A Term Symbol is a shorthand notation that describes the electron distribution in atoms or ions, i.e. the ml and ms quantum numbers.
Spectroscopists most frequently use these.
Finding Term Symbols:
1. Ignore closed shells.
2. Find the maximum possible orbital angular momentum L = Σml for the given electron configuration. Then the possible angular momenta are ML = -L, -L+1, -L+2, ..., L-2, L-2, L
3. Find the maximum possible spin angular momentum S = = Σms for the given electron configuration. The possible angular momenta are MS = -S, -S+1, -S+2, ..., S-2, S-1, S
4. Build a matrix for assignment of microstates that is (2ML+1) rows by (2MS+1) columns.
This will be used to assign each microstate to its appropriate total orbital and spin angular momentum state
5. Fill each entry in the matrix with all appropriate microstates, eliminating Pauli forbidden states, for each ML and MS.
6. Check the totals.
a) pure l state: the number of microstates = Nl!/x!(Nl – x)!
Nl = 2(2l+1)
x = number of electrons
b) mixed l states: the number of microstates = ΠNl where Nl is found for each l state as in a)
7. Start in the upper left of the matrix and work down until the first microstate is encountered. This determines the Russell-Saunders term of the form MS+1ML where 2MS+1 is evaluated numerically and ML is designated with a letter as shown below:
ML 0 1 2 3 4 5 letter S P D F G H etc 8. Each term represents (2ML+1)(2MS+1) microstates (this is the degeneracy of the term) so these need to be eliminated form the matrix. This is done by eliminating one microstate from each matrix entry symmetrically form the current position.
9. Go to 7 and repeat until all microstates are eliminated.
10. The total angular momentum J is found for each Russell-Saunders term by
J = |ML+MS|, |ML+MS-1|,..., |ML-MS| each J value indicates a new term, denoted as a numerical subscript to the right of the R-S term. Each term has a degeneracy of 2J+1 and the sum of all degeneracies for all terms should equal the total number of microstates.
Energies of Terms follow Hund's rules:
1. Lowest energy term is always the one with the highest spin multiplicity and highest orbital multiplicity. If < ½ filled, lowest J. If > ½ filled, highest J. J terms increase in energy sequentially.
2. The rest of the terms follow in order of spin degeneracy and then orbital degeneracy, although there are a lot of exceptions.
Find all the terms for the d3 configuration:
To describe each microstate, use the notation: (ml±, ml±, ml±) where + means ms =+ ½ and - means ms =- ½
maximum possible L = 5 (2+, 2-, 1+), so ML = 5, 4, 3, 2, 1, 0, -1, -2, -3, -4, -5
maximum possible S = 3/2 (2+, 1+, 0+), so MS = 3/2, 1/2, -1/2, -3/2
Build and fill the bookkeeping matrix:
| ML\MS |
|
|
|
|
| 5 |
|
|
||
| 4 |
|
|
||
| 3 |
|
(2+, 1+, 0-) (2+, 2-, -1+) |
(2-, 1-, 0+) (2-, 2+, -1-) |
|
| 2 |
|
(2+, 1+, -1-) (2+, 0+, 0-) (1+, 1-, 0+) (2+, 2-, -2+) |
(2-, 1-, -1+) (2-, 0-, 0+) (1-, 1+, 0-) (2-, 2+, -2-) |
|
| 1 |
|
(2+, 1+, -2-) (2-, 0+, -1+) (2+, 0-, -1+) (2+, 0+, -1-) (1+, 0+, 0-) (1+, 1-, -1+) |
(2-, 1-, -2+) (2+, 0-, -1-) (2-, 0+, -1-) (2-, 0-, -1+) (1-, 0-, 0+) (1-, 1+, -1-) |
|
| 0 |
|
(2+,0+,-2-) (1-,0+,-1+) (1+,0-,-1+) (1+,0+,-1-) (2+, -1+, -1-) (1+, 1-, -2+) |
(2-,0-,-2+) (1+,0-,-1-) (1-,0+,-1-) (1-,0-,-1+) (2-, -1-, -1+) (1-, 1+, -2-) |
|
| -1 |
|
(-2+, -1+, 2-) (-2-, 0+, 1+) (-2+, 0-, 1+) (-2+, 0+, 1-) (-1+, 0+, 0-) (-1+, -1-, 1+) |
(-2-, -1-, 2+) (-2+, 0-, 1-) (-2-, 0+, 1-) (-2-, 0-, 1+) (-1-, 0-, 0+) (-1-, -1+, 1-) |
|
| -2 |
|
(-2+, -1+, 1-) (-2+, 0+, 0-) (-1+, -1-, 0+) (-2+, -2-,2+) |
(-2-, -1-, 1+) (-2-, 0-, 0+) (-1-, -1+, 0-) (-2-, -2+,2-) |
|
| -3 |
|
(-2+, -1+, 0-) (-2+, -2-,1+) |
(-2-, -1-, 0+) (-2-, -2+,1-) |
|
| -4 |
|
|
||
| -5 |
|
|
The total number of microstates = 10!/3!7! = 120 (Nl = 2[2(2)+1] = 10, x = 3), which matches the number in the matrix.
