for the given electron configuration. Then the possible angular momenta
are ML = -L, -L+1, -L+2, ..., L-2, L-2, L
Multielectron atoms: can’t solve the Schroedinger equation exactly
so assumptions must be made; assume the hydrogenic obitals are adequate
and electrons occupy them in some fashion
Two guiding principals used to account for electron configurations
Aufbau Principle: electrons occupy orbitals in such a manner to give the lowest possible total energy
Pauli Exclusion Principle: each electron in an atom is described
by a unique set of quantum numbers (n, l,
ml, ms)
Periodic
Table: based on electron configurations and can be used to predict
them but not absolute (electron configurations are experimental
quantities)
Aids in finding correct electron configurations from the Periodic Table:
Half-filled phenomenon: when d or f electrons are the valence shell, if a shift of 1 electron (occasionally 2 but this is not predictive) from an s orbital to the d or f orbital leads to a filled or half filled d or f orbital, this will stabilize the electron configuration.
anions: add electrons to the neutral atom and follow above rules.
cations: electrons are always removed from the orbitals with
the largest principal quantum number (n); the remaining electrons fill
the orbitals with the lowest n consistent with the Pauli Principle
electron configurations denote n and l quantum numbers
Term symbols: 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 =
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 = 
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 = 2(2l+1)
x = number of electrons
b) mixed l states: the number of
microstates = 
where
Ni 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
where
2MS+1 is evaluated numerically and ML is designated
with a letter as shown below:
| ML |
|
|
|
|
|
|
|
| letter |
|
|
|
|
|
|
|
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.