Lecture 28 Title: Diatomic Molecule : Vibrational and Rotational spectra

Transcription

1 Lctur 8 Titl: Diatomic Molcul : Vibrational and otational spctra Pag- In this lctur w will undrstand th molcular vibrational and rotational spctra of diatomic molcul W will start with th Hamiltonian for th diatomic molcul that dpnds on th nuclar and lctronic coordinat. Thn w will us th Born-Oppnhimr approximation, to sparat th nuclar and lctronic wavfunctions W will driv th ign nrgy valus to undrstand th rotational and vibrational spctra of th ground lctronic stat of diatomic molculs. At th nd w will discuss th rotational and vibrational spctra of som diatomic molculs.

2 Pag-1 For a diatomic molcul A - B with n lctrons as shown in figur-8.1, th Schrodringr quation can b writtn as n i + V Ψ = EΨ µ m i= 1.(8.1) Whr, Ψ is th total lctronic and nuclar wavfunction & E is th total nrgy. 1 st trm: K.E. of th rlativ motion of th nucli with rducd mass mam µ = B m A + m B nd trm: K.E. of all lctrons. r a r b - And th potntial, Z Z ZZ V = + r r r n n n A B A B i > j= 1 i j i= 1 ia i= 1 ib = V V + V n nn A Figur-8.1 B Th molcular wavfunction, Ψ dpnds on both lctron & nuclar coordinats, Ψ=Ψ ( x, y, z, x, y, z,... x, y, z, XYZ,, ) = Ψ ( r, ) n n n w.r.t. origin say nuclus A Sinc th wavfunction dpnds on both lctron and nuclus coordinats, it is difficult to solv this problm vn for a simpl molcul. W would lik to sparat lctronic and nuclar motion. Bcaus of th diffrnt masss of th lctrons and nucli, w can considr th nucli to b stationary to solv th lctronic problm (Born-Oppnhimr approximation).

3 Pag- Th ssnc of th Born-Oppnhimr approximation is to dcompos nuclar and lctronic motions basd on th larg disparity of th masss of nucli and mass of lctron. If w nd to solv th Schrodringr quation givn in quation-8.1 using Born- Oppnhimr approximation, w hav to follow fiv stps. Stp-1 Lt us assum that th nucli ar clampd in fixd positions. This approximation is almost clos to th rality bcaus th lctronic motion is so fast that it will s th nuclar motions almost stationary. This will liminat th nuclar kintic nrgy trm in th Hamiltonian in quation 8.1 Stp- Undr th assumption in stp-1, w can writ quation 8.1 as n i + V r r = E r m i= 1 (, ) φ(, ) φ(, )..(8.) Hr nuclar K.E. is zro, and appars only as a paramtr and φ ( r, ) wavfunction. W can follow th procdur to fix = 1 m Solv + V( r, ) φ( r, ) = E φ( r, ) i is th lctronic Thn, fix = m Solv + V( r, ) φ( r, ) = E φ( r, ) i and do it for th ntir rang of.

4 Pag-3 Stp-3 Having obtaind th lctronic wavfunction φ ( r, ), and molcular wavfunction, E w can writ th total ( r, ) φ( r, ) χ Ψ =..(8.3) whr, χ Nuclar wavfunction Stp-4 Now w can substitut quation 8.3 in quation 8.1 and w gt, i + V r r µ m i (, ) χ (, ) φ(, ) χ = φ χ + i + φ χ µ m i = Eφ r ( r, ) V( r, ) ( r, ) Stp-5 Sinc, oprats only on φ ( r, ) µ i. W can writ, φ χ χ φ χ = + = Now, ( r, ) E E ( r, ) is th diffrnc w.r.t., so can oprat on both ( r, ) φ and χ. As a concqunc, th lctronic wavfunction is rlativly insnsitiv to changs in th nuclar positions and momnta, and is thrfor capabl of adjusting itslf quasistatically to th nuclar motion. This is known as th adiabatic approximation. Assumption: Elctronic wavfunction varis slowly with th intr-nuclar distanc,

5 Pag-4 So w gt, ( r, ) ( r, ) φ χ φ χ µ So, + E χ = Eχ (8.4) This is th Schrödingr quation of nuclar motion of diatomic molcul. It is indpndnt of lctronic motion whos ffct appars only through E, which is lctronic nrgy as a function of and act as a potntial nrgy for th motion of th nucli. This is th cntral approximation of th Born-Oppnhimr approach. Its justification stms from th fact that nuclar vlocitis ar small compard to lctronic vlocitis. Now w ar rady to solv th quation-8.4. This is simpl as w hav solvd it for hydrogn atom. Lt us introduc sphrical polar coordinat (, θφ, ) of on nuclus with rspct to othr as origin in quation 8.4, and w gt 1 1 sin θ E χ(, θφ, ) Eχ(, θφ, ) µ = sinθ θ θ sin θ φ χ(,, ) (, ) Now, θφ = S θφ (8.5) Whr, ( θφ, ) ( θφ, ) S = S JM ar spcifid by th molcular total angular momntum quantum numbr J and Z - componnt M, analogous to H - atom quantum no. & m.

