The Dirac Equation in 2-Spinor Form

In order to analyze the consequences of basis set expansion of the relativistic wave equation for electrons, it is sufficient to consider the one-electron Dirac equation. The only other fundamental complication we foresee when going to a many-electron model is continuum dissolution (Brown-Ravenhall disease), which we have already dealt with in chapter 5 and so need not consider in this context. 

A natural choice for the expansion of the wave function would be the atomic 2-spinors of the form 

We expand the large and small components in a basis of 2-spinors, x  

This we can turn into a matrix equation in the expansion coefficients a, and we get the one-electron Dirac equation in a basis expansion as 

These are the representations of the overlap, the potential energy, and the kinetic energy operators in the expansion basis. Note that = (77 )l, so that 

---

We will work with the Dirac equation in 2-spinor form. 

---