Lecture 2: Rotational and Vibrational Spectra 1. Light-matter interaction 2. Rigid-rotor model for diatomic molecule 3. Non-rigid rotation 4. Vibration-rotation for diatomics 1. Light-matter interaction Possibilities of interaction Permanent electric dipole moment Rotation and vibration produce oscillating dipole (Emission/Absorption) H2O HCl Energy ∆E Absorption = qd Emission What if Homonuclear? Induced polarization Elastic scattering (Raman scattering) (Rayleigh scattering) Virtual State vs s < m or as Inelastic scattering as > 2 1. Light-matter interaction • Line position () is determined by Elements of spectra: difference between energy levels Line position • What determines the energy levels? Line strength • Quantum Mechanics! Line shapes Rotation: Microwave Region (∆J) Internal Energy: Electric dipole moment: qi r i E = E (n)+E ()+ E (J) i int elec vib rot μ + T E rot Erot E C vib μ ∆E x v μ 1/νs O x Eelec Are some molecules Time YES, e.g., H , Cl , CO “Microwave inactive”? 2 2 2 3 1. Light-matter interaction Elements of spectra: Line position Line strength Rotation: Microwave Region (∆J) Line shapes Vibration: Infrared Region (∆v, J) Internal Energy : μ Eint = Eelec(n)+Evib ()+ Erot(J) t= v Erot δ+ μ s C x Evib ∆E μx O δ- Heteronuclear case is IR-active Eelec Are some vibrations “Infra-red inactive”? Yes, e.g., symmetric stretch of CO2 4 1. Light-matter interaction Summary Eint = Eelec(n)+Evib ()+ Erot(J) ∆Erot < ∆Evib < ∆Eelec Energy levels are discrete Optically allowed transitions may occur only in certain cases E rot Absorption/emission spectra are discrete Evib ∆E Current interest Rotation Rigid Rotor Non-rigid Rotor Simple Harmonic Anharmonic Eelec Vibration Oscillator Oscillator 5 2. Rigid-Rotor model of diatomic molecule Rigid Rotor Axes of rotation m 1 m2 + - ~ 10-13cm r1 r2 C: r1m1 = r2m2 Center of mass C -8 r1+r2 = re ~ 10 cm Assume: -13 -8 . Point masses (dnucleus ~ 10 cm, re ~ 10 cm) . re = const. (“rigid rotor”) Relax this later 6 2. Rigid-Rotor model of diatomic molecule Classical Mechanics Quantum Mechanics Moment of Inertia Value of ωrot is quantized 2 2 I miri re I J J 1 h / 2 m m rot 1 2 reduced mass m m 1 2 Rot. quantum number = 0,1,2,… 2-body problem changed Erot is quantized! to single point mass Rotational Energy 2 1 2 1 2 1 2 h Erot Irot Irot J J 1 J J 1 2 2 2I 2I 8 I Convention is to denote rot. energy as F(J), cm-1 E h F J ,cm1 rot J J 1 BJ J 1 J 2 hc 8 Ic hc E E, J h hc rot So energy, cm-1 = (energy, J)/hc J hc 7 2. Rigid-Rotor model of diatomic molecule Absorption spectrum d 2 2m Schrödinger’s Equation: 2 2 E U x x 0 dx Wave function d J 1 Transition probability m n Complex conjugate Dipole moment Selection Rules for rotational transitions ’ (upper) ” (lower) ↓ ↓ ∆J = J’ – J” = +1 Recall: FJ BJ J 1 e.g., J 1J 0 FJ 1 FJ 0 2B 0 2B 8 2. Rigid-Rotor model of diatomic molecule Absorption spectrum Remember that: FJ BJ J 1 E.g., J 1J 0 FJ 1 FJ 0 2B 0 2B 12B 3 JF1st diff = ν 2nd diff = spacing 00 6B 2B 12B 2B 4B Lines every 6B 2 26B 2B 6B 2B! 4B 3 12B 2B 8B 2B 1 4 20B 2B F=0 J=0 In general: J 1J J 'J " BJ"1J"2 BJ"J"1 1 J 'J ",cm 2BJ"1 Let’s look at absorption spectrum 9 2. Rigid-Rotor model of diatomic molecule Absorption spectrum Recall: FJ BJ J 1 E.g., J 1J 0 FJ 1 FJ 0 2B 0 2B 12B 3 1.0 Heteronuclear 6B molecules only! λ ~2.5mm Tλ J”=0 11 6B 2 νrot for J=0→1~10 Hz (frequencies of rotation) 4B 2B 1 0.0 2B 031 2 475 6 ν/2B=J”+1 F=0 J=0 J” 0 1 2 364 5 Note: 1. Uniform spacing (easy to identify/interpret) -1 2. BCO~2cm λJ”=0 = 1/ν = 1/4cm = 2.5mm (microwave/mm waves) 10 11 rot,J=1 = c/λ = 3x10 /.25 Hz = 1.2x10 Hz (microwave) 10 2. Rigid-Rotor model of diatomic molecule Usefulness of rotational spectra Measured spectra Physical characteristics of molecule Line spacing h 2 B I r Accurately! =2B 8 2 Ic e re Example: CO -1 B = 1.92118 cm → rCO = 1.128227 Å 10-6 Å = 10-16 m 11 2. Rigid-Rotor model of diatomic molecule Intensities of spectral lines Equal probability assumption (crude but useful) Abs. (or emiss.) probability per molecule, is (crudely) independent of J Abs. (or emiss.) spectrum varies w/ J like Boltzmann distribution Degeneracy is a QM result associated w/ possible directions of Angular Momentum vector N 2J 1exp E / kT 2J 1 exp J J 1/T Recall: J J r N Qrot T /r E hcFJ hc J BJ J 1 J J 1 k k k r 1 kT 1 T Partition function: Qrot hcB r Symmetric no. (ways of rotating to achieve same orientation) = 1 for microwave active hc CO: σ=1 → microwave active! Define rotational T: r K B k N2: σ=2 → microwave inactive! 