The Two-Minima Problem and the Ammonia Molecule

The system of a particle moving in a potential field containing two equal minima is treated by the Wentzel-Kramers-Brillouin method of approximation. The energy levels are grouped in pairs and the object of the computation is to find the separation between two levels forming a pair. This is accomplished by connecting the oscillatory and exponential approximate solutions of the wave equation by means of the Kramers connection formulae. If $\Delta$ is the separation of a pair and $h\nu$ the distance between two pairs $\frac{\Delta }{h\nu }=\frac{1}{\pi A^2}$ where $A=\exp \left[\right(\frac{2\pi }{h}\left)\int 0^{x_1}{\mathrm{}\left[2m\right(V-E\left)\right]}^{\frac{1}{2}}\mathrm{dx}\right]$. A particular potential curve is chosen consisting of two equal parabolae connected by a straight line. The expression for $\Delta$ may then be evaluated explicitly as a function of the length of the joining line, $2(x_0-\alpha )$ and the distance between two minima, $2x_0$. These formulae may be applied to determining the form of the ammonia molecule. Substituting the experimental values for ${\Delta }_0$ and ${\Delta }_1$, it is found that $x_0=3.161\mathrm{}$ and $\alpha =1.916\mathrm{}$. An exact solution for this particular potential curve may be found by joining Weber's function $D_n(x-x_0)$ and $D_n(x+x_0)$ to a hyperbolic sine or cosine. This process also leads to expressions for $\Delta$ which may be equated to the experimental values yielding $x_0=3.182\mathrm{}$ and $\alpha =1.930\mathrm{}$, in good agreement with the earlier determination. Finally $x_0$ is used to compute $2q_0=0.760\mathrm{}\times {10}^{-8\mathrm{}}$ cm, the distance between the two potential minima, and the following dimensions of the ammonia molecule, H - H = 1.64×${10}^{-8\mathrm{}}$, N - H = 1.02×${10}^{-8\mathrm{}}$ cm.