Now find terms:
First term is ML = 3 , MS =3/2 giving 4F
The degeneracy of the 4F state is [2(3)+1]×[2(3/2)+1] = 28, so 28 microstates must be eliminated.
red denotes states eliminated by 4F:
ML\MS 3/2 1/2 -1/2 -3/2 5 (2+, 2-, 1+) (2-, 2+, 1-) 4 (2+, 2-, 0+) (2+, 1+, 1-) (2-, 2+, 0-) (2-, 1-, 1+) 3 (2+, 1+, 0+) (2-, 1+, 0+) (2+, 1-, 0+) (2+, 1+, 0-) (2+, 2-, -1+)
(2+, 1-, 0-) (2-, 1+, 0-) (2-, 1-, 0+) (2-, 2+, -1-)
(2-, 1-, 0-) 2 (2+, 1+, -1+) (2-, 1+, -1+) (2+, 1-, -1+) (2+, 1+, -1-) (2+, 0+, 0-)
(1+, 1-, 0+) (2+, 2-, -2+)
(2+, 1-, -1-) (2-, 1+, -1-) (2-, 1-, -1+) (2-, 0-, 0+)
(1-, 1+, 0-) (2-, 2+, -2-)
(2-, 1-, -1-) 1 (2+, 1+, -2+) (2+, 0+, -1+) (2-, 1+, -2+) (2+, 1-, -2+) (2+, 1+, -2-) (2-, 0+, -1+)
(2+, 0-, -1+) (2+, 0+, -1-)
(1+, 0+, 0-) (1+, 1-, -1+)
(2+, 1-, -2-) (2-, 1+, -2-) (2-, 1-, -2+) (2+, 0-, -1-)
(2-, 0+, -1-) (2-, 0-, -1+)
(1-, 0-, 0+) (1-, 1+, -1-)
(2-, 1-, -2-) (2-, 0-, -1-) 0 (2+,0+,-2+) (1+,0+,-1+) (2-,0+,-2+) (2+,0-,-2+) (2+,0+,-2-) (1-,0+,-1+)
(1+,0-,-1+) (1+,0+,-1-)
(2+, -1+, -1-) (1+, 1-, -2+)
(2+,0-,-2-) (2-,0+,-2-) (2-,0-,-2+) (1+,0-,-1-)
(1-,0+,-1-) (1-,0-,-1+)
(2-, -1-, -1+) (1-, 1+, -2-)
(2-,0-,-2-) (1-,0-,-1-) -1 (-2+, -1+, 2+) (-2+, 0+,1+) (-2-, -1+, 2+) (-2+, -1-,2+) (-2+, -1+, 2-) (-2-, 0+, 1+)
(-2+, 0-, 1+) (-2+, 0+, 1-)
(-1+, 0+, 0-) (-1+, -1-, 1+)
(-2+, -1-, 2-) (-2-, -1+,2-) (-2-, -1-, 2+) (-2+, 0-, 1-)
(-2-, 0+, 1-) (-2-, 0-, 1+)
(-1-, 0-, 0+) (-1-, -1+, 1-)
(-2-, -1-, 2-) (-2-, 0-,1-) -2 (-2+, 1+, -1+) (-2-, -1+, 1+) (-2+, -1-,1+) (-2+, -1+, 1-) (-2+, 0+, 0-)
(-1+, -1-, 0+) (-2+, -2-,2+)
(-2+, -1-, 1-) (-2-, -1+,1-) (-2-, -1-, 1+) (-2-, 0-, 0+)
(-1-, -1+, 0-) (-2-, -2+,2-)
(-2-, 1-, -1-) -3 (-2+, -1+, 0+) (-2-, -1+, 0+) (-2+, -1-,0+) (-2+, -1+, 0-) (-2+, -2-,1+)
(-2+, -1-, 0-) (-2-, -1+,0-) (-2-, -1-, 0+) (-2-, -2+,1-)
(-2-, -1-, 0-) -4 (-2+, -2-, 0+) (-2+,-1+,-1-) (-2-, -2+, 0-) (-2-,-1-,-1+) -5 (-2+, -2-, -1+) (-2-, -2+, -1-) The next state found is ML =1, MS =3/2, which is 4P, accounting for 12 microstates (green)
ML\MS 3/2 1/2 -1/2 -3/2 5 (2+, 2-, 1+) (2-, 2+, 1-) 4 (2+, 2-, 0+) (2+, 1+, 1-) (2-, 2+, 0-) (2-, 1-, 1+) 3 (2+, 1+, 0+) (2-, 1+, 0+) (2+, 1-, 0+) (2+, 1+, 0-) (2+, 2-, -1+)
(2+, 1-, 0-) (2-, 1+, 0-) (2-, 1-, 0+) (2-, 2+, -1-)
(2-, 1-, 0-) 2 (2+, 1+, -1+) (2-, 1+, -1+) (2+, 1-, -1+) (2+, 1+, -1-) (2+, 0+, 0-)
(1+, 1-, 0+) (2+, 2-, -2+)
(2+, 1-, -1-) (2-, 1+, -1-) (2-, 1-, -1+) (2-, 0-, 0+)
(1-, 1+, 0-) (2-, 2+, -2-)
