Racah Parameters B and C from Electronic Spectra

By Aniket Bhardwaj · 7 November 2026 · Inorganic

Electronic spectra of transition metal complexes appear regularly in CSIR NET, GATE, and IIT-JAM. While crystal field splitting (Δ_oct or 10 Dq) reflects ligand field strength, interelectronic repulsion between d electrons requires Racah parameters B and C. Determining B, C, and the nephelauxetic parameter β from UV-Vis absorption bands is essential for solving coordination chemistry problems.

Origin of Racah Parameters in Coordination Complexes

In free transition metal ions, electron-electron repulsions generate distinct Russell-Saunders terms. Giulio Racah expressed these repulsion energies using three parameters: A, B, and C.

Parameter A shifts all terms equally and cancels out in transition differences. Parameter B measures repulsion between terms of the same spin multiplicity; for instance, the free-ion energy difference between ³P and ³F in d⁸ is 15B. Parameter C accounts for spin-pairing energy and governs transitions between states of different spin multiplicities.

Coordination with ligands expands the d-electron cloud (nephelauxetic effect). This reduces electron repulsion, making B in a complex (B_complex) smaller than in the free ion (B_free). The nephelauxetic ratio is β = B_complex / B_free < 1.

Key Formulas for Octahedral Complexes

For high-spin octahedral complexes with ³F or ⁴F ground states, three spin-allowed transitions occur:

• ν₁: ³T_2g ← ³A_2g (d⁸) or ⁴T_2g ← ⁴A_2g (d³)
• ν₂: ³T_1g(F) ← ³A_2g (d⁸) or ⁴T_1g(F) ← ⁴A_2g (d³)
• ν₃: ³T_1g(P) ← ³A_2g (d⁸) or ⁴T_1g(P) ← ⁴A_2g (d³)

Key Equations for d³ and d⁸ Complexes:

1. Crystal Field Splitting:
10 Dq = Δ_oct = ν₁

2. Racah Parameter B:
B = (ν₂ + ν₃ − 3ν₁) / 15

3. Racah Parameter C (from spin-forbidden band):
E(²E_g ← ⁴A_2g) ≈ 9B + 3C ⇒ C = [E(²E_g) − 9B] / 3

4. Nephelauxetic Ratio (β):
β = B_complex / B_free

Units: ν₁, ν₂, ν₃, 10 Dq, B, and C are in cm⁻¹. β is dimensionless.

Exam details

Source: practice problem written for this article (not a past-paper question)
Relevant exams: CSIR NET Chemical Sciences, GATE Chemistry (CY), IIT-JAM Chemistry
Topic: Racah Parameters and Coordination Spectra
Difficulty: Advanced

Worked Examples with Step-by-Step Solutions

Question: An aqueous solution of the octahedral complex [Ni(H₂O)₆]²⁺ (d⁸) displays three absorption bands at ν₁ = 8,500 cm⁻¹, ν₂ = 13,800 cm⁻¹, and ν₃ = 25,300 cm⁻¹. Given B_free = 1,041 cm⁻¹, calculate: (a) 10 Dq, (b) Racah parameter B, and (c) nephelauxetic ratio β.

Solution:

Step 1: Determine 10 Dq.
For octahedral d⁸, the lowest energy transition equals 10 Dq:
10 Dq = ν₁ = 8,500 cm⁻¹

Step 2: Calculate Racah parameter B.
The diagonal sum rule gives: ν₂ + ν₃ = 15B + 3ν₁.
B = (ν₂ + ν₃ − 3ν₁) / 15
ν₂ + ν₃ = 13,800 + 25,300 = 39,100 cm⁻¹
3ν₁ = 3 × 8,500 = 25,500 cm⁻¹
Numerator = 39,100 − 25,500 = 13,600 cm⁻¹
B = 13,600 / 15 = 906.67 cm⁻¹ ≈ 907 cm⁻¹

Step 3: Calculate β.
β = B_complex / B_free = 906.67 / 1,041 = 0.871

Step 4: Second route check.
Recompute sum: 15B + 3ν₁ = 15 × (906.67) + 3 × (8,500) = 13,600 + 25,500 = 39,100 cm⁻¹. This equals ν₂ + ν₃ = 39,100 cm⁻¹. Verified.

Answer: 10 Dq = 8,500 cm⁻¹, B = 907 cm⁻¹, β = 0.871

Question: An octahedral complex [Cr(H₂O)₆]³⁺ (d³) exhibits three spin-allowed bands at ν₁ = 17,400 cm⁻¹, ν₂ = 24,600 cm⁻¹, and ν₃ = 37,800 cm⁻¹. A weak spin-forbidden band corresponding to ²E_g ← ⁴A_2g occurs at 14,900 cm⁻¹. Calculate: (a) Racah parameter B, and (b) Racah parameter C.

Solution:

Step 1: Calculate Racah parameter B.
B = (ν₂ + ν₃ − 3ν₁) / 15
ν₂ + ν₃ = 24,600 + 37,800 = 62,400 cm⁻¹
3ν₁ = 3 × 17,400 = 52,200 cm⁻¹
Numerator = 62,400 − 52,200 = 10,200 cm⁻¹
B = 10,200 / 15 = 680 cm⁻¹

Step 2: Calculate Racah parameter C.
For the spin-forbidden band: E(²E_g) = 9B + 3C.
3C = E(²E_g) − 9B
9B = 9 × 680 = 6,120 cm⁻¹
3C = 14,900 − 6,120 = 8,780 cm⁻¹
C = 8,780 / 3 = 2,926.67 cm⁻¹ ≈ 2,927 cm⁻¹

Step 3: Check ratio C/B.
C/B = 2,926.67 / 680 = 4.30, matching expected values.

Step 4: Second route check.
Calculate E(²E_g) = 9 × 680 + 3 × 2,926.67 = 6,120 + 8,780 = 14,900 cm⁻¹. Verified.

Answer: B = 680 cm⁻¹, C = 2,927 cm⁻¹

Common Mistakes to Avoid

Frequent Errors in Electronic Spectra Calculations:

Equating 10 Dq to ν₂ or ν₃: Only ν₁ equals 10 Dq in d³ and d⁸ octahedral complexes. In d² and d⁷, configuration mixing shifts ν₁.
Incorrect denominator: Using 10 or 12 instead of 15 in the B formula. The value 15 originates from the 15B term gap between ³P and ³F.
Subtracting ν₁ instead of 3ν₁: Because 30 Dq equals 3 × (10 Dq) = 3ν₁, omitting the factor of 3 gives incorrect B values.
Expecting β > 1: Coordination delocalizes metal d electrons, reducing repulsion; hence β is always less than 1.
Confusing B and C: B applies between terms of the same spin multiplicity, while C governs spin-forbidden transitions.

Exam Relevance and Question Patterns

Coordination chemistry spectra questions test the interpretation of absorption bands and extraction of interelectronic parameters.

Examination	Question Types and Focus Areas	Key Skills Tested
CSIR NET Chemical Sciences	Part C questions on Tanabe-Sugano diagrams and nephelauxetic parameters.	Calculating B and β from band maxima and evaluating ligand covalency.
GATE Chemistry (CY)	Numerical Answer Type (NAT) problems on d-d transitions and electronic spectra.	Exact numerical calculation of 10 Dq, B, and C in cm⁻¹.
IIT-JAM Chemistry & CUET-PG	Fundamental questions on term symbols and Orgel diagram transitions.	Identifying ν₁ as 10 Dq for octahedral d⁸ and d³, and calculating β.

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