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 NH33 NHmolecule 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.