7.3. QUANTUM MECHANICAL TREATMENT OF FLUCTUATIONS: THE ENERGY GAP HAMILTONIAN

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1 Andrei Tokmakoff, MIT Deparmen of Cemisry, 3/5/ QUANTUM MECANICAL TREATMENT OF FLUCTUATIONS: TE ENERGY GAP AMILTONIAN Inroducion In describing flucuaions in a quanum mecanical sysem, we will now address ow ey manifes emselves in an elecronic absorpion specrum by reurning o e Displaced armonic Oscillaor model. As previously discussed, we can also inerpre e DO model in erms of an elecronic energy gap wic is modulaed as a resul of ineracions wi nuclear moion. Wile is moion is periodic for e case coupling o a single armonic oscillaor, we will look is more carefully for a coninuous disribuion of oscillaors, and sow e correspondence o classical socasic equaions of moion. Energy Gap amilonian Now le s work roug e descripion of e Energy Gap amilonian more carefully. Remember a e amilonian for coupling of an elecronic ransiion o a armonic dree of freedom is wrien as = e + Ee + g + Eg (7.48) (7.49) = ω + + g were e Energy Gap amilonian is = e g. (7.5) Noe ow eq. (7.49) can be oug of as an elecronic sysem ineracing wi a armonic ba, were plays e role of e sysem-ba ineracion: = S + SB + B (7.5) We will express e energy gap amilonian roug reduced coordinaes for e momenum, coordinae, and displacemen of e oscillaor p= % ω m pˆ. (7.5) See Mukamel, C. 8 and C. 7.

2 7- q = % mω qˆ (7.53) From (7.5) we ave e mω d = d % ( ) % % % = ω p + q d g = ω p + q % % (7.54) (7.55) = ωdq + ωd % % % = ωdq+ λ % % (7.56) So, we see a e energy gap amilonian describes a linear coupling of e elecronic sysem o e coordinae q. Te slope of versus q is e coupling sreng, and e average value of in e ground sae, (q=), is offse by e reorganizaion energy λ. To obain e absorpion linesape from e dipole correlaion funcion we mus evaluae e depasing funcion. iω () μ e () C = F (7.57) μμ ig ie () e e F = = U U (7.58) g e We now wan o rewrie e depasing funcion in erms of e ime dependence o e energy gap ; a is, if F() = U, en wa is U? Tis involves a ransformaion of e dynamics o a new frame of reference and a new amilonian. Te ransformaion from e DO amilonian o e EG amilonian is similar o our derivaion of e ineracion picure. Noe e mapping = + = + V (7.59) e g Ten we see a we can represen e ime dependence of ime-propagaors are by evoluion under g. Te

3 7-3 ie ig e = e exp dτ ( τ ) U = U U e g + i (7.6) and i g () = e e = U U g g i g. (7.6) Remembering e equivalence beween g and e ba mode(s) B indicaes a e ime dependence of e EG amilonian reflecs ow e elecronic energy gap is modulaed as a resul of e ineracions wi e ba. Ta is Ug = UB. Equaion (7.6) immediaely implies a U i ( τ ) = exp dτ ( τ) i i i + + g e () = e e = exp dτ ( τ) F (7.6) (7.63) Noe: Transformaion of ime-propagaors o a new amilonian If we ave e i A Ae ib and we wan o express is in erms of Ae = Ae, ib A i BA we will now be evolving e sysem under a differen amilonian BA. We mus perform a ransformaion ino is new frame of reference, wic involves a uniary ransformaion under e reference amilonian: new = ref + diff inew iref i e = e exp+ dτ τ = U U diff ref diff ref diff ( τ )

4 7-4 Tis is wa we did for e ineracion picure. Now, proceeding a bi differenly, we can express e ime evoluion under e amilonian of = + B A BA ib i A i e = e exp+ dτ + i A A were τ e e BA BA ( τ ) i = BA. Tis implies: + i A ib i e e = exp+ dτ BA ( τ ) B relaive o A as Using e second-order cumulan expansion allows e depasing funcion o be wrien as i F d () = exp τ ( τ) i τ + dτ dτ ( ) ( ) τ τ τ τ (7.64) Noe a e cumulan expansion is ere wrien as a ime-ordered expansion ere. Te firs exponenial erm depends on e mean value of Tis is a resul of ow we defined = ωd = λ (7.65) %. Alernaively, e EG amilonian could also be defined relaive o e energy gap a Q = : = e g λ. In fac is is a more common definiion. In is case e leading erm in (7.64) would be zero, and e mean energy gap a describes e ig frequency (sysem) oscillaion in e dipole correlaion funcion is ω + λ. Te second exponenial erm in (7.64) is a correlaion funcion a describes e ime dependence of e energy gap ( τ ) ( τ ) ( τ ) ( τ ) = δ τ δ τ (7.66) were δ =. (7.67) Defining e ime-dependen energy gap frequency in erms of e EG amilonian as

