Question

For tetrahedral complex, the correct order of energy of \(d\)-orbitals is:

• A. \(d_{xy} = d_{xz}>d_{x^2-y^2}\)
• B. \(d_{xy} = d_{yz}>d_{zx}\)
• C. \(d_{x^2-y^2}>d_{z^2}>d_{xz}\)
• D. \(d_{x^2-y^2} = d_{z^2}<d_{xy} = d_{yz} = d_{xz}\)

Answer 1

The Correct Option is D

Step 1: Understand crystal field splitting in a tetrahedral complex.
In a tetrahedral complex, the four ligands approach the central metal ion along directions that lie between the Cartesian axes. This is different from an octahedral complex, where ligands approach directly along the axes. Because of this geometry, the repulsion experienced by different \(d\)-orbitals changes.
Step 2: Identify which orbitals face the ligands more strongly.
The \(d_{xy}\), \(d_{yz}\), and \(d_{xz}\) orbitals are oriented more toward the incoming ligands in a tetrahedral field. Therefore, these orbitals experience greater repulsion and their energy increases. These three orbitals form the \(t_2\) set.
On the other hand, the \(d_{x^2-y^2}\) and \(d_{z^2}\) orbitals are directed along the axes, while the ligands in a tetrahedral arrangement do not come directly along these axes. So these orbitals experience comparatively less repulsion and remain at lower energy. These two orbitals form the \(e\) set.
Step 3: Write the splitting pattern.
Thus, for a tetrahedral complex:
\[ e<t_2 \] That means:
\[ d_{x^2-y^2} = d_{z^2}<d_{xy} = d_{yz} = d_{xz} \] So the lower energy set is \(d_{x^2-y^2}\) and \(d_{z^2}\), and the higher energy set is \(d_{xy}\), \(d_{yz}\), and \(d_{xz}\).
Step 4: Compare with the given options.

• (A) Incorrect, because it does not represent the proper tetrahedral splitting pattern.
  
• (B) Incorrect, because in tetrahedral field all three \(t_2\) orbitals are degenerate, not just two of them.
  
• (C) Incorrect, because \(d_{x^2-y^2}\) and \(d_{z^2}\) are degenerate and lower in energy, not higher.
  
• (D) Correct, because it shows the proper tetrahedral splitting: \(e\) set lower and \(t_2\) set higher.
  

Step 5: Conclusion.
Therefore, in a tetrahedral complex, the energy order of orbitals is:
\[ d_{x^2-y^2} = d_{z^2}<d_{xy} = d_{yz} = d_{xz} \] Hence, option (D) is correct.
Final Answer:} \(d_{x^2-y^2} = d_{z^2}<d_{xy} = d_{yz} = d_{xz}\)