J. Phys. Chem. 1992, 96, 8006-8022
8006
which has 9 and 36 degrees of freedom, where xli is the ith rate constant used in the calculation of a mean value at thejth temperature for HNS samples and y, is the corresponding value for the deuterated HNS derivative. ( 3 4 Thornton, E. K.; Thornton, E. R. In Isotope Effects in Chemical Reacrions;Collins, C. J., Bowman, N. S., Eds.;Van Nostrand Reinhold: New York, 1970; ACS Monogr. 167, pp 213-235. (38) Halevi, E. A. In Progress in Physical Organic Chemistry; Cohen, S. G., Streitwieser, A., Jr., Taft, R. W., Eds.; Wiley-Interscience: New York, 1963; Vol. 1, pp 109-221. (39) Saunders, W. H., Jr. In Techniques of Chemistry: Inuestigation of Rates and Mechanisms of Reactiom, 4th 4.; Bernasconi, C. F., Ed.;Wiley: New York. 1986: Vol. 6. Part 1. DO 592-601. (40) Chang, H.; Yang, C.; dden, Y.; Chang, C. Int. Annu. Conf. ICT 1987, 18th, 51.
(41) Crampton, M. R.; Routledge, P. J.; Goldmg, P. J. Chem. Soc., Perkin Trans. 2 1982, 1621. (42) Kinstle, T. M.; Stam, J. G. J . Org. Chem. 1970, 35, 1771. (43) Chin, W. S.;Mok C. Y.; Huang, H. H. J . Am. Chem. Soc. 1990,112,
3n51 ----.
(44) Turner, A. 0.;Davis, L. P. J . Am. Chem. Soc. 1984, 106, 5447. (45) Chow, Y.L. In The Chemistry of Functional Groups. Supplement F The Chemistry of Amino, Nitroso and Nitro Compounds and Their Derivatives; Patai, S., Ed.; Wiley-Interscience: Chichester, 1982; Part 1, p 214. (46) McMillen, D. F.; Golden, D. M. Annu. Rev. Phys. Chem. 1982,493. (47) Batt, L. In The Chemistry of Functional Groups. Supplement F: The Chemistry of Amino, Nitroso and Nitro Compounds and their Drrivarives; Patai, S.,Ed.;Wilcy-Interscience: Chichester, 1982; Part 1, p 418.
Mode Selectivity in the Classical Power Spectra of Highly Vibrationally Excited Molecules Thomas D. Sewell: Donald L.Thompson,* Department of Chemistry, Oklahoma State University, Stillwater, Oklahoma 74078 and R. D. Levine The Fritz Haber Research Centerfor Molecular Dynamics, The Hebrew University, Jerusalem 91 904, Israel, and Department of Chemistry and Biochemistry, University of California, Los Angeles, Los Angeles, California 90024-1569 (Received: December 9, 1991; In Final Form: June 1, 1992)
Classical trajectory computations are used to generate power spectra for polyatomic molecules (C2H4,CH,ONO, and SiF,) at energies both below and above the dissociation threshold. Realistic force fields are employed for each of the three molecules that were studied. A variety of initial conditions were employed, and the trajectories were integrated for 10 ps. The two primary observations are that, on that time scale, spectral mode identity is essentially retained even well above the threshold for dissociation and that mode mixing (or "intramolecular vibrational energy redistribution") is not global. Molecular rotation increases the degree of mode mixing. For ensembles of localized initial conditions the trajectories fail to representatively sample the available phase space, even at energies where significant numbers of trajectories dissociate within the 10-ps time interval.
I. Introduction There is extensive and significant experimental evidence for a rapid loss of the chemical memory of selective initial excitation The evidence comes not only from of polyatomic collisional deactivation of chemically and photophysically activated molec~les'3~9'~3'~ but also from other activation processes including electron impactI5 and multiphoton p ~ m p i n g . ' ~This * ' ~ evidence is further supported by theoretical and computational ~tudies.2"~ The latter are often, but not a l w a y ~ , ~ classical ~ - ~ ' trajectory calculations. The evidence from such studies is less clear-cut, and we shall return to this point below. There have, however, been extensive studies supporting the lack of mode specificity. There are several reasons why, despite the available evidence, there is a continuing interest in the possibility of mode-selective chemistry.+I2 First is the progress in high-resolution spectroscopy of highly excited molecule^."^^^ These studies, which are in the frequency domain, provide a window into the short time intramolecular dynamics.46s1 To be sure, often (but not alwayss2)the information provided is about time intervals short compared to typical dissociation lifetimes of polyatomic molecules in the post-threshold region. Even then, there is increasing evidence for incomplete mode mixing over hundreds of vibrational periods,s3 say of the order of 10 ps. While this time is short it is neither too short for direct experimental probings4nor does it require very high excitation energies and remains in the chemically interesting and accesible range. In other words, it is not obviously the case that photodissociationeither is direct (i.e., within one or very few 'Present address: Department of Physical Chemistry, University of Goteborg, S-41296 GBteborg, Sweden.
