# XIV Spectroscopy: Engel Chapter 18 Vibrational Spectroscopy (Typically 14(Vib Sp · PDF...

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Spectroscopy: Engel Chapter 18 Vibrational Spectroscopy (Typically IR and Raman) Born-Oppenheimer approx. separate electron-nuclear Assume elect-nuclear motion separate, full wave fct. (r,R) = (R) el (r,R) -- product fct. solves sum H Electronic Schrdinger Equation Hel el (r,R) = Uel (R) e (r,R) eigen value Uel(R) parametric depend on R (resolve electronic problem each molecular geometry) Uel(R) is potential energy for nuclear motion (see below) Nuclear Schrdinger Equation Hn (R) = E (R) Hn (R) = -[ /2M

2 h 2 + Vn (R)] (R) = E (R)

Focus: Vn(R) = Uel(R) + , Z Z e2/R

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 momentumcan be quantized

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kinetic energy associated with rotation quantized - no potential, but angular momentum restricted 1. Diatomics (linear) solution YJM (,)

same form as H-atom angular part EJM = 2J(J+1)/2I I=

M R2

(diatomic I= R2) selection rules: J = 1, MJ = 0, 1

transitions: E+ = (J + 1) 2/I - levels spread ~ J2, difference ~ J

Note: 2D problem, no momentum for rotation on z 2. Polyatomics add coordinate (-orientation internal)

and quantum number (K) for its angular momentum previously refer J,MJ 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: EJMrot=J(J+1)2/2I I= Re =MAMB/MA+MB

if Be= h/(82Ic) EJM = J(J + 1) Be in cm-1 or EJMrot = (hc) J (J + 1)Be in Joules Note: levels increase separation as J2 & transitions as J Transitions allowed by absorption (far-IR or -wave)

Also seen in Raman scattering: S = 0 - rot(typical Be< 10 cm-1, from Ic~,light molecules highest)

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Diatomic (linear) Selection rules: J=1 M=0,1 EJJ+1=(J+1)2B

(IR, -wave absorb J = +1, Raman J = 0, 2 ) poly atomic add: K = 0 ( or 1 - vary with Geometry)

IR or -wave

regularly spaced lines, intensity reflect rise degeneracy (J ~ 2J+1) inc. with J (linear) fall exponential depopulation fall with J (exp.) Boltzmann: nJ = J n0 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)

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Stokes=E = -2B (2J + 3) (laser) E = +2B (2J + 3)=AntiStokes

Raman light scattering experiment s = 0 JJ = 0, 1, 2 in general but J = 2 diatomic (K = 0)

diatomic (linear) spacing ~4B

Rotational Raman spect. of N2 (alternate intensity-isotope) (left) anti-Stokes: J = -2 (right) Stokes: J = 2

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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)2 + V(R)

V(R) has all electrons attract all nuclei, in principle could separate, but all nuclei repel, which is coupled - R Harmonic Approximation Taylor series expansion: V(R) = V(Re) + V/R

N3Re(R-Re) +

N3 2V/RRRe(R Re)(R Re) +

Expansion in Taylor Series

1st term constant just add to energy 2nd term zero at minimum 3rd term 1st non-zero / non-constant term

harmonic potential has form of kx2 Problem R, R mixed Hn not separate Solution New coordinates Normal coordinates

Qj = cN3

iij qi where qi = xi/(M)1/2 , yi/(M)1/2 , zi/(M)1/2

normal coordinates mass weighted Cartesian

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analogy actual problem a little different: H = -h

j2/2 2/Qj2+ k

jQiQi

2 = j

hj (Qj)

See this is summed H product =

j j (Qj)

summed E = Ej j

each one is harmonic oscillator (already know solution):

hi (Qi) = Ei (Qi) H = h

jj(Qj) Ej = (j + ) hj

So for 3N 6 dimensions see regular set Ej levels

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 ,) 1 coord. vibration bond

model of harmonic oscillator works E = ( + ) h = (1/2) k k force constant

, = MAMB/(MA + MB)

