Section 5 Exercises, Problems, and Solutions. Exercises:

Transcription

1 Scion 5 Exrciss, Problms, and Soluions Exrciss: 1. Tim dpndn prurbaion hory provids an xprssion for h radiaiv lifim of an xcid lcronic sa, givn by τ R : τ h- R 4 c 4(E i - E f ) µ fi, whr i rfrs o h xcid sa, f rfrs o h lowr sa, and µ fi is h ransiion dipol. a. Evalua h z-componn of h ransiion dipol for h p z 1s ransiion in a hydrognic aom of nuclar charg Z, givn: -Zr 5 -Zr Z ψ 1s 1π a a 1 Z, and ψpz 4 π a r Cosθ a. Exprss your answr in unis of a. b. Us symmry o dmonsra ha h x- and y-componns of µ fi ar zro, i.. <p z x 1s> <p z y 1s>. c. Calcula h radiaiv lifim τ R of a hydrognlik aom in is p z sa. Us h h- rlaion m a o simplify your rsuls.. Considr a cas in which h compl s of sas {φ k } for a Hamilonian is known. a. If h sysm is iniially in h sa m a im whn a consan prurbaion is suddnly urnd on, find h probabiliy ampliuds C k () () and C m () (), o scond ordr in, ha dscrib h sysm bing in a diffrn sa k or h sam sa m a im. b. If h prurbaion is urnd on adiabaically, wha ar C k () () and C m () ()? Hr, considr ha h iniial im is -, and h ponial is η, whr h posiiv paramr η is allowd o approach zro η in ordr o dscrib h adiabaically (i.., slowly) urnd on prurbaion. c. Compar h rsuls of pars a. and b. and xplain any diffrncs. d. Ignor firs ordr conribuions (assum hy vanish) and valua h ransiion ras d d C k () () for h rsuls of par b. by aking h limi η +, o obain h adiabaic rsuls.. If a sysm is iniially in a sa m, consrvaion of probabiliy rquirs ha h oal probabiliy of ransiions ou of sa m b obainabl from h dcras in h probabiliy of bing in sa m. Prov his o h lows ordr by using h rsuls of xrcis, i.. show ha: C m 1 - C k. Problms: k m

2 1. Considr an inracion or prurbaion which is carrid ou suddnly (insananously,.g., wihin an inrval of im which is small compard o h naural priod ω nm -1 corrsponding o h ransiion from sa m o sa n), and afr ha is urnd off adiabaically (i.., xrmly slowly as η ). Th ransiion probabiliy in his cas is givn as: T nm <n m> h- ω nm whr corrsponds o h maximum valu of h inracion whn i is urnd on. This formula allows on o calcula h ransiion probabiliis undr h acion of suddn prurbaions which ar small in absolu valu whnvr prurbaion hory is applicabl. L's us his "suddn approximaion" o calcula h probabiliy of xciaion of an lcron undr a suddn chang of h charg of h nuclus. Considr h racion: 1 H H + + -, and assum h riium aom has is lcron iniially in a 1s orbial. a. Calcula h ransiion probabiliy for h ransiion 1s s for his racion using h abov formula for h ransiion probabiliy. b. Suppos ha a im h sysm is in a sa which corrsponds o h wavfuncion ϕ m, which is an ignfuncion of h opraor H. A, h suddn chang of h Hamilonian occurs (now dnod as H and rmains unchangd). Calcula h sam 1s s ransiion probabiliy as in par a., only his im as h squar of h magniud of h cofficin, A 1s,s using h xpansion: Ψ(r,) ϕ m (r) A nm ψ n (r), whr A nm ϕ m (r)ψ n (r)d r n No, ha h ignfuncions of H ar ψ n wih ignvalus E n. Compar his "xac" valu wih ha obaind by prurbaion hory in par a.. Th mhyl iodid molcul is sudid using microwav (pur roaional) spcroscopy. Th following ingral govrns h roaional slcion ruls for ransiions labld J, M, K J', M', K': J' I <D M'K' ε. µ J DMK >. Th dipol momn µ lis along h molcul's C symmry axis. L h lcric fild of h ligh ε dfin h lab-fixd Z-dircion. 1* a. Using h fac ha Cosβ D, show ha I 8π µε(-1) (M+K) J' 1 J M M J' 1 J K K δ M'M δ K'K b. Wha rsricions dos his rsul plac on J J' - J? Explain physically why h K quanum numbr can no chang.. Considr h molcul BO. a. Wha ar h oal numbr of possibl lcronic sas which can b formd by combinaion of ground sa B and O aoms?

