13. FLUCTUATIONS IN SPECTROSCOPY Fluctuations and Randomness: Some Definitions

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1 3. FLUCTUATIONS IN SPECTROSCOPY Here we will describe how flucuaions are observed in experimenal observables, as is common o experimens in molecular condensed phases. As our example, we will focus on absorpion specroscopy and how environmenally induced dephasing influences he absorpion lineshape. Our approach will be o calculae a dipole correlaion funcion for ransiion dipole ineracing wih a flucuaing environmen, and show how he ime scale and ampliude of flucuaions are encoded in he lineshape. Alhough he descripion here is for he case of a specroscopic observable, he approach can be applied o any such problems in which he deerminisic moions of an inernal variable of a quanum sysem are influenced by a flucuaing environmen. We also aim o esablish a connecion beween his problem and he Displaced Harmonic Oscillaor model. Specifically, we will show ha a frequency-domain represenaion of he coupling beween a ransiion and a coninuous disribuion of harmonic modes is equivalen o a ime-domain picure in which he ransiion energy gap flucuaes abou an average frequency wih a saisical ime scale and ampliude given by he disribuion of coupled modes. Thus an absorpion specrum is merely a specral represenaion of he dynamics experienced by a experimenally probed ransiion. 3.. Flucuaions and Randomness: Some Definiions Flucuaions is my word for he ime-evoluion of a randomly perurbed sysem a or near equilibrium. For chemical problems in he condensed phase we consanly come up agains he problem of random flucuaions o dynamical variables as a resul of heir ineracions wih heir environmen. I is unreasonable o hink ha you will come up wih an equaion of moion for he inernal deerminisic variable, bu we should be able o undersand he behavior saisically and come up wih equaions of moion for probabiliy disribuions. Models of his form are commonly referred o as sochasic. A sochasic equaion of moion is one which includes a random componen o he ime-developmen. When we inroduced correlaion funcions, we discussed he idea ha a saisical descripion of a sysem is commonly formulaed in erms of probabiliy disribuion funcions P. Observables are commonly described by momens of his disribuion, which are obained by inraing over P, for insance x dx x x x dx x x (3.) For ime-dependen processes, we recognize ha i is possible ha he probabiliy disribuion carries a ime-dependence.

2 3-, x dx x x, x dx x x (3.) Correlaion funcions go a sep furher and depend on join probabiliy disribuions (, A;, B) ha give he probabiliy of observing a value of A a ime and a value of B a ime :, ;, (3.3) A B da db AB A B The saisical descripion of random flucuaions are described hrough hese ime-dependen probabiliy disribuions, and we need a sochasic equaion of moion o describe heir behavior. A common example of such a process is Brownian moion, he flucuaing posiion of a paricle under he influence of a hermal environmen. I is no pracical o describe he absolue posiion of he paricle, bu we can formulae an equaion of moion for he probabiliy of finding he paricle in ime and space given ha you know is iniial posiion. Working from a random walk model, one can derive an equaion of moion ha akes he form of he well-known diffusion equaion, here wrien in one dimension: x, D x, (3.4) x Here D is he diffusion consan which ses he ime scale and spaial exen of he random moion. [Noe he similariy of his equaion o he ime-dependen Schrödinger equaion for a free paricle if D is aken as imaginary]. Given he iniial condiion x, x x, he soluion is a condiional probabiliy densiy x x xx,;, exp (3.5) D( 4 ( ) ) D The probabiliy disribuion describes he saisics for flucuaions in he posiion of a paricle averaged over many rajecories. Analyzing he momens of his probabiliy densiy using eq. (3.) we find ha x() x () x D (3.6) where x() x() x. So, he disribuion mainains a Gaussian shape cenered a x, and broadens wih ime as D.

3 3-3 Brownian moion is an example of a Gaussian-Markovian process. Here Gaussian refers o cases in which we describe he probabiliy disribuion for a variable P(x) as a Gaussian normal disribuion. Here in one dimension: x Ae x x x / x (3.6) The Gaussian disribuion is imporan, because he cenral limi heorem saes ha he disribuion of a coninuous random variable wih finie variance will follow he Gaussian disribuion. Gaussian disribuions also are compleely defined in erms of heir firs and second momens, meaning ha a ime-dependen probabiliy densiy P(x,) is uniquely characerized by a mean value in he observable variable x and a correlaion funcion ha describes he flucuaions in x. Gaussian disribuions for sysems a hermal equilibrium are also imporan for he relaionship beween Gaussian disribuions and parabolic free energy surfaces: (3.7) If he probabiliy densiy is Gaussian along x, hen he sysem s free energy projeced ono his coordinae (ofen referred o as a poenial of mean force) has a harmonic shape. Thus Gaussian saisics are effecive for describing flucuaions abou an equilibrium mean value x. Markovian means ha he ime-dependen behavior of a sysem does no depend on is earlier hisory, saisically speaking. Naurally he sae of any one molecule depends on is rajecory hrough phase space, however we are saying ha from he perspecive of an ensemble here is no memory of he sae of he sysem a an earlier ime. This can be saed in erms of join probabiliy funcions as x, ; x, ; x, x, ; x, x, ; x, (3.7) or ; ; ; ; The probabiliy of observing a rajecory ha akes you from sae a ime o sae a ime does no depend on where you were a ime. Furher, given he knowledge of he probabiliy of execuing changes during a single ime inerval, you can exacly describe P for any ime inerval. Markovian herefore refers o ime-dependen processes on a ime scale long compared o correlaion ime for he inernal variable ha you care abou. For insance, he diffusion equaion G x k T ln x B

