INTRODUCTORYINTRODUCTORY LECTURELECTURE onon ROTATIONALROTATIONAL SPECTROSCOPYSPECTROSCOPY CristinaCristina PuzzariniPuzzarini Dip.Dip. ChimicaChimica ““GiacomoGiacomo CiamicianCiamician”” UniversitUniversitàà didi BolognaBologna CODECSCODECS SummerSummer SchoolSchool 20132013 THEORETICALTHEORETICAL SPECTROSCOPYSPECTROSCOPY EE Eel Evib Erot ELECTRONICELECTRONIC VIBRATIONALVIBRATIONAL ROTATIONALROTATIONAL FREQUENCYFREQUENCY REGIONREGION mm/submm waves Rotational Spectroscopy Electronics Photonics “Building”“Building” thethe ROTATIONALROTATIONAL SPECTRUMSPECTRUM intensity (a.u.) 0 20 40 60 80 100 frequency (cm-1) (1)(1) RotationalRotational energyenergy levelslevels (2)(2) SelectionSelection rules:rules: transitionstransitions allowedallowed (3)(3) IntensityIntensity (transitions)(transitions) CoordinateCoordinate SSyystemstemss Molecule-fixed coordinate system Lab-fixed coordinate system ROTATINGROTATING RIGIDRIGID BODYBODY (CLASSIC(CLASSIC VIEW)VIEW) 1 T ωTIω 2 == angolarangolar velocityvelocity II == inertiainertia tensortensor AngolarAngolar VelocitVelocityy vi r i x ω y RIGIDRIGID BODY:BODY: v ωr i i z InertiaInertia TensorTensor I xx I xy I xz n n 2 2 2 2 I xx mi ri xi mi yi zi i1 i1 I I yx I yy I yz n I xy mi xi yi i1 I zx I zy I zz ByBy cconvenonventiontion:: IIc IIb IIa INERTIAINERTIA TENSORTENSOR II Principal Principal I xx I xy I xz I x 0 0 inertiainertia I I yx I yy I yz systemsystem I 0 I y 0 I zx I zy I zz 0 0 I z AngularAngular MomentMoment RIGID BODY J e defined in the rotating coordinate system (CM system) KINETICKINETIC ENERGYENERGY ofof aa ROTANTINGROTANTING RIGIDRIGID BODYBODY (CLASSIC(CLASSIC VIEW)VIEW) 1 1 J 2 T ωTIω 2 2 I == angularangular velocyvelocy II == inertiainertia tensortensor KINETICKINETIC ENERGYENERGY ofof aa ROTANTINGROTANTING RIGIDRIGID BODYBODY 1 1 J 2 T ωTIω 2 2 I 1 J 2 J 2 J 2 E T x y z rot rot 2 I x I y I z •• PotentialPotential energy?energy? •• FromFrom classicclassic mechanicsmechanics toto quantumquantum mechanicsmechanics ClassicClassic view:view: conservationconservation ofof angularangular momentummomentum QuantumQuantum mechanics:mechanics: commutationcommutation ofof operatorsoperators 2 2 2 2 2 2 2 J J x J y J z J X J Y J Z x,y,z molecule-fixed coordinate system X,Y,Z space-fixed coordinate system ˆ 2 ˆ 2 ˆ 2 ˆ 2 ˆ 2 ˆ 2 ˆ 2 J J x J y J z J X JY J Z TheThe spacespace-- andand moleculemolecule--fixedfixed componentscomponents ofof ĴĴ commute!commute! SS == matrixmatrix thatthat relatesrelates thethe coordinatescoordinates ofof thethe atomsatoms inin thethe moleculemolecule--fixedfixed systemsystem toto thosethose inin thethe spacespace--fixedfixed systemssystems Ĵ=SF ĴF where =x,y,z and F=X,Y,Z ĴF= F Ĵ Then: [ĴF,Ĵ] = ĴFSF’ĴF’ –SF’ĴF’ĴF = [ĴF,SF’]ĴF’ + SF’(ĴFĴF’ – ĴF’ĴF) = ieFF’F”(SF”ĴF’ + SF’ĴF”) = 0 !! where eFF’F”=permutation symbol [ĴX,SX] = 0 [ĴX,SY] = iSZ [ĴX,SZ] = –iSY [ĴX,ĴY] = –ieXYZĴZ 22 EIGENVALUESEIGENVALUES ofof ĴĴ ,, ĴĴZZ,, ĴĴzz ˆ 2 ˆ 2 ˆ 2 ˆ 2 Jˆ 2 , Jˆ 2 0 J , J z 0 J , J Z 0 z Z ˆ 2 2 J ,K ,M J J ,K ,M J (J 1) J=0,1,2,3, … ˆ 2 2 2 M=J,J-1 … -J J ,K ,M J Z J ,K ,M M ˆ 2 2 2 J ,K ,M J z J ,K ,M