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 ΔE Absorption = qd Emission What if Homonuclear?
Elastic scattering (Rayleigh), s = / Inelastic scattering (Raman), s =
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 (ΔJ) 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 ΔE 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 (ΔJ) Line shapes Vibration: Infrared Region (Δv, J) Internal Energy : μ Eint = Eelec(n)+Evib ()+ Erot(J)
t= v Erot δ+ μ s C x Evib ΔE μx O δ-
Heteronuclear diatomic 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 (v)+ Erot(J) ΔErot < ΔEvib < ΔEelec
Energy levels are discrete
Optically allowed transitions may occur only in certain cases E rot Absorption/emission spectra are discrete Evib ΔE Current interest Rotation Rigid Rotor Non-rigid Rotor Simple Harmonic Anharmonic Eelec Vibration Oscillator (SHO) Oscillator (AHO)
5 2. Rigid-Rotor model of diatomic molecule
Rigid Rotor Axes of rotation m 1 m2 C + - ~ 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 rotational energy as F(J), cm-1 E h FJ ,cm1 rot J J 1 BJ J 1 hc 8 2 Ic hc Note : E, J h hc( ,cm -1) so (energy,cm-1) = (energy,J)/hc 7 2. Rigid-Rotor model of diatomic molecule Rotational 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) ↓ ↓ ΔJ = J’ – J” = +1
Recall: FJ BJ J 1
e.g., J 1J 0 FJ 1 F J 0 2B 0 2B
8 2. Rigid-Rotor model of diatomic molecule
Rotational spectrum
Remember that: FJ BJ J 1
E.g., J 1J 0 FJ 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 1J J 'J " BJ"1 J"2 BJ" J"1
1 J 'J ",cm 2BJ"1 Let’s look at absorption spectrum 9 2. Rigid-Rotor model of diatomic molecule
Rotational spectrum Recall: FJ BJ J 1
E.g., J 1J 0 FJ 1 F J 0 2B 0 2B
12B 3 1.0
Heteronuclear 6B molecules only! λ ~2.5mm Tλ J”=0 11 6B 2 rot 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~2 cm λJ”=0 = 1/ν = ¼ cm = 2.5mm (microwave/mm waves) 10 11 rot,J=1 = c/λ = 3x10 /0.25 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 hcFJ hc J BJJ 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 dN / 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: B12C16O 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 vv vv Centrifugal distribution constant 3 J 'J ",vv 2B J"1 4D J" 1 v v 4B3 D B Notes: 1. But D is small; where 2 e 2 2 since D B 1.7 6 4 4 310 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 4 310 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 5.810 1/ 2 9 1 e 0.0014De ~ 810 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) (often termed Rovibrational)
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)
Δ/2 rmin
m1 m2
re
Molecule at instant of greatest Equilibrium position (balance between attractive + repulsive compression forces) – i.e. 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 / ,cm1 / c vib 2 s e 1 . Potential Energy U kr 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 Gv ,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 Gv ,cm e v 1 / 2 Gv ,cm e v 1/ 2 e xe v 1/ 2 ... H.O.T.
Decreases energy spacing 1st anharmonic correction
10 GG10 “Fundamental” Band Δν=+1 ee12x real (e.g., 1←0,2←1) 21 e 1 4xe
1st Overtone Δν=+2 20 2 1 3x (e.g., 2←0,3←1) e e
2nd Overtone Δν=+3 30 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 hce Qvib 1 exp exp kT 2kT The same zero energy must
Or choose reference (zero) energy at v=0, so Gv ev be used in specifying 1 molecular energies Ei for hce level i and in evaluating the then 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 vvib /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 cm1 23.0605 kcal/mole 1.602191019 J 5 Force 1 N 10 dynes o Length 1 A 0.1 nm
How many HO levels in a molecule? (Consider CO)
Do 256 kcal
N no. of HO levels 256 kcal/mole 41 2.86 cal/mole cm11 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: Tv, J RR SHO F J G v ΔJ = J' - J" BJ J 1 e v 1/ 2 Selection Rules: v 1 Two Branches: P (ΔJ = -1) J 1 R(ΔJ=+1)
Line Positions: T 'T" T v', J ' T v", J" Aside: Nomenclature for “branches”
Branch O P Q R S J'=J"+1 ΔJ -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" Probabilities Transition -8 -6 -4 -2 0 246 2B 24 4.2. Vib-Rot spectra – simple model
R-branch 1 RJ" ,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) ...
RJ" 0 2B J"1 Note: spacing = 2B, same as RR spectra
J'=J"+1 P-branch v'=1 J'= J" J'= J"-1 PJ" 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 measuring instrument and 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 Transition -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 Rv", J" o v" 2Bv ' 3Bv 'Bv" J" Bv 'B v " J"
2 . P-branch Pv", 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 Transition -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 Transition
P branch J" R branch 4 Bandhead 3 2 1
o -4 -3-2 -1 0 12 3 4 2B dRJ 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 G1 G 0 1 2 x 10 e e , x ee G20 G 2 G 0 2e 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''1/2ee V v B', B" Be, α BB""1/2ee 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 FJ BJ J 1 P(J+1) R(J-1) E FJ 1 F J 1 RJ 1 P J 1 v" J"+1 ΔE J" B" J 1 J 2 B" J 1 J J"- 1 E B"4J 2 B"
v' J'=J"+1 E FJ 1 F J 1 Be , ΔE 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 0.0463.88 0.17cm1
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 3H 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 e10 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 01cm 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)