# Term symbol

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In atomic spectroscopy, a term symbol gives 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:



Here:

• The symbol S is the total spin angular momentum of the state and 2S+1 is the spin multiplicity.
• 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):

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: JL + S. The value J satisfies the triangular conditions:
.

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



The (2S+1)(2L+1) different functions referred to by this 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



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: . Spin angular momentum: S = 1/2. Orbital angular momentum: L = 0. Spin-orbital angular momentum: J = 1/2. Parity: even.
• Carbon atom: . Spin angular momentum: S = 1. Orbital angular momentum: L = 1. Spin-orbital angular momentum: J = 0. Parity even.
• Aluminium atom: . Spin angular momentum: S = 1/2. Orbital angular momentum: L = 1. Spin-orbital angular momentum: J = 1/2. Parity odd.
• Scandium atom: . Spin angular momentum: S = 1/2. Orbital angular momentum: L = 2. Spin-orbital angular momentum: J = 3/2. Parity even.