Atomic Term Symbols

Assigning Term Symbols The ground state of hydrogen atom is one electron in the lowest energy atomic orbital: the 1s. Therefore the total orbital angular momentum off all (one) electrons is L=l=0, and the total electron spin is S=s=1/2.

Applying the 'definition' of the term symbol:

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 2S1/2 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 2D3/2 and 2D5/2 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.

To determine the states(terms) of a given atom or ion:
  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 ML and MS
  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.

Summary Examples
ground state term symbol is 2S1/2

Ground state term symbol is 1S0

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

Term symbol: 2P1/2, 2P3/2
The spin orbit splitting is regular so 2P1/2. is the ground state.

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 ML and MS

Decompose this table into terms

Check this with reality

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

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

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

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