= Spin-orbit splitting of the first excited two-electron oscillator level
{title2=$\Delta E=\lambda\hbar^2[J(J+1)-L(L+1)-S(S+1)]/2$}
For the interaction $\lambda\mathbf S\cdot\mathbf L$ built from the total two-electron <spin> and <orbital angular momentum>, set $\mathbf J=\mathbf L+\mathbf S$. The <spin-dot-product eigenvalue> identity then gives the shift in the title. In the first excited two-electron oscillator manifold, the singlet has $L=1,S=0,J=1$ and zero shift with multiplicity three. The triplet has $L=S=1$, with $J=0,1,2$ shifts $-2\lambda\hbar^2,-\lambda\hbar^2,+\lambda\hbar^2$ and multiplicities $1,3,5$.
The multiplicities sum to twelve, and their weighted shift sums to zero. The specified interaction preserves both $L^2$ and $S^2$; it is distinct from summing separate single-electron spin-orbit interactions. Its shifts are exact within this model's invariant oscillator manifold, with weak coupling ensuring the original level ordering remains useful.
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