Lecture 2: Rotational and Vibrational Spectra

Transcription

1 Lctur : Rotational and Vibrational Spctra 1. Light-mattr intraction. Rigid-rotor modl for diatomic molcul 3. Non-rigid rotation 4. Vibration-rotation for diatomics

2 H O 1. Light-mattr intraction Possibilitis of intraction Prmannt lctric dipol momnt Rotation and vibration produc oscillating dipol (Emission/Absorption) HCl Enrgy ΔE Absorption = qd Emission What if Homonuclar? Elastic scattring (Rayligh), s = Inlastic scattring (Raman), s = / Virtual Stat v s m or as s < Inlastic scattring as >

3 1. Light-mattr intraction Elmnts of spctra: Lin position Lin strngth Lin shaps Intrnal Enrgy: E int = E lc (n)+e vib ()+ E rot (J) Lin position () is dtrmind by diffrnc btwn nrgy lvls What dtrmins th nrgy lvls? Quantum Mchanics! μ Rotation: Microwav Rgion (ΔJ) Elctric dipol momnt: + q r i i i E vib ΔE E rot μ x C O E μ x T rot 1/ν v s E lc Ar som molculs Microwav inactiv? YES,.g., H, Cl, CO Tim 3

4 1. Light-mattr intraction Elmnts of spctra: Lin position Lin strngth Lin shaps Intrnal Enrgy : E int = E lc (n)+e vib ()+ E rot (J) μ Rotation: Microwav Rgion (ΔJ) Vibration: Infrard Rgion (Δv, J) E vib ΔE E rot μ x C O δ+ δ- μ x t= v s E lc Htronuclar diatomic cas is IR-activ Ar som vibrations Infra-rd inactiv? Ys,.g., symmtric strtch of CO 4

5 1. Light-mattr intraction Summary E int = E lc (n)+e vib (v)+ E rot (J) ΔE rot < ΔE vib < ΔE lc E vib ΔE E rot Currnt intrst Enrgy lvls ar discrt Optically allowd transitions may occur only in crtain cass Absorption/mission spctra ar discrt Rotation Rigid Rotor Non-rigid Rotor E lc Vibration Simpl Harmonic Oscillator (SHO) Anharmonic Oscillator (AHO) 5

6 . Rigid-Rotor modl of diatomic molcul Rigid Rotor Axs of rotation m 1 m C + - ~ cm r 1 r C: r Cntr of mass C 1 m 1 = r m r 1 +r = r ~ 10-8 cm Assum: Point masss (d nuclus ~ cm, r ~ 10-8 cm) r = const. ( rigid rotor ) Rlax this latr 6

7 . Rigid-Rotor modl of diatomic molcul Classical Mchanics Momnt of Inrtia I miri r m1m rducd mass m m 1 -body problm changd to singl point mass Rotational Enrgy h Erot Irot Irot J J 1 J J 1 I I 8 I Convntion is to dnot rotational nrgy as F(J), cm -1 F E hc h 8 Ic 1 rot J cm J J 1 BJ J 1, Quantum Mchanics Valu of ω rot is quantizd I rot J 1 h / J Rot. quantum numbr = 0,1,, E rot is quantizd! hc Not : E, J h hc(, cm -1 ) so (nrgy,cm -1 ) = (nrgy,j)/hc 7

8 . Rigid-Rotor modl of diatomic molcul Rotational spctrum Schrödingr s Equation: Transition probability d m dx m d n E U x x 0 J 1 Slction Ruls for rotational transitions (uppr) (lowr) ΔJ = J J = +1 Rcall: F J BJ J 1 Wav function Complx conjugat Dipol momnt.g., J 1 FJ 0 B 0 B J 1 J 0 F 8

9 . Rigid-Rotor modl of diatomic molcul Rotational spctrum Rmmbr that: F J BJ J 1 E.g., J 1 FJ 0 B 0 B J 1 J 0 F J F 1 st diff = ν nd diff = spacing 1B B 6B 3 1B 4 0B B 4B 6B 8B B B B Lins vry B! 6B B F=0 B 4B 6B 1 J=0 In gnral: J 1 J J ' J " BJ" 1J " BJ" J" 1 J ' J ", cm 1 B J" 1 Lt s look at absorption spctrum 9

