THE J O U R N A L O F
PHYSICAL CHEMISTRY Registered in U.S. Patent Office
0 Copyright, 1980, by the American Chemical Society
VOLUME 84, NUMBER 16
AUGUST 7,1980
LETTERS Molecular Overtone Bandwidths from Classical Trajectories Eric J. Heller" and Michael J. Davis Department of Chemistty, University of California, Los Angeies, California 90024 (Received: May 16, 1980)
A simple means of estimating molecular overtone bandwidths from classical trajectory data is given. The method is applied successfully to two-dimensional anharmonic systems with local mode and normal mode overtone spectra.
New experiments capable of detecting very weak transitions in polyatomic mole~ulesl-~ have generated renewed interest in vibrational overtone bandwidths and have made possible a much wider selection of prepared initial states, from which subsequent intramolecular decay can be examined. We report here on a classical method for estimating the vibrational overtone bandwidths caused by intrinsic intramolecular decay. The technique is simple: it relys only upon the uncertainty principle and classical trajectories, and it provides an alternative, intuitive understanding of the dynamics underlying bandwidths in polyatomic molecules. The method is an ad hoc simplification of the formula4i5 for the one-photon absorption cross section do)
preparation of a linear combination of zero-order states (the ul, ..,uN label vibrational quanta in various zero-order modes)
14) =
CCvIU1,~2,... VN) V
= CCVIV)
(2)
V
where relatively few 12;s are nonvanishing and where the zero-order states Iv) which have large Cv lead to identifiable spectral bands. If the bands are indeed resolved from each other, as in the C-H overtone spectra in benzene, we may consider each state Iv) separately, for the fwhm of the band is independent of C,. In fact, a well-chosen displaced Gaussian wave pocket has the expansion 18) = CDvlv)
(3)
V
where I&))is the time-dependent wave function evolving out of I&(O)) 14) = y)xo),wherey is the transition moment and Ixo) is the initial vibrational state. The properties of Ixo), y, and changes in molecular geometry (in the case of a Franck-Condon transition) often lead to the
* Author to whom correspondence should be addressed.
Alfred
P. Sloan Foundation Fellow, Camille & Henrey Dreyfus TeacherScholar.
0022-3654/80/2084-1999$0 1.OO/O
where D, is large for the states Iv) of interest. Provided that the spectral bands produced by 18) are resolved from one another, we can examine the bandwidth of Iv) from eq 1, with (gig@))replacing (+l4(t)). Next, note that, except for a phase (gldt)) = [Tr (~,~,(t))l'/~
(4)
where p,(p,(t)) is the density matrix corresponding to lg) 0 1980 American Chemical Socfety
2000
The Journal of Physical Chemistry, Vol. 84, No. I
1
16, 1980
Letters
I
I
u.1.0
I6
1 7
I
40,05b
Figure 2. Same as Figure 1, except as noted in text. Figure 1. Quantum and classical Franck-Condon spectra for the Hamiltonian and wave packet described in text. The insert shows the full spectrum. Details for the bands v = 4, 7, and 12 are shown for the quantum (top row) and classical (bottom row) cases.
(Ig(t)))and Tr denotes trace. In classical phase space6p is just a Gaussian distribution with parameters determinedi by the uncertainty principle, and p,(t) can be approximately represented by a swarm of classical trajectories chosen so as to mimic pg at t = 0. In this work, 16 trajectories were used such that all moments of the distribution pg ,-irough fifth order were correctly reproduced a t t = 0. (This is the same as using two-point GaussHermite quadrature in each of four dimensions in eq 4, where “Tr” now implies integration over all the phase space.) Other methods for representing pg (such as Monte Carlo sampling) can be used. The neglect of the phase necessarily shifts the spectrum, and the band centered around w = 0 is taken to represent the band of the zeroorder state (with large D,)nearest the average classical energy. To examine successive overtones, the classical distribution pg is moved along the active zero-order mode coordinate to adjust its average energy. Several approximations are involved in the procedure suggested above. They are as follows: (1) Replacing 16) with Ig). This is acceptable so long as there are no unwanted states Iv) with large D, which have bands overlapping those of interest. (2) The neglect of the phase in eq 4. To the extent that the phase is a nonlinear function of time, this approximation can affect band shapes and widths. (3) The replacement of p,(t) with a finite, discrete set of trajectories which are propagated classically. Ideally, we should use the quantum Liouville propagator,6 but classical mechanics is much simpler, and it should be accurate for reasonably long times for well-localized phase space distributions.6 In spite of these approximations, the method seems promising. Figure 1 shows overtone bands for the Hamiltonian % = ps2/2 + p 2 / 2 + w s 2 s 2 / 2 + w,2u2/2 + Xsu’, with w, = 1.0, w, = 1.1,and X = 4.11. The full “FranckCondon” spectrum was generated by displacing the ground state of this Hamiltonian (except X = 0) to s = u = 3.2. This is really a type of local mode displacement, since it involves a linear combination of normal modes. The resulting spectrum (inset) is best labeled by a local mode overtone quantum number u. Various “local mode” overtone bands (u = 4,7,12) are selected for higher resolution scrutiny, and these are shown in the first row of spectra below the insert. The finite width of each individual line is an artifact of truncating the integration time in eq 1. The classical spectra in Figure 2, for example, were trun-
Figure 3. Magnitude of (4 IC$ ( t ) )for the parameters w g = 1.O, w, = 1.1, s = 4.0, u = 0. The period of the symmetric stretch vibration is 2a.
