Group Theory with a Computer

Tetrahedral Symmetry - Methane

Methane has infrared active modes at 1306 and 3019 cm^-1. It has Raman modes at 1306, 1534, 2917, and 3019 cm^-1.

E C₃ C₂ S₄ s_d

E C₃ C₂ S₄ s_d

2z² - x² - y², x² - y²

x² + y² + z²

(R_x, R_y, R_z)

(x, y, z), (xy, xz, yz)

E C₃ C₂ S₄ s_d

Record the number of unmoved atoms under the
symmetry operations of the tetrahedral group.

E C₃ C₂ S₄ s_d

Record the number of unmoved bonds under the
symmetry operations of the tetrahedral group.

E C₃ C₂ S₄ s_d

Calculate the reducible representation
for the total degrees of freedom.

E C₃ C₂ S₄ s_d

Calculate the reducible representation
for the vibrational degrees of freedom.

Calculate the symmetry of
the vibrational modes.

Symmetry of vibrational modes: A₁, E, and 2T₂. The T₂ modes are IR active. All modes are Raman active.

Calculate the symmetry of
the stretching modes.

Symmetry of stretching modes: A₁ and T₂

Determine the symmetry of
the bending modes.

Symmetry of bending modes: E and T₂

Assignment:

IR: bend 1306 cm^-1(T₂); stretch 3019 cm^-1 (T₂).

Raman: stretches 3019 (T₂) & 2917 (A₁); Raman bends 1534 (E) & 1306 (T₂) cm^-1.