10.1 VIBRATIONAL RELAXATION *

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1 Andrei Tokmakoff, MIT Department of Cemistry, 3//009 p VIRATIONAL RELAXATION * Here we want to address ow a quantum mecanical vibration undergoes irreversible energy dissipation as a result of interactions wit oter intra- and intermolecular degrees of freedom. Wy is tis process important? It is te fundamental process by wic non-equilibrium states termalize. As cemists, tis plays a particularly important role in cemical reactions, were efficient vibrational relaxation of an activated species is important to stabilizing te product and not allowing it to recross to te reactant well. We will be looking specifically at vibrational couplings and relaxation, but te principles are te same for spin-lattice relaxation and electronic population relaxation troug electron-ponon coupling. For an isolated molecule wit few vibrational coordinates, an excited vibrational state must relax by interacting wit te remaining internal vibrations or te rotational and translational degrees of freedom. If a lot of energy must be dissipated, radiative relaxation may be more likely. In te condensed pase, relaxation is usually mediated by te interactions wit te environment, for instance, te solvent or lattice. Te solvent or lattice forms a continuum of intermolecular motions tat can absorb te energy of te vibrational relaxation. Quantum mecanically tis means tat vibrational relaxation (te anniilation of a vibrational quantum) leads to excitation of solvent or lattice motion (creation of an intermolecular vibration tat increases te occupation of iger lying states). For polyatomic molecules it is common to tink of energy relaxation from ig lying vibrational states kt << ω ) in terms of cascaded redistribution of ( 0 energy troug coupled modes of te molecule and its surroundings leading finally to termal equilibrium. We seek ways of describing tese igly nonequilibrium relaxation processes in quantum systems.

2 Andrei Tokmakoff, MIT Department of Cemistry, 3//009 p. 0- Classically vibrational relaxation reflects te surroundings exerting a friction on te vibrational coordinate wic damps its amplitude and eats te sample. We ave seen tat a Langevin equation for an oscillator experiencing a fluctuating force f(t) describes suc a process: () + ω γ = ( ) Qt && 0 Q Q & f t/ m (0.) Tis equation ascribes a penomenological damping rate γ to te vibrational relaxation; owever, we also know in te long time limit, te system must termalize and te dissipation of energy is related to te fluctuations of te environment troug te classical fluctuation-dissipation relationsip: ( ) ( 0) γ δ ( ) f t f = m kt t (0.) We would also like to understand te correspondence between tese classical pictures and quantum relaxation. Let s treat te problem of a vibrational system H S tat relaxes troug weak coupling V to a continuum of t states H using perturtion teory. Te eigenstates of H S are a and tose of H are. Altoug our earlier perturtive treatment didn t satisfy energy conservation, ere we can take care of it by explicitly treating te t states. H = H0 + V (0.3) H0 = HS + H (0.4) HS = a Ea a + b Eb b (0.5) H = (0.6) 0 a E ( ) H a = E + E a (0.7) We will describe transitions from an initial state i = a wit energy Ea + E to a final state f = b wit energy Eb + E. Since we expect energy conservation to old, tis undoubtedly requires tat a cange in te system states will require an equal and opposite cange of energy in te t. Initially, we take pa = pb = 0. If te interaction potential is V, Fermi s Golden Rule says te transition from i to f is given by

