J. Phys. Chem. 1983, 87,30-40
30
FEATURE ARTICLE Nonradiative Relaxation and Quantum Beats in the Radiative Decay Dynamics of Large Molecules William Rhodes Dspadment of Chemistry and Institute of Molecular Biophysics, Florida State University, Tallahassee, Florida 32306 (Received: June 2, 1982; In Final Form: September 28, 1982)
The theory of photon emission is discussed for the case in which a molecule is excited by a coherent light pulse of frequency width Au corresponding to a few wavenumbers, or less, and bracketing a number of excited molecular eigenstates. Features of the theory which are considered include (1) conditions whereby the time evolution of a frequency-resolvedemission spectrum may be measured; (2) the nature of quantum beats and the importance of distinguishing quantum beats and oscillations due to quantum mechanical memory effects; (3) the importance of molecular emission channels and their doorway states for radiative decay as a practical (as well as conceptual) basis for understanding excited state dynamics; and (4)the effects of adiabatic modulation of the spectrally active modes of a molecule by the background modes of the molecule and its medium. The role of adiabatic modulation is the most important and far-reaching aspect of the present theory. We consider, in particular, the stochastic modulation (SM) limit of adiabatic modulation which is defined under certain conditions of fast modulation by weak coupling to many background modes. In addition to its effects on kinetic stabilization of quantum transitions and on the tuning of kinetic pathways of molecular processes, SM can erase the undesirable oscillations in photon emission, thereby permitting the emergence of quantum beats in the absence of frequency-resolvable spectral structure. Molecular spectroscopy has undergone revolutionary change during the past 15 years as a result of the development of laser technology and of special techniques, such as the supersonic jet method of preparing molecules under low-temperature, nearly isolated conditions. It is now possible to study processes under highly selective conditions in both the frequency domain (e.g., sub-Doppler spectroscopy') and the time domain (picosecond pulse excitation2). There has been various concommitant lines of theoretical development, many of which focus on the formulation of appropriate molecular state coupling schemes and their associated dynamic^.^ Our theoretical approach in the past has emphasized the role of both the exciting light characteristics and the structure of the molecular absorption spectra in determining the nature of the prepared excited state and its relaxation dynamic^.^ In this paper, however, we develop a different view of the same processes by considering the time evolution of the radiative emission spectrum of a molecule which has been prepared in an arbitrary (but definite) excited state, I$(O)), at t = 0. This involves both radiative and nonradiative relaxation. Our focus, therefore, will be on the rate of photon emission, P(w,t),as a function of both frequency w and time t. In considering the t and w dependence together, we must be careful in treating short-time oscillations that are not
of interest to us here. For example, in a simple atomic line emission there are (low-magnitude) oscillations in P for frequencies lying outside the radiative line width. These can be eliminated by integrating the frequency about w over an interval Aw, x 27t-l. For certain conditions on molecular systems, to be discussed later, we will find that this is unnecessary. Because of radiative decay, P always dies off in the large t limit. The details of the t dependence, however, depend on the structure of I$(O) ), which is usually a superposition of excited molecular eigenstates having definite relative phases. (Thus, we use the term coherent superposition.) For eigenstates of different energies these relative phases are t dependent and can make large nonradiative contributions to P in the form of enhanced decay rates and oscillations. Such oscillations are called quantum beats and are currently of great experimental interest. McDonald et al.5 have studied quantum beats in the microsecond range in various dicarbonyls and Zewail et a1.6 have observed them for anthracene in the nanosecond range. They have also been found recently in other molecule^.^ One of our main purposes here is to consider the conditions required for the appearance of quantum beats both in isolated molecules and in mo!ecules coupled to a background medium. In some cases P(t) depends on the history of the molecule between times zero and t. Such memory
(1) Shimoda, K., Ed. "High-Resolution Laser Spectroscopy"; Springer: New York, 1976; Top. Appl. Phys., Vol. 13. (2) Shapiro, S. L., Ed. 'Ultrashort Light Pulses"; Springer: New York, 1977; Top. Appl. Phys., Vol. 18. (3) (a) Siebrand, W. In 'Dynamics of Molecular Collisions", Miller, W. H., Ed.; Plenum: New York, 1976; Part A. (b) Mukamel, S.; Jortner, J. In "The World of Quantum Chemistry"; Daudel, R.; Pullman, B., Ed.; Reidel: Dordecht, 1974. (4) Rhodes, W. In 'Radiationless Transitions"; Lin, S. H., Ed.; Academis Press: New York, 1980.
(5) (a) Chaiken, J.; Gurnick, M.; McDonald, J. D. J. Chem. Phys. 1981, 74, 106. (b) Ibid. 1981, 74, 117. (c) Ibid. 1981, 74, 123. (6) Lambert, W. R.; Felker, P. M.; Zewail, A. H. J . Chem. Phys. 1981, 75, 5958. (7) (a) Sharfin, W.; Ivanco, M.; Wallace, S. C. J. Chem. Phys. 1982, 76, 2095. (b) Okajima, H.; Saigusa, H.; Lim, E. C. Ibid. 1982, 76, 2096. (c) van der Meer, B. J.; Jonkman, H. T.; ter Horst, G. M.; Kommandeur, Ibid. 1982, 76,2099. (d) Henke, W.; Selzle, H. L.; Hays, T. R.; Lin, S. H.; Schlag, E. W. Chem. Phys. Lett. 1981, 77,448. (e) Schadee, R.; Nonhof, C.; Schmidt, J.; Van der Waals, J. Mol. Phys. 1977, 34, 171. (0 Klein, J.; Voltz, R. Phys. Reu. Lett. 1976, 36, 1214.
