Question
Apply the Taylor expansion to the potential energy given by the Morse equation $\tilde{V}(R)=D_{\mathrm{e}}\left\{1-\exp \left[-a\left(R-R_{0}\right)\right]\right\}^{2}$ to show that the force constant $k$ is given by $k=2 D_{\mathrm{e}} a^{2}$
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A simple analytic function that can be used to approximate the potential energy of the two atoms in a diatomic molecule is the Morse potential $$ U(R)=A\left[\left(e^{\left(R_{0}-R\right) / S}-1\right)^{2}-1\right] \quad(12.42) $$ where $A, R_{0}$, and $S$ are positive constants, with $R_{0} \gg S .$ (a) Sketch this function for $0 \leq R<\infty$. (b) In terms of $A, R_{0}$, and $S$, write down the bond length and binding energy of a molecule whose potential energy is given by (12.42). (c) Write down the first three terms of the Taylor series for $U(R)$ about $R_{0}$ (see Appendix B) and show that they give the parabolic approximation (12.25). (d) Show that the force constant $k$ is $2 A / S^{2}$
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One possible form for the potential energy $(U)$ of a diatomic molecule (Fig. 8$)$ is called the Morse Potential: $\quad U=U_{0}\left[1-e^{-a\left(r-r_{0}\right)}\right]^{2}$ (a) Show that $r_{0}$ represents the equilibrium distance and $U_{0}$ the dissociation energy. $(b)$ Graph $U$ from $r=0$ to $r=4 r_{0}$ assuming $a=18 \mathrm{nm}^{-1}, U_{0}=4.6 \mathrm{eV}, \quad$ and $r_{0}=0.13 \mathrm{nm}$
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