Numrical Problm St for Atomic and Molcular Spctroscopy Yr HT SRM Sction 1: Atomic Spctra 1. For ach of th atomic trm symbols 1 S, P, 3 P, 3 D, 4 D, writ down: a) Th associatd valus of th total spin and orbital angular momntum quantum numbrs, S and L; b) th possibl valus of J, th total angular momntum quantum numbr; and c) th numbr of stats associatd with ach valu of J.. a) How many fin-structur componnts would b obsrvd in th mission lin n = 4 3 of th H- atom if th ffcts of spin-orbit coupling wr fully rsolvd in th spctrum? Illustrat ths transitions on a Grotrian diagram. b) On of th n = 5 trms of hydrogn is split by spin-orbit coupling into two lvls with an nrgy diffrnc of 0.0039 cm -1. Dtrmin th orbital angular momntum quantum numbr, l, for this stat and prdict th analogous splitting in Li +. Th fin structur constant, α = 0.007973. 3. A lin in th mission spctrum of potassium ariss from th transition 4 P 3 D. Upon closr xamination it is found to consist of thr lins at nrgis (in cm -1 ) 8494.13, 8496.45 and 8554.17. a) Draw a diagram rprsnting ths transitions givn that th splitting is rgular (i.., nrgy incrass with J) for th P trms and invrtd for th D trms. b) Calculat th sparation of th individual J componnts of th stats involvd. 4. A numbr of possibl transitions in th bryllium atom ar listd blow. Which ar fully allowd? For thos which ar not allowd, which slction rul would hav to b brokn and what mchanisms which might lad to such a brak-down? (n.b., on trm symbol is invalid, which on and why?) s5s ( 1 S 0 ) s5d ( 1 D ) s5s ( 3 S 1 ) sp ( 1 P 1 ) s5s ( 1 S 0 ) s ( 1 S 0 ) s5p ( 3 P 1 ) s3s ( 3 S 1 ) s5p ( 3 P 1 ) 3s4s ( 3 S 1 ) s3p ( 3 P 1 ) 3p ( 3 D ) s3p ( 3 P 0 ) 3p4p ( 3 D ) s3p ( 3 P 0 ) 3p4p ( 3 P 0 ) 5. a) Sktch th radial distribution function of th 3s, 3p and 3d orbitals of sodium and show in your graph whr th innr shll lctrons li. Explain why ths orbitals hav diffrnt nrgis. b) A transition from th 3s to a 3p lvl in Na is at 16 961cm 1. Transitions from this 3p lvl to th d lvls form a sris of lins of which th first thr ar at 8 75, 16 14 and 19 386cm 1. Sktch th nrgy lvl diagram for th transitions involvd and dduc th ionisation potntial of Na in its ground stat. (lav your answr in cm 1 ). c) Explain th fact that th 3s 3p transition rfrrd to in part (b) is split into a doublt sparatd by 17cm 1 and undr highr rsolution th p to d transitions ar ach sn to consist of thr lins. Explain ths obsrvations.
