LS and jj Coupling; Term Symbols

The central field assigns each electron a configuration such as $1s^{2}2s^{2}2p^2$, but a configuration is a set of states, not one level. Two interactions left over from the central-field split resolve it: the non-central part of the electron-electron repulsion, and the spin-orbit coupling. Their relative size decides how the individual angular momenta combine, and the combination is recorded in a term symbol. This lesson builds the term symbols, counts which ones a configuration allows, and follows the scheme from light atoms to heavy.

The two residual interactions

Write the leftover Hamiltonian as the sum of a residual electrostatic part and a spin-orbit part:¹

where $S(r)$ is the spherical part of the repulsion already absorbed into the central field, so $H_{\text{es}}$ is the anisotropic remainder. The two terms scale oppositely with atomic number:

• $H_{\text{es}}$ is a Coulomb energy, weakly dependent on $Z$; it dominates in light atoms.
• $H_{\text{so}}$ grows steeply, roughly as $Z^4$ for the relevant inner region, because the spin-orbit coefficient $\xi \propto ⟨1/r^3⟩$ and the field an inner electron orbits through both climb with nuclear charge.

Whichever is larger is diagonalized first, and its conserved quantities become the good quantum numbers. Two limiting schemes result.

Russell-Saunders (LS) coupling

When $H_{\text{es}}\gg H_{\text{so}}$ — the light-atom limit — the electrostatic interaction couples the orbital angular momenta among themselves and the spins among themselves. The total orbital and total spin

are separately conserved, and the weaker spin-orbit term then couples them into the grand total $\vec{J}=\vec{L}+\vec{S}$.¹ The good quantum numbers are $L$, $S$, $J$, and $M_J$, and the state is labelled by a term symbol

The superscript $2S+1$ is the multiplicity (the number of $J$ values when $L\ge S$); the letter encodes $L$ by the spectroscopic code $S,P,D,F,G,\dots$ for $L=0,1,2,3,4,\dots$; the subscript is $J$.

jj coupling

When $H_{\text{so}}\gg H_{\text{es}}$ — the heavy-atom limit — the spin-orbit interaction couples each electron's own orbital and spin first,

and only then does the weak residual electrostatic term couple the individual $j_i$ into $\vec{J}={\sum }_i{\vec{j}}_i$. Now $L$ and $S$ are not good quantum numbers; the level is labelled by the pair $(j_1,j_2{)}_J$.² Pure jj coupling is rarely reached even in the heaviest atoms, but the tendency is unmistakable down any column of the periodic table, and the intermediate regime interpolates between the two limits by diagonalizing $H_{\text{es}}+H_{\text{so}}$ together.

Allowed terms of a configuration

Angular-momentum addition alone would let a $p^2$ configuration form every product of $L$ and $S$ values. For equivalent electrons — same $n$ and $\ell$ — the Pauli principle forbids most of them. The clean way to count is to enumerate microstates.¹

A $p$ electron has $m_{\ell }\in \{-1,0,1\}$ and $m_s\in \{\pm \frac{1}{2}\}$, six single-electron states. Two equivalent $p$ electrons occupy two of these six with no repeats and no ordering, giving

microstates. Each microstate has definite $M_L=\sum m_{\ell }$ and $M_S=\sum m_s$. Tabulating them and peeling off complete $(L,S)$ blocks leaves exactly three terms:

with state counts $(2\cdot 0+1)(2\cdot 0+1)=1$, $(2\cdot 2+1)(1)=5$, and $(2\cdot 1+1)(2\cdot 1+1)=9$, summing to $1+5+9=15$. The terms ${}^{3}S$, ${}^{3}D$, and ${}^{1}P$ that non-equivalent electrons would allow are absent: they require two electrons in the same spatial-spin state and violate exclusion.

The same enumeration handles any shell, and a filled subshell always gives a single ${}^1{S}_0$ term (all $M_L$ and $M_S$ sum to zero), so only the open subshells matter. A shell and its complement — $p^2$ and $p^4$, say — yield the identical set of terms, since removing electrons from a full shell leaves holes that couple like electrons.

The fine-structure multiplet and the Landé interval rule

Within a term of given $L$ and $S$, spin-orbit coupling splits the levels by $J$, which runs over

The ${}^{3}P$ term ($L=1$, $S=1$) splits into ${}^3{P}_0$, ${}^3{P}_1$, ${}^3{P}_2$. Treating $H_{\text{so}}=A\vec{L}\cdot \vec{S}$ as a perturbation within the term, and using $\vec{L}\cdot \vec{S}=\frac{1}{2}[J(J+1)-L(L+1)-S(S+1)]$, the level energies are

The spacing between adjacent levels is then proportional to the upper $J$:

the Landé interval rule.¹ For ${}^{3}P$ the gaps ${}^3{P}_2-{}^3{P}_1$ and ${}^3{P}_1-{}^3{P}_0$ stand in the ratio $2:1$. Measuring the ratio tests whether a multiplet is well described by LS coupling; carbon's ground ${}^{3}P$ shows gaps of about $27.0$ and $16.4{\text{cm}}^{-1}$, a ratio near $1.6$ — close to $2:1$, with the deviation flagging the onset of jj mixing.

From LS to jj down a column

The carbon group — C, Si, Ge, Sn, Pb — has the same $n{p}^2$ valence configuration at every step, so the terms are the same ${}^1{S}_0$, ${}^1{D}_2$, ${}^3{P}_{0,1,2}$ throughout. What changes is the coupling. In carbon the spin-orbit splitting of ${}^{3}P$ is a few tens of ${\text{cm}}^{-1}$, far below the $\sim 10^4{\text{cm}}^{-1}$ gaps between terms, so LS is excellent. By lead the spin-orbit energy has grown into the thousands of ${\text{cm}}^{-1}$ and rivals the term separations; the levels regroup according to $(j_1,j_2{)}_J$, and $L$ and $S$ lose their meaning as labels. The level count is conserved across the crossover — the same $J$ values appear at both ends — but the groupings and the labels change.²

The term symbol is the compact record of all of this: multiplicity for the spin, a letter for the orbital, a subscript for the total. What it does not tell you is which level lies lowest — that is fixed by Hund's rules, the subject of the next lesson.