6.3.4 The d-d Splitting Pattern in Octahedral and Tetrahedral Complexes

In the previous section, we learned that the color of transition metal complexes arises from d-d electronic transitions. Now, we will look more closely at how the d orbitals split in energy when ligands are present. The splitting pattern depends on the geometry of the complex. The two most common geometries for transition metal complexes are octahedral and tetrahedral. Understanding the splitting pattern in each case is essential for explaining the magnetic and spectral properties of these compounds.

Main Idea

The main idea is that the presence of ligands causes the five degenerate d orbitals of a transition metal ion to split into two sets of different energies. The way they split depends on the arrangement of the ligands around the metal ion. In an octahedral complex, the d orbitals split into a lower energy set of three (t_2g) and a higher energy set of two (e_g). In a tetrahedral complex, the splitting is reversed: the lower energy set has two orbitals (e) and the higher energy set has three orbitals (t₂), and the energy gap (Δ_t) is generally smaller than in octahedral complexes.

Core Concept

The core concept is that the electrostatic repulsion between the ligand electrons and the metal d electrons determines the energy of each d orbital. Orbitals that point directly toward the ligands experience greater repulsion and are raised to a higher energy level. Orbitals that point between the ligands experience less repulsion and remain at a lower energy level.

Detailed Explanation

Octahedral Complexes

The textbook uses the example of the aqua complex of titanium(III), [Ti(H_2O)_6]³⁺, to explain the splitting in octahedral complexes.

• Geometry: The complex has an octahedral shape, with six water molecules acting as ligands around the titanium ion.
• Splitting:
• The three t_2g orbitals (d_xy, d_yz, and d_xz) lie between the six ligands. Therefore, they experience less repulsion from the ligands' lone pairs and have lower energy.
• The two e_g orbitals (d_x²-y² and d_z²) are aligned directly toward the ligands. They experience more repulsion and have higher energy.
• Energy Gap (Δ E or Δ_o): The energy difference between the t_2g and e_g sets is called the crystal field splitting energy (Δ_o for octahedral).
• Color: When white light strikes [Ti(H_2O)_6]³⁺, the complex absorbs yellow-green light to excite an electron from the t_2g to the e_g set. The complementary color observed is red-violet.

The textbook provides a diagram (Fig. 6.3) showing the splitting of degenerate d orbitals into the lower t_2g set and the higher e_g set when ligands approach.

Tetrahedral Complexes

The textbook also discusses the splitting pattern in tetrahedral complexes, although in less detail.

• Geometry: In a tetrahedral complex, four ligands are arranged around the metal ion at the corners of a tetrahedron.
• Splitting: The splitting pattern is the inverse of the octahedral case.
• The two orbitals that are closer to the ligands (the d_z² and d_x²-y² orbitals, now forming the e set) experience greater repulsion and have higher energy.
• The three orbitals that are further from the ligands (the d_xy, d_yz, and d_xz orbitals, now forming the t₂ set) experience less repulsion and have lower energy.
• Energy Gap (Δ_t): The energy gap between the e and t₂ sets is denoted as Δ_t. Importantly, Δ_t is generally smaller than Δ_o (the octahedral splitting). This is because there are fewer ligands (4 instead of 6) and they are not as directly aligned with the d orbitals. As a result, tetrahedral complexes are often less intensely colored than octahedral complexes.
• Key Point: The textbook states that the splitting pattern is different in octahedral and tetrahedral complexes. For octahedral, it's two higher and three lower; for tetrahedral, it's three higher and two lower.

Summary of Splitting Patterns

Intermediate Level

The d-d splitting pattern depends on the geometry of the complex. In an octahedral complex, the six ligands approach along the x, y, and z axes. The d_x²-y² and d_z² orbitals (the e_g set) point directly at the ligands and experience greater repulsion, so they are higher in energy. The d_xy, d_yz, and d_xz orbitals (the t_2g set) point between the ligands, so they are lower in energy. The energy gap is Δ_o. In a tetrahedral complex, four ligands approach at the corners of a tetrahedron. The splitting is reversed: the e set (two orbitals) is higher in energy, and the t₂ set (three orbitals) is lower. The energy gap, Δ_t, is smaller than Δ_o because there are fewer ligands and they are not as directly aligned with the d orbitals.

