Breit into Green's function

[benroberts999]

Currently, cannot construct Green function with Breit operator.
This means Feynman method cannot be used if Breit is included.
Would be better to be able to include Breit into construction of Green's function.
Requires modification of Dyson equation to include Hartree-Fock-Breit potential.

Currently, we have

$$ \hat G = \hat G_0 + \hat G_0 V_x \hat G = \left[ 1 - \hat G_0 V_x \right]^{-1} , $$

where $\hat G_0$ is the Green's function for the Homogeneous Dirac-Hartree-Fock equation (excluding the non-local exchange potential, $V_x$), and

$$ V_x = - \sum_a^{\rm core} \left| a \right \rangle \hat Q \left \langle a \right | $$

is the non-local exchange potential (with $Q$ the Coulomb operator), which can be easily represented as a coordinate matrix.

The issue to include Breit is

$$ V_x \to V_x + V_{\rm Br} , $$

but the matrix $V_{\rm Br}$ is much more difficult to construct.

Probably, it requires keeping the $g$-parts of the Green's function intact to make sense.

It might be easier to form the matrix $V_{\rm Br} G_0$, rather than directly calculating $V_{\rm Br}$.