vibrational periods) or is p r d e d by an essentially representative sampling of phase space prior to dissociation. Secondly, there is increasing direct evidence for mode selectivity. The earlier results came from van der Waals m0lecules,5~~~ but more recently chemically bound small polyatomicssM2are exhibiting similar selectivity. These studies have been carried out in close cooperation with theoretical a~tivity.~~-~O Finally, the increasing interest in the possibility of c o n t r ~ l l e d dissociation ~ ~ - ~ ~ also requires a better understanding of short time intramolecular dynamics-all of this without mentioning more specialized topics, such as Coulomb explosion^,^^ which have been reviewed within the general framework e l ~ e w h e r e . ~ ~ ~ ~ ~ The concluding discussion at a recent meeting12further showed to us that the (old) question78of possible chemically significant effects of selmive vibrational excitation of polyatomic molecules is not yet resolved. Even the qualitative dependence of the time scale on the excess energy for extensive mode mixing is not fully clear. There are useful indicators such as local instability of the resonance overlap,*23 and avoided c r o s ~ i n g s .Y~et~ ~ ~ computational and theoretical considerations of others as well as our own42*43,76,77 suggest to us that for a realistic polyatomic molecule the energy redistribution can be incomplete even well above the dissociation threshold. Below we present a computational study of three quite different molecules (CH30N0, C2H4,SiF4) and offer the interpretation that the results show that even when the molecules can dissociate they do not sample sisnificant regions of the available phase space. It is probably the case that, for much longer times (say, below or above reaction threshold energy regions), the sampling of phase space will be much more complete. However, on a 10-ps time
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Highly Vibrationally Excited Molecules
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At energies above the threshold for isomerization (e.g., in scale, a scale of interest in both spectroscopy and kinetics, the CH,ONO), there will be new spectral features. Overall, however, exploration of phase space is only partial. the chemical signature of the cold molecule is nowhere fully erased. There are several (related) reasons why the question we raise Pure intramolecular vibrational redistribution (IVR), unaided by has not, thus far, been resolved in a completely satisfactory fashion. power broadening and/or collisional relaxation, is not, in our Presumably the very same reasons will prevent our computational computational study, sufficient to lead to a nearly structureless evidence from being persuasive. We hope, however, that by power spectrum. At the zeroth approximation,almost the opposite focusing attention on what we consider to be important issues, is true. Even above the threshold for dissociation, mode mixing we have a positive contribution to make. is not extensive enough to erase the spectroscopicsignature of the The basic issue appears to us to be the signature of chemically molecule at a chemical level of accuracy. For even a moderately signticant mode mixing. The point is that, despite the tremendous large molecule, the (quantal) zerepoint energy can be comparable recent progress in nonlinear d y n a m i c ~ ? ~it 1does ~ not quite provide to a bond dissociation energy. We further find that, in classical the tools that we need. The first problem is that of time scale. mechania, a molecule excited only to its quantal zerepoint energy Our real interest is in molecules above the threshold for dissowill typically retain its mode structure quite well and be quite far ciation. As is also the case for chaotic behavior in scattering,a5 from chaotic. the system only spends a finte time in the interaction region before dissociation takes place. A representative sampling of phase space The conclusions of a computational study are, of course, only needs to take place on the unimolecular time scale.” As the energy as good as the input. There are several important limitations in our work. The potential energy surfaces used are all taken from is increased, the time available to sample phase space is progressively shorter. Moreover, what constitutes a representative previous studies (except for SiF,). The mode frequencies computed for them are in good accord with experiment or ab initio calcusampling? One clearly does not require that the density in the lations. Yet, one could argue that a more realistic potential will bound part of phase space is uniform on the scale of hours. There include stronger higher order anharmonic couphgs. It is certainly is surelys6 not enough time to establish this finest grained uniformity. the case that when the total angular momentum is not zero, so IVR is possible, we see more facile that rotationally The second essential difficulty is that the measure we need is energy redistribution. Hence one could, presumably, by including neither purely local nor purely global. Our interest is in the additional modemode couplings, obtain a more chemically chaotic dynamics following a selective excitation. The initial state is thus behavior. However, for the present potentials, which are deemed localized. Local instability is however not sufficient. Studies of reasonable even above the dissociation threshold (and have been the survival probability as derived from experimental spectraa7 previously used in that energy range), IVR, while certainly present, clearly establish that while the rate of departure from the initially is not dominant. And note that the potential for the “most excited region can be quite high (up to the order of a vibrational nonchaotic” system, SiF,, has numerous sizable cross terms in the frequency) that is not the same as the rate of a representative force field and switching functions! The second limitation is the sampling of phase space. The latter is typically far slower and use of classical dynamics. It is not easy to assess the direction at most reaches the order of anharmonicityfrequency differences.aa of this error. At lower energies classical mechanics allows for We thus need to follow the initially excited density to other regions a more facile IVR because the zero-point energy is also available of phase space so as to verify that such sampling of phase space for redistribution. At higher energies, various classically foras does occur is not limited to a region small compared to the total bidden9apathways become allowed by quantum dynamic tunavailable volume at that energy (and angular momentum). neling.99 A third point, which is probably not relevant to us, is Neither the Lyapunov exponent^,^^^^^ (which determine the local that quantum evolution has to ultimately cease by about the, rate) nor the Kolmogorov entropy3,@(which d e t e r “ the global so-called, break th1e.8~The last important limitation in the present rate) provides the suitable measures. work is that trajectories were only computed for 10 ps. The The point of view we adopt in this paper is similar in spirit to Fourier transform relation between the descriptions in the time that used by HellerE9in his discussion of intramolecular motion. and frequency domains implies that finer details in the power We ask if, having initiated the system in a spectroscopically spectra will not be resolved. It also means that we do not know localized region of phase space (of dimensions much larger than what happens at longer times. However, at the upper range of hours), do we detect it in a spectroscopically different region. Our energies employed in this study, a si@icant number of trajectories ‘detection” is, of course, on the computer, but in principle (and already dissociate on that time scale. Certainly in the case of sometimes in p r a c t i ~ e ~ * ~it~is) ,an experimentally feasible deC H 3 0 N 0 ,cis-trans isomerization” can be very rapid. Hence, tection. Specifically, if we excite one kind of motion do we discern if IVR is only more extensive at longer times, many molecules in the (computed, power) spectrum another kind of motion. In can dissociate without having significantly sampled a substantial other words, our criterion is a double-resonance fluorescence fraction of the available phase space. “experiment”. Does excitation in one spectral region correspond to a population buildup in another region of phase space which If for chemically distinct localized initial excitations the subcould be detected either by an emission of a different frequency sequent evolution will be quite different, why do we not have more or by a delayed absorption at a different frequency? This is a experimental (or even computational) evidence for mode-selective much more direct and obvious criterion than the interesting and chemistry? One reason may well be that, for obvious and proper useful examination of the fluctuation in the spectral i n t e n ~ i t i e s ~ ~ , ~reasons, ~ what we look for in mode-selective chemistry are difwhich, however, does not immediately translate into simple dyferences in rates. For the dissociations under study, as well as namical considerations. for most other examples, there is one (or, at most, a few) passages into the products region. Even if IVR is very incomplete but the Many computations at a large variety of initial conditions were distribution over phase space settles into a form which is quasiperformed. The results need therefore to be discussed in detail, stationary over the duration of much of the dissociation, then the with proper qualifications. The short summary is, however, that passage across the transition state will become rate determining. while the initially localized excitation does spread, the more so Since different initial excitations will give rise to possibly different as the energy goes up, even well above the dissociation threshold quasi-stationary distributions over phase space, the rates can still many modes will remain relatively cold as judged by the power differ. But, as pointed out long ago,l0’ the rates will not differ spectrum. In a more global sense one can say that there is an by much, since even large differences in the distribution over phase almost surprising retention of the spectral identity of distinct space translate into only moderate differences in the rate of molecular modes. The power spectra of the highly excited molecules we considered, averaged over all possible initial conpassage. To see the technical point, consider for a moment transition-state theory. Say the energetics are perfectly known ditions, are not only not uniform but also bear an obvious and so that there is no error in the barrier height but that the disdiscernible relation to those of the cold molecule. There will, of course, be some level shifts and significant broadening~.’~ Nearly tribution of states at the transition state is not quite thermal. It degenerate modes, (e.g., those of the CH3 stretches) will merge. will require the partition function of that distribution to be deviant
Sewell et al.