0022-3654/83/2087-0030$0 1.50/0
0 1983 American
Chemical Society
The Journal of Physical Chemistty, Vol. 87, No. 1, 1983 31
Feature Article
schematic picture indicating that the density of states increases with energy due to the increasing complexity of electronic, vibrational, and rotational motion. Fortunately, near the ground state electronic motion tends to follow that of the nuclei adiabatically (i.e., without electronic transitions). Thus, to a good approximation the Born-Oppenheimer (BO) approximation is valid, whereby each state, 18) is a product of the ground electronic state with vibrational and rotational states. This is not the case a t higher energies (visible an UV regions) where there is such a high congestion that even a weak coupling of BO 19) states produces large admixtures. There, each molecular eigenstate, Is), is a superposition of BO states of different electronic, vibrational, rotational, and spin multiplicity Flgure 1. Schematlc representation of the molecular eigenstate energy parentage. Of course, the BO states may still provide a levels of a molecule in which the relatively dense region of states Is) good basis for the description of various mechanisms. is in the range normally attributed to electron excitations and the As a reference basis we will use the set of molecular relatively sparse region of states Ig) is in the range normally attributed eigenstates in the upper manifold and the BO states (not to the ground electronic state. The molecule is excited over the necessarily pure spin) in the region of the ground state. frequency interval Aw from which the mean transition frequency to 19) is og. Each upper state, Is), has a radiative line width, ys, due to transitions to the lower states, Ig). We neglect direct effects tend to obscure quantum beats, so resolution of the radiative transitions between different Is) states and belatter depends on the effective erasing of this memory tween different 18) states. Further, we will use the either by the conditions of the measurement or by the Weisskopf-Wigner (WW) approximation which neglects interaction of the molecule with its medium. the contributions to probability amplitudes due to photon The role played by the interaction of a molecule with emission associated with Is) Jg)followed by reabsorption background modes (such as a solvent or other medium) of the same photon associated with 18) Is’), where Is’) in governing the dynamics of processes within the molecule is has recently emerged as a major theme of our r e s e a r ~ h . ~ , ~ different from IS).The validity of the WW approximation depends on the frequency difference w ~between , In particular, we are interested in interactions which do Is) and Is’) being larger than the sum of the contributions not change the state of the molecule (i.e., induce transito ys and ya.arising from those radiative transitions having tions) but, instead, effectively change the phase relations a common set of lg) states. One may wonder how this is among the states of the molecule and thereby alter its possible for a large molecule for which the density of states dynamics. We call this adiabatic modulationg of the Is) may be as large as 1012/cm-’. However, for large molecule (subsystem) by the medium or bath (subsystem). molecules there tends to be a “dilution effect” due to the Actually, this notion applies to any situation in which there combined results of (1)the increased density of lg) states is a spectrally active set of modes (chromophore) coupled along with a tendency for different Is) to decay to different to a set of spectrally inactive modes, such as an aromatic 18) states and (2) the possibility that, because of the ingroup coupled to alkyl side chains. creased number of Is), the value of each ys is ~ma1ler.l~ Another major aspect of this discussion, therefore, will The former effect occurs, for example, when the states deal with the description of adiabatic modulation effects Is) (in a BO basis) contain superpositions of more than one in terms that are conceptually simple and pictorial in excited electronic state, each of which is radiatively coupled nature. to the ground electronic state. The excitation of naphAs with much of our earlier work, it turns out to be thalene in the region of the second excited singlet state, convenient and instructive to use a doorway state apS2, provides a concrete example. Here, each Is) contains p r ~ a c h . ~ JThe ~ ’ ~doorway state from any given state 14) a superposition of S2 (having low vibrational energy) and under an interaction V is defined as the first excited singlet state, S1 (having high vibrational Id) = VI#) (41v%)-1’2 (1) energy). For the isolated molecule, fluorescence tends to occur from the vibrationally hot SI components to various which is simply the (normalized) state which carried all vibrationally hot ground states lg) . of the interaction strength from 14). In practice we are The latter dilution effect occurs, for example, when Is) interested in the doorway state from 14) into a subspace contains a superposition of electronic states, only one of of the spectrum spanned by molecular eigenstates bewhich can decay radiatively. This is the case for a molecule longing to a certain energy range. The doorway state in which Is) lies in the SI region, so that Is) contains S1, approach is especially useful when the various doorway To(lowest triplet), and So (ground state) character. Thus, states can be identified with familiar molecular basis states ys arises only from the S1 component of Is) and the state such as vibronic states. This will be the case for most of S1 is distributed among a very large number of Is) states. the following discussion. In general, the value of ys is the sum of contributions Excited State Relaxation of a n Isolated Molecule ysgdue to radiative decay to the various Ig). On the other A polyatomic molecule has many degrees of freedom hand, the total of all radiative transitions to a given (8) associated with the motion of its electrons and nuclei. is referred to as a radiative decay channel. Accordingly, the density of quantum states tends to be Now, let us assume that, at t = 0, the molecule is premuch larger in any given region of the spectrum than for pared in the state an atom or a diatomic molecule. Figure 1 is a highly
=F
-
(8) Rhodes, W. J . Chem. Phys. 1981, 75, 2588. (9) Rhodes, W. J. Phys. Chem. 1982,86, 2657. (IO) Ziv, A,; Rhodes, W. J . Chem. Phys. 1976, 65, 4895. (11) Cable, R.; Rhodes, W. J . Chem. Phys. 1980, 73, 4736. (12) Rhodes, W. Chem. Phys. 1977, 22, 95.
-
(13) (a) Rhodes, W. J . Chem. Phys. 1969,50, 2885. (b) Rhodes, W., J . Chim. Phys. (Suppl. issue) 1970, 40.
