SCF (Roothan-Hall) Method, why Density Matrix?
When doing the SCF calculation taking the RMSD of the density matrix compared to taking the RMSD of the diagonal of the density matrix takes about the same with respect to the total calculation time. The comparrison of the density matrices is not the time consuming step in the SCF calcultion. When you take the RMSD of the total density matrix you also "keep an eye" on the off diagonal elements, this is a more rigid test for convergence.