There are two neutral atoms set separated at a long distance $R$ and let's consider them phenomenologically through Bohr model. Let's also assume that the nuclei (charged $+q$) of the atoms are fixed and punctate and that their surrounding electrons (totally charged $-q$, mass $m_-$) have perfect circular orbitals at a short distance $r_0$ ($\ll R$). The net electromagnetic (EM) force between the two is approximately zero due to their electrical neutrality. The setup is shown below:
Remark: notice that the term "atoms" may describe electrically neutral entities (i.e. whatever it may be) with a central static charge $+q$ and an orbiting charge $-q$ of opposite sign $-$ consider them as microscopic (i.e. quantum-mechanically) or macroscopic (large, classical) objects as well.
Let's see some different characteristic mutual orientations of the orbitals of electrons which are otherwise statistically randomly set:
From the picture above, one may notice that the force associated with a configuration of orbitals has its symmetrical counterpart which usually cancels it out when treated classically, e.g. like constant electron currents around a fixed nucleus or two electromagnets made of circular wires with identical currents (which repel/attract each other in order to minimize the EM energy in the field). Also, the electrons may have different mutual phases regarding their orbital "motion", especially if their hosting atoms have different charges (thus those electrons have different frequencies orbiting the atoms).
If we involved Special relativity (SR) to these cases, the symmetry would be broken (here shown for the parallel orbitals only):
For example, one electron with velocity $\vec v$ would see the other one which is in phase with the same velocity $\vec v$ as static, because they share the same frame of reference. In the case that their velocities have opposite directions ($\vec v$ vs. $-\vec v$), one has to account for relativistic effects, i.e. to relativistically add $\vec v$ and $-\vec v$.
Is there a calculation of the net force done which treats the system described as above using either SR or QFT ?
or
Does this make sense at all ?
Of course, the net force is assumed to be averaged over time, over all orbital phases, over small differences of the distance $R$ and over all possible mutual orientations of the orbitals. Further, one might calculate it for all possible atoms (on both sides) with their different charges (i.e. atomic numbers) and therefore different orbital speeds of their electrons.
Motivation: to estimate order of strength of the possible EM force asymmetry compared to gravity, therefore it shall be calculated precisely since the strength of the EM force for the electron is about $10^{40}$ times stronger than gravity.
