The Spin-orbit coupling hamiltonian
The magnitude of spin-orbit coupling splitting is measured spectroscopically as
The expression can be modified by realizing that j = l + s.
where A is the magnitude of the spin-orbit coupling in wave numbers. The magnitude of the spin orbit coupling can be calculated in terms of molecule parameters by the substitution
where
a is the fine structure constant (a = 1/137.037) and the carrots indicate that L and S are operators. The fine structure constant is a dimensionless constant, a = e2/ác. Z is an effective atomic number. The spin orbit coupling splitting can be calculated from
This expression can be recast to give an spin-orbit coupling energy in terms of molecular parameters
where
We can evaluate this integral explicity for a given atomic orbital.
For example for
Y210 we have
so that the integral is
which integrates to
Or Z3/24 in atomic units.
Therefore in atomic units we have
Therefore, in general the spin-orbit splitting is given by
Note that the spin-orbit coupling increases as the fourth power of the effective nuclear charge Z, but only as the third power of the principal quantum number n. This indicates that spin orbit-coupling interactions are significantly larger for atoms that are further down a particular column of the periodic table.