Lecture 2: Rotational and Vibrational Spectra

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  I rot  I rot  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 ,cm1   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: FJ  BJ J 1

e.g.,  J 1J 0  FJ 1 FJ  0 2B  0  2B

8 2. Rigid-Rotor model of diatomic molecule

 Absorption spectrum

Remember that: FJ  BJJ 1

E.g.,  J 1J 0  FJ 1 FJ  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 1J  J 'J "  BJ"1J"2  BJ"J"1

1  J 'J ",cm  2BJ"1 Let’s look at absorption spectrum 9 2. Rigid-Rotor model of diatomic molecule

 Absorption spectrum Recall: FJ  BJJ 1

E.g.,  J 1J 0  FJ 1 FJ  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 1exp E / kT  2J 1 exp J J 1/T  Recall: J  J  r N Qrot T /r E hcFJ   hc  J    BJJ 1   J J 1 k k  k  r 1 kT 1 T Q   Partition function: rot   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 dN / N J  0 J  T / 2 1/ 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: B12C16O 1.92118   m13 13.0007 B 13C16O 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 12 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   310  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   310  B NO  e  1900  → D/B smaller for “stiff/hi-freq” bonds

2. v dependence is given by Bv  Be e v 1/ 2 D  D   v 1/ 2 v e e E.g., NO  Aside:  1  2 B 1.7046cm e / Be ~ 0.01  e 8 e xe 5 e e e   / D    1  0.0178  / D ~ 0.001 e e 3 e e e e Be 24Be 6 2  Herzberg, Vol. I De  5.810 1/ 2  9 1 e  0.0014De ~ 810 cm e denotes “evaluated at equilibrium  2 2 e 1904.03 1/ 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  r  r  - Linear force law / Hooke’s law s e  1 . Fundamental Freq.    k /  ,cm1  / c vib 2 s e 1 . Potential Energy U  kr  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 Gv ,cm  e  vvib / cv 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 Gv ,cm  e v 1 / 2  Gv,cm  e v 1/ 2 e xe v 1/ 2 ... H.O.T.

Decreases energy spacing 1st anharmonic correction

 10 GG10   “Fundamental” Band ∆ν=+1 ee12x real (e.g., 1←0,2←1)  21  e 1 4xe 

1st Overtone ∆ν=+2  20  2 1 3x  (e.g., 2←0,3←1) e e

2nd Overtone ∆ν=+3  30  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    hc e    hce  Qvib  1 exp  exp    kT   2kT  The same zero energy must

Choose reference (zero) energy at v=0, so Gv  ev be used in specifying 1 molecular energies Ei for    hce  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

O2 2270 2.1 Nvib gvib exp vvib /T   N Qvib N2 3390 2.9

 v vib   vib  NO 2740 2.5  exp 1 exp   T   T    Cl2 808 0.351

where gvib 1

21 4.1. Diatomic Molecules

 Some typical values (Banwell, p.63, Table 3.1)

Molecular Vibration ωe Anharmonicity Force constant ks Internuclear Dissociation Gas -1 Weight [cm ] constant xe [dynes/cm] distance re [Å] energy Deq [eV] CO 28 2170 0.006 19 x 105 1.13 11.6 NO 30 1904 0.007 16 x 105 1.15 6.5

† 5 H2 2 4395 0.027 16 x 10 1.15 6.5 † 5 Br2 160 320 0.003 2.5 x 10 2.28 1.8

† . Not IR-active, use Raman spectroscopy!

