Harmonic oscillator calculation

The thermodynamic functions, including the effects of available vibrational anharmonicities, are calculated using the computer program of McDowell ( 3). Compared to a harmonic oscillator calculation using the same data the present results are... 

A single particle of mass 2.30 X 10 kg held by a spring with force constant 150 N m undergoes harmonic oscillations. Calculate the zero point energy of this oscillator. 

Fig. 8.7. The Barkas term, L, as deduced from the energy loss measurements of equivelocity antiprotons and protons by Medenwaldt et al. [8.24] is shown as open symbois. These data are compared with the following theoretical calculations, indicated by curves. Full curve Jackson and McCarthy [8.7] dashed curve two times this result (Lindhard [3.60]) dotted curve harmonic oscillator calculation of Mikkelsen and Sigmund [8.12], with one frequency given by the mean ionization potential of Si dot-long-dashed curve static homogeneous electron gas calculation by Esbensen and Sigmund [8.13] dot-short-dashed curve Lenz-Jensen atom-target calculation using local density approximation and the electron gas model (Esbensen and Sigmund [8.13]).

This com poTicii 1 is oficn approximated as a harmonic oscillator and can be calculated using Hooke s law. 

These are all empirical measurements, so the model of the harmonic oscillator, which is pur ely theoretical, becomes semiempirical when experimental information is put into it to see how it compares with molecular vibration as determined spectroscopically. In what follows, we shall refer to empirical molecular models such as MM, which draw heavily on empirical information, ab initio molecular models such as advanced MO calculations, which one strives to derive purely from theory without any infusion of empirical data, and semiempirical models such as PM3, which are in between (see later chapters). 

Using MMd. calculate A H and. V leading to ATT and t his reaction has been the subject of computational studies (Kar, Len/ and Vaughan, 1994) and experimental studies by Akimoto et al, (Akimoto, Sprung, and Pitts. 1972) and by Kapej n et al, (Kapeijn, van der Steen, and Mol, 198.V), Quantum mechanical systems, including the quantum harmonic oscillator, will be treated in more detail in later chapters. 

For very-high-accuracy ah initio calculations, the harmonic oscillator approximation may be the largest source of error. The harmonic oscillator frequencies... 

This is a check on the reasonableness of the method chosen. For example, it would not be reasonable to select a method to investigate vibrational motions that are very anharmonic with a calculation that uses a harmonic oscillator approximation. To avoid such mistakes, it is important the researcher understand the method s underlying theory. 

We proceed now to the calculation of B, following [Benderskii et al. 1992a]. The denominator in (4.11) (apart from normalization) is equal to the harmonic-oscillator partition function [2 sinh(ico+ )] The numerator is the product of the s satisfying an equation of the Shrodinger type... 

Fig. 7. Model calculations for the reflectivity (a) and the optical conductivity (b) for a simple (bulk) Drude metal and an effective medium of small metallic spherical particles in a dielectric host within the MG approach. The (bulk) Drude and the metallic particles are defined by the same parameters set the plasma frequency = 2 eV, the scattering rate hr = 0.2 eV. A filling factor/ = 0.5 and a dielectric host-medium represented by a Lorentz harmonic oscillator with mode strength fttOy, 1 = 10 eV, damping ftF] = I eV and resonance frequency h(H = 15 eV were considered for the calculations.

In the infinite sum each successive term is smaller than the previous by a constant factor ( -hujVT which is <1), and can therefore be expressed in a closed form. Only the vibrational frequency is needed for calculating the vibrational partition function for a harmonic oscillator, i.e. only the force constant and the atomic masses are required. 

In the case of simple diatomic molecules it is possible to calculate the vibrational frequencies by treating the molecule as a harmonic oscillator. The frequency of vibration is given by ... 

It is important to note that in all these methods, the first term in the series solution constitutes the so-called approximation of zero order. This is generally the solution of a simple linear problem e.g., the harmonic oscillator the second term appears as the first approximation, and so on. The amount of labor increases very rapidly with the order of approximation, but the additional information obtained from approximations of higher orders (beginning with the second) does not increase our knowledge from the qualitative point of view. It merely adds small quantitative corrections to the first approximation, and in most applied problems, these corrections are scarcely worth the considerable complication in calculations. For that reason the first approximation is generally sufficient in exploring a new problem, or in investigating the qualitative aspect of a phenomenon. 

Figure 10.3 Configurations leading to the distribution of five units of energy among five equivalent harmonic oscillators. The number of different ways that each configuration can be achieved are not shown. Equation (10.14) calculates this number. For example, we show that configuration (a) can be obtained in five different ways.

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.

Under most circumstances the equations given in Table 10.4 accurately calculate the thermodynamic properties of the ideal gas. The most serious approximations involve the replacement of the summation with an integral [equations (10.94) and (10.95)] in calculating the partition function for the rigid rotator, and the approximation that the rotational and vibrational partition functions for a gas can be represented by those for a rigid rotator and harmonic oscillator. In general, the errors introduced by these approximations are most serious for the diatomic molecule." Fortunately, it is for the diatomic molecule that corrections are most easily calculated. It is also for these molecules that spectroscopic information is often available to make the corrections for anharmonicity and nonrigid rotator effects. We will summarize the relationships... 

By starting with this partition function and going through considerable mathematical manipulation, one arrives at the following equations for calculating the corrections to the rigid rotator and harmonic oscillator values calculated from Table 10,4, U... 

E10.6 For the diatomic molecule Na2, 5 = 230.476 J-K-1-mol" at T= 300 K, and 256.876 J-K-,-mol-1 at T= 600 K. Assume the rigid rotator and harmonic oscillator approximations and calculate u, the fundamental vibrational frequency and r, the interatomic separation between the atoms in the molecule. For a diatomic molecule, the moment of inertia is given by l pr2, where p is the reduced mass given by... 

Table A4.5 summarizes the equations for calculating anharmonicity and nonrigid rotator corrections for diatomic molecules. These corrections are to be added to the thermodynamic properties calculated from the equations given in Table A4.1 (which assume harmonic oscillator and rigid rotator approximations).

The sequence of levels shown in Figure 2 closely resembles the level diagram found by Mayer and Jensen by analysis of observed nuclear properties, with the help of the calculated level sequences for harmonic-oscillator and square-well potential func-... 

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