term symbol

In Physics, the term symbol is an abreviated description of the angular momentum quantum numbers in a multi-electron atom. It is related with the energy level of a given electron configuration.

The term symbol has the form

{}^{2S+1}\!L_{J}

where

S is the total spin quantum number;

2S+1 is the multiplicity, i.e., the number of different possible states of J for a given L—S combination;

L is the total orbital quantum number in spectroscopic notation,

and J is the total angular momentum quantum number.

The term symbols assumes LS coupling. The ground state term symbol is predicted by the Hund's rules.

Ground state term symbol

It is relatively easy to calculate the term symbol for the ground state of an atom. It corresponds with a state with maximal S and L.

Start with the most stable electron configuration. Full shells and subshells do not contribute to the overall angular momentum, so they are discarted.

If all shells and subshells are full then the term symbol is ${}^1\!S_0$ .

Distribute the electrons in the available orbitals, following Pauli exclusion principle. First, we fill the orbitals with highest m_l value with one electron each, and assign a maximal m_s to them (i.e. +1/2). Once all orbitals in a subshell have one electron, add a second one (following the same order), assigning m_s = −1/2 to them.
The overall S is calculated by adding the m_s values for each electron. That is the same as multiplying times the number of unpaired electrons. The overall L is calculated by adding the m_l vslue for each electron (so if there are two electrons in the same orbital, then we add twice that orbital's m_l).
Calculate J as:

if less than half of the subshell is ocupied, take the minimum value: $J = |L-S|$
if more than half-filled, take the maximum value $J = L + S$ .
if the subshell is half-filled, then L will be 0, so J = S.

As an example, in the fluorine case the electronic configuration is: 1s² 2s² 2p⁵. 1. Discart the full subshells and keep the 2p⁵ part. So we have five electrons to place in subshell p (l = 1). 2. There are three orbitals (m_l = 1, 0, −1) that can hold up to 2(2l+1) = 6 electrons. The first three electrons can take m_s = 1/2 (↑) but the Pauli exclusion principle force the next two to have m_s = −1/2 (↓) because they go to already ocupied orbitals

width=30px rowspan=2 \|	colspan=3 align=center \| m_l
align=center width=30px \| +1	align=center width=30px \| 0	align=center width=30px \| −1
m_s:	align=center \| ↑↓	align=center \| ↑↓	align=center \| ↑

3. S = 1/2 + 1/2 + 1/2 − 1/2 − 1/2 = 1/2; and L = 1 + 0 − 1 + 1 + 0 = 1, which is "P" in spectroscopic notation; 4. As fluorine 2p subshell is more than half filled, J = L + S = 3/2. Its ground state term symbol is then

{}^{2S+1}\!L_J = {}^2\!P_{3/2}

Term Symbol

Ground state term symbol

See also