Model harmonic oscillator

In many situations, atoms vibrate inside molecules, in solids, and on surfaces, for example. To a first approximation, such atoms act as if they were cormected to other atoms by springs. Here s the simplest model of such bonds. 

202 Chapter u. The Statistical Mechanics of Simple Gases Solids 

You can get the single particle partition function by substituting the energy Equation (11.22) into the partition function Equation (10.23), 

EXAMPLE 11.3 The vibrational partition function of O2. Oxygen molecules have a vibrational frequency of 1580cm To convert from cm to v, which has units of s multiply by the speed of electromagnetic radiation (hght) to get V = (1580cm-i)(2.997 x 

At room temperature, dvibration/T = 2 2 74 K/300 K = 7.58. For the partition function at room temperature. Equation (11.26) gives vibration = 1.0005. Most oxygen molecules are in their vibrational ground states at this temperature. Even at lOOOK, most oxygen molecules are still in their vibrational ground states 0vibratlon/T = 2.27 and vibration = 1.11. 

This potential permits the extention of the simple formalism of the harmonic oscillator model to the region of the liquid state (Prigogine, Trappeniers and Mathot [1953]). 

In paragraph 6 we shall briefly consider the smoothed potential model. We shall see that these simplified versions give very similar results. 

Using the potential (16.4.6), where the restoring coefficient kr is defined 

From (16.5.1) and (16.5.3) we can derive explicit expressions for the free energy and all other thermodynamic quantities. The equation of state is 

The first two terms in (16.5.4) are due to the volume dependence of the cell partition function, while the last term is due to the lattice energy. This equation of state can be written in a reduced form. Let us introduce the reduced variables T and (cf. 2.4.11) 

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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...

Discuss how to compute vibrational frequencies using a simple harmonic oscillator model of nuclear motion. 

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. 

RSE values can also be calculated from experimentally measured X - H bond dissociation energies or heats of formation (where available). In order to be directly comparable to the RSE values calculated at the ROMP2 or G3(MP2)-RAD level described above, this requires thermochemical data for the species in Eqs. 1-4 at 0 K. One straightforward approach is the back correction of experimentally measured heats of formation at 298.15 K to 0 K values using thermochemical corrections calculated using the rigid ro-tor/harmonic oscillator model in combination with scaled DFT or UMP2 frequencies [19,23]. 

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... 

Presented below are three examples designed to give the reader some idea of what one can expect from the theoretical analysis of vibrational spectra based on the simple harmonic oscillator model. Systems have been chosen whose structures have been know for many years and, in fact, were known prior to the availability of IR spectroscopy. Hence their spectra have previously been well characterized and these serve as a test of the method . 

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)... 

Electrons in metals at the top of the energy distribution (near the Fermi level) can be excited into other energy and momentum states by photons with very small energies thus, they are essentially free electrons. The optical response of a collection of free electrons can be obtained from the Lorentz harmonic oscillator model by simply clipping the springs, that is, by setting the spring constant K in (9.3) equal to zero. Therefore, it follows from (9.7) with co0 = 0 that the dielectric function for free electrons is... 

However, this simplification obviously does not apply to real electrochemical reactions. This harmonic oscillator model is described here because it is so well known. It will be compared with experiment in Section 9.6.3. 

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... 

For higher temperatures, one may consider a pure harmonic oscillator model. Here one obtains particularly transparent results when the polarizability term is neglected, that is, when a = 0. In this case one has [7]... 

One might suspect that the discrepancy between the thermal rigidity factors of Eqs. (24) and (36) is due to inadequacies of the SACM treatment in general or of the pure harmonic oscillator model. However, after having corrected the analytical representation of the trajectory results through Eqs. (28)—(31), for x 11, Eq. (28) leads to... 

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