(2-, 1-, -1-) 1 (2+, 1+, -2+) (2+, 0+, -1+) (2-, 1+, -2+) (2+, 1-, -2+) (2+, 1+, -2-) (2-, 0+, -1+)
(2+, 0-, -1+) (2+, 0+, -1-)
(1+, 0+, 0-) (1+, 1-, -1+)
(2+, 1-, -2-) (2-, 1+, -2-) (2-, 1-, -2+) (2+, 0-, -1-)
(2-, 0+, -1-) (2-, 0-, -1+)
(1-, 0-, 0+) (1-, 1+, -1-)
(2-, 1-, -2-) (2-, 0-, -1-) 0 (2+,0+,-2+)(1+,0+,-1+) (2-,0+,-2+)(2+,0-,-2+) (2+,0+,-2-) (1-,0+,-1+)
(1+,0-,-1+) (1+,0+,-1-)
(2+, -1+, -1-) (1+, 1-, -2+)
(2+,0-,-2-)(2-,0+,-2-) (2-,0-,-2+) (1+,0-,-1-)
(1-,0+,-1-) (1-,0-,-1+)
(2-, -1-, -1+) (1-, 1+, -2-)
(2-,0-,-2-)(1-,0-,-1-) -1 (-2+, -1+, 2+) (-2+, 0+,1+) (-2-, -1+, 2+) (-2+, -1-,2+) (-2+, -1+, 2-) (-2-, 0+, 1+)
(-2+, 0-, 1+) (-2+, 0+, 1-)
(-1+, 0+, 0-) (-1+, -1-, 1+)
(-2+, -1-, 2-) (-2-, -1+,2-) (-2-, -1-, 2+) (-2+, 0-, 1-)
(-2-, 0+, 1-) (-2-, 0-, 1+)
(-1-, 0-, 0+) (-1-, -1+, 1-)
(-2-, -1-, 2-) (-2-, 0-,1-) -2 (-2+, 1+, -1+) (-2-, -1+, 1+) (-2+, -1-,1+) (-2+, -1+, 1-) (-2+, 0+, 0-)
(-1+, -1-, 0+) (-2+, -2-,2+)
(-2+, -1-, 1-) (-2-, -1+,1-) (-2-, -1-, 1+) (-2-, 0-, 0+)
(-1-, -1+, 0-) (-2-, -2+,2-)
(-2-, 1-, -1-) -3 (-2+, -1+, 0+) (-2-, -1+, 0+) (-2+, -1-,0+) (-2+, -1+, 0-) (-2+, -2-,1+)
(-2+, -1-, 0-) (-2-, -1+,0-) (-2-, -1-, 0+) (-2-, -2+,1-)
(-2-, -1-, 0-) -4 (-2+, -2-, 0+) (-2+,-1+,-1-) (-2-, -2+, 0-) (-2-,-1-,-1+) -5 (-2+, -2-, -1+) (-2-, -2+, -1-) The next state found is ML =5, MS =1/2, which is 2H, accounting for 22 microstates (blue)
ML\MS 3/2 1/2 -1/2 -3/2 5 (2+, 2-, 1+) (2-, 2+, 1-) 4 (2+, 2-, 0+) (2+, 1+, 1-) (2-, 2+, 0-) (2-, 1-, 1+) 3 (2+, 1+, 0+) (2-, 1+, 0+) (2+, 1-, 0+) (2+, 1+, 0-) (2+, 2-, -1+)
(2+, 1-, 0-) (2-, 1+, 0-) (2-, 1-, 0+) (2-, 2+, -1-)
(2-, 1-, 0-) 2 (2+, 1+, -1+) (2-, 1+, -1+) (2+, 1-, -1+) (2+, 1+, -1-) (2+, 0+, 0-)
(1+, 1-, 0+) (2+, 2-, -2+)
(2+, 1-, -1-) (2-, 1+, -1-) (2-, 1-, -1+) (2-, 0-, 0+)
(1-, 1+, 0-) (2-, 2+, -2-)
(2-, 1-, -1-) 1 (2+, 1+, -2+) (2+, 0+, -1+) (2-, 1+, -2+) (2+, 1-, -2+) (2+, 1+, -2-) (2-, 0+, -1+)
(2+, 0-, -1+) (2+, 0+, -1-)
(1+, 0+, 0-) (1+, 1-, -1+)
(2+, 1-, -2-) (2-, 1+, -2-) (2-, 1-, -2+) (2+, 0-, -1-)
(2-, 0+, -1-) (2-, 0-, -1+)
(1-, 0-, 0+) (1-, 1+, -1-)
(2-, 1-, -2-) (2-, 0-, -1-) 0 (2+,0+,-2+)(1+,0+,-1+) (2-,0+,-2+)(2+,0-,-2+) (2+,0+,-2-) (1-,0+,-1+)
(1+,0-,-1+) (1+,0+,-1-)
(2+, -1+, -1-) (1+, 1-, -2+)
(2+,0-,-2-)(2-,0+,-2-) (2-,0-,-2+) (1+,0-,-1-)
(1-,0+,-1-) (1-,0-,-1+)
(2-, -1-, -1+) (1-, 1+, -2-)
(2-,0-,-2-)(1-,0-,-1-) -1 (-2+, -1+, 2+) (-2+, 0+,1+) (-2-, -1+, 2+) (-2+, -1-,2+) (-2+, -1+, 2-) (-2-, 0+, 1+)
