# Relativistic corrections to first-principles Hamiltonian?

In quantum treatments of solids it is common to start off discussions by writing down the "full" first-principles Hamiltonian for a group of electrons and nuclei as

$$H = \sum_i \frac{\hat{p}_i^2}{2m_i}+\frac{e^2}{2}\sum_{i,j} \frac{Z_i Z_j}{\vert \mathbf{r}_i - \mathbf{r}_j \vert}$$

This Hamiltonian can in principle be solved for a many body wavefunction $$\Psi(\mathbf{r}_1,\dots, \mathbf{r}_N)$$ and contains all the physics of a solid.

There are relativistic corrections that can be made to this Hamiltonian. The simplest is replacing the kinetic energy term with higher order terms in $$p$$. But there are also spin-dependent terms (like the Zeeman term), and spin orbit coupling as discussed in this question.

Let's assume we still want to stay in the comfort of non-relativistic quantum mechanics (particle number is conserved, fixed Coulomb gauge, etc.), but want a more accurate first principles Hamiltonian.

My question is: what is the correct expression for the many-body Hamiltonian of a group of electrons/nuclei that includes all important relavistic corrections to leading order?

If you let me rephrase that question in a more hand-wavy way, how do we explicitly include relavistic corrections (Zeeman spin term, spin-orbit, Darwin, etc.) into the full many body Hamiltonian above to make sure it doesn't leave out important SR physics?

The reason I ask this question is that it is common in solids state texts to include relavistic corrections at the single particle level, but rarely at the level of the "full" many body treatment.