Vibration Modes: Normal modes

Normal Modes - vibration modes in molecules: molecular topology - classified under Group Theory, distinguishing features: orthogonal to each other, symmetry, and frequencies often predicting wavelength range of active in sample (IR and/or Raman active).

Degrees of Freedom (3N where N is the #nuclei per molecule)

3 coordinates translation motion

3 coordinates rotation motion

3n - m (where m is 6 or 5, pending on if it's non-linear or linear, respectively)

Molecule	vmodes
CO₂	4	n=3; 3n - 5 = 3(3) - 5
CO	1	n=2; 3n - 5 = 3(2) - 5

v = stretching ∆ bond length

δ = bending ∆ in bond angle

ρ_r = rocking ∆ in angle between group of atoms

ρ_w= wagging ∆ in angle between plane of group of atoms

ρ_t = twisting ∆ in angle between planes of two groups of atoms

π = out of plane

s = symmetric

as = symmetric

d = degenerate

A rotation about principal axis = symmetrical

B rotation about principal axis = asymetrical

E & T doubly and triply degenerate

inversion center: g = no change in sign, u = change in sign w/ respect to inversion.

CO₂ - cyclic C_nv Group ( n-fold proper rotation axis Cn )

E, C2, σ (xz) & σ (yz)

C2v	E	C2	σ (xz)	σ (yz)
v1	1	1	1	1	a1
v2	1	1	1	1	a2
v3	1	-1	-1	1	b2

A_1g, b_1g, e_u = stretching vibrations

b_2g, a_2u, b_2u, e_u = bending vibrations

s + l

J and branches: P ∆J = -1 Branch, Q - ∆J = 0 Branch, R - ∆J = 1 Branch