Important Points

• - Octahedral Splitting: Lower t_2g (3 orbitals), higher e_g (2 orbitals), gap = Δ_o.
• - Tetrahedral Splitting: Lower t₂ (3 orbitals), higher e (2 orbitals), gap = Δ_t.
• - Magnitude: Δ_t < Δ_o (tetrahedral splitting is smaller).
• - Repulsion: Orbitals pointing directly at ligands have higher energy.
• - Color: The size of the energy gap determines the color absorbed.
• - Magnetic Properties: The splitting pattern affects the arrangement of electrons and thus the magnetic properties.

Remember

• - Octahedral: t_2g (lower, 3), e_g (higher, 2).
• - Tetrahedral: t₂ (lower, 3), e (higher, 2).
• - Splitting is inverse for tetrahedral vs. octahedral.
• - Δ_t < Δ_o.
• - Orbitals pointing at ligands = higher energy.
• - Orbitals between ligands = lower energy.
• - Example: [Ti(H_2O)_6]³⁺ is octahedral, absorbs yellow-green, appears red-violet.

Common Mistakes

• - Thinking that the d orbitals split the same way in all geometries. The splitting pattern is different for octahedral and tetrahedral complexes.
• - Forgetting which set is higher in energy. In octahedral, the e_g set is higher; in tetrahedral, the e set is higher.
• - Assuming that Δ_t is larger than Δ_o. It is generally smaller.
• - Confusing the labels t_2g and t₂. The subscript "g" (for gerade) is used in octahedral complexes (which have a center of symmetry), but not in tetrahedral complexes.
• - Not relating the splitting to the color. The energy gap determines the color of light absorbed and, consequently, the observed color.

Quick Revision

The d orbitals of a transition metal ion split into two sets of different energies in a ligand field. In an octahedral complex, the three t_2g orbitals (which point between the ligands) are lower in energy, and the two e_g orbitals (which point directly at the ligands) are higher in energy. The energy gap is Δ_o. In a tetrahedral complex, the splitting is inverted: the three t₂ orbitals are lower, and the two e orbitals are higher, with a smaller energy gap Δ_t. This splitting pattern is responsible for the color and magnetic properties of the complexes.

60-Second Revision

In an octahedral complex, the d orbitals split into a lower energy t_2g set (3 orbitals) and a higher energy e_g set (2 orbitals), with a gap of Δ_o. In a tetrahedral complex, the splitting is inverted: lower t₂ (3 orbitals) and higher e (2 orbitals), with a smaller gap Δ_t. The splitting pattern determines the color and magnetic properties of the complex.

Frequently Asked Questions (FAQs)

Q1. What are degenerate and non-degenerate 3d orbitals? Answer: Degenerate orbitals are orbitals that have the same energy. In an isolated transition metal ion, the five 3d orbitals are degenerate. Non-degenerate orbitals are orbitals that have different energies. When ligands approach the metal ion to form a complex, the d orbitals become non-degenerate due to the different repulsions they experience. They split into two sets: the lower energy t_2g set and the higher energy e_g set (in an octahedral complex).

Q2. Why do degenerate 3d orbitals split into two sets of orbitals in some complexes while not in others? Answer: The splitting of d orbitals occurs because the ligands create an asymmetric electric field around the metal ion. The d orbitals that point directly toward the ligands experience greater repulsion than those that point between the ligands. This difference in repulsion causes the orbitals to have different energies, i.e., to split. In a completely spherical field (which is not the case in a complex), the orbitals would remain degenerate. The splitting occurs in all complexes, but the pattern depends on the geometry.

Q3. Explain the d-d splitting pattern in an octahedral complex. Answer: In an octahedral complex, six ligands approach the metal ion along the x, y, and z axes. The two d orbitals that point directly along these axes (d_x²-y² and d_z², the e_g set) experience greater repulsion and are raised to a higher energy level. The three d orbitals that point between the axes (d_xy, d_yz, and d_xz, the t_2g set) experience less repulsion and remain at a lower energy level. The energy difference between the two sets is Δ_o.

