Consider the coupling of two electrons. From eq205, we have
where and .
This implies that the state corresponding to is a coupled representation of and (orbitorbit coupling), and that the state corresponding to is a coupled representation of and (spinspin coupling). The overall state can be either a spinorbit coupled representation , or a spinorbit uncoupled representation . As mentioned in a previous article, if we neglect spinorbit coupling, we can construct atomic terms using the uncoupled representation.
If the two electrons are in the same open subshell, they are called equivalent electrons. An example of such a system is the carbon atom. Even though carbon has six electrons, four of them are in closed shells with zero angular momentum and are therefore ignored when determining atomic terms.
The first step is to tabulate all possible microstates of p^{2 }(arrangements of p^{2} electrons)^{ }that do not violate the Pauli exclusion principle:
Groups 

+1 
0 
1 

All up 
u 
u  1 
1 

u 
u  0  1  
u 
u  1 
1 

All down  d  d  1 
1 

d  d  0 
1 


d  d  1  1  
One up, one down 
ud 
2  0  

ud  0 
0 


ud  2 
0 

u  d  1 
0 

u  d  0 
0 


u  d  1 
0 

d 
u  1 
0 

d 
u  0 
0 


d  u  1 
0 
where u and d represent and respectively.
The above table is reorganised as:
+1 
0 
1 

+2 
1 


+1 
1  2 
1 

0 
1  3 
1 

1 
1  2 
1 

2 
1 

where each green number represents the number of microstates corresponding to each combination.
The only microstate with and when expressed in the uncoupled representation of is because . It must belong to the term ^{1}D, since ^{1}D is when and , with degenerate states of , , , and . For accounting purposes, we refresh the above table by removing these 5 states of ^{1}D, giving:
+1 
0 
1 

+2 

+1 
1  1 
1 

0 
1  2 
1 

1 
1  1 
1 

2 
Similarly, the only state with must be one of the 9 degenerate states of ^{3}P. Again, we refresh the above table by removing these 9 states of ^{3}P, leaving behind one state (), which corresponds to the term ^{1}S. Therefore, the atomic terms for the p^{2} configuration of carbon are ^{1}D, ^{3}P and ^{1}S.
Question
Why are the 9 degenerate states of^{ 3}P, combinations of and ?
Answer
Recall that states with same and same have the same energy and are grouped into a term. For the term ^{3}P, and with degeneracy 9 (^{3}P_{2} has 5, ^{3}P_{1} has 3 and ^{3}P_{0} has 1). For , we have three spincoupled basis vectors associated with the quantum numbers (formed by the coupling of and ). For , we have another three orbitcoupled basis vectors associated with the quantum numbers (formed by the coupling of and ). The total uncoupled microstates are the number of ways to form Kronecker products of basis vectors from the two vector spaces and hence combinations of and .
For the configuration 1s^{2}2s^{2}2p^{3}, e.g. nitrogen, we have
Groups  
+1  0  1  
All up  u  u  u  0  3/2 
All down  d  d  d  0  3/2 
One up two down  ud  d  2  1/2  
ud  d  1  1/2  
d  ud  1  1/2  
ud  d  1  1/2  
d  ud  1  1/2  
d  ud  2  1/2  
u  d  d  0  1/2  
d  u  d  0  1/2  
d  d  u  0  1/2  
Two up one down  ud  u  2  1/2  
ud  u  1  1/2  
u  ud  1  1/2  
ud  u  1  1/2  
u  ud  1  1/2  
u  ud  2  1/2  
u  u  d  0  1/2  
u  d  u  0  1/2  
d  u  u  0  1/2 
We can reorganise the above table as:
+3/2  1/2  1/2  3/2  
+2  1  1  
+1  2  2  
0  1  3  3  1  
1  2  2  
2  1  1 
The only state with must be one of the 4 degenerate states of ^{4}S. For accounting purposes, we retabulate the above table by removing these 4 states of ^{4}S, giving:
+3/2  1/2  1/2  3/2  
+2  1  1  
+1  2  2  
0  2  2  
1  2  2  
2  1  1 
A good guess for the next term is ^{2}D with a degeneracy of 10, because we are left with states of , spanning from +2 to 2. The remaining states are:
+3/2  1/2  1/2  3/2  
+2  
+1  1  1  
0  1  1  
1  1  1  
2 
which obviously belong to the term ^{2}P.
Therefore, the terms for the configuration of nitrogen are ^{4}S, ^{2}D, ^{2}P.
For a system with nonequivalent electrons (electrons in different subshells) in open shells, the way to construct atomic terms is similar to a system with equivalent electrons, except that we do not have to apply the Pauli exclusion principle in the construction process. The easiest method of construction is to use the ClebschGordan series by coupling the values to find out the possible values, and then coupling the values to give the possible values, and finally combining the possible and values to produce the atomic terms. For example, the terms for the 2p^{1}3p^{1} system are ^{3}D, ^{1}D, ^{3}P, ^{1}P, ^{3}S and ^{1}S.