8008 The Journal of Physical Chemistry, Vol. 96, No. 20, 1992 TABLE I: Comparison of Experimentd rad Calcul.tcd Normal-Mode Freppeaekd (mi1)of C2H, CalCd
exptl
freq 824 88 1 1011 1072 1181 1343
fred
freq
1444 1630 3021 3026 3103 3105b
1461 1656 2995 3015 3112 3129
WlCd
"From Herzberg, ref 104. 'Duncan, J. L.; McKean, D. C.; Mallison, P. D. J. Mol. Specrrosc. 1973, 45, 221. from its value at thermal equilibrium by a factor of 10, to get a rate constant which is deviant by the same factor. In other words, reaction rates for (quasi) stationary distributions over phase space are not very sensitive measures of the deviance from equilibrium of that distribution. If you are interested in having reaction rates that may be good news. Since reaction rates are the primary observables, our preoccupation with IVR may be misdirected. On the other hand, if we are interested in the detailed dynamics, mode-selective excitation can still be quite interesting, even if it can have only a limited effect on the reaction rate. There are two ways whereby mode-selective excitation can, all the same, have an appreciable effect on the reaction rate. One, which is not the case here, is when a particular excitation can give rise to a rather direct dissociati~n.~~ The other is if the distribution over phase space is not quasi-stationary on the time scale of interest. The point is that on the one hand the lifetime of the excited molecule can be varied by changing its energy content. On the other hand, we are acquiring a fair body of evidence, including the present study, that the sampling of phase space does not occur with a nearly time-independent rate. Rather, there appears to be a hierarchy of time scales such that the distribution is quasi-stationary in betweenss (i.e., on the previous scales). It is when the lifetime for dissociation is comparable to the time scale for the exploration of a new region in phase space that one can expect the role of selective initial excitation to be most dominant. We do not have an example in hand of this effect as the survival probabilities we compute are all nearly quasi-stationary on the time scales of present interest, as will be illustrated in some detail later on. We are, however, looking for this effect and hope to report on it in the future. The technical presentation of the results begins in section I1 with a discussion of the potential energy surfaces used. The essential details of the method of computation are given in section 111. The results for the three molecules, CH,ONO, C2H4,and SiF4, which were studied in detail are presented in section IV, where they are arranged both according to molecule and with reference to special features such as the role of overall rotation. The discussion is resumed in section V with a summary in section VI.
11. Potential-Energy Surfaces A. Methyl Nitrite. The potential-energy surface for methyl nitrite is taken from Preiskorn and Thompson.Ioo Specifically, it corresponds to "PES-111'' (for the trans conformer) used by Preiskorn and Thompson in their calculation of the rate of trans cis isomerization in CH30N0. The potential is diagonal and consists of Morse stretches for the bonds, harmonic functions for the angles, and a six-term cosine series for the dihedrals. This force field is a simplification of an experimentally derived potential-energy surface published by Ghosh and Gllnthard.'O* See ref 100 for details of the potential-energy surface. B. Etbene. The ptential-energy surface for ethene is the same as used by Sewell and Thompson1o3in their study of the unimolecular dissociation of 2-chloroethyl radical, for which CzH4 is one of the reaction products. It consists of five Morse stretches, six harmonic bends, two harmonic wag-angle bends, and four dihedral potentials. The force field was adjusted to yield frequencies that compare favorably with experimental values. Details of the potential for ethene are given in ref 103.
-
TABLE U Ab Iaitioo Structure 8od Fmoencied for SIF. modes geometric parameters daignation frm (em-9 E(v7&) 271.6 (2) rs+F = 1.56150 A 6F-Si-F
= 109.47122'
403.0 (3) 842.4 F A Y I ~ ~ V M , Y 1099.5 ~J (3)
F2(Yg,Y1o1Yll)
Ai(viJ
"Gordon, M. S. Private communication.
TABLE Ilk Copprriao~~ of rb Initio 8od Empirid H a s h Matrix EkmesQ for SiF, ?f!J 7fli rel. i i (ab initio)' (empincal)' 96 error magnitude 1 1 1 4 4 4 4 4 4
1 4 5 4 5 7 8 9 13
0.688 406 -0.172 099 0.127 091 0.180 987 -0.144939 -0.008 334 0.017 845 0.005 204 0.007 778
0.735 766 -0,183942 0.118935 0.173 076 -0.135411 -0.005 386 0.016 477 -0.000 225 0.021 638
-6.897 676 -6.881 269 6.417844 4.371 316 6.573 863 35.377 580 7.669 272 104.319659 -178.197 711
1.OOOOOO -0.249 996 0.184 616 0.262 908 -0.210543 -0.012 107 0.025922 0.007 559 0.011 298
"Gordon, M. S. Private communication. bRedundant force field. The correspondence between the experimental and calculated normal-mode assignments is given in Table I. Note that some of the normal modes are switched relative to the those given in Herzberg.lW The agreement between the calculated and experimental out-of-plane frequencies is not very good; however, the out-of-plane modes do correspond to the three lowest frequencies in both cases. According to Herzberg, the symmetry of out-ofplane modes u7 and us are reversed. The in-plane modes are in better agreement, except that modes u j and Y6 are reversed. The correspondence between the results obtained using this empirical force field and the experimental data could be improved (e.g., by including nondiagonal terms in the force field and/or using a least-squaresoptimization to give closer agreement with the experimental frequencies). The original purpose of the potential-energy surface was to describe one of the reaction products of 2-chloroethyl radical decomposition, and inclusion of the nondiagonal terms would have likely resulted in undesirable behavior. Thus, the approximate force field was deemed acceptable for its role in the reaction dynamics study. We think that it should be suitable for the work we describe here as well, since we are only trying to answer a qualitative question concerning the vibrational dynamia in "large" molecules. C. SiF,. The force field for SiF4 is based on unpublished results105of a dculation performed using the GAMESS molecular orbital package.1o6 Starting with an ab initio structure and Hessian matrix,lo5a (3N- 6) X (3N- 6) force field was obtained in terms of a set of nonredundant internal coordinates (four bonds and five angles). (Thisis a standard option in GAMESS.)However, since the resulting force field only includes five of the six valence angles in SiF4, it is unsuitable for dynamics calculations at high energies-for example, the F-Si-F angles do not all experience the same restoring forces for large displacements from equilibrium. Also, many of the offdiagonal interaction force constants are quite large. This can lead to aphysical behavior in dynamics calculations. The ab initio structure and frequencies are given in Table 11. For the present purpose, all of the bond-angle and some of the angleangle interactions were removed, and the remaining force constants were then adjusted to give agreement with the ab initio results using a least-squares optimization. This procedure leads to a nearly exact match between the ab initio and empirical frequencies but introduces some error into the empirical Hessian. However, the magnitude of the deviation between ab initio and empirical Hessian matrix elements was found to be rather small. This is demonstrated in Table 111, where we compare representative Hessian matrix elements obtained using ab initio to the
The Journal of Physical Chemistry, Vol. 96, No. 20, I992
Highly Vibrationally Excited Molecules
8009
TABLE Tv: R e d h t Force Field for SiF, Used h Dynamics CdcUl.tioa9
