Potential energy simple 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. 

These potential energy terms and their attendant empirical parameters define the force field (FF). More complicated FFs which use different and/or more complex functional forms are also possible. For example, the simple harmonic oscillator expression for bond stretching can be replaced by a Morse function, Euorse (3), or additional FF terms may be added such as the stretch-bend cross terms, Estb, (4) used in the Merck molecular force field (MMFF) (7-10) which may be useful for better describing vibrations and conformational energies. 

The potential energy of such an oscillator can be plotted as a function of the separation r, or, for a normal mode in a polyatomic molecule, as a function of a parameter characterizing the phase of the oscillation. For a simple harmonic oscillator, the potential energy function is parabolic, but for a molecule its shape is that indicated in Figure 2.6. The true curve is close to a parabola at the bottom, and it is for this reason that the assumption of simple harmonic motion is justified for vibrations of low amplitude. 

Near the equilibrium bond length qe the potential energy/bond length curve for a macroscopic balls-and-spring model or a real molecule is described fairly well by a quadratic equation, that of the simple harmonic oscillator (E = ( /2)K (q — qe)2, where k is the force constant of the spring). However, the potential energy deviates from the quadratic (q ) curve as we move away from qc (Fig. 2.2). The deviations from molecular reality represented by this anharmonicity are not important to our discussion. 

There are a number of ways to describe this FC term. An early way of describing the nuclear position and free energy-dependent FC term was proposed in Nobel prize-winning work by Marcus [9, 10]. Marcus approximated the reactant and product, before and after electron transfer, as simple harmonic oscillators with intersecting parabolic potential surfaces. As the driving force of the reaction increases and the product potential surface drops further down in energy, the barrier that must be crossed in going from the bottom of the reactant parabola to the bottom of... 

The one-dimensional quadratic potential V = kx2 has been used for the description of covalent binding. The ground-state wave functions for a simple harmonic oscillator, /t and iR, have been used to describe the proton in the left and right wells. The force constant k has been determined from the stretch-mode vibrational transitions for water occurring at 3700 cm-1. The ground-state energy for the proton is 0.368 x 10-19 J. The tunneling barrier is AE = 4 x 10-19 J. 

This is an aromaticity index based on the energy deformation derived from a simple harmonic oscillator potential and defined as [Krygowski and Wieckowski, 1981 Krygowski, Anulewicz et al, 1983, 1995 Bird, 1997] ... 

Here, we provide formulas that will enable the calculation of the Condon locus in terms of molecular constants for parabolic potential energy fiinctions. Figure 8.1 shows schematically the parabolic energy curves of two simple harmonic oscillators and their discrete vibrational energy levels. 

The energy of deformation has been derived from a simple harmonic oscillator potential... 

We can explain the qualitative aspects of the nonlinear response of organic materials by analogy to the anharmonic oscillator problem [3-5]. For an electron bound to a molecule, a small perturbation will result in simple harmonic oscillation. The potential energy as a function of electronic displacement can be represented by the graph shown in Fig. 1. 

The potential-energy curve for a simple harmonic oscillation, derived from Equation 16-4, is a parabola, as depicted in Figure l6-3a. Notice that the potential energy is a maximum when the spring is stretched or compressed to its maximum amplitude A. and it decreases to zero at the equilibrium position. 

The equations of ordinary mechanics that we have used thus far do not completely describe the behavior of particles of atomic dimensions. For example, the quantized nature of molecular vibrational energies, and of other atomic and molecular energies as well, does not appear in these equations. We may, however, invoke the concept of the simple harmonic oscillator to develop the wave equations of quantum mechanics. Solutions of these equations for potential energies have the form... 

Also in chemistry artificial neural networks have found wide use. They have been used to fit spectroscopic data, to investigate quantitative structure-activity relationships (QSAR), to predict deposition rates in chemical vapor deposition, to predict binding sites of biomolecules, to derive pair potentials from diffraction data on liquids, " to solve the Schrodinger equation for simple model potentials like the harmonic oscillator, to estimate the fitness function in genetic algorithm optimizations, in experimental data analysis, to predict the secondary structure of proteins, to predict atomic energy levels, " and to solve classification problems from clinical chemistry, in particular the differentiation between diseases on the basis of characteristic laboratory data. ... 

FIGURE 8.2 The simple harmonic oscillator potential curve with energy levels and wavefunctions 17 (y) shown to v = 5. 

In Chapter 2 we examined several systems with discontinuous potential energies. In this chapter we consider the simple harmonic oscillator—a system with a continuously varying potential. There are several reasons for studying this problem in detail. First, the quantum-mechanical harmonic oscillator plays an essential role in our understanding of molecular vibrations, their spectra, and their influence on thermodynamic properties. Second, the qualitative results of the problem exemplify the concepts we have presented in Chapters 1 and 2. Finally, the problem provides a good demonstration of mathematical techniques that are important in quantum chemistry. Since many chemists are not overly familiar with some of these mathematical concepts, we shall deal with them in detail in the context of this problem. 

In order to evaluate the contribution from the nuclear component it is necessary to know the vibrational wavefunctions. For a simple harmonic oscillator, the potential energy well for intemuclear distances is given by a parabolic curve where the vibrational energy levels are quantized (Fig. 1.4). The curves for v = 0,1,2, and 3 represent the wavefunctions for each vibrational level. When a transition occurs from the ground to an excited state, it is necessary to consider the potential energy wells of both states. Since the electronic structure of the excited state is different from that of the ground state, the vibrational energy profiles will be different. In... 

Show that the kinetic and potential energy contributions to the energy of a simple harmonic oscillator are equal. Show explicitly (i.e., not by invoking the analogy to a set of independent harmonic oscillators) that the same holds for the kinetic and potential energy contributions to the energy of a collection of phonons, as given in Eq. (6.42). 

Fig. 3.3.2 Energy levels of a simple harmonic oscillator. The potential energy for harmonic motion is parabolic. The internuclear distance, r, is the equilibrium separation.

A simple eigenvalue problem can be demonstrated by the example of two coupled oscillators. The system is illustrated in fug. 2. It should be compared with the classical harmonic oscillator that was treated in Section 5.2.2. Here also, the system will be assumed to be harmonic, namely, that both springs obey Hooke s law. The potential energy can then be written in the form... 

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