Vibrational Spectroscopy


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1 Vibrational Spctroscopy armonic scillator Potntial Enrgy Slction Ruls V( ) = k = R R whr R quilibrium bond lngth Th dipol momnt of a molcul can b pandd as a function of = R R. µ ( ) =µ ( ) µ ( ) =µ dipol momntum at quilibrium, prmannt dipol momnt. chang in dipol momnt along vibrational coordinat ffct of strtching, comprssion, bnding, tc on dipol momnt. Rcall that transition is possibl only whn transition dipol matri lmnt is nonzro, that is f µ ˆ i For th dipol oprator in th transition dipol matri lmnt, substitut th sris approimation abov. 3 3 ˆ f µ i = f µ + i f 3 i µ + 6 f µ + i = f µ i + f i =µ f i + fi A transition implis that th final stat must b diffrnt from th initial stat fi = Thus fi for a transition to occur. Also, for a transition to occur. This rquirmnt mans that th vibration that is bing citd must caus th dipol momnt of th molcul to chang. This rquirmnt is tru for transitions in infrard vibrational spctroscopy.
2 Rcall that N ( ) ξ ν = ξ whr ν ν µ k 4 ξ= ħ Carful: µ is rducd mass. ξ 4 ξ 4 f i fi N f f ( ) ħ ħ µ = = ξ ξ N i i( ξ) dξ µ k µ k ħ ħω = f ξ i = f ξ i µ k k Rcall that ξ ν =ν ν + ν+ ξν =νν + ν+ ħω ħω f µ i = f ξ i = fii + f i+ k k ħω = ifi + fi+ k r prhaps statd a littl mor clarly. ħω f µ i = νi f i + fi+ k Th first intgral is nonzro only whn ν f =ν i ν= Th scond intgral is nonzro only whn ν f =ν i+ ν=+ Thrfor th slction rul for a harmonic oscillator in vibrational spctroscopy is ν=± Anharmonic scillator Gnral Potntial Enrgy Th actual potntial nrgy (not idalizd) can b writtn as a Taylor sris about th point R = R (i.., = ). V V V V V( ) = V( ) Th origin of th potntial nrgy can b st arbitrarily (and snsibly) to zro. That is, V() = by dfinition. V Also, sinc th slop of th potntial nrgy curv at R is zro =
3 Thus th first nonzro cofficint of th potntial nrgy sris abov is This cofficint can b rlatd to th forc constant. V 3 V V = k k = Th nt nonzro cofficint of th potntial nrgy sris causs th oscillator to b anharmonic. Th nonzro cofficints can b manipulatd to yild to yild th cubic anharmonicity constant, χ ν, th quartic anharmonicity constant, y ν, tc, Enrgy lvls of th anharmonic oscillator. 3 Eν = hν ν+ χν ν+ + yν ν+ + Not: Th nrgy lvls for th harmonic oscillator ar vnly spacd. ( E= hν ) owvr th nrgy lvls for th anharmonic oscillator ar not vnly spacd. ( E< hν ). Th nrgy lvls for th anharmonic oscillator gt closr togthr as th quantum numbr ν incrass. Mors Potntial Elctronic potntial nrgy curv is not purly quadratic, i.., V( ) k A bttr approimation of an actual potntial nrgy curv is th Mors potntial. a ( ) ( ) V = D whr µω a = D and D is th potntial wll dpth. V() harmonic (quadratic) D D anharmonic (Mors) D dissociation nrgy D D = ħ ω zro point nrgy
4 Using th Mors potntial, th spctroscopic trm bcoms 4 G ( ν ) =ν ν+ χν ν+ whr hν χ = 4D This prssion is an act solution for th Mors potntial. Worth mphasizing is that th Mors potntial is only an approimation (albit, a good on). If th abov spctroscopic trm, cannot rproduc th primntal spctra, thn a mor accurat potntial nrgy function is ndd. (Usually basd on Taylor sris pansion.) Whn th polynomial found from th Taylor pansion is usd, a mor gnral spctroscopic trm can b writtn. 3 4 G( ν ) =ν ν+ χν ν+ + yν ν+ + zν ν+ + Slction Ruls of th Anharmonic scillator Nonzro transition dipol matri lmnts of th anharmonic oscillator hav two origins.) Anharmonic wavfunctions ar combinations of wavfunctions..) Dipol can hav quadratic (cubic, tc ) dpndnc 3 4 i.., thus f µ i and f i, f i, f i, tc Th rsultant slction ruls ar complicatd, but transitions whr ν is a singl digit ar all allowd. That is, ν=±, ±, ± 3, ± 4, ± 5, Consquncs of Anharmonicity.) Transitions whr ν> ar allowd  Such transitions ar calld ovrtons.  transition is calld th fundamntal..) Mor than on vibration can b citd with a singl transition.  Such transitions ar combination bands.  Considr watr with its thr mods of vibration. symmtric strtch  ν s antisymmtric strtch  ν a bnding  ν b
5  Th total vibrational wavfunction is product of individual vibrational wavfunctions. νs νa ν b = νs νa ν b  vibrational ground stat  fundamntal for symmtric strtching mod  fundamntal for antisymmtric strtching mod  fundamntal for bnding mod 5  Eampls of combinations bands would b Miscllanous 3 A hot band is an absorptiv transition from an citd stat. Vibration Rotation Spctra Analysis for th Cl/DCl primnt is part of this sction. Th discussion won t b rpatd hr. Rfr to th lab handout. Combination Diffrncs Th tchniqu of combination diffrncs is usd to find th rotational constant for an citd stat. Th analysis is similar (but not idntical) to th analysis for th band spctrum of nitrogn. Considr th simplst trm for rotovibrational spctroscopy  harmonic oscillator and rigid rotor only. S ν,j =ν ν+ + BJ J+ ( ) ( ) owvr, th rotational constant changs for ach vibrational stat. W can find th rotational constant for th citd vibrational stat by taking th diffrnc of spcific spctral lins to rmov th influnc of th rotation of th ground vibrational stat.
