Wave properties of electrons are constrained by certain boundary conditions:
1) the electron exists somewhere in space
2) the electron is continuous
3) the electron is finite
When these conditions are imposed upon Schroedingers equation, the
result is quantization
1) each electron can only have certain energies (energy quantization)
2) each electron can only occupy certain volumes of space (spatial quantization)
quantization is described by a set of numbers called quantum numbers:
n = principal quantum number, found only in R, distance dependence
l = orbital angular momentum, found in R and Y, orbital shape
ml = magnetic quantum number, found only in Y, orientation in space
The quantum numbers are mathematically related
n = 1, 2, 3, 4, ...
l = n-1, n-2, n-3, ..., 0
ml = -l, -l+1, -l+2, ..., l-2, l-1, l
The l quantum number is usually designated by a letter:
Experimental evidence (and relativistic theory) indicates the presence
of a fourth quantum
number, ms, the spin quantum number = ±1/2
General nodal properties:
total number of nodes = n-1
total number of planar (or angular) nodes = l
total number of radial nodes = n-l-1
Multielectron atoms: can not 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,
Table: based on electron configurations and can be used to predict
them but not absolute (electron configurations are experimental
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
Term symbols: a shorthand notation that describes the electron
distribution in atoms or ions, i.e. the ml and ms
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 formwhere 2MS+1 is evaluated numerically and ML is designated with a letter as shown below:
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 Hunds 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.