Exactly solvable Schrödinger equation with double-well potential for hydrogen bond

Abstract

We construct a double-well potential for which the Schrödinger equation can be exactly solved via reducing to the confluent Heun's one. Thus the wave function is expressed via the confluent Heun's function. The latter is tabulated in Maple so that the obtained solution is easily treated. The potential is infinite at the boundaries of the final interval that makes it to be highly suitable for modeling hydrogen bonds (both ordinary and low-barrier ones). We exemplify theoretical results by detailed treating the hydrogen bond in KHCO 3 and show their good agreement with literature experimental data.

Generalized confluent hypergeometric solutions of the Heun confluent equation

“…, (9), and (12). A concluding remark is that the system of the algebraic equations at hand leads to the following generalization of equations (13), (14):…”

“…The Heun confluent equation [1][2][3] is a second order linear differential equation widely encountered in contemporary physics research ranging from hydrodynamics, polymer and chemical physics to atomic and particle physics, theory of black holes, general relativity and cosmology, etc. (see, e.g., [4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22] and references therein). This equation has two regular singularities conventionally located at points 0 z  and 1 z  of complex z -plane, and an irregular singularity of rank 1 at z   .…”

Analytic calculation of ground state splitting in symmetric double well potential

“…Earlier CHF was used for obtaining the exact solution of the Smoluchowski equation for reorientational motion in Maier-Saupe DWP [41], [42] that gives the probability distribution function in the form convenient for application to NNR [43]. In the present article we develop similar approach initiated in [34], [35] for SE with trigonometric DWP. It should be stressed that the reduction of SE with trigonometric DWP to the Coulomb (generalized) SF requires integer m and thus in this form SE with trigonometric DWP belongs to quasi-exact type.…”

“…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].…”

Analytic treatment of IR-spectroscopy data for double well potential

“…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).…”