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J. Phys. Chem. 1981, 85,3592-3599
possible with turning points at -1 and -2 A,26 Such large must considerably changes in the size and shape of 02-* change the polarization energy for large v’which is contrary to formation of a quantized state. Strong coupling of 02-* with phonons provides the only mechanism for stabilizing the anion. The comparative efficiency of attachment in xenon cannot be accounted for if an acceptor state with large v’in strongly polarized media is forbidden. In gross terms the kinetics fit type v, but the details are not understood.
polar media: electron transport to and from an encounter or collision pair (e-, A) and ion formation. One must consider also the parameter Vo, distinguish between localized and delocalized electrons, and also allow for different transport mechanisms. The kinetics of detachment may provide a more sensitive probe for identifying the state of the reactive electron. There may be about five distinguishable types of overall kinetics. Within this general model, k,(Vo), k,(E), and k,(p) are important isothermally, but k,(T) is not useful.
Conclusions A minimum of three mechanistic steps are required to describe the specific rate of electron attachment in non-
Acknowledgment. I am indebted to K. Funabashi for extensive discussions and to A. Mozumder and D. M. Chipman for comments. The research described herein was supported by the Office of Basic Energy Sciences of the Department of Energy. This is Document No. NDRL-2151 from the Notre Dame Radiation Laboratory.
(26) Das, G.; Wahl, A. C.; Stwalley, W. C. J. Chem. Phys. 1978, 68, 4262.
Intramolecular Vibrational Energy Redistribution Paul R. Stannard+ and William M. Gelbart*t Department of Chemistry, University of California,Los Angeles, California 90024 (Received: M y 12, 198 I; In Final Form: July 15, 1981)
A simple quantum-mechanicalscheme is presented which describes the flow of vibrational energy from an initially prepared distribution into the rest of the molecule. Strong and weak couplings lead to at least two time scales for decay of the initial state. Computational studies for highly vibrationally excited water (H,O) and benzene (C6H6)are used to illustrate the differences between large- and small-molecule behaviors within this theoretical scheme. In both cases we discuss also the nature of the breakdown, with increasing energy, of “local-mode’’ separability. This context allows us to examine the dynamical consequences of “overtone” vs. “combination” distributions of vibrational energy. Some recent experiments-in particular, near-infrared overtone studies of benzene, and transient hot-band electronic spectra of glyoxal-are treated in light of these results.
I. Introduction Much attention has recently been focused on the problem of vibrational energy redistribution in isolated polyatomic molecules. Experimental studies have included time-resolved fluorescence spectra via “cold” molecular beam1 and “chemical timing”2 techniques, multiphoton infrared3 and single-photon near-infrared4excitation, and chemical a~tivation.~ Most theoretical work has involved stability and ergodicity analysis of the classical and semiclassical dynamics associated with model HamiltoniansPg Numerical simulations (classical trajectory analyses) of “real” molecules have been carried out on only a few threeand four-atom species.1° Quantum-mechanical studies of model systems have also been reported.l1-l2 In the present work we present results of our numerical solutions to the quantum-mechanical equations of motion for some realistic vibrational Hamiltonians. In section I1 we outline briefly the basic phenomenology relating vibrational eigenstates and energies to the time evolution of prepared nonstationary states. We find that the time evolution of the initial state is characterized by several time scales and modes of behavior. These general features are illustrated in sections I11 and IV by two numerical examples, H20 and C6H6, which exhibit the ‘Systems, Science and Software, P.O. Box 1620, L a Jolla, CA 92038. Camille and Henry Dreyfus Foundation Teacher-Scholar.
*
0022-3654/81/2085-3592$01.25/0
“small”- and “large”-molecule limits of vibrational energy redistribution dynamics. In the closing section (V) we discuss qualitatively the following: multiple-time-scale relaxation, the role of energy delocalization (i.e., combi(1) J. B. Hopkins, D. E. Powers, and R. E. Smalley, J. Chem. Phys., 72, 2905, 5039, 5049 (1980); 73, 683 (1980). (2) R. A. Coveleskie, D.A. Dolson, and C. S. Parmenter, J. Chem. Phys.,72,5774 (1980); D.A. Dolson, C. S. Parmenter, and B. M. Stone in “Proceedines of the NATO Advanced Studv Institute on Fast Reactions in Energetic Systems”, Ioannina, Greece, July 1980. (3) (a) P. A. Schulz, Aq. S. Sudbs, D. J. Krajnovich, H. S. Kwok, Y. R. Shen, and Y. T. Lee, Annu. Reu. Phys.Chem., 30,379 (1979); (b) N. Bloembergen and E. Yablonovitch, Phys.Today,31,23 (1978); (c) J. C. Stephenson and D. S. King, J. Chem. Phys.,69,1485 (1978). (4) (a) R. G. Bray and M. J. Berry, J. Chem. Phys.,71,4909 (1979); (b) R. L. Swofford, M. E. Long, and A. C. Albrecht, ibid.,65, 179 (1976); (c) J. S. Wong and C. B. Moore in “Proceedings of the Sergio Porto Memorial Conference”, Rio de Janeiro, Brazil, July 1980. (5) See the review discussion by I. Oref and B. S. Rabinovitch, Acc. Chem. Res., 12, 166 (1979). (6) E. J. Heller, E. B. Stechel, and M. J. Davis, J. Chem. Phys.,73, 4720 (1980). (7) Y. Weissman and J. Jortner, submitted for publication in J.Chem. Phys.and Phys.Lett. A. (8) D.W. Noid, M. L. Koszykowski, and R. A. Marcus, J. Chem. Phys., 71, 2864 (1979). (9) C. Jaffe and P. Brumer, J. Chem. Phys.,73, 5646 (1980); R. T. Lawton and M. S. Child, Mol. Phys.,37, 1799 (1979). (10) R. J. Wolf and W. L. Hase, J. Chem. Phys., 72, 316 (1980); 73, 3779 (1980), and references contained therein. (11) (a) S. Nordholm and S. A. Rice, J. Chem. Phys.,61, 203, 768 (1974); (b) K. G. Kay, ibid.,72,5955 (1980), and references cited therein. (12) E. Thiele, M. F. Goodman, and J. Stone, Chem. Phys.Lett., 69, 18 (1980).