K K=J,J-1 … -J ROTATIONALROTATIONAL HAMILTONIANHAMILTONIAN 1 Jˆ 2 Jˆ 2 Jˆ 2 Hˆ x y z rot 2 I I I x y z ˆ H rotrot Erot rot ROTATIONALROTATIONAL ENERGYENERGY LEVELSLEVELS ClassificationClassification Examples CO CO2 CH4 SF6 NH3 H2O ByBy cconvenonventiontion:: IIc IIb IIa Let’s consider the simplest case m1 m2 R DIATOMIC/LINEARDIATOMIC/LINEAR MOLECULE:MOLECULE: RIGIDRIGID ROTORROTOR (approx)(approx) R z m1 CM m2 r1 r2 2 I = 0 I = I = I z x y I miri i m m 2 1 2 where I R m1 m2 reduced mass rr1212 rr2323 1 2 3 1 2 2 2 I m1m2r12 m2m3r23 m1m3 r12 r23 M Iz = 0 Ix = Iy = I ROTATIONALROTATIONAL ENERGYENERGY LEVELS:LEVELS: DiatomicDiatomic andand LinearLinear moleculesmolecules ˆ 1 ˆ 2 1 ˆ 2 1 ˆ 2 H rot J x J y J 2I x 2I y 2I Iz = 0 Ix = Iy = I makingmaking useuse ofof thethe eigenvalueseigenvalues ofof ĴĴ2 2 BB == rotationalrotational constantconstant Erot J (J 1) BJ (J 1) 2I JJ == 0,1,2,3,….0,1,2,3,…. ROTATIONALROTATIONAL ENERGYENERGY LEVELS:LEVELS: DiatomicDiatomic andand LinearLinear moleculesmolecules E =12B J=3 rot Erot BJ (J 1) E =6B J=2 rot E =2B J=1 rot E =0 J=0 rot ROTATIONALROTATIONAL ENERGYENERGY LEVELS:LEVELS: DiatomicDiatomic andand LinearLinear moleculesmolecules E =12B J=3 rot E(J 1 J ) 2B(J 1) E =6B J=2 rot E =2B J=1 rot E =0 J=0 rot ROTATIONALROTATIONAL ENERGYENERGY LEVELSLEVELS ˆ 2 ˆ 2 ˆ 2 ˆ 2 ˆ 2 ˆ 2 ˆ 2 J J x J y J z J X JY J Z x,y,z molecule-fixed coordinate system X,Y,Z space-fixed coordinate system ˆ 2 ˆ 2 ˆ 2 ˆ 2 J , J z 0 J , J Z 0 ˆ 2 2 J ,K ,M J J ,K ,M J (J 1) ˆ 2 2 2 J ,K ,M J Z J ,K ,M M M=J,J-1 … -J RotationalRotational energyenergy levels:levels: (2(2JJ+1)+1) foldfold degeneratedegenerate inin MM “Building”“Building” thethe ROTATIONALROTATIONAL SPECTRUMSPECTRUM intensity (a.u.) 0 20 40 60 80 100 frequency (cm-1) (1)(1) RotationalRotational energyenergy levelslevels (2)(2) SelectionSelection rules:rules: transitionstransitions allowedallowed (3)(3) IntensityIntensity (transitions)(transitions) SELECTIONSELECTION RULESRULES TransitionTransition moment:moment: 00 Approx BO: tot=rotvibele ele vib rot rot vib eled d d f f f i i i rot vib ele dipole moment in the space-fixed coordinate system X Xy Xx Xz Z=direction cosines F cosF F where =x,y,z (molecule-fixed) F F=X,Y,Z (space-fixed) F SELECTIONSELECTION RULESRULES where =x,y,z F F=X,Y,Z F F=direction cosines rot rot d ele vib vib eled d f F i rot f f i i vib ele F (1)(1) (2)(2) molecular dipole moment components (1)(1) SelectionSelection rulesrules (2)(2) NonNon--vanishingvanishing permanentpermanent dipoledipole momentmoment SELECTIONSELECTION RULESRULES “Rotational” transition moment Rij: Rij f F i where: F JKM F J' K' M' J F J' JK F J' K' JM F J' M' (1)(1) (2)(2) (3)(3) The direction-cosine matrix elements are known: (1)(1) JJ == 11 (2)(2) KK == 00 (3)(3) MM == 0,0, 11 “Building”“Building” thethe ROTATIONALROTATIONAL SPECTRUMSPECTRUM intensity (a.u.) 0 20 40 60 80 100 frequency (cm-1) (1)(1) RotationalRotational energyenergy levelslevels (2)(2) SelectionSelection rules:rules: transitionstransitions allowedallowed (3)(3) IntensityIntensity (transitions)(transitions) RotationalRotational