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 is the separation of a pair and h the distance between two pairs h=1{A^2} where A=exp (2h) 0^{x₁}[2m (V-E) ^1{2}dx]. A particular potential curve is chosen consisting of two equal parabolae connected by a straight line. The expression for may then be evaluated explicitly as a function of the length of the joining line, 2 (x₀-) 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 ₀ and ₁, it is found that x₀=3. 161 and =1. 916. An exact solution for this particular potential curve may be found by joining Weber's function D₍ (x-x₀) and D₍ (x+x₀) to a hyperbolic sine or cosine. This process also leads to expressions for which may be equated to the experimental values yielding x₀=3. 182 and =1. 930, in good agreement with the earlier determination. Finally x₀ is used to compute 2q₀=0. 76010^-8 cm, the distance between the two potential minima, and the following dimensions of the ammonia molecule, H - H = 1. 6410^-8, N - H = 1. 0210^-8 cm.