For a one$-$electron atom, the contribution of the spin$-$orbit interaction to the energy of the

state labeled by $n,l,$ and $j$ is given by

$E^{SO}(nlj)=\frac{1}{2}[j(j+1)-l(l+1)-s(s+1)]{}_{nl}(Z),$

where $s=\frac{1}{2}$ because we have only one electron and ${}_{nl}(Z)$ is given by

${}_{nl}(Z)=\frac{Z^4(g_e-1)}{2n^{3}l(l+\frac{1}{2})(l+1)}\frac{e^2{}^2}{4{}_0{m}_e^2{c}^2{a}_0^3}$

For a multielectron atom in a given configuration $($which I'll label by c$),$ and term

$($identified by $L$ and $S),$ the spin$-$orbit contribution to the energy for the level $J$ is given in

first$-$order perturbation theory as

$E^{SO}(c,L,S,J)=\frac{1}{2}[J(J+1)-L(L+1)-S(S+1)](c,L,S).$

Here, $(c,L,S)$ is a zeta$-$parameter that is specific to the overall configuration and term, and

is a combination of the ${}_{nl}$ values for the partially occupied orbitals. As an example, for an

$s^1{p}^2$ configuration $($as occurs in the excited $1s^{2}2s^{2}2p^{6}3s^{1}3p^2$ configuration of aluminum$),$

the terms that result are ${?}^2{S}_g,{?}^2{P}_g,{?}^2{D}_g,$ and ${?}^4{P}_g.$ The $(c,L,S)$ values for these terms are

simple multiples of the orbital parameter, ${}_{3p}(Al),$ which has the approximate value of

$75c{m}^{-1}.$ Specifically, for the $s^1{p}^2$ configuration, $(c,L,S)$ takes the values $(s^1{p}^2,{?}^{2}S)=0,$

$(s^1{p}^2,{?}^{2}P)=\frac{2}{3}{}_{np},(s^1{p}^2,{?}^{2}D)=0,$ and $(s^1{p}^2,{?}^{4}P)=\frac{1}{3}{}_{np}.$

Now finally, a very simple question:

From the formula for $E^{SO}(c,L,S,J),$ show that the energy difference $E(L,S,J)-E(L,S,J-1)$

is given by

$E(L,S,J)-E(L,S,J-1)=J(c,L,S).$

This expression is called the Land$$ interval rule.