5 7-5 δ δω (7.68) we obain (, ) C τ τ = δω τ τ δω (7.69) i F d d C τ () = exp λ τ τ ( τ τ ) So, e dipole correlaion funcion can be expressed as () iee Eg+ λ / g C = μ e e μμ (7.7) (7.7) τ () = τ τ δω ( τ τ ) δω. (7.7) g d d Tis is e correlaion funcion expression a deermines e absorpion linesape for a imedependen energy gap. I is a perfecly general expression a is poin. Te only approximaion made for e ba is e second cumulan expansion. Now, le s look specifically a e case were e ba we are coupled o is a single armonic mode. Evaluaing e energy gap correlaion funcion () = δω δω C p n n n n = n p ne e n i g i g n δ δ iω + iω ( ) = ω D n + e + ne (7.73) ere, as before, D = d, and n is e ermally averaged occupaion number for e oscillaor % Noe a C is a complex quaniy wi x x x x ere co ( x) ( e e ) ( e e ) ( ) n p n n n a a n e β ω = = (). (7.74) C = C + ic (7.75) () ω co ( βω ) cos( ω ) () ω sin ( ω ) C = D C = D (7.76) = +. As e emperaure is raised well beyond e frequency of e oscillaor, C becomes real, C >> C, and C ( ) ~cosω. Tis is e simple classical limi in wic e energy gap is modulaed a e frequency of e oscillaor.

6 7-6 Evaluaing (7.7) gives e linesape funcion () = co ( βω /)( cosω ) + ( sinω ω ) g D i = g + ig (7.77) We also ave real ( g ) and imaginary ( g ) conribuions o F( ). Alernaively, we can wrie is in a form a more closely parallels our earlier DO expressions () iω + iω iω ( ) ( ) iω + iω = ( + )( ) + ( ) g = D n e + e + e idω D n e n e idω Te leading erm gives us a vibraional progression, e second erm leads o o bands, and e final erm is e reorganizaion energy. Looking a e low emperaure limi for is expression, ( β ω ) n, we ave Combining wi we ave our old resul: obain () = [ cos + sin ] g D i i ω ω ω i e ω ω = D i F() = e = e ω iλ/ g id g () exp ( i = D e ω ) In e ig emperaure limi ( β ω ) (7.78) co / and. (7.79) (7.8) F. (7.8) co β ω and g >> g. From eq. (7.77) we D βω ( ) () = exp cos( ω ) F j DkT / ω DkT = e j= j! ω cos ( ω ) j (7.8) wic leads o an absorpion specrum wic is a series of sidebands equally spaced on eier side of ω.

7 7-7 Specral represenaion of energy gap correlaion funcion Since ime- and frequency domain represenaions are complemenary, and one form may be preferable over anoer, i is possible o express e frequency correlaion funcion in erms of is specrum. We define a Fourier ransform pair a relaes e ime and frequency domain represenaions: + + iω iω ( ω) = () = Re () C % e C d e C d. (7.83) + iω C () = e C% ( ω) d π (7.84) * Te second equaliy in eq. (7.83) follows from C ( ) C ( ) Were C % ( ω) and C ( ω ) C (), respecively. Noe a C ( ω) =. Also i implies a ( ω) = ( ω) + ( ω) C% C% C% (7.85) % are e Fourier ransforms of e real and imaginary componens of % is an enirely real quaniy. Wi ese definiions in and, we can e specrum of e energy gap correlaion funcion for coupling o a single armonic mode specrum (eq. (7.73)): C % ( ω) = ωd( ω) ( n + ) δ( ω ω) + nδ( ω+ ω). (7.86) Tis specrum caracerizes e ermally averaged balance beween upward energy ransiion of e sysem and downward in e ba δ ( ω ω ) and vice versa in δ ( ω+ ω ). Tis is given by e deailed balance expression C% ω = C% ω. (7.87) e β ω Te balance of raes ends oward equal wi increasing emperaure. Fourier ransforms of eqs. (7.76) gives wo oer represenaions of e energy gap specrum C % ( ω) = ωd( ω) co ( β ω ) δ( ω ω) + δ( ω+ ω) (7.88) ( ω ) ω D( ω ) δ( ω ω ) δ( ω ω ) C % = + +. (7.89) Te represenaions in eqs. (7.86), (7.88), and (7.89) are no independen, bu can be relaed o one anoer roug e deailed balance expression:

8 7-8 ( ω ) = co ( β ω ) C ( ω ) C% % (7.9) ( ω ) = ( + co( β ω ) ) C ( ω) Due o is independence on emperaure, C ( ω ) C% % (7.9) from eqs. (7.7) and (7.84) we obain e linesape funcion as % is a commonly used represenaion. Also. ( ω) + C% () g = dω exp ( iω) iω π ω +. (7.9)

