Potential energy harmonic oscillation

As a second example we analyze the anisotropic charge-permanent dipole potential where SACM and PST differ from each other. Here, for demonstration, we only consider the low-energy perturbation and the high-energy harmonic oscillator limits. In the former limit, the adiabatic channel potential curve for the lowest channel j = m = 0 has the form... 

Energy levels for a particle in a parabolic potential the harmonic oscillator... 

Fig. 1. Optimization of the Onsager-Machlup action for the two dimensional harmonic oscillator. The potential energy is U(x,y) = 25i/ ), the mass is 1...

Most of the molecules we shall be interested in are polyatomic. In polyatomic molecules, each atom is held in place by one or more chemical bonds. Each chemical bond may be modeled as a harmonic oscillator in a space defined by its potential energy as a function of the degree of stretching or compression of the bond along its axis (Fig. 4-3). The potential energy function V = kx j2 from Eq. (4-8), or W = ki/2) ri — riof in temis of internal coordinates, is a parabola open upward in the V vs. r plane, where r replaces x as the extension of the rth chemical bond. The force constant ki and the equilibrium bond distance riQ, unique to each chemical bond, are typical force field parameters. Because there are many bonds, the potential energy-bond axis space is a many-dimensional space. 

Figure 4-3 Potential Energy as a Function of Compression or Stretching of a One-Dimensional Harmonic Oscillator.

The Morse oscillator model is often used to go beyond the harmonic oscillator approximation. In this model, the potential Ej(R) is expressed in terms of the bond dissociation energy Dg and a parameter a related to the second derivative k of Ej(R) at Rg k = ( d2Ej/dR2) = 2a2Dg as follows ... 

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...

The reason that does not change with isotopic substitution is that it refers to the bond length at the minimum of the potential energy curve (see Figure 1.13), and this curve, whether it refers to the harmonic oscillator approximation (Section 1.3.6) or an anharmonic oscillator (to be discussed in Section 6.1.3.2), does not change with isotopic substitution. Flowever, the vibrational energy levels within the potential energy curve, and therefore tq, are affected by isotopic substitution this is illustrated by the mass-dependence of the vibration frequency demonstrated by Equation (1.68). 

Figure 1.13 shows the potential function, vibrational wave functions and energy levels for a harmonic oscillator. Just as for rotation it is convenient to use term values instead of energy levels. Vibrational term values G(v) invariably have dimensions of wavenumber, so we have, from Equation (1.69),... 

Figure 6.4 Potential energy curve and energy levels for a diatomic molecule behaving as an anharmonic oscillator compared with those for a harmonic oscillator (dashed curve)...

Exercise 3.4. Evaluate the free-energy difference between two one-dimensional harmonic oscillators with potentials U1 = (X- l)2 and U2 = 0.3(X —... 

Molecules possess discrete levels of rotational and vibrational energy. Transitions between vibrational levels occur by absorption of photons with frequencies v in the infrared range (wavelength 1-1000 p,m, wavenumbers 10,000-10 cm , energy differences 1240-1.24 meV). The C-0 stretch vibration, for example, is at 2143 cm . For small deviations of the atoms in a vibrating diatomic molecule from their equilibrium positions, the potential energy V(r) can be approximated by that of the harmonic oscillator ... 

The simple harmonic oscillator picture of a vibrating molecule has important implications. First, knowing the frequency, one can immediately calculate the force constant of the bond. Note from Eq. (11) that k, as coefficient of r, corresponds to the curvature of the interatomic potential and not primarily to its depth, the bond energy. However, as the depth and the curvature of a potential usually change hand in hand, the infrared frequency is often taken as an indicator of the strength of the bond. Second, isotopic substitution can be useful in the assignment of frequencies to bonds in adsorbed species, because frequency shifts due to isotopic substitution (of for example D for H in adsorbed ethylene, or OD for OH in methanol) can be predicted directly. 

Thus, from equations (4.1) and (4.4), we see that for a harmonic oscillator the potential energy is given by... 

The harmonic oscillator may be generalized to three dimensions, in which case the particle is displaced from the origin in a general direction in cartesian space. The force constant is not necessarily the same in each of the three dimensions, so that the potential energy is... 

As illustrations of the application of perturbation theory we consider two examples of a perturbed harmonic oscillator. In the first example, we suppose that the potential energy V of the oscillator is... 

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