4 3-4 only holds afer he paricle has experienced sufficien collisions wih is surroundings ha i has no memory of is earlier posiion and momenum:. Readings. Nizan, A., Chemical Dynamics in Condensed Phases. Oxford Universiy Press: New York, 6; Ch..5 and Ch. 7. c

5 Line-Broadening and Specral diffusion We will invesigae how a flucuaing environmen influences measuremens of an experimenally observed inernal variable. Specifically we focus on he specroscopy of a chromophore, and how he chromophore s ineracions wih is environmen influence is ransiion frequency and absorpion lineshape. In he absence of ineracions, he resonance frequency ha we observe is. However, we have seen ha ineracions of his chromophore wih is environmen can shif his frequency. In condensed maer, ime-dependen ineracions wih he surroundings can lead o ime-dependen frequency shifs, known as specral diffusion. How hese dynamics influence he line widh and lineshape of absorpion feaures depends on he disribuion of frequencies available o your sysem ( and he ime scale of sampling varying environmens (c). Consider he following cases of line broadening: ) Homogeneous. Here, he absorpion lineshape is dynamically broadened by rapid variaions in he frequency or phase of dipoles. Rapid sampling of a disribuion of frequencies acs o average he experimenally observed resonance frequency. The resul in a moionally narrowed line widh ha is narrower han he disribuion of frequencies available and proporional o he rae of flucuaion induced dephasing. ) Inhomogeneous. In his limi, he lineshape reflecs a saic disribuion of resonance frequencies, and he widh of he line represens he disribuion of frequencies,, which arise, from differen srucural environmens available o he sysem. 3) Specral Diffusion. More generally, every sysem lies beween hese limis. Given a disribuion of configuraions ha he sysem can adop, for insance an elecronic chromophore in a liquid, an equilibrium sysem will be ergodic, and over a long enough ime any molecule will sample all configuraions available o i. Under hese circumsances, we expec ha every molecule will have a differen insananeous frequency which evolves in ime as a resul of is ineracions wih a dynamically evolving sysem. This process is known as specral diffusion. The homogeneous and inhomogeneous limis can be described as limiing forms for he flucuaions of a frequency hrough a disribuion of frequencies. If homogeneously broadened. If i i i i evolves rapidly relaive o -, he sysem is evolves slowly he sysem is inhomogeneous. This behavior can be quanified hrough he ransiion frequency ime-correlaion funcion C (3.8) Our job will be o relae he ransiion frequency correlaion funcion correlaion funcion ha deermines he lineshape, C. C wih he dipole

6 3-6

7 Gaussian-Sochasic Model for Specral Diffusion We will bin wih a classical descripion of how random flucuaions in frequency influence he absorpion lineshape, by calculaing he dipole correlaion funcion for he resonan ransiion. This is a Gaussian sochasic model for flucuaions, meaning ha we will describe he ime-dependence of he ransiion energy as random flucuaions abou an average value hrough a Gaussian disribuion. (3.9) (3.) The flucuaions in allow he sysem o explore a Gaussian disribuion of ransiions frequencies characerized by a variance: (3.) The ime scales for he frequency shifs will be described in erms of a frequency correlaion funcion C () (3.) Furhermore, we will describe he ime scale of he random flucuaions hrough a correlaion ime c. The absorpion lineshape is described wih a dipole ime-correlaion funcion. Le s rea he dipole momen as an inernal variable o he sysem, whose value depends on ha of. Qualiaively, i is possible o wrie an equaion of moion for by associaing he dipole momen wih he displacemen of a bound paricle (x) imes is charge, and using our inuiion rarding how he sysem behaves. For a unperurbed sae, we expec ha x will oscillae a a frequency, bu wih perurbaions, i will vary hrough he disribuion of available frequencies. One funcion ha has his behavior is x () i () xe (3.3) If we differeniae his equaion wih respec o ime and muliply by charge we have In many figures he widh of he Gaussian disribuion is labeled wih he sandard deviaion (here ). This is mean o symbolize ha is he parameer ha deermines he widh, and no ha i is he line widh. For Gaussian disribuions, he full line widh a half maximum ampliude (FWHM) is.35.

8 3-8 i (3.4) Alhough i is a classical equaion, noe he similariy o he quanum Heisenberg equaion for he dipole operaor: ih() / hc.. The correspondence of ( ) wih H ( ) / offers some insigh ino how he quanum version of his problem will look. The soluion o eq. (3.4) is exp i d (3.5) Subsiuing his expression and eq. (3.9) ino he dipole correlaion funcion gives or i C e F where F exp i d (3.6) (3.7) The dephasing funcion here is obained by performing an equilibrium average of he exponenial argumen over flucuaing rajecories. For ergodic sysems, his is equivalen o averaging long enough over a single rajecory. The dephasing funcion is a bi complicaed o work wih as wrien. However, for he case of Gaussian saisics for he flucuaions, i is possible o simplify F() by expanding i as a cumulan expansion of averages (see Appendix) i F i d d d! (3.8) exp In his expression, he firs erm is zero, since. Only he second erm survives for a sysem wih Gaussian saisics. Now recognizing ha we have a saionary sysem, we have F d d exp (3.9) We have rewrien he dephasing funcion n erms of a correlaion funcion ha describes he flucuaing energy gap. Noe ha his is a classical excepion, so here is no ime-ordering o he exponenial. F() can be rewrien hrough a change of variables ( ): F exp d (3.) So he Gaussian sochasic model allows he influence of he frequency flucuaions on he lineshape o be described by C () a frequency correlaion funcion ha follows Gaussian saisics. Noe, we are now dealing wih wo differen correlaion funcions C and C. The frequency correlaion funcion encodes he dynamics ha resul from molecules ineracing wih