9 7-9 Disribuion of Saes: Coupling o a armonic Ba More generally for condensed pase problems, e sysem coordinaes a we observe in an experimen will inerac wi a coninuum of nuclear moions a may reflec molecular vibraions, ponons, or inermolecular ineracions. We conceive of is coninuum as coninuous disribuion of armonic oscillaors of varying mode frequency. Te Energy Gap amilonian is readily generalized o e case of a coninuous disribuion of moions if we saisically caracerize e densiy of saes and e sreng of ineracion beween e sysem and is ba. Tis meod is also referred o as e Spin-Boson Model used for reaing a spin wo-level sysem ineracing wi a quanum armonic ba. Following our earlier discussion of e DO model, e generalizaion of e EG amilonian o e mulimode case is = ω + + (7.93) B B = ω p + q % % = ωd q + λ % % (7.94) (7.95) λ = ω d % (7.96) Noe a e ime-dependence o resuls from e ineracion wi e ba: ib B = e e i (7.97) Also, since e armonic modes are normal o one anoer, e depasing funcion and linesape funcion are readily obained from F () = F( ) g( ) = g( ) (7.98) For a coninuum, we assume a e number of modes are so numerous as o be coninuous, and a e sums in e equaions above can be replaced by inrals over a coninuous disribuion of saes caracerized by a densiy of saes W ( ω ). Also e ineracion wi modes of a paricular frequency are equal so a we can simply average over a frequency dependen coupling consan D ( ω ) d ( ω ) = %. For insance, eq. (7.98) becomes

10 7- () ω ( ω ) ( ω ) g = d W g, (7.99) Coupling o a coninuum leads o depasing a resuls from ineracions of modes of varying frequency. Tis will be caracerized by damping of e energy gap frequency correlaion funcion C () ere C ( ω, ) δω ( ω, ) δω ( ω,) () ω ( ω, ) ( ω ) C = d C W. (7.) = refers o e energy gap frequency correlaion funcion for a single armonic mode given in eq. (7.73). Wile eq. (7.) expresses e modulaion of e energy gap in e ime domain, we can alernaively express e coninuous disribuion of coupled ba modes in e frequency domain: + iω ( ω ) = ω ( ω ) ( ω, ) C% d W e C d. (7.) = dω W ( ω ) C% ( ω ) An inral of a single armonic mode specrum over a coninuous densiy of saes provides a coupling weiged densiy of saes a reflecs e acion specrum for e sysem-ba ineracion. We evaluae is wi e single armonic mode specrum, eq. (7.86). We see a e specrum of e correlaion funcion for posiive frequencies is relaed o e produc of e densiy of saes and e frequency dependen coupling ( ω) = ω ( ω) W ( ω)( + ) C% D n (7.) Tis is an acion specrum a reflecs e coupling weiged densiy of saes of e ba a conribues o e specrum. More commonly, e frequency domain represenaion of e coupled densiy of saes in eq. (7.) is expressed as a specral densiy ( ω) C% ρ ω πω = dω W ω D ω δ ω ω π = W ( ω) D( ω) π From eqs. (7.7) and (7.) we obain e linesape funcion in wo forms (7.3)

11 7- ( ω) + C% g() = dω exp ( iω) iω π ω +. (7.4) βω = dω ρ( ω) co ( cosω) + i( sinω ω) In is expression e emperaure dependence implies a in e ig emperaure limi, e real par of g() will dominae, as expeced for a classical sysem. Te reorganizaion energy is obained from e firs momen of e specral densiy d λ = ωωρ ω. (7.5) Tis is a perfecly general expression for e linesape funcion in erms of an arbirary specral disribuion describing e ime-scale and ampliude of energy gap flucuaions. Given a specral densiy ρ(ω), you can calculae specroscopy and oer ime-dependen processes in a flucuaing environmen. Now, le s evaluae e linesape funcion for wo special cases of e specral densiy. To keep ings simple, we will look specifically a e ig emperaure limi, kt ( β ω ) >> ω. ere co β ω and we can nlec e imaginary par of e frequency correlaion funcion and linesape funcion: ) Wa appens wen C ( ω) % grows linearly wi frequency? Tis represens a sysem a is coupled wi equal sreng o a coninuum of modes. Seing C% ω = Γω and evaluaing ( ω) ( cosω ) + % g() = dω πβ ω ω C. (7.6) =Γ A linearly increasing specral densiy leads o a omogeneous Lorenzian linesape wi wid Γ. Tis case corresponds o a specral densiy a linearly decreases wi frequency, and is also referred o as e wie noise specrum. ) Now ake e case a we coose a Lorenzian specral densiy cenered a ω=. Specifically, le s wrie e imaginary par of e Lorenzian linesape in e form

12 7- % Λω ω = λ. (7.7) ω +Λ C ere, in e ig emperaure (classical) limi kt >> Λ, nlecing e imaginary par, we find: λkt g() exp ( Λ ) + Λ Λ Tis expression looks familiar. If we equae λkt Δ = and (7.8) (7.9) τ =, (7.) Λ we obain e same linesape funcion as e classical Gaussian-socasic model: () c g =Δτc exp / τc + / τc (7.) So, e ineracion of an elecronic ransiion wi a armonic ba leads o line broadening a is equivalen o random flucuaions of e energy gap.