5.80 Small-Molecule Spectroscopy and Dynamics

Transcription

1 MIT OpnCoursWar Small-Molcul Spctroscopy and Dynamics Fall 008 For information about citing ths matrials or our Trms of Us, visit:

2 Lctur # 3 Supplmnt Contnts 1. Anharmonic Oscillator, Vibration-Rotation Intraction Enrgy Lvls of a Vibrating Rotor Rfrncs Anharmonic Oscillator, Vibration-Rotation Intraction Th Hamiltonian is P R k J H = + x +ax 3 + (1.1) µ µr }{{}}{{ } harmonic oscillator non rigid rotor First w must r-xprss R 1 in trms of a quantity whos matrix lmnts w know such as th displacmnt x. whr w truncat xpansion aftr (x/r ). x = R R x R = R + x = R 1 + (1.) R ( ) 1 1 x x R R powr sris xpansion (1.3) R R Th Hamiltonian oprator bcoms [ ] P R k J x 3x H = + x + ax (1.4) µ µr R R Lt us choos a basis { v J } whr v is th harmonic oscillator basis and J is th rigid rotor basis. Th rotational matrix lmnts w nd ar J J J = J(J + 1)δ JJ (1.5) J constant J = δ JJ constant (1.6) J vibr.coord. J = δ JJ vibr.coord. (1.7) ħ and B (in nrgy units) = J. (1.8) µr 1

3 5.76 Lctur # 3 Supplmnt Pag Thus th rotational xpctation valus of th Eq.(1) Hamiltonian bcom [ ] P H + k x + ax 3 x 6µB J J = + B J(J + 1) 1 + x (1.9) µ R ħ 1 whr was rplacd by µb R. ħ Now w nd som x matrix lmnts. Lt kµ k γ = hν = ħω = (1.10) ħ γ 1/ v + 1 x v,v+1 = (1.11) γ 1 x v,v + = [(v + 1)(v + )] 1/ (1.1) γ v + 1 x v,v = (1.13) γ [ ] 1/ (v + 1)(v + )(v + 3) x 3 v,v+3 = (1.14) 8γ 3 [ ] 3/ 3 x v,v v = 3 (1.15) γ Rmmbr that th harmonic oscillator part of our Hamiltonian is diagonal in th harmonic oscillator basis w hav chosn. [ ] 1 3 6µB vj H v J = hν v + δ JJ δ vv + ax vv δ jj + B J(J + 1) δ vv δ JJ x vv δ JJ + R ħ x vv δ JJ (1.16) Not that th Hamiltonian matrix is compltly diagonal in J. Th rmaining problm is how to arrang th H matrix now that w hav two indics J and v. Th Hamiltonian matrix is a supr-matrix consisting of a v, v matrix of J, J matrics. Howvr, sinc thr ar no matrix lmnts off-diagonal in J, it is convnint to altr our prspctiv and think of a st of singl v, v matrics, on for ach valu of J. Thus th Hamiltonian matrix is givn by v H v = hν v + 1 6µ + B J(J + 1) + B v + 1 J(J + 1) (1.17) γħ [ ] 3/ 1/ v + 1 B v + 1 v H v + 1 = 3a J(J + 1) (1.18) γ R γ [ ] 3/ 1/ v v H v 1 = 3a B v J(J + 1) (18a) γ R γ v H v + = 3 B µ γħ J(J + 1)[(v + 1)(v + )] 1/ (1.19) v H v = 3 B µ γħ J(J + 1)[(v 1)v] 1/ (19a) [ ] 1/ (v + 1)(v + )(v + 3) v H v + 3 = a 8γ 3 (1.0) [ ] 1/ v(v 1)(v ) v H v 3 = a 8γ 3 (0a)