9 3-9 he surroundings, whereas he dipole correlaion funcion describes how he sysem ineracs wih a ligh field and hereby he absorpion specrum. Now, we will calculae he lineshape assuming ha C decays wih a correlaion ime c and akes on an exponenial form Then eq. (3.) gives C exp / c (3.) F exp c exp / c / c exp which is in he form we have seen earlier F g g (3.) exp / / (3.3) c c c To inerpre his lineshape funcion, le s look a is limiing forms: ) Long correlaion imes c. This corresponds o he inhomogeneous case where C we can perform a shor ime expansion of exponenial, a consan. For c and from eq. (3.3) we obain (3.4) / c e c c / g (3.5) A shor imes, he dipole correlaion funcion will have a Gaussian decay wih a rae given by : F exp /. This has he proper behavior for a classical correlaion funcion, i.e., even in ime C C. In his limi, he absorpion lineshape is: d e i e i g i d e e (3.6) / exp We obain a Gaussian inhomogeneous lineshape cenered a he mean frequency wih a widh dicaed by he frequency disribuion.

10 3- ) Shor correlaion imes c can approximae C gives If we define he consan. This corresponds o he homogeneous limi in which you / c. For c we se e, / c and eq. (3.3) g we see ha he dephasing funcion has an exponenial decay: (3.7) c c (3.8) exp F (3.9) The lineshape for shor correlaion imes (or fas flucuaions) akes on a Lorenzian shape Re d e i e (3.3) This represens he homogeneous limi. Even wih a broad disribuion of accessible frequencies, if he sysem explores all of hese frequencies on a ime scale fas compared o he inverse of he disribuion (c > ), hen he resonance will be moionally narrowed ino a Lorenzian line. More generally, he envelope of he dipole correlaion funcion will look Gaussian a shor imes and exponenial a long imes. The correlaion ime is he separaion beween hese rimes. The behavior for varying ime scales of he dynamics (c) are bes characerized wih respec o he disribuion of accessible frequencies (). So we can define a facor c (3.3) << is he fas modulaion limi and >> is he slow modulaion limi. Le s look a how C, change as a funcion of. F, and abs

11 3- We see ha for a fixed disribuion of frequencies he effec of increasing he ime scale of flucuaions hrough his disribuion (decreasing c) is o gradually narrow he observed lineshape

12 3- from a Gaussian disribuion of saic frequencies wih widh (FWHM) of.35 o a moionally narrowed Lorenzian lineshape wih widh (FWHM) of c. This is analogous o he moional narrowing effec firs described in he case of emperaure dependen NMR specra of wo exchanging species. Assume we have wo resonances a A and B associaed wih wo chemical species ha are exchanging a a rae kab A kab kba If he rae of exchange is slow relaive o he frequency spliing, kab <<AB, hen we expec wo resonances, each wih a linewidh dicaed by he molecular relaxaion processes () and ransfer rae of each species. On he oher hand, when he rae of exchange beween he wo species becomes faser han he energy spliing, hen he wo resonances narrow ogeher o form one resonance a he mean frequency. B Anderson, P. W. A mahemaical model for he narrowing of specral lines by exchange or moion. J. Phys. Soc. Japan 9, 36 (954).; Kubo, R. in Flucuaion, Relaxaion, and Resonance in Magneic Sysems (ed. Ter Haar, D.) (Oliver and Boyd, London, 96).

13 3-3 Appendix: The Cumulan Expansion For a saisical descripion of he random variable x, we wish o characerize he momens of x: x, x, Then he average of an exponenial of x can be expressed as an expansion in momens e ikx ik n n! n x n (3.3) An alernae way of expressing his expansion is in erms of cumulans cn(x) n ikx ik e exp cn x n n! (3.3) where he firs few cumulans are: c x x mean (3.33) c x x x variance (3.34) c x x x x x skewness (3.35) An expansion in cumulans converges much more rapidly han an expansion in momens, paricularly when you consider ha x may be a ime-dependen variable. Paricularly useful is he observaion ha all cumulans wih n > vanish for a sysem ha obeys Gaussian saisics. We obain he cumulans above by expanding eq. (3.3) and (3.3), and comparing erms in powers of x. We sar by posulaing ha, insead of expanding he exponenial direcly, we can insead expand he exponenial argumen in powers of an operaor or variable H F exp c c c c ch c H (3.36) (3.37) Insering eq. (3.37) ino eq. (3.36) and collecing erms in orders of H gives ch c c H F c H c H c H c H Now comparing his wih he expansion of he exponenial F exp fh fh fh (3.38) (3.39) allows one o see ha

14 3-4 c f c f f (3.4) The cumulan expansion can also be applied o ime-correlaions. Applying his o he imeordered exponenial operaor we obain: F exp i d exp c c (3.4) (3.4) c i d c d d d d For Gaussian saisics, all higher cumulans vanish. Readings (3.43). Kubo, R., A Sochasic Theory of Line-Shape and Relaxaion. In Flucuaion, Relaxaion and Resonance in Magneic Sysems, Ter Haar, D., Ed. Oliver and Boyd: Edinburgh, 96; pp McHale, J. L., Molecular Specroscopy. s ed.; Prenice Hall: Upper Saddle River, NJ, Mukamel, S., Principles of Nonlinear Opical Specroscopy. Oxford Universiy Press: New York, Schaz, G. C.; Raner, M. A., Quanum Mechanics in Chemisry. Dover Publicaions: Mineola, NY, ; Secions 7.4 and W. Anderson, P., A Mahemaical Model for he Narrowing of Specral Lines by Exchange or Moion. Journal of he Physical Sociey of Japan 954, 9, Wang, C. H., Specroscopy of Condensed Media: Dynamics of Molecular Ineracions. Academic Press: Orlando, 985.

