Spectroscopy: Tinoco Chapter 10 (but vibration, Ch.9)
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1 Spectroscopy: Tinoco Chapter 10 (but vibration, Ch.9) XIV 67 Vibrational Spectroscopy (Typical for IR and Raman) Born-Oppenheimer separate electron-nuclear motion ψ (rr) = χ υ (R) φ el (r,r) -- product fct. solves sum electronic Schrödinger Equation el φ el (r,r) = U el (R) φ e (r,r) eigen value U el (R) parametric on R potential energy for nuclear motion (see below) Nuclear Schrödinger Equation n χ (R) = E υ χ υ (R) n = - /2Mα 2 α + V n (R) α 2 h V n (R) = U el (R) + Zα Z β e 2 /R αβ α,β Solving this is 3N dimensional N atom 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 kinetic energy associated with rotation quantized - no potential, but angular momentum restricted 1. Diatomics (linear) sol'n Y JM (θ,φ) same as -atom E JM = ћ 2 J(J+1)/2I I= M α α R α 2 (diatomic I=µ R α 2 ) (spectra J = ±1, M J = 0, ±1 E + = (J + 1) ћ 2 /I ) 2. Polyatomics add coordinate (ω) and quantum number (K) for 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 Note: levels increase separation as J inc.
2 XIV 68 Transitions allowed by absorption (far-ir or µ-wave / typical B e <10cm -1 ). Selection rules: J=±1 M=0,±1 E J J+1 =(J+1)2B (absorb J=+1) IR or µ-wave regularly spaced lines, intensity reflect rise degeneracy (δ J ) fall exponential depopulation 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)
3 XIV 69 E = -2B (2J + 3) (laser) E = +2B (2J + 3) 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 (left) anti-stokes: J = -2 (right) Stokes: J = 2
4 Condensed phase these motions wash out (bio-case) (still happen no longer free translation or rotation phonon and libron in bulk crystal or solution) XIV 70 Vibration: Internal coordinates solve problem V(R) not separable 3N 6 coordinates (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 armonic Approximation Taylor series expansion: 3N V(R) = V(R e ) + V/ Rα R e (R α-r e ) + α 3N 1/2 2 V/ R α R β R e (R α R e )(R β R e ) + 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 1/2 kx 2 Problem R α, R β mixed n not separate Solution New coordinates Normal coordinates Q j = 3 N cij q i where q i = x iα /(M α ) 1/2, y iα /(M α ) 1/2, z iα /(M α ) 1/2 i normal coordinates mass weighted Cartesian analogy actual problem a little different: = -h 2 /2 2 / Q 2 j + 1/2 kqi Q i 2 = j j j See this is summed product χ = j summed E = E j j each one is harmonic oscillator (know solution): h i χ (Q i ) = E i χ (Q i ) = hj(q j ) E j = (υ j + 1/2) hν j j χ j (Q j ) h j (Q j )
5 So for 3N 6 dimensions see regular set E j levels υ j = 0, 1, 2, but for each 3N-6 coordinate j XIV 71 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 + 1/2) hν ν = 1/2π k µ k force constant, µ = M A M B /(M A + M B ) heavier molecules lower frequency 2 ~4000 cm -1 F cm -1 Cl ~2988 cm -1 Cl cm -1 F ~4141 cm -1 I I ~214 cm -1 C ~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 cm N =& -1 O 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. O C C O C X O C O C N
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7 XIV 73
8 Transitions spectra measure change in energy levels These are caused by interaction of light & molecule XIV 74 E electric field B magnetic field --in phase and vibration (correspond to E = hν = E i E j ) then oscillating field drives the transition leads to absorption or emission E interacts with charges in molecules - like radio antenna if frequency of light = frequency of 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 ] = qjr j sum over all charge α i j Depends on position operator r, R electron & nuclei armonic 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
9 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 XIV 75 Raman: υ = ±1 J = 0, ±2 ( α/ R) 0 all molec. change polarizability
10 XIV 76 Diatomic -must be hetero so have dipole Most common υ = 0 υ = 1 E = h ν if harmonic υ = 1 υ = 2 also E = hν if anharmonic. υ = 1 υ = 2 lower freq. 0-->1 C O IR overtone 0-->2 e.g. O=C=O Linear: υ = 2 υ = 3 even lower get series of weaker transition lower ν Also n = ±1, ±2, ±3, possible --> overtones Polyatomics same rules: υ = ±1 J = ±1 But include J = 0 for non-linear vibrations (molec.) O=C=O O = C = O sym. stretch asym stretch bend (2) Raman IR IR r s r O = C = O O = C = O O = C = O symmetric (1354) non-linear asymmetric (2396) bend (673)
11 XIV 77 O O O (IR and Raman ) Bends normally ~ 1/2 ν e of stretches If 2 coupled modes, then there can be a big difference eg. CO 2 symm: 1354 asym: 2396 bend: O symm: 3825 asym: 3936 bend: 1654 Selection rules armonic 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 Raman ( α/ Q i ) 0 must change dipole in norm.coord. dislocate charge charge polarizability typical expand electrical charge Rotation: J = ±1 (linear vibration) --> P & R branches J = 0, ±1 bent or bend linear molec. --> Q-band CN linear stretch (no Q) CN bend mode (Q-band) 3N Polyatomic χ = 6 χυ0 j= 1 (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 Dipole selects out certain modes Allow molecules with symmetry often distort to dipole
12 XIV 78 dipole no dipole e.g. IR Intensity most intense if move charge e.g. O >> C C O >> C C, etc. In bio systems: -COO, -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
13 Characteristic C- modes - clue to structure XIV 79
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18 Bio-Applications of Vibrational Spectroscopy Biggest field proteins and peptides a) Secondary structure Amide modes I II III XIV 84 O C N O C N O C N ~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 Depends on ψ angle, characterize 2nd struct. b) Active sites - structurally characterize, selective i) difference spectra e.g. flash before / after - kinetic amides COO - / COO functional group ii) Resonance Raman intensify modes chromophore (e.g. heme) coupled to Nucleic Acids less a) monitor ribose conformation b) single / duplex / triplex / quad -bond Sugars little done, spectra broad, some branch appl. Lipids monitor order self assemble - polarization
19 Peptide Primary, Secondary, Tertiary, Quaternary Structure diagrams (full page) XIV 85
20 Amino Acids and Characteristic Amide Vibrations diagrams (full page) XIV 86
21 IR absorbance and Raman spectra (full page) XIV 87
22 Protein/ 2 O IR spectra examples (full page) XIV 88
23 DNA Base IR and FTIR Spectroscopy of nucleic acids examples (full page) XIV 89
24 Optical Spectroscopy Processes example (full page) XIV 90
25 XIV 91 Discussed Diatomic Vibrations at length Polyatomics a) expand V(q) = V(q e ) + i V/ q i q i + i,j V/ q i q j qi qi b) diagonalize V(q) V(Q) Q i = cij q j linear combination x y z on each atom; α, β i,j Normal coordinates 6 Transitions, rotation 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) O O O
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/2Mα 2 α + V n (R)] χ (R) = E υ χ υ (R)
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