Problem $2$

The parameters of a Morse potential for $H_2$ are $D_e=458\mathrm{.}3k\frac{J}{m}ol,r_e=0\mathrm{.}74$ and $=1\mathrm{.}83{}^{-1},$ where the Morse potential is given by the expression:

$V_M(r)=D_e$exp$(-2(r-r_e))-2D_e$exp$(-(r-r_e))$

Part a$.$ The Morse potential may be approximated using the following cubic potential,

$V(r)=A+B(r-r_e)+C(r-r_e{)}^2+D(r-r_e{)}^3$

Derive equations for the constants where $A,B,C$ and $D$ expressed in terms of the Morse potential parameters.

Part b$.$ Determine an equation for the force between atoms interacting by the cubic potential. Based on the force, determine the maximum bond extension $(r-r_e{)}_{\text{m}\text{a}\text{x}}$ above which the cubic approximation must become invalid. Express the maximum bond extension only in terms of the Morse parameters. Clearly explain your answer to receive credit. Bond extension means that $r>r_e.$

Part c$.$ Determine the approximate harmonic oscillator state of $H_2,$ specified by quantum number $n,$ for which the maximum bond length is $r=1\mathrm{.}455r_e.$ Note that a harmonic oscillator state is determined by a harmonic potential.

Part d$.$ How much energy would need to be transferred to a ground state $D_2$ molecule to excite the molecule to the quantum state determined in Part c$.$