  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 $0 \leq B \leq 100 a.u.$ For both states with gerade parity we
observe a monotonous decrease in the dissociation energy with increasing field
strength up to $B = 0.1 a.u.$ 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 $B \approx0.39
a.u.$, respectively. The transition between the $^3\Sigma_u$ and $^3\Pi_u$
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.
