Chapter 8. The vibrational and rotational spectroscopy of diatomic molecules General features nteraction of electromagnetic field with atoms/molecules Spectral range: Wave number ~ (frequency ) Wavelength Radio MW R VS UV X-ray rot. vib. elec. (valence) elec. (core) Absorption and emission spectroscopy Photon is absorbed or emitted when molecule makes transition h E E emission absorption Raman Raman spectroscopy: Light scattered by molecule ( h event) h E E s i
A spectrometer typically contains i. radiation source and dispersion element ii. sample chamber iii. detector eer-lambert law: log 0 [ C] L, C] L 0 0 [, 0 : intensity [C]: concentration of species L: length : absorption coefficient (L mol- cm-) A [ C] L: absorbance T / 0 : transmittance A logt Very useful in analytical chemistry to measure concentration. Transition probabilities There are three types of transitions: stimulated absorption, stimulated emission, spontaneous emission. Absorption rate (single molecule) w
Emission rate w A Rate coefficients: A: Einstein coefficient of spontaneous emission. ':...... stimulated emission. :........ absorption. is energy density of radiation, for black-body 3 8h (Planck s dist.) 3 h / kt c e Total rates of emission and absorption: W Nw, W Nw N, N': numbers of molecules in E and E. Populations of states At a given temperature T, the population of states is given by the oltzmann factor: N f e e N E / kt hc / kt where k is the oltzmann constant and T is the temperature. 3
At 5 o C, electronic transitions ( ~ 0,000 cm-) f e 48 9.4 0 % Vibrational transitions ( ~,000 cm-) f e 4.8 0.8% Rotational transitions ( ~ cm-) 0.0048 f e 99% (not including degeneracy) At thermal equilibrium, W W or N N( A ) Rearrange NA N N A N N N'/N replaced by oltzmann distribution e h / kt : A e h / kt 4
Compare with Planck's distribution 8h, A 3 c 3 Stimulated absorption = emission Spontaneous emission 3, (important only at high ) The Einstein coefficient is an inherent property of the molecule: fi 0 6 where fi f ˆ d i is the transition dipole. For vib. and rot. spectra, spontaneous emission can be ignored. Net rate of absorption: W N N ( N N) At equilibrium, W N N N N( e h / kt ) More population on lower state, smaller energy gap higher rate of absorption. 5
Vibration spectroscopy Harmonic oscillator (H.O.) V k( R R e ) Vibrational energy: E, 0,,,... Vibrational frequency: k, where is the reduced mass. Anharmonicity (deviations from H.O.) Morse potential: V ( RRe ) D e e where D e is well depth and D e 6
Vibrational energy E x e where the anharmonicity constant : x e Generally, vibrational term (cm - ): G( ) ~ 3 ~ x ~ e y e... where x e and y e can be fitted to experimental data. Selection rules: active if dipole changes during vibration. = ±, ( = ±, ±3,..., are possible due to anharmonicity) H, N are vibrationally inactive NO, OCS, H O, CO, C 6 H 6 are vibrationally active. Example. Calc. k of CO from vib. spectrum, if ~ = 6 cm -..50 6 kg 7
k c ~ k c ~ (.50 907N / m 6 kg)( 3.4 3.0 0 0 cm/ s 6cm ) Force constant reflects the bonding strength (k(h )=50 N/m). Vibrations of polyatomic molecules Number of vibrational modes: 3N-6 (3N-5 for linear molecules) H has vib. mode. H O has 3 vib. modes. CO has 4 vib. modes (linear). Normal modes: combined, synchronous vib. motion. Normal modes for CO and H O: symmetric stretch antisymmetric stretch bend Total energy: E (,,..., 3N 6 ) i i i 8
Pure rotational spectra Moment of inertia: m i x i x i : distance to the axis m : atomic mass i i Example. H For axis along H-H bond: // 0 (true for all linear molecules) For axis perpendicular to H-H bond m H r (.670.87 0 m 47 H 7 / )(7.40 kgm r r m) Rotational energy levels E classical x x y y z z E classical ( ) E quantum 9
Spherical rotors: CH 4, SF 6 x y z x y z Energy: E classical x y z ( ) E ( ) hc Rotational constant (cm-): hc Rotational term (cm - ): F ( ) ( ) Transition frequency for : F( ) F( ) [( )( ) ( )] ( ) Pure rotational spectra consist of nearly equally spaced lines separated by. 0
ntensity is determined by population: f g e ( ) e E / kt ( ) hc / kt Linear rotors: CO x y, 0 z x y Energy: E classical x y ( ) E ( ) hc Rotational constant (cm-): hc Rotational terms and spectra are the same as spherical rotors.
Gross selection rule: ntensity () Active if permanent dipole () is not zero. H, O, CO, (D h ), CH 4 (spherical rotor) have no rot. spectra. Specific selection rules: = ± Example. Rot. spec. of CO consists of lines separated by 3.86 cm-. Calc. its bond length. = 3.86/ =.93 cm- 6 7.660 kg.50 6 7 kg hc r r hc 4c 4 3.4.50.0 0.050 7 m 0.nm 34 s kg 3.00 0 cm/ s.93cm
Vibration-rotation spectra: Vibrational excitation inevitably involves rotational excitation. Vibration-rotation term (linear rotor): S (, ) G( ) F( ) ~ ( ) Combined selection rules (linear rotor): = ±, = 0, ± = -, 0, + correspond to the P, Q, R branches: ~ P( ) ~, - ~ ( ) ~ Q, ~ ( ) ~ ( ), + R n highly excited ro-vibrational states, both the rigid rotor and harmonic oscillator approximations break down. 3
Raman spectra Two-photon process, involving incident and scattered photons. Selection rules polarizability should change with vibration., 0, H, N, O are Raman active. ~ ~ ~ ~ O( ) i 4, - ( ) ~ ~ Q i, ~ ( ) ~ ~ 6, + S i 4 Lines are separated by 4. 4