P3/, 3 P 0, S 1/, 1 S 0, 4 S 3/, P 1/ 6. Th following trm symbols rfr to th ground stats of crtain first row atoms. Which atoms? 7. Th 3s 3p4p configuration of an xcitd stat of th Si atom lads to a stat with orbital angular momntum L = 1 and which is split by spin-orbit coupling into th lvls shown blow. Dduc possibl trm symbols for th thr lvls and indicat th dipol allowd transitions which would aris in absorption from ths lvls whn th 4p lctron is xcitd into a 5s orbital. iii ii i 160 cm -1 8. a) Draw up a tabl showing th microstats allowd by th Pauli principl for an nd lctronic configuration and hnc driv th prmittd trm symbols assuming Russll-Saundrs coupling. b) Us angular momntum coupling argumnts to show that th lvls drivd in j-j coupling for an npnf configuration ar [ 1 5 1 7 3 7 3 5 ] 3,, [ ] 4, 3, [ ] and [ ] 5, 4, 3, 4 3,, 1,. 9. a) Th lowst nrgy configuration of th N + ion is s p. Th first xcitd configuration is obtaind by promoting and lctron from th s to th p orbital. Us angular momntum coupling argumnts to driv th stats that aris from th xcitd configuration, noting that a p 3 configuration, as in th nutral N atom, givs ris to trms 4 S, P and D. An allowd transition has bn dtctd in N + btwn on of th lvls of its ground trm and on of th lvls associatd with th first xcitd configuration. b) Th g-factors of both lvls wr masurd to b 1.50. Assuming th Landé g-factor formula, suggst an assignmnt for th transition. 10. Explain th following obsrvations: a) Th four lowst nrgy lctronically xcitd stats of H (at low rsolution) li at 159850 cm -1, 16671 cm -1 169083 cm -1 and 17119 cm -1 abov th ground stat. Th absorption spctrum of H, howvr, displays a strong lin at only on of ths nrgis. b) H gas shows ngligibl absorption of radiation in th infra-rd, but strong absorption bands ar obsrvd at 4858 cm -1 and 933 cm -1 whn th gas is xcitd in an lctric discharg. c) Th first two xcitd lctronic stats of gasous carbon and oxygn atoms occur at th following xcitation nrgis: Spcis xcitation nrgis / cm -1 Carbon 16.4, 43.3 Oxygn 158.5, 6.5 Howvr, th first two xcitd stats for gasous nitrogn atoms li at mor than 19000 cm -1 abov th ground stat. d) A 1 D 1 P 1 transition obsrvd in th mission spctrum of Cd consists of a singl lin. In th prsnc of a magntic fild thr lins ar obsrvd. For th P 1/ S 1/ mission lin in sodium th singl lin is obsrvd to split into four lins whn a magntic fild is applid.
11. Shown blow is th PES spctrum of mrcury vapour. Th x-axis givs th kintic nrgy of th jctd lctrons, as dtrmind by th instrumnt, rathr than th mor usual ionization nrgis. Th incidnt photons usd to ioniz th sampl ar from a hlium rsonanc lamp and hav wavlngth 584 Å. a) What is th nrgy of th incidnt radiation in V? b) What ar th approximat ionization nrgis lading to ach of th paks in th spctrum? c) Th paks in th spctrum corrspond to rmoving lctrons from th valnc shll of th mrcury atom, i.. from th 5d shll or th 6s shll. Giv th trm symbols for th possibl ions arising from th jction of a singl lctron from ithr th 5d or 6s shll in Hg. d) By simply considring th dgnracis of ach of ths ionic lvls, what should th xpctd ratios of th pak intnsitis in th spctrum b? Assign appropriat trm symbols to ach of th paks in th spctrum. Sction : Molcular Rotational (microwav) Spctroscopy 1. Which of th following molculs xhibit pur rotational (microwav) spctra? HF, NH 3, CH 4, CH 3 F, BF 3, H O, C F H, O 3, CO, tolun, Argon---HCl 13. Th rotational constant of H 35 Cl is 10.5909 cm -1. Calculat th rotational constants of H 37 Cl and H 35 Cl. 14. Without calculating th spcific valus, arrang th following molculs in ordr of incrasing valu of thir rotational constants, B : HF, DF, H C C C C C N, HD, 1 C 16 O, 13 C 16 O, 1 C 18 O, CICN.
