Constructing atomic terms (coupled representation)

The construction of term symbols in the spin-orbit coupled representation (i.e. levels) is an extension of the procedure for deriving term symbols in the spin-orbit uncoupled representation (i.e. terms).

Russell-Saunders (LS) coupling

For the p² configuration of the carbon atom, we have shown in the previous article that the terms are ¹D, ³P and ¹S with degeneracies $(2L+1)(2S+1)$ of 5, 9 and 1 respectively. The total degeneracy of 15 corresponds to 15 microstates in the uncoupled representation, which are the basis states for the coupled representation. Since the total number of microstates describing a system is independent of the chosen representation, we have 15 coupled states after linearly combining the uncoupled basis states. The procedure to construct levels is to use the Clebsch-Gordan series, which describes the allow values of the total angular momentum number $J$ for a given value of $L$ and a given value of $S$ , while ensuring that the total number of microstates (degeneracy) of each term (e.g. ³P) is equal to the total number of microstates of the associated levels (e.g. ³P_2,³P₁, ³P₀), i.e. $(2L+1)(2S+1)=\sum_{allowed\: J}(2J+1)_{allowed\: J}$ .

Term	Degeneracy	$\boldsymbol{\mathit{J}}=\boldsymbol{\mathit{L}}+\boldsymbol{\mathit{S}}$	Level	Degeneracy
¹D	5	2 = 2 + 0	¹D₂	5
³P	9	2 = 1 + 1	³P₂	5
			³P₁	3
			³P₀	1
¹S	1	0 = 0 + 0	¹S₀	1

For the p³ configuration of nitrogen, the terms are ⁴S, ²D, ²P and the levels are:

Term	Degeneracy	$\boldsymbol{\mathit{J}}=\boldsymbol{\mathit{L}}+\boldsymbol{\mathit{S}}$	Level	Degeneracy
⁴S	4	3/2 = 0 + 3/2	⁴S_3/2	4
²D	10	5/2 = 2 + 1/2	²D_5/2	6
²D	10	5/2 = 2 + 1/2	²D_3/2	4
²P	6	3/2 = 1 + 1/2	²P_3/2	4
²P	6	3/2 = 1 + 1/2	²P_1/2	2

You’ll notice that the degeneracies for both carbon and nitrogen are not fully lifted (i.e. each level may contain more than one state) when spin-orbit interactions are considered. Other effects, like an external magnetic field, are required to completed lift the degeneracy of an atom.

jj-coupling

In jj-coupling, each electron of the p² configuration has $l=1$ and $s=\frac{1}{2}$ , which combine to give $j=\frac{3}{2}$ or $j=\frac{1}{2}$ . For each value of $j$ , there are $2j+1$ allowed $m_j$ values (see this article for details). Thus, each electron is described by the one-electron wavefunctions $\vert j=\frac{3}{2},m_j=\frac{3}{2}\rangle$ , $\vert j=\frac{3}{2},m_j=\frac{1}{2}\rangle$ , $\vert j=\frac{3}{2},m_j=-\frac{1}{2}\rangle$ , $\vert j=\frac{3}{2},m_j=-\frac{3}{2}\rangle$ , $\vert j=\frac{1}{2},m_j=\frac{1}{2}\rangle$ and $\vert j=\frac{1}{2},m_j=-\frac{1}{2}\rangle$ , giving a total of six one-electron states.

According to the Pauli exclusion principle, no two electrons can can be described by the same set of quantum numbers. Therefore, two-electron wavefunctions like $\vert J,M_J\rangle=\vert j_1=\frac{3}{2},m_{j1}=\frac{3}{2}\rangle\otimes\vert j_2=\frac{3}{2},m_{j2}=\frac{3}{2}\rangle$ are forbidden.

The 15 Pauli-allowed microstates can be grouped as follows:

For the $\vert j_1=\frac{3}{2}\rangle\otimes\vert j_2=\frac{3}{2}\rangle$ group, the microstates with $M_J$ spanning from +2 to -2 form the term $\biggr$\frac{3}{2},\frac{3}{2}\biggr$_2$ , with the remaining microstate with $M_J=0$ belonging to $\biggr$\frac{3}{2},\frac{3}{2}\biggr$_0$ . Similarly, for the $\vert j_1=\frac{1}{2}\rangle\otimes\vert j_2=\frac{3}{2}\rangle$ group, the microstates correspond to the terms $\biggr$\frac{1}{2},\frac{3}{2}\biggr$_2$ and $\biggr$\frac{1}{2},\frac{3}{2}\biggr$_1$ . Therefore, the jj-coupling term symbols for the p²configuration are $\biggr$\frac{1}{2},\frac{1}{2}\biggr$_0$ , $\biggr$\frac{1}{2},\frac{3}{2}\biggr$_1$ , $\biggr$\frac{1}{2},\frac{3}{2}\biggr$_2$ , $\biggr$\frac{3}{2},\frac{3}{2}\biggr$_0$ and $\biggr$\frac{3}{2},\frac{3}{2}\biggr$_2$ .