Final answer:
The px, dxy, and dz² orbitals belong to different point groups based on their symmetries—C∞v, D2h, and D∞h, respectively. They feature unique geometric arrangements and have signs indicating the phase of the wave functions associated with their lobes.
Step-by-step explanation:
Point Groups of Atomic Orbitals
Identifying the point groups of atomic orbitals involves examining the symmetries present within each orbital's shape and nodal planes. The point group of an orbital determines the types of symmetry operations (like rotations and reflections) that leave the orbital unchanged.
px Orbital
The px orbital has a nodal plane along the yz axis and lobes along the x-axis. The signs on the lobes are positive on the side of the positive x-axis and negative on the side of the negative x-axis. The px orbital belongs to the C∞v point group due to its infinite-fold rotation symmetry around the x-axis and a vertical mirror plane.
dxy Orbital
The dxy orbital is composed of four lobes lying in the xy plane paired two by two with opposite signs. This orbital's point group is D2h because it has a twofold rotation axis along the z-axis and mirror planes corresponding to the xy, xz, and yz planes.
dz² Orbital
The dz² orbital has a distinctive shape with a doughnut (torus) around the waist and two lobes pointing along the z-axis. This orbital is part of the D∞h point group, signifying infinite-fold rotation symmetry around the z-axis and a horizontal mirror plane on the xy plane. The signs of the phases are positive for the lobe along the positive z-axis and negative for the lobe along the negative z-axis as well as the torus region.