Ideal harmonic oscillator

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. 

However, as seen in Fig. 3.2, this idealized harmonic oscillator (Fig. 3.2b) is satisfactory only for low vibrational energy levels. For real molecules, the potential energy rises sharply at small values of r, when the atoms approach each other closely and experience significant charge repulsion furthermore, as the atoms move apart to large values of r, the bond stretches until it ultimately breaks and dissociation occurs (Fig. 3.2c). 

When exposed to electromagnetic radiation of the appropriate energy, typically in the infrared, a molecule can interact with the radiation and absorb it, exciting the molecule into the next higher vibrational energy level. For the ideal harmonic oscillator, the selection rules are Av = +1 that is, the vibrational energy can only change by one quantum at a time. However, for anharmonic oscillators, weaker overtone transitions due to Av = +2, + 3, etc. may also be observed because of their nonideal behavior. For polyatomic molecules with more than one fundamental vibration, e.g., as seen in Fig. 3.1a for the water molecule, both overtones and... 

FIGURE 3.2 (a) Vibration of diatomic molecule, HC1, (b) potential energy of an ideal harmonic oscillator, and (c) an anharmonic oscillator described by the Morse function. 

Figure 1 Potential energy diagrams, (a) Plot of energy versus distance for an ideal harmonic oscillator, (b) Plot of energy versus distance for a nonideal or real-life oscillator.

Fairly good agreement exists between the calculated value of 1682 cm-1 and the experimental value of 1650 cm1. Direct correlation does not exist because Hooke s law assumes that the vibrational system is an ideal harmonic oscillator and, as mentioned before, the vibrational frequency for a single chemical moiety in a polyatomic molecule corresponds to the vibrations from a group of atoms. Nonetheless, based on the Hooke s law approximation, numerous correlation tables have been generated that allow one to estimate the characteristic absorption frequency of a specific functionality (13). It becomes readily apparent how IR spectroscopy can be used to identify a molecular entity, and subsequently physically characterize a sample or perform quantitative analysis. 

Calculations of band positions using Equation 1.18 will more closely approximate observed band positions than those calculated from the ideal harmonic oscillator expression found in Equation 1.16. Eor a rule of thumb, the first overtone (2v) for a fundamental can be calculated as 1% shift due to anharmonicity, or Xj = (0.01). Thus, the expression using wavenumbers is... 

The model of the anharmonic oscillator or local mode approximation more closely follows the actual condition for molecular absorption than that of the harmonic oscillator. Figure 1.15 illustrates the differences between the ideal harmonic oscillator case that has been discussed in detail for this chapter vs. the anharmonic oscillator model (better representing the actual condition of molecules). Unlike the ideal model illustrated by the harmonic oscillator expression, the anharmonic oscillator involves considerations such that, when two atoms are in close proximity (minimum distance), they repel each other when two atoms are separated by too large a distance, the bond breaks. The anharmonic oscillator potential energy curve is most useful to predict behavior of real molecules. 

FIGURE 1.15 Illustration of the differences in potential eneigy curves between the ideal harmonic oscillator model and the harmonic oscillator model (better representing the actual condition of molecules). (From Workman, J., Interpretive spectroscopy for near-infrared, Appl. Spectmsc. Revs., 31 (3) 251-320, 1996. With permission.)... 

Classically, the behavior of the ideal harmonic oscillator is well known. The position of the oscillator versus time, x t), is... 

FIGURE 11.1 A plot of the potential energy diagram V(x) = for for an ideal harmonic oscillator. 

FIGURE 11.2 A diagram of the energy levels of an ideal harmonic oscillator, as predicted by the solutions to the Schrodinger equation. Note that the lowest quantized level, (n = 0), does not have zero energy. 

Show that the energy separation between any two adjacent energy levels for an ideal harmonic oscillator is hv, where is the classical frequency of the oscillator. 

Substitute into the complete expression for the Hamiltonian operator of an ideal harmonic oscillator and show that = 3/2hv. 

FIGURE 14.25 Potential energy diagram for an ideal harmonic oscillator. Usually, this diagram is applicable only for low-energy (that is, low qrantum number) vibrations. 

By making this ideal harmonic oscillator assumption, we make the wavefunctions and energies for the ideal harmonic oscillator directly applicable to the diatomic molecule s vibrations In particular, because spectroscopy deals with differences in the energy states, we are particularly interested in the fact that... 

Assuming that the vibrational frequency of 2886 cm (8.652 X 10 hydrogen chloride is H Cl, predict the vibrational frequencies for H Cl and H Cl. Assume that the molecule is an ideal harmonic oscillator and that the force constant does not change upon isotopic substitution. (Such assumptions are common in vibrational spectroscopy.)... 