0 1981 American Chemical Society
The Journal of Physical Chemistry, Vol. 85, No. 24, 198 1
Intramolecular Vlbrational Energy Redistribution
“F/Is>!
3593
t r
Is>
-
Flgure 2. Time evolution of an initially prepared quantum state associated with the line shape In Figure 1.
E > P’(E) Flgure 1. Coupling and resultant line shape for an initially prepared quantum state
Is).
nation vs. overtone initial states), and “TI”vs. ‘‘Ti’ decay. These several questions are considered within the context of the benzene near-infrared overtone studies of Bray and Berry4 and the related glyoxal beam experiments by Naaman, Lubman, and Zare.13
11. General Qualitative Picture We begin, as is inevitable, by writing the full vibrational Hamiltonian 7f as the sum of a zero-order Hoand a coupling term V: 7f=Ho+V (1) The eigenstates of Hocomprise a zero-order basis which is usually described by products of one-dimensional-oscillator wave functions (e.g., normal or local-mode states). These states can be partitioned into three groups as follows (see Figure 1): (i) One of the basis states, Is), corresponds to the initially prepared state of the syptem. (ii) A small subset of levels (11))is coupled via (ual)to Is) and forms a “sparse manifold”. (iii) The remaining states comprise a which is weakly coupled (via (v,) “quasi-continuum”, (Is)), and (ulq)) to both Is) and (11)). Green’s function (resolvent operator) or effective Hamiltonian techniques can be used to determine both the absorption profile as a function of energy and the time evolution of the initially prepared state. The solution for the line shape, which is outlined el~ewhere,’~ is characterized by two main features: (A) The weak coupling of Is) and (11))to the quasi-continuum broadens each of these levels into a Lorentzian-like envelope with width y. (B) The Is)+ interaction leads to a distribution of Is) probability over an energy range r. More explicitly, if V is diagonalized within the subspace of Is) and (Il),then a new set of molecular states, )In)),is obtained. Is) is distributed among the set (In))according to 1s) = CC;ln) n
(2)
Conversely, In) = Cnsls) + .... Since each state In) has a different eigenenergy, E,,, a plot of lCna12vs. E,, defines the spreading out (width r) of Is) due to its interaction with {ll)l-
-
(13)R. Naaman, D.M. Lubman, and R. N. Zare, J . Chem. Phys., 71, 4192 (1979). (14)P.R. Stannard, Ph.D. Thesis, University of California, Los Angeles, CA, 1980.
Is)
The overall result of contributions A and B described above is the modulated envelope shown on the right-hand side of Figure 1. Each “resonance” corresponds to an In) state containing Is) character; its broadening, y, is deterand the mined by coupling to the quasi-continuum (Is)), spacing between them, E, is on the order of the inverse density of 11) levels. Each of the basic quantities y, I”, and E leads to a well-defined feature in the time evolution of Is) as follows. For the state function at time t, we can write l\k,(t)) = e-(i/h)Ht(S)= CCmae-(i/h)Ht Im ) = CCma(Em)e-(i/h)Emtlm ) (3) where the (Im))are the full eigenstates of H. The distriis not quite the same as the (lC;12) described bution (ICma12) earlier, even though the same notation is used-recall that {In))come from diagonalizing 7f in the discrete basis Is) and (ll)),whereas the (Im))are the full (quasi-continuous) eigenstates which include coupling to {Iq)). (lCna12) correspond to a series of “spikes”, separated on the average by an energy E, and with heights having a Lorentzian-like envelope of width I’. (ICma12), on the other hand, correspond to taking (lCns12)and broadening each spike into a Lorentzian-like feature with width y. Since the eigenenergies E, are (quasi) continuous, we can replace Emby JdEm p(Em)and write ~ , ( t=) 1(s1~~(t))12 = l J d ~ , p(Em)IC,a(Em)12e-(i/h)Emt2 I
(4)
P,(t) shows at least three distinct types of time dependence, each characterized by one of the quantities y, r, and E (see Figure 2): (1) The Lorentzian-like envelope ICma(Em)12, with width r in energy space, transforms into an exponential decay with rate constant F/ ti-this accounts for the fast (short-time) falloff in Pa(t).(2) The set of resonances, separated on the average by an energy t, transform into a set of oscillations whose periods are roughly multiples of E / ti. Only if the spacings were constant would the constructive interference of these oscillations be complete at T = h / ~ .For any realistic distribution of spacings, only partial recurrences occur on this time scale, and we must wait until much longer times for the P&) pattern to repeat it~e1f.l~(3) The envelope of each individual resonance also transforms into an exponential-like decay, characterized by the rate y / h . This occurs on a much longer time scale than the initial r / h falloff and corresponds to “leakage” from the sparse Is) - (11)) subspace into the quasi-continuum (1s)).The re(15)W.M.Gelbart, D. F. Heller, and M. L. Elert, Chem. Phys., 7,116 (1975).
