/2Mα 2 α + V n (R)] χ (R) = E υ χ υ (R)
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1 Spectroscopy: Engel Chapter 18 XIV 67 Vibrational Spectroscopy (Typically IR and Raman) 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 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) is potential energy for nuclear motion (see below) Nuclear Schrödinger Equation H n χ (R) = E υ χ υ (R) H n χ (R) = -[ h 2 /2Mα 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 a) Translation like atoms no impact on spectra since continuous (no potential plane wave) b) Rotation also no potential but have angular momentum can be quantized
2 XIV 68 kinetic energy associated with rotation quantized - no potential, but angular momentum restricted 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: J = ±1, M J = 0, ±1 transitions: E + = (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 molecules highest)
3 XIV 69 Diatomic (linear) Selection rules: J=±1 M=0,±1 E J J+1 =(J+1)2B (IR, µ-wave absorb J = +1, Raman J = 0, ±2 ) poly atomic add: K = 0 ( or ±1 - vary with Geometry) IR or µ-wave 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=5 --> J=6 (I think)
4 XIV 70 Stokes= E = -2B (2J + 3) (laser) E = +2B (2J + 3)=AntiStokes Raman light scattering experiment ν s = ν 0 ν J J = 0, ±1, ±2 in general but J = ±2 diatomic (K = 0) diatomic (linear) spacing ~4B Rotational Raman spect. of N 2 (alternate intensity-isotope) (left) anti-stokes: J = -2 (right) Stokes: J = 2
5 XIV 71 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
6 XIV 72 analogy actual problem 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 = each one is harmonic oscillator (already know solution): h i χ (Q i ) = E i χ (Q i ) H = j hj(q j ) j E j E j = (υ j + ½) hν 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 go back to 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 )
7 XIV 73 heavier molecules lower frequency H 2 ~4000 cm -1 F cm -1 HCl ~2988 cm -1 Cl 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. C O OH C OH C X C O C NH
8 XIV 74
9 XIV 75
10 XIV 76 Transitions spectra measure change in energy levels These are 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 E = hν = E i E j ) then oscillating field drives the transition leads to absorption or emission Probability of induce transition P i j ~ ψ i * µ ψ j d τ 2 where µ is electric dipole op. µ = [(Zαe)R α +er i ] = q j r j sum over all charge α i j Depends on position operator r, R electron & 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. υ = ±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
11 XIV 77 Allowed transitions Diatomic: IR: υ = ±1 J = ±1 (i.e. also rotate) µ/ R 0 hetero atomic observe profiles series of narrow lines separate 2B spacing yields geometry: B e I e R e
12 Raman: υ = ±1 J = 0, ±2 ( α/ R) 0 all molec. change polarizability XIV 78 Diatomic -must be hetero so have dipole Most common: υ = 0 υ = 1 E = h ν
13 XIV 79 if harmonic υ = 1 υ = 2 also E = hν if anharmonic. υ = 1 υ = 2 lower freq. υ = 2 υ = 3 even lower get series of weaker transition lower ν Also n = ±1, ±2, ±3, possible --> overtones 0-->1 C O IR overtone 0-->2 Polyatomics same rules: υ = ±1 J = ±1 But include J = 0 for non-linear vibrations (molec.) e.g. O=C=O O=C=O Linear: O = C = O sym. stretch asym stretch bend (2) Raman IR IR r O = C = O O s r C = O symmetric (1354) asymmetric (2396) = O = C = O non-linear bend (673) H O O O H H H H 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
14 XIV 80 Selection rules Harmonic υ i = ±1, υ j = 0 i j Anharmonic υ i = ±2, ±3, overtones υ j = ±1 υ 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: J = ±1 (linear vibration) --> P & R branches J = 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 υ j = ±1 υ i = 0 i j ( µ/ Q j ) 0
15 XIV 81 Dipole selects out certain modes Allow molecules with symmetry often distort to dipole 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 υ = ±1 these tend to complement IR α / Q j 0 homo nuclear diatomic symmetrical modes -- aromatics, -S-S-, large groups -- most intense
16 Characteristic C-H modes - clue to structure XIV 82
17 XIV 83
18 XIV 84
19 XIV 85
20 XIV 86
21 Bio-Applications of Vibrational Spectroscopy Biggest field proteins and peptides a) Secondary structure Amide modes I II III XIV 87 O C N H O C N H O C N H ~1650 ~1550 ~1300 IR coupling changes with conform (typ. protein freq.) I II helix ~ sheet ~ coil ~ Raman -see I, III III has characteristic mix with C α H Depends on ψ angle, characterize 2nd struct. b) Active sites - structurally characterize, selective i) difference spectra e.g. flash before / after - kinetic amides COO - / COOH functional group ii) Resonance Raman intensify modes coupled to chromophore (e.g. heme) Nucleic Acids less a) monitor ribose conformation b) single / duplex / triplex / quad H-bond Sugars little done, spectra broad, some branch appl. Lipids monitor order self assemble - polarization
22 Peptide Primary, Secondary, Tertiary, Quaternary Structure diagrams (full page) XIV 88
23 Amino Acids and Characteristic Amide Vibrations diagrams (full page) XIV 89
24 IR absorbance and Raman spectra (full page) XIV 90
25 Protein/H 2 O IR spectra examples (full page) XIV 91
26 DNA Base IR and FTIR Spectroscopy of nucleic acids examples (full page) XIV 92
27 Optical Spectroscopy Processes example (full page) XIV 93
28 XIV 94 Discussed Diatomic Vibrations at length Polyatomics a) expand V(q) = V(q e ) + V/ q i q i + i i,j b) diagonalize V(q) V(Q) Q i = cij q j linear combination x y z i,j on each atom; α, β V/ q i q j qi qi Normal coordinates 6 Translations, rotations no potential E eigen value 0 {diagonalize potential (3N 6) vibrations internal nuclear motion examples: Triatomics linear bent O = C = O symmetric (1354) r s r O = C = O asymmetric (2396) O = C = O bend (673) H O H H O H H O H
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30 XIV 96
31 XIV 97
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