Two-component formalism and four-component formalism When deriving the Dirac equation for spin-1/2 particles, we realize that the wave function must be four-component. In some works, people use two-component wave function for calculation. 
So, my question is that, what are advantages of the two-component formalism compared to the four-component one?
 A: Four component formalism is the "right" formalism, but it has negative energy eigenstates corresponding to the antiparticles.  Most chemists and solid state physics are not interested in the antiparticles, and such negative energy solution causes trouble for conventional variational methods, where you might end up falling to negative infinity energy.  It is also a larger problem.
One may therefore, do a transformation to remove the negative energy part, but this is an approximation.  Other treatments are available to allow one to use directly a four component formalism, and some claim that the cost is in fact similar in order of magnitude, but the implementation can be more difficult.
So to sum this up, when solving Dirac equation, one has to construct the wave function on 4 components, including two more "small component" wave functions. This requires more resource, causing problems where the non-relativistic problem is already often too large to handle.  One would choose various ways to handle this, and doing a two component transformation is merely one way of doing this while making the solution procedure more similar to non-relativistic case.