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currences described above in type 2 are damped out by this slow decay (see Figure 2). Experimentally, the observed time-dependent behavior depends, of course, upon the time scale and the nature of the measurement. If the experiment is fast compared with T, then only the initial decay will be seen. Similarly, if y >> t, the recurrences will be damped out and only the initial decay will be observed. Correspondingly, no resonances will appear in the line shape-this is reiminiscent of the “statistical-limit” behavior of radiationless transition theory.16 If, on the other hand, the experiment is carried out on a time scale comparable to T , oscillations (e.g., quantum beats) will be observed; on longer time scales, only a time-averaged behavior showing the slow y decay will be seen. In specific numerical calculations and experiments treated below, we shall determine each of the characteristic features shown in Figure 2. 111. H20.The Small-Molecule Limit For our first computational study we consider the example of the water molecule. Recently, we presented’l a diagonalization of an H20 vibrational Hamiltonian, using local-mode states as a basis. After defining a semiempirical potential energy surface, we “sliced” it along each of the three valence coordinates: the two 0-H bond stretches r and P, and the HOH angle bond 0. These cuts imply complete sets of one-dimensional-oscillator wave functions whose products comprise the eigenstates of Ho. Diagonalization of the r / ? and r/O coupling terms in the potential and kinetic energy provides the eigenfunctions of N as linear combinationsof these local-mode states. The results show that the product-state basis is a good description of the vibrationally excited molecule up to energies of 20 000 cm-l; above this, however, the local-mode separability begins to break down. Onset of basis-state mixing was examined earlier,ll with specific reference to its effect on the near-infrared overtone/combination spectra. Here we report its effect on vibrational energy redistribution. In anticipation of these dynamical consequences, however, it is useful first to summarize the nature of the breakdown of vibrational-mode separability in the case of H20. Suppose we construct local-mode product states by solving numerically for the eigenfunctions of the one-dimensional anharmonic “slices” of the potential energy surface described in ref 17. In particular, we consider the 100 lowest-lying product states, in order of increasing enfunctions are then used as a basis set for ergy. These (Ho) representing the full (H, + V) Hamiltonian, which includes both potential and kinetic energy coupling of the stretches and bend. Diagonalization of N leads to eigenfunctions Jr, as linear combinations of the product states c$~: fpi = CCniJrn (5)
-
n
(Here Hofpi= ti& and %Jrn = EnJr,). For low zero-order energies ti,the product states 9 i are almost eigenstates of N ,i.e., C,” = ani and I(C,”)max12= 1. At higher energies, however,,each & begins to be spread out over many Jr,, and I(C,”)max12becomes significantly smaller than unity. This behavior is shown quantitatively by the dotted curve in Figure 3, for the case of the water molecule. A hint of the dynamical consequences of such vibrational-mode breakdown is seen in Figure 4, where IC,’I2 is plotted schematically versus n, for two close-lying (16)P.Avouris, W.M. Gelbart, and M. A. El-Sayed, Chem. Rev., 77, 793 (1977). (17)P.R. Stannard, M. L. Elert, and W. M. Gelbart, J. Chem. Phys., 74,6050 (1981).
Stannard and Gelbart
O 0.6 .*l
0.21
--- Icci, )rnox)-
-
t
. 2
IC:,
t
I
I
n0-
(B)
i
i’
Figure 4. (A) lCA12 vs. nfor two low-qstates. (B) lC,’I2 vs. nfor two high-el states.
4’s. Figure 4A depicts the situation for two low-energy @s, while Figure 4B shows the distinctly different behavior which obtains at higher energy. In the first case the states are “isolated” (suggesting no exchange between i and i?, whereas in the second the states are “overlapping” (implying stochasticity6). Another hint of the time-dependent characteristics of the vibrational-mode breakdown - is contained in the time-averaged probabilities Pi(t). Here Pi(t) is the probability of finding the system in the zero-order state & a t time t , given that it was prepared initially (t = 0) in &. Writing pi&)=
I
ICIc,’p-(i/h)E”t 2 n
(6)
and plotting the long-time averages (7)
vs. ti, we obtain the results shown by the solid curve in Figure 3. As is transparent in these plots, the separability-breakdown index ((C,”),12relates directly with the long-time average probability Pi(t)of being in any product state It is important to note that, inthe region 10000 < E < 20000 cm-l where I(Cf),I2 (and Pi(t))appear to nose-dive, the local-mode separability remains quite strong. Consider,
Intramolecular Vibrational Energy Redistribution
“-1
’
0.0
0.2
’
The Journal of Physical Chemistry, Vol. 85, No. 24, 1981 3505
1
c
-0
Flgure 5.
‘Y 0.1
0.2
0.4 0.5 TIME (psec.)
0.3
0.6
0.7
Time evolution of the (3,1)+0 combination state of water
(W. for example, the fifth stretching overtone state (6,0,0) = 1(6,0)+0),at E i= 17000 cm-l. (C~’6,0~o)max is as large as 0.7; i.e., this zero-order local-mode state is still very close to an exact vibrational eigenstate-see eq 5, with $6,0,0 = q9*6,0,?9 + little bits of other \kn)s. And, of the few other q n 7 sinvolved, one of them-q.5,1,0n-makes a dominant contribution. Thus, the fifth overtone remains considerably more pure than one would guess from a quick superficial look at Figure 3. This is consistent with results recently obtained by Heller et a1.6 in which high-energy states of the Henon-Heiles Hamiltonian are shown to be highly localized in individual bonds, even in the face of nonnegligible mixing of bond-mode overtones and their close-lying combinations (e.g., (6,0)+ and (5,1)+). Consider now Is) = 1(4,0)+0),Le., the initial state is taken to be the u = 4 OH local-mode overtone-one of the stretches is excited with four quanta, the other with none, and the bend with none. (The + subscript denotes our having taken a symmetric linear combination of the (4,O)O and (0,4)0zero-order states; we do not know after all which bond is excited, but only that one of them is.le To make contact with our general discussion in sections I1 and I11 above, we stress here that water is a small molecule; i.e., the vibrational density of levels is so low that only the states Is) and (11)) can be identified. No quasi-continuum (1s)) exists. We simply need to take into account all zero-order states (11))which couple to Is) through V. These s-1 interactions lead to 7# eigenstates (In))whose energies (E,] and Is) weights (lC,S1z]determine the time-dependent Is) probability according to
p 8- ( t ) =
I
ICIC$l2e--(i/h)Ent 2 n
(6’)
Using the diagonalization ((E,] and (IC$lz)) from our earlier work,17we have displayed in Figure 5 the P,(t) for Is) = 1(4,0)+0).The fast, initial decay of Is) is associated with the width of the (IC,S1z),as discussed in section 11. The decay is highly nonexponential since Is) is spread out over so few states In), and because the weights (lC,SI2)have such a non-Lorentzian envelope. Similarly, the partial recurrences occur at times h/c where E is the average spacing between the levels which are coupled strongly to Is). No damping (on time scale h l y ) of the partial recurrences is seen, since there are no other states into which the In) can “leak”. In the “real world”, of course, there are radiative and collisional continua which interact with the (In));but the associated irreversible decays occur on time scales (18)W. M. Gelbart, P. R. Stannard, and M. L. Elert, Znt. J.Quantum Chem., 14, 703 (1978).