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heavier molecules lower frequency

H2 ~4000 cm-1 F2 892 cm-1HCl ~2988 cm-1 Cl2 564 cm-1HF ~4141 cm-1 II ~214 cm-1

CH ~2900 cm-1 ICl ~384 cm-1CD ~2100 cm-1 stronger bonds higher k - higher frequency CC ~2200 cm-1 O=O 1555 cm-1

C=C ~1600 cm-1 N =& O 1876 cm-1CC ~1000 cm-1 NN 2358 cm-1

CO 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

OH

OC OH C X

C O C NH

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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 = h = Ei Ej )

then oscillating field drives the transition leads to absorption or emission Probability of induce transition Pij ~ i* j d 2 where is electric dipole op. = [(Z

i e)R+eri] =

jqjrj sum over all charge

Depends on position operator r, R electron & nuclei Harmonic oscillator: transform = (Qj) (norm. coord.) *(Qj) j k(Qj)d Qj 0 if k = 1 in addition: /Qj 0 i.e. = 1 Normal mode must change dipole moment to have dipole transition occur in IR Most observations are Absorption: = 0 = 1

population nj = ni e-(Ej Ei)/kT Boltzmann

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Allowed transitions Diatomic: IR: = 1 J = 1 (i.e. also rotate)

/R 0 hetero atomic observe profiles series of narrow lines separate 2B spacing yields geometry: Be IeRe

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Raman: = 1 J = 0, 2 (/R) 0 all molec. change polarizability

Diatomic -must be hetero so have dipole Most common: = 0 = 1 E = h

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if harmonic = 1 = 2 also E = h if anharmonic. = 1 = 2 lower freq. = 2 = 3 even lower get series of weaker transition lower Also n = 1, 2, 3, possible --> overtones

CO IR overtone

0-->2

0-->1

Polyatomics same rules: = 1 J = 1 But include J = 0 for non-linear vibrations (molec.)

e.g. O=C=O O=C=O

== OCO

sym. stretch asym stretch bend (2) Raman IR IR

Linear: (1354) symmetric

OCO ==(2396) asymmetric

OCO rsr

== non-linear )673( bend

OCO

==

3 8 2 5 3 9 3 6 1 6 5 4

O

H HH

O

H

O

H H

(IR and Raman )

Bends normally ~ e of stretches

If 2 coupled modes, then there can be a big difference eg. CO2 symm: 1354 asym: 2396 bend: 673 H2O symm: 3825 asym: 3936 bend: 1654

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Selection rules Harmonic i = 1 , j = 0 i j Anharmonic i = 2, 3, overtones j = 1 i = 1 combination band IR (/Qi) 0 must change dipole in norm.coord. dislocate charge Raman (/Qi) 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 =

=

6N3

1j0 (Qj)

Due to orthogonality only one Qj can change j j = 1 i = 0 i j (/Qj) 0

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Dipole selects out certain modes Allow molecules with symmetry often distort to dipole

e.g.

dipole

no dipole

IR Intensity most intense if move charge e.g. OH >> CH CO >> CC , etc. In bio systems: -COOH, -COO- , amide C=O, -PO2- Raman light scattering s = 0 vib caused by polarizability = 1 these tend to complement IR /Qj 0 homo nuclear diatomic symmetrical modes -- aromatics, -S-S-, large groups -- most intense

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Characteristic C-H modes - clue to structure

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Bio-Applications of Vibrational Spectroscopy Biggest field proteins and peptides a) Secondary structure Amide modes

C NO

H

C NO

HC N

O

H

I II III

~1650 ~1550 ~1300 IR coupling changes with conform (typ. protein freq.) I II helix ~1650+ 1550 sheet ~1630- 1530 coil ~1640-50 1520-60 Raman -see I, III III has characteristic mix with CH Depends on 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 inte

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