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 Δ is the separation of a pair and hν the distance between two pairs Δhν=1πA² where A=exp[(2πh)∫0^x₁[2m(V-E)]1/2dx]. A particular potential curve is chosen consisting of two equal parabolae connected by a straight line. The expression for Δ may then be evaluated explicitly as a function of the length of the joining line, 2(x₀-α) and the distance between two minima, 2x₀. These formulae may be applied to determining the form of the ammonia molecule. Substituting the experimental values for Δ₀ and Δ₁, it is found that x₀=3.161 and α=1.916. An exact solution for this particular potential curve may be found by joining Weber's function Dₙ(x-x₀) and Dₙ(x+x₀) to a hyperbolic sine or cosine. This process also leads to expressions for Δ which may be equated to the experimental values yielding x₀=3.182 and α=1.930, in good agreement with the earlier determination. Finally x₀ is used to compute 2q₀=0.760×10^-8 cm, the distance between the two potential minima, and the following dimensions of the ammonia molecule, H - H = 1.64×{}10^-8, N - H = 1.02×{}10^-8 cm.