6.4 ELECTRONIC SPECTROSCOPY: DISPLACED HARMONIC OSCILLATOR MODEL 1

Transcription

1 Andri Tokmakoff, MIT Dpartmnt of Chmistry, 3/1/ ELECTRONIC SPECTROSCOPY: DISPLACED HARMONIC OSCILLATOR MODEL 1 Hr w will start with on approach to a class of widly usd modls for th couplin of nuclar motions to an lctronic transition that taks many forms and has many applications. W will look at th spcific xampl of lctronic absorption xprimnts, which lads to insiht into th vibronic structur in absorption spctra. Spctroscopically, it is also usd to dscrib wavpackt dynamics; couplin of lctronic and vibrational stats to intramolcular vibrations or solvnt; or couplin of lctronic stats in solids or smiconductors to phonons. Furthr xtnsions of this modl can b usd to dscrib fundamntal chmical rat procsss, intractions of a molcul with a dissipativ or fluctuatin nvironmnt, and Marcus Thory for non-adiabatic lctron transfr. Two-lctronic stats as displacd harmonic oscillators W ar intrstd in dscribin th lctronic absorption spctrum for th cas that th lctronic nry dpnds on nuclar confiuration. Th simplifid modl for this is two idntical harmonic oscillators potntials displacd from on anothr alon a nuclar coordinat, and whos - nry splittin is E E. W will calculat th lctronic absorption spctrum in th intraction pictur ( H H V () t = + ) usin th tim-corrlation function for th dipol oprator. Th Hamiltonian for th mattr rprsnts two Born- Oppnhimr surfacs H = G HG G + E HE E (6.1) whr th Hamiltonian dscribin th round and xcitd stats hav contributions from th nuclar nry and th lctronic nry HG = E + H. (6.) H = E + H E

2 Andri Tokmakoff, MIT Dpartmnt of Chmistry, 3/1/ Th harmonic vibrational Hamiltonian has th sam curvatur in th round and xcitd stats, but th xcitd stat is displacd by d rlativ to th round stat. p 1 H = + mω q m (6.3) p 1 H = + mω ( q d) m (6.4) Now w ar in a position to valuat th dipol corrlation function iht/ h iht/ h () n n= E, G Cμμ t = p n μ μ n. (6.5) with th tim propaator ih ( + E ) t/ h ih ( + E) t/ h = G G + E E iht/ h (6.6) W bin by makin two approximations: 1) Born-Oppnhimr Approximation. Althouh this is implid in q. (6.) whn w writ th lctronic nry as indpndnt of q, spcifically it mans that w can writ th stat of th systm as a product stat in th lctronic and nuclar confiuration: G =, n (6.7) ) Condon Approximation. This approximation stats that thr is no nuclar dpndnc for th dipol oprator. It is only an oprator in th lctronic stats. μ = μ + μ (6.8) Undr all rasonabl conditions, th systm will only b on th round lctronic stat at quilibrium, n,, and with th xprssion for th dipol oprator (6.8), w find: ( ) C () t = μ μμ ie E t/ h iht/ h iht/ h (6.9) Hr th oscillations at th lctronic nry ap ar sparatd from th nuclar dynamics in th final factor, somtims known as th dphasin function: () F t = = ih t/ h iht/ h U U (6.1)

