William Rhodes Department of Chemistry and Institute of Molecular Biophysics Florida State University Tallahassee, FL 32306
Molecular Excited State Relaxation Processes
Most molecular processes undergo a monotonic relaxation toward equilihrium. For example, almost all known chemical reactions proceed toward chemical (thermodynamic) equilibrium in a uniform, asymptotic manner. The same is true of a svstem suhiected to a small but sudden temperature . iumo . or a i Y > t c i',Irnulecules in condensed pha!c subjected to a s h m pulse of light f r m a wnventional light sourci.. All of these cases correspond to linear displacemeits from equilihrium (1,2). This means simply that the displacement from equilihrium is sufficiently small and that the effective number of degrees of freedom of the system is so large that the energies involved in the nrocess auicklv become dissinated: i.e... thev " become distributed over many hegrees of freedom tither than beine localized in a few modes. This is usuallv discussed in terms of entropy production. There are orocesses. however. which do not nroceed hv monotonic relaxation toward equilihrium. Prigogine and coworkers (2) have made extensive studies of thermodvnamics in the noniinear regime in which systems may be s b far removed from equilihrium that local nonequilihrium "structures" are formed. Such structures may he maintained for long times by interactions within the system, but if the system is isolated, it eventually relaxes toward equilibrium. ~rigogine refers to these as dissipative structures, but we shall refer to them as nonequilihrium structuresaI; opposed M equilihrium structures. A yund example is the oscillating chemical reartion, discovered hv Belousov. oromoted hvZhahotinskv. and studied extensiveiy by ~ o y e s ' e tal. (3),i n which t h i n o n eauilihrium structure consists of chemical concentration g r d i r n t s which osrillatc spatially and twnpornllg. Another examole. . . closer ru the suhirct this .oaoer. . ia thr hroad clirss of laser-driven nonlinear, coherent optical phenomena whereby a molecular system is driven so far from equilibrium by strong interaction with intense, coherent light that oscillatory processes are established in the molecular svstem. Here the no"equilihrium structure consists of microBcopic electric-dipole polarization which extends coherently over many molec;les. u n d e r these conditions each molecule oscillates (nutates) hetween its ground and an excited state (4). The ahove two cases are mentioned as examples of processes in which a few modes or states (i.e., degrees of freedom) dominate in determining the observed properties of the system. These few modes or states couple strongly among each other so that the energy tends toremain localized among them rather than being dissipated among the many remaining degrees of freedom which are only weakly coupled. This freWilliam Rhodes is Professor of Chemistry and Executive Director of the Institute of Molecular Biophysics at the Florida State University. He received his A.B. degree from Howard College in Birmingham, Alabama, in 1954 and his Ph.D. from Johns Hopkins University in 1958. His research interests include: theoretical moleculnr ph)sic< and muierular hio. physics: dynamic arpmtr of rnrrlerular exrilntmn by light; excited slaw relaxation processes; optical and electronic response properties of biological macroI molecules.
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562 / Journal of Chemical Education
quently results in an oscillatory-type hehavior which may persist for a long period of time while the system slowly relaxes toward equilihrium. The discussion thus far points out the existence of the following fundamental dichotomy in molecular processes. (1) Those processes in which the coupling among a few modes or states predominates tend to give oscillatory relaxation toward equilibrium. (2) Those processes in which the distribution of energy among many modes or states predominates tend to give dissipative (monotonic) relaxation toward equilihrium. The usage of the term dissipative here should not he confused with that of Prieoeine e t al. The existence of oscillatory relaxation is not limited to those orocesses which are driven (or orenared) far from eauilihrium . . in II nonlinear thermudyni~micsenic.It may exist alsoin ivs. t(ms in u,hich the efti:vti\e numher of d e u r e s of frecviom or modes have been either reduced or isolated by some means. An example is a system of small molecules a t very low pressure in which the behavior of the molecules may be observed during time intervals in which the molecules are effectively isolated. Such systems should he capable of exhibiting oscillatory fluorescence decay. Another kind of system consists of electron or nuclear spins which tend to he only weakly coupled to the remaining degrees of freedom of the molecular system and for which coherent, nonlinear (oscillatory) phenomena is a common occurrence (5). With the ahove discussion serving as an heuristic basis we concept of oscillatory will proceed to further develop versus dissipative limits as it applies to electronic excited state processes in molecular systems. We are interested particularly in the radiative and nonradiative dynamics of the excited state of a molecule (or system of molecules) prepared by interaction with light or some other excitation source. The degrees of freedom, referred to above, are the quantum mechanic2 states of the system.'According to the basic tenets of quantum theory the time dependence of all quantum mechanical phenomena is determined by the time dependence of the amplitudes of all basis states comnrisine the state of the svstem. Thus. the dynamics of processes are governed by the interference of time-de~endentamolitudes and it is imnortant to understand the role of phase-relations among the components of the system.
