What types of energies can be determined by molecular dynamics in gas phase or explicit solvents?
First, whether you use DFT or molecular mechanics to get your energies and forces doesn't really matter. If you have equations of motion, forces, and energies then you have the dynamics necessary to sample an ensemble and from there you should be able to get any physical observable you desire.

It is often very difficult to determine free energy changes using straight up molecular dynamics. The reason for this is that even very fast reactions are rare events when compared to typical molecular motions such as vibrations and rotations. For this reason, many methods have been developed to determine free energies without having to run simulations long enough to get good sampling of reactions. Usually, this is not possible anyways.

First, you may wish to read about umbrella sampling. Umbrella sampling, as a general idea, does not sample a trajectory on the potential energy surface determined the potential energy of the system (your output from DFT if you like), but rather, a harmonic potential energy is added to the system as well. This has the effect of lifting the molecule out of the potential well it may be stuck in so that sampling of rare events like reactions becomes much more common.

It is not obvious at all that any physically meaningful quantities should be able to be derived from this new system where the potential energy is constantly being modified. Yet, surprisingly and rather beautifully, it is possible. You should probably see ref. [1] and the citations therein.

The second way to do this I will mention is so-called meta-dynamics. The conceptual idea here is to first define the so-called "collective variables" of the system. These are literally any variables which can be used to describe the properties of a system. Typically one would choose some set of internal coordinates, particularly for a gas-phase system as you describe. However, my understanding is that it is also possible to choose various other coordinates such as intermolecular distances. One could imagine this being quite useful if you are interested in something like radial distribution functions in some hard-to-sample region.

Using these collective variables, again, a biasing potential is applied to the system, but this potential is said to be "history-dependent" in the sense that the form of the biasing potential might change based on what parts of the free energy landscape have already been seen. This differs from umbrella sampling where the potential is always harmonic and the rate at which the potential is biased and how extreme the bias is, is chosen by the user. You should see ref. [2] and references therein for more details on metadynamics.

So, yes, it is possible and even fairly routine to calculate free energies, enthalpies, entropies and pretty much anything else from molecular dynamics simulations. However, just because it is routing does not mean these methods are perfect. It can be very hard to measure error with these sorts of methods because one does not generally know the correct answer particularly for dynamical processes. That is, if your aim is to study details of dynamics which can't be observed experimentally, then it is very hard to know if the dynamics under your biased potential mean much beyond the fact they should give you pretty good ensemble averages. Also, there is the inherent subjectivity in metadynamics involved in choosing collective variables which has been known to cause problems. How many variables are enough and which ones?

Hopefully that is about what you were looking for.

I don't address the differences between explicit solvent and gas-phase much because these methods will work in any context (even implicit solvent) for which forces and energies can be supplied.



[1]: Kästner, J. (2011). Umbrella sampling. Wiley Interdisciplinary Reviews: Computational Molecular Science, 1(6), 932-942.

[2]: Barducci, A., Bonomi, M., &amp; Parrinello, M. (2011). Metadynamics. Wiley Interdisciplinary Reviews: Computational Molecular Science, 1(5), 826-843.