3 Andri Tokmakoff, MIT Dpartmnt of Chmistry, 3/1/ Not that physically th dphasin function dscribs th tim-dpndnt ovrlap of th initial nuclar wavfunction on th round stat with th tim-volution of th sam wavpackt on th whn initially projctd onto th xcitd stat ( ) ϕ ( ) ϕ ( ) F t = t t. (6.11) This is a prfctly nral xprssion that dos not dpnd on th particular form of th potntial. If you hav knowld of th nuclar and lctronic instats or th nuclar dynamics on your round and xcitd stat surfacs, this xprssion is your rout to th absorption spctrum. To valuat F(t), it hlps to raliz that w can writ th nuclar Hamiltonians as ( a a 1 ) H = h ω + (6.1) ˆ H ˆ = D HD. (6.13) Hr D is th spatial displacmnt oprator D ˆ = xp( ipd h ) (6.14) which shifts an oprator in spac: ˆ DqDˆ = q+ d. (6.15) This allows us to xprss th xcitd stat Hamiltonian in trms of a shiftd round stat Hamiltonian in q. (6.13), but also allows us to rlat th tim-propaators on th round and xcitd stats ˆ ih / ih t/ t h = D h Substitutin q. (6.16) into q. (6.1) allows us to writ ( ) F t = U U = idp/ h idp/ h idp t ()/ h idp( ) sinc ( ) ( ) Dˆ. (6.16) / h (6.17) p t = U p U. (6.18) Up to now, vrythin w v writtn is nral to any form of th potntial, but hr w will continu by valuatin th rsults for th spcific cas of a harmonic nuclar potntial. Th tim-volution of p is obtaind by valuatin q. (6.18) by applyin q. (6.1) to Rmmbrin aa= n, w find i p= ( a a) mh ω. (6.19)

4 Andri Tokmakoff, MIT Dpartmnt of Chmistry, 3/1/ inω t inω t in ( 1) ωt inω t iω t U au = a = a = a U a U = a + iωt (6.) which ivs mhω iωt iωt () = ( ) p t i a a So for th dphasin function w now hav () ( iωt iωt F t = xp d a a ) xp d( a a) % % whr w hav dfind a dimnsionlss displacmnt variabl mω Sinc idntitis This lads to. (6.1), (6.) d = d % h. (6.3) a and a do not commut ( a, a = 1 ) (), w split th xponntial oprators usin th 1 ˆ ˆ Aˆ+ B ˆ Aˆ B ˆ A, B = (6.4) 1 λa + μa λa μa λμ ω =. (6.5) F t = i t i t 1 xp da xp da xp d % % % 1 xp da xp[ d a ] xp d % % % ω (6.6) Now to simplify our work, lt s spcifically considr th low tmpratur cas in which w ar only in th round vibrational stat at quilibrium n =. Sinc a = and a =, and λa = λ a = d i t () = % ω (6.7) F t xp da xp da. (6.8) % % Sinc th oprator dfind throuh an xpansion in raisin oprators, this xprssion in a bit touh to valuat, as is. Howvr, th valuation bcoms as asy as th prvious stp if w can xchan ordr of oprators. Sinc w writ ˆ ˆ ˆ ˆ BA ˆ, ˆ A B B A =, (6.9)

5 Andri Tokmakoff, MIT Dpartmnt of Chmistry, 3/1/ () d % F t = da da xp d % % % iωt = xp d ( 1) % So finally, w hav th dipol corrlation function: iωt iωt xp xp () xp i t = ( 1 + ) C t i t D ω μμ μ ω (6.3) (6.31) D is known as th Huan-Rhys paramtr, and is a dimnsionlss factor rlatd to th man squar displacmnt dmω D= d = (6.3) % h It rprsnts th strnth of couplin to th nuclar drs of frdom. Not w can writ our corrlation function as Hr (t) is our linshap function i mnt () t () n μmn n Ct = p ω. (6.33) i t () D( ω 1) t =. (6.34)

6 Andri Tokmakoff, MIT Dpartmnt of Chmistry, 3/1/ Absorption Linshap and Franck-Condon Transitions Th absorption linshap is obtaind by Fourir transformin q. (6.31) σ abs If w now xpand th final trm as + iωt ( ω) = dt C ( t) = μ xp + μμ i t iω t D ω dt xp D j= iωt. (6.35) iω 1 t j j iωt D = D ( ), (6.36) j! th linshap is D 1 j σ ( ω) = μ D δ ( ω ω jω ). (6.37) abs j= j! Th spctrum is a prorssion of absorption paks risin from ω, sparatd by ω with a Poisson distribution of intnsitis. This is a vibrational prorssion accompanyin th lctronic transition. Th amplitud of ach of ths paks ar ivn by th Franck-Condon cofficints for th ovrlap of vibrational stats in th round and xcitd stats v D 1 v = D (6.38) v! Th intnsitis of ths paks ar dpndnt on D, which is a masur of th couplin strnth btwn nuclar and lctronic drs of frdom.

