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

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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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