15 The Energy Gap Hamilonian Inroducion In describing flucuaions in a quanum mechanical sysem, we describe how an experimenal observable is influenced by is ineracions wih a hermally agiaed environmen. For his, we work wih he specific example of an elecronic absorpion specrum and reurn o he Displaced Harmonic Oscillaor model. We previously described his model in erms of he eigensaes of he maerial Hamilonian H, and inerpreed he dipole correlaion funcion and resuling lineshape in erms of he overlap beween wo wave packes evolving on he ground and excied surfaces E and G. ie eeg/ C e g e (3.43) I is worh noing a similariy beween he DHO Hamilonian, and a general form for he ineracion of an elecronic wo-level sysem wih a harmonic oscillaor bah whose drees of freedom are dark o he observaion, bu which influence he behavior of he sysem. Expressed in a slighly differen physical picure, we can also conceive of his process as nuclear moions ha ac o modulae he elecronic energy gap. We can imagine rewriing he same Hamilonian in a form wih a new physical picure ha describes he elecronic energy gap s dependence on q, i.e., is variaion relaive o. If we define an Energy Gap Hamilonian: we can rewrie he DHO Hamilonian H He Hg (3.43) H e Ee e g Eg g He Hg (3.44) as an elecronic ransiion linearly coupled o a harmonic oscillaor: Noing ha we can wrie his as a sysem-bah Hamilonian: H e E e g E g H H (3.44) e g g p Hg m q (3.44) m H H S H B H (3.44) SB where HSB describes he ineracion of he elecronic sysem (HS) wih he vibraional bah (HB). Here HS e Ee e g Eg g, HB Hg and

16 - HSB H m qd m q mdq md cq (3.44) The Energy Gap Hamilonian describes a linear coupling beween he elecronic ransiion and a harmonic oscillaor. The srengh of he coupling is c and he Hamilonian has a consan energy offse value given by he reorganizaion energy. Any moion in he bah coordinae q inroduces a proporional change in he elecronic energy gap. In an alernae form, he Energy Gap Hamilonian can also be wrien o incorporae he reorganizaion energy ino he sysem: H H H H S B SB H e E e g E g S e g p H B m q m H m d q SB (3.44) This formulaion describes flucuaions abou he average value of he energy gap, however, he observables calculaed are he same From he picure of a modulaed energy gap one can bin o see how random flucuaions can be reaed by coupling o a harmonic bah. If each oscillaor modulaes he energy gap a a given frequency, and he phase beween oscillaors is random as a resul of heir independence, hen ime-domain flucuaions and dephasing can be cas in erms of a Fourier specrum of couplings o oscillaors wih coninuously varying frequency.

17 -7 Energy Gap Hamilonian Now le s work hrough he descripion of elecronic specroscopy wih he Energy Gap Hamilonian more carefully. Working from eqs. (3.43) and (3.44) we express he energy gap Hamilonian hrough reduced coordinaes for he momenum, coordinae, and displacemen of he oscillaor p pˆ ( m) q qˆ ( m ) d d( m ) / / / He p qd Hg p q From eq. (3.43) we have H dq d m dq (3.45) (3.46) (3.47) (3.48) (3.49) The energy gap Hamilonian describes a linear coupling of he elecronic sysem o he coordinae q. The slope of H versus q is he coupling srengh, and he average value of H in he ground sae, H(q=), is offse by he reorganizaion energy. We noe ha he average value of he energy gap Hamilonian is H =. To obain he absorpion lineshape from he. dipole correlaion funcion we mus evaluae he dephasing funcion. i C e F (3.5) ihg ihe e e F U U (3.5) We wan o rewrie he dephasing funcion in erms of he ime dependence o he energy gap ; ha is, if F U, hen wha is U? This involves a uniary ransformaion of he dynamics o a new frame of reference. The ransformaion from he DHO Hamilonian o he EG Hamilonian is similar o our derivaion of he ineracion picure. g e H

18 -8 Transformaion of ime-propagaors If we have a ime dependen quaniy of he form ih A ihb e Ae (3.5) we can also express he dynamics hrough he difference Hamilonian HBA HB HA Ae Ae (3.53) ihbha ihba using a commonly performed uniary ransformaion. If we wrie HB HA HBA (3.54) we can use he same procedure for pariioning he dynamics in he ineracion picure o wrie where ihb ih A i e e exp d H BA ih A A H e H e BA (3.55) ih BA (3.56) Then, we can also wrie: ih A ihb i e e exp d H BA (3.57) Noing he mapping o he ineracion picure H H H H H V (3.58) e g we see ha we can represen he ime dependence of he elecronic energy gap H using ih ih e g i e e exp d H U U U e g (3.59) where ih g e e H H U H U g g ihg (3.6) Remembering he equivalence beween he harmonic mode Hg and he bah mode(s) HB indicaes ha he ime dependence of he EG Hamilonian reflecs how he elecronic energy gap is modulaed as a resul of he ineracions wih he bah. Tha is Ug UB. Equaion (3.59) immediaely implies ha

