In multielectron atoms, the transition an electron from a ground state to an excited state involves energy changes that include the difference in the orbital energies and the changes in electron-electron repulsion. This makes understanding electronic spectroscopy of transition metal complexes more complicated than just moving an electron from a t2g orbital to an eg and saying the energy required is 10Dq.
Consider the free ions: the electron occupation of the orbitals lead to different energies in most cases.
d1 _|_ ___ ___ ___ ___ = ___ _|_ ___ ___ ___ = ___ ___ _|_ ___ ___ etc – this is called a 2D state (the superscript refers to the spin quantum number = 2S + 1 = the spin degeneracy)
d2 _|_ _|_ ___ ___ ___ = ___ _|_ _|_ ___ ___ = _|_ ___ _|_ ___ ___ etc – this is the 3F state
_||_ ___ ___ ___ ___ = ___ _|| ___ ___ ___ etc – this is the 1D state
There are a total of 5 different states for d2
For free ions, the states are
d2, d8 3F, 3P, 1G, 1D, 1S
d3, d7 4F, 4P, 2H, 2G, 2F, 2D, 2D, 2P
d4, d6 5D, 3H, 3G, 3F, 3F, 3D, 3P, 3P + singlet states
d5 6S, 4G, 4F, 4D, 4P, + doublet states
The energies of all the states can be found using Condon-Shortley parameters (F0, F2, F4) or Racah parameters (A, B, C). Racah parameters are usually preferred because the difference in energies of the lowest energy states of the same spin degeneracy can written in terms of just B.
In an Oh field these state split into more states, which can be found using group theory.
D becomes T2g + Eg
F becomes T1g + T2g + A2g
P becomes T1u
S becomes A1g
The states in an Oh field also retain the spin degeneracy. For example, in a d2 case the free ion state 3F becomes 3T1g + 3T2g + 3A2g in an Oh field.
Transitions between are allowed between states of the same spin degeneracy plus the selection rules based on group theory: the direct product of the ground state, the excited state, and the electric field operator (the irreducible rep(s) for x, y, or z) must contain the totally symmetric representation (a1g in Oh). For true Oh complexes this never happens so all electronic spectra in Oh complexes are relatively weak.
Usually done spectroscopically, move electron from t2g to eg orbital with no spin change
hν = 10Dq in this case
Because of electron–electron repulsion, the lowest energy transition is not always equal to 10Dq.
Configuration Lowest Energy Spin–Allowed Transition d1 10Dq d2 8Dq d3 10Dq d4(hs) 10Dq d4 (ls) ~ 9Dq d5 (hs) No spin–allowed transition d5 (ls) ~ 8.5 Dq d6(hs) 10Dq d6 (ls) ~ 9Dq d7 (hs) 10Dq d7 (ls) ~ 9Dq d8 8Dq d9 10Dq
Complication : charge transfer transitions
Metal to Ligand Charge Transfer (MLCT)
Ligand to Metal Charge Transfer (LMCT)
CT transitions are usually much more intense than d–d transitions so can be distinguished by molar absorptivity
ε (CT) ~ 103 – 104 L/mol–cm
ε (d–d spin allowed) ~ 101 – 102 L/mol–cm
ε (d–d spin forbidden) ~ 10–1 – 100 L/mol–cm
Exact formulas for some cases have been published: A. B. P. Lever, J. Chem. Ed., 1968, 45, 711 – 712; Y.-S. Dou, J. Chem. Ed., 1990, 67, 134.
For d3 and d8 complexes three spin allowed transitions are predicted at energies given by
ν1 = 10Dq
ν2 = 7.5B + 15Dq – 0.5(225B2 + 100Dq2 -180BDq)½
ν3 = 7.5B + 15Dq + 0.5(225B2 + 100Dq2 -180BDq)½
If all three transitions are observed, then B can easily found from 15B = ν1 + ν2 – 3ν3
If ν3 is obscured by a CT band, which is often the case, then B can be found by 3B = [(ν2 – 2ν1)(ν2 – ν1)]/[5ν2 – 9ν1]
Likewise, for d2 and d7(hs) complexes three spin allowed transitions are predicted at energies given by
ν1 = –7.5B + 5Dq + 0.5(225B2 + 100Dq2 -180BDq)½
ν2 = –7.5B + 15Dq + 0.5(225B2 + 100Dq2 -180BDq)½
ν3 = (225B2 + 100Dq2 -180BDq)½
If all three transitions are observed, then 10Dq = ν2 – ν1 and 15B = ν3 + ν2 – 3ν1
If ν3 is obscured by a CT band, then B can be found by 3B = [(ν2 – 2ν1)ν1]/[9ν1 – 4ν2]
When simple formulas don't work or there are spin-forbidden transitions observed, then Tanabe-Sugano diagrams are used