energyenergy levelslevels ++ SelectionSelection rulesrules RotationalRotational transitiontransition frequenciesfrequencies (rotational(rotational spectrum:spectrum: xx axis)axis) E =12B J=3 rot Erot BJ (J 1) E =6B ++ J=2 rot J 1 E =2B J=1 rot E =0 J=0 rot E(J 1 J ) 2B(J 1) (B in energy units) h rot 2B(J 1) ??? ??? JJ=1=1--00 JJ=2=2--11 JJ=3=3--22 JJ=4=4--33 JJ=5=5--44 y y t t i i s s en en t t n n I I 2B/h 2B/h 2B/h 2B/h 2B/h 2B/h 4B/h 6B/h 8B/h frequency LINELINE STRENGTHSSTRENGTHS (1)(1) BoltzmannBoltzmann distributiondistribution E N g rot J J e kT N0 g0 (2)(2) degeneracydegeneracy 2J+1 E N rot J (2J 1)e kT N0 g /g =2J+1 f 0 exp(-E /kT) rot N /N =(2J+1)exp(-E /kT) f 0 rot J max 0 / N f N J E N rot J (2J 1)e kT N0 IntensityIntensity ofof RotationalRotational TransitionsTransitions 1 2 I N 2 n μm T i mn NJ/N0 Iabs 012345678910 J Intensity Population “Building”“Building” thethe ROTATIONALROTATIONAL SPECTRUMSPECTRUM intensity (a.u.) 0 20 40 60 80 100 frequency (cm-1) (1)(1) RotationalRotational energyenergy levelslevels (2)(2) SelectionSelection rules:rules: transitionstransitions allowedallowed (3)(3) IntensityIntensity (transitions)(transitions) 0Intens 20ity (u.a.) 40 60 80 100 Rotational spectrum ofCO wavenumbers ( cm -1 ) LinearLinear Rotor:Rotor: EIGENFUNCTIONSEIGENFUNCTIONS M |M | iM , YJ , PJ cos e SPHERICALSPHERICAL HARMONICSHARMONICS ^ Eigenvalues of J2: ħ2J(J+1) with J = 0, 1, 2, … ^ Eigenvalues of Jz : ħM with -J ≤ M ≤ J SPHERICALSPHERICAL HARMONICSHARMONICS J M VectorVector RappresentaRappresentationtion ofof AngularAngular MomentumMomentum JJ == 22 55 valuesvalues forfor MM Costant length (J) - 5 orientations (M) OneOne stepstep furtherfurther …..….. MoleculesMolecules areare NOTNOT rigid:rigid: centrifugalcentrifugal distortiondistortion SEMISEMI--RIGIDRIGID ROTORROTOR withwith CENTRIFUGALCENTRIFUGAL DISTORTIONDISTORTION perturbation theory ˆ ˆ 0 ˆ ' H rot H rot H dist rigidrigid--rotorrotor ˆ ' DJ ˆ 4 H dist 4 J 3 4B e >> 00 !!!! DJ 2 ' 2 2 Edist DJ J (J 1) 2 2 Erot / h BJ (J 1) DJ J (J 1) E =12hB J=3 rot/h J=3 centrifugalcentrifugal distortiondistortion E =6hB J=2 rot J=2 Erot=2hB J=1 J=1 E =0 J=0 rot J=0 3 rot 2B(J 1) 4DJ (J 1) [B, DJ in frequency units] y t i s JJ=1=1--00 JJ=2=2--11 JJ=3=3--22 JJ=4=4--33 JJ=5=5--44 en t n I 2B 2B 2B 2B 2B 2B 4B 6B 8B frequency AnotherAnother stepstep furtherfurther …..….. OtherOther typestypes ofof rotorrotor ClassificationClassification Examples CO CO2 CH4 SF6 NH3 H2O ByBy cconvenonventiontion:: IIc IIb IIa thusthus CC BB AA SPHERICALSPHERICAL TOPSTOPS CH4, SF6 , … Ia = Ib = Ic = I Erot = B J(J+1) Each level: (2J + 1)2 fold degenerate (K,M) = 0 !!! SYMMETRICSYMMETRIC TOPSTOPS I x I y I I z I // (z = symmetry axis) 1 Jˆ 2 1 1 ˆ ˆ 2 H rot J z 2 I I // I SYMMETRICSYMMETRIC TOPSTOPS 2 J (J 1) 1 1 2 Erot K 2 I I // I K=J,J-1 … -J SYMMETRICSYMMETRIC TOPSTOPS 2 J (J 1) 1 1 2 Erot K 2 I I // I CH F 3 prolateprolate oblateoblate II >> II BCl3 II << II II// == IIa II// == IIc >0 2 Prolate:Prolate: EErot == BJBJ((JJ+1)+(+1)+(AA––BB))KK wherewhere AA>>B=CB=C 2 Oblate:Oblate: EErot == BJBJ((JJ+1)+(+1)+(CC––BB))KK