19 -9 F ih g/ ihe/ i exp e e d H (3.6) Now he quanum dephasing funcion is in he same form as we saw in our earlier classical derivaion. Using he second-order cumulan expansion allows he dephasing funcion o be wrien as i F exp d H i d d H H H H (3.6) Noe ha he cumulan expansion is here wrien as a ime-ordered expansion. The firs exponenial erm depends on he mean value of H (3.63) This is a resul of how we defined H. Alernaively, he EG Hamilonian could have been defined relaive o he energy gap a Q : H He Hg. In his case he leading erm in (3.6) would be zero, and he mean energy gap ha describes he high frequency (sysem) oscillaion in he dipole correlaion funcion is. The second exponenial erm in (3.6) is a correlaion funcion ha describes he ime dependence of he energy gap where H H H H H H H H H m dq (3.64) (3.65) Defining he ime-dependen energy gap ransiion frequency in erms of he EG Hamilonian as we can wrie he energy gap correlaion funcion I follows ha H H ˆ (3.66), ˆ ˆ C (3.67) d i g F e e, g d d C (3.68) (3.69)

20 - and he dipole correlaion funcion can be expressed as ieeeg / g C e e (3.7) This is he correlaion funcion expression ha deermines he absorpion lineshape for a imedependen energy gap. I is a general expression a his poin, for all forms of he energy gap correlaion funcion. The only approximaion made for he bah is he second cumulan expansion. Now, le s look specifically a he case where he bah we are coupled o is a single harmonic mode. The energy gap correlaion funcion is evaluaed from Noing ha he bah oscillaor correlaion funcion Cqq () q() q() n e ne m we find Here, as before, D d m oscillaor i i i i C D n e ne (3.7) (3.7) (3.73), n is he hermally averaged occupaion number for he (3.74) and = /kbt. Noe ha he energy gap correlaion funcion is a complex funcion. We can separae he real and imaginary pars of C as (3.75) C Dsin (3.76) where we have made use of he relaion n( ) coh (3.77) x x x x and coh( x) e e e e ˆ ˆ C p n n n n ih g ihg n e e n p n H H n n p n a a n e n n C C ic coh cos C D. We see ha he imaginary par of he energy gap correlaion funcion is emperaure independen. The real par has he same ampliude a T=, and rises wih emperaure. We can analyze he high and low emperaure limis of his expression from

21 Looking a he low emperaure limi, - (3.78) coh / and n, we see ha eq. (3.8) reduces o eq. (3.84). In he high emperaure limi, kt coh kt kt, and we recover he expeced classical resul. The magniude of he real componen dominaes he imaginary par C C, and he energy gap correlaion funcion C () becomes real and even in ime. Similarly, we can evaluae (3.69), he lineshape funcion, i i g D n e n e id (3.79) The leading erm in eq. (3.79) gives us a vibraional progression, he second erm leads o ho bands, and he final erm is he reorganizaion energy ( id i/ ). The lineshape funcion can be wrien in erms of is real and imaginary pars g g ig (3.8) coh / cos sin g D g D (3.8) Because hese ener ino he dipole correlaion funcion as exponenial argumens, he imaginary par of g() will reflec he bah-induced energy shif of he elecronic ransiion gap and vibronic srucure, and he real par will reflec damping, and herefore he broadening of he lineshape. Similarly o C(), in he high emperaure limi g g. Now, using eq. (3.68), we see ha he dephasing funcion is given by i i exp F D n e n e. (3.8) exp Dcoh cosisin (3.83) Le s confirm ha we ge he same resul as wih our original DHO model, when we ake he low emperaure limi. Seing n x x lim coh x lim coh x x in (3.83), we have our original resul i exp FkT D e In he high emperaure limi g g, and from eq. (3.78) we obain (3.84)

22 - DkT exp cos F DkT j j! which leads o an absorpion specrum which is a series of sidebands equally spaced on eiher side of. j cos j (3.85) Specral represenaion of energy gap correlaion funcion Since ime- and frequency-domain represenaions are complemenary, and one form may be preferable over anoher, i is possible o express he frequency correlaion funcion in erms of is specrum. For a complex specrum of vibraional moions composed of many modes, represening he nuclear moions in erms of a specrum raher han a bea paern is ofen easier. I urns ou ha calculaion are ofen easier performed in he frequency domain. To sar we define a Fourier ransform pair ha relaes he ime and frequency domain represenaions: i C e C d (3.86) i C e C d (3.87) * Since he energy gap correlaion funcion has he propery C C (3.86) ha he energy gap correlaion specrum is enirely real: or i Re, i also follows from C e C d (3.88) C C C (3.89) Here C and C C, respecively. C and C are even and odd in frequency. Thus while C are he Fourier ransforms of he real and imaginary componens of is enirely real valued, i is asymmeric abou =. Wih hese definiions in hand, we can wrie he specrum of he energy gap correlaion funcion for coupling o a single harmonic mode specrum (eq. (3.7)): C D n n (3.9)

23 -3 This is a specrum ha characerizes how bah vibraional modes of a cerain frequency and hermal occupaion ac o modify he observed energy of he sysem. The firs and second erms in (3.9) describe upward and downward energy shifs of he sysem, respecively. Coupling o a vibraion ypically leads o an upshif of he energy gap ransiion energy since energy mus be pu ino he sysem and bah. However, as wih ho bands, when here is hermal energy available in he bah, i also allows for down-shifs in he energy gap. The ne balance of upward and downward shifs averaged over he bah follows he deailed balance expression C e C (3.9) The balance of raes ends oward equal wih increasing emperaure. Fourier ransforms of eqs. (3.76) gives wo oher represenaions of he energy gap specrum C D coh (3.9) C D. (3.93) The represenaions in eqs. (3.9), (3.9), and (3.93) are no independen, bu can be relaed o one anoher hrough coh C coh C C (3.94) C (3.95) Tha is, given eiher he real or imaginary par of he energy gap correlaion specrum, we can predic he oher par. As we will see, his relaionship is one manifesaion of he flucuaiondissipaion heorem ha we address laer. Due o is independence on emperaure, he specral C is he commonly used represenaion. densiy

