$1-24.$ In this problein, we investigate the harmonic nscillator potential as the leading term in a Taylor expansun of the actual internuclear potential, $V(R),$ about its equilibrium position, $R_f.$ Accordilig $(1)$ Problent $(\text{'}1.?,$ the tirst few terms in this expansion are

$])$

$([dR$

If $R$ is always close to $R_e,$ then $R-R_e$ is always small. Consequently, the terms on the right side of Equation $1$ get smaller and smaller. The first term in Equation I is a cunstant and depends upon where we choose the zero of energy. It is comvenient to shouse the zers of energy such that $V(R_k)$ equals zero and to relate $V(R)$ to this convention. Explain why the linear term in the displacement vanishes in Equation I. $($Note that $-d\frac{V}{d}R$ is the force acting between the two nuclei.$)$

Denote $R-R_c$ by $x,(d^2\frac{V}{d}{R}^2{)}_k=k_e$ by $k_1$ and $(d^3\frac{V}{d}{R}^3{)}_k=R_r$ by $$ to write Eyuation $1$ as

$V(x)=\frac{1}{2}k(R-k_c{)}^2+\frac{1}{6}(R-R_c{)}^3+$cdots

$=\frac{1}{2}k{x}^2+\frac{1}{6}{x}^3+$cdots

Argue that if we restrict ourselves to small displacements, then $x$ will be small and we can neglect the terms beyond the quadratic term in Equation $2,$ showing that the general potential cnergy function $V(R)$ can be approximated by a harmonic$-$nscillator potential. We can consider corrections or extensions of the harmonic$-$oscillator model by the higher$-$order terms in Equation $2.$ These terms are called antiannonic terms.

An analytic expression that is a good approximation to an intermolecular potential energy curve is a Morse potentiat

where $l_e$ and $$ are parameters that depend upon the mulecule. The parameter $D_s$ is the ground$-$state clectronic energy of the molecule measured from the minimum of $V(R),$ and $$ is hat$(x)$ meusure of the curvature of $V(R)$ at is minimum. Derive a relation hetween the force constan! and the parameters $D_e$ and $.$ Given that $l_e=7\mathrm{.}31{10}^{N}J*molecul{e}^{-1}.=0\mathrm{.}0181n^{-1},$ and $R_{}=127\mathrm{.}5pm$ for $HC{l}_{(g)},$ calculate the force constant of $HCl(g).$ Plot the Morse potential for $HCl(g),$ and plot the corresponding harmonic oscillator potential on the same graph $($cf$.$ Frigure $1\mathrm{.}5).$