[QUE/QM-08001]

The potential energy of two protons in hydrogen molecule ion in a model is given below \begin{eqnarray}
   V(x) &=& |E_1| f(x) \\
   f(x) &=& - 1 + \frac{2}{x}\left[ \frac{(1-(2/3)x^2) e^{-x} + (1+x) e^{-2x}}
   {1+(1+x+x^2/3) e^{-x} }\right], \qquad x=R/a
\end{eqnarray}
$E_1= 13.6 \text{ eV}$ is the ground state energy of H atom and $a$ is the Bohr  radius $\hbar^2/me^2$. The graph of this  function $f(x)$ is reproduced below. Find numerical values of the bond length in ${A^o}$, the zero point energy and spacing of vibrational spectrum, both energies in electron volts.

NoteThe expression for $V(x)$ is taken from an approximate variational calculation of energy of the H molecule ion in Born Oppenheimer approximation.