24 -4 Also from eqs. (3.69) and (3.87) we obain he lineshape funcion as C g d exp i i C. (3.96) d coh cos isin The firs expression relaes g() o he complex energy gap correlaion funcion, whereas he second separaes he real and he imaginary pars and relaes hem o he imaginary par of he energy gap correlaion funcion. Coupling o a Harmonic Bah More generally for condensed phase problems, he sysem coordinaes ha we observe in an experimen will inerac wih a coninuum of nuclear moions ha may reflec molecular vibraions, phonons, or inermolecular ineracions. We describe his coninuum as coninuous disribuion of harmonic oscillaors of varying mode frequency and coupling srengh. The Energy Gap Hamilonian is readily generalized o he case of a coninuous disribuion of moions if we saisically characerize he densiy of saes and he srengh of ineracion beween he sysem and his bah. This mehod is also referred o as he Spin-Boson Model used for reaing a wolevel spin-½ sysem ineracing wih a quanum harmonic bah. Following our earlier discussion of he DHO model, he generalizaion of he EG Hamilonian o he mulimode case is H H H (3.97) H B HB p q d q d (3.98) (3.99) (3.) Noe ha he ime-dependence o H resuls from he ineracion wih he bah: ihb ihb e e H H (3.) Also, since he harmonic modes are normal o one anoher, he dephasing funcion and lineshape funcion are obained from F F g g (3.)

25 -5 For a coninuum, we assume ha he number of modes are so numerous as o be coninuous, and ha he sums in he equaions above can be replaced by inrals over a coninuous disribuion of saes characerized by a densiy of saes W. Also he ineracion wih modes of a paricular frequency are equal so ha we can simply average over a frequency dependen coupling consan D d. For insance, eq. (3.) becomes (3.3) Coupling o a coninuum leads o dephasing resuling from ineracion o a coninuum of modes of varying frequency. This will be characerized by damping of he energy gap frequency correlaion funcionc Here C,,,, C d C W. (3.4) refers o he energy gap frequency correlaion funcion for a single harmonic mode given in eq. (3.7). While eq. (3.4) expresses he modulaion of he energy gap in he ime domain, we can alernaively express he coninuous disribuion of coupled bah modes in he frequency domain: C C d W. (3.5) An inral of a single harmonic mode specrum over a coninuous densiy of saes provides a coupling weighed densiy of saes ha reflecs he acion specrum for he sysem-bah ineracion. We evaluae his wih he single harmonic mode specrum, eq. (3.9). We see ha he specrum of he correlaion funcion for posiive frequencies is relaed o he produc of he densiy of saes and he frequency dependen coupling C D W n ( (3.6) C D W n ( (3.7) This is an acion specrum ha reflecs he coupling weighed densiy of saes of he bah ha conribues o he specrum. g d W g, In pracice, he unusually symmery of C and is growh as make i difficul o work wih. Therefore we choose o express he frequency domain represenaion of he couplingweighed densiy of saes in eq. (3.6) as a specral densiy, defined as

26 -6 C d W D W D This expression is real and defined only for posiive frequencies. Noe C in, and herefore () is also. (3.8) is an odd funcion Example of specral densiy using an ohmic densiy of saes, W ( ) exp[ / C ] and a linearly varying frequency dependen coupling. The reorganizaion energy can be obained from he firs momen of he specral densiy d (3.9) Furhermore, from eqs. (3.69) and (3.5) we obain he lineshape funcion in wo forms C g d exp i i. (3.) i d coh cos isin In his expression he emperaure dependence implies ha in he high emperaure limi, he real par of g() will dominae, as expeced for a classical sysem. This is a perfecly general expression for he lineshape funcion in erms of an arbirary specral disribuion describing he ime scale and ampliude of energy gap flucuaions. Given a specral densiy (), you can calculae various specroscopic observables and oher ime-dependen processes in a flucuaing environmen. Now, le s evaluae he behavior of he lineshape funcion and absorpion lineshape for differen forms of he specral densiy. To keep hings simple, we will consider he high emperaure limi, kt B coh and we can nlec he imaginary par of he frequency correlaion funcion and lineshape funcion. These examples are moivaed. Here by he specral densiies observed for random or noisy processes. Depending on he frequency

27 7 range and process of ineres, noise ends o scale as -n, where n =, or. This behavior is ofen described in erms of a specral densiy of he form (3.) s s / c c e where c is a cu-off frequency, and he unis are inverse frequency. These specral densiies have he desired propery of being an odd funcion in, and can be inraed o a finie value. The case s = is known as he Ohmic specral densiy, whereas s > is super-ohmic and s < is sub-ohmic. ) Le s firs consider he example when drops as / wih frequency, which refers o he Ohmic specral densiy wih a high cu-off frequency. This is he specral densiy ha corresponds o an energy gap correlaion funcion ha decays infiniely fas: C ~. To choose a definiion consisen wih eq. (3.9), we se where is a finie high frequency inraion limi ha we enforce o keep well behaved. has unis of frequency, i is equaed wih he inverse correlaion ime for he fas decay of C(). Now we evaluae Then we obain he dephasing funcion / (3.) kt B i g d cos ktcos B i d kt B i F (3.3) e (3.4) where we have defined he exponenial damping consan as kt (3.5) From his we obain he absorpion lineshape abs ( ) i (3.6) Thus, a specral densiy ha scales as / has a rapidly flucuaing bah and leads o a homogeneous Lorenzian lineshape wih a half-widh.