DiagonalForce Constants
k,,,Ji=1,4) = 994.747 kcal mol-' A-2 kb&'=1,6) = 177.600 kcal mol-' rad-*
Bond-Bond Interaction Force Constants j) = 49.496 kcal mol-' A-2
k,,(i= 1,4;i= I&#
AngleAngle Interaction Force Constants
kb,+,(i=1,6;i= 1,6;i# j) = 40.4 kcal mol-' rad-2
corresponding ones based on the empirical force field. We also show the percent error in the Hessian elements (assuming the ab initio values to be "correct") and the magnitude of the individual elements relative to the largest element in the matrix. The largest errors occur in the elements having the smallest magnitude. Although some of the elements are in error by as much as 178%, the relative size of such terms is quite small. The larger terms tend to have an error of roughly f61. Anharmonicity was included in the bond stretching terms by replacing the harmonic oscillator terms used to fit the force field with Morse oscillators. A typical value of 135 kcal/m01'~~ was used for the well depth of the Si-F bonds and the range parameter, a,was obtained using a = (kr/2D,)'/'
(1)
where k, is the diagonal bond-stretching force constant and 0, is the well depth. The force field used in the dynamics calculations reported here is given in Table IV. The potential-energy surfaces for C H 3 0 N 0 and C2H4completely neglect attenuation of the forcefield parameters as reaction occurs (through dissociation and, in the case of CH30N0, isomerization). The force fields for SiF4 are attenuated as a function of the four Si-F bond lengths. Attenuation of the geometrical parameters has been neglected. The appropriate diagonal angle-bending force constants and off-diagonal bond-bond, angl-ngle, and bondangle interaction constants are attenuated to zero as one (or more) of the Si-F bond lengths becomes highly extended. The attenuation is accomplished by writing a given force constant as where qj and qj are the internal coordinates explicitly involved in a given potential-energy term (diagonal bond-stretching terms are not attenuated) and S(rk)is a switching function dependent on the length of bond k. The switching function used is
(3) The product of switching functions appearing in eq 2 includes factors for all Si-F bonds involving F atoms that are involved in the definition of the internal coordinates qj and q,. Thus, while the attenuation of the potential-energy surface does not account for changes in the equilibrium geometry or force constants in the product radical, SiF3, forces that are present in SiF4but absent in SiF3 F diminish to zero as one of the Si-F bonds dissociates.
+
III. Computational Methods A. Trajectory Integration. Classical trajectories were computed in a spacefixed Cartesian coordinate system using a fourth-order RungeKutta4ill integrator with a fixed step size of 1.5 X s, except a step size of 7.5 X lo-'' s was used for the C2H4 trajectories at 155 kcal/mol total energy. In most cases,ensembles of 20 trajectories (each of 10-ps duration) were computed for a given distribution of energy. The internal coordinates were recorded after every 3.0 fs of trajectory time. The GenDyn" classical trajectory code was used to propagate the trajectories. In the case of SiF4,the standard potential-energy functions included in GenDyn were modified in order to include the attenuation of the potential-energy surface. B. Normal-ModeAnalysis. Normal-mode analysis was accomplished by taking analytical second derivatives of the potential-energy surface (another standard option in GenDyn).
0.0
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Figure 1. Composite power spectrum for CH30N0 vibrating at the zerc+pointenergy (29 kcal/mol-approximately twice the classical energy
required for isomerization about the CON0 dihedral angle). The spectrum is a superposition of the power spectra for the 16 internal coordinates of the molecule. The results correspond to an ensemble of 20 trajectories initiated by partitioning zero-point energy into each normal mode (with appropriate phase averaging). The peaks are broadened compared to the analogous result computed for a total energy of 0.06 kcal/mol (not shown), but most of the peaks are still quite sharp and can be mapped onto particular normal-mode frequencies (with allowance for minor red shifts in some of the modes). There is a triplet of C-H mode in CH30N0at very low energy; they have broadened into a doublet at this energy. Switching functions were not explicitly included in the normalmode analysis, but since the second derivatives of the switching functions (evaluated at equilibrium) are zero, this poses no problem. C. Selection of Initial C ~ n d i t i Two ~ ~ . general methods were used to select initial conditions for trajectories. In the fmt method, the normal-mode frequencies are used to define the mode energies (assuming harmonic oscillator energy levels)lo9
Ej = (nj
+ t/z)hvj
(4)
where ni and vi are the number of quanta of energy to be partitioned into mode i and the normal-mode frequency of mode i, respectively. Then, Ei of energy is partitioned into mode i with the phase of the normal mode selected randomly from the a p propriate distribution.'I0 In some cases it was desired to excite a high-frequency C-H "local mode". Under such circumstances, zerepoint energy was partitioned into each normal mode using the above prescription and the C-H bond was then excited to an energy consistent with the allowed energy levels of a nonrotating Morse (assuming that it is separable from the remainder of the molecule). Following the initial partitioning of energy, the total linear and angular momentum is removed and the energy is scaied exactly to the desired value, resulting in a nonrotating molecule vibrating with the center of mass located at the origin of the lab-fixed coordinate system. The details of these methods are described elsewhere"*"2 and are not reproduced here. Initial conditions obtained using these methods are generally called "quasi-classical" initial conditions since the molecules are prepared in a loosely defined "state". The other method for selecting initial conditions is based on a random walk over the reactant phase space followed by selection of atomic momenta (just prior to trajectory integration) so as to ensure a microcanonical ensemble.' 13,' l 4 Initial conditions generated using this procedure are "random" in that no attempt is made to characterize the system as being in a particular state. The only specified constants are the total energy and (sometimes) the angular momentum. The algorithm for selecting initial conditions using this method is detailed in refs 113 (for L = 0) and 114 (for no restrictions on L). Starting with the molecule at equilibrium, we performed
8010 The Journal of Physical Chemistry, Vol. 96, No. 20, 1992
Sewell et al. r -
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Figure 2. Power spectra for the individual internal coordinates for the case described in Figure 1. The intensitiesin this figure are relative to the internal coordinate giving the largest intensity (the N = O bond). In cast8 for which there are more than one chemically equivalent internal coordinate (e.g., the C-H bonds), only one spectrum will be shown. The panels are (a) C-H bond, (b) C-0 bond, (c) 0-N bond, (d) N - O bond, (e) HCH angle, (f) HCO angle, (g) CON angle, (h) O N 0 angle, (i) CONO dihedral angle, and (j) HCON dihedral angle. Of particular interest in the context of the present work is the existence of several spectra that contribute intensity in only one region of the spectrum (and for which they are the major contributor of intensity in that region). Among these are the C-H bonds (panel a), the N 4 bond (panel d), the HCH angle, and the CONO dihedral angle (panel i). Many of these bands persist for energies up to and in excess of the bond-dissociation threshold.
