Hamiltonian: accounting for entropy
One definition of entropy is that the number of microstates that makes up a macrostate influences the probability of observing the macrostate.  So your wavefunction needs to be multi-configurational and include both $\mathrm{H}_2^+$ and $\mathrm{H}^+ + \mathrm{H}$. The usual approach is to invoke the Born-Oppenheimer approximation, which makes the wavefunction depend on a single static set of nuclear coordinates, i.e. single-configurational. This approach will just give you the energy and you must compute reaction probability using statistical mechanics.

Multi-configurational SCF does include entropy in the following sense.  Let's say you have a molecule with a HOMO-LUMO gap that is sufficiently small to allow for significant mixing. You run a CASSCF(2,2) to determine the LUMO contribution and is 5%.  Now suppose you have the same system, same HOMO-LUMO gap, but the LUMO is doubly degenerate.  Your CASSCF(2,3) would now have a larger LUMO contribution, though I am not sure it would be 10%.