. e  k /  ←   m / 2 for homonuclear molecules

. De  e / 4xe ← large k, large D . Weak, long bond → loose spring constant → low frequency

22 4.1. Diatomic Molecules

 Some useful conversions

 Energy 1 cal  4.1868 J 1 cm-1  2.8575 cal/mole 1 eV  8065.54 cm1  23.0605 kcal/mole 1.602191019 J 5  Force 1 N 10 dynes o  Length 1 A  0.1 nm

How many HO levels? (Consider CO)

Do  256 kcal

N no. of HO levels 256 kcal/mole 41 2.86 cal/mole cm11 2170 cm

Actual number is ?GREATER as AHO shrinks level spacing 23 4.2. Vib-Rot spectra – simple model

 Born-Oppenheimer Approximation

 Vibration and Rotation are regarded as independent → Vibrating rigid rotor

Energy: Tv, J  RR  SHO  FJ  G v   ∆J = J' - J"  BJ J 1 e v 1/ 2 Selection Rules: v  1 Two Branches: P (∆J=-1) J  1 R(∆J=+1)

Line Positions:   T 'T" T v', J 'T v", J" Aside: Nomenclature for “branches”

 Branch O P Q R S J'=J"+1  ∆J -2 -1 0 +1 +2 v'=1 J'= J" Null Gap J'= J"-1 R(2) P branch R branch P R R(0) J"+1 P(1) v"=0  o

J" Transition Probabilities -8 -6 -4 -2 0 246 2B 24 4.2. Vib-Rot spectra – simple model

 R-branch 1  RJ" ,cm  Gv' Gv" BJ"1J"2 BJ"J"1 

o  vo = Rotationless transition wavenumber   e (SHO)   e 1 2xe (AHO,1  0)   e 1 4xe (AHO,2  1)  ...

RJ"  0  2BJ"1 Note: spacing = 2B, same as RR spectra

J'=J"+1  P-branch v'=1 J'= J" J'= J"-1  PJ"  0  2BJ" Note: ωo =f(v")forAHO P R J"+1 v"=0 _ 8BkT J"  P-R Branch peak separation   hc Larger energy 25 4.2. Vib-Rot spectra – simple model

 Absorption spectrum (for molecule in v" = 0)

Width, shape depends on instrument, experimental conditions

Null Gap R(2) P branch R branch Line (sum of all lines is a “band”) R(0) P(1)  o Transition Probabilities -8 -6 -4 -2 0 246 2B

. Height of line ∝ amount of absorption ∝ NJ/N . “Equal probability” approximation – independent of J (as with RR)

What if we remove RR limit? → Improved treatment

26 4.3. Vib-Rot spectra – improved model

 Breakdown of Born-Oppenheimer Approximation

 Allows non-rigid rotation, anharmonic vibration, vib-rot interaction

T v, J  G v  Fv, J  B(v) 2 2 2  e v 1/ 2 e xe v 1/ 2  Bv J J 1  Dv J J 1

SHO Anharm. corr. RR(v) Cent. dist. term

2 . R-branch Rv", J"  o v" 2Bv '3Bv 'Bv"J"Bv 'Bv "J"

2 . P-branch Pv", J"  o v" Bv 'Bv"J"Bv 'Bv"J"

Bv  Be e v 1/ 2 Bv ' Be 'e v'1/ 2 B ' B " Bv" Be"e v"1/ 2 v v Spacing ↑ on P side, ↓ on R side Bv 'Bv" e  0 Null Gap R(2) P branch R branch

R(0) P(1)    o 2B

Transition Probabilities -8 -6 -4 -2 0 246 27 4.3. Vib-Rot spectra – improved model

 Bandhead

Null Gap R(2) P branch R branch

R(0) P(1)    o -8 -6 -4 -2 0 246 2B Transition Probabilities

P branch J" R branch 4 Bandhead 3 2 1

   o -4 -3-2 -1 0 12 3 4 2B dRJ 2B' B   e  3B'B"  2 B'B" J" 0 J"bandhead   dJ   2 e e 2B' e e Increasing Decreasing spacing spacing B 1.9 E.g., CO  106 → not often observed e 0.018 28 4.3. Vib-Rot spectra – improved model

 Finding key parameters: Be, αe, ωe, xe st  1 Approach: Use measured band origin data for the fundamental and first

overtone, i.e., ΔG1←0, ΔG2←0, to get ωe, xe G  G1 G0  1 2x  10 e e  , x     ee G20  G 2  G 0  2e 1 3xe

nd  2 Approach:

