$H_2^{+}$ molecular ion in a strong magnetic field: ground state
  A detailed quantitative analysis of the system $(ppe)$ placed in magnetic field ranging from $0 - 4.414 \times 10^{13} G$ is presented. The present study is focused on the question of the existence of the molecular ion $H_2^{+}$ in a magnetic field. As a tool, a variational method with an optimization of the form of the vector potential (optimal gauge fixing) is used. It is shown that in the domain of applicability of the non-relativistic approximation the system $(ppe)$ in the Born-Oppenheimer approximation has a well-pronounced minimum in the total energy at a finite interproton distance for $B \lesssim 10^{11} G$, thus manifesting the existence of $H_2^{+}$. For $B \gtrsim 10^{11} G$ and large inclinations (of the molecular axis with respect to the magnetic line) the minimum disappears and hence the molecular ion $H_2^{+}$ does not exist. It is shown that the most stable configuration of $H_2^{+}$ always corresponds to protons situated along the magnetic line. With magnetic field growth the ion $H_2^{+}$ becomes more and more tightly bound and compact, and the electronic distribution evolves from a two-peak to a one-peak pattern. The domain of inclinations where the $H_2^{+}$ ion exists reduces with magnetic field increase and finally becomes $0^o - 25^o$ at $B = 4.414 \times 10^{13} G$. Phase transition type behavior of variational parameters for some interproton distances related to the beginning of the chemical reaction $H_2^{+} \leftrightarrow H + p$ is found. 
J.C. Lopez Vieyra and A.V. Turbiner