32
Rhodes
The Journal of Physical Chetnisfry, Vol. 87, No. 1, 1983
all other states Is’), and (c) the off-resonance memory oscillations due to transitions from all other Is’) into 18) as well as into other channels. Now, the memory oscillaleaving the tions (type c) tend to die off for t >> 27r~-~,,,, more interesting q-beats (type b) intact. However, simple analysis will show that it is not practical in most cases (at present) to look for q-beats at w values corresponding to individual eigenstates w,,. There are two reasons for this. First, the spacings of effective states producing beats that are currently observable are less than 1cm-l. This presents a spectral resolution problem. Second, because of the decay of the (radiative) excited state, it is usually necessary to measure the beats within the first few periods, which means t 2~w-~,, or less. This brings in the uninteresting memory oscillation terms. One way around this problem is to integrate eq 5 over w , giving the total t-dependent emission intensity. This produces the function 27r 6(tl - t ) in the integrand of eq 5, thereby removing all memory effects (including the undesirable oscillations) and leaving the coherent q-beats intact. On the other hand, we are interested in the ~+,(t))= e-’7fo’l+(o))= CIS)Cse-i(u~-L(~~~2)f (3) emission for each spectrally resolvable channel Ig), so we 9 defeat this purpose by integrating over all w. As is often which is being damped due to radiative decay into the the case, the answer lies in compromise. We shall simply ground manifold lg). Henceforth, energy is expressed in integrate w over a range Aw, which is large enough to units of frequency, so h E 1. remove the “bad” oscillations but is small enough to What is the nature of the emission spectrum from maintain identity of the channe!. Suppose, for example, ( t )) ? The probability amplitude for the electric-dipole that we want to start measuring P at a time corresponding emission of a photon of frequency w via channel 18) is to 1110 the period of the highest frequency beat composimply nent. The maximum possible beat frequency is Aw, the spread of excited eigenstates. Thus, a reasonable choice A(g,w,t) = -ia(w) dt, (glp~+e(tl))ei(w+w~)tl (4) is Au, = lOAw. Now, Aw, is a frequency range which brackets symmetrically the average transition frequency, where w, is the frequency (energy) of Ig), p is the electric LJ,,for transitions into channel lg) (Figure 1). dipole operator, and a ( w ) is a frequency-dependent factor Integrating eq 5 over Aw, gives, to good approximation accounting for the degeneracy and polarization of photons w. By taking the time derivative of the absolute value squared of A , it follows that the total rate of photon The understanding is that we will not consider t < 2aAw;’ emission for channel J g ) is and we are neglecting the (weak) memory effects within the interval 2aAw;l prior to t which were not washed out P(g,w,t) = 2 Re la(w)12Jtdtl e - i ( w + w g ) ( t - t l ) ( glpp(t,,t)plg) by the local w integration. If A u is of order 1 cm-’, this 0 critical time interval is only -3 ps. For that price we are (5) able to resolve all channel spacings >10 cm-’. In practice this means that we do not attempt to resolve the emission Here, Re refers to the real part and p ( t l , t ) is shorthand for spectrum with better than 10-cm-’ resolution. I+,(tl)) ( + e ( t ) l . This is a generalization of the density operator form of expressing the pure state I+e). The adEquation 6 contains all of the propensity for quantum vantage of using p is that it permits a later extension to beats. It is important that such beats are related simply to the oscillatory t dependence of the appropriate diagonal a statistical ensemble of molecules for a distribution of excited states Thus, along with the beauty of its element of p ( t ) . This remarkable feature results from the symmetry and simplicity, eq 5 is quite general in form. time constriction of the tl amplitudes relative to the t There are two very important physical aspects of eq 5: amplitudes caused by local frequency integration. We will see later that this kind of time constriction in the am(1)The existence of a definite phase relation among the plitudes (memory erasure) can be caused by the naturally molecular eigenstates contained in leads to interferoccurring (intrinsic) interactions within a complex moences between photon emission amplitudes from different eigenstates 1s) and Is’). Each pair of such states contriblecular system. utes a term to P which oscillates with frequency wge,= Iwg The conditions necessary for quantum beats has been - wS$ Quantum beats (q-beats), as well as other nonraa subject of confusion in the literature. Freed and Nitzan14 diative transition manifetations, arise from the pairwise have emphasized the point that frequency integration is additive contributions of these terms. (2) The integration necessary. A similar conclusion was implied in one of our on t , provides a memory effect whereby amplitudes in p earlier papers.13b This turns out to not be the case, howfor times prior to t contribute to P. This amounts to ever, since q-beats are an intrinsic manifestation of exunequal-time interferences of emission amplitudes and istence of a definite phase relation (coherence) among produces oscillations in P associated with the emission excited states. To be sure, the detailed nature of the probability from each Is). These memory related oscillaq-beats can depend strongly on w and they can be entantions depend strongly on Jw - w,,l for the various Is). gled with memory oscillations, but they are indeed present To see these effects more clearly, let w lie within the line at each w (if at all). In summary, the important points are width ys of transition frequency wSg. The contributions (a) for any given channel, Ig), q-beats can be “tuned in” to P consist of (a) a strong on-resonance component, (b) strong quantum beats due to coherence between 1s) and (14) Freed, K. F.; Nitzan, A. J. Chem. Phys. 1980, 73, 4765. where Aw is a small frequency interval which typically is of order 1 cm-’, or less. This value would correspond to a 30-ps coherent pulse excitation, for example. The precise details of the light pulse re its shape and exactly when it passes the molecule are not important here since we do not ask questions about the behavior of the molecule until well after the pulse has passed. Also, the use of a fully excited state means that any coherence relations with the ground state resulting from the excitation process are ignored. These details could have been included in the linear excitation limit by using a convolution relation. The important thing is that energy has been deposited in a relatively narrow region of the molecular spectrum and we assume (for present discussion) that the molecular eigenstates have definite phase relations. Our problem is to determine the conditions which govern the time dependence of the emission intensity into the various channels Ig), as well as the nonradiative channeling of the energy. For t > 0, the excited state of the molecule is
s,‘
-
The Journal of Physical Chemistty, Vol. 87, No. 1, 1983 33
Feature Article
R ,cs,2 Figure 2. The frequency region, Aw, over which the molecule is excited, showing the profile of the probability distribution, lC81z,of excited molecular eigenstates Is).
more clearly by w integration; (b) such tuning for each channel requires only a local o integration in the neighborhood of a,; and (c) for the total integrated emission intensity into all channels, it is possible that beats for different channels can (at least partially) cancel each other. The latter point will be discussed further below. Before considering some of the practical ways of developing eq 6, it is instructive to examine its structure in terms of the molecular eigenstate basis, Is). By expressing p ( t ) in this basis we obtain Aw
P(g,a,,t) = z:Ys~8,g(SIP(t)lS’)
(7)
L ICSl2
t Figure 3. Upper left. Absorption spectrum containing two molecular resonances, each composed of many (unresolved) molecular eigenstates (not shown). Upper right. Coupling scheme for the associated molecular resonance states which are coupled radiatively to ground state 19). Lower. The timedependent fluorescence intensity resulting from coherent pulse excitation of the two resonances.