6 Pag-5 Th angular part, ( θφ, ) ( 1 ) ( θφ, ) JM ( + 1) 1 1 J J sin θ + SJM ( θφ, ) = S JM ( θφ, ) µ sinθ θ θ sin θ φ µ 1 1 sin θ + S, 1, JM θφ = J J + SJM θφ sinθ θ θ sin θ φ M S = J J + S JM This is nothing but th rotational motion as shown in figur-8. Putting it in th quation, (8.6) 1 1 sin θ E S, JM µ sinθ θ θ sin θ φ = + + µ = E υ, J J J E S JM S ( θφ, ) JM ( 1 ) ( θφ, ) ( θφ) θ,φ otation whr, υ Vibrational quantum numbr J otational quantum numbr Vibration ( + 1) J J E E + + = υ, µ µ.(8.7) This dpnds only on intr-nuclar distanc and is calld th radial quation for th nuclar motion. Vibrational Wavfunction SJM (, ) θφ otational Wavfunction Figur-8.

7 Pag-6 otational Spctra of Diatomic Molcul Simpl modl is rigid rotor i.. is fixd as at quilibrium. In classical mchanics, th magnitud of angular momntum J of such a molcul rotating about cntr of mass with angular vlocity ω. J = µ υ = I ω From 8.6, rotational nrgy: EJ E J = + µ J J 1 = J J + I ( 1) = putting, µ = I This is in nrgy unit joul. W will convrt it to cm -1. EJ 1 h F ( J ) = cm = J ( J + 1) = hc hci 8π cµ ( 1) = B J J + Enrgy Lvls as shown in figur 8.3 J = ; F = J = 1 ; F = B J = ; F = 6B J = 3 ; F = 1B..(8.8) ( cosθ ) M imφ SJM = PJ J = 3 F(J) = 1B J = J = 1 J = otational nrgy lvls Figur-8.3 F(J) = 6B F(J) = B F(J) =

8 Pag-6 Now w will driv th slction rul for th transitions. W know that w hav to calculat th lctric dipol transition momnt intgrals to driv th slction rul. W tak µ l as dipol momnt of th molcul. So th transition momnt intgral is = SJM µ SJ M dτ Th componnts of th dipol momnt µ l ( ) ( ) ( ) µ = µ sinθ cosφ x µ = µ sinθ sinφ dτ = sinθ dθ dφ y µ = µ cosθ z π M cos M π sin im φ im φ Z = J J µ P θ P θ dθ dφ Sphrical Harmonics M M J1 J M 1 = M M J + M M J M + 1 M cosθ PJ ( cosθ) = PJ 1 + PJ+ 1 J + 1 J + 1 J + M M M J M + 1 M M Z = πµ PJ 1 PJ sinθ dθ + PJ+ 1 PJ sinθ dθ J + 1 J + 1 P P sinθ dθ = if J J J 1= J J =± 1; M = J + 1= J =± 1.(8.9)

9 Pag-7 Using th slction rul drivd in quation 8.9, th transitions ar shown in th following figur-8.4. J = 3 F(J) = 1B J = J = 1 J = otational nrgy lvls Figur-8.4 F(J) = 6B F(J) = B F(J) = Th transition nrgis ar J = ± 1 ( + 1) = ( + 1)( + ) = B J( J + 1) ( J) = F( J + 1) F( J) = B ( J + 1)[ J + J] = B ( J + 1) F J B J J F J ν Whr J =, 1,,.. Intnsity B 4B 6B 8B B B B B Figur-8.5 υ cm 1 Th xpctd rotational spctrum is shown in figur-8.5 So w s that undr non-rigid approximation, th transition ar quidistant with valu B and dpnds on th valu of B.

10 Pag-8 Exampl: otational Spctrum of HCl molcul shown in figur-8.6 B = 1.18 cm B 1 1 = 1.59 cm = = Ihc 8π µ h = 8π µ c h hc Intnsity H = 1.79 amu Cl = amu 1 amu = gm B υ cm B B B Figur µ = gm sc.. = rg cm gm = cm o cm 1.7 A = = cm sc So th quilibrium bond lngth of th HCl molcul is 1.7Å. Thus from th rotational spctrum of a molcul, w can driv th bond lngth and structur. Not: For molculs without prmannt dipol momnts such as H, th lctric dipol pur rotation transition probability is zro. In that systm th momnt of inrtia & intrnuclr distanc can b found out from analysis of rotational structur of lctronic absorption bands or rotational aman Spctra. W will discuss ths two spctra in latr.

