Harmonic oscillator energy spectrum

Experimental. The vibrational spectrum of an ideal harmonic oscillator would consist of one line at frequency v corresponding to A = hv, where A is the distance between levels on the vertical energy axis in Fig. 10-la. In the harmonic oscillator, AE is the same for a transition from one energy level to an adjacent level. A selection rule An = 1, where n is the vibrational quantum number, requires that the transition be to an adjacent level. 

The most simple way to accomplish this objective is to correct the external field operator post factum, as was repeatedly done in magnetic resonance theory, e.g. in [39]. Unfortunately this method is inapplicable to systems with an unrestricted energy spectrum. Neither can one use the method utilizing the Landau-Teller formula for an equidistant energy spectrum of the harmonic oscillator. In this simplest case one need correct... 

We make use of the assumption which is conventional in kinetic theory of the harmonic oscillator [193] as well as in energy-corrected IOS [194]. All the transition rates from top to bottom in the rotational spectrum are supposed to remain the same as in EFA. Only transition rates from bottom upwards must be corrected to meet the demands of detailed balance. In the same way the more general requirements expressed in Eq. (5.21) may be met ... 

This result determines the energy spectrum of the harmonic oscillator through... 

For a spectroscopic observation to be understood, a theoretical model must exist on which the interpretation of a spectrum is based. Ideally one would like to be able to record a spectrum and then to compare it with a spectrum computed theoretically. As is shown in the next section, the model based on the harmonic oscillator approximation was developed for interpreting IR spectra. However, in order to use this model, a complete force-constant matrix is needed, involving the calculation of numerous second derivatives of the electronic energy which is a function of nuclear coordinates. This model was used extensively by spectroscopists in interpreting vibrational spectra. However, because of the inability (lack of a viable computational method) to obtain the force constants in an accurate way, the model was not initially used to directly compute IR spectra. This situation was to change because of significant advances in computational chemistry. 

Fermi resonance occurs in C02. We note from (6.100) that 2vl e (= 1346 cm-1) is very close to e (= 1354 cm-1). Hence harmonic-oscillator levels of C02 for which 2v + v 2 = 2vx +1>2 (where the primed and unprimed quantum numbers refer to different vibrational levels) are quite close together and we have Fermi resonance. For example, (6.99) and (6.100) predict the levels (10°0), (02°0), and (0220) to lie 1335, 1339, and 1335 cm-1, respectively, above the ground vibrational level the observed spectrum (Table 6.2) shows these levels actually lie 1388, 1285, and 1335 cm-1, respectively, above the ground level (Problems 6.20 and 6.21). Clearly, the (10°0) and (02°0) levels have interacted with each other, thus shifting their energies. The (0220) level is unaffected because the matrix element H j is zero if states i and j have different values of / (Problem 6.22) Fermi resonance occurs only between states of the same symmetry. Fermi resonance between two levels increases the energy of the upper level and decreases the energy of the lower level the levels repel each other. 

Besides the sum-over-states and time-dependent models for the resonance Raman cross-section, other models can be used to calculate resonance Raman cross-sections, such as the transform and time correlator models. In the transform model, the resonance Raman cross-sections as a function of excitation energy, the excitation profiles, can be calculated from the absorption spectrum within the separable harmonic oscillator approximation directly by the following relationship [85-87]... 

We know from Section 4.7 that the vibrational energy is quantized, and we treat it as a simple harmonic oscillator E = nhvyn, where n = 0, 1,2, 3,. .. is the vibrational quantum number. As a result of the vibrational excitation, the peak i in the spectrum is actually a series of narrower peaks the separation between adjacent peaks depends on the vibrational frequency of the diatomic ion ... 

We are going to give the frequency spectrum of radiation and explain the natural width of the spectrum line. As an atom emits photons, its energy drops and the amplitude of transition decreases over time. Therefore, the emission is not harmonic, and a spectrum occurs. We shall see that the natural width of the spectral line can be connected to the attenuation coefficient of the damped oscillator. Inversely, from the width of the spectral line, we might determine the attenuation coefficient of the oscillator. 

Figure 1. Upper inset tilted potential U(q) [Eq. (5)] solid maximal tilt, t = 1 dashed minimal tilt, e = 0 [Eq. (4)]. Lower inset U (e = 0) is approximated by a binding potential for energies below Urn, since tunneling is negligible U (e = 1) is divided into the same binding potential and a perturbation V < 0, allowing tunneling. Main figure (a) the coupling spectrum Gn i2 ( ) and the modulation function J t= 4r0 (oo) (multiplied by 2) with n = I /ojo and To = 5n, where loo is the fundamental (harmonic) oscillation frequency in the well (b) idem, with Gn=is(u>), Ti = 0.3/loo, and Ft=iTO(uo) (times 4).

---