(-2+, 0-, 1+) (-2+, 0+, 1-)
(-1+, 0+, 0-) (-1+, -1-, 1+)
(-2+, -1-, 2-) (-2-, -1+,2-) (-2-, -1-, 2+) (-2+, 0-, 1-)
(-2-, 0+, 1-) (-2-, 0-, 1+)
(-1-, 0-, 0+) (-1-, -1+, 1-)
(-2-, -1-, 2-) (-2-, 0-,1-) -2 (-2+, 1+, -1+) (-2-, -1+, 1+) (-2+, -1-,1+) (-2+, -1+, 1-) (-2+, 0+, 0-)
(-1+, -1-, 0+) (-2+, -2-,2+)
(-2+, -1-, 1-) (-2-, -1+,1-) (-2-, -1-, 1+) (-2-, 0-, 0+)
(-1-, -1+, 0-) (-2-, -2+,2-)
(-2-, 1-, -1-) -3 (-2+, -1+, 0+) (-2-, -1+, 0+) (-2+, -1-,0+) (-2+, -1+, 0-) (-2+, -2-,1+)
(-2+, -1-, 0-) (-2-, -1+,0-) (-2-, -1-, 0+) (-2-, -2+,1-)
(-2-, -1-, 0-) -4 (-2+, -2-, 0+) (-2+,-1+,-1-) (-2-, -2+, 0-) (-2-,-1-,-1+) -5 (-2+, -2-, -1+) (-2-, -2+, -1-) The next state found is ML =4, MS =1/2, which is 2G, accounting for 18 microstates (pink)
ML\MS 3/2 1/2 -1/2 -3/2 5 (2+, 2-, 1+) (2-, 2+, 1-) 4 (2+, 2-, 0+) (2+, 1+, 1-) (2-, 2+, 0-) (2-, 1-, 1+) 3 (2+, 1+, 0+) (2-, 1+, 0+) (2+, 1-, 0+) (2+, 1+, 0-) (2+, 2-, -1+)
(2+, 1-, 0-) (2-, 1+, 0-) (2-, 1-, 0+) (2-, 2+, -1-)
(2-, 1-, 0-) 2 (2+, 1+, -1+) (2-, 1+, -1+) (2+, 1-, -1+) (2+, 1+, -1-) (2+, 0+, 0-)
(1+, 1-, 0+) (2+, 2-, -2+)
(2+, 1-, -1-) (2-, 1+, -1-) (2-, 1-, -1+) (2-, 0-, 0+)
(1-, 1+, 0-) (2-, 2+, -2-)
(2-, 1-, -1-) 1 (2+, 1+, -2+) (2+, 0+, -1+) (2-, 1+, -2+) (2+, 1-, -2+) (2+, 1+, -2-)(2-, 0+, -1+)
(2+, 0-, -1+) (2+, 0+, -1-)
(1+, 0+, 0-) (1+, 1-, -1+)
(2+, 1-, -2-) (2-, 1+, -2-) (2-, 1-, -2+)(2+, 0-, -1-)
(2-, 0+, -1-) (2-, 0-, -1+)
(1-, 0-, 0+) (1-, 1+, -1-)
(2-, 1-, -2-) (2-, 0-, -1-) 0 (2+,0+,-2+)(1+,0+,-1+) (2-,0+,-2+)(2+,0-,-2+) (2+,0+,-2-)(1-,0+,-1+)
(1+,0-,-1+) (1+,0+,-1-)
(2+, -1+, -1-) (1+, 1-, -2+)
(2+,0-,-2-)(2-,0+,-2-) (2-,0-,-2+) (1+,0-,-1-)
(1-,0+,-1-) (1-,0-,-1+)
(2-, -1-, -1+) (1-, 1+, -2-)
(2-,0-,-2-)(1-,0-,-1-) -1 (-2+, -1+, 2+) (-2+, 0+,1+) (-2-, -1+, 2+) (-2+, -1-,2+) (-2+, -1+, 2-) (-2-, 0+, 1+)
(-2+, 0-, 1+) (-2+, 0+, 1-)
(-1+, 0+, 0-) (-1+, -1-, 1+)
(-2+, -1-, 2-) (-2-, -1+,2-) (-2-, -1-, 2+) (-2+, 0-, 1-)
(-2-, 0+, 1-) (-2-, 0-, 1+)
(-1-, 0-, 0+) (-1-, -1+, 1-)
(-2-, -1-, 2-) (-2-, 0-,1-) -2 (-2+, 1+, -1+) (-2-, -1+, 1+) (-2+, -1-,1+) (-2+, -1+, 1-) (-2+, 0+, 0-)
(-1+, -1-, 0+) (-2+, -2-,2+)
(-2+, -1-, 1-) (-2-, -1+,1-) (-2-, -1-, 1+) (-2-, 0-, 0+)
(-1-, -1+, 0-) (-2-, -2+,2-)
(-2-, 1-, -1-) -3 (-2+, -1+, 0+) (-2-, -1+, 0+) (-2+, -1-,0+) (-2+, -1+, 0-) (-2+, -2-,1+)
(-2+, -1-, 0-) (-2-, -1+,0-) (-2-, -1-, 0+) (-2-, -2+,1-)