The Journal of Physical Chemistry, Vol. 96, No. 20, 1992 8011
Highly Vibrationally Excited Molecules a warm-up walk of 1OOO00 steps and then selected a set of atomic momenta prior to calculation of the first trajectory. After the first trajectory, the random walk was reinstated with 1OO00 steps taken between successive trajectories. D. Power Spectra Power spectra were computed as the Fourier transforms of the ensemble averages of the autocorrelation functions of the time histories of some variable q. The Fourier transform is115 Z(v) = x:dT
( Cqq(7))e-**'"'
(5)
where Y is the frequency at which the intensity Z(v) is being evaluated and ( Cqq(7))is the ensemble averaged autocorrelation function of the variable q:
(Cqq(4) =
1
- WI
lim i S T d t [ q ( t ) - ( q ) ~ [ q ( t + T )
N ~ - m
T
0
(6)
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T+=
T
(7)
The positive and negative limits on the integral result from the fact that ( C q q ( ~is) )an even function. In practice, the autocorrelation function for an individual trajectory was approximated by the summation N-m
C[dtj)- (q)12
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The total length of the trajectory is T,and T is a time lag (which may be both positive and negative). The average value of q(t), ( q ) , is over the values of q for an individual trajectory. The autocorrelation function ( Cqq(7))is an even function of T having a maximum value at T = 0. The normalization in q 6 results in a function having the properties C(0) = 1.0 and lC(~)1 I C(0). Since the autocorrelation function is an even function of T , the complex Fourier integral can be reduced to a cosine transform
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where (j m) I N. The approximation to Cqq(f)was inserted into the cosine transform, and the integral was evaluated numerically using a Simpson's rule algorithm.l16 In this work we have chosen to transform the time histories of the internal coordinates. It is equally valid to transform, say, the Cartesian momenta or the normal coordinates. The reason for selecting the internal coordinates is that they provide a relatively "intuitive" description of the molecular motions (bond lengths, bond angles, etc.) and are independent of any spacefmed reference frame. By contrast, the Cartesian momenta are not readily associated with particular vibrational modes. The normal coordinates do not provide a convenient alternative since we are, in some cases,examining the influence of angular momentum on the power spectra,and we would have to set up a rotating frame with respect to which the normal modes could be calculated for rotating molecules. In interpreting the results we use the conventional interpretation of the power spectrum in electrical engineering and other applications of stochastic processes;"' that is, the power spectrum is the frequency density in the motion of the coordinate q. The point is that this interpretation is equally valid in quantum me~ h a n i ~ s ,when ~ ~an~average ~ ~ . over ~ ~initial , ~ conditions ~ ~ ~ ~has ~ been carried out, as is the case here. The time autocorrelation functions provide a complementary point of view of the dynamics. Particularly important to us is their interpretation as "survival probabilities". That is, we regard C,(t) as the probability to be in the region initially sampled (with each point in phase space having the weight q) at the time t. The decline
Figure 3. Composite power spectrum for CHJONO vibrating with 71 kcal/mol total energy. Initial conditions were selected as states in a Markov chain. The total angular momentum is zero.
in the time autocorrelation function with t is thus indicative of the exploration in phase space.51 Of course, to know where the initial density goes, we need also to study the cross-correlation functions and we intend to return to this point. Here we will consider the self-correlation functions of the different modes. If the density in phase space beannes quasi-stationary,then so should these correlation functions. We reiterate however that they can become quasi-stationary due to the density being about uniform but over a restricted portion of phase space.88
IV. Results Power spectra and time autocorrelation functions ("survival* probabilities) have been computed for a variety of initial conditions, both at increasing total energy and for different, mode selective, initial conditions at about the same total energy. An ensemble of 20 classical trajectories was typically employed to mimic the distribution of classical initial conditions over a specified initial region in phase space. With very few trajectories in the ensemble there can be, particularly so at lower total energies, more structure in the power spectra than in the results presented below. Beyond a dozen or so trajectories in the ensemble, the resulting spectra is not sensitive to the number of trajectories employed. The energy range shown spans from the mepoint energy (zpe) to well over the threshold for dissociation. Not shown are the power spectra at total energies well below the zero point. The resulting spectra are very sharp (Le., they are Fourier transform limited) and serve to verify the harmonic frequencies of the different modes. For each molecule we display both the aggregated power spectra obtained by adding the contribution from the different modes and the power spectra corresponding to the time autocorrelation function of particular modes. The computational details are given in section 111. Also discussed therein is the interpretati~n'~-~' of the power spectra. Since this interpretation is important and provides a central motivation for examining the results,we reiterate the point here. The power spectra, Z(v), that are displayed show the probability density of a transition at the frequency v apart from a given transition. Z(v) will be small either if there are hardly any transitions at the frequency interval v, v + dv or if such frequencies have only very weak intensities. Our main points derive from the following: general features of the types of power spectra: (i) While even at a total energy as low as zpe, the power spectra are broadened (as compared to their limiting width at a very low total energy), the spectral identity of the different modes is clearly evident. Further broadening surely occurs as the energy is increased, but only to a quite moderate extent. Even well above the dissociation threshold, the main spectral features remain distinct. (ii) The power spectra of in-
Sewell et al.
8012 The Journal of Physical Chemistry, Vol. 96, No. 20, 1992
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Figure 4. Power spectra for the individual internal coordinate for the case shown in Figure 3. The panels are (a) C-H bond, (b) C-O bond, (c) 0-N bond, (d) N I O bond, (e) HCH angle, (0 HCO angle, (g) CON angle, (h) ON0 angle, (i) CONO dihedral angle, and (j)HCON dihedral angle. The N - 0 bond is still quite localized in frequency space and, once again, is asentially the only source of intensity for the sharp band centered at 1660 cm-'(panel d). This is particularly intercsting in light of the fact that the band has shifted about 85 cm-' due to mechanical anharmonicity. An intriguing behavior is that the CONO dihedral angle a h gives a welldefined rpcctrum in spite of the fact that the total energy is nearly 5 times the isomerization barrier (panel i). The increased complexity in the low-frequency range is evidently due to the heavy-atom bending mod@ (panels g and h).