cated after approximately 10 periods. Using the same truncation time, we show the classical spectra, obtained by using the replacement eq 4 and the method described above, below the corresponding quantum spectra. The bandwidth narrowing as energy increases is correctly reproduced by the classical spectra, and even the spacing of the combination lines under the bands is well represented. The bands narrow in this case because of the presence of nearly periodic trajectories near E N 12 with s = u,p, = p , = 0 initial condition. The nearly periodic behavior of [Tr ( p g p , ( t ) ) ] 1 / 2 at E = 12 is reflected in the Fourier transform spectra as a narrowed band. Figure 2 shows an analogous and “normal mode” spectrum for w, = 1.818181,w, = 1.0, X = 0.1, and s = 2.7, u = 0 initially. Here, the band envelopes broaden at higher energies. The classical trajectories, though nearly periodic in the vicinity of the symmetric stretch, do escape the symmetric stretch region at a rate which increases with increasing symmetric stretch energy. This escape in turn causes a decay in [Tr ( ~ , p , ( t ) ) ] l / ~= I(d+#~(t))l (dashed line in Figure 3) which gives rise to a wid% which may be associated with each overtone band. In ref 5 , a close correspondence is noted between the classical stability parameter (the classical rate of escape) and the quantum bandwidth. The classical spectra become negative for certain frequencies (no more than 10% of the peak maximum), and it seems best to set such regions to zero. The negative values occur because of the square root in eq 4 and because only 16 trajectories represent the distribution P,(t).
The method works because wave packets follow classical-like paths for some time, and, for example, when (gJg(t)) is large, the trajectories will likewise be found near p,(p,q). Thus, the qualitative features of the amplitude
J. Pbys. Cbem. 1980, 8 4 , 2001-2004
of the integrand in eq 1 are reproduced in the approximation embodied in eq 4 and the approximate spectrum. In order for a vibrational structure to be seen in a specrum, one must have classical trajectories which make return visits to the vicinity of their initial conditions (the reciprocal of this return period will give the spacings between overtone bands), but the swarm of trajectories representing p,(t) gets increasingly filamentary, or its center moves away from its original position upon repeated passes, or both, with the result that the periodic maxima in the trace (right-hand side of eq 4) decay from one to the next, resulting in broadened overtone bands. Typically, relatively small displacements in phase space of p,(t) will cause significant reduction of the overlap [Tr ( p p,(t))l1l2. Thus, the bandwitlth-determining decay may take place without much energy relaxation out of the prepared mode, for a displacement may affect the overlap with little effect on the mode energy. This provides a semiclassical means of understanding energy relaxation effects as separate from dephasing or overlap effects. Thus, the result7 that T1 relaxation is classically very much longer than the 50 fs implied by the bandwidths in benzene is indeed indication of a T,dominated mechanism. Following the original “dephasing”, which gives rise to the bandwidths of the overtone structures, the trajectories
2001
may return en masse to their original vicinity, causing structure to be seen under the bands, as in Figure 1. When this return time is correctly predicted by the classical mechanics, the classical spectrum will show structure under the overtone bands with the correct spacing between the “combination” lines. Acknowledgment. The authors acknowledge Dr. E. B. Stechel for use of a computer program for obtaining spectra and time dependence from the wave functions and for helpful. conversations. This work was supported by NSF Grant CHE77-13305. References and Notes (1) R. L. Swofford, M. E. Long, and A. C. Albrecht, J. Cbem. fbys., 85, 179 (1976); R. L. Swofford, M. E. Long, M. S. Burberry, and