(-2-, -1-, 0-) -4 (-2+, -2-, 0+) (-2+,-1+,-1-) (-2-, -2+, 0-) (-2-,-1-,-1+) -5 (-2+, -2-, -1+) (-2-, -2+, -1-) The next state found is ML =3, MS =1/2, which is 2F, accounting for 14 microstates (turquoise)
ML\MS 3/2 1/2 -1/2 -3/2 5 (2+, 2-, 1+) (2-, 2+, 1-) 4 (2+, 2-, 0+) (2+, 1+, 1-) (2-, 2+, 0-) (2-, 1-, 1+) 3 (2+, 1+, 0+) (2-, 1+, 0+) (2+, 1-, 0+) (2+, 1+, 0-)(2+, 2-, -1+)
(2+, 1-, 0-) (2-, 1+, 0-) (2-, 1-, 0+)(2-, 2+, -1-)
(2-, 1-, 0-) 2 (2+, 1+, -1+) (2-, 1+, -1+) (2+, 1-, -1+) (2+, 1+, -1-)(2+, 0+, 0-)
(1+, 1-, 0+) (2+, 2-, -2+)
(2+, 1-, -1-) (2-, 1+, -1-) (2-, 1-, -1+)(2-, 0-, 0+)
(1-, 1+, 0-) (2-, 2+, -2-)
(2-, 1-, -1-) 1 (2+, 1+, -2+) (2+, 0+, -1+) (2-, 1+, -2+) (2+, 1-, -2+) (2+, 1+, -2-)(2-, 0+, -1+)
(2+, 0-, -1+) (2+, 0+, -1-)
(1+, 0+, 0-) (1+, 1-, -1+)
(2+, 1-, -2-) (2-, 1+, -2-) (2-, 1-, -2+)(2+, 0-, -1-)
(2-, 0+, -1-) (2-, 0-, -1+)
(1-, 0-, 0+) (1-, 1+, -1-)
(2-, 1-, -2-) (2-, 0-, -1-) 0 (2+,0+,-2+)(1+,0+,-1+) (2-,0+,-2+)(2+,0-,-2+) (2+,0+,-2-)(1-,0+,-1+)
(1+,0-,-1+) (1+,0+,-1-)
(2+, -1+, -1-) (1+, 1-, -2+)
(2+,0-,-2-)(2-,0+,-2-) (2-,0-,-2+)(1+,0-,-1-)
(1-,0+,-1-) (1-,0-,-1+)
(2-, -1-, -1+) (1-, 1+, -2-)
(2-,0-,-2-)(1-,0-,-1-) -1 (-2+, -1+, 2+) (-2+, 0+,1+) (-2-, -1+, 2+) (-2+, -1-,2+) (-2+, -1+, 2-) (-2-, 0+, 1+)
(-2+, 0-, 1+) (-2+, 0+, 1-)
(-1+, 0+, 0-) (-1+, -1-, 1+)
(-2+, -1-, 2-) (-2-, -1+,2-) (-2-, -1-, 2+) (-2+, 0-, 1-)
(-2-, 0+, 1-) (-2-, 0-, 1+)
(-1-, 0-, 0+) (-1-, -1+, 1-)
(-2-, -1-, 2-) (-2-, 0-,1-) -2 (-2+, 1+, -1+) (-2-, -1+, 1+) (-2+, -1-,1+) (-2+, -1+, 1-) (-2+, 0+, 0-)
(-1+, -1-, 0+) (-2+, -2-,2+)
(-2+, -1-, 1-) (-2-, -1+,1-) (-2-, -1-, 1+) (-2-, 0-, 0+)
(-1-, -1+, 0-) (-2-, -2+,2-)
(-2-, 1-, -1-) -3 (-2+, -1+, 0+) (-2-, -1+, 0+) (-2+, -1-,0+) (-2+, -1+, 0-) (-2+, -2-,1+)
(-2+, -1-, 0-) (-2-, -1+,0-) (-2-, -1-, 0+) (-2-, -2+,1-)
(-2-, -1-, 0-) -4 (-2+, -2-, 0+) (-2+,-1+,-1-) (-2-, -2+, 0-) (-2-,-1-,-1+) -5 (-2+, -2-, -1+) (-2-, -2+, -1-) The next state found is ML =2, MS =1/2, which is 2D, accounting for 10 microstates (yellow)
ML\MS 3/2 1/2 -1/2 -3/2 5 (2+, 2-, 1+) (2-, 2+, 1-) 4 (2+, 2-, 0+) (2+, 1+, 1-) (2-, 2+, 0-) (2-, 1-, 1+) 3 (2+, 1+, 0+) (2-, 1+, 0+) (2+, 1-, 0+) (2+, 1+, 0-)(2+, 2-, -1+)
(2+, 1-, 0-) (2-, 1+, 0-) (2-, 1-, 0+)(2-, 2+, -1-)
(2-, 1-, 0-) 2 (2+, 1+, -1+) (2-, 1+, -1+) (2+, 1-, -1+) (2+, 1+, -1-)(2+, 0+, 0-)
(1+, 1-, 0+) (2+, 2-, -2+)
(2+, 1-, -1-) (2-, 1+, -1-) (2-, 1-, -1+)(2-, 0-, 0+)
(1-, 1+, 0-) (2-, 2+, -2-)
