Analytic calculation of ground state splitting in symmetric double well potential

Abstract

The exact solution of the one-dimensional Schrödinger equation with symmetric trigonometric double-well potential (DWP) is obtained via angular oblate spheroidal function. The results of stringent analytic calculation for the ground state splitting of ring-puckering vibration in the 1,3-dioxole (as an example of the case when the ground state tunneling doublet is well below the potential barrier top) and 2,3dihydrofuran (as an example of the case when the ground state tunneling doublet is close to the potential barrier top) are compared with several variants of approximate semiclassical (WKB) ones. This enables us to verify the accuracy of various WKB formulas suggested in the literature: 1. ordinary WKB, i.e., the formula from the Landau and Lifshitz textbook; 2. Garg's formula; 3. instanton approach. We show that for the former case all three variants of WKB provide good accuracy while for the latter one they are very inaccurate. The results obtained provide a new theoretical tool for describing relevant experimental data on IR spectroscopy of ring-puckering vibrations.

Analytic treatment of IR-spectroscopy data for double well potential

“…In [24] the ground-state splitting was calculated by the instanton approach to double Morse potential. For low-barrier HB in CrOOH and KH 2 PO 4 the ground state doublet is close to the barrier top and the instanton method is known to yield very inaccurate results [9] in this case. Our approach provides a more stringent and adequate tool for calculating the ground state splitting.…”

“…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 ′′…”

Lambert W function methods in double square well and waveguide problems

“…Analytic descriptions of molecules which have inversional vibration modes and ground state splitting are developed in two interesting papers by Sitnitsky [14,15]. These are not square well potentials, but they are symmetric dual well potentials, and have analytic expressions for their properties.…”