Q4. Explain the d-d splitting pattern in a tetrahedral complex. Answer: In a tetrahedral complex, four ligands approach the metal ion at the corners of a tetrahedron. The splitting pattern is the inverse of the octahedral case. The two d orbitals that are closer to the ligands (the e set) experience greater repulsion and are higher in energy. The three d orbitals that are further from the ligands (the t₂ set) experience less repulsion and are lower in energy. The energy gap is Δ_t, which is generally smaller than Δ_o.

Q5. Which different ligands cause different splitting energy (Δ E)? Give one example. Answer: Different ligands cause different degrees of splitting. Strong-field ligands (like CN^-) cause a larger splitting (Δ E), while weak-field ligands (like H_2O or Cl^-) cause a smaller splitting. For example, the CN^- ligand causes a much larger splitting than the H_2O ligand, which is why complexes with CN^- are often more intensely colored or have different colors than aqua complexes.

Q6. What is the difference between Δ_o and Δ_t? Answer: Δ_o is the crystal field splitting energy in an octahedral complex (the energy difference between the t_2g and e_g sets). Δ_t is the crystal field splitting energy in a tetrahedral complex (the energy difference between the t₂ and e sets). In general, Δ_t is smaller than Δ_o (approximately 49 of Δ_o).

Q7. Why are tetrahedral complexes often less intensely colored than octahedral complexes? Answer: Tetrahedral complexes have a smaller splitting energy (Δ_t < Δ_o). This means the energy gap between the lower and higher d orbitals is smaller, and the absorption of light occurs at a lower energy (longer wavelength). The smaller gap also means that the color is often less intense because the absorption band is broader and the transition is less probable (due to the lack of a center of symmetry in tetrahedral complexes).

Q8. What is the color of the [Ti(H_2O)_6]³⁺ complex and why? Answer: The [Ti(H_2O)_6]³⁺ complex is red-violet. It absorbs yellow-green light, and the complementary color of yellow-green is red-violet. The absorption corresponds to the excitation of an electron from the lower t_2g set to the higher e_g set.

Multiple Choice Questions (MCQs)

Q1. In an octahedral complex, which d orbitals are higher in energy? A. t_2g B. e_g C. Both are equal D. It depends on the metal Correct Answer: B Explanation: The e_g set (d_x²-y² and d_z²) points directly at the ligands and experiences greater repulsion, so it is higher in energy than the t_2g set.

• Q2. What is the name of the lower energy set of d orbitals in a tetrahedral complex?
• A. e_g
• B. t_2g
• C. t₂
• D. e
• Correct Answer: C
• Explanation: In a tetrahedral complex, the lower energy set is the t₂ set, which consists of the d_xy, d_yz, and d_xz orbitals.

• Q3. Which of the following statements about the splitting in tetrahedral complexes is correct?
• A. Δ_t > Δ_o
• B. Δ_t = Δ_o
• C. Δ_t < Δ_o
• D. There is no splitting in tetrahedral complexes.
• Correct Answer: C
• Explanation: The splitting energy in tetrahedral complexes (Δ_t) is generally smaller than that in octahedral complexes (Δ_o).

• Q4. In an octahedral complex, the d_x²-y² and d_z² orbitals belong to the:
• A. t_2g set
• B. e_g set
• C. t₂ set
• D. e set
• Correct Answer: B
• Explanation: The d_x²-y² and d_z² orbitals point along the axes and belong to the higher energy e_g set in an octahedral complex.

• Q5. The energy gap between the t_2g and e_g orbitals is called:
• A. Crystal field splitting energy (Δ_o)
• B. Ionization energy
• C. Lattice energy
• D. Hydration energy
• Correct Answer: A
• Explanation: The energy gap is called the crystal field splitting energy (Δ_o for octahedral complexes).

