Vibrational spectroscopy

Elucidation of the structure-function relationships of molecules is one of the most challenging tasks in molecular physics.

Vibrational (Infrared and Raman) Spectroscopy:

provides high structural sensitivity

is not restricted by the size of the sample

probes molecular events under conditions that are closely related to the physico-chemical environment of the reaction

can be a time-resolved method Diatomic molecules: harmonic oscillator

(1)

(2)

Vibrational energy: (3)

(4)

(5) Vibrational terms: Diatomic molecules: molecules : harmonic oscillator

where (6)

Hermite polynomial Diatomic molecules: molecules : Harmonic oscillator

Selection rule: the electric dipole moment of the molecule must change when the vibrational transition occur. dµ / d x ≠ 0 Diatomic molecules: molecules : Anharmonic oscillator

(7)

D0

De

Morse potential: (8)

Vibrational terms: (9)

Selection rule: Diatomic molecules: molecules : Vibrotor A vibrational transition is accompanied by a simultaneous rotational transition (10)

(11)

Selection rules : (12)

J = -1 J = +1

IR active vibration–rotation fundamental Vibration–rotation spectrum of the H-O-H bands of carbon dioxide: antisymmetric bending mode of water vapor stretching mode NHNH 33 molecule

µ The vibrations ofofpolyatomic molecules

Reality Normal modes

Potential energy for a polyatomic molecule

(13)

3N cartesian displacement coordinates for the N atoms, i = 1, 2, . . ., 3N mass-weighted coordinates : (14)

(16)

Potential energy: (17)

fij = force constants

Kinetic energy: (18) Normal modes

Lagrangian : (19)

(20)

(21) Solutions:

(22)

Secular determinant

(23) Normal modes

Amplitudes A ik : (24)

In a given vibr . mode k all atoms vibrate in -phase and with the same frequency λ1/2, but with different amplitudes . group frequency

(25) qi are converted into normal coordinates ( orthogonalization):

(26)

normal modes Normal coordinates

CO 2 vibrations: linear combinations individual CO vibrations Normal modes (G. M. Barrow – Introduction to molecular spectroscopy – Mc GrawHill)

Total energy: (27)

Hamiltonian: (28)

Wavefunction: (29) 3N -6 (5) factors

Schrödinger eq.: (30)

Solutions: (31)

(32) Normal modes

(33)

Vibrational state j: 1, 2,.. modes (34)

Ground state:

-1 Ex: N = 50 atoms, 144 modes, = 300 cm , E 0 = 2.7 eV

Selection rules: (35)

Transition dipole moment µ i`i

(36)

≠ 0 (37) Normal modes

Normal modes of CO 2 (4 modes = 9-5):

s (inactive)

as

parallel bands perpendicular bands

μ μ μ ε (ν) Normal modes

Asymmetric 3 2349 cm-1:

Symmetric: 1388 cm-1 1 Normal modes

Normal modes of H 2O (3 modes = 9-6): The effects ofof anharmonicity

(38)

overtones : νi, 2 νi, 3 νi,….

combination bands: νi+ νj, νi- νj, νi+ 2νj…. Fermi resonance Normal modes:modes : classification Skeletal vibrations (fingerprint) 1400-200 cm -1 : involve significant displacements of almost all atoms in the molecule. Far IR: C-Cl, C-Br, C-I bending modes below 400 cm -1; restricted rotational oscillations (librations) of molecules in liquid state, ring breathing benzene 260 cm-1 Group frequencies 4000-1400 cm -1: stretching, bending modes involve only a small group of the molecule, the rest is almost stationary. ν (O-H) : 3590-3650 cm -1 -1 ~ 2050 cm -1 ~ 1650 cm -1 ~ 900 cm rock twist scissors

wag torsion