5.80 Small-Molecule Spectroscopy and Dynamics

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1 MIT OpeCurseWare Small-Mlecule Spectrscpy ad Dyamics Fall 8 Fr ifrmati abut citig these materials r ur Terms f Use, visit:

2 5.8 Lecture #33 Fall, 8 Page f pages Lecture #33: Vibric Cuplig Last time: H CO A A X A Electrically frbidde if A -state is plaar vibrically allwed t alterate v vibratial levels if A -state is plaar iertial defect says A -state is t plaar expect t see all v if t plaar staggerig f v level spacigs iversi thrugh lw barrier t plaarity dyamic vs. rigid mlecule symmetry classificati: mlecular symmetry grup Hw des vibric cuplig really wrk? What are the vibratial itesity factrs aalgus t Frack-Cd factrs i the case f vibrically allwed rather tha electrically allwed trasiti? See T. Azumi ad K. Matsuzaki, Phtchemistry ad Phtbilgy 5, 35 (977) fr a extremely readable review article. Outlie: Crude Adiabatic Apprximati Crrecti f ψ fr effect f eglected ff-diagal matrix elemets H CO A A example What happes t Frack-Cd factrs fr a vibrically allwed trasiti? Tw electric basis sets prediagalize symmetry-breakig vibric iteracti Chages i shapes f ptetial curves (deperturb t a simpler, mre atural shape) K. K. Ies mdel fr vibratial bad itesities ad level staggerig Recall Br-Oppeheimer r clamped uclei apprximati. We use this prcedure t defie cmplete sets f electric ad uclear mti wavefuctis with which we ca FORMALLY expad exact ψ s ad cmpute (r parametrize) all prperties f exact eigestates. The simplest basis set is called CRUDE ADIABATIC () ψ jt (r,q) = ψ j (r,q ) χ jt ( Q) electric state vibratial state fixed uclear lcatis! Q is a cveiet referece structure (usually the equilibrium gemetry r a high-symmetry ptetial eergy maximum r saddle pit). ψ j is the electric wavefucti i the j-th electric state cmputed at the chse ad explicitly specified set f fixed uclear crdiates Q.

3 5.8 Lecture #33 Fall, 8 Page f pages (Q) is the vibrati-rtati wavefucti cmputed frm a apprximate uclear Schrödiger χ jt Equati. eigevalue f clamped uclei uclear ptetial electric U(r,Q) = U(r,Q) U(r,Q ) kietic eergy f Schrödiger chage i e uclear ad e e eergy bare uclei Equati at Q Culmb eergy T N(Q) + V(Q) + ε j (Q ) + ψ j (r,q ) U(r,Q) ψ j (r,q ) χ jt ( ) Q = E jt χ jt ( Q) effective ptetialeergy surface Nte that the U itegral is evaluated usig ψ j (r, Q ) thus cat ctai the exact effect f distrti f mlecule frm Q. T get a better represetati f the distrti frm Q, we must use perturbati thery. We have explicitly excluded the effects f ff-diagal matrix elemets. I rder t get a better apprximati t the exact ψ, we must use perturbati thery t crrect ψ jt. ψ jt (r,q) = ψ jt (r,q) + kr jt {ψ U jt E jt E kr ψ kr } ψ kr (r,q) ψ k U ψ j = ψ j (r,q )χ jt (Q)+ E kr k j r (χ kr E jt ψ k (r,q )χ kr (Q) call this a vibric mixig cefficiet { } meas itegrate ver r ad Q (bth electric ad uclear) ( ) meas itegrate ver Q (uclear) meas itegrate ver r (electric) χ ) jt This frm f ψ jt (r,q) is called the Herzberg-Teller expasi. Nw expad U(r,Q) i pwer series abut Q i each f the rmal crdiates.