many orders of magnitude longer than those of the recurrences. Even in the absence of (apparent) irreversible decay, we can, of course, still speak of vibrational energy redistribution. For example, we can solve for the probability a t time t of finding the water molecule in the state (3,1)+0, having started in (4,0)+0. We find that (3,1)+0“grows in” on essentially the same time scale that (4,0)+0 “falls off’ (see Figure 5); this pattern is repeated with a frequency e/ fi characteristic of the recurrences discussed above. This behavior describes the exchange of energy between the two 0-H bonds. Whether or not this kind of flow can be considered “stochastic” or “ergodic” depends, of course, on the definition of stochasticity and ergodicity! We mentioned above that the nonstationarity of the state (4,0)+0comes about from its mixing with only a few, widely spaced, neighboring, zero-order states. Thus, its initial decay is highly nonexponential,and the recurrences occur on a very short time scale. In fact, so few states interact significantly with (4,0)+0 that we can retain only these in the diagonalization and time-evolution scheme; keeping as few as five 11) states gives a P8-(t) which is virtually identical with that obtained in our full (73-state) calculation. This is true for all other Is) states below 20000 cm-l and follows from the small-molecule nature of HzO in this energy region.
IV. C6H6.The Large-Molecule Limit The C-H-stretching overtone spectrum of benzene has received considerable attention r e ~ e n t l y .The ~ basic features have been known since 1929;l9 despite a very high density of vibrational states, a simple progression of broad resonances is observed at near-infrared energies through 20000 cm-’. The positions of these lines follow the Birge-Sponer relationship20 characteristic of transitions up to n = 7 of a one-dimensional (diatomic) Morse oscillator. Partial deuteration and other substitution have little effect on the line positions, and the intensities scale as the number of C-H bonds. This fact, in conjunction with the closeness of the transition frequencies to those of the diatomic CH radical, s ~ g g e s t ~that l ? ~the ~ C-H bonds in benzene behave as six essentially independent (uncoupled) oscillators. In addition to the line positions, much has been made of the fact that the collisionless (low-pressure, gas-phase) line shapes have widths on the order of 100 cm-1,2z Furthermore, the lines corresponding to the u = 5,6, and 7 overtones are nearly perfectly Lorentzian in shape, with widths which decrease from 109 to 94 to 87 cm-l. Making contact with the basic model outlined in section I1 above, it is natural to consider as the initially prepared state Is), the overtone level u = 5,6 or 7 in which there are u quanta excited in one C-H bond and none in the others. This state is coupled, via stretch-stretch interactions, to a small number of sparsely spaced levels (11))which involve small changes in quantum numbers (and small changes in the number of quantum numbers that change). Finally, the quasi-continuum(14))comprises the “bath” states in which the remaining 24 vibrational degrees of freedom (notably the bending modes which are knownz3to interact most strongly with the C-H stretches) have exchanged energy with the C-H bonds. (19)J. W.Ellis, Trans. Faraday Soc., 25, 888 (1929). (20)R. T.Birge and H. S. Sponer, Phys. Reu., 28,259 (1926). (21)B.R.Henry and W. Siebrand, J. Chem. Phys., 49,5369 (1968); B. R. Henry, Acc. Chem. Res., 10,207 (1977). (22)D. F. Heller and S. Mukamel, J. Chem. Phys., 70,463 (1979). (23) A. C.Albrecht, private communication.
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The Journal of Physical Chemistty, Vol. 85, No. 24, 1981
More explicitly, the C-H stretches are considered in zero order to be six, independent, Morse oscillators (with the usual a and D constants).u For u = 6, for example, there are as many as 462 different ways in which these six quanta can be distributed over the six C-H bonds. The lowest energy level of these 462 is the overtone (i.e., I6,0,0,0,0,0)) state which lies at 16500 cm-l. The highest is the combination state ~l,l,l,l,l,l) at -18 200 cm-l. In between are states like 13,2,1,0,0,0) (at -17 750 cm-l) in which the excitation is neither localized completely in one bond (viz., 16,0,0,0,0,0))nor spread out evenly over all of them (viz., ~l,l,l,l,l,l)). We begin our numerical study of local-mode spectra and vibrational energy redistribution in benzene by restricting ourselves to those C-H-stretch states. That is, we start with the molecule prepared in one of the C-H overtone or combination states and monitor the effects of the couplings between them on the ensuing time evolution. Exchange of energy with the remaining 24 degrees of freedom is neglected at first (see, however, the Discussion in section V). Furthermore, we confine attention to a single value of u, i.e., only Iul,u2,U3,u4&,ug) states with Ci.?ui = u are included. Recall that for u = 6 there are 462 different ways to distribute the six quanta over the six C-H bonds; 462 sets of phase factors can be chosen so that each of the corresponding 462 linear combinations of product states transform according to one of the irreducible representations of the D* group. Thus, for example, we can construct 80 linear combinations which are of Al symmetry, 75 of El,symmetry, and so on. Since % = do + V transforms according to Alg, an initially prepared Is) can couple only to those other zero-order states which have its symmetry. To simplify bookkeeping procedures, we consider only the At linear combinations for our zero-order basis, even though these are not the states which are excited in the near-infrared one-photon e ~ p e r i m e n t .The ~ qualitative dynamical behavior of the coupled Al, states is expected to be essentially the same as that, for example, of the El, states prepared optically; roughly equal numbers of states are involved (80 vs. 75), as are similar distributions of quantum numbers, To couple the C-H-stretching states of benzene, we need to define our V %-Ho-here Ho %localmode; i.e., Ho includes separable terms involving powers of the stretch displacements, summed to all orders. We take V to include the “ortho”, “meta”, and “para” interactions in the potential energy, retaining however only the quadratic contributions; ko~ho{rlr2]kme&1r3) + kwa(r1r4). Here ri denotes the displacement of the ith C-H stretch from its equilibrium bond length and
Stannard and Gelbart
tO.,yl
-
+
(r1r2)= (rlr2+ r2r3
+ r3r4+ r4r5+ r5r6+ r6r1)/6~/’
Diagonalization of this interaction in the 80 X 80 basis of Al states allows us in the usual way to obtain the distrihtion (lC,S12)for an arbitrary initial state Is). With the corresponding eigenvalues (E,) we compute Ps*(t ) according to eq 6’. Figure 6 shows P,,(t) for the case Is) = 13,1,2,0,0,0)-A1,. This state lies -1200 cm-l above the u = 6 overtone, 16,0,0,0,0,0). The force constants kodo,k,, and ,k have been set equal to 6000,4000, and 2000 crn-’/A2, respecti~ely.’~Note that the u = 6 excited molecule, initially prepared in a 3,1,2,0,0,0distribution, decays away from this state on a time scale as short as a few hundredths of 1ps. (24) M. L. Sage, Chem. Phys. Lett., 35,375 (1978). (25) These values were chosen as “upper limit” estimates based on couplings in other hydrides. Recent estimates deriving from analysis of the benzene spectrumz3indicate that even smaller values are appropriate.