tee
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Types of Processes
lntrarnolecular Radiationless Transitions Prior to 1960 most exoeriments in molecular soectroscoov were carried out in condensed phase, wherein the energy'df excitation becomes channeled into ( 1 ) luminescence (fluorescence and phosphorescence), (2) thermal energy of the medium (heat), and (3) photochemistry (photoionization, photofragmentation, photoisomerization). In such systems the prepared excited state of the molecule tends to decay by dissinative relaxation accordine to unimolecular kinetics (exponential decay), with quantum yields for the ahove processes determined by the rate constants. Since the dynamics is governed by sequential decay kinetics and associated branchine ratios. it is hardlv necessarv to use nnantum theow a t all, except to consider discrete states of the molecule or understand the mechanistic asoects of the orocesses ( 6 ) . Of course, the mechanistic aspects of traditional mole&lar
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spectroscopy is couched in terms of quantum theory and the framework for this is provided hy the Born-Oppenheimer approximation (7). In this scheme, because the nuclei move much more slowly than the electrons, it is assumed that the electronic energy surfaces of the molecule act as potential energy surfaces for the motion of the nuclei. By this approximation the eigeustates of the molecular Hamiltonian are obtained as vibronic states. each of which is a nroduct of an electronic state and a vihrational state. The traditional view (8) is that light absorption prepares the molecule is an excited Born-Oppenheimer (B-0) vihronic state. This vibronic state decays by (a) radiative emission, (b) vihrational relaxation (transitions involving one electronic state), and (c) electronic relaxation (transitions involving two or more electronic states; viz., internal conversion and intersystem crossing). The molecule tends to relax radiationlesslv to the lowest excited vibronic level of each multiplicity, with the branching ratio depending on the degree of intersystem crossing. Radiative and radiationless decay occurs from these levels (the Kasha Rule (9)).The discovery of deuterium isotopic sul)stit~~tion effects un phuiphort:scence quantum yicldi (1111has o n ~ l d l an d imi~orrnntfoul fur s~udvinr " " the mechanistic aspects of these processes. According to the above picture, molecular electronic Spectroscopy is conccrnerl only with dissipative relnxarion proresses. However, during the 19fi0's tau kinds oiexocrimentd developments set the stage for developing a different view of molecular excitation processes. (1) Isolated molecule (low-pressure gas phase) experiments on large (11) and small (12) molecules emphasized and enhanced the intramolecular aspects of radiationless nrocesses. It was shown that large moleches (e.g., naphthalenej can have a sufficiently large number of vibrational degrees of freedom that the molecule can act as its own energy sink, so that electronic relaxation is irreversible in the time range for which the molecule is isolated (up to 10-Ssec). On the other hand, small molecules, such as SO2 or NOz, have such few vihrational degrees of freedom that electronic relaxation tends to he reversible (i.e., oscillatory in character). (2) The development of laser techniques has led to a class of optical phenomena so different from traditional molecular spectroscopy that they will be discussed separately in a section below. During the past 20 years there have been several important facets of theoretical research on molecular excited state processes. Robinson and coworkers earlv emphasized the importance of Frauck-Condon factorsin the mechanism of electronic relaxation (13). Lin discovered the distinction hetween promuting modes and accepting modes in the mechanism oielectnnuc rthxnrion (14) Siehraud first ~ o i n t e dout the importance of the energy gap hetween the mikma of the electronic energy surfaces involved in electronic relaxation (15). Jortner and coworkers (Chicago-Tel Aviv School) stressed the use of Green's operator methods for analyzing large and small molecule limits of isolated molecule behavior (16-23). Similar approaches have been used by other researchers. including erouos in France (24-26). Rhodes and coworkers have em&asized the impor&nce ofthe properties of the exciting liaht and the structure of the molecular absorption spec&; in determining the nature of the prepared excited state (27-33). There are several topical and comprehensive reviews of radiationless transitions in molecules (3445). As a result of $be experimental and theoretical research on molecular excited state processes during the past decade there has been a revolution in the approach to molecular spectroscopy. The traditional line of thinking in terms of the BornOppenheimer scheme has branched into two lines of thinking. (1) The dynamic line regards the overall process of liaht absorption and emission, with associated relaxation pro~esses, within a unified framework. It is not assumed that the prepared excited state is necessarily a B-0 vibronic state (31,33).