15. Givn that th CO bond lngth in th molcul OCS is 0.1165 nm and th CS bond lngth is 0.1558 nm, dtrmin its momnt of inrtia. At which frquncis (units Hz or GHz) do th J = 1 0 and 1 transitions occur in th rotational spctrum of OCS? 16. Classify th following molculs as sphrical, symmtric or asymmtric tops, and stat which will giv pur rotational spctra. For th symmtric tops, sktch th principal axs and indicat th uniqu axis. SF 6, BrF 5, NH 3, NO, CS, CH, N O, CF 3 I, BH 3, BH, SO, C 6 H 6, XF 6, IF 7 17. Th rotational trms of a diatomic molcul (th nrgy lvls xprssd as wavnumbrs) ar givn to a good approximation by F J = BJ ( J + 1) DJ ( J + 1). i) Explain th maning of J, B and D in this xprssion. ii) What slction ruls apply to pur rotational spctroscopy? iii) Driv an xprssion for th nrgy of transitions obsrvd in a high rsolution rotational spctrum. iv) In a high rsolution microwav study of H 19 F, th first four lins in th spctrum wr obsrvd at.0180 cm -1 44.018 cm -1 65.9970 cm -1 87.995 cm -1 By drawing a suitabl straight-lin graph, dduc th valus of B and D for H 19 F. v) Hnc dtrmin th H 19 F bond lngth 18. Ammonia, NH 3, is an oblat symmtric top. a) Stat th slction ruls which apply to this molcul undrgoing changs in rotational nrgy. Th origin of ths ruls. Hnc dtrmin an xprssion for th frquncis of th allowd transitions (ignor cntrifugal distortion); draw a lablld sktch of th spctrum you xpct. b) Th idalizd microwav spctrum of ammonia shows absorptions at th following frquncis, in GHz 1798.9 398.6 998. Assign th transitions (giving your rasons), and dtrmin all you can about th molcul. 19. Th molcul H 35 Cl xhibits rotational absorption lins in th far infrard at th following wavnumbrs (in cm -1 ): 83.3, 104.13, 14.73, 145.37, 165.89, 186.3, 06.6, 6.86. (Not that thr may b othr lins in th microwav rgion, too.) a) Idntify th transitions and us a graphical procdur to dtrmin th rotational and cntrifugal distortion constants. Calculat th bond lngth of HCl. b) Prdict th rotational constants for DCl. c) Dtrmin th most populatd rotational lvl in HCl at T = 300 K.
0. Show that th quantum numbr of th most highly populatd rotational lvl of a rigid diatomic molcul is givn by kt 1 J = max hbc. a) Which stat in HCl (B= 10.6 cm -1 ) would b th most populatd at 700, 800, 900, 1000, 1100 and 100 K? b) If it could b dtrmind xprimntally, would J max b a good way to dtrmin th tmpratur of a sampl? Why? c) How could on bttr dtrmin th tmpratur spctroscopically? d) How is th intnsity of th transition J+1 J rlatd to th population of stat J. 1. Th nrgy lvls of a symmtric top molcul including cntrifugal distortion ar givn (in cm -1 ) by F(J,K) = BJ(J+1) + (A B)K D J J (J+1) D JK J(J+1)K D K K 4 whr D J, D JK and D K ar cntrifugal distortion constants. a) Driv an xprssion for th wavnumbrs of th allowd transitions in th rotational spctrum (ΔJ = ±1, ΔK = 0) In th rotational spctrum of CH 3 F, absorption maxima wr found at wavlngths of 1.958 mm and.937 mm. At highr rsolution, ths paks split into thr and two componnts, rspctivly. b) Explain th origin of ths obsrvations, assign th transitions and dtrmin as many rotational and cntrifugal distortion constants of CH 3 F as you can. Giv your constants in cm 1 and in MHz.. Show that for a homonuclar diatomic molcul composd of atoms with nuclar spin quantum numbrs I, that th ratio of th numbr of symmtric nuclar spin functions to antisymmtric nuclar spin wavfunctions is (I+1/ I). Commnt on th valu of this quantity for I =0 atoms. 3. a) Draw a diagram showing th ffct of an lctric fild on th nrgy lvls (J =, K = 1) and (J = 3, K = 1) of a prolat symmtric top. Indicat th splittings (in trms of μ and E fild) of th M J lvls. Idntify th allowd transitions and hnc prdict th xpctd form of th spctrum. b) For CH 3 I th rotational constant, B, is 153 MHz and th dipol momnt is 1.0 Dby. Using your rsults from part i), prdict th frquncis (in MHz) of th lins you would xpct to s from th (J =, K = 1) (J = 3, K = 1) transition whn: (a) no lctric fild is applid; (b) whn an lctric fild of 10 4 Vm 1 is applid paralll to th lctric fild vctor of th radiation. 4. Whn an lctric fild, of magnitud 10 4 V, is applid paralll to th lctric-fild vctor of th microwav radiation, th rotational transition (J =, K = 1) (J = 1, K = 1) of CH 3 Cl at 1.67360 cm 1 is split into thr componnts, with lins at 1.67569, 1.67360 and 1.67151cm 1. Dtrmin th rotational constant, B, of CH 3 Cl and its dipol momnt, in Dby.