10 . Rigid-Rotor modl of diatomic molcul Rotational spctrum 1B Rcall: E.g., F J BJ J 1 0 FJ 1 FJ 0 B 0 B J 1 J B B F=0 B 4B 6B 1 J=0 T λ Htronuclar molculs only! λ J =0 ~.5mm rot for J=0 1 ~10 11 Hz (frquncis of rotation) J ν/b=j +1 Not: 1. Uniform spacing (asy to idntify/intrprt). B CO ~ cm -1 λ J =0 = 1/ν = ¼ cm =.5mm (microwav/mm wavs) rot,j=1 = c/λ = 3x10 10 /0.5 Hz = 1.x10 11 Hz (microwav) 10

11 . Rigid-Rotor modl of diatomic molcul Usfulnss of rotational spctra Masurd spctra Lin spacing = B Physical charactristics of molcul h B I r r Accuratly! 8 Ic Exampl: CO B = cm -1 r CO = Å 10-6 Å = m! 11

12 . Rigid-Rotor modl of diatomic molcul Intnsitis of spctral lins Equal probability assumption (crud but usful) Abs. (or miss.) probability pr molcul is (crudly) indpndnt of J Abs. (or miss.) spctrum varis w/ J lik Boltzmann distribution Rcall: J 1xp E kt N J J / N Q E J k hcf k Partition function: J Q rot Dgnracy is a QM rsult associatd w/ possibl dirctions of Angular Momntum vctor rot hc BJ J k 1 kt hcb hc k Dfin rotational T: K B r 1 J 1xp J J 1 r J 1 T r J 1 Symmtric no. (ways of rotating to achiv sam orintation) = 1 for microwav activ T r / r / T CO: σ=1 microwav activ! N : σ= microwav inactiv! 1

13 . Rigid-Rotor modl of diatomic molcul Intnsitis of spctral lins Rotational Charactristic Tmpratur: K Spcis θ rot [K] O.1 N.9 NO.5 Cl N J J 1 xp r J J 1 / T N T / r Strongst pak: occurs whr th population is at a local maximum dn J / N 1/ 0 J T / 1/ f T / dj max rot r B hc k hc k 1.44K / cm rot 1 13

14 . Rigid-Rotor modl of diatomic molcul Effct of isotopic substitution h Rcall: B 8 Ic Changs in nuclar mass (nutrons) do not chang r 0 r dpnds on binding forcs, associatd w/ chargd particls Can dtrmin mass from B Thrfor, for xampl: B B 1 13 C C O O m m 13 C C Agrs to 0.0% of othr dtrminations 14

15 3. Non-Rigid Rotation Two ffcts; follows from B 1/ r Vibrational strtching r(v) v r B Cntrifugal distortion r(j) J r B Effcts shrink lin spacings/nrgis Nots: Rsult: J ", v v v 3 4B 1. But D is small; whr D B sinc J B J J 1 D J J 1 F v v v D B NO J" 1 4D J" 3 J ' B 1 B Cntrifugal distribution constant D/B smallr for stiff/hi-frq bonds 15

16 3. Non-Rigid Rotation Nots: E.g., NO B 6 1. D is small;.g., D B. v dpndnc is givn by 1 3 4B D B B D/B smallr for stiff/hi-frq bonds NO / D ~ D cm D x 13.97cm ; / 1 1/ ~ 810 / B 9 cm 1 ~ / B D v v Asid: B D v v 8 x 1/ 1/ / D 3 B 4B Hrzbrg, Vol. I 5 1 dnots valuatd at quilibrium intr-nuclar sparation r 16

17 4. Vibration-Rotation Spctra (IR) (oftn trmd Rovibrational) 1. Diatomic Molculs Simpl Harmonic Oscillator (SHO) Anharmonic Oscillator (AHO). Vibration-Rotation spctra Simpl modl R-branch / P-branch Absorption spctrum 3. Vibration-Rotation spctra Improvd modl 4. Combustion Gas Spctra Vibration-Rotation spctrum of CO (from FTIR) 17

18 4.1. Diatomic Molculs Simpl Harmonic Oscillator (SHO) Δ/ r min m 1 m r Molcul at instant of gratst comprssion Equilibrium position (balanc btwn attractiv + rpulsiv forcs) i.. min nrgy position As usual, w bgin w. classical mchanics + incorporat QM only as ndd 18