(2-, 1-, -1-) 1 (2+, 1+, -2+) (2+, 0+, -1+) (2-, 1+, -2+) (2+, 1-, -2+) (2+, 1+, -2-)(2-, 0+, -1+)
(2+, 0-, -1+)(2+, 0+, -1-)
(1+, 0+, 0-) (1+, 1-, -1+)
(2+, 1-, -2-) (2-, 1+, -2-) (2-, 1-, -2+)(2+, 0-, -1-)
(2-, 0+, -1-) (2-, 0-, -1+)
(1-, 0-, 0+) (1-, 1+, -1-)
(2-, 1-, -2-) (2-, 0-, -1-) 0 (2+,0+,-2+)(1+,0+,-1+) (2-,0+,-2+)(2+,0-,-2+) (2+,0+,-2-)(1-,0+,-1+)
(1+,0-,-1+)(1+,0+,-1-)
(2+, -1+, -1-) (1+, 1-, -2+)
(2+,0-,-2-)(2-,0+,-2-) (2-,0-,-2+)(1+,0-,-1-)
(1-,0+,-1-)(1-,0-,-1+)
(2-, -1-, -1+) (1-, 1+, -2-)
(2-,0-,-2-)(1-,0-,-1-) -1 (-2+, -1+, 2+) (-2+, 0+,1+) (-2-, -1+, 2+) (-2+, -1-,2+) (-2+, -1+, 2-) (-2-, 0+, 1+)
(-2+, 0-, 1+) (-2+, 0+, 1-)
(-1+, 0+, 0-) (-1+, -1-, 1+)
(-2+, -1-, 2-) (-2-, -1+,2-) (-2-, -1-, 2+) (-2+, 0-, 1-)
(-2-, 0+, 1-) (-2-, 0-, 1+)
(-1-, 0-, 0+) (-1-, -1+, 1-)
(-2-, -1-, 2-) (-2-, 0-,1-) -2 (-2+, 1+, -1+) (-2-, -1+, 1+) (-2+, -1-,1+) (-2+, -1+, 1-) (-2+, 0+, 0-)
(-1+, -1-, 0+) (-2+, -2-,2+)
(-2+, -1-, 1-) (-2-, -1+,1-) (-2-, -1-, 1+) (-2-, 0-, 0+)
(-1-, -1+, 0-) (-2-, -2+,2-)
(-2-, 1-, -1-) -3 (-2+, -1+, 0+) (-2-, -1+, 0+) (-2+, -1-,0+) (-2+, -1+, 0-) (-2+, -2-,1+)
(-2+, -1-, 0-) (-2-, -1+,0-) (-2-, -1-, 0+) (-2-, -2+,1-)
(-2-, -1-, 0-) -4 (-2+, -2-, 0+) (-2+,-1+,-1-) (-2-, -2+, 0-) (-2-,-1-,-1+) -5 (-2+, -2-, -1+) (-2-, -2+, -1-) The next state found is another ML =2, MS =1/2, which is 2D, accounting for 10 microstates (teal)
ML\MS 3/2 1/2 -1/2 -3/2 5 (2+, 2-, 1+) (2-, 2+, 1-) 4 (2+, 2-, 0+) (2+, 1+, 1-) (2-, 2+, 0-) (2-, 1-, 1+) 3 (2+, 1+, 0+) (2-, 1+, 0+) (2+, 1-, 0+) (2+, 1+, 0-)(2+, 2-, -1+)
(2+, 1-, 0-) (2-, 1+, 0-) (2-, 1-, 0+)(2-, 2+, -1-)
(2-, 1-, 0-) 2 (2+, 1+, -1+) (2-, 1+, -1+) (2+, 1-, -1+) (2+, 1+, -1-)(2+, 0+, 0-)
(1+, 1-, 0+)(2+, 2-, -2+)
(2+, 1-, -1-) (2-, 1+, -1-) (2-, 1-, -1+)(2-, 0-, 0+)
(1-, 1+, 0-)(2-, 2+, -2-)
(2-, 1-, -1-) 1 (2+, 1+, -2+) (2+, 0+, -1+) (2-, 1+, -2+) (2+, 1-, -2+) (2+, 1+, -2-)(2-, 0+, -1+)
(2+, 0-, -1+)(2+, 0+, -1-)
(1+, 0+, 0-) (1+, 1-, -1+)
(2+, 1-, -2-) (2-, 1+, -2-) (2-, 1-, -2+)(2+, 0-, -1-)
(2-, 0+, -1-)(2-, 0-, -1+)
(1-, 0-, 0+) (1-, 1+, -1-)
(2-, 1-, -2-) (2-, 0-, -1-) 0 (2+,0+,-2+)(1+,0+,-1+) (2-,0+,-2+)(2+,0-,-2+) (2+,0+,-2-)(1-,0+,-1+)
(1+,0-,-1+)(1+,0+,-1-)
(2+, -1+, -1-) (1+, 1-, -2+)
(2+,0-,-2-)(2-,0+,-2-) (2-,0-,-2+)(1+,0-,-1-)
(1-,0+,-1-)(1-,0-,-1+)
(2-, -1-, -1+) (1-, 1+, -2-)
(2-,0-,-2-)(1-,0-,-1-) -1 (-2+, -1+, 2+) (-2+, 0+,1+) (-2-, -1+, 2+) (-2+, -1-,2+) (-2+, -1+, 2-) (-2-, 0+, 1+)
(-2+, 0-, 1+) (-2+, 0+, 1-)