4 5.8 Lecture #33 Fall, 8 Page 3 f pages U(r,Q) U = U(r,Q ) + Q + U = by defiiti Q ges it,m Q Q m Q Q m etc. uclear f U electric factr matrix elemet t be iitially eglected Nw defie the mixig cefficiet. γ kr, jt U Q Q ψ k (r,q ) ψ j (r,q ) χ kt χ jt E jt everythig cllected it sigle parameter E kt ψ jt (r,q) = ψ (r,q )χ j jt (Q) + But we ca see that γ kr, jt must vaish if γ kr, jt ψ (r,q )χ k kr k j r te vibratial Electric Prmtig wavefucti fr states vibratial mde k-th, states (Q) NOT j-th electric state! Γ k Γ j Γ Q OR Γ r Γ t Γ Q (required by defiiti f U abve) electric selecti vibratial selecti rule rule which is equivalet t requirig that Γ kr Γ jt Γ ttally symmetric (ad Q is t ttally symmetric). S w we are ready t csider the specific case f the H CO A A state. Out-f-plae Bedig mde as prmter b A = B b vibrati vibric symmetry S we are csiderig vibric cuplig t the B state. N-Lecture This is a simplified versi f Ies mdel, t be discussed later. Let s make a really crude mdel fr the ut-f-plae bedig levels f bth A ad B states. This is a example whe ature is t careless. Deperturb back t a simpler picture. * bth are harmic (NB assume that the A state is NOT a duble miimum -plaar state!!)

5 5.8 Lecture #33 Fall, 8 Page f pages * bth have same frequecy ω * cuplig is exclusively via U Q term. Q 3a a b 5a b O atm π i-plae σ π π σ b b 6a X A ev elect. frbidde b b A A π* 3.5 ev elect. allwed (b-type) 6a b B B σ* 7. ev elect. allwed (a-type) b b A π* π 8. ev elect. allwed (c-type) b 5a B π* σ 9.5 ev A X trasiti ca brrw scillatr stregth by vibric cuplig with B via b vibrati because A b = B A via a vibrati because A a = A (a vibrati des t exist) B via b vibrati because A b = B I will w shw, via a simple mdel, that vibric cuplig accuts fr bth the scillatr stregth f the vibratial bads i the A X trasiti ad the staggerig f ν vibratial levels i A -state. Assume ν i A ad B states is harmic - t a duble miimum -plaar state cveiet fr same ω ad t displaced (ecessarily t displaced if miimum r maximum is calculatig vibratial matrix plaar the high-symmetry pit) elemets U cuplig is exclusively via Q Q term

6 5.8 Lecture #33 Fall, 8 Page 5 f pages B v = A 3 v = ψ A v = ψ A χ A v + Similarly fr ψ B v v γ B v,a v ψ B actually Q is mass weighted mde #, t th pwer γ B v,a v ψ B χ B v (perturbati thery) U v Q v ψ A Q E A v E B v a mass-idepedet electric factr β B A (µω ) / v + δ v,v + + v δ v B A T ω (v v ) (T A +v ω ) (T B +v ω ) retaiig ly levels f B state i the Herzberg-Teller expasi,v cmes frm mdelig ν as the same i bth B ad A states.

7 5.8 Lecture #33 Fall, 8 Page 6 f pages Summary f -zer matrix elemets 3 v = 3 3 matrix elemet S we have ψ A v ψ B v = ψ B = ψ A χ A v + βψ B χ B v 3 v = lump everythig it this adjustable cstat + v + χ B v + v χ B v βψ A v χ A v same cstat but ppsite sig because f the eergy demiatr. + v + χ A v +

8 5.8 Lecture #33 Fall, 8 Page 7 f pages Trasiti prbability fr A v X v ( B ad A states have same ptetial surface but X is differet). I A v,x v = ψ A v µ ψ X v Oly the B state is assumed t ctribute I A v,x v = ψ µ / + (v + ) / b B ψ χ X Bv χ Xv χ Bv + χ β v Xv B X v q v,v = β M b,b X F-C factr ( + v + )q B X / v +,v + v ( v + ) B v X v X v B v + psitive squared terms either sig crss terms Nte that this is mre cmplicated tha usual FRANCK-CONDON expressi fr allwed trasitis. It is expressed i terms f Frack-Cd factrs fr B X NOT A X!!!! We still have a symmetry selecti rule fr the ν vibratial mde because it is -ttally symmetric. Frm v = we ca ly reach B v = eve (because ν is t ttally symmetric) which requires A v = dd. Nte that the itesity expressi abve vaishes fr v = ad v = because q, B X (by symmetry). This ca be expressed mre geerally, fr ay vibratial bad i the A X system that is made allwed by vibric cuplig t the B B state prmted by ν. idividual mde F-C factrs (prduct ver all mdes except the prmtig mde) I A V,X V = β M b,b X q B v i v i X v A Q v B v B v X v i v v B symmetry selecti B v v X = eve b-type rule A v v B = dd v A v X = dd K. K. Ies J. Ml. Spectrsc. 99, 9 (983) perfrmed a vibric cuplig calculati which t ly reprduced the mde- itesity prmti factrs, but als explaied the level staggerig i the A -state.

