What you are looking at are The Crystal Field Splitting of Russell-Saunders Terms. The crystal field states corresponding to the Russell-Saunders states have been calculated by Bethe (Cotton 1990, p. 264). For other symmetry environments of , the crystal field states can be found in Appendix IIB of Cotton (1990, p. 437).
The Crystal Field Splitting of Russell-Saunders Terms :
The effect of a crystal field on the different orbitals (s, p, d, etc.) will result in splitting into subsets of different energies, depending on whether they are in an octahedral or tetrahedral environment. The magnitude of the d orbital splitting is generally represented as a fraction of Δ
oct or 10D
q.
The ground term energies for free ions are also affected by the influence of a crystal field and an analogy is made between orbitals and ground terms that are related due to the angular parts of their electron distribution. The effect of a crystal field on different orbitals in an octahedral field environment will cause the d orbitals to split to give t2g and eg subsets and the D ground term states into T
2g and E
g, (where upper case is used to denote states e.g., T = tripely degenerate and lower case orbitals). f orbitals are split to give subsets known as t
1g, t
2g and a
2g. By analogy, the F ground term when split by a crystal field will give states known as T
1g, T
2g, and A
2g.
Note that it is important to recognise that the F ground term here refers to states arising from d orbitals and not f orbitals and depending on whether it is in an octahedral or tetrahedral environment the lowest term can be either A2g or T1g.
Microstates: For the L = 2 configuration, where m
l = 2,1,0,-1,-2, the five microstates are 2,1,0,-,-2. Basically, microstates denote/designate all the distinguishable states that would have different energies in the presence of an external magnetic field.
So for a 2P
2 configuration you have different configurations that fit that description. They are called microstates and they have different energies because of inter-electronic repulsions.
Electronic Configurations of multi-electron atoms (Russell-Saunders or LS Coupling): M
L/M
S define microstates and L/S define states (collections of microstates). Groups of microstates with the same energy are called "Terms."
For example, take the P
2, then build the M
L/M
S, then build the MICROSTATE table, then determine the STATES (S, P, D) spin multiplicity, then find the TERMS
3P,
1D,
1S; where, the Ground State Term is
3P.
For a d
1-d
10 electron structure you must consider
3F,
3P,
1G,
1D,
1S. This is a long arduous/tedious process for 3 or more electrons. They are tabled.
The LaPorte Rules:
Transitions between states of the same parity are forbidden
g -----> g forbidden (that is d-d, forbidden)
u ----->u forbidden
g -----> u allowed (that is d-p, allowed)
Spin Rules:
Transitions between states of different multiplicities are forbidden
Transitions between states of the same mutiplicity are allowed
Singlet -----> Singlet is allowed
Singlet -----> Triplet is forbidden
Δl = + 1
Allowed
s ---> p, p--->d, d --->f..
Forbidden
s ---> s, p ---> p, d ---> d, f ---> f, s --->d, p ---> f…etc
These rules are relaxed by molecular vibrations and spin-orbit coupling.
It would appear that most transitions for metal complexes are forbidden; but, several mechanisms exist that relax these rules:
1. vibronic coupling
Change the symmetry of octahedral complexes.
d-d transitions have molar absorp. 10-50 L/mol·cm
2. metal-ligand sigma bonding
Ligand p orbital (u) mixing with d orbitals (g)
3. spin-orbit coupling
Absorp. Less than 1 L/mol·cm
Please see the attached word document which gives a full explanation 
I have also attached 2 useful pdfs.
Polyelectronic Atoms:For polyelectronic (i.e. real) atoms, a direct solution of the Schrodinger Equation is not possible.
When we construct polyelectronic atoms, we use the hydrogen-atom orbital nomenclature to discuss in which orbitals the electrons reside.
This is an approximation (and it is surprising how well it actually works).
Aufbau is German for “Building Up”. We are building a multielectronic atom from the rules for the 1 electron atom (so a few things get modified), but it works pretty well.
Basically, you are extending the 1-electron results (AO shapes, energy equation, quantum numbers, etc...) to polyelectron atoms; which, are atoms with more than 1 electron.
I have attached links to two powerpoint presentations that address this:
LINK:
http://www.chem.umd.edu/groups/vedernikov/Lecture2005-02n.ppt#256,1,Lecture 2 THE ELECTRONIC STRUCTURE OF THE POLYELECTRONIC ATOM. PART I
LINK:
http://www.d.umn.edu/~btsai/Chem1151spring2007/PowerPoint%20Lect/PPTCh7%20at%20struc%20-%20part%202.ppt#282,2,POLYELECTRONIC ATOMS
Please also see the attached link to a pdf that discusses The Franck-Condon Principle anda few other important concepts.
http://w3.rz-berlin.mpg.de/~jentoft/lehre/clocishner_electronvibrational_optical_bands_151206.pdfAdditionally:
Cotton, F. A. "The Crystal Field Theory." Chemical Applications of Group Theory, 3rd ed. New York: Wiley, pp. 282-287, 1990.
Cotton, F. A.; Wilkinson, G.; and Gaus, P. Ch. 23 in Basic Inorganic Chemistry, 3rd ed. New York: Wiley, 1995.
Huheey, J. E.; Keiter, E. A.; and Keiter, R. L. Inorganic Chemistry: Principles of Structure and Reactivity, 4th ed. Reading, MA: Addison-Wesley, 1993.
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