3 b. Wha lcron configuraions of h molcul ar likly o b low in nrgy? Considr all rasonabl ordrings of h molcular orbials. Wha ar h sas corrsponding o hs configuraions? c. Wha ar h bond ordrs in ach of hs sas? d. Th ru ground sa of BO is Σ. Spcify h +/- and u/g symmris for his sa.. Which of h xcid sas you drivd abov will radia o h Σ ground sa? Considr lcric dipol, magnic dipol, and lcric quadrupol radiaion. f. Dos ionizaion of h molcul o form a caion lad o a srongr, wakr, or quivaln bond srngh? g. Assuming ha h nrgis of h molcular orbials do no chang upon ionizaion, wha ar h ground sa, h firs xcid sa, and h scond xcid sa of h posiiv ion? h. Considring only hs sas, prdic h srucur of h phoolcron spcrum you would obain for ionizaion of BO cm -1 ν (HCN) ν (HCN) 17 cm -1 ν (HCN) 6 cm -1 8 cm cm 15 cm -1 cm cm Th abov figur shows par of h infrard absorpion spcrum of HCN gas. Th molcul has a CH srching vibraion, a bnding vibraion, and a CN srching vibraion. a. Ar any of h vibraions of linar HCN dgnra? b. To which vibraion dos h group of paks bwn 6 cm -1 and 8 cm -1 blong? c. To which vibraion dos h group of paks bwn cm -1 and 4 cm -1 blong? d. Wha ar h symmris (σ, π, δ) of h CH srch, CN srch, and bnding vibraional moions?. Saring wih HCN in is,, vibraional lvl, which fundamnal ransiions would b infrard aciv undr paralll polarizd ligh (i.., z-axis polarizaion):

4 i. 1? ii. 1? iii. 1? f. Which ransiions would b aciv whn prpndicular polarizd ligh is usd? g. Why dos h 71 cm -1 ransiion hav a Q-branch, whras ha nar 17 cm -1 has only P- and R-branchs? Exrciss: 1. a. Evalua h z-componn of µ fi : Cosθ -Zr a. Soluions µ fi <p z r Cosθ 1s>, whr ψ 1s 5 1 Z µ fi 4 π a Z 1π a -Zr a, and ψpz -Zr -Zr Z 1π a <r Cosθ a r Cosθ a > -Zr -Zr 1 4π Z a 4 <r Cosθ a r Cosθ a > π π 4π Z a 4 r dr Sinθdθ dϕ -Zr -Zr r a a Cos θ 4π π Z a 4 -Zr π r 4 a dr SinθCos θdθ Using ingral quaion 4 o ingra ovr r and quaion 17 o ingra ovr θ w obain: 4π π Z a 4 π 4! -1 Z 5 Cos θ a 4π π Z a 4 5 a 5 4! 5 Z 5-1 ( (-1) - (1) ) 8 a 5 Z a Z a Z 5 1 Z 4 π a r