( 2 ) The mechanistic line is a continuation of the traditional view in terms of the B - 0 scheme and emphasizes the mechanisms of radiative and radiationless processes in terms of molecular structure and interactions, such as vihronic coupling and spin-orbital coupling. Level Crossing It is well e&hlished that applied electric and magnctic fields have dtfinitive ~+frcrson molecular spectral oro~enies. Most of these effects are related to the poskions and intensities of absorption bands and can he described hy low-order perturbation theory. These usually do not involve anything which is conceptually new. However, there is one property, referred to as level crossine. which does involve concentual features of qualitative impo-&nce for dynamics of the excited state. S u ~ o o s ea molecule has two or more excited enerw lev& which lie very close together and all of which are allow; for absor~tionfrom the around state bv the same photon DOlarizatioi. ~urthermore;suppose eachbf these excited 1e;els can give photon emission of the same polarization (nossihlv difle-rent from ahiorption) upon rransitk,n to a conmi& lower level. It the absorption lines are d~stinctin the absence of a field, it is possiblefor an applied field to decrease the spacing between the 1e.velsto the point that the spacing is less than the radiative linewidths, whereupon there is a dramatic increase in the intensity of luminescence. Such a property is known as the Hanle effect (46). The theoretical basis was first formulated by Breit and Lowen (47) and later by Franken (48). A good example of level crossing is with the triplet levels of atoms which exhibit zero-field splitting (48). An applied maenetic field can brine the levels to near dezeneracv. The level crossing effect involves the excitation of a coherent suoer~osition of the states for the different levels. This . . superposition then decays radiatively with constructive interference oroducine intensitv enhancement. Such a ohenokenon is bf i m p o r t k e to thk subject of this paper because it provides a prototwe .. model for most aspects of excited state dynamics in more complex molecular systems.
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Intermolecular Excitation Transfer The distinction between oscillatory and dissipative relaxation limits is manifested very clearly in the phenomenon of excitation transfer (or delocalization) in molecular systems. We are interested here in excitation transfer induced by Coulomb interaction between molecules sufficiently far apart that their wavefunctions do not overlap. The two well-established limiting cases are as follows. 1) The Exciton Mechanism involves excited states of ordered systems (e.g., crystals, polymers, molecular dimers) of identical molecules in which the excitation is delocalized with amplitudes having definite phase relations among the molecules (49-50). If the system is excited in a manner that the excitation initially is localized on one molecule, the excitation will migrate over the system by a coherent spreading. For the particular case of a dimer the excitation oscillates between the molecules with a frequency proportional to R-3, where R is the distance between molecular centers. While the excitation oscillates the dimer decays via radiative and radiationless relaxation to its -around state. This is the oscillatorv relaxation case. 2) The Forster Mechanism is operative in both ordered and disurdered sy+tcmsof nonidentical as w 1 1 as i d e n r i ~ dmolerulrh (51-521. The excitntiun is nor delocalizcd cohcrentlv: rather, it hops from molecule to molecule. The rate of transfer from one molecule to another is proportional to R-6. This is basically an irreversible process which represents the dissipative relaxation limit. In a later section we will see how the exciton and Forster mechanisms can be descrihed in terms of two limiting cases of the same model. Coherent-Nonlinear Optical Phenomena The properties of laser light which distinguish it from ordinary light include the tendency for it to be coherent, intense, Volume 56, Number 9, September 1979 1 563