88‘
where yst8,is the coherent radiative damping matrix, de(glpls). In termined ky eq 6 to be ySts,,= 2~la(a,)(~(s’lplg) this special basis the existence of q-beats depends on the nondiagonal (coherence) terms in p. For the case that p(t) is a pure state corresponding to eq 3, we have the simple form P(g,a,,t) =
Aw Cy8,8,g~a~*s,e-i[w~’-i(y,+y,’)/21t (8) 88‘
The diagonal terrms represent the first-order radiative decay rates from each state Is) and the nondiagonal terms produce t-dependent interferences. The variety of possibilities for the dynamics of P arising from a band of states Aw 1 cm-’ is enormous when one considers the complexity of spectral structure available in polyatomic molecules. Following is a summary of types of behavior possible. (1)Each excited state has a different channel (g) (de: termined by the structure of ysts,). For each channel P is a single exponential decay, whie the full w-integrated emission is the sum of exponential decays. (2) Several states Is) couple to the same channel 18) and the distribution of with respect.to w8 has no definite pattern. This is the condition for P to contain a q-beat pattern. (3) Many states Is) are coupled to the same channel 18) and the profile of the distribution IC8I2is a Lorentzian of width rn. If the initial state, (#&O)), were prepared by pulse excitation from (g),then the phase relations among the C,is such that the interference among the Is) leads to nonradiative exponential decay of P (with rate constant I?,) rather than q-beats. As far as channel lg) is concerned this corresponds to a nonradiative channeling of energy within the excited state manifold of the molecule. Such nonradiative decay occurs for values of t less than the average density of states Is). For larger t, there is a possibility for recurrences in P but these will be ignored here. (4) A more general case is given by a composite of the previous two, whereby the distribution of (C812is a set of distinct Lorentzian profiles, as shown in Figure 2. Each Lorentzian contains many states Is). Again, we suppose
-
that I#,(O)) has been prepared by pulse excitation from lg), so that ICJ2 gives the structure of the absorption spectrum from lg). The resulting P(g,o,,t) is oscillatory decay describing q-beats along with competitive radiative and nonradiative relaxation. The .nonradiative contribution to the overall relaxation of P is determined by the widths of the individual Lorentzian profiles. Each such Lorentzian manifold defines a molecular resonance involving Ig). We propose that this is the typical situation for polyatomic molecule emission containing q-beats; namely, q-beats are associated primarily with (interferences between) molecular resonances containing many eigenstates Is), rather than with individual eigenstates. The excited state associated with a molecular resonance involving Ig) may thus be defined as A% Ian)
= C CJs) s
(9)
where Ao, is the interval containing the (approximate) Lorentzian distribution As a simple example of the application of eq 6-9, consider the case that a molecule is excited by a coherent pulse of width Aw, (centered a t the molecule at t = 0) which brackets two distinct molecular resonances separated by Aw2’, as is shown in Figure 3. If Aw, >> Awzl, the state prepared by such a pulse is a coherent superposition, I#@)) = allal) + a21a2). Each molecular resonance state is a nonstationary superposition of eigenstates (eq 9) which decays radiatively to (g) and nonradiatively into its associated manifold of molecular states (Figure 3). The superposition of the two radiative amplitudes to (8) undergoes a time-dependent pattern of constructive and destructive interference which (in this case) is sinusoidal with frequency Amzl. The resulting quantum beat pattern of P is a simple oscillatory decay. Obviously, if more than two resonances are conherently excited, the beat pattern becomes more complicated. Doorway State Formulation Thus far, we have been interested in the most elementary description of the principles governing the t-de-
34
The Journal of Physical Chemistry, Vol. 87, No. 1, 1983
Rhodes
pendent relaxation of a prepared excited state and for this purpose the molecular eigenstate basis Is) is ideal. In practice, however, we often know very little about the structure of the individual Is) or of the molecular resonance states (a,) in any given region Aw. Therefore, in order to develop a theory with productive as well as descriptive potential, it is necessary to consider other bases as well. On such basis consists of the radiative doorway states from the ground manifold {lg))[cf. eq 11. For each state 18) these are defined by4J&12 Irg) E
P k ) (gIP2lg)-li2
(10)
so Ir,) is the (normalized) state which carries all of the electric dipole strength from Jg). The radiative (doorway) state ITg) determines the structure of the entire (electricdipole) absorption spectrum from Ig); i.e., in the expansion Ir,) = Cd,,ls) the probability distribution ldSgl2gives the structure of the absorption spectrum. Using eq 10, we obtain
P(g,a,,t) = Yg(ag)(rgMt)Irg)
(11)
-
where yg(ag)= is the effective radiative decay constant for the transition (rg) (8) defined as though it occurs at frequency a,. Thus, it turns out that the t dependence of the emission intensity for channel 18) is simply given by the t dependence of the probability of the doorway state from Ig). Remember that for a pure excited state the latter is given by l(rgl+e(t))12.If l+,(t)) is localized within a small region Aw 1 cm-', then (rglp(t)lrg) is a very small number, indeed, but it is compensated by the large ygwhich is on the order of the total oscillator strength of the molecule. In reducing P to the simple form of eq 11, it may seem that we are paying too great a price in practicability since Ir,) contains amplitudes for the entire absorption spectrum of the molecule and is thus generally not determined completely. However, the fact that p(t) is taken to be localized in a (small) subspace spanned by the eigenstates of the molecule lying in the range Aw provides us with a great deal of flexibility. Instead of using the full lrg), it is possible (without loss of vigor) to use the projection of Ir,) onto any subspace containing p. This includes all subspaces which bracket Aw. Let P denote the projection operator for such a subspace. Then p(t) = Pp(t)P and it is easy to see that (rglp(t)lrg) (r,lrgP)(rgPlp(t)lrpP) (r where IrgP) is the normalized projection of ITg) by P. how, eq 11 is replaced by the equivalent (but more practical) form
-
P(g,ag,t)= ygp(a,) (r,Plp(t)lr,P)
(12)
where y g P ( ~ , )= yg(Og) I ( rgPlrg)l2
-
