Could you clarify your question? Are you just asking whether enthalpy is the same as energy at T = 0 K, or are you asking how you could include T > 0 K effects? I'm also slightly unclear as to your notation; H is usually reserved for enthalpy, and G for the Gibbs free energy. I think your "H" is just the computed energy, so it would be better to call it "E" - or am I wrong?

My main question is how do we know if a reaction will happen or not. For which I have described my procedure. And I do agree that it is not technically 'H' as in enthalpy but the energy 'E' calculated using SCF calculation. Now considering H = E + PV how do I calculate this term. And is it necessary to calculate this term in order to see the feasibility of the reaction, like do I need to know the $\Delta H$(and thereby $\Delta G$) or is $\Delta E$ enough. @PhilHasnip Thanks for taking the time out !!

So actually I am trying to sort of replicate the procedures performed in nature.com/articles/s41524-019-0252-6 and pubs.acs.org/doi/10.1021/acs.chemmater.9b04021 where in the first one they say that they have evaluated enthalpies using DFT and in the second one the calculate the difference in formation energies of the products and reactants. Now the first paper does not mention how they have tackled the pressure and I am not sure whether the second one calculates the formation energy specifically or the energy from scf calculation.

OK I looked at the first one, and they appear to mean "energies" not "enthalpies". From the SI: "ΔH, which is defined as the total energy difference between the decomposed producers and the... compounds". There's no reason you couldn't include pressure, and it would probably be relevant for microphase formation, but for true thermodynamic stability, you should calculate each phase at its own equilibrium volume (where P=0).

So I was actually looking in to how to calculate the pressure term thereby actually calculating the gibbs free energy. But the methods that I found are a bit complicated researchgate.net/post/… and the other idea was to use VASPKIT from where I can get thermal corrections but that idea did not turn up because that would work for ideal gases or for absorbate materials. @PhilHasnip do you know of any direct way to calculate gibbs free energy using some code or something

and @PhilHasnip in your opinion don't you think calculating $\Delta E$ like they have in the paper, is not a good enough idea for the checking stability. As the reactions could be exothermic or endothermic. So is $\Delta E$ enugh ??

Pressure is easy, you just apply the pressure you want and get your DFT code of choice to compute the equilibrium volume at that pressure for you. Entropy is the difficult part, and it can be really difficult; a phonon calculation can get you the vibrational contribution, albeit within the quasiharmonic approximation, but configurational entropy is probably impractical to calculate for most systems.

As for whether the energy alone is "good enough", that depends on what you're trying to accomplish. It's enough to say something about the likely stability at low temperature, but you won't know generally whether "low temperature" includes the temperature range you're actually interested in. It's certainly "good enough" to be published, provided you explain that's what you're doing, but IMO it's more satisfying to at least be able to estimate some of the remaining terms in the free energy.

I don't know for certain, but I think the use of Birch-Murnaghan EOS for these purposes dates back to software which couldn't optimise the lattice vectors. You'll notice that almost none of the people doing Birch-Murnaghan use the computed stress tensor, even though that would halve the number of calculations they have to do, and they almost always do a (possibly incorrect) isotropic strain.