nonstationary, and of minimum frequency width (53). The idealization of a laser mode is a classical, sinusoidal electromagnetic wave with frequency wo. An ideal laser pulse is then a coherent (definite dhasej superposition of such modes having frequencies in the neighborhood of wn. This is referred to as a mi&num (uncertain&) width pulse,~andit is characterized by wo, field strength Eo, and width Aw. In the electric-dipole approximation the important coupling between the molecule and field is given by h w = ~ Evflsowhere w~ is the Rabi frequency and p,o is the electric-dipole transition moment connecting the ground state and some excited state s. The distinction between oscillatorv excitation (nutation) and dissipative excitation (monotomic increase in excited state prohabilitv) depends on the maenitude of w a r t h t l w 10 bofh l ( r *and;,, th'e spnntilneot1sdecay llnewidtl; 01 the excited stare. If WH IS larger than hoth I& and 3 . . the molecules of the system tend tooscillate in phase hetween the pound and excited states, thus creating.a microsco~icelectric polarization in the medium. There are a number of dramatic optical phenomena associated with the nonlinear, coherent (oscillatory) excitation limit (4,53-56). Superradiance is very intense fluorescence arising from the fact that the molecules are excited with definite relative phases. Photon Echo (56) is a recurrence of super-radiant emission ~ u l s e shroueht about hv successive coherent excitation pulses. Self-Induced 7hnspa~ency(SIT) results when a minimum width Duke of . Droner . Eo and Aw pn.;rs through an ahsorhing mrditlm. The intensity and shape o f t h e pulse is such that ir drives the molecul~sthrvueh one cycle o? oscillation and thereby passes through the medium unattenuated. Dynamic Stark Splitting is exhibited by fluorescence from molecules excited in the oscillatory limit. This involves the concept of molecule-field dressed states (57). In recent years nonlinear, coherent optical phenomena have been observed and intensively studied in large molecule systems, such as aromatic hydrocarbons (58,59). This shows that certain modes can he selectively "driven" in systems having so many degrees of freedom that one might expect dissipative hehavior to always predominate. Elementary Theoretical Description
The remainder of this paper will he devoted to a discussion of elementary quantum theoretical concepts and models which are useful for understanding the basic dichotomy of dynamical hehavior (viz., oscillatory versus dissipative relaxation) observed in the broad range of phenomena mentioned ahove. For simplicity we will assume discrete energy levels (hound states), since the results are easily generalized to include continua. In general, the state of a molecule or system of molecules may he nonstationary, in which case i t can he expressed as a superposition of molecular (system) stationary states, Wt) =
x c,(t)+.
(1)
where I.I.." is a stationarv state and c.(t) " . . is the time-denendent expansion cmfficient. Actually, we are interested for the most part in the nonstationary hehavior (relaxation) of a molecule, or system, after the perturbation (e.g., light interaction) has been removed. For these cases the Hamiltonian operator is independent of time, which results in each c, having the time dependence c,(t) = e,e-'Esll" (2) where c, is independent o f t and E, is the energy eigenvalue for state $*. It is important to note that the prohahility of state $, is given by lc,(t)12, which is independent o f t . Similarly, the ~robahilitvdistribution of anv- nhvsical auantitv which . , wmmutw with the Hamiltvnian ?r is independenr of time. However. for non.;talionarv states the ~rohahditieioi~hviical which do not c&nmute with H generally'depend 564 1 Journal of Chemical Education
on t. The reason lies in the fact that the eigenfunctions for such quantities are superpositions of the stationary states; namely, 6k = X Thus, the probability amplitude for 4k is ($al*(t))
=
a;,~,e-'~-'l~
(3)
(4)
so the corresponding probability 1 (@k[*(t)) l 2 depends on t by way of the cross terms exp(-iw,,,t), where US,, = (E, -E.,)lh The ahove simple result shows the remarkable feature that the time deoendence of all . nhvsical auantities in a svstem whose H a m i l r u n ~ ais~indep.ndent ~ of I is give11by the timedependent interferences o f a m ~ l i t u d efor s the difterent statiinary states. ~lternatively; it should he noted that (&I *(t)) is a sum of complex numbers, each of which is roI in the complex plane with tating on a circle of radi~s-la;~c, a frequency EJh. The time-dependent prohability of @k is determined by the phase matching of all these points. Thus, it might be said that the time dependence of anv isolated system results from the phasing and dephilsing of the \.ariuus stationary itatt:s cnrnprising the state uirhe system.
