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The harmonic potential, V(x) = ¼kx?, is useful start for modelling molecular vibrations, but it has limitations. A realistic potential between to atoms should accurately represent the sharp increase in the potential as two nuclei come in close proximity, and also have the ability for a bond to break: that is, an asymptote V → 0 as x →o. One option, as shown in the figure, is the Morse potential: V(r) = D(1 – e-a(r=re))2

The harmonic potential, V(x) = ¼kx?, is useful start for modelling molecular vibrations, but it has limitations. A realistic potential between to atoms should accurately represent the sharp increase in the potential as two nuclei come in close proximity, and also have the ability for a bond to break: that is, an asymptote V → 0 as x →o. One option, as shown in the figure, is the Morse potential: V(r) = D(1 – e-a(r=re))2

College Physics
College Physics
11th Edition
ISBN:9781305952300
Author:Raymond A. Serway, Chris Vuille
Publisher:Raymond A. Serway, Chris Vuille
Chapter1: Units, Trigonometry. And Vectors
Section: Chapter Questions
Problem 1CQ: Estimate the order of magnitude of the length, in meters, of each of the following; (a) a mouse, (b)...
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1) Take the derivative of this function to find the force function associated with it. 2) Demonstrate that for values of r close to \( r_e \), the potential is close to harmonic: i.e., the force is proportional to displacement and opposite in direction. (Suggestion: expand the exponential function as a power series.) 3) Show that for large amplitudes, the vibrational frequency of the oscillator is less than the frequency of an equivalent harmonic oscillator. (Suggestion: include higher order terms in the expansion.)
Transcribed Image Text:1) Take the derivative of this function to find the force function associated with it. 2) Demonstrate that for values of r close to \( r_e \), the potential is close to harmonic: i.e., the force is proportional to displacement and opposite in direction. (Suggestion: expand the exponential function as a power series.) 3) Show that for large amplitudes, the vibrational frequency of the oscillator is less than the frequency of an equivalent harmonic oscillator. (Suggestion: include higher order terms in the expansion.)
# The Morse Potential The harmonic potential, \( V(x) = \frac{1}{2}kx^2 \), is a useful start for modeling molecular vibrations, but it has limitations. A realistic potential between atoms should accurately represent the sharp increase in the potential as two nuclei come in close proximity, and also have the ability for a bond to break: that is, an asymptote \( V \rightarrow 0 \) as \( x \rightarrow \infty \). One option, as shown in the figure, is the Morse potential: \[ V(r) = D(1 - e^{-\alpha(r - r_e)})^2 \] ### Graph Explanation The graph displays the Morse potential as a curve. The horizontal axis represents the interatomic distance \( r \), while the vertical axis represents the potential energy \( V(r) \). The curve starts high as \( r \) approaches zero (indicating strong repulsion at short distances), dips down to a minimum value (indicative of the bond energy), and then gradually rises toward zero as \( r \) increases, representing the gradual dissociation of the bond. ### Parameter Description - The parameter \( D \) is the well depth (or binding energy) of the potential. - \( r_e \) is the bond length. - \( \alpha \) is the anharmonicity constant.
Transcribed Image Text:# The Morse Potential The harmonic potential, \( V(x) = \frac{1}{2}kx^2 \), is a useful start for modeling molecular vibrations, but it has limitations. A realistic potential between atoms should accurately represent the sharp increase in the potential as two nuclei come in close proximity, and also have the ability for a bond to break: that is, an asymptote \( V \rightarrow 0 \) as \( x \rightarrow \infty \). One option, as shown in the figure, is the Morse potential: \[ V(r) = D(1 - e^{-\alpha(r - r_e)})^2 \] ### Graph Explanation The graph displays the Morse potential as a curve. The horizontal axis represents the interatomic distance \( r \), while the vertical axis represents the potential energy \( V(r) \). The curve starts high as \( r \) approaches zero (indicating strong repulsion at short distances), dips down to a minimum value (indicative of the bond energy), and then gradually rises toward zero as \( r \) increases, representing the gradual dissociation of the bond. ### Parameter Description - The parameter \( D \) is the well depth (or binding energy) of the potential. - \( r_e \) is the bond length. - \( \alpha \) is the anharmonicity constant.
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