Hydrogen molecule in a magnetic field: The lowest states of the $\Pi$ manifold and the global ground state of the parallel configuration

The electronic structure of the hydrogen molecule in a magnetic field is investigated for parallel internuclear and magnetic field axes. The lowest states of the $\Pi$ manifold are studied for spin singlet and triplet ${(M}_s=-1)$ as well as gerade and ungerade parity for a broad range of field strengths $0<~B<~100$ a.u. For both states with gerade parity we observe a monotonic decrease in the dissociation energy with increasing field strength up to $B=0.1\mathrm{}$ a.u. and metastable states with respect to the dissociation into two H atoms occur for a certain range of field strengths. For both states with ungerade parity we observe a strong increase in the dissociation energy with increasing field strength above some critical field strength $B_c.$ As a major result we determine the transition field strengths for the crossings among the lowest ${}^1{\Sigma }_g,$ ${}^3{\Sigma }_u,$ and ${}^3{\Pi }_u$ states. The global ground state for $B\lesssim 0.18$ a.u. is the strongly bound ${}^1{\Sigma }_g$ state. The crossings of the ${}^1{\Sigma }_g$ with the ${}^3{\Sigma }_u$ and ${}^3{\Pi }_u$ state occur at $B\approx 0.18$ and $B\approx 0.39$ a.u., respectively. The transition between the ${}^3{\Sigma }_u$ and the ${}^3{\Pi }_u$ state occurs at $B\approx 12.3$ a.u. Therefore, the global ground state of the hydrogen molecule for the parallel configuration is the unbound ${}^3{\Sigma }_u$ state for $0.18\lesssim B\lesssim 12.3$ a.u. The ground state for $B\gtrsim 12.3$ a.u. is the strongly bound ${}^3{\Pi }_u$ state. This result is of great relevance to the chemistry in the atmospheres of magnetic white dwarfs and neutron stars.