Second order relativistic corrections to <PERSON> equation from Dirac equation
I'm trying to derive the full and correct Hamiltonian for spin$\frac{1}{2}$ particles from Dirac equation up to second order in $v/c$. For a potential and magnetic field constant in time.
In particular, there should be <PERSON> term, relativistic kinetic term, spin-orbit term and <PERSON> term:
$$\hat{H}=\frac{\mathbf{P}^2}{2m}+U-e \hbar \mathbf{ \sigma} \mathbf{B}-\frac{\mathbf{P}^4}{8m^3 c^2}- \frac{e \hbar}{4 m^2 c^2} \mathbf{ \sigma} (\mathbf{F} \times \mathbf{P})- \frac{e \hbar^2}{8 m^2 c^2} (\nabla \mathbf{F}) \tag{1}$$
Here:
$$\mathbf{P}=- i \hbar \nabla-e \mathbf{A} \ \mathbf{B}= \nabla \times \mathbf{A} \ \mathbf{F}=- \frac{1}{e} \nabla U$$
This is taken from various sources, for example, <PERSON>, <PERSON>, <PERSON> "Quantum electrodynamics", (33.12) though I have added the <PERSON> term and changed some notation.
Now, assuming the above is correct, I want to derive this expression from Dirac equation directly.
I use the following form of the equation as a starting point:
$$\begin{cases} \displaystyle \Psi_+ = - \frac{\sigma \mathbf{P}}{mc} \Psi_-+\frac{E}{mc^2} \Psi_+ \ \displaystyle \Psi_- = \frac{\sigma \mathbf{P}}{mc} \Psi_+ -\frac{E}{mc^2} \Psi_- \end{cases} \tag{2}$$
Where $\Psi_{\pm}$ are two component spinors and $\sigma$ is the vector of Pauli matrices.
Introducing a new energy:
$$E=mc^2+\mathcal{E} \ \mathcal{E}=i \hbar \frac{\partial}{\partial t} -U $$
We obtain:
$$\begin{cases} \displaystyle \frac{\mathcal{E}}{mc^2} \Psi_+= \frac{\sigma \mathbf{P}}{mc} \Psi_- \ \displaystyle \left(2+ \frac{\mathcal{E}}{mc^2} \right)\Psi_- = \frac{\sigma \mathbf{P}}{mc} \Psi_+ \end{cases} \tag{3}$$
Assuming $$\Vert \mathcal{E} \Vert \ll mc^2$$
We obtain, up to first order in $\Vert \mathcal{E} \Vert / mc^2$:
$$\mathcal{E} \Psi_+ = \frac{\sigma \mathbf{P}}{2m} \left(1- \frac{\mathcal{E}}{2mc^2} \right) \sigma \mathbf{P} \Psi_+ \tag{4}$$
I assume that this should directly give us (1).
However, after some simplifications, while I get most of the terms above right (including the most important SO term), I get an incorrect <PERSON> term (twice as large) and also a bunch of extra terms.
My full derivation is lengthy, so I wanted to ask first if I missed something? Is (4) correct (up to second order in $v/c$) and should it lead to (1) without any extra terms?
I have extensively used the identity:
$$(\sigma \mathbf{a}) (\sigma \mathbf{b})= \mathbf{a} \mathbf{b}+i \sigma (\mathbf{a} \times \mathbf{b})$$
though it's not always clear how to apply it to operators. In any case, it gives a lot of extra terms not present in (1) when applied to (4).
There's a related question, but only about Pauli equation, which is simple enough to derive. 955