7 Andri Tokmakoff, MIT Dpartmnt of Chmistry, 3/1/ Lt s plot th normalizd absorption linshap σ ( ω) σ ( ω) abs abs = as a function of D. D μ For D < 1, th dpndnc of th nry ap on q is wak and th absorption maximum is at ω with n =, with th amplitud of th vibronic prorssion fallin off at D n. For D >> 1 (stron couplin), th transition with th maximum intnsity is found for pak at n D. So D corrsponds rouhly to th man numbr of vibrational quanta xcitd from q = in th round

8 Andri Tokmakoff, MIT Dpartmnt of Chmistry, 3/1/ stat. This is th Franck-Condon principl, that transition intnsitis ar dictatd by th vrtical ovrlap btwn nuclar wavfunctions in th two lctronic surfacs. To invstiat th nvlop for ths transitions, w can prform a short tim xpansion of th corrlation function applicabl for t < 1 ω. If w approximat th first trm with dampin thn th linshap is abs ( ) σ ω = μ ( iω t) iω t ω t xp 1, (6.39) μ = μ dt This can b solvd by compltin th squar, ivin iωt iω t i( ω ωt) dt σabs ( ω) = π μ xp ( ) dt ( xp( ω ) 1) D i t 1 D iωt ωt i ω ω Dω 1 t Dω t ( ω ω ) Dω Dω (6.4). (6.41) Th nvlop has a Gaussian profil which is cntrd at Franck-Condon vrtical transition ω = ω + Dω. (6.4) Thus w can quat D with th man numbr of vibrational quanta xcitd in E on absorption from th round stat. Also, w can dfin th vibrational nry vibrational nry in E on xcitation at q = 1 λ = Dh ω = mω d. (6.43) λ is known as th roranization nry. This is th nry that must b dissipatd for vibrational rlaxation on th xcitd stat surfac.

9 Andri Tokmakoff, MIT Dpartmnt of Chmistry, 3/1/9 6- minimum q Not also that λ is th vibrational nry on th round stat surfac at th xcitd stat = d. This is th point for vrtical transitions for mission from th xcitd stat minimum to th round stat. Sinc vibrational nry on is dissipatd quickly, w xpct fluorscnc to b rd-shiftd by λ and hav mirror symmtry with rspct to th absorption. In fact, whn you solv th problm in which th dipol corrlation function is obtaind by avrain ovr th round vibrational lvl of th lctronic xcitd stat, () μ( ) C =, μ t,, on can stablish that σ σ abs fluor ( ω) ( ω) t = + i( ω ω ) t () t dt + * i( ω ω λ) t () t = dt (6.44) iωt = D 1 () ( )

10 Andri Tokmakoff, MIT Dpartmnt of Chmistry, 3/1/9 6-1

11 Andri Tokmakoff, MIT Dpartmnt of Chmistry, 3/1/ DISPLACED HARMONIC OSCILLATOR MODEL: COUPLING TO A BATH AND TEMPERATURE DEPENDENCE Couplin to a Harmonic Bath It is worth notin a similarity btwn th Hamiltonian for this displacd harmonic oscillator problm, and a nral form for th couplin of an lctronic systm which is obsrvd, and a harmonic oscillator bath whos drs of frdom ar dark to th obsrvation, but which influnc th bhavior of th systm. This is a prviw of th concpts that w will dvlop mor carfully latr for th dscription of fluctuations in spctroscopy. W dmonstratd th lctronic absorption linshap drivs from a dipol corrlation function which dscribs th ovrlap btwn two wav packts volvin on th round and xcitd surfacs E and G. () iht iht C t = G G μμ μ μ = ihgt ih Et G μ μ G () () μ ϕ t ϕ t ( ) ie E t (6.45) ϕ ih t () ϕ ( ) iht t t = = (6.46) This is a prfctly nral xprssion, which indicats that th absorption spctrum is th Fourir transform of th tim-dpndnt ovrlap btwn xcitd and round stat nuclar wav packts. Exprssd in a slihtly diffrnt physical pictur, w can also conciv of this procss as nuclar motions that act to modulat th lctronic nry ap ω. W can imain r-writin th sam problm in trms of a Hamiltonian that dscribs th lctronic nry ap s dpndnc on q, i.. its variation rlativ to ω. Dfinin an Enry Gap Hamiltonian: H = H H h ω = H H (6.47) E G W can s that this lads to a problm for an lctronic transition linarly coupld to a harmonic oscillator: Notin that H = H+ E + H + E. (6.48) = hω + H + H