4 5.76 Lctur # 3 Supplmnt Pag 3 W ar now in a position to us prturbation thory to dtrmin th contributions of th various trms in th Hamiltonian to th ignvalus. Not that whn facd with an infinit Hamiltonian matrix it will always b ncssary to us th Van Vlck transformation in ordr to truncat th matrix. W act as if part of th matrix isn t thr, yt w know that th Van Vlck transformd nrgis will b a vry good approximation to th ignvalus of th infinit matrix. In this xampl, th matrix will b tratd ntirly by first and scond ordr prturbation thory and no diagonalization will b carrid out. For our choic of basis, th zro-ordr Hamiltonian is diagonal by dfinition. [ ] (0) H (0) = hν v B J(J + 1) δ JJ δ vv = E vj (1.1) Th prturbation trms ar what is lft ovr B 6µB H (1) = ax 3 J(J + 1)x + R ħ J(J + 1)x δ JJ (1.) Th first ordr corrctions to th nrgy ar givn by µ 1 but γħ = hν. Thus v, J H (1) v, J = B 6µ γħ From th v H v ± 3 matrix lmnts w gt [ ] E () = a v(v 1)(v ) (v + 1)(v + )(v + 3) ±3 3hν 8γ 3 ( v + 1 ) J(J + 1) (1.3) E (1) = 6 ( B v + ) 1 hν J(J + 1) (1.4) This is a harmonic vibrational corrction to th rotational constant. Compar with th lading trm in Dunham s xprssion for Y 11 α. Not that this trm causs B(v) to incras as v incrass. This is puzzling bcaus vryon knows that B(v) dcrass for ral diatomics. Th scond ordr corrctions ar = v, J H (1) v, J v, J H (1) v, J E () (0) (0) v = v E (v,j) E (v,j) (1.5) Not that for ach of th thr allowd off-diagonal matrix lmnts of quations 18-0, thr will b two nonzro trms in th summation ovr v. (1.6) () a E ±3 = 8γ3 hν [(3v + 3v + )] (1.7) (0) (0) Th 3hν in th dnominator of (6) coms from E v,j E v±3,j. From th v H v ± matrix lmnts w gt () 9B 4 µ J (J + 1) E ± = γ ħ 4 [(v 1)v (v + 1)(v + )] hν () 9B 4 µ J (J + 1) 18B 4 J (J + 1) v + 1 E ± = γ ħ 4 [v + 1] = hν (hν) 3 (1.8)

5 5.76 Lctur # 3 Supplmnt Pag 4 This is a harmonic vibrational corrction trm to a cntrifugal distortion constant. This will not agr with Dunham s rsult bcaus w only kpt trms in th xpansion of B in powrs of x R through th scond powr. This scond x ordr corrction to th nrgy is actually a = 4 th powr corrction in R. Trms in th B xpansion through 4 x R also contribut. From th v H v ± 1 matrix lmnts w obtain () 9a (v 3 (v + 1) 3 ) 4B J (J + 1) [v (v + 1)] 1aB J(J + 1)[v (v + 1) ] E ±1 = + (γ) 3 hν γr hν R (γ) hν () 9 a B J (J + 1) 1 a B J(J + 1) v + E ±1 = (3v + 3v + 1) 8 hνγ 3 γr + 6 hν hνγ (1.9) R Th first trm of (9) should b addd to a similar trm which occurs in quation (7). Th sum will b th first trm in quation (30) blow. Th scond trm may b simplifid using µ 1 1 B µ = and = γħ hν R ħ giving 4B3 J (J+1) (hν) which should b compard with Dunham s Y 0. This trm is th harmonic oscillator contribution to th cntrifugal distortion constant. In summary E () a ( 15 ) [ ] v a 6BJ(J+1)(v+ + 1 ) 4B 3 J (J+1) 18B = hνγ hνγ R (hν) (hν) 3 4 J (J+1) (v+ 1 ) (1.30) Th first trm is an anharmonic contribution to ω x ( v + 1 ) and to th zro point nrgy. In fact, this is why ω x is calld th anharmonicity constant. Th scond trm is an anharmonic corrction to th rotational nrgy. Not that this corrction is of th sam sign as th harmonic corrction w obtaind in first-ordr in quation (4). B = B α v + 1. Howvr th sign of a is ngativ for ralistic potntial curvs. If a is larg nough, th ngativ anharmonic contribution to α will b largr than th positiv harmonic contribution and B(v) will dcras as v incrass. Th third trm is th harmonic contribution to cntrifugal distortion. E J = BJ(J + 1) DJ (J + 1). Th fourth trm is a harmonic vibrational corrction to th cntrifugal distortion constant D = D β v Enrgy Lvls of a Vibrating Rotor: Dunham s Exprssion for E(v, J) Drivd from E(r) Following is an xcrpt from Microwav Spctroscopy by C. Towns and A. Schawlow, pags 9 11, which dscribs th rsults of Dunham s invrsion of th potntial nrgy, V (r), xprssd as a powr sris in th dimnsionlss displacmnt coordinat ξ, into a powr sris in th rotational and vibrational quantum numbrs, J(J + 1) and (v + 1/). Th Rydbrg-Klin-Rs procdur is xactly th rvrs of this, convrting E(v, J) into V (r). Dunham s Solution for Enrgy Lvls Dunham 1 has calculatd th nrgy lvls of a vibrating rotor, by a Wntzl-Kramrs-Brillouin mthod, for any potntial which can b xpandd as a sris of powrs of (r r ) in th nighborhood of th potntial minimum. This tratmnt shows that th nrgy lvls can b writtn in th form

