Numerical simulation of Bohr-like and Thomson-like dark atoms with nuclei

The puzzles of direct dark matter searches can be solved in the scenario of dark atoms, which bind hypothetical, stable, lepton-like particles with charge $-2n$, where $n$ is any natural number, with $n$ nuclei of primordial helium. Avoid experimental discovery because they form with primary helium neutral atom-like states $OHe$ ($X~-$ helium), called "dark" atoms. The proposed solution to this problem involves rigorous proof of the existence of a low-energy bound state in the dark atom interaction with nuclei. It implies self-consistent account for nuclear attraction and Coulomb repulsion in such an interaction. We approach the solution of this problem by numerical modeling to reveal the essence of the processes of dark atom interaction with nuclei. We start with the classical three-body problem, to which the effects of quantum physics are added. The numerical model of the dark atom interaction was developed for $O^{--}$ having a charge of $-2$, bound with He in Bohr-like $OHe$ dark atom and for $-2n$ charged $X$ bound with $n$ $\alpha$-particle nucleus in the Thomson-like atom $XHe$. The development of our approach should lead to the solution of the puzzles of direct dark matter searches in the framework of dark atom hypothesis.