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dividual modes remain relatively distinct. The lack of overlap of spectral features in the aggregated power spectra, as noted in point i, is thus a result of lack of sufficient mode mixing. Exciting, say, the CH3 stretch modes of C H 3 0 N 0 does not result, on the 10-ps time scale of the present computation, in intensity buildup in, say, the region of ON stretch. (iii) The nonuniformity in the sampling of phase space is equally well demonstrated by the survival probabilities computed for different modes at the same total energy. A. CH,ONO. The power spectrum for C H 3 0 N 0 vibrating with zero-point energy (zpe)is shown in Figure 1. This plot is an aggregate of the contributions from individual modes some of which are shown in Figure 2. Note that only one member of each type of mode (e.g., CH stretch) is shown in Figure 2, but all modes are included in making up Figure 1. These two figures, while at a rather low total energy (29 kcal mol-I), are typical of the results at much higher energies: the spectral features are broadened by the intramolecular coupling, but they are quite distinct. Apart from obvious further broadening (e.g., the CH3 triplet will merge into one envelope), the spectra will not change in an essential way. The individual power spectra in Figure 2 also serve to identify which modes do mix at the higher energy. Figures 3 and 4 are analogous to Figures 1 and 2 except that the molecule now contains an additional 42 kcal mol-' of energy (for a total energy of 71 kcal mol-'). Initial conditions were selected by using a Markov walk. The three CH stretch modes have merged in the spectrum, but the other stretches are still fairly isolated. The torsional modes, which participate in the cis-trans isomerization (whose rate is above 0.7 ps-' for such an excitao@ int',)' are mixed, several stretches are weakly mixed with the bends, and the ON stretch is strongly mixed. Overall, however, the additional 42 kcal mol-' of energy has not resulted in a qualitatively different extent of mode mixing beyond that already present at the zpe level. The picture, Figure 5, remains the same when the 42 kcal mol-' of excitation energy is initially partitioned into a single C-H local mode. The aggregated power spectrum shown in Figure 5 is rather similar to that of Figure 3. There is a somewhat different partitioning of excitation strength between the ON0 and CON bends (a feature which occurs also at higher energies), but otherwise, the degree of mode mixing remains similar and limited. The power spectra for excitation to well above the dissociation energy (excitation to 144 kcal mol-' by elevating all modes to v
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FREQUENCY [l/cml Figure 5. Composite power spectrum for CH30N0 vibrating with 71 kcal/mol total energy. Initial conditions were selected by first partitioning zero-point energy in the normal modes (as in Figure 1) and then exciting a C-H local mode to the fifth overtone ( u = 6). The total angular momentum is zero. Comparison of the power spectrum to the isoenergetic result in Figure 3 indicates qualitatively similar behavior in both band positions and relative intensities. The only obvious differenw between this spectrum and that of Figure 3 is the relative decrease in intensity of the peak centered at 270 cm-l when the C-H local mode is excited.
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Figure 6. Composite power spectrum for CHJONOvibrating with 144 kcal/mol total energy. Initial conditions were selected as states in a Markov chain. The total angular momentum is zero. At this energy, all bond dissociation channels are energetically accessible (with at least mepoint energy to spare);all of the trajectories in the ensemble reacted within the 10-ps time cutoff (via either C-0 or 0-N bond fission). The spectrum is broadened relative to the results for lower energies. Interestingly, the congestion that was noted in the low-frequency range for Figure 3 has disappeared. The two very sharp peaks between 300 and 700 cm-' are due to CON and ON0 bending modes. The spectra for these two modes are substantially different than at lower energies.
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Figure 7. Composite autocorrelation function for CH30N0vibrating with a total energy of 71 kcal/mol. The initial conditions are as described in Figure 3. The autocorrelation function is plotted on a semilogarithmic scale as In J(C(r))lversus T . (The plot is for the natural logarithm as are all the other plots of autocorrelation functions presented.) Note the initially rapid decay of the autocorrelation function. This reflects the initial departure of the density in phase space from the region sampled at t = 0. Note that in about 1 ps the envelop has assumed an only very slowly changing value.
= 2) are shown in Figure 6. All stretch modes are now quenched but still quite evident. The skeletal stretches, except for the CH's, are also overlapping, but much of that overlap is found to be due to coupling to the torsion and the bends. The time autocorrelation function corresponding to the power spectrum of Figure 5 is shown in Figure 7. The total energy is 71 kcal mol-' and is initially randomly distributed. There is clearly a quasi-stationary distribution over phase space during much of the cis-trans isomerization stage. The ON0 bending which is strongly coupled to the ON stretch is the only region in phase space which is exceptional. Ultimately, additional mode coupling does set in, but the isomerization is effectively over and equilibrated at such times. Many other plots and results for this and the other two molecules are available from the authors upon request. R GH,. Power spectra of individual modes at three different energies are displayed in the panels of Figures 8-10. Only one
Sewell et al.
8014 The Journal of Physical Chemistry, Vol. 96, No. 20, 1992
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F l p e 8. Power spectra for C2H, vibrating with zero-point energy (31 kcal/mol). Comments in Figure 1 pertaining to initial conditions selection and ensemble averaging apply here as well. At this energy, no reaction or isomerization about the 0 - C bond is possible. The spectra shown are (a) composite spectrum of the 17 internal coordinates, (b) C--C bond, (c) C-H bond, (d) HCH angle, (e) CCH angle, (fj H 2 C 4 wag angle, (g) dihedral angle (“cis”), (h) dihedral angle (-trans”). As was the case for CH,ONO, the zero-point energy spectrum consists of sharp peaks. The sharp peak at 1050 cm-’ in panel a derives its intensity from the spectrum of the CCH angle (panel e) and the out-of-plane modes (panels f-h). The remaining (non C-H stretch modes) are all complicated combinations of the internal coordinatts. Note the very narrow peak at 1173 cm-’. The fwhm of this peak is on the order of 10 cm-’ or less.
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Figure 9. Power spectra for C2HI vibrating with 77 kcal/mol of energy. Initial conditions were aclccted from states of a Markov chain. The total angular momentum is zero. The spectra shown are (a) composite spectrum of the 17 internal coordinates, (b) C 4 bond, (c) C-H bond, (d) HCH angle, (e) CCH angle, ( f ) H2C< wag angle, (g) dihedral angle ("cis"), and (h) dihedral angle ("trans"). At this energy, rotation about the DIc bond is possible (but improbable as the barrier is 63.5 kcal/mol). All of the bands that were prescnt in the zero-point energy spectrum arc still pruettl and rtsolued (compare panel a of Figures 9 and 10). Moreover, the spectra for the individual coordinates are only moderately affected. One can ersily pick out the spectrum for a given type of internal coordinate. We think that the dynamics are not chaotic!
Sewell et al.