28 -8 ) Now ake he case ha we choose a Lorenzian specral densiy cenered a =. To keep he proper odd funcion of and definiion of we wrie: (3.7) Noe ha for frequencies, his has he ohmic form of eq. (3.). This is a specral densiy ha corresponds o an energy gap correlaion funcion ha drops exponenially as ~exp C. Here, in he high emperaure (classical) limi kt, nlecing he imaginary par, we find kt g exp This expression looks familiar. If we equae and Readings kt (3.8) (3.9) (3.) c we obain he same lineshape funcion as he classical Gaussian-sochasic model: g c exp / c / c (3.) So, he ineracion of an elecronic ransiion wih a harmonic bah leads o line broadening ha is equivalen o random flucuaions of he energy gap. As we noed earlier, for he homogeneous limi, we find c.. Mukamel, S., Principles of Nonlinear Opical Specroscopy. Oxford Universiy Press: New York, 995; Ch. 7 and Ch. 8.

29 Correspondence of Harmonic Bah and Sochasic Equaions So, why does he mahemaical model for coupling of a sysem o a harmonic bah give he same resuls as he classical sochasic equaions of moion for flucuaions? Why does coupling o a coninuum of bah saes have he same physical manifesaion as perurbaion by random flucuaions? The answer is ha in boh cases, we really have imperfec knowledge of he behavior of all he paricles presen. Observing a small subse of paricles will have dynamics wih a random characer. These dynamics can be quanified hrough a correlaion funcion or a specral densiy for he ime-scales of moion of he bah. In his secion, we will demonsrae a more formal relaionship ha illusraes he equivalence of hese picures. To ake our discussion furher, le s again consider he elecronic absorpion specrum from a classical perspecive. I s quie common o hink ha he elecronic ransiion of ineres is coupled o a paricular nuclear coordinae Q which we will call a local coordinae. This local coordinae could be an inramolecular normal vibraional mode, an inermolecular raling in a solven shell, a laice vibraion, or anoher moion ha influences he elecronic ransiion. The idea is ha we ake he observed elecronic ransiion o be linearly dependen on one or more local coordinaes. Therefore describing Q allows us o describe he specroscopy. However, since his local mode has furher drees of freedom ha i may be ineracing wih, we are exracing a paricular coordinae ou or a coninuum of oher moions, he local mode will appear o feel a flucuaing environmen a fricion. Classically, we describe flucuaions in Q as Brownian moion, ypically hrough a Langevin equaion. In he simples sense, his is an equaion ha resaes Newon s equaion of moion F=ma for a flucuaing force acing on a paricle wih posiion Q. For he case ha his paricle is confined in a harmonic poenial, R mq m Q mq f (3.) Here he erms on he lef side represen a damped harmonic oscillaor. The firs erm is he force due o acceleraion of he paricle of mass m ( Facc ma). The second erm is he resoring force of he poenial, Fres V Q m. The hird erm allows fricion o damp he moion of he coordinae a a rae. The moion of Q is under he influence of f ( ), a random flucuaing force R exered on Q by is surroundings. Under seady-sae condiions, i sands o reason ha he random force acing on Q is he origin of he damping. The environmen acs on Q wih sochasic perurbaions ha add and remove kineic energy, which ulimaely leads o dissipaion of any excess energy. Therefore, he Langevin equaion is modelled as a Gaussian saionary process. We ake f ( ) o have a imeaveraged value of zero, R

30 3-3 R and obey he classical flucuaion-dissipaion heorem: f (3.3) fr fr mk T (3.4) B This shows explicily how he damping is relaed o he correlaion ime for he random force. We will pay paricular aenion o he Markovian case m k T f f (3.5) R R B which indicae ha he flucuaions immediaely lose all correlaion on he ime scale of he evoluion of Q. The Langevin equaion can be used o describe he correlaion funcion for he ime dependence of Q. For he Markovian case, eq. (3.) leads o kt B CQQ cos sin m / e (3.6) where he reduced frequency. The frequency domain expression, obained by Fourier ransformaion, is 4 kt C QQ (3.7) m Remembering ha he absorpion lineshape was deermined by he quanum mechanical energy gap correlaion funcion q q, one can imagine an analogous classical descripion of he specroscopy of a molecule ha experiences ineracions wih a flucuaing environmen. In essence his is wha we did when discussing he Gaussian sochasic model of he lineshape. A more general descripion of he posiion of a paricle subjec o a flucuaing force is he Generalized Langevin Equaion. The GLE accouns for he possibiliy ha he damping may be ime-dependen and carry memory of earlier configuraions of he sysem: mq m Q m d Q f (3.8) The memory kernel,, is a correlaion funcion ha describes he ime-scales over which he flucuaing force reains memory of is previous sae. The force due o fricion on Q depends on he hisory of he sysem hrough, he ime preceding, and he relaxaion of. The Nizan, A., Chemical Dynamics in Condensed Phases. Oxford Universiy Press: New York, 6.

31 3-3 classical flucuaion-dissipaion relaionship relaes he magniude of he flucuaing forces on he sysem coordinae o he damping mk T f f (3.9) R R B As expeced, for he case ha, he GLE reduces o he Markovian case, eq. (3.). To demonsrae ha he classical dynamics of he paricle described under he GLE is relaed o he quanum mechanical dynamics for a paricle ineracing wih a harmonic bah, we will ouline he derivaion of a quanum mechanical analog of he classical GLE. To do his we will derive an expression for he ime-evoluion of he sysem under he influence of he harmonic bah. We work wih a Hamilonian wih a linear coupling beween he sysem and he bah H H ( P, Q) H ( p, q ) H ( Q, q) (3.3) HB S B SB We ake he sysem o be a paricle of mass M, described hrough variables P and Q, whereas m, p and q are bah variables. For he presen case, we will ake he sysem o be a quanum harmonic oscillaor, P H M Q M s (3.3) and he Hamilonian for he bah and is ineracion wih he sysem is wrien as p m c HB HSB q Q (3.3) m m This expression explicily shows ha each of he bah oscillaors is displaced wih respec o he sysem by an amoun dependen on heir muual coupling. In analogy o our work wih he Displaced Harmonic Oscillaor, if we define a displacemen operaor 3 ˆ i Dexp pˆ (3.33) c where Q (3.34) m hen H H Dˆ H Dˆ (3.35) B SB B Nizan, A., Chemical Dynamics in Condensed Phases. Oxford Universiy Press: New York, 6.; Mukamel, S., Principles of Nonlinear Opical Specroscopy. Oxford Universiy Press: New York, Calderia, A. O.; Lge, A. J., Ann. Phys 983, 49,