Sction 3: Molcular vibrational (infrard) and vibration-rotation spctroscopy 5. Th forc constant of 79 Br is 40 Nm -1. Calculat th fundamntal vibrational frquncy and th zropoint nrgy of 79 Br. 6. What is mant by a normal mod of vibration? How many normal mods of vibration do th following molculs hav? NH 3, HCN, SO, C H, C 10 H 8, C 60. Sktch th normal mods for HCN and SO 7. What is mant by zro-point nrgy? Calculat th nrgy chang for th raction HD + HCl H + DCl assuming that ach molcul is in its ground vibrational stat. Ignor anharmonicity ffcts. [ω (H ) = 4395 cm 1, ω (HD) = 3817 cm 1, ω (HCl) = 990 cm 1, ω (DCl) = 091 cm 1.] 8. a) Th Mors potntial can b writtn V ( r) = D[1 xp( βx)], whr x, th displacmnt, is (R R ). Expand th xponntial to first ordr in x (i.., x 1 + x) and show that, in this limit, th potntial is harmonic with forc constant, k Mors, givn by k = β. Mors D Th vibrational frquncy, ω, is dfind in trms of this forc constant, ω (in rad s -1 ) = kmors μ, whr µ is th rducd mass. Us this dfinition of ω, togthr with th xprssion abov for k Mors to find an xprssion for β in trms of µ, D and ω. Givn th anharmonicity paramtr for a Mors oscillator is β x = show that D in J = ω μω 4x b) Using th Mors nrgy lvls, show that th wavnumbr of a transition from th ground stat (v = 0) to th v th vibrational lvl is givn by v ~ (0 v) = vω v( v + 1) ω x, whr ω is th vibrational constant (a wavnumbr givn byω = ω / πc ). In th low rsolution IR spctrum of 1 H 79 Br a strong absorption is obsrvd at 558.5 cm 1 and a wakr absorption at 506.5 cm 1. Us th Mors oscillator nrgy lvls to account for ths obsrvations; dtrmin th paramtrs ω, x, D and β, clarly stating th units of your answrs. [Not that all of th formula givn in part (a) ar in SI, so to us thm to comput β you will nd to mak sur that ω is in rad s 1 and D in J. Us intgr atomic masss.] 9. Vibrational absorption lins for H 35 Cl li at th following wavnumbrs: 885.9, 5668.0, 8347.0, 109.9 cm -1 Show that ths valus ar in agrmnt with thos xpctd for a Mors oscillator. Driv valus for th forc constant, zro-point nrgy and dissociation nrgy of th molcul.