8016 The Journal of Physical Chemistry, Vol. 96, No. 20, 1992
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Figure 10. Power spectra for C@, vibrating with 155 kcal/mol of energy. Initial conditions were selected from statee of a Markov chain. The total angular momentum is zero. Thespectra shown are (a) composite spectrum of the 17 internal coordinates, (b) C-C bond, (c) C-H bond, (d) HCH
angle, (e) CCH angle, (6H 2 H wag angle, (8) dihedral angle (“cis”), and (h) dihedral angle (“trans”). At this energy, any one of the bonds can break (although scission of the C< bond would leave laur than zero-point energy in the products). The composite spectrum is highly diffuse (panel a) and h~ sigmficanfintensity over the entire range for which the Fourier transform was performed. The spectrum for the c--C bond is still rdatively intact (panel b). Intensity due to the C-I4 boa& spans over lo00 cm-’.The low-energy fingerprints of the other io-plane internal coordimtcs are recognizable except that the domains of the btoad bands asaodated witb the C-C-H and H-C-H coordinates differ somewhat. Spectra for the aut-of-phe mrdinatc~are, for all practical puq”,inditinguhhable (aside from the fact that the spectrum of the wag angle is comidcrably brooder than those for the dihedral angles). It is clear that the extent of mode mixing is greater than is the case for the lower energice, particularly so for the out-of-plane modes.
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Figure 11. Autocorrelation functions for the internal coordinates of C2H4plotted on a semilogarithmic scale. The results are for the zerqoint energy and correspond to the spectra shown in Figure 9. Panel (a) C==Cbond, (b) C-H bond, (c) HCH angle, (d) CCH angle, (e) H 2 C 4wag angle, ( f )
dihedral angle ("cis"). Some modes decay quite slowly (DCbond, HCH angle, and CCH angle (panels a, c, and d, respectively)) while the others decay more rapidly. Note that, in the case of the modes that do decay quickly, the initially fast decay occurs within about the first picosecond and that in some cam there are significant recurrences in the autocorrelation functions. example for each type of mode is given as the others are generally quite similar. This molecule is more rigid than C H 3 0 N 0 with the result that, apart from the CH stretches, the other modes are much more similar in their spectra as the energy is increased. One would intuitively expect more mode mixing in this case, and at an excitation energy of 155 kcal mol-' (Figure lo), this is indeed borne out. One the other hand, at the ground state (zpe), the spectra are quite sharp. The time autocorrelation functions for the five types of modes (C=€ stretch,CH stretch, HCH angle, CCH angle, and dihedral angle) at the zero point of energy are shown in Figure 1 1. These. serve to illustrate a rather nonuniform sampling of phase space and stand in contrast to the corresponding result (not shown) at an excitation energy of 155 kcal mol-]. At this energy it is almost impossible to distinguish individual modes. C. SiF,. We have considered this molecule as a prototype of those often studied with infrared multiphoton pumping techniques. As discussed in section 11, the potential energy surface used in
most of the computations was of a more sophisticated kind where the cross force terms which involve an F atom are attenuated as the Si-F bond distance increases during dissociation. This attenuation does make for a discernible difference in the power spectra at the higher energies. This should serve as a caveat that differences in potential energy surfaces that cannot be discerned in the fits to the lower energy spectra or to the thermochemistry can affect the present conclusions. Figure 12 is a panel of aggregated power spectra at a series of increasing total energies, starting at zpe. The spectra at the zero-point energy (8 kcal mol-') is rather sharp and remains 80 at 15 kcal mol-' excitation which corresponds to six quanta in u9. At 27 kcal mol-', corresponding to six quanta in vis, there is a broadening which is distinct from that at the same energy when 24 quanta are placed in u7. The initial states at 42 and 76 kcal mol-' correspond to all modes at u = 2 and u = 4, respectively. Note that the broadening in the spectra affects primarily the stretch modes. The angle modes remain relatively sharp. This
Sewell et al.
8018 The Journal of Physical Chemistry, Vol. 96, No. 20, 1992
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nplln 12. Composite power spectra for SiF, for various energies and initial excitations. All cases corrcspond to zero angular momentum. Panel (a) zero-point energy in all normal modes (E = 8 kcal/mol). In the other panels, zero-point energy in all normal modes except the following: (b) v9 is excited to u = 6 (E = 15 kcal/mol); (c) v15is excited to u = 6 (E = 27 kcal/mol); (d) v, is excited to u = 24 (E = 27 kcal/mol); (e) all normal modes excited to u = 2 (E = 42 kcal/mol); (f) all normal modes excited to u = 2 (E = 42 kcal/mol); (f) all normal modes excited to u = 4 (E = 75.4 kcal/mol). The zero-point energy spectrum shown in panel a consists of four very sharp peaks (fwhm = 5 an-'). The two low-frquency peaks correspond primarily to bending modes. The peak at 842 cm-' arises from a totally symmetric mode and thus has zero intensity due to angle deformation. The peak at 1099 cm-I has a small contribution from angle-bending motion. Excitation of one of the F2modes at 402 cm-' results in a small broadening of that peak as well as the F2modes at 1099 an-'(panel b). Also, an overtone of the fundamental appears in the spectrum. Panels c and d show that different isanergetic excitations give rise to different spectral responses in SiF,. Panel c corresponds to &citation of the highest frquency stretching mode w h m a s panel d corresponds to isoenergetic excitation of the lowest frequency bending mode. The differences in the spectra indicate mode-specific IVR pathways. (This is corroborated in plots of the ensemble-averaged normal-mode energies for the two different excitations.) Panels e and f demonstrate that the bond stretching modes are less regular than are those principally associated with angle bending. The modes at 271 cm-' exhibit only minor broadening over the energy range 42-74.8 kcal/mol. The remaining bands broaden somewhat more, particularly the asymmetric stretching modes at 1099 cm-I.
can also be seen in figure 13 which shows the time autocorrelation function for the bonds and the angles for an energy of 75.80 kcal mol-'. It is evident that even at that energy the distribution in phase space is quite far from the uniform. Not shown are the corresponding results at lower energies where the differences in the survival probability for distinct initial region in phase space are even more extensive. That even at a significantly high energy SiF4 is not a strongly mode-mixed molecule and hence not what one can call 'in a quasi-continuum" must have a bearing on the question of uppumpingin an infrared multiphoton experiment. Possibly, in many real experiments much longer time scales are involved. It follows that picosecond uppumping can lead to the, so far somewhat elusive, goal of mode-selective chemistry in a multiphoton dissociation. Even at 143 kcal mol-', which is above the dissociation limit, the power spectrum shown in Figure 14 implies that the angles
are still somewhat isolated. The survival probabilities of the bonds and the angles (Figure 15) take about 7 ps to coincide. Of the three molecules studied, SiF, is the most nonchaotic, so much so that we feel that there is a real case to be made for a return to the search for mode selectivity but using faster up-pumping. The results discussed so far are for zero total angular momentum. We turn next to the role of vibration-rotation coupling. D. Rotationally-MediatedIVR. Allowing the molecules to rotate does provide for additional (Coriolis) coupling of the vibrations, and this enhanced mixing is evident in the power spectrum. Our most extreme example is C H 3 0 N 0at an energy well above its dissociation. Even for this energy rich (144 kcal mol-') molecule, the power spectra of individual modes overlap only those of their immediate neighbors. That mode mixing is really incomplete is also shown by the survival probability plots for individual modes (Figure 16). While the different CH related modes have all relaxed to a common value, the CO- and NO-
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Figure 13. Autocorrelation functions of the bonds and angles of SiF,
Figure 15. Autocorrelation functions of the bonds and angles of SiF,
vibrating with 75.8 kcal/mol of energy. The autocorrelation functions correspond to the spectrum shown in Figure 14f. Panel a is for the bonds and panel b is for the angles. The time scale for decay of the bond autocorrelation function is much shorter than that for the angle autocorrelation function. Also, the qualitative behavior differs. The bond function decay rapidly over the first one-half to three-quarters of a picosecond. The angle autocorrelation function decays nearly linearly (on the log scale) for essentially the entire time for which results were generated.