6 5.76 Lctur # 3 Supplmnt Pag 5 ( 1 ) l E vj = Y lj v + J j (J + 1) j (.1) l,j whr l and j ar summation indics, v and J ar, rspctivly, vibrational and rotational quantum numbrs, and Y lj ar cofficints which dpnd on molcular constants. Th ffctiv potntial function of th vibrating rotor may b writtn in th form U = a 0 ξ (i + a 1 ξ + a ξ +... ) + B J(J + 1)(1 ξ + 3ξ 4ξ ) (.) whr ξ = (r r )/r, B = h/8π µr. Th trm involving B J(J + 1) allows for th influnc of th rotation on th ffctiv potntial. Examination of th Harmonic Oscillator part of th potntial nrgy, U, will giv a valu for a 0. U = 1/k(r r ) = π µω (r r ) = π µω ξ r h hω if B thn U = ξ 8π µr 4B ω a 0 = h 4B Dunham 1 shows that th first 15 Y li s ar Y 00 = B /8(3a 7a 1/4) Y 10 = ω [1 + (B /4ω )(5a 4 95a 1 a 3 / 67a / a 1 a /8 1155a 1 4 /64)] Y 0 = (B /)[3(a 5a 1/4) + (B /ω )(45a a 1 a 5 / 885a a 4 / 1085a 3 / a 1 a 4 / a 3 / a 1 a a 3 /4 3, 865a 1 a 3 /16 6, 013a 1 a /3 (.3) , 985a 1 a /18 09, 055a 6 1 /51)] Y 30 = (B /ω )(10a 4 35a 1 a 3 17a / + 5a 1 a /4 705a 4 1 /3) Y 40 = (5B 3 /ω )(7a 6 / 63a 1 a 5 /4 33a a 4 /4 63a 3/ a 1 a 4 / a 3 3 / a 1 a a 3 /8 1953a 1 a 3 /3 4989a 1 a 4 /64 + 3, 65a 1 a /56 3, 151a 6 1 /104)