5 b. Examin h symmry of h ingrands for <p z x 1s> and <p z y 1s>. Considr rflcion in h xy plan: Funcion Symmry p z -1 x +1 1s +1 y +1 Undr his opraion h ingrand of <p z x 1s> is (-1)(1)(1) -1 (i is anisymmric) and hnc <p z x 1s>. Similarly, undr his opraion h ingrand of <p z y 1s> is (-1)(1)(1) -1 (i is also anisymmric) and hnc <p z y 1s>. c. τ h- R 4 c 4(E i - E f ) µ fi, E i E pz Z a E f E 1s -Z a E i - E f 8 a Z Making h subsiuions for E i - E f and µ fi in h xprssion for τ R w obain: τ h- R 4 c 4 8 a Z a 8, Z 5 h - 4 c 8 a 8 Z 4 8, h- Insring m a w obain: h- 4 c a Z6 a Z 16 () 1 τ R h - 4 c 8 a m 4 a 4 8 c a 5 m 4 h-8 Z h-4 Z c a 5 m 4 h-4 Z 4,

6 5,689 1 Z 4 x (.998x1 1 cm sc -1 ) (.59177x1-8 cm) 5 (9.19x1-8 g) 4 (1.546x1-7 g cm sc -1 ) x1-9 sc x 1 Z 4 So, for xampl: Aom H H + Li + B + N +9 τ R ns 99.7 ps 19.7 ps 6. ps 159 fs. a. H H + λh ' (), H'() θ(), H ϕ k E k ϕ k, ω k E k /h- ih- ψ Hψ l ψ(r,) ih- c ()ϕ -iω and insr ino h abov xprssion: ih- c - iω c -iω ϕ ih- c () -iω (H + λh ' ()) ϕ So, ih- c + E c - c E - c λh' -iω ϕ ih- c <m > - c λ<m H' > -iω ih- c m -iω m c λh' m -iω 1 ih- c λh' m -i(ω m) Going back a fw quaions and muliplying from h lf by ϕ k insad of ϕ m w obain: ih- c <k > - c λ<k H' > -iω ih- c k -iω k c λh' k -iω So,

7 c k 1 ih- c λh' k -i(ω k) Now, l: c m c m () + c m (1) λ + c m () λ +... c k c k () + c k (1) λ + c k () λ +... and subsiuing ino abov w obain: () + c m (1) λ + c m () λ ih- [c () + c (1) λ + c () λ +...] firs ordr: () c m () 1 scond ordr: (1) 1 ih- c () H' m -i(ω m) λh' m -i(ω m) (n+1) s ordr: (n) 1 ih Similarly: firs ordr: c k () c k m () scond ordr: c k (1) 1 ih- c (n-1) H' m -i(ω m) c () H' k -i(ω k) (n+1) s ordr: c k (n) 1 c (n-1) H' k ih- -i(ω k) So, (1) 1 ih- c m () H' mm -i(ω mm) 1 ih- H' mm c m (1) () 1 ih- d' mm mm ihand similarly, c k (1) 1 ih- c m () H' km -i(ω mk) 1 ih- H' km -i(ω mk) c k (1) () 1 ih- km d' -i(ω mk)' km [ ] h- -i(ω mk) - 1 ωmk

8 () 1 ih- c (1) H' m -i(ω m) () 1 m [ ih- h- -i(ω m) - 1 ] H'm -i(ω m) 1 + mm H' mm ωm ih- ih c m () 1 m m ih- h- d' -i(ω m)' [ -i(ω m)' - 1 ] - mm mm 'd' ωm h- m m ih d'[ 1 - -i(ω m)' ] - mm ω m h- m m ih ω m - -i(ω m) - 1 -iω - mm m h- ' m m h- ( -i(ω m) - 1 ) + ' m m - mm ω m ih ω m h- Similarly, c k () 1 c (1) H' k ih- -i(ω k) 1 m [ ] ih- h- -i(ω m) - 1 H'k -i(ω k) + ωm c k () () ' m k ih ω m 1 mm ih- ih- d' -i(ω k)' [ -i(ω m)' - 1] - mm km h- H' km -i(ω mk) 'd' -i(ω mk)' ' m k ih ω m -i(ω m+ωm) i(ωk) - 1 -iω mk -iω k - mm km h- -i(ω mk)' ' 1 - -iω mk -(iω mk ) ' m k h- ω m -i(ω mk) i(ωk) - 1 ω mk ω k