• Q6. In a tetrahedral complex, the d_xy, d_yz, and d_xz orbitals belong to the:
• A. e_g set
• B. t_2g set
• C. t₂ set
• D. e set
• Correct Answer: C
• Explanation: In a tetrahedral complex, the d_xy, d_yz, and d_xz orbitals belong to the lower energy t₂ set.

• Q7. Why is the splitting energy in tetrahedral complexes smaller than in octahedral complexes?
• A. Because there are fewer ligands and they are not as directly aligned with the d orbitals.
• B. Because the ligands are always weaker.
• C. Because the metal ion is always larger.
• D. Because there is no d-d splitting in tetrahedral complexes.
• Correct Answer: A
• Explanation: The smaller splitting is due to the fewer number of ligands (4 instead of 6) and their less direct alignment with the d orbitals.

• Q8. In an octahedral complex, which set of orbitals is lower in energy?
• A. e_g
• B. t_2g
• C. Both are equal
• D. It depends on the ligands
• Correct Answer: B
• Explanation: The t_2g set is lower in energy because these orbitals point between the ligands and experience less repulsion.

• Q9. The [Ti(H_2O)_6]³⁺ complex absorbs yellow-green light. What is its color?
• A. Blue
• B. Green
• C. Red-violet
• D. Orange
• Correct Answer: C
• Explanation: The complementary color of yellow-green is red-violet.

• Q10. In a tetrahedral complex, which set of orbitals is higher in energy?
• A. e
• B. t₂
• C. Both are equal
• D. It depends on the metal
• Correct Answer: A
• Explanation: In a tetrahedral complex, the e set (two orbitals) is higher in energy, and the t₂ set (three orbitals) is lower.

Multiple Choice Questions (MCQs)

• Q1. In an octahedral complex, which d orbitals are higher in energy?
• A. t_2g
• B. e_g
• C. Both are equal
• D. It depends on the metal
• Correct Answer: B
• Explanation: The e_g set is higher in energy.

• Q2. What is the name of the lower energy set of d orbitals in a tetrahedral complex?
• A. e_g
• B. t_2g
• C. t₂
• D. e
• Correct Answer: C
• Explanation: In a tetrahedral complex, the lower energy set is the t₂ set.

• Q3. Which of the following statements about the splitting in tetrahedral complexes is correct?
• A. Δ_t > Δ_o
• B. Δ_t = Δ_o
• C. Δ_t < Δ_o
• D. There is no splitting in tetrahedral complexes.
• Correct Answer: C
• Explanation: The splitting energy in tetrahedral complexes (Δ_t) is generally smaller than that in octahedral complexes (Δ_o).

• Q4. In an octahedral complex, the d_x²-y² and d_z² orbitals belong to the:
• A. t_2g set
• B. e_g set
• C. t₂ set
• D. e set
• Correct Answer: B
• Explanation: The d_x²-y² and d_z² orbitals belong to the e_g set.

• Q5. The energy gap between the t_2g and e_g orbitals is called:
• A. Crystal field splitting energy (Δ_o)
• B. Ionization energy
• C. Lattice energy
• D. Hydration energy
• Correct Answer: A
• Explanation: The energy gap is called the crystal field splitting energy (Δ_o).

• Q6. In a tetrahedral complex, the d_xy, d_yz, and d_xz orbitals belong to the:
• A. e_g set
• B. t_2g set
• C. t₂ set
• D. e set
• Correct Answer: C
• Explanation: In a tetrahedral complex, these orbitals belong to the lower energy t₂ set.

• Q7. Why is the splitting energy in tetrahedral complexes smaller than in octahedral complexes?
• A. Because there are fewer ligands and they are not as directly aligned with the d orbitals.
• B. Because the ligands are always weaker.
• C. Because the metal ion is always larger.
• D. Because there is no d-d splitting in tetrahedral complexes.
• Correct Answer: A
• Explanation: The smaller splitting is due to the fewer number of ligands and their less direct alignment.

• Q8. In an octahedral complex, which set of orbitals is lower in energy?
• A. e_g
• B. t_2g
• C. Both are equal
• D. It depends on the ligands
• Correct Answer: B
• Explanation: The t_2g set is lower in energy.

