---
title: LS and jj Coupling; Term Symbols
module: Many-Electron Atoms
moduleNumber: 5
lessonNumber: 5
order: 505
summary: >
  A configuration is not a single energy level. The residual electrostatic
  repulsion and the spin-orbit interaction split it, and which one dominates fixes
  the coupling scheme. In light atoms the electrostatic term wins: orbital and spin
  angular momenta couple separately into L and S, then into J, giving Russell-
  Saunders term symbols. In heavy atoms spin-orbit wins and each electron's j
  forms first. The Pauli principle prunes the allowed terms of equivalent
  electrons, the Landé interval rule spaces the fine-structure multiplet, and the
  scheme crosses over from LS to jj down a column.
topics: [Many-Electron Atoms]
sources:
  - book: Foot
    ref: "Ch. 5 — The LS-Coupling Scheme; §5.5 Coupling Schemes and Term Symbols"
  - book: Bransden & Joachain
    ref: "Ch. 7 — Many-Electron Atoms; §7.4–7.5 LS and jj Coupling"
  - book: Demtröder
    ref: "Ch. 6 — Atoms with Many Electrons; Coupling of Angular Momenta"
draft: false
---

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](/atomic-physics/many-electron-atoms/central-field-self-consistent) 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:[^foot-55]

$$
H_1 = \underbrace{\sum_{i<j}\frac{e^2}{4\pi\varepsilon_0\,r_{ij}} - \sum_i S(r_i)}_{H_{\text{es}}}
\;+\; \underbrace{\sum_i \xi(r_i)\,\vec\ell_i\cdot\vec s_i}_{H_{\text{so}}},
$$

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 $\xi \propto \langle 1/r^3\rangle$ 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

$$
\vec L = \sum_i \vec\ell_i,
\qquad
\vec S = \sum_i \vec s_i
$$

are separately conserved, and the weaker spin-orbit term then couples them into
the grand total $\vec J = \vec L + \vec S$.[^foot-55] The good quantum numbers are
$L$, $S$, $J$, and $M_J$, and the state is labelled by a **term symbol**

$$
^{2S+1}L_J.
$$

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

$$
% caption: Anatomy of a term symbol. The left superscript is the spin
% multiplicity 2S+1, the letter names the total orbital number L, and the right
% subscript is the total angular momentum J.
\begin{tikzpicture}[scale=1.0, font=\footnotesize, >=stealth]
\definecolor{acc}{HTML}{4A6FA5}
\node[black!85, scale=2.4] at (2.4,1.4) {$^{3}\!P_{2}$};
\draw[black, ->] (1.55,2.7) -- (1.95,1.95);
\node[black, anchor=south] at (1.4,2.7) {spin $(2S+1)$};
\draw[black, ->] (2.4,0.1) -- (2.4,0.75);
\node[black, anchor=north] at (2.4,0.05) {orbital $L$ (code letter)};
\draw[black, ->] (3.35,2.5) -- (2.95,1.7);
\node[black, anchor=south] at (3.7,2.5) {total $J$};
\end{tikzpicture}
$$

$$
% caption: LS vs jj coupling. In LS the orbital momenta add to L and spins to S,
% then L and S add to J. In jj each electron's l and s add to j first, then the j
% add to J.
\begin{tikzpicture}[scale=1.0, font=\footnotesize, >=stealth,
  nd/.style={draw, minimum width=8mm, minimum height=6mm, inner sep=1pt}]
\definecolor{acc}{HTML}{4A6FA5}
% --- LS scheme ---
\node[black!70] at (1.9,3.5) {LS coupling};
\node[nd] (l1) at (0.4,2.6) {$l_1$};
\node[nd] (l2) at (1.6,2.6) {$l_2$};
\node[nd] (s1) at (2.8,2.6) {$s_1$};
\node[nd] (s2) at (4.0,2.6) {$s_2$};
\node[nd] (L) at (1.0,1.3) {$L$};
\node[nd] (S) at (3.4,1.3) {$S$};
\node[nd, thick] (J) at (2.2,0.2) {$J$};
\draw[->] (l1) -- (L); \draw[->] (l2) -- (L);
\draw[->] (s1) -- (S); \draw[->] (s2) -- (S);
\draw[->] (L) -- (J); \draw[->] (S) -- (J);
% --- jj scheme ---
\begin{scope}[xshift=6.4cm]
\node[black!70] at (1.9,3.5) {jj coupling};
\node[nd] (a1) at (0.4,2.6) {$l_1$};
\node[nd] (b1) at (1.6,2.6) {$s_1$};
\node[nd] (a2) at (2.8,2.6) {$l_2$};
\node[nd] (b2) at (4.0,2.6) {$s_2$};
\node[nd] (j1) at (1.0,1.3) {$j_1$};
\node[nd] (j2) at (3.4,1.3) {$j_2$};
\node[nd, thick] (Jj) at (2.2,0.2) {$J$};
\draw[->] (a1) -- (j1); \draw[->] (b1) -- (j1);
\draw[->] (a2) -- (j2); \draw[->] (b2) -- (j2);
\draw[->] (j1) -- (Jj); \draw[->] (j2) -- (Jj);
\end{scope}
\end{tikzpicture}
$$