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The Two-State Model
Suppose we have a model system consisting of two states @b which are eiaenfunctions of a Hamiltonian Hn . having eigenvalues c, and cb, as shown in Figure 1.Now consider the addition of interaction V to the system, giving a total Hamiltonian H = Ho V, which is time independent. In the Ho basis H is a 2 X 2 matrix with diagonal matrix elements (a = t,, + Vooand ib = €6 Vbb. and off-diagonal elements Vob
6, and
=
v;?
+ +
I t 1s easily shown that the eigenvalues of H are given E l ='/deb+ € d l(11/2A.b12+ 1V.b12)112 (5) where Aab e i - cb. In general, the eigenfunctions of H are complex, orthogonal combinations of 4. and rpb, hut for the case that Vb. is real (which holds for many physical systems) the eigenfunctions can be expressed in the form (52) ++ = & sina
+ 6b cosa
+- = 60 cosa - $a sina,
(6)
in which a is defined hv
This form of the solution shows clearly how the admixtures of 4. and 4b depend on the relative values of Vb, and Aab. which are the two parameters that completely define the energetics of system. We are interested in the time dependence of the system when i t is prepared in one of the zeroth-order states, say b,, a t t = 0.Equations (I),(2) and (6) give
+
*(t) = $ ~ e - ' ~ + sine ~ / * + -e-iE-t/h cma Use of eqns. (5)and (6) yields W(t) = [(cosbt + i sinat eosZa)6.,
(7)
- i sinat s i n Z a a ] e ~ ' ~(8)~
with
Figure 1. Two-state mcdel in which the sties are coupled by Ihe mahix element V*&.
.I.(O)= &, A = (2h)-I
(ib+ 6;)
and = h-'l(1h.A.a)2
+ V&l"2
Equation (8) is our important result for the two-state model. It shows that, starting with state @., the system tends to undergo oscillations to @b with a frequency and amplitude that depends on the values of V.band A,b. For the special case of degeneracy to first order (f, = f a ) the system undergoes complete oscillations from 4. to with frequency h-'V.6. The Picket-Fence Model
This is a model (Fig. 2) which has been used widely in many areas of physics for describing exponential decay in systems having a high density of states. It consists of a single state 4, coupled to a manifold of equally-spaced states @/ which are uniformly spaced by energy A. The states are not coupled to each other and the couoline . .. of each to the state L is the same value \'. Fur the special case that thr mnnifold exrends toinfinitv in borh dirertiunsand tht~eneravoithestateo,is degenerate with one of the @/ the solutions have heen formulated exactly by Lemmer (60) in the context of nuclear physics and by Bixon and Jortner (16) in the context of radiationless transitions in molecules. The eigenvalues, E,, are intercalated between the zeroth-order levels y . The state &, may be expanded in terms of stationary states, +a =
Zc..**
(9)
and the probability distrihution of & among the stationary states turns out to be Lorentzian; namely,
The width of the Lorentzian a t half maximum is Physical processes associated with this model are usually such that the system is prepared at t = 0 in state &. The state fort > 0 is then given by .I.(t) =
x e-'E*LJhe,.J.,
(11)
respectively. All systems interact at least weakly with their surroundings, however, so ultimately they are dissipative (relax toward equilibrium). Therefore, a more appropriate model, which serves as a prototype for distinguishing oscillatory and dissipative limits is one which combines the twostate and picket-fence models, as shown in Figure 3. It depends on 6 parameters: Vob, Aobr Vr, A?. VI, and A? We are usually interested in the dynamics of a system prepared in state 4, a t t = 0. Thus, the probability distribution, lc,. 1 2, of @. among the (stationary) eigenstates of the system (cf. eqn. 9) is very important. This distribution corresponds to an observable spectrum for those spectroscopic processes for which 4, carries all of the intensity for transitions from some other state (not shown), . . such as the around s&e (cf. spectruin on left in Fig. 4). For example, for elekronic transitions it would he the ahsorotion soectrum for a certain frequency region. The shape of the spectrum of & depends on the values of the 6 parameters. For strong coupling V.6 dominates and Ic,, 12 forms a bimodal spectrum with 2 bands; but for weak coupling, where ri = 2 a 1 Vfl AT' dominates, lc,. 