19 4.1. Diatomic Molculs Simpl Harmonic Oscillator (SHO) Classical mchanics Forc k r - Linar forc law / Hook s law Fundamntal Frq. Potntial Enrgy Quantum mchanics v = vib. quantum no. = 0,1,,3, Vibration nrgy G=U/hc G Slction Ruls: s r 1 v, cm v / cv 1/ v v' v" 1 vib only! 1 1 vib k s /, cm / c 1 U kr r Parabola cntrd at distanc of min. potntial nrgy Equal nrgy spacing ral Zro nrgy = diss. nrgy 19

20 4.1. Diatomic Molculs Anharmonic Oscillator (AHO) SHO AHO 1 1 Gv, cm v 1/ Gv, cm v 1/ x v 1/... H. O. T. Dcrass nrgy spacing 1 st anharmonic corrction ral Δν=+1 Fundamntal Band (.g., 1 0, 1) 1 10 G 1 G 0 1x 1 4x Δν=+ 1 st Ovrton (.g., 0,3 1) 0 1 3x Δν=+3 nd Ovrton (.g., 3 0,4 1) x In addition, brakdown in slction ruls 0

21 4.1. Diatomic Molculs Vibrational Partition Function hc hc Qvib 1 xp xp kt kt Or choos rfrnc (zro) nrgy at v=0, so Gv v thn Vibrational Tmpratur Q vib 1 hc 1 xp kt 1 Th sam zro nrgy must b usd in spcifying molcular nrgis E i for lvl i and in valuating th associatd partition function hc k vibk N N vib g vib xp xp v Q v T vib vib vib / T 1 xp T whr g vib vib 1 Spcis θ vib [K] θ rot [K] O 70.1 N NO Cl

22 4.1. Diatomic Molculs Som typical valus (Banwll, p.63, Tabl 3.1) Gas Molcular Wight Vibration ω [cm -1 ] Anharmonicity constant x Forc constant k s [dyns/cm] Intrnuclar distanc r [Å] Dissociation nrgy D q [V] CO x NO x H x Br x Not IR-activ, us Raman spctroscopy! k / m / for homonuclar molculs D / 4x larg k, larg D Wak, long bond loos spring constant low frquncy

23 4.1. Diatomic Molculs Som usful convrsions Enrgy Forc Lngth 1cal J 1cm cal/mol 1V cm 1 N 10 1A o 5 0.1nm dyns kcal/mol J How many HO lvls in a molcul? (Considr CO) D o 56 kcal N no. of HO lvls 56 kcal/mol 1.86 cal/mol cm 170 cm 1 41 Actual numbr is?greater as AHO shrinks lvl spacing 3

24 4.. Vib-Rot spctra simpl modl Born-Oppnhimr Approximation Vibration and Rotation ar rgardd as indpndnt Vibrating rigid rotor Enrgy: Slction Ruls: Lin Positions: v'=1 v"=0 P R T v, J RR SHO FJ Gv BJ J 1 v 1/ v 1 J 1 T ' T" T J'=J"+1 J'= J" J'= J"-1 J"+1 J" v', J ' T v", J" Transition Probabilitis P branch Two Branchs: P (ΔJ = -1) R(ΔJ=+1) Asid: Nomnclatur for branchs Branch O P Q R S ΔJ Null Gap R() ΔJ = J' - J" R branch R(0) P(1) o B 4

25 4.. Vib-Rot spctra simpl modl R-branch R 1 J", cm G v' Gv" BJ" 1J " BJ" J" 1 R v o o (SHO)... 1 x 1 4x J" 0 BJ" 1 = Rotationlss transition wavnumbr (AHO,1 0) (AHO, 1) Not: spacing = B, sam as RR spctra P-branch P J" 0 BJ" Not: ω o =f(v")foraho v'=1 P R J'=J"+1 J'= J" J'= J"-1 P-R Branch pak sparation _ 8BkT hc v"=0 J"+1 J" Largr nrgy 5

26 4.. Vib-Rot spctra simpl modl Absorption spctrum (for molcul in v" = 0) Width, shap dpnds on masuring instrumnt and xprimntal conditions Transition Probabilitis Null Gap R() P branch R branch Lin (sum of all lins is a band ) R(0) P(1) o B Hight of lin amount of absorption N J /N Equal probability approximation indpndnt of J (as with RR) What if w rmov RR limit? Improvd tratmnt 6