30. Adjacnt vibration-rotation lins nar th cntr of th ν 3 strtching fundamntal band in th infrard spctrum of CO occur at th following wavnumbrs: 351.64, 350.08, 347.74, 346.18 cm 1. Calculat th band origin of this fundamntal vibration and th C O bond lngth. 31. Th four cntral lins in th high rsolution v=1 v=0 infra-rd spctrum of H 37 Cl occur at 837.6, 858.8, 899. and 918.6 cm -1. Dduc as much as possibl about th molcul. Would th corrsponding lins in H 35 Cl li at th sam spctral positions? [ 1 H = 1.0078 amu, 37 Cl = 36.9659 amu] 3. a) Sktch th normal mods of dichloroactyln (dichlorothyn), Cl C C Cl, indicating thir symmtris and which ar dgnrat. c) For ach normal mod, dtrmin th symmtry of th v = 0, 1, and vibrational lvls, and hnc dtrmin whthr or not th v = 0 1 transition for ach mod is allowd. For th allowd transitions, stat th dirction of th transition dipol momnt. Mor difficult: do th sam for th v = 1 transitions. d) If th vibrations ar anharmonic, transitions with v = ± may b wakly allowd. Dtrmin which (if any) of th v = 0 transitions ar allowd by symmtry considrations. ) Idntify a combination lin which is symmtry allowd. 33. Th vibration-rotation spctrum of CS shows strong bands at 397 cm -1 (IR activ, prpndicular band) 685 cm -1 (Raman activ) and 1510 cm -1 (IR activ, paralll band). Th corrsponding transitions in OCS li at 50, 859 and 06 cm -1 but appar in both th IR and Raman spctra. i) Explain what is mant by paralll and prpndicular bands. ii) Assign th fundamntal mods of vibration and discuss th gomtry of ths molculs. iii) Th lin spacing in th rotational Raman spctrum of CO and CS ar 3.1 and 0.873 cm -1 rspctivly. Calculat th C-O bond lngth in CO and th C-S bond lngth in CS. iv) Explain fully why th lin spacing in th rotational Raman spctrum of OCS is vry clos to that of CS n.b., Assum intgral rlativ molcular masss. Th nuclar spins of 16 O and 3 S =0. 34. a) In th idalizd IR spctrum of actyln (thyn) a strong absorption is sn at 390 cm 1 ; this is attributd to th fundamntal (v = 0 1) of mod 3 (for th normal mods, s th lctur nots). A scond strong absorption at 730 cm 1 is attributd to th fundamntal of mod 5. Combination lins in which only two mods ar involvd ar obsrvd at th following wavnumbrs (in cm 1 ): 1340 700 3900 4100 560 6660 Dtrmin th wavnumbrs of th othr thr normal mods (it is not ncssary to idntify which is which; anharmonicity is ignord throughout). b) Account for th following combination lins, which involv mor than two mods 1950 and 3310 cm 1
35. Th tabl blow lists th vibrational wavnumbrs and infrard/raman activitis for cyanogn (C N ). What dos this information tll us about th structur of cyanogn? Suggst an assignmnt for ach band. 6 cm 1 Infrard activ xhibiting PQR branchs 506 cm 1 Raman activ 848 cm 1 Raman activ 149 cm 1 Infrard activ xhibiting PR branchs only 3 cm 1 Raman activ Th rotational Raman spctrum of th molcul shows a sris of anti-stoks lins sparatd by 0.63 cm 1. If th CN bond lngth is 0.1165 nm, dtrmin th C C bond lngth. Sction 4: Molcular lctronic spctroscopy 36. Th v = 0 0 vibrational band in th A 1 Π X 1 Σ + lctronic transition in BO shows a band had in th R-branch. a) Driv a gnral xprssion for th wavnumbr of a lin in th R-branch (J +1 J ) and show that th wavnumbr is a maximum whn B 3B 0 0 J " = ( B B 0 0) b) Th highst wavnumbr lins in th R- branch ar thos corrsponding to transitions out of J = 4 and J = 5 which li at ssntially th sam wavnumbr. Th microwav spctrum of th ground stat givs B = 1.64 cm -1. Obtain an approximat valu for B, th rotational constant in th A- 0 0 stat giving an stimat of uncrtainty in your answr. c) Calculat th bond lngths of BO in both lctronic stats and account qualitativly for any diffrnc btwn thm. (n.b. BO is isolctronic with C ) d) Which lctronic stat would you xpct to find lying lowr in nrgy than th A 1 Π stat? 37. For Br, th quilibrium intrnuclar distancs of th ground stat and of an xcitd lctronic stat ar R = 8 pm and R = 66 pm, rspctivly. Th vibrational wavnumbr, ω = 1 πc k, for th xcitd stat is 168 cm -1. μ Assuming that th uppr stat potntial nrgy curv may b approximatd as a simpl harmonic oscillator, us th classical vrsion of th Franck-Condon principl to prdict which is th most probabl (vrtical) chang in vibrational quantum numbr for a transition in absorption from v = 0 of th ground stat.