0.2c
OO
1
\ 0.00
0.16
0.24
0.40
0.32
TIME (psec.)
Figure 6. Time evolution of the (3,1,2,0,0,0) stretching comblnation of benzene (C,H,).
As discussed at length in section 11, this corresponds to the inverse of I’/h where I’ is the width in energy of the IC2I2distribution vs. E,. (Accordingly,we expect an absorption line broadened to a width of several hundreds of 1 cm-l.) At longer times, on the order of 1 ps, a partial recurrence is seen-this corresponds to the inverse of E/ ti where l/c is the mean number of states per cm-’ in the energy region of I3,1,2,0,0,0). In our actual calculations these partial recurrences are not damped, but this is only because we have failed to include coupling to the bath degrees of freedom. Because of the interaction between C-H stretches and, e.g., the lower-frequency bending motions, energy will leak out of the C-H stretches before the initial state is “forced (by the small number of strongly coupled CH states) to recur. This leakage will occur on a time scale y/ti where y is the width of the states (11))due to their interaction with the quasi-continuum(1s)). When we take the u = 6 overtone 16,0,0,0,0,0)-A1 to be the initial state, we find essentially no decay of &&), This is because the overtone is nondegenerate and lies as much as 500 cm-l from the closest u = 6 combinationhence, it remains essentially isolated or “uncontaminated” by the local-mode couplings (which are on the order of tens of cm-l). The 13,1,2,0,0,0)-A1, state, on the other hand, is manyfold degenerate (with, for example, 13,2,1,0,O,0)-Alg involving a single-quantum exchange induced by the term korthor2r3and so on). Furthermore, it lies very close in energy to several other, manyfold degenerate combination states (e.g., ~3,1,1,1,0,0)-A1 only 55 cm-’ above). These are precisely the states whica are excited during the decay of the initially prepared )3,1,2,0,0,0)-A1,. The above-described “isolation” of overtone states, compared to combinations, is a general phenomenon. Thus, we can expect that combination states-in which the energy is spread out over several bond modes-will decay faster than the highly localized overtone states. Heller and MukameP2have come to a similar conclusion in their phenomenological study of the benzene overtone spectra. They suggested in particular that the width of the 16,0,0,0,0,0) state is “borrowed” from that of its nearest neighbor 15,1,0,0,0,0). Crudely speaking, rS = gl(u,l/ hE$Tl, where gl is the degeneracy of the 5,l... combination and rl is its decay width; usl is the coupling between the 5 , l...and 6,O... states, and aESlis their energy separation. Note that it follows immediately that I?, can decrease with increasing u. This is because us? = u while hE,1 u2; gl is essentially constant, whereas rl increases slightly with u. The overall result is a slower-than-linear decrease of rswith u, as ~ b s e r v e d . ~ But this mechanism appears to require too large a value for the strength of C-H-stretch couplings. In particular, for rsto be on the order of 100 cm-l, and for rl no larger ... be as large as -100 than -1000 cm-l, usl = ~ 5 , 1 . . . 6 , ~must
-
Intramolecular Vibrational Energy Redistribution
cm-‘. If one assumes that the CH-CH coupling arises entirely from the quadratic force constant kortho, this suggests (from harmonic oscillator matrix element relations) a value for korth,, which is considerably larger than that inferred independently by spectroscopic analysis of the benzene valence force field.23 This is consistent with the fact that, when we use “reasonable” (Le., K 5 5000 cm-’/A2) estimates of the C-H-stretch couplings, we find no decay of the u = 6 overtone on a subpicosecond time scale. It is important to note, however, that the relaxation of highly vibrationally excited (e.g., u = 6) states can not be attributed solely to the quadratic force constants. Instead, higher-order anharmonic couplings-which are not easily estimable at present-will surely make significant contributions. We believe that the observed width for the u = 6 overtone is indeed borrowed through coupling to close-lying combination states. But instead of considering only the CH-combination states-which leads to the inconsistencies described immediately above-it is necessary to include explicitly the bath degrees of freedom. Thus, we include a collective quantum number Vbath in writing out the zero-order states: IUl,UZ,U3,uq,Ug,Ug;Vbath). In this way it is ~ be E much ~ ) ~ possible for the “borrowing” strength ( U ~ / to larger, because of a decreased AE, and an increased u,~. Consider the benzene bending modes whose frequencies are 1500 cm-l (ca. half those of the C-H stretches). It is known23 that, much as in the H 2 0 example, the stretch-bend interactions are more significant than those between stretching motions alone; i.e., usl is enhanced by dowing 11) to include changes in vbth (=vbnd) arising from stretch-bend coupling terms in u. Furthermore, the energy “gap” aEslfor Is) = CH overtone is greatly decreased as soon as 11) includes Vbend changes. For example, Is) = 16,0,0,0,0,0;0bnd) lies at 16500 cm-l, whereas the 11) states ICi=lu,= 5;u1600-cm-1bnd = 1)range from -15800 to 16700 cm-l. Thus, many of these 11) states will lie within a few hundred cm-’ of Is). The 11) states ICi=tui= 6;vbnd = 0) considered earlier, on the other hand, all lie more than 500 cm-l away. The involvement