# Term symbol

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 **L**^{2} and **S**^{2} 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 2*S*+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:**J**≡**L**+**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 (2*S*+1)(2*L*+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 2*J*+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 2*S*+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.