If the molecule is acting like an ideal harmonic oscillator and the force constant is not changing, then for the classical frequency of the H Cl oscillator we have... 

This selection rule is applicable to each normal mode of vibration. In absorption spectroscopy, the change is +1. This assumes that the normal mode acts as an ideal harmonic oscillator. Real molecules do not act as ideal harmonic oscillators, so in some cases it is not uncommon to detect Av = 2, 3,..., transitions. Such observations are part of what is called overtone spectroscopy. Partly because of the selection rule in equation 14.33, detection of overtones is difficult because many such absorptions are only weakly represented in a vibrational spectrum. Lasers, with their high intensities, are frequently utilized in overtone spectroscopy. However, the majority of vibrational spectroscopy deals with transitions following equation 14.33. 

If the normal mode is acting as an ideal harmonic oscillator, then we can use the quantum-mechanical expressions that describe its energy. Recall that for an ideal harmonic oscillator,... 

A vibrational spectrum is composed of absorptions that correspond to hv, where v is the classical frequency of the vibration. Note the very broad applicability of equation 14.34 It is independent of the quantum number v. For an ideal harmonic oscillator, the allowed transitions occurring for any one normal mode all have the same AE, and so all will absorb the same frequency of light... 

Changes in vibrational energy should be exact multiples of the Av = 1 transition. However, real normal vibrations are not ideal (which is why such transitions are observed occasionally in the first place), so absorptions due to overtone transitions are usually less than an integral number of hv. This deviation is a measure of anhar-monicity, which we will consider in the next section. Table 14.3 lists the absorptions due to the fundamental and overtone vibrational transitions for HCl (g). Also listed are the various multiples of the fundamental vibrational frequency, and the variance from the multiple as shown by experiment. Note how the overtone absorptions get farther and farther from ideal. The fact that Av > 1 is possible (although to a much lesser extent than Av = 1) and the variance from exact multiples of the fundamental vibrational frequency are both reminders that molecules are not true harmonic oscillators. They are anharmonic oscillators. The use of the ideal harmonic oscillator system in describing molecular vibrations is an approximation—but a good approximation. 

FIGURE 14.27 For an ideal harmonic oscillator, the potential energy equals and the quantized energy levels are equally spaced. 

FIGURE 14.28 A more realistic potential energy surface for the vibration of a molecule is superimposed on the ideal harmonic oscillator curve. Only at low vibrational quantum numbers does the ideal potential energy curve adequately approximate the real system. Note how the vibrational energy levels get closer and closer together as the vibrational quantum number increases. 

This potential is called the Morse potential and is plotted in Figure 14.29, along with the potential curve for the ideal harmonic oscillator. is the molecular dissociation energy as measured to the bottom of the potential energy curve, as shown in Figure 14.29. The constant a is related to the force constant k of the molecule by the expression... 

Are deviations from an ideal harmonic oscillator more likely to be seen at low energies or high energies Explain your answer. 

We will consider a simple diatomic molecule first, then generalize our final equations for a polyatomic molecule that has 3N — 6 (or 3N — 5 for linear molecules) vibrational motions, where N is the number of atoms in the molecule. If we make the assumption that the single vibration of a diatomic molecule is an ideal harmonic oscillator, then quantum mechanics gives us an equation for the quantized energy of that harmonic oscillator ... 

For the ideal harmonic oscillator only the fundamental vibrations are allowed and there would be no NIR spectrum. An important consequence of the an-harmonic nature of molecular vibrations is that transitions between more than one energy levels are allowed. These transitions give rise to overtone absorption bands. The near-IR bands result from transitions between the ground state and second or third excited vibrational states. The near-IR region of the spectrum thus contains mainly overtones and combination bands of fundamental mid-IR absorption bands cfr. Fig. 1.5). The intensity of the overtones depends on the anharmonicity of the vibration. Near-IR intensities are some 10 to 100 times lower than the corresponding fundamentals in mid-IR to compensate this, samples are 0.1 to 1 mm thick, which is a large virtue in comparison to mid-IR. There is no special theory of near-IR spectroscopy. 

This means there is a set of levels spaced in energy by hvg or in wavenumber by Vq. The selection rule for an ideal harmonic oscillator allows transitions where Av = , giving a single, fundamental vibrational absorption peak. 

From the dimensional analysis, the vibration frequency is proportional to the square root of the bond stiffness. Equaling the vibration energy of an ideal harmonic oscillator to the corresponding term in the Taylor series of the interatomic potential around its equilibrium, one can find. 

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