of bath states in the decay of C-Hstretch excitations can also be clarified by studying the effects of deuteration. The main effect of complete deuteration is to “compress” the zero-order spectrum of ICi=lui = 6;O) states by a factor of 2 (since the bond anharmonicity varies inversely with the reduced mass). In addition, all coupling energies are decreased, since the deuterated species are forced to undergo smaller vibrational displacements. The result is that the (lC,a12] are spread out over a much narrower range of E, and Pa,@) decays more slowly than in C6H6. This decrease in I?, is indeed observed4for each of the CH overtones. But the decrease observed experimentally reflects a balance between several competing effects. We have left out of our calculations, for example, the kinetic energy coupling of the C-H stretches, since it is so small compared to the potential energy interactions. (Recall that each H atom of mass 1 is vibrating against an essentially rigid carbon skeleton of mass 72 and that the coupling elements of the G matrix are thus 1/72 times smaller than those on the diagonal.) Its inclusion would lead to a small increase in relaxation of the C6D6 overtones. We have also left out coupling to the lower-frequency bends, as discussed earlier. As with the kinetic energy,this interaction mechanism is also enhanced by deuteration, since the CD frequencies are no longer as mismatched (as the CH’s) from those of the bends. Nagy and Hase,26in their classical trajectory studies of CH en-
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(26) P. J. Nagy and W. L. Hase, Chem. Phys. Lett., 54, 73 (1978).
The Journal of Physical Chemistry, Vol. 85, No. 24, 198 1 3597
ergy relaxation in benzene, have left out the potential energy couplings involving the C-H stretch. Instead, they include only the much smaller kinetic energy interactions-accordingly, they find a very slow relaxation of C-H bond excitation which is greatly enhanced by deuteration. Furthermore, they have considered only the (classical analogue of) overtone states (e.g., 50 kcal/mol in a single C-H bond). Their finding that relaxation of the C-H bond is enhanced by putting additional energy into the other degrees of freedom does not mean that combinations decay faster than overtones, but rather that a single C-H bond (excited with a fixed amount of energy) loses its energy faster when other vibrational displacements are also large.
V. Discussion In the above we have considered a fully quantum-mechanical description of vibrational energy redistribution in isolated polyatomic molecules. For HzO we found typical small-molecule behavior; that is, few enough vibrational states are coupled to each other (even at high energies) so that the system runs through them all, periodically, on a time scale short compared to experimental resolution. For C6H6,on the other hand, the time evolution of highly vibrationally excited C-H stretches was seen to proceed on a t least two different time scales; first, the prepared CH-overtone or -combination state decays (in 10-14-10-13s) into other combination states, and then (after s) the CH excitation leaks into the remaining degrees of freedom (notably the bending motions). These two behaviors are expected to be typical of small- and large-moleculevibrational energy redistribution dynamics. Recent experimental studies by Perry and ZewaiP’ provide a nice example of how these two behaviors can be present in the same molecule. They study the “intermediate” case comprising the aromatic and aliphatic C-H stretches in durene (2,3,5,6-tetramethylbenzene).The Au = 5 CH spectral region was recorded in the low-temperature (2 K) crystal, with bands due to aliphatic (methyl) vs. aromatic (benzene) stretches assigned by selective deuteration. The fourth overtone of the methyl CH is split into three lines, separated by 90 and 200 cm-l, each with widths as small as 20-25 cm-l. The aromatic CH band, on the other hand, has a line width of 100 cm-’. In terms of our general qualitative picture of section 11, the three methyl bands correspond to the In) resonances that derive from diagonalizing V in the Is) (11)jsubspace. The large splittings suggest a strong coupling of the CH’s among themselves, whereas the small widths indicate a weak inassociated with teraction with the quasi-continuum (1s)) the remaining molecular vibrations. Conversely, the broad aromatic bands correspond to the fast decay of excitation in the individual C-H stretches. Another general feature emerging from our analysis concerns the relative characteristics of overtone and combination states. In a previous paper17 we showed how anharmonicities in the potential energy and in the electronic ground-state dipole moment can result in combinations being excited (by infrared absorption) more strongly than neighboring overtones. Similarly, in another communication28it has been shown how combinations can be preferentially excited by emission to the ground state from higher-lying electronic configurations. The question remained: in a given energy region, will combination or overtone states decay faster via intramolecular vibrational energy redistribution? Our numerical studies of the six-
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(27) J. W. Perry and A. H. Zewail, J. Phys. Chem., in press. (28) E. J. Heller and W. M. Gelbart, J. Chem. Phys., 73, 626 (1980).
