From what I understand the Dirac equation is supposed to be an improvement on the Schrödinger equation in that it is consistent with relativity theory. Yet all methods I have encountered for doing actual ab initio quantum mechanical calculations uses the Schrödinger equation. If relativistic effects are important one adds a relativistic correction. If the Dirac equation is a more correct description of reality, shouldn't it give rise to easier calculations? If it doesn't, is it really a more correct description?
Think it with an example, Einstein's field equations are much more precise than Newton's law of gravity, but it's much more complicated to solve a Classical Mechanics problem with General Relativity.
More fundamental and precise doesn't mean that it will give easier calculations. If it did, then then chemistry, medicine, etc... wouldn't exist because they can be described almost completely using Dirac's equation.
If the Dirac equation is a more correct description of reality, shouldn't it give rise to easier calculations?
It is true that the Dirac equation takes into account theory of relativity, so in this respect it is more correct than the Schroedinger equation.
However, the problem with the Dirac equation is that it involves function $\psi$ defined on space-time, not on configuration space, and it describes naturally one particle under action of electromagnetic field. For more than one particle, this means a big difference. For example, for two interacting particles, like two electrons in a field of fixed nuclei, we have the Schroedinger equation for function $\psi$ on 6-dimensional space, but it is not clear how to do similar thing with the Dirac equation, because if we want to claim it is more accurate, we need to describe the interaction between the particles in a better way than just by electrostatic potential.
This is partially accomplished by the Breit equation, which is kind of modified Schroedinger equation containing relativistic corrections, but still not entirely consistent with relativity and having some problems with the new terms; some quantities diverge which should not so it is not a satisfactory equation.
This and other problems lead people to reinterpret the Dirac equation as describing a kind of "quantum field", not a distinct particle. Unfortunately the resulting theory seems too difficult and problematic to be used regularly for complicated calculations of properties of molecules. The non-relativistic theory is much more developed for this purpose and people that worked on it (for example, John Slater, David Cook) say that in its basic form it works quite well for common atoms and molecules (I think unless one wants to include more subtle details like those relativistic corrections).
IMO, the reason why the Dirac equation is not used a lot is because we have a better theory of relativistic quantum mechanics called Quantum Field Theory.
The Dirac equation is one of the key equations of QFT but the calculations in QFT do not rely on solving the Dirac equation explicitly. Rather properties of the solutions of the Dirac equation (normalization and orthogonality of spinors, projection operators on positive/negative "energy" states) are used.
The Dirac equation is indeed widely used in ab initio calculations in quantum chemistry (you may wish to google "relativistic quantum chemistry"). For example, the use of the Dirac equation is especially important where you have heavy nuclei: one requires the Dirac equation to explain operation of even such mundane devices as lead-acid batteries (Phys. Rev. Lett. 106, 018301 (2011)).