7 5.76 Lctur # 3 Supplmnt Pag 6 Y 01 = B {1 + (B /ω )[ a 1 9a + 15a 3 3a 1 a + 1(a + a 3 )/]} 1 1 Y 11 = (B /ω ){6(1 + a 1 ) + (B /ω )[ a 1 335a / + 190a 3 5a 4 / + 175a a 1 /8 459a 1 a + 145a 1 a 3 /4 795a 1 a 4 / a /8 715a a 3 / a 3 1 /4 9639a 1 a / a 1 a 3 / a 1 a 3 /8 14, 59a 1 a / (.4) + 31, 185(a 4 + a 5 )/18]} Y 1 = (6B 3 /ω )[5 + 10a 1 3a + 5a 3 13a 1 a + 15(a 1 + a 3 1 )/] Y 31 = (0B 4 /ω 3 )[7 + 1a 1 17a / + 14a 3 9a 4 / + 7a 5 + 5a 1 /8 45a 1 a + 105a 1 a 3 /4 51a 1 a 4 / + 51a /8 45a a 3 / + 141a 3 1 /4 945a 1 a / a 1 a 3 / a 1 a /8 1509a 3 1a / (a a 5 1)/18] Y 0 = (4B 3 /ω ){1 + (B /ω )[ a 1 119a + 90a 45a 4 07a 1 a + 05a 1 a 3 / 333a 1 a / + 693a 3 1 /4 + 46a + 16(a a 4 1 /)]} ( / ) Y 1 = 1B ω + 9a 1 + 9a 1 / 4a (.5) Y = (4B 5 /ω 4 )[ a 1 61a + 30a 3 15a a 1 /4 117a 1 a + 6a + 95a 1 a 3 / 07a 1 a / (a 1 + a 4 1 /)] Y 03 = 16B 5 4 (3 + a 1 )/ω Y 13 = (1B 6 /ω 5 )( a a a 1 88a 1 a 10a + 80a 3 /3) (.6) Y 04 = (64B 7 /ω 6 )(13 + 9a 1 a + 9a 1 /4) / It should b notd that B is gnrally much smallr than ω. For most molculs th ratio B ω is of th ordr of 10 6, although for light molculs such as H it approachs mor narly to In such cass mor trms ar rquird in th xprssions for th various cofficints. If B /ω is small, th Y s can b rlatd to th ordinary band spctrum constants as follows: Y 10 ω Y 0 ω x Y 30 ω y Y 01 B Y 11 α Y 1 γ Y 0 D Y 1 β Y 40 ω z Y 03 H (.7)

8 5.76 Lctur # 3 Supplmnt Pag 7 whr ths symbols rfr to th cofficints in th Bohr thory xpansion for th molcular nrgy lvls: F vj = ω v + ω x v + + ω y v + + ω z v + +B v J(J + 1) D J (J + 1) + H J 3 (J + 1) (.8) whr B v = B α v γ v (cf., p. 9, pp ). Sandman 3 has xtndd Dunham s tratmnt to includ othr trms of th sam ordr of magnitud which involv highr powrs of th vibrational quantum numbr. For th spcial cas of th Mors potntial function, Dunham shows that all th Y l0 s xcpt Y 10 and Y 0 vanish and all but th first trms in th xprssions for Y 10 and Y 0 ar zro. Bcaus of th simplicity of th xprssions obtaind with th Mors function, and bcaus it dos giv a quit good fit to th actual potntial in th rgion of r = r, th Mors function has bn widly usd. Svral important rlationships btwn constants hav bn drivd for Mors potntial functions. Ths ar oftn usful for stimating othrwis unknown paramtrs. B 3. Th Kratzr rlation D = 4 ω Th Pkris rlation α = ω 6 [ ( 3 ω3 x B ) ] 1/ B Th constant Y 00 is xactly zro for Harmonic and Mors oscillators, but an approximat valu for gnral oscillators is B ω x α ω α ω 1 Y 00 = B 1B B 3. Rfrncs 1. J. L. Dunham, Phys. Rv. 41, 71 (193). Also s ibid pag G. Hrzbrg, Spctra of Diatomic Molculs, D. Van Nostrand, Nw York (1950). 3. I. Sandman, Proc. Roy. Soc. Edinburgh 60, 10 (1940). Othr Important Rfrncs P. M. Mors, Phys. Rv. 34, 57 (199). C. L. Pkris, Phys. Rv. 45, 90 (1934). A. Kratzr, Zits. f. Physik 3, 89 (190).