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The Journal of Physical Chemistry, Vol. 85, No. 24, 1987
quanta C-H-stretch combinations and overtones in benzene suggest strongly that the delocalized distributions of energy (e.g., the 13,1,2,0,0,0) state) decay considerably faster than those (e.g., (6,0,0,0,0,0))in which only a single bond is excited. This is essentially due to the combination states being more highly degenerate and having so many close-lying neighboring states. Note that, in our analysis of the benzene C-H-stretch states, we did not explicitly include the remaining 24 degrees of freedom (i.e., the bath). We wrote each C-HE IvCH). Extending our stretch state as Iu1u2u3u4~5u6) zero-order description to include the bath vibrations, we can write IvCH) 1VCH;Vbath). We allowed for relaxation of each state I V c ~ ( 6 ) ; V b ~ t h )by coupling it only to other , states which involve no change states I V ’ c ~ ( 6 ) ; V b ~ t h ) i.e., in the bath and only a redistribution of the original quanta over the C-H stretches. (The 6 in parentheses reminds us that we are dealing only with, e.g., vCH combinations and overtones for which Ci=?ui= 6.) By constraining vbth to stay the same, we have ruled out the possibility of resonance-type interactions with states of the form 1 ~ ’ ~ (6);v’ba*)-here the bath adjusts itself to help energy be conserved as the C-H stretches undergo the transition vcH(6) ~ ’ ~ ~ (This 6 ) . mechanism requires, of course, coupling terms in the potential energy which mix C-H stretches and bath modes. As soon as it is included, we must allow for the possibility of the C-H stretches losing (to the bath) a quantum or more of excitation, e.g., IvCH-
Stannard and Gelbart
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(6);vbath)
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IVCH(5);V’bath).
(In this connection it is interesting to mention the probable existence of at least two sets of bath states, one which interacts strongly with the CH’s and the other which is only weakly coupled. Note that, if all of the bath states were involved directly in the decay of the initially prepared overtone, the measured line widths would be expected to grow dramatically with u, following the sharp increase in bath-state density. But this is not observed to be the case: the overtone line width “saturates” and is essentially constant for u R 5. For earlier discussion of these linewidth saturation effects, see the work of Heller and Mukame122and Stone et al.29 Our model also implicitly assumes that the system has a “temperature” of 0 K. If we allow for a thermal distribution of vibrational energy, particularly over the lowfrequency bath modes, then the initial state Is) is an incoherent superposition of different bath states, even before the effect of coupling is considered, Le., “ls)” = Cicilv(6),vibath). This adds extra width, or inhomogeneous broadening, to the observed line shape of Is), in just the same way as a thermal distribution of translational energies leads to Doppler broadening. This additional source of broadening will be negligible at room temperature since the lowest-frequencymode in benzene is -600 cm-l. The pressure-independent (hence, assumed collisionless) Lorentzian features, observed by Bray and Berry4 for the absorption overtones of the C-H stretch, are thus most probably homogeneously broadened. If the measurement of overtone line widths could be carried out at 0 K, then one could, of course, rule out completely the inhomogeneous broadening arising from “hot bands”. In the case of tetramethyldioxetane, in fact, West et al.30have shown that the line widths observed in a supersonic nozzle beam are essentially the same as those measured in a room-temperature, low-pressure, photoa(29)J. Stone, E.Thiele, and M. F. Goodman, Chem. Phys. Lett., 71, 171 (1980). (30)G. West, R. Mariella, J. Pete, D. Heller, and W. Hammond, to be submitted for publication.
~
coustic absorption experiment. Thus, it appears that the broadening associated with these overtones is indeed homogeneous. In principle it is possible to separate the basis states coupling to a pure initial state Is) = Iv(6),vbath) into two groups: those where the C-H quantum state v(6) changes-with or without changes in the bath-and those where it (v(6)) remains the same. From the point of view of the C-H-stretching state only, coupling to the former group leads to “T1”-type decay, because the stretch state 146)) is destroyed (via mixing with other v = 6 vcHstates, or with u = 5, etc.). Coupling to the latter group leads to homogeneous T2-typedecay, since lv(6)) remains the same. This T2decay mechanism will be insignificant because all states in the second group are at least 600 cm-l off-resonance when Is) has Vbath = 0. Of course, from the point of view of the ouerull Is) quantum state, coupling to either group of states leads to T1decay.31 No measurement has yet been reported which distinguishes between these several T1and T2contributions to -the dynamics of the initially prepared state. One possibility would involve a direct probe of final states via a double resonance e ~ p e r i m e n t .Our ~ ~ theoretical approach is also able, in principle, to resolve this question, but only upon explicit inclusion of the bath degrees of freedom. Finally, we mention a closely related experimental probe of intramolecular vibrational energy redistribution, which has been suggested recently by Naaman, Lubman, and Zare.13 Here “h0t7’ ground electronic states (So*) are prepared via internal conversion from the excited singlet (S,) prepared by a monochromatic “pump” laser. Starting 0.5 ps later the transient So* S1* absorption spectrum is monitored as a function of time. Up to several microseconds the fluorescence spectrum excited by the “probe” laser remains constant with time and exhibits considerable structure. Since the molecules (glyoxal) are contained in an effusive beam with a density of less than lOl1/cm3under conditions such that the average time between collisions is on the order of 1s, it is reasonable to assume that only intramolecular relaxation can occur. But the constant structure observed in the transient absorption appears to indicate that it does not. This conclusion is based on the following analysis. Because of fairly strict selection rules for the FranckCondon factors governing the internal conversion, only a few So* states are initially prepared. I f these highly vibrationally excited levels are not mixed by potential energy anharmonicity with other So* states, then the So* S1* absorption-governed by the same strong Franck-Condon selection rules-will show considerable structure. Indeed, this spectrum can be straightforwardly calculated. We simply made educated estimates of the geometry and frequency changes (So-Sl)associated with each of the 12 normal modes of glyoxal. In the harmonic approximation, then, all of the relevant Franck-Condon factors can be evaluated analytically. In this way we find that there are roughly 100 So* levels populated significantly by internal So*) at 22712 cm-l (the SI origin). conversion (S, Calculating the So*-S1* Franck-Condon factors associated with these So* states, we find a spectrum in the 22 712 f 200 cm-’ region which has far more structure than is observed. ‘Our calculated So* S1*spectrum also shows much more structure than is contained in the computations reported by Naaman et al.13 This is because we have taken
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(31)S. Mukamel, J. Chem. Phys., 70,5834 (1978). (32)R. A. Gottscho, J. B. Koffend, R. W. Field, and J. R. Lombardi, J. Chem. Phys., 68,4110 (1978).