is the radiative decay constant for the transition IrgP) Jg) defined at 0,. Note that ygP I yg. Equations 11 and 1 2 demonstrate an important equivalence relation, whereby the structure of the t dependence of the probability of all projected doorway states is the same (for a given 18) and for projections spanning Aw). Going from one lrgP) to another merely shifts a weight factor from the p term to the y term. A practical choice of P depends on the molecule and the region of excitation. Suppose that, for example, we have excited a large molecule like anthracene in the middle of a vibronic band of a low-lying allowed electronic transition with Aw 1-2 cm-'. Now, h w may contain many molecular eigenstates whose vibronic structure arises from electronic mixing (associated with interaction such as vi-
-
bronic coupling, Born-Oppenheimer breakdown, spinorbital coupling, etc.) or from anharmonic mixing of normal modes (associated with one parent electronic state). For such a case it would not be practical to let P span only the region Aw because the structure of the states contained therein is unknown. It would be better to let 'i3 span a subspace for which we can make an association between spectral structure and molecular structural dynamics, Then IrgP) can be identified with known molecular states. As an example, let us assume that the ground manifold 118))is sufficiently isolated that the BO approximation is valid. Then we have (8) = leo)lxog),a product of the ground electronic state leo) and various vibrational states (xog).We further assume that leo) is independent of all vibrational coordinates (Condon approximation). This means that the full doorway state from (each) Jg) is simply vibronic with the structure
1%)
= Ir0)lxog)
(13)
where Ira) is the electronic doorway state from leo). Because of the orthogonality of the different l ~ ~ , ) these , doorway states form an orthogonal set. Since the vibrational state IxoB)is the same for each J g ) and Jrg),we say that Ixog)is a self-doorway state for electric dipole transitions from Ig). The only requirement is that k) be a BO state in the Condon approximation! Thus, for such a molecule the doorway states from the various channel states Ig) have a common electronic doorway state Ir,) and are orthogonal by virtue of their self-doorway vibrational components Ixog). Insofar as eq 13 is valid it is desirable to expand the prepared excited state in terms of an electronic basis which either contains Ira) or can be simply related to it. Such an expansion has the general (and exact) form I$(t)) = Clei(Q))xj(Q,t)C,(t)
(14)
L
where lei(&)) denotes any dependence of the electronic basis on nuclear coordinates, Q. The (vibrational) wave functions xi(Q,t) are normalized but are not necessarily mutually orthogonal, the normalization being determined by the coefficients Ci(t). Equation 14 describes the time-dependent correlation of the motion of the electrons and the nuclei in a remarkably simple way, whereby lCi(t)I2 is the probability that the molecule is in electronic state lei(Q))and lxi(Q,t)I2is a wavepacket for the motion of the nuclei on an electronic energy surface for state lei(Q)). Thus, all aspects of the motion are contained therein, including vibronic coupling, vibrational energy redistribution, spin-orbital coupling effects, etc. Let us now consider the structure of eq 14 for several key phenomena. ( I ) Vibronic Stealing of Intensity. This involves the Q dependence of lei) which can be expanded in terms of Q to give lei(Q)) = lei(o)) f Iro)briQ1 + ... where we have tacitly assumed that lei(0))does not contain any of the electronic doorway state Ira) and Qi is a normal mode which produces mixing of Ira) (i.e., bil # 0). There may be more than one such mode. It follows from eq 11 that P(gtag,t)=
rg(ag)I(XogIQiIX(t) )B(t)12
(15)
where Ix(t))B(t) Cilxi(t)bil. This is the most general expression for vibronic induction of emission intensity. The effective excited vibrational state Ix(t))B(t)may undergo complicated time dependence. A typical, simpler
The Journal of Physical Chemistry, Vol. 87, No. 1, 1983 35
Feature Article
I
I
\
Flgure 4. Energy level diagrams for vibronic intensity stealing. (a) Actual radiative emission from the prepared vibronic state IedQ)) (b) The same process described in terms of vibronic coupling via the electronic doorway state Ira), defined for Q , = 0.
I
/
Ix,,).
situation, however, involves the preparation of a single whereby vibronic BO state lei(Q))lxiv),
corresponding to exponential decay, since lCi(t)I2is presumed to contain only radiative decay effects. A good example is the excitation of a single vibronic state of the B,, electronic state of benzene (-2600 A). The doorway state from the ground Alg state has El, symmetry. The lowest-lying electronic state of this symmetry is 1800 A, so evaluations of eq 16 for benzene usually involve calculating yg and bil with only this state. Figure 4 shows the dynamics of vibronic coupling as it would apply to the benzene case. The actual two-state process is given in Figure 4a, emphasizing the fact that it is really a simple radiative relaxation. However, Figure 4b shows how the same process may be viewed mechanistically as a “coherent tunneling” through the doorway state due to vibrational motion. (2) Intramolecular Vibrational Redistribution (IVR). For this case this simplest structure of eq 14 is
-
I$(t)) = lei)xi(Q,t)Ci(t)
(17)
which describes a vibrational wavepacket moving on the electronic energy surface for lei). Henceforth, any Q dependence of the electronic states will be neglected. I t is natural to let the subspace P contain only lei) as far as the electronic subsystem is concerned. Thus, we have
-
where ygis defined for the electronic transition lei) leo) as though it occurred at frequency a,. Again, lCi(t)I2describes the radiative relaxation of lei). The interesting dynamics for our present studies is contained in the time-dependent (dynamic) Franck-Condon (FC) factors I(XOplXi(t))12, describing the overlap of the self-doorway vibrational state Ixog)with the moving wavepacket xi(t). For a bound state these FC factors will be (quasi) periodic, although for a large molecule having many modes the path of xi(t) may be very complicated. A schematic representation of the essential idea is given in Figure 5. The experimental observation of quantum beats in anthracene by Zewail et a1.6 may be interpreted in terms of such vibrational motion on the electronic energy surface of the lowest excited singlet state. Although eq 18 is quite general regarding the effects of vibrational motion on an excited electronic energy surface,
Figure 5. The dynamics of photon emission into the ground vibrational channel Ixoo)for the case that the excited state is a nonstationary vibrational wavepacket, x#), moving on the electronic surface w,. The conformation region defined by the stationary Ixoo)is like a “hole” in the w, surface into which the itinerant packet wanders.