32 3-3 Eqn. (3.3) is merely a differen represenaion of our earlier harmonic bah model. To see his we wrie (3.3) as HB HSB p ( q c Q) where he coordinaes and momena are wrien in reduced form (3.36) Q Q m q q m p p m Also, he reduced coupling is of he sysem o he h oscillaor is (3.37) c c mm (3.38) Expanding (3.36) and collecing erms, we find ha we can separae erms as in he harmonic bah model HB p q HSB dq B The reorganizaion energy due o he bah oscillaors is (3.39) (3.4) B d and he uniless bah oscillaor displacemen is d Qc For our curren work we rroup he oal Hamilonian (eq.(3.3)) as P HHB M Q p q cqq M (3.4) (3.4) (3.43) where he renormalized frequency is c (3.44) To demonsrae he equivalence of he dynamics under his Hamilonian and he GLE, we can derive an equaion of moion for he sysem coordinae Q. We approach his by firs expressing hese variables in erms of ladder operaors

33 3-3 Pˆ iaˆ aˆ pˆ ib ˆ bˆ (3.45) Qˆ aˆ aˆ qˆ b ˆ bˆ (3.46) Here â, â are sysem operaors, ˆb and ˆb are bah operaors. If he observed paricle is aken o be bound in a harmonic poenial, hen he Hamilonian in eq. (3.3) can be wrien as ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ H HB a a bb a a c bb (3.47) The equaions of moion for he operaors in eqs. (3.45) and (3.46) can be obained from he Heisenberg equaion of moion. i aˆ H, ˆ HB a (3.48) from which we find aˆ i aˆ i c b ˆ bˆ (3.49) ˆ ˆ b i b i c ( aˆ aˆ ) (3.5) To derive an equaion of moion for he sysem coordinae, we bin by solving for he imeevoluion of he bah coordinaes by direcly inraing eq. (3.5), bˆ e e i c a a d bˆ e i () i i ˆ ˆ () and inser he resul ino eq. (3.49). This leads o (3.5) ˆ ˆ ˆ ˆ ˆ ˆ a i ai c ( a a) i d( ) a ( ) a ( ) if (3.5) where (3.53) () ccos( ) i F () c b() c a() a() e hc.. (3.54) and ˆ ˆ ˆ Now, recognizing ha he ime-derivaive of he sysem variables is given by ˆ P i a ˆ a ˆ Qˆ a ˆ a ˆ and subsiuing eq. (3.5) ino (3.55), we can wrie an equaion of moion (3.55) (3.56)

34 3-3 c P () Q d ( ) Q ( ) F F (3.57) Equaion (3.57) bears a sriking resemblance o he classical GLE, eq. (3.8). In fac, if we define () () M c m cos fr () M F F p () c q()cos sin m hen he resuling equaion is isomorphic o he classical GLE () () ( ) ( ) () R (3.58) (3.59) P M Q M d Q f (3.6) This demonsraes ha he quanum harmonic bah acs a dissipaive environmen, whose fricion on he sysem coordinae is given by eq. (3.58). Wha we have shown here is an ouline of he proof, bu deailed discussion of hese relaionships can be found elsewhere. 4 4 Weiss, U. Quanum Dissipaive Sysems. 3rd ed.; World Scienific: Hackensack, N.J., 8; Lge, A. J.; Chakravary, S.; Dorsey, A. T.; Fisher, M. P. A.; Garg, A.; Zwerger, W. Dynamics of he dissipaive wo-sae sysem. Reviews of Modern Physics 987, 59 (), -85; Yan, Y.; Xu, R. Quanum Mechanics of Dissipaive Sysems. Annual Review of Physical Chemisry 5, 56 (), 87-9.

35 3-3 Readings. Calderia, A. O.; Lge, A. J., H.O.-bah model;heory. Ann. Phys 983, 49, Fleming, G. R.; Cho, M., Chromophore-Solven Dynamics. Annual Review of Physical Chemisry 996, 47 (), Lge, A.; Chakravary, S.; Dorsey, A.; Fisher, M.; Garg, A.; Zwerger, W., Dynamics of he dissipaive wo-sae sysem. Reviews of Modern Physics 987, 59 (), Mukamel, S., Principles of Nonlinear Opical Specroscopy. Oxford Universiy Press: New York, 995; Ch Nizan, A., Chemical Dynamics in Condensed Phases. Oxford Universiy Press: New York, 6; Ch Schaz, G. C.; Raner, M. A., Quanum Mechanics in Chemisry. Dover Publicaions: Mineola, NY, ; Secions 6.5, 8,., Weiss, U., Quanum Dissipaive Sysems. 3rd ed.; World Scienific: Hackensack, N.J., Yan, Y. J.; Xu, R. X., Quanum mechanics of dissipaive sysems. Annual Review of Physical Chemisry 5, 56, 87-9.