• Q9. The [Ti(H_2O)_6]³⁺ complex absorbs yellow-green light. What is its color?
• A. Blue
• B. Green
• C. Red-violet
• D. Orange
• Correct Answer: C
• Explanation: The complementary color of yellow-green is red-violet.

• Q10. In a tetrahedral complex, which set of orbitals is higher in energy?
• A. e
• B. t₂
• C. Both are equal
• D. It depends on the metal
• Correct Answer: A
• Explanation: In a tetrahedral complex, the e set is higher in energy.

Multiple Choice Questions (MCQs)

• Q1. In an octahedral complex, the t_2g set consists of how many orbitals?
• A. 2
• B. 3
• C. 4
• D. 5
• Correct Answer: B
• Explanation: The t_2g set consists of three orbitals: d_xy, d_yz, and d_xz.

• Q2. In a tetrahedral complex, the e set consists of how many orbitals?
• A. 2
• B. 3
• C. 4
• D. 5
• Correct Answer: A
• Explanation: The e set in a tetrahedral complex consists of two orbitals: d_x²-y² and d_z².

• Q3. Which of the following is true about the splitting pattern in octahedral complexes?
• A. The t_2g set is higher in energy than the e_g set.
• B. The e_g set is higher in energy than the t_2g set.
• C. Both sets have the same energy.
• D. The splitting depends on the metal.
• Correct Answer: B
• Explanation: In octahedral complexes, the e_g set is higher in energy than the t_2g set.

• Q4. Which of the following is true about the splitting pattern in tetrahedral complexes?
• A. The t₂ set is higher in energy than the e set.
• B. The e set is higher in energy than the t₂ set.
• C. Both sets have the same energy.
• D. The splitting depends on the metal.
• Correct Answer: B
• Explanation: In tetrahedral complexes, the e set is higher in energy than the t₂ set.

• Q5. The color of a transition metal complex is due to:
• A. The metal ion only.
• B. The ligands only.
• C. d-d electronic transitions.
• D. The solvent.
• Correct Answer: C
• Explanation: The color is due to d-d electronic transitions, which depend on the splitting of the d orbitals.

• Q6. The energy gap Δ in a complex is the energy difference between:
• A. The t_2g and e_g sets.
• B. The t₂ and e sets.
• C. The two sets of d orbitals formed by splitting.
• D. All of the above (depending on the geometry).
• Correct Answer: D
• Explanation: The energy gap Δ is the difference between the lower and higher energy d orbital sets, which depends on the geometry.

• Q7. In an octahedral complex, the d_xy orbital is:
• A. Higher in energy than the d_z² orbital.
• B. Lower in energy than the d_z² orbital.
• C. The same energy as the d_z² orbital.
• D. Not involved in the splitting.
• Correct Answer: B
• Explanation: The d_xy orbital is part of the lower energy t_2g set, while the d_z² orbital is part of the higher energy e_g set.

Q8. In a tetrahedral complex, the d_xy orbital is: A. Higher in energy than the d_z² orbital. B. Lower in energy than the d_z² orbital. C. The same energy as the d_z² orbital. D. Not involved in the splitting. Correct Answer: B Explanation: In a tetrahedral complex, the d_xy orbital is part of the lower energy t₂ set, while the d_z² orbital is part of the higher energy e set.

• Q9. The magnitude of the splitting energy Δ depends on:
• A. The nature of the metal ion.
• B. The oxidation state of the metal ion.
• C. The nature of the ligands.
• D. All of the above.
• Correct Answer: D
• Explanation: The splitting energy depends on the metal ion, its oxidation state, and the ligands (spectrochemical series).

• Q10. Which of the following complexes would have the largest splitting energy?
• A. [Ti(H_2O)_6]³⁺
• B. [Ti(CN)_6]^3-
• C. [TiCl_6]^3-
• D. [Ti(NH_3)_6]³⁺
• Correct Answer: B
• Explanation: CN^- is a strong-field ligand and causes a larger splitting than H_2O, Cl^-, or NH_3.