A. C. Albrecht, ibid., 66, 664 (1977); R. L. Swofford, M. S. Blurberry, J. A. Monell, and A. C. Albrecht, ibld., 66, 5245 (1977); M. S. Burbeny and A. C. Albrecht, ibid., 71, 4631 (1979). (2) J. W. Perry and A. H. Zewail, J. Chem. Pbys., 70, 582 (1979). (3) R. G. Bray and M. J. Berry, J. Cbem. Pbys., 71, 4909 (1979). (4) E. J. Heller, J. Cbem. Pbys., 88, 2077, 3891 (1978). (5) E. J. Heller, E. 8. Stechel, and M. J. Davis, J. Cbem. Pbys., 71, 4759 (1979); ibid., to be submitted for publication. (6) See, e.g., K. Imre, E. Ozizmir, M. Rosenbaum, and R. F. ,Zwelfel, J. Math. fbys., 8, 1097 (1967); E. J. Heller, J. Cbem. Pbys., 68, 2066 (1978). (7) P. J. Nagy and W. L. Hase, Chem. Phys. Lett., 54, 73 (1978); cmatum, ibid., 58, 482 (1978).
Site-Dependent Vibronic Line Widths and Relaxation in the Mixed Molecular Crystal Pentacene in p-Terphenyl R. W. Olson and M.
D. Fayer”
Department of Chemistry, Stanford University, Stanford, California 94305 (Received: May 29, 1980)
The different line widths and shapes observed in absorption spectra of the first vibronic transition of the four sites of pentacene in p-terphenyl are examined. A model involying vibrationally induced line narrowing and lattice coupling is suggested to explain the differences. The iiportance of local environment in vibrational relaxation is discussed. Introduction Intramolecular vibrational relaxation is an area of substantial interest for molecules in gas, liquid, and solid phast3s.l In all phases the density of final states and the nature of the interactions with the final states are important in determining relaxation and dephasing rates. In solid-state systems, coupling to lattice phonons, either directly or via intervening intramolecular vibrations, can affect the relaxation and dephasing processes. Here we present an example in which pentacene molecules in different sites of a p-terphenyl host lattice, hence experiencing the same bulk phonon environment, exhibit substantially different vibrational dephasing properties. This demonstrates that subtle details of the local environment must be considered to obtain a clear understanding of vibrational dephasing. Pentacene in the low temperature p-terphenyl lattice exhibits four distinct sites.3 These sites result from the p-terphenyl phase transition at -190 K in which p-terphenyl ring rocking freezes The final ring distribution causes the low temperature unit cell to contain four conformationally different p-terphenyl molecule^.^ Each pentacene molecule presumably replaces one p-terphenyl *Address correspondence to this author. Alfred P. Sloan Fellow. 0022-3654/80/2084-2001$01 .OO/O
molecule in the lattice, yielding four conformationally different pentacene environments. Since these crystals are grown well above the phase transition temperature, pentacenes populate the four sites equally. Each of the four So S1origin transitions is Gaussian, characteristic of inhomogeneousl) broadened lines. The first vibronic transition of each site might be expected to exhibit a line shape which is the convolution of the inhomogeneous Gaussian width of the origin and a vibrational-lifetime-broadened Lorentzian. Although the three lower energy sites’ vibronic lines are of about the same ___________ width, only one exhibits the ext3ected shaDe. while one is a pure Gaussian and one is a i u r e Lorentzian. The first vibronic transition of the fourth (highest energy) site differs strikingly from the others. It is Lorentzian in shape, much narrower than the other sites’ vibronic transitions, and considerably narrower than its origin. We wish to report and discuss these four vibronic transitions and propose a static and a dynamic model to explain the observed behaviors.
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Experimental Section Absorption spectra were redorded at 1.4 K by using a xenon arc lamp, a 3/4-mmonochromator (slits set for 0.1-A resolution), and a 1P28 phototube. Numerical convolution 0 1980 American Chemical Soclety