(-1+, 0+, 0-) (-1+, -1-, 1+)
(-2+, -1-, 2-) (-2-, -1+,2-) (-2-, -1-, 2+) (-2+, 0-, 1-)
(-2-, 0+, 1-) (-2-, 0-, 1+)
(-1-, 0-, 0+) (-1-, -1+, 1-)
(-2-, -1-, 2-) (-2-, 0-,1-) -2 (-2+, 1+, -1+) (-2-, -1+, 1+) (-2+, -1-,1+) (-2+, -1+, 1-) (-2+, 0+, 0-)
(-1+, -1-, 0+) (-2+, -2-,2+)
(-2+, -1-, 1-) (-2-, -1+,1-) (-2-, -1-, 1+) (-2-, 0-, 0+)
(-1-, -1+, 0-) (-2-, -2+,2-)
(-2-, 1-, -1-) -3 (-2+, -1+, 0+) (-2-, -1+, 0+) (-2+, -1-,0+) (-2+, -1+, 0-) (-2+, -2-,1+)
(-2+, -1-, 0-) (-2-, -1+,0-) (-2-, -1-, 0+) (-2-, -2+,1-)
(-2-, -1-, 0-) -4 (-2+, -2-, 0+) (-2+,-1+,-1-) (-2-, -2+, 0-) (-2-,-1-,-1+) -5 (-2+, -2-, -1+) (-2-, -2+, -1-) The next state found is ML =1, MS =1/2, which is 2P, accounting for 6 microstates (violet)
ML\MS 3/2 1/2 -1/2 -3/2 5 (2+, 2-, 1+) (2-, 2+, 1-) 4 (2+, 2-, 0+) (2+, 1+, 1-) (2-, 2+, 0-) (2-, 1-, 1+) 3 (2+, 1+, 0+) (2-, 1+, 0+) (2+, 1-, 0+) (2+, 1+, 0-)(2+, 2-, -1+)
(2+, 1-, 0-) (2-, 1+, 0-) (2-, 1-, 0+)(2-, 2+, -1-)
(2-, 1-, 0-) 2 (2+, 1+, -1+) (2-, 1+, -1+) (2+, 1-, -1+) (2+, 1+, -1-)(2+, 0+, 0-)
(1+, 1-, 0+)(2+, 2-, -2+)
(2+, 1-, -1-) (2-, 1+, -1-) (2-, 1-, -1+)(2-, 0-, 0+)
(1-, 1+, 0-)(2-, 2+, -2-)
(2-, 1-, -1-) 1 (2+, 1+, -2+) (2+, 0+, -1+) (2-, 1+, -2+) (2+, 1-, -2+) (2+, 1+, -2-)(2-, 0+, -1+)
(2+, 0-, -1+)(2+, 0+, -1-)
(1+, 0+, 0-)(1+, 1-, -1+)
(2+, 1-, -2-) (2-, 1+, -2-) (2-, 1-, -2+)(2+, 0-, -1-)
(2-, 0+, -1-)(2-, 0-, -1+)
(1-, 0-, 0+)(1-, 1+, -1-)
(2-, 1-, -2-) (2-, 0-, -1-) 0 (2+,0+,-2+)(1+,0+,-1+) (2-,0+,-2+)(2+,0-,-2+) (2+,0+,-2-)(1-,0+,-1+)
(1+,0-,-1+)(1+,0+,-1-)
(2+, -1+, -1-) (1+, 1-, -2+)
(2+,0-,-2-)(2-,0+,-2-) (2-,0-,-2+)(1+,0-,-1-)
(1-,0+,-1-)(1-,0-,-1+)
(2-, -1-, -1+) (1-, 1+, -2-)
(2-,0-,-2-)(1-,0-,-1-) -1 (-2+, -1+, 2+) (-2+, 0+,1+) (-2-, -1+, 2+) (-2+, -1-,2+) (-2+, -1+, 2-)(-2-, 0+, 1+)
(-2+, 0-, 1+) (-2+, 0+, 1-)
(-1+, 0+, 0-) (-1+, -1-, 1+)
(-2+, -1-, 2-) (-2-, -1+,2-) (-2-, -1-, 2+) (-2+, 0-, 1-)
(-2-, 0+, 1-) (-2-, 0-, 1+)
(-1-, 0-, 0+) (-1-, -1+, 1-)
(-2-, -1-, 2-) (-2-, 0-,1-) -2 (-2+, 1+, -1+) (-2-, -1+, 1+) (-2+, -1-,1+) (-2+, -1+, 1-) (-2+, 0+, 0-)
(-1+, -1-, 0+) (-2+, -2-,2+)
(-2+, -1-, 1-) (-2-, -1+,1-) (-2-, -1-, 1+) (-2-, 0-, 0+)
(-1-, -1+, 0-) (-2-, -2+,2-)
(-2-, 1-, -1-) -3 (-2+, -1+, 0+) (-2-, -1+, 0+) (-2+, -1-,0+) (-2+, -1+, 0-) (-2+, -2-,1+)
(-2+, -1-, 0-) (-2-, -1+,0-) (-2-, -1-, 0+) (-2-, -2+,1-)