9 + mm km h- ω mk -i(ω mk)' ' i - 1 ω mk ' m k -i(ω mk) - 1 E m - E E m - E - -i(ω k) - 1 k E - E k + mm km h- (Em - E k ) -i(ω mk) i ω mk ω mk So, h ovrall ampliuds c m, and c k, o scond ordr ar: c m () 1 + mm + ' m m ih- ih- (Em - E ) + ' m m h- ( -i(ω m) ) mm (E m - E ) h- c k () km -i(ω mk) (E m - E k ) [ ] mm km (E m - E k ) [ 1 - -i(ω mk) ] + mm km (E m - E k ) h- -i(ω mk) + i ' m k -i(ω mk) - 1 E m - E E m - E - -i(ω k) - 1 k E - E k b. Th prurbaion quaions sill hold: (n) 1 c (n-1) H' m ih- -i(ω m) ; c k (n) 1 ih So, c m () 1 and c k () (1) 1 ih- H' mm c m (1) 1 ih- mm d' η mm η - ih- η c k (1) 1 ih- H' km -i(ω mk) c (n-1) H' k -i(ω k) c k (1) 1 ih- km d' -i(ω mk+η)' km [ - ih- -i(ω mk+η)] (-iωmk +η) km [ ] E m - E k + ih- -i(ω mk+η) η () ' 1 m ih- E m - E + ih- -i(ω m+η) m η -i(ω m) + η

10 1 mm η mm ih- ih- η η c m () ' 1 m m ih- E m - E + ih- η' d' - mm η - h- η' d' η - ' m m ih- η(em - E + ih- η - mm η) h- η η c k () ' 1 m ih- E m - E + ih- -i(ω m+η) H'k -i(ω k) + η 1 mm η H' km ih- ih- -i(ω mk) η c k () ' 1 m k ih- E m - E + ih- -i(ω mk+η)' d' - η - mm km h- η -i(ω mk+η)' d' - ' m k -i(ω mk+η) - mm km -i(ω mk+η) (E m - E + ih- η)(em - E k + ih- η) ih- η(em - E k + ih- η) Thrfor, o scond ordr: c m () 1 + mm η + m m ih- η ih- η(em - E + ih- η η) c k () km [ ih- -i(ω mk+η)] (-iωmk +η) + m k -i(ω mk+η) (E m - E + ih- η)(em - E k + ih- η) c. In par a. h c () () grow linarly wih im (for mm ) whil in par b. hy rmain fini for η >. Th rsul in par a. is du o h suddn urning on of h fild. ' d. c k () m k -i(ω mk+η) (E m - E + ih- η)(em - E k + ih- η) k k' m 'm -i(ω mk+η) i(ωmk+η) (E m -E +ih- η)(em -E ' -ih- η)(em -E k +ih- η)(em -E k -ih- η)

11 k k' m 'm 4η ' [(E m -E )(E m -E ' )+ih- η(e -E ' )+h- η ]((E m -E k ) +4h- η ) d d c k() 4η k k' m 'm ' [(E m -E )(E m -E ' )+ih- η(e -E ' )+h- η ]((E m -E k ) +4h- η ) Now, look a h limi as η + : d d c k() whn E m E k lim 4η η + ((E m -E k ) +4h- η ) α δ(e m-e k ) So, h final rsul is h nd ordr goldn rul xprssion: d d c k() π h- δ(e m-e k ) lim η + m k (E - E m - ih- η). For h suddn prurbaion cas: c m () 1 + ' c m () 1 + ' c k () 1 - k m m m (E m - E ) [ ] -i(ω m) i(ωm) O( ) m m (E m - E ) [ ] -i(ω m) + i(ωm) - + O( ) km mk (E m - E k ) [ - -i(ω mk) - i(ωmk) + ] + O( ) ' c k () 1 - ' km mk (E m - E k ) [ ] k - -i(ω mk) - i(ωmk) + + O( ) 1 + ' km mk (E k m - E k ) [ -i(ω mk) + i(ωmk) - ] + O( ) o ordr, c m () 1 - ' c k (), wih no assumpions mad rgarding mm. k For h adiabaic prurbaion cas: c m () 1 + ' m m η + m m η + O( ih- η(em - E + ih- η) -ih- η(em - E - ih- ) η) ' ih- η (E m m m -E +ih- η) (E m -E -ih- η + O( ) η) 1 -ih- η 1 + ' ih- η (E m -E ) +h- η m m η + O( )