J. Phys. Chem. 1981, 85,3599-3603
into account the So-SI geometry changes in the totally symmetric modes. Also, we have used local-mode arguments to estimate the S1 frequencies of the nontotally symmetric (b,) C-H-stretching and C-0 stretching motions. In the ground electronic state these modes are known to differ in frequency by less than 1% from their ag counterparts. This is con~istent’~ with the C-H and C-0 stretches being good local modes. Thus, we expect that in S1(where strong progressions in the b, modes are not available for determining the excited-state frequencies) ahCH = CJ CH and = osco. In the case of for example, %is implies a So-S1 frequency change of almost 20%. Naaman et al., however, have chosen-in the absence of direct observation of ohcoand o CH in Sl-to put wb,S1 = wBsO. This procedure, along w i h the neglect of Sl-So geometry changes for the v;s, leads to a loss of much structure in the transient absorption spectrum. In any case, a possible conclusion suggested by the Naaman et al. experiment is that So anharmonicity must mix in many more vibrationally hot ground electronic states than are prepared initially by the internal conversion. Rather than the lack of time dependence, it is the structure in the probe (transient absorption) spectrum
3599
which is significant. As discussed in sections I11 and IV, the high-energy So* states are linear superpositions of large numbers of (zero-order) product states. The breakdown of mode separability corresponds to the mixture of the (- 100) S1 So* prepared So* levels with a much larger number of states which do not interact directly with SI. But, via the potential energy anharmonicity, the “new” So*%contribute to the So* S1* absorption. Accordingly, we must enumerate the So*-S1* Franck-Condon factors for all of these states, just as we did for those few which were initially prepared by the internal conversion. In particular-as discussed earlier in this section-we need to know how to partition the bath states into two seta, one which couples strongly with the S1 So* populated states and the other which is weakly coupled. But a quantummechanical calculation along these lines becomes unfeasible since too many transitions (Le., too many So*’s) need to be included to account for the little bit of structure observed. Analysis of this experiment, then, remains an open challenge.
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Acknowledgment. This work was supported in part by
NSF Grants CHE79-02983 and CHE80-24270.
Static and Time-Resolved Fluorescence Quenching for the Study of Solubilization and Transmembrane Transport in Vesicles Mats Almgren Department of Physical Chemistry I, Chemical Center, University of Lund, S-220 07 Lund 7, Sweden (Received: June 12, 1981; In Flnal Form: August 6, 1981)
Fluorescence spectroscopy, static and time-resolved fluorescence quenching, and fluorescence stopped-flow measurements have been utilized in the study of interactions between vesicle membranes and fluorescent molecules and/or quenchers. The fluorescenceof pyrenebutyric acid in a vesicle with positive charge, didodecyldimethylammoniumbromide, is strongly quenched by a negative quencher like iodide ion. At low quencher concentrations quenching occurs only at the outer vesicle surface, leaving -30% of the probes protected. At high iodide concentrations also the inner surface was affected. Such an inside-outside discrimination was not evident in the quenching behavior for R~(bpy),~+, quenched by methylviologen in dicetylphosphate containing vesicles. Characteristic in this case was instead a site-competition behavior. Ru(bpy)BzC evidently binds to two different types of sites in these vesicles. One is ionic; at this site quenching, but also competition, occurs, so that at high quencher concentrations Ru(bpy)32+retreats to the second, more hydrophobic site, where it is well protected from the quencher. Stopped-flowmeasurements in this case revealed no measurable relaxation that could give evidence for a penetration of the membrane. Transmembrane transport was measured for cetylpyridinium chloride by using its quenching of pyrene fluorescence as monitor. The time for the “flip-flop” process was on the order of 100 s.
Introduction Photophysical and photochemical phenomena in aggregate systems have turned into a very active research area.’ Just as in the case of micelles, the interest for photoprocesses in vesicles derives both from the possible uses of these partially ordered structures in the control of photochemical processes-in particular, photochemical solar energy conversion2-and from the use of the photophenomena in elucidating the dynamic organization of the
We have for some time been interested in both of these interconnected aspects of photophenomena in vesicle sol u t i o n ~ .In ~ the course of these studies, we have noted on some occasion unusual types of fluorescence quenching. In some cases only part of the fluorescence was quenched although no protective action was expected; in other cases an expected protection was absent. The fluorescence quenching is more varied and more complex than in micelles. In this paper three different types of fluorescence quenching behavior will be exemplified and shown to yield
(1) The literature is overwhelming. General reference is here made only to some recent review~.~J (2) (a) Gratzel, M. Ber. Bunsenges. Phys. Chern. 1980,84,981-91. (b) Matauo, T.;Nagamura, T.; Itoh, K.; Nishijima, T.; Mern. Fac. Eng., Kyushu Uniu. 1980,40, 25-36. (c) Calvin, M. Int. J.Energy Res. 1979, 3, 73. (d) Thomas, J. K. Chern. Rev. 1980,80,283-99.
(3) (a) J. Fendler, “Membrane Mimetic Chemistry”;Wiley-Interscience, in press. (b) Radda, G. Methods Membr. Biol. 1975,4,97-188. (c) Sackmann, E. Ber. Bunsenges. Phys. Chern. 1978,82,891-909. (4) (a) Almgren, M. Phys. Chern. Lett. 1980, 71,539. (b) Almgren, M. J. Am. Chern. SOC. 1980, 107, 7882-7. (c) Almgren, M.;Thomas, J. K. Photochern. Photobiol. 1980, 31, 329.
structure^.^
0022-3654/81/2085-3599$0 1.25/0
0 1981 American Chemical Society
