For the formaldehyde molecule, H₂CO having C_2v symmetry with the character table as given below,

C2v	E	C2	σv (xz)	σv (yz)
A1	1	1	1	1	z
A2	1	1	-1	-1	Rz
B1	1	-1	1	-1	x, Ry
B2	1	-1	-1	1	y, Rx

the reducible representation Γ_3N (or Γ_tot) is Γ_3N = 4A₁ + A₂ + 4B₁ + 3B₂. The reducible representation for the vibrational modes alone, namely Γ_vib will be

Concept:-

• The general form of a character table is

(a) Gives the Schonflies symbol for the point group.

(b) Lists the symmetry operations by class) for that group.

(c) Lists the characters, for all irreducible representations of each class of operation.

(d) Shows the irreducible representation for which six vectors x, y, z, and Rx, Ry, and Rz provide the basis.

(e) Shows how functions that are binary combinations of x,y, and z (xy or z2) provide bases for certain irreducible representations.

(f) Γ3N or Γtot = Γ_trans + Γrot + Γvib

(g) The irreducible representations along the x, y, and z axis represent translational motion, along Rx, Ry, and Rz represent translational motion, and the vibrational modes will be

Γvib = Γtot - (Γtrans + Γrot)

Explanation:

• For the formaldehyde molecule, H2CO having C2v symmetry with the character table as given below,

• Given, the reducible representation Γ3N (or Γtot) is

Γ3N = 4A1 + A2 + 4B1 + 3B2

• The reducible representation for the translational motion are A1 + B1 + B2 
• Now, the reducible representation for the rotational motion are A2 + B₁ + B2
• Thus, the reducible representation for the vibrational modes alone, namely Γvib will be

Γvib = Γ3N - (Γtrans + Γrot)

Γvib = 4A1 + A2 + 4B1 + 3B2 - (A1 + B1 + B2 + A2 + B1 + B2)

= 4A1 + A2 + 4B1 + 3B2 - (A1 + 2B1 + 2B2 + A2)

= 3A1 + 2B1 + B2  

Conclusion:-

Hence, Γvib will be 3A1 + 2B1 + B2