12 1 - ' m m η + O( (E m -E ) +h- ) η c k () km mk (E m -E k ) +h- η η + O( ) o ordr, c m () 1 - ' c k (), wih no assumpions mad rgarding mm for his k cas as wll. Problms: 1. a. T nm <n m> h- ω nm valuaing <1s s> (using only h radial porions of h 1s and s wavfuncions sinc h sphrical harmonics will ingra o uniy) whr (,r): -Zr <1s s> Z a a 1 r 1 Z a 1 - Zr -Zr a a r dr <1s s> Z a -Zr -Zr r a dr - Zr a a dr Using ingral quaion 4 for h wo ingraions w obain: <1s s> Z a 1 Z - Z a Z a a <1s s> Z a a Z - a Z <1s s> Z a () a - a 8Z Z 7a Now, E n - Z n, E 1s - Z a a, E s - Z 8a, E s - E 1s Z 8a So, 8Z T nm 7a Z 6 Z 6 a () 8 a Z (for Z 1) Z 8a -Zr b. ϕ m (r) ϕ 1s Z a a Y Th orhogonaliy of h sphrical harmonics rsuls in only s-sas having non-zro valus for A nm. W can hn drop h Y (ingraing his rm will only rsul in uniy) in drmining h valu of A 1s,s.

13 ψ n (r) ψ s 1 Z a 1 - Zr -Zr a a Rmmbr for ϕ 1s Z 1 and for ψ s Z -Zr A nm Z a a 1 Z+1 a 1 - (Z+1)r -(Z+1)r a a r dr A nm -(Z+1)r Z a Z+1 a a 1 - (Z+1)r a r dr A nm Z a -(Z+1)r Z+1 a r a (Z+1)r -(Z+1)r dr - a a dr Evaluaing hs ingrals using ingral quaion 4 w obain: A nm Z a Z+1 a Z+1 - Z+1 ()() a Z+1 4 a a A nm Z a Z+1 a 4 a (Z+1) - (Z+1) ()4 a (Z+1) 4 A nm Z a Z+1 a -5 a (Z+1) 4 A nm - [ Z(Z+1) ] (Z+1) 4 Th ransiion probabiliy is h squar of his ampliud: T nm - [ Z(Z+1) ] (Z+1) 4 11 Z (Z+1) (Z+1) 8.5 (for Z 1). Th diffrnc in hs wo rsuls (pars a. and b.) will bcom ngligibl a larg valus of Z whn h prurbaion bcoms lss significan as in h cas of Z 1.. ε is along Z (lab fixd), and µ is along z (h C-I molcul fixd bond). Th angl bwn Z and z is β: So, J' I <D M'K' ε. µ εµcosβ εµd 1* (αβγ) ε. µ J DMK J' > DM'K'. µ J DMK ε Sinβdβdγdα εµ J' 1* J D M'K' D DMK Sinβdβdγdα. Now us:

14 o obain: Now us: J'* D M'n' D 1* <J'M'1 m> * * D mn <n J'K'1> *, mn I εµ <J'M'1 m> * <n J'K'1> * * J D mn D MK Sinβdβdγdα. mn D * J mn D MK Sinβdβdγdα 8π J+1 δ Jδ Mm δ Κn, o obain: I εµ 8π J+1 <J'M'1 m> * <n J'K'1> * δ J δ Mm δ Κn mn εµ 8π J+1 <J'M'1 JM><JK J'K'1>. W us: <JK J'K'1> J+1(-i) (J'-1+K) J' 1 J K' K and, <J'M'1 JM> J+1(-i) (J'-1+M) J' 1 J M' M o giv: I εµ 8π J+1 J+1(-i)(J'-1+M) J' 1 J M' M J+1(-i) (J'-1+K) J' 1 J K' K εµ8π (-i) (J'-1+M+J'-1+K) J' 1 J M' M J' 1 J K' K εµ8π (-i) (M+K) J' 1 J M' M J' 1 J K' K Th -J symbols vanish unlss: K' + K and M' + M. So, I εµ8π (-i) (M+K) J' 1 J M M J' 1 J K K δ M'M δ K'K. b. J' 1 J and M M J' 1 J vanish unlss J' J + 1, J, J - 1 K K J ±1, Th K quanum numbr can no chang bcaus h dipol momn lis along h molcul's C axis and h ligh's lcric fild hus can xr no orqu ha wiss h molcul abou his axis. As a rsul, h ligh can no induc ransiions ha xci h molcul's spinning moion abou his axis.. a. B aom: 1s s p 1, P ground sa L 1, S 1, givs a dgnracy ((L+1)(S+1)) of 6. O aom: 1s s p 4, P ground sa L 1, S 1, givs a dgnracy ((L+1)(S+1)) of 9. Th oal numbr of sas formd is hn (6)(9) 54. b. W nd only considr h p orbials o find h low lying molcular sas:

15 6σ p Which, in raliy look lik his: 6σ π 1π 5σ p π 5σ 1π This is h corrc ordring o giv a Σ + ground sa. Th only low-lying lcron configuraions ar 1π 5σ or 1π 4 5σ 1. Ths lad o Π and Σ + sas, rspcivly. c. Th bond ordrs in boh sas ar 1. d. Th Σ is + and g/u canno b spcifid sinc his is a hronuclar molcul.. Only on xcid sa, h Π, is spin-allowd o radia o h Σ +. Considr symmris of ransiion momn opraors ha aris in h E1, E and M1 conribuions o h ransiion ra Elcric dipol allowd: z Σ +, x,y Π, h Π Σ + is lcric dipol allowd via a prpndicular band. Magnic dipol allowd: R z Σ -, R x,y Π, h Π Σ + is magnic dipol allowd. Elcric quadrupol allowd: x +y, z Σ +, xy,yz Π, x -y, xy h Π Σ + is lcric quadrupol allowd as wll. f. Sinc ionizaion will rmov a bonding lcron, h BO + bond is wakr han h BO bond. g. Th ground sa BO + is 1 Σ + corrsponding o a 1π 4 lcron configuraion. An lcron configuraion of 1π 5σ 1 lads o a Π and a 1 Π sa. Th Π will b lowr in nrgy. A 1π 5σ configuraion will lad o highr lying sas of Σ -, 1, and 1 Σ +. h. Thr should b bands corrsponding o formaion of BO + in h 1 Σ +, Π, and 1 Π sas. Sinc ach of hs involvs rmoving a bonding lcron, h Franck-Condn ingrals will b apprciabl for svral vibraional lvls, and hus a vibraional progrssion should b obsrvd.

16 4. a. Th bnding (π) vibraion is dgnra. b. H---C N bnding fundamnal c. H---C N srching fundamnal d. CH srch (ν in figur) is σ, CN srch is σ, and HCN (ν in figur) bnd is π.. Undr z (σ) ligh h CN srch and h CH srch can b xcid, sinc ψ σ, ψ 1 σ and z σ provids coupling. f. Undr x,y (π) ligh h HCN bnd can b xcid, sinc ψ σ, ψ 1 π and x,y π provids coupling. g. Th bnding vibraion is aciv undr (x,y) prpndicular polarizd ligh. J, ±1 ar h slcion ruls for ransiions. Th CH srching vibraion is aciv undr (z) polarizd ligh. J ±1 ar h slcion ruls for ransiions.