27 4.3. Vib-Rot spctra improvd modl Brakdown of Born-Oppnhimr Approximation Allows non-rigid rotation, anharmonic vibration, vib-rot intraction T v, J Gv Fv, J B(v) v 1/ x v 1/ B J J 1 D J J 1 R-branch P-branch SHO Anharm. corr. RR(v) Cnt. dist. trm v", J" v" Bv ' 3Bv ' Bv" J" Bv ' Bv" J" v", J" v" B ' B " J" B ' B " J R o P o v v v v v " Bv B v 1/ Bv ' B ' v' 1/ B " " v" 1/ Transition Probabilitis P branch B v v B ' B " v 0 Null Gap P(1) R(0) R() v B ' B " v v Spacing on P sid, on R sid R branch o B 7

28 4.3. Vib-Rot spctra improvd modl Bandhad Transition Probabilitis P branch Null Gap R() P branch R branch R(0) P(1) J" R branch 4 Bandhad 3 1 o B dr dj J B' B" B' B" J" 0 B' E.g., CO B o B B J" ' bandhad B not oftn obsrvd Incrasing spacing Dcrasing spacing 8

29 4.3. Vib-Rot spctra improvd modl Finding ky paramtrs: B, α, ω, x 1 st Approach: Us masurd band origin data for th fundamntal and first ovrton, i.., ΔG 1 0, ΔG 0, to gt ω, x nd Approach: G G 10 0 G G 1 G0 1 x G0 1 3x Fit rotational transitions to th lin spacing quation to gt B and α B' B v' 1/ B" B v" 1/ o V V B' B" m B' B" m m J 1 in R - branch m J in P - branch B', B", x B, α 9

30 4.3. Vib-Rot spctra improvd modl Finding ky paramtrs: B, α, ω, x 3 rd Approach: Us th mthod of common stats v' v" ΔE P(J+1) R(J-1) J'= J" J'= J"-1 J"+1 J" J"- 1 Common uppr-stat In gnral E F R B FJ BJ J 1 J 1 FJ 1 J 1 PJ 1 " J 1J B" J 1J E B" 4J B" v' ΔE P(J) R(J) J'=J"+1 J'= J" J'= J"-1 E E F B J 1 FJ 1 ' J 1J B' J 1J B' 4J B' B, v" J"+1 J" Common lowr-stat 30

31 4.3. Vib-Rot spctra improvd modl Isotopic ffcts 1 B I ks 1 1 Lin spacing changs as μ changs Band origin changs as μ changs 1 st Exampl: CO Isotop 13 C 16 O 13 1 C C O O B 13 C C O 16 O B 13 C 16 O C 16 O B cm / 50cm 1 31

32 4.3. Vib-Rot spctra improvd modl Isotopic ffcts CO fundamntal band Not vidnc of 1.1% natural abundanc of 13 C 3

33 4.3. Vib-Rot spctra improvd modl Isotopic ffcts 1 B I ks 1 1 Lin spacing changs as μ changs Band origin changs as μ changs nd Exampl: HCl Isotop H 35 Cl and H 37 Cl H Cl 3H Cl / / / Shift in ω is.00075ω =.cm -1 Small! 33

34 4.3. Vib-Rot spctra improvd modl Isotopic ffcts HCl fundamntal band Not isotropic splitting du to H 35 Cl and H 37 Cl 34

35 4.3. Vib-Rot spctra improvd modl Hot bands Whn ar hot bands (bands involving xcitd stats) important? v v g xp Nv T v v v xp 1 xp N Q vib T T 10 N 1 E.g. v, CO 3000K 1 1 N 1 Hot bands bcom important whn tmpratur is comparabl to th charactristic vibrational tmpratur hc / kt 1 Gas 01 cm N1/ N0 300K 1000K H x x 10-3 HCl x x 10 - N x x 10 - CO x x 10 - O x x 10-1 S x x 10-1 Cl x x 10-1 I x x

36 4.3. Vib-Rot spctra improvd modl Exampls of intnsity distribution within th rotation-vibration band B = 10.44cm -1 (HCl) B = cm -1 (CO) 36

37 4.4. Absorption Spctra for Combustion Gass TDL Snsors Provid Accss to a Wid Rang of Combustion Spcis/Applications Small spcis such as NO, CO, CO, and H O hav discrt rotational transitions in th vibrational bands Largr molculs,.g., hydrocarbon fuls, hav blndd spctral faturs 37

38 Nxt: Diatomic Molcular Spctra Elctronic (Rovibronic) Spctra (UV, Visibl)