Group Theory And Symmetry Mock Tests

The point group of sulphur hexafluoride (SF₆) is O_h. Using group theory, how many infrared-active vibrational modes does SF₆ possess? (SF₆ has 6 atoms, giving 3N − 6 = 12 vibrational degrees of freedom distributed among the irreducible representations of O_h.)

The point group of sulfur hexafluoride (SF₆) is:

The point group of allene (H₂C=C=CH₂) in its equilibrium geometry, assuming the two CH₂ groups lie in perpendicular planes, is:

The number of IR-active vibrational modes of carbon dioxide (CO₂) is:

The water molecule (H₂O) belongs to the C₂v point group. How many irreducible representations does the C₂v character table have, and what are their symmetries?

The point group of xenon difluoride (XeF₂) is:

The point group of allene (H₂C=C=CH₂) is:

Ammonia (NH₃) belongs to the C₃v point group. The character table for C₃v has three irreducible representations: A₁, A₂, and E. The z coordinate transforms as A₁, the (x, y) pair transforms as E, and the rotations (R_x, R_y) transform as E while R_z transforms as A₂. Based on selection rules for spectroscopic transitions, which vibrational modes of NH₃ are infrared-active? (NH₃ has 3N − 6 = 6 vibrational modes with symmetry species 2A₁ + 2E.)