9 5.8 Lecture #33 Fall, 8 Page 8 f pages I rder t defie cmplete basis sets, we slve a apprximate Schrödiger equati by eglectig specified terms i the exact H, r by igrig ff-diagal elemets f these terms. I the crude adiabatic apprximati, we defie ptetial curves by igrig terms f the frm ψ jt H(r,Q) ψ kr. We shwed that, by expadig U as pwer series i Q (the rmal mde displacemets), we get (H electric ) jk = U ψ j (r,q ) Q ψ k (r,q ) Q = γ jk Q. We ca g t a ew electric basis set by diagalizig H + γ jk Q. Suppse we have tw harmic zer-rder ptetial curves, V (Q ), fr mde f electric states j ad k. The we have the fllwig zer-rder ad diagalized ptetial curves. V k (Q ) V k (Q ) H = γ jk Q V j (Q ) Q = V j (Q )

10 5.8 Lecture #33 Fall, 8 Page 9 f pages Upper curve gets arrwer. Lwer curve turs it a duble miimum curve. Q = pits f bth curves d t shift because H at Q = Q by defiiti. Vibratial eigestates f lwer curve will exhibit the patter f a symmetric duble miimum ptetial. V k (Q ) = ω k Q V j (Q ) = ω Q j H ij (Q ) = γ jk Q here we are allwig harmic frequecies t be differet Secd-rder perturbati thery: V k = ω k Q + ( ω k ω ) j γ) jk Q ( Q = ω k Q + αq + T ek T ej (We d pwer series expasi f the secd term abut Q =. There ca be cstat term because V k vaishes at Q =. Nte that the cefficiet f Q chages because it is ω k plus a Q term frm the pwer series.) V j = ω j Q αq (same α because it is the same expasi but with ppsite sig eergy demiatr) ( γ jk ) ω k = ω k + frm pwer series expasi T T ek ( (ω ω ) = ω ω k k j j ej ) = T ( γ ek jk ) T ej get ppsite sig shifts i the effective harmic frequecy (γ jk ) α (ω k ω j ) get a quartic term that depeds differece i ω's fr j ad k. (T ek T ej ) The quartic term vaishes if ω j = ω k. This shws that upper state ω icreases ad lwer state ω decreases. Exact deperturbati treatmet fr the ptetial curves V E H / V ± = V k + V j ± V + H V H E V = (ω ω k j )Q + T T ek ej (icludes differece betwee miima f V k ad V j ) V k + V j ω k + ω j (ω k ω j ) / V ± = + Q ± Q + (γ jk ) Q

11 5.8 Lecture #33 Fall, 8 Page f pages (the T e / term seems t have bee mitted frm the [ ] / term) Fr large γ, secd term i [ ] / will dmiate at small Q but first term will evetually dmiate at large Q. We btai tw perturbed ptetial eergy curves. Nw g back t the rigial vibric Hamiltia ad get a degeerate perturbati expressi fr the eergy levels. A secd-rder perturbati treatmet f this kid f -state iteracti i the picture cat give this type f level stagger. It is ecessary t set up ad diagalize tw large dimesi matrices H I dd quata f upper state eve quata f lwer state H II eve quata f upper state dd quata f lwer state because f dd-eve symmetry f a symmetric (t ecessarily harmic) ptetial, there ca be cuplig matrix elemets betwee these tw matrices. The level shifts are larger fr the lwer states i H II tha thse i H I. Fr example, the lwer state v = level is pushed dw by v = ad f the upper state, but the lwer state v = level is ly pushed dw by v =. This prduces level staggerig. K. K. Ies [J. Ml. Spectrsc. 99, 9-3 (983)] reprduced A X itesity ad A -state level patter with T B A = 835 cm ω B A = ω = 5 cm H A va,b v B = v A + = β (v A + ) / / H A va,b v B = v A = βv A β = 338 cm