atomic physics - Interaction between electrons in which the magnetic dipole moments interact more strongly than their electric fields

Asking a question Has anyone tried to incorporate the electrons magnetic dipole moment into the atomic orbital theory?, I was curious whether anyone has attempted to relate the intrinsic property of the magnetic moment of the electron to the above-mentioned properties of spin.

In the extremely detailed answer (thanks to the author, who took the time despite the pointlessness of such a question) it is clarified that

The effects are weak, and they are secondary to all sorts of other interactions that happen in atoms,...

Also, in case you're wondering just how weak: this paper calculates the energy shifts coming from electron spin-spin coupling for a range of two-electron systems. The largest is in helium, for which the coupling energy is of the order of $\sim 7 \:\mathrm{cm}^{-1}$, or about $0.86\:\rm meV$, as compared to typical characteristic energies of $\sim 20\:\rm eV$, some five orders of magnitude higher, for that system.

Now there is a new question about Electron to electron interaction.

There is a critical distance

$$d_\text{crit}=\sqrt\frac{3\epsilon_0\mu_0\hbar^2}{2m^2}=\sqrt{\frac{3}{2}}\frac{\hbar c}{m}=\sqrt{\frac{3}{2}}\overline\lambda_C,$$

where $\overline\lambda_C$ is the reduced Compton wavelength of the electron, at which the two forces are equal in magnitude.

Since the Compton wavelength is a standard measure of where quantum effects start to be important, this classical analysis can't be taken too seriously. But it indicates that spin-spin interactions are important at short distances.

I wonder how these two points of view can be related.

Answer

They can be related by the fact that the Bohr radius of hydrogen is $1/\alpha\approx 137$ times larger than the reduced Compton wavelength of the electron. (Here $\alpha$ is the fine-structure constant. For helium, divide by 2 to get 68.5.) At this large a separation between the proton and electron, the magnetic interaction that I calculated is small compared to the electrostatic interaction.