22/ Breakdown of the Born-Oppenheimer approximation. Selection rules for rotational-vibrational transitions. P, R branches.

Transcription

1 Subjct Chmistry Papr No and Titl Modul No and Titl Modul Tag 8/ Physical Spctroscopy / Brakdown of th Born-Oppnhimr approximation. Slction ruls for rotational-vibrational transitions. P, R branchs. CHE_P8_M

2 TABLE OF CONTENTS 1. Larning Outcoms. Introduction 3. Brakdown of th Born-Oppnhimr approximation. 4. Th Intraction of Rotations and Vibrations 5. Summary

3 1. Larning Outcoms Aftr going through this modul, you should b abl to: (a) Us th Born-Oppnhimr approximation to undrstand th ffct of rotational transitions on vibrational spctra (b) Undrstand th brakdown of this approximation and intrprt th obsrvd spctra. (c) Us data from ral vibrational spctra to obtain fundamntal vibrations and rotational constants.. Introduction Th vibrational spctra of diatomic molculs is xpctd to show a fw lins corrsponding to a fundamntal and a fw ovrtons, as wll as a hot band corrsponding to transitions from th xcitd lvl. Howvr, th gas phas spctra ar mor complicatd than this, and th distinctiv shap of th contours givs us mor information about th molcul. 3. Brakdown of th Born-Oppnhimr Approximation Whn considring pur vibrational or pur rotational transitions, w assum that th two ar indpndnt. This is ssntially th Born-Oppnhimr approximation, which stats that sinc th nrgis involvd ar vry diffrnt, thy do not intract. Considr a molcul that is simultanously undrgoing vibration and rotation. Th rotational nrgy lvl sparations ar of th ordr 1-10 cm -1 whil th vibrational nrgy lvl sparations ar of th ordr of 3000 cm -1. Thus, to a good approximation, a molcul xcuts rotational and vibrational motions indpndntly. According to th Born-Oppnhimr approximation, th total nrgy of such a molcul can b writtn as th sum of th rotational and vibrational nrgis: E = E + E (1) Total rot vib In wavnumbr trms, th total nrgy would b as

4 S( υ, J) = BJ( J + 1) DJ υ + ω υ + ( J + 1) ω χ + HJ 3 ( J + 1) Th rotational constants ar much smallr than th vibrational frquncis and th cntrifugal distortion constants ar still smallr and can b ignord in comparison with th vibrational nrgis. Thus w hav: S( υ, J ) = BJ ( J + 1) + υ + ω υ + ωχ () Th slction ruls ar th sam as thos for th individual transitions, i.. Δυ = ± 1, ± ΔJ = ± 1,... (3) An analytical xprssion for th spctrum may b obtaind by applying th slction ruls to quation (). Considring only th υ = 0, J = J υ = 1, J = J transition, w can hav ΔS( J, υ) = S( J',1) S( J",0) = BJ ʹ ( J ʹ + 1) + (3/ ) ω (9 / 4) ω χ = ω ( 1 ) + B( J ʹ J ʹ ʹ )( J ʹ + J ʹ ʹ + 1) χ = ω 0 + B( J ʹ J ʹ ʹ )( J ʹ + J ʹ ʹ + 1) whr ω = ω (1 χ ). 0 In th cas whr ΔJ = +1, i.. J = J +1, w obtain ΔS( J", υ ) = ω0 + B( J ʹ ʹ + 1), J ʹ ʹ = 0,1,,... In th cas whr ΔJ = -1, i.. J = J +1, w gt ( BJ ʹ ʹ ( J ʹ ʹ + 1) + (1/ ) ω (1/ 4) ω χ ) ΔS( J', υ ) = ω0 B( J ʹ + 1), J' = 0,1,,...

5 Ths may b combind to yild ΔS( J, υ ) = ω0 + Bm, m = ± 1, ±,... (4) whr ω 0 is known as th band origin or band cntr. Equation (4) rprsnts th combind vibration-rotation spctrum. It will consist of vnly spacd lins (spacing = B ) on ach sid of th band origin ω 0 but sinc m 0, th lin at ω 0 will not appar. Lins to th low sid ofω 0(whn m is ngativ, i.. J = -1) ar rfrrd to as th P branch, whil thos to th high frquncy sid (m positiv, J = +1 ) ar rfrrd to as th R branch (Fig. 1). Figur 1: Schmatic diagram showing th P and R branchs of a diatomic molcul. Lins on th lft sid ar th P branchs and thos on th right sid ar R branchs. Th band cntr is missing.

