SME
Inorganic Chemistry
Electronic spectra of complexes (Tanabe–Sugano)
Use this for “[Cr(H₂O)₆]³⁺ has its first band at 575 nm, and its two bands put it at Δₒ/B = 23 on the Tanabe–Sugano diagram. Find Δₒ, B and β (free-ion B₀ = 918 cm⁻¹).”
Diagram
Δₒ = 17,391 cm⁻¹ < P = 30,000 cm⁻¹: high spin, every orbital takes one electron before any pairs; t2g³.
3
unpaired;
μ
=
√
3 × 5
=
3.87
BM.
CFSE = (−0.4 × 3 + 0.6 × 0) × 17,391 + 0 × 30,000 = −20,870 cm⁻¹.
Each pair beyond the free ion's 0 costs P: 0 extra.
This is an example. Type your own values to start a new problem.
λ₁
First band
example · nm
ν₁
First band wavenumber
calculated · cm⁻¹
Δₒ
Splitting
calculated · cm⁻¹
Δₒ/B
Splitting over B, read from the diagram
example
B
Racah parameter in the complex
calculated · cm⁻¹
B₀
Free-ion Racah parameter
example · cm⁻¹
β
Nephelauxetic ratio
calculated
Formulas
ν₁
=
10
7
λ₁
17,391
=
10
7
575
Δₒ
=
ν₁
17,391 = 17,391
B
=
Δₒ
Δₒ/
B
756.1
=
17,391
23
β
=
B
B₀
0.8237
=
756.1
918
Assumptions
•
Octahedral d³ or d⁸: the first spin-allowed band lifts one electron from t₂g to eg and nothing else changes, so its energy is Δₒ.
•
The ratio of the second band to the first fixes Δₒ/B on the Tanabe–Sugano diagram: read it there, then B = Δₒ ÷ (Δₒ/B).
•
β = B ÷ B₀ compares the Racah B in the complex with the free ion’s: below 1, the d electrons spread onto the ligands and repel less.
•
The picture draws a d³ ion; its P bar is a stand-in at 30,000 cm⁻¹, since d³ fills t₂g alone at any pairing energy.
Step-by-step
WE KNOW
λ₁ = 575 nm, Δₒ/B = 23, B₀ = 918 cm⁻¹
FIND
first band wavenumber (ν₁), splitting (Δₒ), Racah parameter in the complex (B), nephelauxetic ratio (β)
Step 1 · Find first band wavenumber (ν₁)
Use ν₁ = 10⁷ ÷ λ₁
The
wavenumber
is
1
λ
;
with
λ
in
nm,
10
7
nm
make
1
cm.
ν₁
=
10
7
575
ν₁
=
10,000,000
575
ν₁
=
17,391.3
cm⁻¹
Step 2 · Find splitting (Δₒ)
Use Δₒ = ν₁
For
d
3
and
d
8
the
first
band
moves
one
electron
from
t₂g
to
eg,
so
it
sits
at
Δₒ.
Δₒ = 17,391.3 cm⁻¹
Step 3 · Find Racah parameter in the complex (B)
Use B = Δₒ ÷ (Δₒ/B)
The
diagram
gives
Δₒ
in
units
of
B
;
divide
Δₒ
by
that
ratio.
B
=
17,391.3
23
B
=
756.14
cm⁻¹
Step 4 · Find nephelauxetic ratio (β)
Use β = B ÷ B₀
Compare the repulsion between d electrons in the complex with the free ion’s.
β
=
756.14
918
β
=
0.8237
→
β
below
1:
the
d
cloud
spreads
a
little
onto
the
ligands,
so
its
electrons
repel
less.
Check
Put the numbers back in. Both sides should match.
17,391.3
=
10
7
575
✓
17,391.3 = 17,391.3 ✓
756.14
=
17,391.3
23
✓
0.8237
=
756.14
918
✓
Topic check
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