## 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,

$$
\vec j_i = \vec\ell_i + \vec s_i,
$$

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 $(j_1, j_2)_J$.[^bj-74] 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.[^foot-55]

A $p$ electron has $m_\ell \in \{-1,0,1\}$ and $m_s \in \{\pm\tfrac12\}$, six
single-electron states. Two equivalent $p$ electrons occupy two of these six with
no repeats and no ordering, giving

$$
\binom{6}{2} = 15
$$

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:

$$
p^2:\quad {}^1S,\ {}^1D,\ {}^3P,
$$

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 $^3S$, $^3D$,
and $^1P$ that non-equivalent electrons would allow are absent: they require two
electrons in the same spatial-spin state and violate exclusion.

$$
% caption: The 15 microstates of p² arranged by M_L (horizontal) and M_S
% (vertical), partitioned into the three Pauli-allowed terms 1-S, 1-D, and 3-P.
\begin{tikzpicture}[scale=1.0, font=\footnotesize, >=stealth]
\definecolor{acc}{HTML}{4A6FA5}
% axes
\draw[->, black] (-2.6,0) -- (2.8,0) node[right, black!70] {$M_L$};
\draw[->, black] (0,-1.7) -- (0,1.9) node[above, black!70] {$M_S$};
\foreach \x in {-2,-1,0,1,2}{\node[black, anchor=north, scale=0.8] at (\x*0.9,-0.05) {\x};}
% 3P block: M_L in -1..1, M_S in -1..1 (9 dots)
\foreach \x in {-1,0,1}{\foreach \y in {-1,0,1}{\fill[acc] (\x*0.9,\y*0.7) circle (2.4pt);}}
% 1D block: M_L in -2..2, M_S=0 (5 dots) — offset slightly to avoid 3P overlap at MS=0
\foreach \x in {-2,-1,0,1,2}{\fill[black] (\x*0.9,0) circle (1.3pt);}
% 1S: M_L=0, M_S=0 (1 dot)
\fill[black!80] (0,0) circle (0.7pt);
\node[acc, anchor=west] at (1.6,1.35) {$^3P$ (9)};
\node[black, anchor=west] at (1.6,0.55) {$^1D$ (5)};
\node[black!80, anchor=west] at (1.6,-0.25) {$^1S$ (1)};
\end{tikzpicture}
$$

The same enumeration handles any shell, and a filled subshell always gives a
single $^1S_0$ 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

$$
J = |L-S|,\ |L-S|+1,\ \ldots,\ L+S.
$$

The $^3P$ term ($L=1$, $S=1$) splits into $^3P_0$, $^3P_1$, $^3P_2$. Treating
$H_{\text{so}} = A\,\vec L\cdot\vec S$ as a perturbation within the term, and using
$\vec L\cdot\vec S = \tfrac12[J(J+1)-L(L+1)-S(S+1)]$, the level energies are

$$
E_J = \frac{A}{2}\big[J(J+1)-L(L+1)-S(S+1)\big].
$$

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

$$
E_J - E_{J-1} = A\,J,
$$

the **Landé interval rule**.[^foot-55] For $^3P$ the gaps $^3P_2\!-\!{}^3P_1$ and
$^3P_1\!-\!{}^3P_0$ stand in the ratio $2:1$. Measuring the ratio tests whether a
multiplet is well described by LS coupling; carbon's ground $^3P$ 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.

