1. Why does [Cr(NH3)6]3+ have two absorptions for the eg*t2g transition?

2. A standard spectroscopy problem

3. To answer the question
The actual physical movement of an electron from one orbital to another will be affected by the distance the electron has to move to get to the new orbital and any electronic repulsions it encounters along the way
dz2 (eg*)  dxy (t2g) is far and encounters many repulsion (higher energy for the transition)
dz2 (eg*)  dxz (t2g) is not so far and fewer repulsions

4. d2 ion
Electron-electron repulsion eg eg z2
x2-y2 z2
x2-y2
t2g
t2g xy xz yz xy xz yz
xy + z2
xz + z2 z z y y x x
lobes overlap, large electron repulsion
lobes far apart, small electron repulsion
These two electron configurations do not have the same energy

5. How to classify different types of transitions within eg*  t2g
d orbital electrons differ in quantum number; l = 2, ml = +2, +1, 0 , –1, or –2; s = ± ½
In fact, picking an ml value and a spin state is sufficient to specify a particular electronic state: 2+ means a spin-up electron in the ml = 2 orbital

6. More than one d electron
But one electron cases are dull – no electronic repulsion
With more than one electron, Hund’s Rule (no two electrons on an atom can share the same quantum numbers) kicks in
For d2 configurations, use ordered pairs to specify electronic states: (2+, 2–) or (–1–, 1–) to give two examples — these are called microstates

7. Microstates
So a microstate is a snapshot description of the complete quantum state of an atom’s valence electrons. Define ML as the total angular momentum quantum number for all the valence electrons and S as the total spin of all the valence electrons.
Thus, for the microstate (2+, 2–), ML = = 4 and S = +1/2 –1/2 = 0

8. Why microstates?
Slight digression: why worry about just the angular orbital momentum and spin?
The energy of a particular microstate is a sum of (from most important to least) spin-spin coupling, orbital-orbital coupling and spin-orbital coupling – this is a technique called Russell-Saunders coupling, and all these coupling terms are actually reasonably complicated integrals of wavefunctions.

9. Clebsch-Gordan series
Named after mathematicians R.F. Alfred Clebsch and Paul Gordan (University of Giessen, Germany) who published Theorie der Abelschen Funktionen (1866), a study on purely theoretical spherical harmonics.
Applied to molecules in the 1940s by numerous researchers.
For us, the series provides a way to organize microstates into degenerate energy sets without doing any integrals.

10. p2 microstate table

11. Term symbols
The microstate table can be decomposed into groups of degenerate energy microstates; each group is symbolized by a term symbol which has the structure nX, where n is the multiplicity, which is a whole number, and X is the term itself, which is a capital letter.
The multiplicity is calculated simply as 2S +1.
The term is designated by the largest value of ML: if ML = 0, the term is S; if 1, P; if 2, D; if 3, F; if 4, G and so forth alphabetically (excluding J).

12. The decomposition
The algorithm for finding the term symbols for a given dn electron configuration begins with writing the full microstate table, and identifying the largest value of ML row and determining the number of S columns that have entries within that row – this allows you to write the term symbol
Next, eliminate one entry from each of the “cells” under each S value covered by that term symbol. So your top row might have had only one entry in one column; one entry would be crossed out in every cell below that first cell.

13. The decomposition, part 2
Continue with this process, generating new term symbols and removing entries in the microstate table, until all microstates are accounted for.
The ground state term symbol is the one that first has the highest multiplicity, then next has the letter that corresponds to the largest ML.

14. Free-ion Terms in Oh SA1g PT1g DEg + T2g FA2g + T1g + T2g
GA1g + Eg + T1g + T2g
HEg + 2T1g + T2g
I A1g + A2g + Eg + T1g + 2T2g

15. Tanabe-Sugano diagrams
Yukito Tanabe and Satoru Sugano (University of Tokyo) published their paper On the absorption spectra of complex ions (1954), a quantitative diagram showing the splitting of molecular terms in weak and strong ligand fields

16. Tanabe-Sugano diagram for d2
The x-axis is the ligand field strength; the y-axis is the energy of the orbital
B is the Racah parameter; both axes are in terms of B, which has units of energy
Giulio Racah (Hebrew University of Jerusalem) Irreducible Tensorial Sets (1959)

17. The Racah parameters
There are three of them A, B and C, used to calculate the energy of various molecular term orbitals. The parameters themselves are various measures of electron-electron repulsion.
For d2, E(3P) = A + 7B and E(3F) = A – 8B, so DE = 15B (the A parameter cancelled), which can be calculated in cm–1 units.
Since 3F3P is the egt2g transition, 10 Dq = DO = 15B, so it gives a way to calculate the LFSE.

18. Tanabe-Sugano diagram for d5
Weak field, high spin side
Strong field, low spin side

19. Orgel diagrams
Leslie Orgel (Cambridge University) developed a simplified qualitative (and easily memorized) diagram in Spectra of Transition Metal Complexes (1955) that shows how atomic microstates split into molecular orbital (term) microstates in the presence of a strong ligand field, which is one of its limitations. It also shows only allowed transitions, which is another of its limitations.

20. An Orgel diagram d1, d6 tetrahedral d4, d9 octahedral

21. The other Orgel diagram
d2, d7 octahedral
d3, d8 tetrahedral
d2, d7 tetrahedral
d3, d8 octahedral
tetrahedral
octahedral

22. So to answer the question:
For [Cr(NH3)6]3+, which is d3, the Tanabe-Sugano diagram shows that the ground state (t2g)3 term is 4A2g.
Since the intensity of the bands is moderate, the transitions must be spin-allowed, so the multiplicity of the excited state(s) must be “4”.
However, the excited state is (t2g)2(eg)1, so the transition is gg, which is Laporte forbidden, so the absorption intensity is moderate, not high.

23. The answer:
The lower energy (lower wavenumber, longer wavelength) transition must be 4T2g  4A2g
The higher energy (higher wavenumber, shorter wavelength) transition must be 4T1g  4A2g
Two absorption bands, one transition!

24. Band intensities
The height of an absorption peak is related to the molar extinction coefficient (ε) for a particular electronic transition,
There are two types of electronic transitions: charge-transfer (CT) bands and the d-d transitions on the metal.

25. Different bands
CT bands represent electron density moving from either the metal to the ligand (ML) or the ligand to the metal (LM). Either way, these are both high-intensity, high-energy absorptions.
The d-d transitions are subject to selection rules, which yield allowed transitions. All other transitions are forbidden.

26. Selection rules
Spin-selection rule: Electronic transitions with a change in multiplicity are forbidden.
Laporte selection rule: On a centrosymmetric molecule and ion, the only allowed transitions are those accompanied by a change in parity (e.g., ug or gu)

27. Transition e complexes
Selection Rules
Transition e complexes
Spin forbidden 10-3 – 1 Many d5 Oh cxs
Laporte forbidden [Mn(OH2)6]2+
Spin allowed
Laporte forbidden 1 – 10 Many Oh cxs
[Ni(OH2)6]2+
10 – 100 Some square planar cxs
[PdCl4]2-
100 – coordinate complexes of low symmetry, many square planar cxs particularly with organic ligands
Spin allowed 102 – 103 Some MLCT bands in cxs with unsaturated ligands
Laporte allowed
102 – 104 Acentric complexes with ligands such as acac, or with P donor atoms
103 – 106 Many CT bands, transitions in organic species

28. Fluorescence and phosphorescence
singlet
FAST
SLOW
singlet