6 From Figur 1, it is clar that th band cntr is th avrag of P (1) and R (0). Tabl 1 givs part of th IR spctrum of CO. From th tabl w s that th band cntr is at cm -1, whil th avrag lin sparation nar th cntr is 3.83 cm -1. This immdiatly givs: B = 3.83 cm -1 and B = cm -1. Tabl 1: Part of th IR spctrum of CO Lin cm -1 sparation, cm -1 Lin cm -1 sparation, cm -1 P (1) R (0) P () R (1) P (3) R () This is in satisfactory agrmnt with th mor prcis valu of B = cm -1 drivd from microwav spctroscopy. From B, th momnt of inrtia, I, and th bond lngth can b dtrmind. --- Exampl 1: Th four cntral lins in th high rsolution υ =1 υ = 0 infrard spctrum of H 37 Cl occur at 837.6, 858.8, 899. and cm -1. Dduc as much as possibl about th molcul. Would th corrsponding lins in H 35 Cl li at th sam spctral positions? Solution Th band cntr is at th avrag of th two cntral lins, i cm -1. This is qual to th fundamntal frquncy, ω 0. Th 4 B sparation at th cntr is 40.4 cm -1, giving a valu of 10.1 cm -1 for B. Th momnt of inrtia is h I = = 8π Bc 47 kg m

7 Th rducd mass is givn by 3 mh mcl µ = = = kg 3 ( m + m ) N H Cl A Thrfor, I r = µ = = m = nm Sinc th positions of th lins dpnd on th rducd mass, th lins for H 35 Cl will b at a diffrnt position. Th rducd mass of H 35 Cl is smallr and hnc its momnt of inrtia is smallr and rotational constant largr, so th lins will hav largr sparation. Th fundamntal vibration frquncy will also b highr sinc it is invrsly proportional to th squar root of th rducd mass Th Intraction of Rotations and Vibrations According to th prvious discussion, th rotational spacing should b th sam. Howvr, this is not th cas. A crowding of lins is sn in th R branch with incrasing J, whras th spacing incrass in th P branch. Apparntly, th assumption of indpndnc of vibration and rotation is in rror and thr is an intraction btwn th two. Sinc th frquncy of vibration is highr than that of rotation, a molcul vibrats 10 3 tims during th cours of a singl rotation and so th bond lngth (and hnc th momnt of inrtia I and B ) also changs continually during th rotation. Th rotational constant B dpnds on th 1 avrag valu of. Furthr, an incras in vibrational nrgy is accompanid by an incras in r vibrational amplitud bcaus of th widning of th potntial wll, and hnc th valu of B will dpnd on th vibrational quantum numbr. This incras is vn mor pronouncd for anharmonic vibrations. An incras in vibrational nrgy will lad to an incras in th avrag bond lngth. Th vibrational constant thn varis vn mor with vibrational nrgy. In gnral, sinc r av incrass with th vibrational nrgy,

8 B is smallr in th uppr vibrations than in th lowr ons. W may xprss this dpndnc in trms of an mpirical quation 1 B υ = B α υ + (5) whr B υ is th rotational constant in th vibrational lvl υ, B is th quilibrium rotational constant at th potntial minimum, and α is a small positiv constant. Considr th fundamntal vibrational chang υ = 0 υ = 1. Th rspctiv B valus ar B0 and B. For this transition, w will hav 1 ν = G( J', υ = 1) G( J", υ = 0) = ω0 + B1 J ʹ ( J ʹ + 1) B0 J ʹ ʹ ( J ʹ ʹ + 1) (6) whr W thn hav th two cass: Cas 1 ω = ω (1 χ ) 0 ΔJ = + 1, ν = ω J ʹ = J ʹ ʹ + 1 ʹ ʹ 0 + ( B1 + B0 )( J + 1) + ( B1 B0 )( J + 1) ʹ ʹ (7) whr J = 0, 1,, Cas ΔJ = 1 ν = ω, J ʹ ʹ = J ʹ + 1 ʹ 0 ( B1 + B0 )( J + 1) + ( B1 BO )( J + 1) ʹ (8) whr Jꞌ = 0, 1,,

9 Ths two quations may b combind to yild: ν m (9) P, R = ω0 + ( B1 + B0 ) m + ( B1 B0 ) with m = ±1, ±, ±3, Th positiv m valus rfr to R branchs and th ngativ ons to P branchs. Equation (9) allows us to fit th obsrvd transitions to a quadratic quation, from which th various trms can b dtrmind. Sinc B 1 < B0 bcaus th intrnuclar sparation is highr in highr vibrational stats, th spacing in th R branchs dcrass with incrasing J and th opposit is obsrvd for th P branchs. This givs a distinctiv shap to th spctrum. 5. Summary An approximation that can b usd to undrstand vibration-rotation spctra is th Born- Oppnhimr approximation, which stats that sinc th nrgy scals ar diffrnt, th two motions can b tratd indpndntly. Th two slction ruls also rmain th sam. Application of th slction ruls prdicts two branchs in th spctra, but th band cntr itslf is missing. Howvr, analysis of th spctra yilds information about th vibrational frquncis and rotational constants and all corrsponding molcular paramtrs can b dtrmind. Howvr, analysis of ral spctra shows a crowding of lins at th highr wavnumbr R branchs, and a spacing out of th P branchs with incrasing J. This implis that th rotational constant is not indpndnt of th vibrational quantum numbr and dcrass with incrasing υ sinc th avrag bond lngth incrass with incrasing amplitud of vibration.