38. Th figur blow shows th v = 0 progrssion for a wakly allowd, 3 Π X 1 Σ +, lctronic transition in ICl. Th wavnumbrs markd corrspond to th band origin, ν 00 and th onst of a dissociativ continuum. a) What conclusions can on draw from th fact that so fw vibrational lvls ar supportd in th xcitd stat? b) What is th dissociation nrgy, D, of th xcitd stat? 0 Th ground stat of ICl dissociats to two ground-stat (i.., P 3/ ) atoms. Thr ar two possibilitis for th dissociation of th xcitd stat: ICl ( 3 Π) I ( P 1/ ) + Cl ( P 3/ ) Or ICl ( 3 Π) I ( P 3/ ) + Cl ( P 1/ ) c) Givn that th ground-stat spin-orbit splitting for I is 7603 cm -1 and for Cl is 881 cm -1, dduc possibl valus for th dissociation nrgy of th ground stat of ICl. d) Th dissociation nrgis of I and Cl ar 145 and 1997 cm -1, rspctivly. Us ths to dduc th likly dissociation products of ICl ( 3 Π). ) Sktch th potntial nrgy curvs for th two stats and mark on th quantitis you hav calculatd. 39. Th tabl blow givs data for th ground and a low-lying xcitd stat of CN T ω / cm -1 ω x / cm -1 R / nm X Σ + 0 068.6 13.1 0.11718 A Π 945.3 181.5 1.6 0.1333 a) Sktch an MO diagram for CN and show th occupancis for th X and A stats and us it to account for th obsrvd changs in vibrational constant and quilibrium bond-lngth. b) Us a Mors potntial modl to stimat th numbr of accssibl vibrational lvls and th dissociation nrgis for ach stat. c) At what wavnumbr is th band origin (th 0-0 transition) obsrvd? d) Combin your rsults for parts b) and c) to show that th potntial nrgy curvs must cross at larg sparation. ) Sktch th two potntial nrgy curvs and show mark all th quantitis you hav calculatd. f) Thr is a highr nrgy stat, B Σ + for which T = 5, 75.0 cm -1. Th dominant faturs of th B X lctronic band systm ar th 0 0 and 1 0 vibrational bands which hav rlativ intnsitis of 11 : 1. Dduc what you can about th potntial nrgy curv for th B stat and propos an lctronic configuration consistnt with your conclusion.
Sction 5: Raman Scattring 40. Th wavnumbr of th incidnt radiation in a Raman spctromtr is 0487 cm -1. What is th wavnumbr of th scattrd Stoks radiation for th J = 0 in 15 N (B = 1.9896 cm -1 )? 41. Th rotational Raman spctrum of H was rcordd at 350K. Th displacmnts of th first 5 lins of th Stoks branch from th xciting lin ar listd blow, togthr with thir rlativ intnsitis. Displacmnt / cm -1 Rlativ Intnsity 364.8 1.0 608.0 5.46 851. 1.1 1094.4 1.05 1337.6 0.06 a) Explain quantitativly th magnitud of both th displacmnts and intnsitis. b) Prdict what diffrncs in displacmnts and intnsitis you would xpct for D. 4. Explain th following obsrvations: Th ratio of th frquncis (masurd from th Rayligh lin) of th first and scond pur rotational Stoks Raman lins of 16 O is 5/9. This ratio is 3/7 for C 16 O and 3/5 for 17 O. [Th nuclar spin of 16 O is 0 and that of 17 O is 5/] 43. a) Th rotational Raman spctrum of 35 Cl shows a sris of lins sparatd by 0.975 cm 1 in both Stoks and anti-stoks branchs. Dtrmin th bond lngth of Cl. b) Th CO and CS bond lngths in CO and CS ar 0.116 nm and 0.1555 nm, rspctivly. Explain why th lin spacing in th rotational Raman spctrum of OCS is vry clos to that for CS.