12 Andri Tokmakoff, MIT Dpartmnt of Chmistry, 3/1/9 6-3 w s p 1 H = + mω q (6.49) m 1 ( ) 1 H = mω q d mω q 1 = mω dq+ mωd = cq+λ (6.5) Th Enry Gap Hamiltonian dscribs a linar couplin btwn th lctronic transition and a harmonic oscillator. Th strnth of th couplin is c and th Hamiltonian has a constant nry offst valu ivn by th roranization nry. This discussion illustrats how th displacd harmonic oscillator and Enry Gap Hamiltonian ar isomorphic with a Hamiltonian for an lctronic systm coupld to a harmonic oscillator bath : H = HS + HB + HSB (6.51) ( ) H = E +λ + E S p 1 HB = + mω q m H = mω d q SB (6.5) Hr H SB dscribs th intraction of th lctronic systm (H S ) with th vibrational bath (H B ). It is a linar couplin Hamiltonian, manin that it is linar in th bath coordinat has a strnth- of-couplin trm ( mω d ). Couplin to Multipl Vibrations or a Continuum Th Hamiltonians w hav writtn so far dscrib couplin to a sinl bath dr of frdom, but th rsults can b nralizd to many vibrations or a continuum of nuclar motions. This approach is usd to trat th spctroscopy of dissipativ systms, throuh th intraction of a systm with a continuum of stats that ar dark to th fild, and which w trat in a statistical mannr, in addition to dscribin fluctuations in spctroscopy. So, what happns if th lctronic transition is coupld to many vibrational coordinats, ach with its own displacmnt? Th xtnsion is straihtforward if th mods ar indpndnt,

13 Andri Tokmakoff, MIT Dpartmnt of Chmistry, 3/1/9 6-4 i.. w can conciv of th bath vibrations as harmonic normal mods. W imain an lctronic transition coupld to a st of normal mods for th molcul or lattic. Thn w writ th stat of th systm as product stats in th lctronic and nuclar occupation, i.. G = ; n1, n,..., ni. Th dipol corrlation function is thn iωt () = μ () () L () C t F t F t F t μμ () 1 N iωt iωit = μ xp Di ( 1) i= 1 = μ iωt t N (6.53) with i it ( ) = Di ( ω 1) t (6.54) i For indpndnt mods, th dipol corrlation function is just a product of multipl dphasin functions that charactriz th tim-volution of th diffrnt vibrations. In th tim-domain this would lad to a complx batin pattrn, which in th frquncy domain appars as a spctrum with svral suprimposd vibronic prorssions that follow th ruls dvlopd abov. Takin this a stp furthr, th nralization to a continuum of nuclar stats should b apparnt. This approach dscribs th absorption linshap that rsults from dphasin or irrvrsibl rlaxation inducd by couplin to a continuum. Givn that w hav a continuous frquncy distribution of normal mods charactrizd by a dnsity of stats, W ( ω ), and a frquncy dpndnt couplin, D(ω), w can chan th sum in q. (6.54) to an intral ovr th distribution Hr th product W ( ) D( ) i t () = ω ( ω) ( ω)( 1) t d W D ω. (6.55) ω ω can b considrd a couplin-wihtd dnsity of stats, somtims rfrrd to as a spctral dnsity. What this tratmnt dos is provid a way of introducin a bath of stats that th spctroscopically intrroatd transition coupls with. You can s that if th distribution of stats is vry broad and couplin is a constant, w can associat ( t) with a constant Γ, and w obtain a Lorntzian linshap. So couplin to a continuum or bath provids a way of introducin rlaxation ffcts or dampin of th lctronic cohrnc in th absorption spctrum. Mor

