Spin-orbit coupling is the interaction between a particle’s spin angular momentum and orbital angular momentum. An electron orbiting around the nucleus ‘sees’ the nucleus circling it, just as a person on earth perceives the sun circling the earth while the latter orbits around the sun.
This apparent nuclear orbit creates a magnetic field 
that exerts a torque on the electron’s spin magnetic dipole moment 
, resulting in an additional term of 
(where 
) in the multi-electron Hamiltonian. To derive this term, we consider a 1-electron atom.
Let 
be the orbital angular momentum of the electron and 
be the proton’s current loop, which generates a magnetic field of magnitude 
given by the Biot-Savart law. Since 
, we have
Substitute eq259 in , we have, 
or 
, where 
and 
are unit vectors. Since 
and 
point in the same direction, 
. Multiplying both sides of 
by 
, we have 
, which we substitute in eq65 (where 
is the spin analogue of eq61) to give 
.
Substituting eq164 and 
in 
yields
For a 1-electron atom, 
and so
Eq260 can be written in terms of the Larmor frequency of the electron. From eq149, 
. So, 
. Swapping 
with the Thomas precession rate 
, we obtain the correction term of 
.
The total spin-orbit Hamiltonian is
For a multi-electron atom,