12 2. Rigid-Rotor model of diatomic molecule Intensities of spectral lines hc Rotational Characteristic Temperature: r K B k Species θrot [K] O2 2.1 hc 1 N 2.9 1.44K / cm 2 k NO 2.5 Cl2 0.351 N 2J 1 exp J J 1/T J r N T /r Strongest peak: occurs where the population is at a local maximum dN / N 1/ 2 J 0 J T / 2 1/ 2 f T / dJ max rot rot 13 2. Rigid-Rotor model of diatomic molecule Effect of isotopic substitution h Recall: B 8 2 Ic Changes in nuclear mass (neutrons) do not change r0 → r depends on binding forces, associated w/ charged particles → Can determine mass from B Therefore, for example: B12C16O 1.92118 m13 13.0007 B13C16O 1.83669 C m12C 12.00 Agrees to 0.02% of other determinations 14 3. Non-Rigid Rotation 2 Effects shrink line spacings/energies Two effects; follows from B 1/ r Vibrational stretching r(v) v↑ r↑ B↓ Centrifugal distortion r(J) J↑ r↑ B↓ Result: F J B J J 1 D J 2 J 12 v v v Centrifugal distribution constant 3 J 'J ",v 2Bv J"1 4Dv J"1 4B3 1. D is small; where Notes: D 2 B e since, 2 2 D B 1.7 6 4 4 310 B NO e 1900 → D/B smaller for “stiff/hi-freq” bonds 15 3. Non-Rigid Rotation 4B3 1. D is small; Notes: D 2 B e e.g., 2 2 D B 1.7 6 4 4 310 B NO e 1900 → D/B smaller for “stiff/hi-freq” bonds 2. v dependence is given by Bv Be e v 1/ 2 Dv De e v 1/ 2 E.g., NO Aside: 1 2 B 1.7046cm e / Be ~ 0.01 e 8e xe 5e e e / D 1 0.0178 / D ~ 0.001 e e 3 e e e e Be 24Be 6 2 Herzberg, Vol. I De 5.810 1/ 2 9 1 e 0.0014De ~ 810 cm e denotes “evaluated at equilibrium 2 2 e 1904.031/ 2 ;1903.68 3/ 2 inter-nuclear separation” re 1 e xe 13.97cm 16 4. Vibration-Rotation Spectra (IR) 1. Diatomic Molecules . Simple Harmonic Oscillator (SHO) . Anharmonic Oscillator (AHO) 2. Vibration-Rotation spectra – Simple model . R-branch / P-branch . Absorption spectrum 3. Vibration-Rotation spectra – Improved model Vibration-Rotation spectrum of CO 4. Combustion Gas Spectra (from FTIR) 17 4.1. Diatomic Molecules Simple Harmonic Oscillator (SHO) ∆/2 rmin m1 m2 re Molecule at instance of greatest Equilibrium position (balance between attractive + repulsive compression forces – min energy position As usual, we begin w. classical mechanics + incorporate QM only as needed 18 4.1. Diatomic Molecules Simple Harmonic Oscillator (SHO) Classical mechanics . Force k s r r e - Linear force law / Hooke’s law 1 . Fundamental Freq. k / ,cm1 / c vib 2 s e 1 . Potential Energy U kr r 2 2 e Parabola centered at distance Quantum mechanics of min. potential energy . v = vib. quantum no. = 0,1,2,3,… real . Vibration energy G=U/hc = diss. 1 energy Gv ,cm e vvib / cv 1/ 2 . Selection Rules: Equal energy spacing v v'v"1 only! Zero energy 19 4.1. Diatomic Molecules Anharmonic Oscillator (AHO) SHO AHO 1 1 2 Gv ,cm e v 1/ 2 Gv,cm e v 1/ 2 e xe v 1/ 2 ... H.O.T. Decreases energy spacing 1st anharmonic correction 10 GG10 “Fundamental” Band ∆ν=+1 ee12x real (e.g., 1←0,2←1) 21 e 1 4xe 1st Overtone ∆ν=+2 20 2 1 3x (e.g., 2←0,3←1) e e 2nd Overtone ∆ν=+3 30 3 1 4x (e.g., 3←0,4←1) e e In addition, breakdown in selection rules 20 4.1. Diatomic Molecules Vibrational Partition Function 1 hce hce Qvib 1 exp exp kT 2kT The same zero energy must Choose reference (zero) energy at v=0, so Gv ev be used in specifying 1 molecular energies Ei for hce level i and in evaluating the Qvib 1 exp kT associated partition function Vibrational Temperature hc vib K e Species θ [K] θ [K] k vib rot O 2270 2.1 N g exp v /T 2 vib vib vib N Qvib N2 3390 2.9 vvib vib NO 2740 2.5 exp 1 exp T T Cl2 808 0.351 where gvib 1 21 4.1.