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 .