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 | |
---|---|---|

CO_{2} | 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

https://chem.libretexts.org/Reference/Reference_Tables/Group_Theory_Tables/Character_Tables

https://en.wikipedia.org/wiki/Molecular_symmetry

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.

Character Tables:

CO_{2} - cyclic C_{nv} Group ( n-fold proper rotation axis C*n* )

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

J total angular momentum quantum number

s + l

https://en.wikipedia.org/wiki/Total_angular_momentum_quantum_number

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