Hamiltonian: accounting for entropy I might have a bit of a conceptual misunderstanding here, but how do we account for the entropy of a system when constructing a Hamiltonian? All entropy should be accounted, because the Hamiltonian includes the total energy, including the entropic contribution. However, when considering e.g. $\ce{H2+}$, the fact that the dissociated state is more entropically favoured doesn't go from the Hamiltonian. 

Any suggestion that would clean up my understanding of the relationship between entropy and energy in this and other cases would be very helpful. Thanks!
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%.