6 Considr a portion of a molcular nrgy lvl diagram. 6 ν = ν = ν P (J+) ν R (J) ν P (J) J+ J J ν R (J) J+ J J R P ( J) S(,J ) S(,J) ν = ν= ν= 3 = ν + B( J J ) BJJ ( ) ν + + =ν B + B J+ B B J ( ) ( ) ( J) S(,J ) S(,J) ν = ν= + ν= 3 = ν + B( J+ )( J+ ) ν + BJJ ( + ) ( B B )( J ) ( B B )( J ) =ν B rotational constant of th ν = stat B rotational constant of th ν = stat B ν gnral rotational constant (has dpndnc on ν) Rmmbr P branch J= Rmmbr R branch J=+ Sinc th frquncis of th spctral lins dpnd on B and B, on can tak combination diffrncs of th appropriat lins to liminat ithr B or B, thus solving for th othr rotational constant. ν R( J ) ν P( J+ ) = 4B J+ ν R( J) ν P( J) = 4B J+ Thus, both rotational constants can b found.
7 Polyatomic Vibrational Spctroscopy 7 Normal Vibrational Mods Vibrations in a polyatomic molcul involv movmnt of all th atoms in a molcul.  In contrast, to only two atoms vibrating at a tim Mods of vibration that ar indpndnt of ach othr (i.., don t affct ach othr) ar calld normal mod vibrations. For a molcul with N atoms, 3N total mods of motion ist. 3 translational mods 3 rotational mods mods for linar molcul 3N6 vibrational mods 3N5 mods for linar molcul Symmtry of Polyatomic Vibrations To find symmtry of a vibration, attach vctors to atoms to indicat atomic movmnts. Th symmtry labl of th vibration is found by noting how ths vctors transform whn thy ar opratd upon by symmtry oprators. Eampl: Considr puckring mod of BF 3. F F B abov plan F blow plan BF 3 blongs to th D 3h symmtry group D 3h E σ h C 3 S 3 3 C 3 σ v A A   A A E   E   Not th ffct of symmtry oprations on vctors. E vctors sam χ = σ h vctors rvrsd χ =  C 3 vctors sam χ = vibration has A symmtry S 3 vctors rvrsd χ =  C vctors rvrsd χ =  σ v vctors sam χ =
8 Raman Spctroscopy 8 Raman Scattring Raman spctra com from fundamntally diffrnt procss than normal absorption/mission spctra. For Raman spctra, light is scattrd by molcul, not absorbd or mittd. Such scattring of light can b considrd as a collision btwn th photon and th molcul..) If th collision is lastic (lastic scattring), th frquncy of th scattrd photon is th sam as th incidnt photon  Procss is known as Rayligh scattring..) If th collision is inlastic (inlastic scattring), th frquncy of th scattrd photon is diffrnt than th incidnt photon. That is, th photon has gaind or lost nrgy.  Procss is known as Raman scattring. Vibrational Raman Spctroscopy An incidnt photon has an lctric fild that can intract with th charg distribution of a targt molcul. Th applid lctric fild dforms th lctron cloud and th ffct of th dformation is proportional to th strngth of th applid lctric fild.  In othr words, th dipol momnt of th molcul changs in rspons to th lctric fild. µ=µ +α E α  polarizability citd lctronic stat virtual lctronic stat ground lctronic stat Rayligh Stoks antistoks ν = 4 ν = 3 ν = ν = ν = Th dformd lctron cloud can rturn to its original stat to yild Rayligh scattring. r th dformd lctron cloud can rturn to anothr stat..) If th final stat has highr nrgy than th initial stat, th scattring vnt (and subsqunt spctral lin) is known as a Stoks shift (or Stoks scattring)..) If th final stat has lowr nrgy than th initial stat, th scattring vnt (and subsqunt spctral lin) is known as an antistoks shift (or antistoks scattring).
9 Slction Ruls 9 α α f µ ˆ i = f µ +α Ei = f µ + α + + Ei + α α f µ + α + E i = f µ i + f α Ei + f Ei α α = + + Efi = Efi Thus for vibrational Raman spctroscopy, th slction rul is ν=± for th harmonic oscillator. For th anharmonic oscillator, th slction rul ar complicatd as for vibrational absorption spctroscopy. ν=±, ±, ± 3, ± 4, ± 5, Also important to not is that not all vibrations can participat in Raman scattring. nly thos vibrations whr th polarizability of th molcul changs during th cours of a α vibration will hav a Raman spctrum (i.., ) Rotational Raman Spctroscopy Rmmbr that an applid lctric fild dforms th lctron cloud of a molcul. As th molcul rotats, th molcular dipol changs bcaus of th applid lctric fild Th inducd dipol changs at doubl th rat of rotation.  Th inducd dipol changs aftr half of a rotation. Th consqunc of th changing dipol is that th slction rul for rigidrotor Raman spctroscopy is J =±. S branch Q branch (Rayligh) branch ν
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