1 is a spectrum with one peak. The former is the prototype model for oscillatory relaxation while the latter corresponds to dissipative relaxation. Figure 4 shows the spectra associated with the strong coupling case for the basic model. The spectrum with 2 hands on the left is the distrihution, Ic,, 1 2, of state 4, among the system eigenstates $, as was discussed above. The spectrum on the right is the (Lorentzian) distribution, (cat 1 2, of state +b among the eigenstates 4; obtained by diagonalizing the interaction VI of 4b with the manifold 41. The width of this distribution is rf.The weighted spectrum I V.6 1 21cbl12 is referred to as the Interaction Density Spectrum (36)for the state 4.. It completely determines the right half of Figure 3 and can therefore renlace that oortion of our basic model. i t is imporiant to establish the rate constants for the transitions from &. for the soecial case in which AA = 0. It is easy to show the fillowing ocder of inequalities (by starting with
1vabls ri)
w *IVoa[ * r, rf
(14)
The upper inequalities hold for the oscillatory limit, where the frequency of oscillation is h-' I V.b I. The lower inequalities hold for the dissipative limit, where the decay constant
The probability amplitude of @, is then given by I t turns out that, for t < 2ahA-I, eqn. (12) is identically an exponential function (28). Thus, the probability of 4, is pot = I ( $ . I q ( t ) ) l 2 = e - r t " ( t
and results in the excited molecule oscillating between the allowed vihronic state #. and the forbidden vihronic manifold 4;. Relaxation results from (1)radiativedecay into 6, and (2) time-dependent phase changes in the #I leading to irreversibility regarding 4.. For the large molecule (prototype) limit I Vb. I < Pi and the molecule undergoes competitive decay kinetics into 4, and &. In manv cases the sDectrum of interaction I, Vn,I -.. Le.. . the ~nteractionDensity Spectrum) has considerahle resolved structure corresponding to real small (62) and large (63) molecule limits. The associated absorption spectra have
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rf
586 1 Journal of Chemical Education
considerahle resolvable structure and it is possible to selectively excite individual resonances, rather than the vihronic state &, resulting in a completely different kind of description of excited state dynamics (33,361. (2) A hydrogen atom in a n electric field. Consider the ns and n p orbitals of an H atom in a constant, uniform electric field applied along the z-axis. Since the atomic orbitals are degenerate (zero field), this corres~ondsto the first-order ~ t h effect k and the resulting eigen&ites in t h prrwnce ~ of the tield are sp directed a l m -r the z-axis, as shown in . hvhrids . Figure 6. The basic theoretical model (Fig. 3) applies faithfully to this system. State 4. is the np orbital which is electric-dipole allowed from theground state. States&c are the radiative decay channels to lower-lyings and d orbitals (decay constant, l',). State 4 b is the ns orbital which is electric-dipole allowed with respect to p orbitals. Consequently, the manifold #/ corresponds to radiative decay channels to lower-lying p orbitals (decay constant, Pi). The interaction Vb, is due to interaction with the external E field. A number of interesting thought experiments can he carried out with this system. (a) Suppose the atom, initially in its ground state, is excited by a light pulse, of duration 7, such l';', and l'il. Under these condidons the that ~ / < h I Vob I pulse prepares the molecule in the np orbital, which in the presence of E is not a stationary state. The excited atom tends to oscillate between n p and ns while irradiating from each. Of course, the condition for oscillation is I V,b I > r,, rP The net quantum yields for emission from n p and ns depends on the relative value of these 3 interaction parameters. This is a prototype level-crossing experiment. (b) Suppose I V,bl >> F, and Ff,which is the oscillatory limit. To a very good approximation it is possible to selectively excite oneof the hybrid eigenstates by using a narrow-hand pulse on resonance, for example, a pulse centered on E+ and of duration h ~ - 1
