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

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.