# Exactly solvable Schr\"odinger equation with double-well potential for hydrogen bond

## TL;DR

This paper presents an exactly solvable double-well Schr"odinger equation model for hydrogen bonds, using confluent Heun functions, and demonstrates its effectiveness by modeling hydrogen bonds in KHCO3 with good experimental agreement.

## Contribution

The authors develop a new exactly solvable double-well potential model for hydrogen bonds using confluent Heun functions, facilitating analytical solutions and practical applications.

## Key findings

- Analytical solutions for hydrogen bond potentials are obtained.
- Model shows good agreement with experimental data for KHCO3.
- The potential is suitable for both ordinary and low-barrier hydrogen bonds.

## Abstract

We construct a double-well potential for which the Schr\"odinger 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 {\sl {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.