Term symbol

In atomic spectroscopy, a term symbol indicates the total spin-, orbital-, and spin-orbital angular momentum of an atom in a certain quantum state (often the ground state). The simultaneous eigenfunctions of L2 and S2 labeled by a term symbol are obtained in the Russell-Saunders coupling (also known as LS coupling) scheme.

A term symbol has the following form:

^{2S+1}\!L_{J} .\; $$

Here:
 * The symbol S is the total spin angular momentum of the state and 2S+1 is the spin multiplicity (number of linearly independent states with S).


 * The symbol L represents the total orbital angular momentum of the state. For historical reasons L is coded by a letter as follows (between brackets the L quantum number):

S(0), \; P(1),\; D(2),\; F(3),\; G(4),\; H(5),\; I(6),\; K(7), \dots, $$
 * and further up the alphabet (excluding P and S).


 * The subscript J in the term symbol is the quantum number of the spin-orbital angular momentum: J &equiv; L + S. The value J satisfies the triangular conditions:

J = |L-S|,\, |L-S|+1, \, \ldots,\, L+S-1, \, L+S,\quad\hbox{that is,}\quad |L-S| \le J\le L+S. $$.

A term symbol is often preceded by the electronic configuration that leads to the L-S coupled functions, thus, for example,

(ns)^k \, (n'p)^{k'}\, (nd)^{k}\,\,\, ^{2S+1}L , $$ which indicates an electron configuration with k electrons occupying an ns orbital (0 &le; k &le; 2), k&prime; electrons occupying an n&prime;p orbital (0 &le; k&prime; &le; 6), and k&prime;&prime; electrons occupying an n&prime;&prime;d orbital (0 &le; k&prime;&prime; &le; 10). The k+k&prime;+k&prime;&prime; electrons are coupled to a spin state of quantum number S that has multiplicity 2S+1; and the electrons are coupled to an orbital state characterized by the letter L (where the letter is in one-correspondence with the quantum number L) that has multiplicity 2L+1. The (2S+1)(2L+1) different functions referred to by a term symbol form a term. When the quantum number J is added (as a subscript) the symbol refers to an energy level, comprising 2J+1 components.

Sometimes the parity of the state is added, as in

^{2S+1}L_{J}^o, \, $$ which indicates that the state has odd parity. This is the case when the sum of the one-electron orbital angular momentum numbers in the electronic configuration is odd.

For historical reasons, the term symbol is somewhat inconsistent in the sense that the quantum numbers L and J are indicated directly, by a letter and a number, respectively, while the spin S is indicated by its multiplicity 2S+1.

Examples
A few ground state atoms are listed.
 * Hydrogen atom: $$\scriptstyle 1s\,\,\, ^2S_{\frac{1}{2}}$$. Spin angular momentum: S = 1/2. Orbital angular momentum: L = 0. Spin-orbital angular momentum: J = 1/2. Parity: even.


 * Carbon atom: $$\scriptstyle (1s)^2\,(2s)^2\, (2p)^2\,\,\, ^3P_{0}\,$$. Spin angular momentum: S = 1. Orbital angular momentum: L = 1. Spin-orbital angular momentum: J = 0. Parity even.


 * Aluminium atom: $$\scriptstyle (1s)^2\,(2s)^2\,(2p)^6\,(3s)^2\,3p\,\,\, ^2P_{\frac{1}{2}}^o\,$$. Spin angular momentum: S = 1/2. Orbital angular momentum: L = 1. Spin-orbital angular momentum: J = 1/2. Parity odd.


 * Scandium atom: $$\scriptstyle (1s)^2\,(2s)^2\,(2p)^6\,(3s)^2\, (3p)^6\, 3d\, (4s)^2 \,\,\, ^2D_{\frac{3}{2}}\,$$. Spin angular momentum: S = 1/2. Orbital angular momentum: L = 2. Spin-orbital angular momentum: J = 3/2. Parity even.