Analytic description of inversion vibrational mode for ammonia molecule

Abstract

The one-dimensional Schrödinger equation with symmetric trigonometric doublewell potential is exactly solved via angular prolate spheroidal function. Although it is inferior compared with multidimensional counterparts and its limitations are obvious nevertheless its solution is shown to be analytic rather than commonly used numerical or approximate semiclassical (WKB) one. This comprises the novelty and the merit of the present work. Our exact analytic description of the ground state splitting can well be a referee point for comparison of the accuracy of numerous WKB formulas suggested in the literature. The approach reasonably well suits for the inversion mode in the ammonia molecule N H 3 and thus yields a new theoretical tool for its description. The results obtained provide good quantitative description of relevant experimental data on microwave and IR spectroscopy of N H 3 .

Analytic treatment of IR-spectroscopy data for double well potential

“…The one-dimensional time-independent Schrödinger equation (SE) for a quantum particle moving in some double-well potential (DWP) is ubiquitous in physics and chemistry (see [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16] and refs. therein).…”

“…On the other hand the exact analytic solutions [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [16] remain an important tool for understanding the reality. Such solutions are expressed via confluent Heun's function (CHF) [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [14] (implemented in Maple), spheroidal function (SF) [7], [8], [9] or derived via the functional Bethe ansatz [16]. Unfortunately these solutions can be obtained only for some particular DWPs and lack the universality of numerical methods.…”

“…where −π/2 ≤ x ≤ π/2 is a particular type of DWPs expressed via trigonometric functions for which SE can be exactly solved via SF (implemented in Mathematica) or CHF [7], [8], [9]. In the symmetric form a = 0 (see examples in Fig.1, Fig.2 and Fig.5) it contains two parameters {m, p} allowing one to model the most important characteristics of DWP (barrier height and the distance between the minima of the potential).…”

“…Then SE with trigonometric DWP in dimensionless units (see Appendix 1 for details) takes the form [8], [9] ψ ′′…”

Analytic calculation of ground state splitting in symmetric double well potential

“…Recently the reduction of SE with DWP to the confluent Heun's equation (CHE) enabled one to obtain quasi-exact (i.e., exact for some particular choice of potential parameters) [25], [26], [27] and exact (those for an arbitrary set of potential parameters) [28], [34], [35] solutions. A plenty of potentials for SE are shown to be exactly solvable via CHF [36].…”