11 Pag-9 Vibrational Spctra of Diatomic Molcul adial quation 8.7, d d + + = µ d d Whr EOT E Eυ, J ( + 1) J( J + 1) J J µ µ = E OT Th total nrgy, Eυ, J = EOT + Evib = Enucl d d + + = + µ d d EOT E EOT Evib d d + = µ d d Putting = E Evib 1 ξ d + = µ E E E vib d ξ ξ ξ is th lctronic nrgy as shown in th following figur-8.7 which bhavs as th potntial nrgy of th nuclar motion. Enrgy Figur-8.7 E() Bonding ab

12 Pag-1 Expand this function in powr sris about =, E = E de 1 d E d d Choos zro of th nrgy at E( ) = minimum bcaus this point is minimum de d =. 3 1 de 1 de 3 = So, E ( ) ( ) 3 d 3! d Putting = ρ Enrgy E 1 = +... ( ρ) K ρ Harmonic potntial Whr K d E = d If nglct th highr powr d 1 ξ ρ + ρ ξ ρ = ξ ρ µ dρ vib K E Harmonic oscillator quation with potntial as shown in figur-8.8, so th nrgy valu 1 Evib = E υ = υ+ ω υ = vibrational quantum numbr =,1,,3,... ν Classical frquncy 1 K = π µ 1 E() Bonding Figur-8.8 ab

13 Pag-11 Wav Function ξ 1 β x υ = υ υ 4 β = π µν os ( β ) N H x h whr, H ( x) υ β Hrmit polynomial Th wavfunctions ar shown in th figur-8.9 with dashd curvs. Th solid curvs ar th Dipol momnt µ = µ + µ x 1 ξ υ υ = υ = 1 Intrnuclar distanc whr, µ Prmannt Slction rul : x µ 1 Inducd du to vibration = µ ξ ξ dx + µ xξ ξ dx * * υ υ 1 υ υ υ υ ; υ = υ E = ω 3 E ( 1) = ω E = ω ( 1) E υ = υ 1 So only on transition is prdictd undr harmonic oscillator approximation Figur- ω 8.9). Howvr, many transitions ar sn in Figur-8.9 th infra rd vibrational spctra of diatomic molcul. So harmonic approximation is not th good approximation. Not: Dipol momnt of molcul with qual nuclus is always zro. So for N,O,... no infrard spctrum. υ = 3 υ = υ = 1 υ = ω ω ω ω

14 Pag-1 Anharmonic Oscillator Introducing anharmonicity 3 E = f r r g r r +... Condition: g f Mors Potntial as shown in th figur-8.1 : ( ) ( 1 ) r r E = D β Enrgy lvls: Eυ = ω υ+ ωx υ+ + ωy υ Convrt in cm G( υ) = ω υ+ ωx υ+ + ωy υ+ Enrgy whr ω, ω x, ω y in trms of υ Vibrational quantum numbr ω > ω x > ω y Hr 1 cm. Mors potntial a Slction rul, υ =± 1, ±, ± 3 Zro point nrgy: G = ω ω x + ω y 4 8 If th nrgy rfrrd to th lowst lvl, ( υ) = ( υ) G G G = ω υ+ ωx υ+ + ωy υ ω ωx + ωy = ω ωx + ωy υ ωx ωy +... υ + ( ωy +...) υ 4 3 = ωυ ωxυ + ωyυ 3 ω = ω ωx + ωy 4 3 whr, ωx = ωx ωy +... ω y = ω y +... G() E() Bonding Figur-8.1

15 Pag-13 Transitions btwn th vibrational lvls ν ( υ) ( υ ) abs = G G = G Thus th obsrvd absorption givs dirctly th positions of th nrgy lvls. Exampl: Obsrvd Vibration Frquncy of HCl -1 (cm ) Cal. Harmonic Approx. υ υabs G G G G G υ ωυ ω xυ = = ω ω x = = ω 4ω x = ω 4ω x = ω ω x = ( ) ( + ) ( ) ω x = 13.8 ω x = ( υ) ( υ 1) ( υ) G = G + G ( 1) x ( 1) x = ω υ+ ω υ+ ωυ+ ω υ = ωυ + ω ω x υ ωxυ ωx ωυ + ω x υ = ω ω x ω xυ ( υ) ( υ 1) ( υ) = + G G G = ω ω x ω x υ+ 1 ω + ω x + ω xυ = ω x which is dirctly th masurd anharmonicity.

16 Pag-14 cap In this lctur w hav larnt th origin of vibrational and rotational spctra. W startd with th Schrodingr quation for th diatomic molcul. Thn w usd th Born-Oppnhimr approximation, to sparat th nuclar and lctronic wavfunctions. W drivd th ign nrgy valus of th rotational and vibrational motions of th ground lctronic stat of diatomic molculs. At th nd w hav discussd th rotational and vibrational spctra of som diatomic molculs.