Two of the most severe limitations of the harmonic oscillator model, the lack of anharmonicity (i.e., non-uniform energy level spacings) and lack of bond dissociation, result from the quadratic nature of its potential. By introducing model potentials that allow for proper bond dissociation (i.e., that do not increase without bound as x=>°°), the major shortcomings of the harmonic oscillator picture can be overcome. The so-called Morse potential (see the figure below)... 

The harmonic oscillator model does not take into account the real nature of chemical bonds, which are not perfect springs. The force constant k decreases if the atoms are pulled apart and increases significantly if they are pushed close together. The vibrational levels, instead of being represented by a parabolic function as in equation (10.3), are contained in an envelope. This envelope can be described by the Morse equation (Fig. 10.5) ... 

A further refinement of the harmonic oscillator model is possible, in which the lattice is put into contact with a heat bath at temperature T and remains in contact with the heat bath, so that the initial correlations decay not only through mutual interactions but also through random collisions with an external fluctuating field. Although it might appear that such a case would contain features of both the independent particle case and the harmonic oscillator model just analyzed, the resulting formalism is much closer to that required for the latter, and the results differ only in detail. The model to be discussed is specified by the equations of motion... 

The most important chemical applications of the harmonic oscillator model are to the vibrations of molecules. Figure 3.7 shows how we can regard a diatomic molecule as two nuclei held together by a spring which represents the effects of the electrons forming the chemical bond. There are two difficulties we need to discuss, before the results of the previous section can be applied. 

In the main group of molecular models studied here the dielectric response of a linear molecule, characterized by a moment of inertia /, is examined. A molecule librates/rotates in a conservative intermolecular potential. Besides, in the harmonic oscillator model such a response is determined by the elastic constant k and by the masses of two vibrating particles. 

This mle will be illustrated in Section VII—for example, for the harmonic oscillator model. 

In Section V the reorientation mechanism (A) was investigated in terms of the only (hat curved) potential well. Correspondingly, the only stochastic process characterized by the Debye relaxation time rD was discussed there. This restriction has led to a poor description of the submillimeter (10-100 cm-1) spectrum of water, since it is the second stochastic process which determines the frequency dependence (v) in this frequency range. The specific vibration mechanism (B) is applied for investigation of the submillimetre and the far-infrared spectrum in water. Here we shall demonstrate that if the harmonic oscillator model is applied, the small isotope shift of the R-band could be interpreted as a result of a small difference of the masses of the water isotopes. 

We use the harmonic oscillator model [18], (VIG, p. 27), which will be described in detail in Section VII.C. The vibration spectral function is given by the formula... 

We conclude The submillimeter spectra calculated in terms of the harmonic oscillator model substantially differ from the spectra typical for the low-frequency Debye relaxation region. Such a fundamental difference of the spectra, calculated for water in microwave and submillimeter wavelength ranges, evidently reveals itself in the case of the composite hat-curved-harmonic oscillator model. 

Validity of our formulas for the resonance lines, which express the complex susceptibility through the spectral function, could be confirmed as follows. We have obtained an exact coincidence of the equations (353), (370), (371), which were (i) directly calculated here in terms of the harmonic oscillator model and (ii) derived in GT and VIG (see also Section II, A.6) by using a general linear-response theory. 

Similar reasoning concerns also the R-band, if we shall apply for calculations the harmonic oscillator model (we mean now a rough qualitative description). 

In a more recent study, the problem of n delocalization in porphyrins and metalloporphyrins was reinvestigated using two different aromaticity indexes, HOMA andNICS [37], The harmonic oscillator model of aromaticity (HOMA) [31] quantifies the aromaticity of a system on the basis of the calculated deviation of its... 

Another theoretical criterion applied to estimation of aromaticity of homo- and heteroaromatic ring system is aromatic stabilization energy (ASE). Based on this approach, the aromatic sequence of five-membered ring systems (ASE in kcal mol-1) is pyrrole (20.6) > thiophene (18.6) > selenophene (16.7) > phosphole (3.2) [29], According to geometric criterion HOMA, based on the harmonic oscillator model [30-33], thiophene is more aromatic than pyrrole and the decreasing order of aromaticity is thiophene (0.891) > pyrrole (0.879) > selenophene (0.877) > furan (0.298) > phosphole (0.236) [29],... 

The calculated modes at 1282, 1296, and 1306 cm-1 represent motions of the diazene moiety. The assignment of these vibrations in (101) as overtones of the modes at about 650 cm-1, which was confirmed by the experimental spectra of the isotopomers, was well reproduced by the calculations (103), in which we used the harmonic oscillator model as a zeroth-order approximation for the estimation of the overtone wavenumbers. 

The harmonic oscillator model of aromaticity (HOMA) index and Bird aromaticity indices (/5,15 6, and /A) for selected heterocycles are shown in Table 35 and Figure 15. The theoretical background to these indices is discussed in Section 2.2.4.2.3. To facilitate direct comparison between ring systems, Bird introduced a unified aromaticity index /A that is related to the indices for five- and six-membered rings and fused rings by the expression ... 

Several aspects of aromaticity have been studied <2002JOC1333> using statistical analyses of quantitative definitions of aromaticity. ASEs, REs, magnetic susceptibility exaltation (A), nucleus-independent chemical shift (NIGS), the harmonic oscillator model of aromaticity (HOMA), (/j) and (Aj), evaluated for a set of 75 five-membered 7t-electron systems and a set of 30 ring-monosubstituted compounds (aromatic, nonaromatic, and antiaromatic systems) revealed statistically significant correlations between the various aromaticity criteria, provided the whole set of compounds is used. The data in Table 9 have been found for arsole (AsH) 1 (E = As, R = H), its anion (As ), and protonated species (AsH2 ). 

From the harmonic-oscillator model of quantnm mechanics, the term valne G for the vibrational energy levels for a linear polyatomic molecnle can be written as ... 

Potential Functions. Near the minimum in the potential-energy curve of a dia-tomie moleeule, the harmonic-oscillator model is usually quite good. Therefore the foree constant h can be calculated from the relation... 

In this model we consider only one vibrational mode, viz., the so-called breathing mode in which the surrounding lattice pulsates in and out around the dopant ion (symmetrical stretching mode). This mode is assumed to be described by the harmonic oscillator model. The configurational coordinate (Q) describes the vibration. In our approximation it represents the distance between the dopant ion and the surrounding ions. In ruby this Q would be the Cr -0 distance. 

Kruszewski, J. and Krygowski, T.M. (1972). Definition of Aromaticity basing on the Harmonic Oscillator Model. Tetrahedron Lett., 36,3839-3842. 

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