Question

The lobes of which $d$ orbitals point directly between the ligands in (a) octahedral geometry, (b) tetrahedral geometry?

Short answer

(a) In octahedral geometry, the $d_{xy},d_{yz}$, and $d_{xz}$ orbitals point directly between the ligands. (b) In tetrahedral geometry, the $d_{x^2-y^2}$ and $d_{z^2}$ orbitals point directly between the ligands.

Step-by-step solution

Understand the geometries of the d orbitals

There are five d orbitals: $d_{xy},d_{yz},d_{xz},d_{x^2-y^2},$ and $d_{z^2}$. The $d_{xy},d_{yz},$ and $d_{xz}$ orbitals have four lobes in the xy, yz, and xz planes, respectively. The $d_{x^2-y^2}$ orbital has four lobes in the x and y directions, and the $d_{z^2}$ orbital has a doughnut-shaped lobe in the xy plane and two additional lobes along the z-axis. Step 2: Analyze the octahedral geometry

Analyze the octahedral geometry

In octahedral geometry, there are six ligands surrounding the central atom, with each ligand located at the corners of an octahedron. The x, y, and z axes bisect the angles formed by the ligands, which means the ligands are placed along the diagonals of the axes. Step 3: Identify the d orbitals for octahedral geometry

Identify the d orbitals for octahedral geometry

Since the ligands are located along the diagonals of the axes in octahedral geometry, the d orbitals that point directly between the ligands are the ones that have lobes between these diagonals. In this case, the $d_{xy},d_{yz}$, and $d_{xz}$ orbitals have lobes pointing directly between the ligands in octahedral geometry. Step 4: Analyze the tetrahedral geometry

Analyze the tetrahedral geometry

In tetrahedral geometry, there are four ligands surrounding the central atom, with each ligand located at the corners of a tetrahedron. The ligands are not located along the x, y, and z axes but rather between these axes. Step 5: Identify the d orbitals for tetrahedral geometry

Identify the d orbitals for tetrahedral geometry

Since the ligands are located between the axes in tetrahedral geometry, the d orbitals that point directly between the ligands are the ones that have lobes along the axes. In this case, the $d_{x^2-y^2}$ and $d_{z^2}$ orbitals have lobes pointing directly between the ligands in tetrahedral geometry. #Answer#: (a) For octahedral geometry, the $d_{xy},d_{yz}$, and $d_{xz}$ orbitals have lobes pointing directly between the ligands. (b) For tetrahedral geometry, the $d_{x^2-y^2}$ and $d_{z^2}$ orbitals have lobes pointing directly between the ligands.