3 Andrei Tokmakoff, MIT Department of Cemistry, 3//009 p. 0-3 π k = p i V f δ ( E Ei ) (0.8) fi i f i, f π = pa, a V b δ (( Eb + E) ( Ea + E (0.9) a,, b, + i( ( Eb Ea ) + ( E E )) t = dt p a, a V b b V a e (0.0) a, b, Equation (0.0) is just a restatement of te time domain version of (0.8) kfi = () ( 0) + dt V t V ih0t 0 () e V e V t ih t )) (0.) =. (0.) Now, te matrix element involves bot evaluation in bot te system and t states, but if we write tis in terms of a matrix element in te system coordinate Vab Ten we can write te rate as = av b : a V b = V ab (0.3) + + ie t ie t iωt = e ab e e, k dt p V V (0.4) k dt V () t V ( ) e ω (0.5) + i t = 0 ab ab () iht ab ih t e e V t = V (0.6) Equation (0.5) says tat te relaxation rate is determined by a correlation function ( ) ( ) ( 0) C t = V t V (0.7) ab wic describes te time-dependent canges to te coupling between b and a. Te timedependence of te interaction arises from te interaction wit te t; ence its time-evolution under H. Te subscript L means an equilibrium termal average over te t states L = p L. (0.8) Note also tat eq. (0.5) is similar but not quite a Fourier transform. Tis expression says tat te relaxation rate is given by te Fourier transform of te correlation function for te fluctuating coupling evaluated at te energy gap between te initial and final state states.

4 Andrei Tokmakoff, MIT Department of Cemistry, 3//009 p. 0-4 Alternatively we could tink of te rate in terms of a vibrational coupling spectral density, and te rate is given by its magnitude at te system energy gapω. were te spectral representation C ( ω ) k = C % ( ω ) ab ab. (0.9) % is defined as te Fourier transform of C t. () Vibration coupled to a armonic t To evaluate tese expressions, let s begin by consider te specific case of a system vibration coupled to a armonic t, wic we will describe by a spectral density. Imagine tat we prepare te system in an excited vibrational state in v = and we want to describe relaxation to 0 v =. HS = ω0 ( P + ) % % Q (0.0) ( ) ω ( ) H = ω p + q = a a % % + (0.) We will take te system-t interaction to be linear in te system and t coordinates: V = H = ξ q Q. (0.) % % Here ξ is a coupling constant tat describes te strengt of te interaction S between system and t mode. Note, tat tis bilinear coupling form suggests tat te system vibration is a local mode interacting wit a set of normal vibrations of te t. In principle, tis form of te coupling can be diagonalized to obtain a complete set of normal modes in wic te system vibration is mixed sligtly into eac of te t modes. For te case of single quantum relaxation from a = to b = 0, we can write te coupling matrix element as ( ) V = ξ a + a (0.3) Evaluating eq. (0.7) is now muc te same as problems we ve ad previously:

5 Andrei Tokmakoff, MIT Department of Cemistry, 3//009 p. 0-5 () ( 0) V t V = e V e V ih t ih t ab ab ( ) = ξ + + iωt + iωt n e n e (0.4) ω ere n = ( e ) is te termally averaged occupation number of te t mode at ω. In evaluating tis we take advantage of relationsips we ave used before ih t ih t iω t e a e = a e e a e = ae ih t ih t + iω t aa = n + aa = n So, now by Fourier transforming (0.4) we ave te rate as k = [ ξ ] ( n + ) δ ( ω + ω ) + n δ ( ω ω ) ab (0.5) (0.6) (0.7) Tis expression describes two relaxation processes wic depend on temperature. Te first is allowed at T = 0 K and is obeys ω = ω. Tis implies tat Ea > E b, and tat a loss of energy in te system is lanced by an equal rises in energy of te t. Te second term is only allowed for elevated temperatures. It describes relaxation of te system by transfer to a iger energy state Eb > E a, wit a concerted decrease of te energy of te t. Naturally, tis process vanises if tere is no termal energy in te t. To more accurately model te relaxation due to a continuum of modes, we can replace te explicit sum over t states wit an integral over a density of t states W k = dω W( ω ) ξ ( ω ) n( ω ) + δ ω + ω + n ω δ ω ω ( ) ( ) ( ) ( ) (0.8)