it is applicable in particular to the redistribution of vibrational energy among normal modes.15 For example, ~ ~ (could 0 ) be one of the normal-mode eigenstates, (xi,,), which is nonstationary because of anharmonic coupling on the lei) surface. The resulting anharmonic potential causes Ixi,) to undergo transitions among the various harmonic eigenstates. The motion of xi(t) contains all of the information concerning this redistribution which could be extracted by expanding xi(t) in the harmonic basis of various combination and overtone states. The above description is, in fact, what is meant by the term IVR. It should be pointed out here that an approach similar to the one in this section has been developed extensively by Heller,16who has emphasized a time-dependent formulation of molecular spectroscopic processes in terms of vibrational wavepacket motion. While Heller’s approach is similar in spirit to ours, his focuses and objectives are quite different (as will be shown in the following sections); so, it seems quite appropriate to say that his work is complementary to ours. (3) Electronic Relaxation. This involves transitions between electronic energy surfaces due to such interactions as spin-orbital coupling and BO breakdown. Insofar as the ground state leo) is pure singlet (to good approximation), then so is its doorway state Ir,). The prepared excited state can then be expressed accurately as
where If) is a convenient basis orthogonal to Ira). Again, our problem is to choose the optimum subspace within which lro) is the projected doorway state. This choice is a matter of practicability and experience. For example, if the excitation range, Aw, lies within one of the low-lying, allowed electronic absorption bands of a molecule such as anthracene and if Ixog)is a low-lying channel state, it is natural to define Ir,) as the allowed (singlet) electronic state lying in the neighborhood of Am. The states If) then become the various singlet and triplet states to which Ira) is coupled by intramolecular interaction. The coefficients C, and Cf depend on t because of transitions among the excited states as well as radiative decay. (15) Covaleskie, R. A.; Colson, D. A.; Parmenter, C. S. J. Chem. Phys. 1980, 72, 5774. (16) Heller, E. J. Acc. Chem. Res. 1981, 14, 368.
36
The Journal of Physical Chemistty, Vol. 87, No. 1, 1983
Rhodes
Now, the emission spectrum becomes simply Rg,ag,t) = TroI (XogIXr(t))121cr(t)Iz
(20)
the same form as eq 18. The difference is that lC,(t)I2may oscillate or relax due to nonradiative transitions into the manifold If). But, this is the feature which distinguishes the IVR and electronic relaxation mechanism of q-beats. For the former, q-beats arise from oscillations in the FC factor. The quantum beats observed in dicarbonyls by McDonald et al.5 are interpreted in terms of singlet-triplet electronic state coupling. How do we experimentally distinguish quantum beats due to electronic relaxation from those due to IVR? One way is to sum the emission over all channels associated with the ground electronic state. In eq 18 and 20 the sum over all FC factors gives unity, so we have
P ( t ) = rrolcr(t)12
(21)
showing that the total intensity summed over all ground vibronic channels is proportional to lCr(t)I2,the probability of the electronic doorway state. Thus, q-beats in the "frequency integrated" emission into the ground electronic manifold are the result of excited electronic state oscillations! This depends, of course, on neglecting any Q dependence of the electronic states (Condon approximation). It is interesting that the summation over channels can produce a "washing out" of q-beats in the frequency-integrated spectrum.6
Adiabatic Modulation In many of the states of a molecule the motion of the nuclei tends to be much slower than that of the electrons. This causes a "one-sided" relationship in the sense that the electrons can follow the motions of the nuclei but not vice versa. This means that the motion of the nuclei tends to not induce transitions between electronic states. We say that the electrons are adiabatic with respect to the nuclei or, equivalently, that the nuclei adiabatically modulate the motion of the electrons. Such behavior is the rationale for using the Born-Oppenheimer (BO) approximation. the procedure is to formulate the electronic Schrodinger equation He(Q)lei(Q)) = oi(Q)lei(Q)) (22) for each configuration Q of the nuclei. For each electronic state lei(@) there is an electronic energy surface, w i ( Q ) , which is then used as the potential for the motion of the nuclei. Thus, each electronic surface provides a unique Hamiltonian for the nuclei, having eigenvectors Ixi,). The BO approximation consists of neglecting the coupling between different electronic states by the nuclear momentum and kinetic energy operators. This is valid insofar as the motion of the electrons is adiabatic. Regardless of the extent of validity of the BO approximation, however, it is very useful to use an adiabatic formulation. This can be accomplished by expressing the Hamiltonian of the molecule in the form %, = H, + V , in which V , contains all of the BO breakdown terms (together with possibly other intramolecular interactions) and H,,, is the adiabatic part, having the Schrodinger equation Hmlei)lxiv) = Eivlei)lXiv) (23) We assume that any effects due to the Q dependence of lei) can be accounted for by appropriate terms in V,, so
>
Q
Flgure 6. The effective potential difference between electronic energy surfaces w, and wo, referred to as the sudden potential, Y . This determines the change in force on the nuclei when the molecule switches surfaces.