Fit rotational transitions to the line spacing equation to get Be and α  2  o  B'B"m  B'B"m m  J 1 in R - branch m  J in P - branch

BB''1/2ee V v   B', B" Be, α BB""1/2ee V v  

29 4.3. Vib-Rot spectra – improved model

 Finding key parameters: Be, αe, ωe, xe rd  3 Approach: Use the “method of common states”

v' J'= J" ← Common upper-state J'= J"-1 In general FJ  BJ J 1 P(J+1) R(J-1) E  FJ 1 FJ 1  R J 1  P J 1 v" J"+1  ∆E J"  B" J 1 J  2  B" J 1 J J"- 1 E  B"4J  2 B"

v' J'=J"+1 E  FJ 1 FJ 1 Be , ∆E J'= J"  B' J 1 J  2  B' J 1 J J'= J"-1 E  B'4J  2 B' P(J) R(J) v" J"+1 J" ← Common lower-state 30 4.3. Vib-Rot spectra – improved model

 Isotopic effects 1 1 B   → Line spacing changes as μ changes  I  k 1  s  → Band origin changes as μ changes e 

1st Example: CO Isotope 13C16O

 13 16 B13 16 C O 1.046 . C O B13C16O   12C16O 1.046 2B 0.0463.88  0.17cm1

e 13C16O .  13 16  e C O 1.046 1 e  0.046 2200 / 2  50cm

31 4.3. Vib-Rot spectra – improved model

 Isotopic effects

CO fundamental band Note evidence of 1.1% natural abundance of 13C

32 4.3. Vib-Rot spectra – improved model

 Isotopic effects 1 1 B   → Line spacing changes as μ changes  I  k 1  s  → Band origin changes as μ changes e 

2nd Example: HCl Isotope H35Cl and H37Cl

. H 35Cl 3H 37Cl  37.1/ 38 . 37 / 35   1.0015 35.1/ 36

-1 Shift in ωe is .00075ωe=2.2cm → Small!

33 4.3. Vib-Rot spectra – improved model

 Isotopic effects

HCl fundamental band

Note isotropic splitting due to H35Cl and H37Cl

34 4.3. Vib-Rot spectra – improved model   Hot bands When are hot bands (bands involving excited states) important?   v  g exp v  Nv  T   v v   v    exp 1 exp  N Qvib  T   T  N  e10  0 @300K 1  E.g. v,CO  3000K  1 1 N e 1 e  0.23 @3000K “Hot bands” become important when temperature is comparable to the characteristic vibrational temperature

hc / kT 1 NN/  e Gas  01cm  10 300K 1000K -9 -3 H2 4160.2 2.16 x 10 2.51 x 10 HCl 2885.9 9.77 x 10-7 1.57 x 10-2 -5 -2 N2 2330.7 1.40 x 10 3.50 x 10 CO 2143.2 3.43 x 10-4 4.58 x 10-2 -4 -1 O2 1556.4 5.74 x 10 1.07 x 10 -2 -1 S2 721.6 3.14 x 10 3.54 x 10 -2 -1 Cl2 566.9 6.92 x 10 4.49 x 10 -1 -1 I2 213.1 2.60 x 10 7.36 x 10 35 4.3. Vib-Rot spectra – improved model

 Examples of intensity distribution within the rotation-vibration band

B = 10.44cm-1 (HCl) B = 2cm-1 (CO)

36 4.4. Absorption Spectra for Combustion Gases

 TDL Sensors Provide Access to a Wide Range of Combustion Species/Applications

 Small species such as NO,

CO, CO2, and H2O have discrete rotational transitions in the vibrational bands

 Larger molecules, e.g., hydrocarbon fuels, have blended spectral features

37 Next: Diatomic Molecular Spectra

 Electronic (Rovibronic) Spectra (UV, Visible)