6 Andrei Tokmakoff, MIT Department of Cemistry, 3//009 p. 0-6 We can also define a spectral density, wic is te vibrational coupling-weigted density of states: Ten te relaxation rate is: ( ) W ( ) ( ) ρ ω ω ξ ω (0.9) k = dω W( ω ) ξ ( ω ) n( ω ) + δ ω + ω + n ω δ ω ω = ( n( ω ) + ) ρ ( ωab ) + n( ω ) ρ ( ωab ) ( ) ( ) ( ) ( ) (0.30) We see tat te Fourier transform of te fluctuating coupling correlation function, is equivalent to te coupling-weigted density of states, wic we evaluate atω or ω depending on weter we are looking at upward or downward transitions. Tis is a full quantum expression, and obeys detailed lance between te upward and downward rates of transition between two states: k = exp( ω ) k. (0.3) ab ab From our description of te two level system in a armonic t, we see tat ig frequency relaxation ( kt << ω0 ) only proceeds wit energy from te system going into a mode of te t at te same frequency, but at lower frequencies ( kt ω0 ) tat energy can flow bot into te t and from te t ck into te system. Wen te vibration as energies tat are termally populated in te t, we return to te classical picture of a vibration in a fluctuating environment tat can dissipate energy from te vibration as well as giving kicks tat increase te energy of te vibration. Note tat in a cascaded relaxation sceme, as one approaces kt, te fraction of transitions tat increase te system energy increase. Also, note tat te bi-linear coupling in eq. (0.) and used in our treatment of quantum fluctuations can be associated wit fluctuations of te t tat induce canges in energy (relaxation) and sifts of frequency (depasing).

7 Andrei Tokmakoff, MIT Department of Cemistry, 3//009 p. 0-7 Multiquantum relaxation of polyatomic molecules 3 Vibrational relaxation of polyatomic molecules in solids or in solution involves anarmonic coupling of energy between internal vibrations of te molecule, also called IVR (internal vibrational energy redistribution). Mecanical interactions between multiple modes of vibration of te molecule act to rapidly scramble energy deposited into one vibrational coordinate and lead to cascaded energy flow toward equilibrium. For tis problem te bilinear coupling above doesn t capture te proper relaxation process. Instead we can express te molecular potential energy in terms of well defined normal modes of vibration for te system and te t, and tese interact weakly troug small anarmonic terms in te potential. Ten we can extend te perturtive approac above to include te effect of multiple accepting vibrations of te system or t. For a set of system and t coordinates, te potential energy for te system and system-t interaction can be expanded as V V V V V Q Q q q Q Q q 3 3 S + S = a + a + a b a Qa 6 a,, Qa q q 6 a, b, Qa Qb q L (0.3) Focussing explicitly on te first cubic expansion term, for one system oscillator: ( 3) VS + VS = mω Q + V Qq q % % % % (0.33) Here, te system-t interaction potential describes te case for a cubic anarmonic coupling tat involves one vibration of te system Q interacting weakly wit two vibrations of te t q ( 3) % and q, so tat Ω>>V. Energy deposited in te system vibration will dissipate to te two % vibrations of te t, a tree quantum process. Higer-order expansion terms would describe interactions involving four or more quanta. Working specifically wit te cubic example, we can use te armonic t model to calculate te rate of energy relaxation. Tis picture is applicable if a vibrational mode of frequency Ω relaxes by transferring its energy to anoter vibration nearby in energy ( ω ), and te energy difference ω being accounted for by a continuum of intermolecular motions. For tis case one can sow k = ( n( ω ) + ) ( n( ω ) + ) ρ ( ωab ) + ( n( ω) + ) n( ω) ρ ( ωab ) (0.34)