14 Andri Tokmakoff, MIT Dpartmnt of Chmistry, 3/1/9 6-5 nrally th linshap function will b complx, whr th ral part dscribs dampin and th imainary part modulats th primary frquncy and lads to fin structur. Couplin to Sinl undampd vibration Couplin to a continuum Stron dampin

15 Andri Tokmakoff, MIT Dpartmnt of Chmistry, 3/1/9 6-6 Displacd Harmonic Oscillator Modl at Finit Tmpratur If you solv th problm for couplin to a sinl vibrational mod at finit tmpraturs, whr xcitd vibrational lvls in th round stat ar initially populatd, you find i () ω t i xp ( 1)( ) ( ) t i 1 t ω + = μ + + ω 1 Cμμ t D n n. (6.56) ( h 1) n β ω = 1 (6.57) n is th thrmally avrad occupation numbr of th harmonic vibrational mod. Now, lt s calculat th linshap. Expandin xponntials in th dphasin function and Fourir transformin ivs j+ k D( n+ 1) D k σ abs ( ω) = μ + δ ω ω ω j= k= jk!! ( ) j ( n 1) n ( j k) (6.58) Th first summation ovr j (sttin all k to zro) looks as bfor, but th scond summation now includs hot bands : transitions upward from thrmally populatd vibrational stats with a nt dcras in vibrational quantum numbr on xcitation. Not thir amplituds dpnd on th thrmal occupation. W can xtnd this dscription to dscrib couplin to a many indpndnt nuclar mods or couplin to a continuum. W writ th stat of th systm in trms of th lctronic stat and th nuclar quantum numbrs, i.. E = ; n1, n, n3k, and from that:

16 Andri Tokmakoff, MIT Dpartmnt of Chmistry, 3/1/9 6-7 F t D n n (6.59) j or chanin to an intral ovr a continuous frquncy distribution of normal mods charactrizd by a dnsity of stats, ( ) W ω iω () xp ( 1)( 1) ( 1) j t + iωjt = j j + + j () ( ) ( ) ( ) iωt iωt ( )( ) ( )( ) F t = xp dω W ω D ω n ω n ω 1 (6.6) D ( ω ) is th frquncy dpndnt couplin. Lt s look at th nvlop of th nuclar structur on th transition by doin a short-tim xpansion on th complx xponntial as in q. (6.39) ω t F( t) = xp dωd( ω) W( ω) iωt ( n + 1). (6.61) Th linshap is calculatd from i( ω ω ) t 1 σabs ( ω) = + dt xp i ω t xp ω t (6.6) whr w hav dfind th man vibrational xcitation on absorption and ( ) ( ) ω = dωw ω D ω ω = λ / h ( 1) ( ) ( ) ( ) ω dω W ω D ω ω n ω (6.63) = +. (6.64) ω rflcts th thrmally avrad distribution of accssibl vibrational stats. Compltin th squar, q. (6.6) ivs abs ( ) σ ω = μ π ω ( ω ω ) ω xp ω (6.65) Th linshap is Gaussian, with a transition maximum at th lctronic rsonanc plus roranization nry. Th width of th Gaussian is tmpratur-dpndnt and ivn by q. (6.64).

17 Andri Tokmakoff, MIT Dpartmnt of Chmistry, 3/1/9 6-8 Radins 1. S also: Mukaml, S. Principls of Nonlinar Optical Spctroscopy (Oxford Univrsity Prss, Nw York, 1995), p. 17, also p Nitzan, A. Chmical Dynamics in Condnsd Phass (Oxford Univrsity Prss, Nw York, 6). Chaptr 1, Sc. 5.. For furthr on this s: Chaptr 9 of Schatz, G. C. & Ratnr, M. A. Quantum Mchanics in Chmistry (Dovr Publications, Minola, NY, ). Also, Rimrs, JR, Wilson, KR, Hllr, EJ Complx tim dpndnt wav packt tchniqu for thrmal quilibrium systms: Elctronic spctra. J. Chm. Phys. 79, 4749 (1983).