vibrating with 143 kcal/mol of energy. The autocorrelation functions correspond to the spectrum shown in Figure 16. Panel a is for the bonds and panel b is for the angles. The time scale for decay of the bond autocorrelation function is much shorter than that for the angle autocorrelation function. In this case, the autocorrelation functions for both the angles and the bonds experience rapid “short-term” decay followed by long-term limiting behavior. However, what we mean by short term depends on which coordinate is of interest. The initial decay for the bonds occurs within about 1 p (panel a). The angles require about 3 p (panel b). Also. the limiting values of the envelopes differ noticeably, with the angle modes remaining at a higher value.
V. Conclusions
5z W
c f, I
We have addressed the issue of mode selectivity, and that of chaos, in vibrationally excited polyatomic molecules at energies below and above disoociation thresholds. We selected threediverse molecules of moderate size, C2H4, CH30N0, and SiF,, and computed autocorrelation functions and power spectra to determine the nature of their dynamics. It is often assumed that the dynamics of molecules, in general, at the dissociation limits are chaotic. This assumption stems from studies of models and small molecules. However, for most polyatomic molecules, unless the total energy is quite high (well above the dissociation limit), the energy per mode is far too low to expect chaotic behavior and, of course, the energy per mode is even lower as the system scales the exit valley toward dissociation. Therefore, it is reasonable to conjecture that, unlike the well-documented ~ 8 8 of ~ very 8 few coupled anharmonic oscillators, an initially localized nmtationary distribution of ex= energy does not representatively distribute over all the available modes prior to reaction. Our computationd evidence is that on times of chemical interest (Le., reaction) this is often the case.118,119 While there is delocalization of an initially l o c a l i i excitation, typically only very few modes participate in the energy exchange. The results are documented in both the This is most strikingly illustmtd frequency and the time do”. by power spectra of individual modes. When mode mixing occurs, such spectra should contain fr@uencies of the modes involved. The computational evidence is that mode mixing is quite selective and is extensive only at the higher energies (well a h dissociation thresholds). A similar conclusion is reached when the dynamics
1
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Figure 14. Composite power spectrum for SiF, vibrating with 143 kcal/mol of energy (zpe + DJ. Initial conditions were picked from states of a Markov chain. The total angular momentum is zero. Bands associated with angle-bending modes are still relatively well defined (fwhm a 15 and 35 cm-l for the bands at 271 and 403 cm-l, respectively). The asymmetric stretching mode (a. 1100 cm-I) is rather broad. The symmetric stretch (ca.850 cm-’) band is only about twice as high as the low-intensity “noise” present at nearly all the other frequencies in the spectrum.
related modes have not. The results for rotating SiF, at the high energies are not essentially different. While the individual power spectra do show stretch-bend mixing, it requires over 2 ps for the time correlation functions to approach a nearly common value (Figure 17).
Sewcll et al.
8020 The Journal of Physical Chemistry, Vol. 96, No. 20, 1992 0.0
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Eprr 16. Autocorrelation functions of the individual internal coordinates for CHIONO rotating and vibrating with a total energy of 144 kcal/mol. Initial conditierre were ~lclectcdas states from a Markov cham. The panels are (a) C-H bond, (b) (2-0 bond, (c) 0-N bond, (d) N - O bond, (e) HCH
angle, (0HCO an&le, (g) CON angle, (h) O N 0 angle, (i) C O N 0 dihedral angle, and (j)HCON dihedral angle. Some of the autocorrelation function decay quite rapidly. e.g., those corresponding to the CH modes, the HCH angles, the HCO anglee, and the HCON dihedral angle. The functions d a t e d with coordinates that directly involve the 0-N b n d dccay slowly and exhibit an interesting structure (panels b, c, g, and h). The autaxmlation function for the 0-N bond is very interesting (panel c). It is very smooth and decays slowly for about 3 ps after which it d a y s swiftly; but thm it experiews a sharp,strong recurrence that persists until the finitetime approximation that W M usbd in computing the autocomlation functions begins to dominate. This is also evident in the autocorrelation functions for some of the other coordinates (e+, the C-0 and N 4 bonds and the CON and O N 0 angles). The origin of this behavior is not completely understood but may have to do with the fact that the e N bond becomes a translation upon dissociation (with a constant center-of-mass velocity between the two fragments).
Highly Vibrationally Excited Molecules
The Journal of Physical Chemistry, Vol. 96,No. 20, 1992 8021
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Figme 17. Autocorrelation functions for the individual internal coordinates of SiF, for the conditions described in the caption for Figure 21: (a) bond; (b) angle. Compare to Figure 17, which is for the same conditions except that there is no angular momentum. The behavior ( r a t a of dcdly and limiting behavior) for the bonds is similar in both cam. The difference in the spectra for the bending modes due to the presence or absence of angular momentum is also evident in the autocorrelation functions. The initial rate of decay of the angle autooorrelation functions for rotating SiF, is noticeably faster than for nonrotating SiF+ The difference persists for about 2 ps, after which the two results are rather similar. In contrast to CH30N0excited to a similar energy, angular momentum plays a clear role in the initial states of IVR in SiF,.
are considered in the time domain. To a remarkable degree the identities of distinct modes of vibration are preserved even above the dissociation thresholds for times comparable or even longer than the dissociation times.
Acknowledgment. This work was supported by the U.S.Army Research Office and the US. Air Force Office of Scientific Rcsearch (Grants AFOSR-90-0048 and AFOSR-89-0158). We thank Professor Mark S.Gordon for providing the ab initio results for SiF4. Registry No. C2H4, 74-85-1; CH30N0, 624-91-9; SiF,, 7783-61-1.
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