all lei) will be taken to be independent of Q. The basic idea of adiabatic modulation is that transitions between electronic energy surfaces which are driven by V,,, and any external potentials such as radiative coupling, V,, become modulated by the motion of the nuclei on the relevant energy surfaces. Consider, for example, the electronic energy surfaces for leo)and its radiative doorway state Ira), depicted in Figure 6. If the molecule switches energy surfaces at t , due to the interaction V (i.e., Vr or Vm), then the nuclei undergo a sudden change in force which affects the motion of the vibrational wave function. This force change is due to the effective potential difference between the two surfaces, which we term the sudden V . We may obtain Yab(Q)for all pairs of p0tential,8>~ energy surfaces by using a reference value of Q, for example, the minimum of wo, and displacing the other surface vertically, as shown in Figure 6. The details of the associated components of the Hamiltonian are given else-
here.^^^ Suppose that at t = 0 the molecule is in the state le,)x,(O), where x,(O) may be a vibrational eigenstate of surface or. As the molecule evolves under the full H , the effects of V are to cause transitions between the surfaces which, in turn, generate a "churning" effect,on the nuclear motion. The result is the evolution of a correlated state of the form in eq 19 (with an leo)term added). The extent to which the correlated vibrational components, xi(t), become different depends on the ability of Y to drive vibrational transitions and this depends partly on the closeness of the spacings of the vibrational energy levels. Whereas electronic motion tends to be adiabatic with respect to nuclear motion, the latter tends to be nonadiabatic with respect to electronic motion. The nuclei do not follow electronic transitions smoothly. To see the effects of adiabatic modulation on spectroscopic properties, let us return to the full frequency-dependent formula, eq 5, which contains memory effects. For the moment, we are interested in the total emission into all channels. The summation over g involves closure and by expressing the t dependence in eq 5 in the interaction picture we easily obtain the simple result
giving the time evolution of the frequency-dependent emission spectrum in terms of the electric-dipole correlation function for the prepared state l$e(0)). The structure
The Journal of Physical Chemistry, Vol. 87, No. 1, 1983
Feature Article
of eq 24 is analogous to that commonly used for the absorption spectrum.16 Rather than working with the dipole correlation function explicitly, we prefer to use the vibronic doorway state formulation in the spirit of the preceding sections. In eq 5, we have d t l , t ) = (#&)I, where I#&)) is given by eq 19 The summation of eq 5 over all channels (Le., all vibrational states) to obtain the totalemission spectrum into the ground electronic state amounts to taking the trace (reduction) with respect to all vibrational degrees of freedom (as was done in obtaining eq 21). By doing this and using eq 19, eq 5 yields the result P(w,t) = t a-ly,o Re dt, e-i(e%)(t-ti)Cr' ( t,)cr'(t)* (Xr(t)IXor(t,tl)) (25) where C,' is the probability amplitude (in the interaction picture) of the electronic doorway state Ira). This is a remarkable formula which describes the dynamics of evolution of the emission spectrum in terms of the relative motion of vibrational wavepackets on the two surfaces w, and wo. The vibrational wavepacket overlap function (X,(t)lXo,(t,tl))is the overlap of the vibrational states x,(t) and xo,(t,t1)where x,(t) results from the evolution of x,(O) (for t = 0 in eq 19) under the Hamiltonian 7fo = H - Vr and xo,(t,t1) is the vibrational wavepacket resulting from the evolution of xr from time 0 to tl under go, followed by evolution from tl to t under 7fo on the ground state surface. The latter wavepacket results from the fact that (in eq 5) the molecule emits a photon at tl under the action of VI. For the simplest case of only two electronic states we have the result that the emission depends on the overlap of a wavepacket moving from 0 to t only on surface w, with a corresponding wavepacket which moves from 0 to tl on surface w,, at which time it jumps to surface wo under which potential it moves from tl to t. This is shown in Figure 7. Obviously, the evolution of the emission spectrum depends on the complexity of the vibrational degrees of freedom and their interactions. Consider a diatomic molecule having a harmonic vibration of frequency w, and x,(O) corresponding to the ground vibrational state for the ground electronic surface, as in Figure 7. Then x,(t) is periodic with t , = 2 aw.'; For t < '/$, the "ground state packet" xo,(t,t1) overlaps strongly with z,(t) only if tl is near t and the value of w for maximum P depends on the interchange between potential and kinetic vibrational energy. The gross features of the emission spectrum evolve during the first period t,. For t >> t , there are recurring values of t l for which the overlap is large and this leads to the detailed vibronic features of the emission spectrum. On the other hand, for a large molecule having many vibrational modes and anharmonicities, x,(t) may have a highly irregular (chaotic) trajectory" which could mean a lack of fme structure in the large t limit of P. At the same time, however, this may imply that tl must be in the neighborhood of t for there to be nonvanishing overlap. This suggests that adiabatic modulation may be operative in producing a time constriction or memory loss in P. The above comments are intended to be heuristic and to set the stage for the next step in the theoretical development. Consider a large molecule having a broad spectrum of vibrational frequencies. Of particular interest are molecules having a distinct chromophoric part such as the alkyl substituted benzenes studied by Smalley et (17) Tabor, M. Adu. C h e n . Phys. 1981, 46, 73.
37
I
Flgure 7. Depiction of how the excited vibrational wavepacket motion affects the total rate of photon emission from excited electronic surface, a,,to the ground electronic surface, ao.The rate depends on the overlap at time t of the excited state wavepacket, X , ( t ) , with its "partner", Xo,(t,t,), which dropped to surface a. at t , upon emission of a photon.
al.,18 but the following applies to large molecules, in general. Now, we assume that the vibrational modes of the molecule can be divided into two classes: a set of primary modes whose states we wish to identify in the emission dynamics and a set of secondary (or background) modes the states of which will not be considered explicitly. The essential feature here is that any molecular system is composite in nature, consisting of subsystems (modes or sets of modes), and it is possible (in prinicple) to "measure" the properties of any given subsystem (our primary subsystem). These properties are then degenerate with respect to all of the states of the secondary subsystems. The probability of any given primary subsystem state (or property) automatically invokes the summation over all secondary subsystem states. Doing this "reduces" the primary subsystem with respect to the secondary ones. By maintaining a set of primary vibrational modes we are preserving some of the channel identity in the emission spectrum. In order to avoid cumbersome notation we shall continue to use the notation 18) for those (ground) vibronic states involving the primary modes and Ir,) for their corresponding vibronic doorway states.
The Role of Adiabatic Modulation by the Medium The secondary vibrational modes may be regarded as part of the background medium of the molecule. In addition, however, it is important to include with these any additional modes associated with the coupling of the molecule to its medium (e.g., solvent). Again, we make an adiabatic formulation in terms of the energy surfaces for the various molecule state^.^,^ This does not preclude nonadiabatic dissipative effects such as quenching of excited molecule states, but we will incorporate such interactions in the general molecular potential V. The result is that there are many molecule-medium modes, q, having (18) (a) Hopkins, J. B.; Powers,D. E.; Smalley, R. E. J. Chem. Phys. 1980, 73, 683. (b)Beck, S. M.; Hopkins, J. B.; Powers, D. E.; Smalley, R. E. Ibid. 1981, 74, 43.
