Vibrational Spectra (IR and Raman)- 2010 update Tinoco has very little, p.576, Engel Ch. 18, House Ch. 6 Born-Oppenheimer approx. separate electron-nuclear Assume elect-nuclear motion separate, full wave fct. ψ (r,r) = χ υ (R) φ el (r,r) -- product fct. solves sum H Electronic Schrödinger Equation next section: Notes 15 H el φ el (r,r) = U el (R) φ e (r,r) eigen value U el (R) parametric depend on R (resolve electronic problem each molecular geometry) U el (R)+V nn is potential energy for nuclear motion (see below) Nuclear Schrödinger Equation H n χ (R) = E υ χ υ (R) H n χ (R) = -[ ħ 2 /2M α ) 2 α + V n (R)] χ (R) = E υ χ υ (R) α Focus: V n (R) = U el (R) + Z α Z β e 2 /R αβ α, β Solving this is 3N dimensional N atom, each has x,y,z Simplify Remove (a) Center of Mass (Translate) (b) Orientation of molecule (Rotate) Results in (3N 6) coordinates - called internal coord. motion of nuclei w/r/t each other vibration a) Translation like atoms no impact on spectra since continuous (no potential plane wave) b) Rotation also no potential but have quantized angular momentum States are Y JM (θ,φ) no potential, but kinetic energy assoc. with rotation quantize energy levels, ΔE in μ-wave, far-ir, ΔJ=±1
(Skip rotations) little bio-impact, not solve, particle on sphere 1. Diatomics (linear) solution Y JM (θ,φ) same form as H-atom angular part E JM = ћ 2 J(J+1)/2I I= M 2 α R α α (diatomic I=μ R 2 α ) selection rules: ΔJ = ±1, ΔM J = 0, ±1 transitions: ΔE + = (J + 1) ћ 2 /I - levels spread ~ J 2, difference ~ J Note: 2 D problem, no momentum for rotation on z 2. Polyatomics add coordinate (ω-orientation internal) and quantum number (K) for its angular momentum previously refer J,M J to a lab axis, now complex (this K is projection of angular momentum onto molecular axis, so internal orientation of molecule) Rotational Spectra (aside little impact on Biology) Diatomic: E JM rot =J(J+1)ћ 2 /2I I=μ R e μ=m A M B /M A +M B if B e = h/(8π 2 I c ) E JM = J(J + 1) B e in cm -1 or E rot JM = (hc) J (J + 1)B e in Joules Note: levels increase separation as J 2 & transitions as J Transitions allowed by absorption (far-ir or μ-wave) Also seen in Raman scattering: ν S = ν 0 - ν rot (typical B e < 10 cm -1, from I c ~μ, light molecule highest) Diatomic (linear) Selection rules: ΔJ=±1 ΔM=0,±1 ΔE J J+1 =(J+1)2B (IR, μ-wave absorb ΔJ = +1, Raman ΔJ = 0, ±2 ) poly atomic add: ΔK = 0 ( or ±1 - vary with Geometry)
IR or μ-wave all ΔJ=+1 regularly spaced lines, intensity reflect rise degeneracy (δ J ~ 2J+1) inc. with J (linear) fall exponential depopulation fall with J (exp.) Boltzmann: n J = δ J n 0 exp [ J(J+1)B/kT] Pure Rotational Far-IR spectrum of CO -- note 1st transition (23 cm -1 ) is for J=6 --> J=7 (I think) Stokes=ΔE = -2B (2J + 3) (laser) ΔE = +2B (2J + 3)=AntiStokes
Raman light scattering experiment ν s = ν 0 ν J ΔJ = 0, ±1, ±2 (Stoles +, anti-stokes -) in general but ΔJ = ±2 diatomic (K = 0) (or linear) and spacing ~4B Rotational Raman spect. of N 2 (alternate intensity-isotope) (left) anti-stokes: ΔJ = -2 (right) Stokes: ΔJ = 2 Condensed phase these motions wash out (bio-case) (still happen no longer free translation or rotation phonon and libron in bulk crystal or solution)
Vibration: Internal coordinates solve problem V(R) not separable 3N 6 coordinates H(R) = - (h 2 /2M α ) 2 α + V(R) α V(R) has all electrons attract all nuclei, in principle could separate, but all nuclei repel, which is coupled - R αβ Harmonic Approximation Taylor series expansion: 3N V(R) = V(R e ) + V/ Rα R e(r α -R e ) + α ½ 3 2 V/ R α R β R e(r α R e )(R β R e ) + N α Expansion in Taylor Series 1 st term constant just add to energy 2 nd term zero at minimum 3 rd term 1 st non-zero / non-constant term harmonic potential has form of ½ kx 2 Problem R α, R β mixed H n not separate Solution New coordinates Normal coordinates Q j = 3 N i cij q i where q i = x iα /(M α ) 1/2, y iα /(M α ) 1/2, z iα /(M α ) 1/2 normal coordinates mass weighted Cartesian set up separated harmonic oscillator problem, actually a little different: H = -h 2 /2 2 / Q 2 j +½ kqi Q i 2 = h j (Q j ) j j j
See this is summed H product χ = χ j (Q j ) j summed E = E j j each χ j (Q j ) is harmonic oscillator (already know solution): h i χ (Q i ) = E i χ (Q i ) H = hj(q j ) E j = (υ j + ½) hν j j So for 3N 6 dimensions see regular set E j levels υ j = 0, 1, 2, but for each 3N-6 coordinate j Interpret - diatomic N = 2 3N = 6 coordinates remove translation 3 coordinates left remove rotation (just θ,φ) 1 coord. vibration bond model of harmonic oscillator works E = (υ + ½) hν ν = (1/2π) k/μ k force constant μ = M A M B /(M A + M B ) heavier molecules bigger μ - lower frequency H 2 ~4000 cm -1 F 2 892 cm -1 HCl ~2988 cm -1 Cl 2 564 cm -1 HF ~4141 cm -1 I I ~214 cm -1 C H ~2900 cm -1 I Cl ~384 cm -1 C D ~2100 cm -1