8 Andrei Tokmakoff, MIT Department of Cemistry, 3//009 p. 0-8 were ( ) ( ) ( ( ) ( ) 3 ρ ω W ω V ω ). Here we ave taken Ω, ω >> ω. Tese two terms describe two possible relaxation patways, te first in wic anniilation of a quantum of Ω leads to a creation of one quantum eac of ω and ω. Te second term describes te dissipation of energy by coupling to a iger energy vibration, wit te excess energy being absorbed from te t. Anniilation of a quantum of Ω leads to a creation of one quantum of ω and te anniilation of one quantum ofω. Naturally tis latter term is only allowed wen tere is adequate termal energy present in te t. Rate calculations using classical vibrational relaxation In general, we would like a practical way to calculate relaxation rates, and calculating quantum correlation functions isn t practical. How do we use classical calculations for te t, for instance drawing on a classical molecular dynamics simulation? Is tere a way to get a quantum mecanical rate? * Te first problem is tat te quantum correlation function is complex Cab () t = Cab ( t) and te classical correlation function is real and even C ( t) C ( t) Cl =. In order to connect tese two correlation functions, one can derive a quantum correction factor tat allows one to predict te quantum correlation function on te sis of te classical one. Tis is sed on te assumption tat at ig temperature it sould be possible to substitute te classical correlation function wit te real part of te quantum correlation function Cl ( ) ( ) Cl C t C t (0.35) To make tis adjustment we start wit te frequency domain expression derived from te detailed lance expression C% ( ) e ω = ω C% ( ω) C% ( ω) = C% ( ω) + exp ω (0.36) ( )

9 Andrei Tokmakoff, MIT Department of Cemistry, 3//009 p. 0-9 Here C ( ω) % is defined as te Fourier transform of te real part of te quantum correlation function. So te vibrational relaxation rate is k 4 + ( exp( ω kt )) () t V ( ) iωt = dte V ab 0 Re 0 (0.37) Now we will assume tat one can replace a classical calculation of te correlation function ere as in eq. (0.35). Te leading term out front can be considered a quantum correction factor tat accounts for te detailed lance of rates encoded in te quantum spectral density. In practice suc a calculation migt be done wit molecular dynamics simulations. Here one as an explicit caracterization of te intermolecular forces tat would act to damp te excited vibrational mode. One can calculate te system-t interactions by expanding te vibrational potential of te system in te t coordinates V V V V V Q Q S + S = Q Q L = V0 + FQ+ GQ + L (0.38) Here V represents te potential of an interaction of one solvent coordinate acting on te excited vibrational system coordinate Q. Te second term in tis expansion FQ depends linearly on te system Q and t coordinates, and we can use variation in tis parameter to calculate te correlation function for te fluctuating interaction potential. Note tat F is te force tat molecules exert on Q! Tus te relevant classical correlation function for vibrational relaxation is a force correlation function Cl ( ) ( ) ( 0) C t = F t F (0.39) kcl = dtcosωt F() t F( 0) kt. (0.40) 0

10 Andrei Tokmakoff, MIT Department of Cemistry, 3//009 p. 0-0 Readings and Notes * Oxtoby, D. W. Vibrational population relaxion in liquids. Adv. Cem. Pys. 47, 487 (98); Skinner, J. L. Semiclassical approximations to golden rule rate constants. J.Cem.Pys.07, 877 (997); Egorov, S. A., Rani, E. & erne,. J. Nonradiative relaxation processes in condensed pases: Quantum versus classical ts. J. Cem. Pys.0, 538 (999). Note tat we are using an equilibrium property, te coupling correlation function, to describe a non-equilibrium process, te relaxation of an excited state. Underlying te validity of te expressions are te principles of linear response. In practice tis also implies a time-scale separation between te equilibration of te t and te relaxation of te system state: Te t correlation function sould work fine if it as rapidly equilibrated, even toug te system may not ave. An instance were tis would work well is electronic spectroscopy, were relaxation and termalization in te excited state occurs on picosecond time scales, wereas te electronic population relaxation is on nanosecond time scales. Tere is an exact analogy between tis problem and te interaction of matter wit a quantum radiation field. Te interaction potential is instead a quantum vector potential and te t is te poton field of different electromagnetic modes. Equation (0.7) describes as two terms tat describe emission and absorption processes. Te leading term describes te possibility of spontaneous emission, were a material system can relax in te absence of ligt by emitting a poton at te same frequency. 3 V. M. Kenkre, A. Tokmakoff and M. D. Fayer, Teory of vibrational relaxation of polyatomic molecules in liquids, J. Cem. Pys., 0, 068 (994).