Applying the 'definition' of the term symbol:

^2S+1L_J

Note: In the symbol L must be replaced with its alphabetic 'code': L=0 is S, L=1 is P, L=2 is D, L=3 is F, L=4 is G, L=5 is H...

J is the vector sum of L and S: J = L+S, L+S-1, L+S-2, ....L-S

results in a ²S_1/2 atomic term for the ground state of the H atom.

Exciting the single hydrogenic electron to higher orbitals results in different atomic states or 'terms' of the atom. Note that an H atom with the electron in a 3d and 10d orbital both result in ²D_3/2 and ²D_5/2 terms, but at different energies.

The lowest electron configuration of He is 1s². The ground state of the neutral helium atom is therefore ¹S₀
In fact, any electron configuration (orbital population) that consists of any combination of closed shells or subshells will result in this (totally symmetric ¹S₀ term.
Therefore, in the designation of atomic terms, the contribution from closed subshell electrons may be neglected.

1. Write down the electronic configuration (ignore closed subshell electrons)
2. Determine the number of distinct microstates that can represent that configuration. If you have e electrons in a single open subshell of 2l+1 orbitals, this value is
   #microstates = (2(2l+1))!/e!(2(2l+1)-e)!
3. Tabulate the number of microstates that have a given M_L and M_S
4. Decompose your table into terms by elimination
5. Test the total degeneracy of the resultant terms to account for all the microstates counted in parts 2 and 3
6. Determine the lowest term for the configuration by Hunds Rules.

He(1s²)
Ground state term symbol is ¹S₀
 

He(1s¹2s¹) An Excited State Configuration
Terms: ¹S₀  , ³S₁
{ There is no     ³S₀     Nor    ³S_-1    Term }

B(1s²2s²2p¹)
Term symbol: ²P_1/2, ²P_{3/2
The spin orbit splitting is regular so ²P1/2. is the ground state.}
 
 

C(1s²2s²2p²)
Calculate the number of possible electron arrangements in the given configuration

There are 6!/4!2! = 15 microstates expected

Write down all these possibilities

Tabulate the total numbers by M_L and M_S

Decompose this table into terms

Check this with reality

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What are the atomic state term symbols resulting from the lowest energy configuration of N (1s² 2s² 2p³) ?
This configuration has 6!/3!3! = 20 microstates

Draw all the possibilities

 
 

Tabulate the totals

Assign Terms

 
 
  Check this with reality

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What are the atomic state term symbols resulting from the lowest energy configuration of O (1s² 2s² 2p⁴) ?

Aha! We don't haver to do this because the terms arising from p² and p⁴ are exactly the same.
Note however the ordering of the J levels is now inverted
 
 
  Check this with reality

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If you think you have mastered the process of determining the states arising from a given electronic configuration, you should try Ti(1s²2s²2p⁶3s²3p⁶4s²3d²)
Here is the answer depicted graphically
 
 
  Check this with reality

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The terms resulting from a single open subshell are tabulated below for your amusement.

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