stronger bonds higher k - higher frequency C C ~2200 cm -1 O=O 1555 cm -1 C=C ~1600 cm -1 N =& O 1876 cm -1 C C ~1000 cm -1 N N 2358 cm -1 C O 2169 cm -1 This is key to structural use of IR frequency depends on mass (atom type) bond strength (type) Thus frequencies characteristic of structural elements Called group frequencies: Typical frequencies for given functional groups e.g. formaldehyde, below, and table next page and orgo books Unique local vibrations are ~same for diff. molec. characteristic Identical vibs couple, delocalize normal mode, repres. bond character Animation of benzene modes to see coupling: http://faculty.physics.tamu.edu/allen/benzene-modes/index.html breathing CC stretch (Kekule) CCC bend ~1120 cm -1 ~1020 cm -1 ~1400 cm -1
Transitions spectra measure energy level change caused by interaction of light & molecule E electric field B magnetic field --in phase and E interacts with charges in molecules - like radio antenna if frequency of light = frequency of vibration (correspond to ΔE = hν = E i E j ) then oscillating field drives the transition leads to absorption or emission Probability of induce transition-result of time-varying field P i j ~ ψ i * μ ψ j d τ 2 where μ is electric dipole operator μ = [(Zαe)R α +er i ] = q j r j sum over all charges α i j Depend on position operator r, R (vector) elect. & nuclei Harmonic oscillator: transform μ = μ(q j ) (norm. coord.) χ υ l*(q j ) μ j χ υk(q j )d Q j 0 if υ k = υ l ± 1 in addition: μ/ Q j 0 i.e. Δυ = ±1 Normal mode must change dipole moment to have dipole transition occur in IR Most observations are Absorption: υ = 0 υ = 1 -(Ej Ei)/kT population n j = n i e Boltzmann
Allowed transitions (for those of you in Phys Chem Lab!!) Diatomic: IR: Δυ = ±1 ΔJ = ±1 (i.e. rotate-gas phase) μ/ R 0 hetero atomic observe profiles series of narrow lines separate 2B spacing yields geometry: B e I e R e
Raman: Δυ = ±1 ΔJ = 0, ±2 ( α/ R) 0 all molec. change polarizability Diatomic -must be hetero so have dipole Most common: υ = 0 υ = 1 ΔE = h ν
if harmonic υ = 1 υ = 2 also ΔE = hν if anharmonic. υ = 1 υ = 2 lower freq. υ = 2 υ = 3 even lower get series of weaker transition lower ν Also Δn = ±1, ±2, ±3, possible --> overtones 0-->1 C O IR overtone 0-->2 Polyatomics same rules: Δυ = ±1 ΔJ = ±1 But include ΔJ = 0 for non-linear vibrations (molec.) = C = O e.g. O=C=O O=C=O coupled Linear: non-linear sym. stretch asym stretch bend (2) local Raman IR IR vibs r s r O = C = O O = C = O symmetric (1354) asymmetric (2396) O O = C = O bend (673) H O O O H H H H 3825 3936 1654 H (IR and Raman ) Bends normally ~ ½ ν e of stretches If 2 coupled modes, then there can be a big difference eg. CO 2 symm: 1354 asym: 2396 bend: 673 H 2 O symm: 3825 asym: 3936 bend: 1654
Selection rules Harmonic Δυ i = ±1, Δυ j = 0 i j Anharmonic Δυ i = ±2, ±3, overtones Δυ j = ±1 Δυ i = ±1 combination band IR ( μ/ Q i ) 0 must change dipole in norm.coord. dislocate charge Raman ( α/ Q i ) 0 charge polarizability typical expand electrical charge Rotation: ΔJ = ±1 (linear vibration) --> P & R branches ΔJ = 0, ±1 bent or bend linear molec. --> Q-band HCN linear stretch (no Q) HCN bend mode (Q-band) Polyatomic χ = 3N 6 j= 1 χυ0 (Q j ) Due to orthogonality only one Q j can change υ j Δυ j = ±1 Δυ i = 0 i j ( μ/ Q j ) 0 Dipole selects out certain modes Allow molecules with symmetry often distort to dipole
Benzene example: dipole no dipole e.g. IR Intensity most intense if move charge e.g. O H >> C H C O >> C C, etc. In bio systems: -COOH, -COO -, amide C=O, -PO 2 - Raman light scattering ν s = ν 0 ν vib caused by polarizability Δυ = ±1 these tend to complement IR α/ Q j 0 Modes tend to be those that expand e - density homo nuclear diatomic symmetrical modes, e.g. Benzene breathing -- aromatics, -S-S-, large groups -- most intense Examples, typically IR presented as Transmission (T=I/I 0 ) Historical, modern FTIR present Absorbance (A = -logt) Means these %T spectra have baseline at top, and transitions correspond to negative going (down) peaks Raman are presented as intensity of scattering (positive) If corrected, can ratio to internal standard (eg PO 4-2...) Also include polarization, parallel and perpendicular to I 0 This can identify type of mode, discriminate overlaps
Characteristic C-H modes - clue to structure