(-2-, -1-, 0-) -4 (-2+, -2-, 0+) (-2+,-1+,-1-) (-2-, -2+, 0-) (-2-,-1-,-1+) -5 (-2+, -2-, -1+) (-2-, -2+, -1-) This is all of the terms: 4F (28), 4P (12), 2H (22), 2G (18), 2F (14), 2D (10), 2D (10), 2P (6)
28 + 12 + 22 + 18 + 14 + 10 +10 +6 = 120, so all microstates are accounted for.
To find the J values:
4F ML= 3, MS =3/2 so J = 9/2, 7/2, 5/2, 3/2 and the terms become 4F9/2, 4F7/2, 4F5/2, 4F3/2
4P ML= 1, MS =3/2 so J = 5/2, 3/2, 1/2 and the terms become 4P5/2, 4P3/2, 4P1/2
2H ML= 5, MS =1/2 so J = 11/2, 9/2 and the terms become 2H11/2, 2H9/2
2G ML= 4, MS =1/2 so J = 9/2, 7/2 and the terms become 2G9/2, 2G7/2
2F ML= 3, MS =1/2 so J = 7/2, 5/2 and the terms become 2F7/2, 2F5/2
2D ML= 2, MS =1/2 so J = 5/2, 3/2 and the terms become 2D5/2, 2D3/2
(this is the same for both 2D terms)
2P ML= 1, MS =1/2 so J = 3/2, 1/2 and the terms become 2P3/2, 2P1/2
Energy order predicted by Hund’s rules (less than ½ filled):
4F3/2< 4F5/2< 4F7/2< 4F9/2<4P1/2< 4P3/2< 4P5/2< 2H9/2< 2H11/2< 2G7/2< 2G9/2< 2F5/2< 2F7/2< 2D3/2~ 2D3/2< 2D5/2~2D5/2<2P1/2< 2P3/2
For a more quantitative consideration of multielectron systems requires better wavefunctions or the introduction of fudge factors. We convert multielectron systems to hydrogen-like systems by introducing the effective nuclear charge, Z*:
Z* = Z - S
Z* = effective nuclear charge
Z = true nuclear charge
S = screening or shielding constant
Can find S a number of ways: Clementi and Raimondi used accurate numerical calculations to fit S to experiment.
Easier, less quantitatively accurate method is Slater's: he used the radial (n) and nodal (l) characteristics of hydrogenic wavefunctions to establish shielding constants:
Slater's rules:
1. Write electron configurations according to principal quantum number, grouping s and p orbitals: (1s)(2s2p)(3s3p)(3d)(4s4p)(4d)(4f)...
2. Any electron to the right of the selected electron is ignored (contributes 0 to S)
3. If the electron under consideration is in an (ns np) group:
a) all other electrons in the same group contribute 0.35/e to S (except 1s, 0.30)
b) all electrons in the n-1 shell contribute 0.85/e to S
c) all electrons in n-2 and lower shells contribute 1.00/e to S
4. If the electron under consideration is in an (nd) or (nf) group:
a) all other electrons in the same group contribute 0.35/e to S
b) all electrons to the left of the selected group contribute 1.00/e to S
Periodic Trends in Z* : Z* increases to the right on the Periodic Table, increases as go down the Periodic Table.