$$
% caption: The 3-P fine-structure multiplet under the Landé interval rule. The
% spacing E(J) − E(J−1) grows as J, so the 2-to-1 gap sits above the 1-to-0 gap
% in the ratio 2:1.
\begin{tikzpicture}[scale=1.0, font=\footnotesize, >=stealth]
\definecolor{acc}{HTML}{4A6FA5}
% unsplit term
\draw[black, dashed] (0,1.4) -- (1.4,1.4);
\node[black, anchor=east] at (0,1.4) {$^3P$};
% split levels: J=0 lowest, then +A, then +2A above
\draw[black, thick] (2.4,0.4) -- (4.0,0.4); \node[black, anchor=west] at (4.05,0.4) {$^3P_0$};
\draw[black, thick] (2.4,1.4) -- (4.0,1.4); \node[black, anchor=west] at (4.05,1.4) {$^3P_1$};
\draw[black, thick] (2.4,3.4) -- (4.0,3.4); \node[black, anchor=west] at (4.05,3.4) {$^3P_2$};
% interval markers
\draw[acc, <->] (2.9,0.45) -- (2.9,1.35); \node[acc, anchor=east] at (2.85,0.9) {$A$};
\draw[acc, <->] (2.9,1.45) -- (2.9,3.35); \node[acc, anchor=east] at (2.85,2.4) {$2A$};
\draw[black] (1.4,1.4) -- (2.4,0.4);
\draw[black] (1.4,1.4) -- (2.4,3.4);
\end{tikzpicture}
$$

## From LS to jj down a column

The carbon group — C, Si, Ge, Sn, Pb — has the same $np^2$ valence configuration
at every step, so the terms are the same $^1S_0$, $^1D_2$, $^3P_{0,1,2}$
throughout. What changes is the coupling. In carbon the spin-orbit splitting of
$^3P$ 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.[^bj-74]

$$
% caption: LS-to-jj correlation for np². The five levels keep their J values (0,
% 1, 2, 2, 0) across the diagram; on the left they group by LS term, on the right
% by the pair (j1, j2).
\begin{tikzpicture}[scale=1.0, font=\footnotesize, >=stealth]
\definecolor{acc}{HTML}{4A6FA5}
% left (LS) labels
\node[black, anchor=east] at (0.2,0.4) {$^3P_0$};
\node[black, anchor=east] at (0.2,1.1) {$^3P_1$};
\node[black, anchor=east] at (0.2,1.8) {$^3P_2$};
\node[black, anchor=east] at (0.2,2.9) {$^1D_2$};
\node[black, anchor=east] at (0.2,3.8) {$^1S_0$};
% right (jj) labels
\node[black, anchor=west] at (6.4,0.4) {$(\tfrac12,\tfrac12)_0$};
\node[black, anchor=west] at (6.4,1.1) {$(\tfrac12,\tfrac32)_1$};
\node[black, anchor=west] at (6.4,1.9) {$(\tfrac12,\tfrac32)_2$};
\node[black, anchor=west] at (6.4,3.0) {$(\tfrac32,\tfrac32)_2$};
\node[black, anchor=west] at (6.4,3.8) {$(\tfrac32,\tfrac32)_0$};
% connecting correlation lines (J conserved)
\draw[black] (0.6,0.4) -- (6.2,0.4);
\draw[black] (0.6,1.1) -- (6.2,1.1);
\draw[black] (0.6,1.8) -- (6.2,1.9);
\draw[black] (0.6,2.9) -- (6.2,3.0);
\draw[black] (0.6,3.8) -- (6.2,3.8);
\node[black, anchor=south] at (1.4,4.1) {light (LS)};
\node[black, anchor=south] at (5.4,4.1) {heavy (jj)};
\end{tikzpicture}
$$

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](/atomic-physics/many-electron-atoms/hund-rules-ground-terms), the
subject of the next lesson.

[^foot-55]: **Foot**, _Atomic Physics_, §5.5 — the LS-coupling scheme, the term-symbol notation $^{2S+1}L_J$, microstate counting for equivalent electrons giving the $p^2$ terms $^1S,{}^1D,{}^3P$, and the Landé interval rule $E_J-E_{J-1}=AJ$. <https://global.oup.com/academic/product/atomic-physics-9780198506959>
[^bj-74]: **Bransden & Joachain**, _Physics of Atoms and Molecules_, 2nd ed., §7.4–7.5 — the relative $Z$-scaling of the electrostatic and spin-orbit interactions, jj coupling and the $(j_1,j_2)_J$ labels, and the LS-to-jj correlation with $J$ conserved along a column. <https://www.pearson.com/en-gb/subject-catalog/p/physics-of-atoms-and-molecules/P200000005386>