38
The Journal of Physical Chemistry, Vol. 87, No. 1, 1983
Y l a A,> Figure 8. The coupling pattern characteristic of a molecular resonance invoking states la) and Ib). The sudden potential, ‘v, drives transitions between bath states, while v represents dissipative coupling involving transitions of the two states to other primary subsystem states, Is,) and 1g2). The bath states 11,) are eigenstates for the molecule bath energy surface o, corresponding to l a ) .
adiabatic molecular energy surfaces and associated sudden potentials like those in Figure 6. The secondary vibrational modes together with all relevant medium modes will be considered collectively as the bath. The observed properties of the molecule of interest to us are determined by its set of primary modes and their associated states. The states of the primary modes are coupled by V which contains radiative interactions, intramolecular coupling (e.g., spin-orbital coupling, BO breakdown), and dissipative interactions with the medium. Transitions involving only states of the bath are driven by the sudden potential, Y. The above comments are relevant to the concept of a molecular resonance, which may be defined as the coupling of two molecule states by V in which the resulting transitions are modulated by transitions (motions) in the bath due to ‘v. Figure 8 represents this idea. As a result of the effects of V and Y ,any pure state of the molecule-bath system can be expressed in the form (26) IiNt)) = Cli) ddt)Ci(t) I
where the states li) are the orthonormal states of the primary subsystem (molecule), which are here taken to be vibronic, and the @Jt)are normalized, but not necessarily orthogonal states of the bath. The t-dependent coefficients, Ci(t), are the probability amplitudes for the states li). The set li) includes the ground vibronic manifold lg) and the appropriate set of excited vibronic states. In considering a particular emission channel lg), the excited manifold li) is defined to contain its (appropriately projected) vibronic doorway state Irg). We are now in a position to give an important formula for the time evolution of the emission spectrum of a molecule which involves particular emission channels and the effects of adiabatic modulation by the relevant background modes (bath): P(g,w,t) = H - ~ YRe ~ ~
tdt e-‘(w-w~g)(t-t~)ur’(t l,t)(
I&(t ,t ) (27)
Here u; = C;(t,)C,l*(t) (the prime denotes interaction picture) for the present case of a pure state of the system. We define the bath wavepacket overlap function Fgr(t,tI) E (+r(t)l@gr(t,tl)) (28) which is analogous to the vibrational overlap function in eq 25. Figure 7 also shows its qualitative features. Equation 27 can easily be modified for the most general case of a statistical state of the molecule-bath system. How do the terms in eq 27 affect the evolution of spectral structure? The quantity Cr’(t) is the probability amplitude for Irg) and its time dependence arises from exponential radiative decay and from interactions with other states If) of the primary subsystem (molecule). The latter couplings can produce both nonradiative decay and oscillations in the probability of Irg). In absence of such
Rhodes
-
interactions, the tendency is for the transition Ir,) 18) to be atomiclike in character. But, this transition is further modulated by the bath function F,(t,t,). We are especially interested here in the situation of relatively weak modulation by very many bath modes, q. The molecule energy surfaces for most of these modes have bound (discrete) states so, strictly speaking, the motion of the bath wave functions tends to be (quasi) periodic in character. However, for a sufficiently large hyperspace of q’s the motion of 4i(t)tends to be irreversible in the sense that recurrences be far outside the time range of interest. This is related to the properties of the correlation functions for the sudden potentials Yij. The requirement is that these bath correlation functions behave as d functions of time, which is referred to as the fast modulation limit.g For any given system the properties of Fgrdepend on the bath wave function 4r(0). The initial state l$(O)) is presumably prepared by coherent pulse excitation of a molecule in its ground state. We next make the Ansatz that the properties of the bath-molecule system are such that F gr (t t 1) = e-rrgIt-t~I (29) 9
where rrgis a bath relaxation constant associated with states Irg) and (g). We have used the term stochastic modulation (SM) to refer to the fast modulation limit together with the associated form for the bath relaxation in eq 29.8,9 This term is used because of the seeming random character introduced by the numerous weakly coupled bath modes, which in other theories is treated by use of a stochastic variabie.1933 Now, suppose the prepared excited state of the molecule is such that C,‘ varies little over the time interval rrL1. This is the condition of strong bath modulation. Then, eq 27 can be integrated to give
where Pr(t) 5 lC,(t)I2 = cr,(t) is the probability of lrg). Adiabatic modulation of the molecule in the stochastic limit has produced a remarkable effect on the dynamics of the molecule. The associated bath relaxation function in eq 29 tends to produce time constriction like that produced by frequency integration of eq 5. The strong modulation limit, whereby rrilis less than the time of variation of u,’(At), is analogous to using Awg > Aw (cf. eq 2) in the formulation of eq 6 and 11(Le., it is sufficent that rrg> Aw). The result in both cases is that P is proportional to the probability of lrg). The difference, however, is that SM produces the time evolution of a homogeneously broadened spectral line for each channel (g). The precise Lorentzian structure is, of course, due to neglect of any residual memory effect in d,so the actual spectrum would be a perturbed Lorentzian whose detailed shape could be t dependent. There are two very important aspects of the role of SM in producing eq 30: (1) The propensity for quantum beats is maintained without having to (locally) integrate the photon emission frequency. The homogeneous broadening rrgmay be attributed to the “virtual excitation” of states of the background associated with the molecule transition; i.e., the (19) (a) Takagahara, T.; Hanamura, E.; Kubo, R. J . Phys. SOC.Jpn. 1977, 43, 802. (b) Ibid. 1977, 74, 811. (c) Ibid. 1977, 43, 1522. (20) (a) Hochstrasser, R. M.; Novak, F. A. Chem. Phys. Lett. 1978,53. (b) Ibid. 1977,48,1. (c) Hochstrasser, R. M.; Novak, F. A,; Nyi, C. A. Isr. J . Chem. 1977, 16, 250.
Feature Article
-
transition Irg) Ig) becomes clothed by a cloud of virtual phonons, the states of which are summed over in eq 28 and 30. This summation replaces the need for frequency integration, but at the expense of broadening the spectrum. Quantum beats occur if there are oscillations in PI@).A very important point, however, is that the validity of eq 30 requires that any existing quantum beats result from molecular eigenstates that are completely buried within the homogeneous line width. This means that, if rrswere zero (negligible) due to a change in conditions, there would be resolved structure in the absorption spectrum from 18) to which the beats can be attributed (cf. Figure 2). Such spectral structure is “washed out” by rrg, however, so it is possible to observe quantum beats due to spectral components which are completely buried by homogeneous broadening! (2) The bath relaxation function, eq 29, describes the effective loss of a definite phase relation between Ir,) and Ig). To see this, consider the structure of FBrin eq 28. A bath wavepacket on surface w, is suddenly switched to wg at tl and propagates on that surface from tl to t. This wavepacket is compared with its partner wavepacket which is identical a t tl but, instead of jumping to surface w,, it continues to propagate on 0, from tl to t. The stochastic limit implies that the two partners tend toward orthogonality with first-order rate constant ITrg. The effect is that quantum amplitudes of Ir,) and lg) can contribute (as interference) to a physical process only if their times are within the range r,
