Dynamics of photoinduced reactions in hydrogen ... - ACS Publications

Dec 1, 1993 - A. B. McCoy, Y. Hurwitz, R. B. Gerber. J. Phys. Chem. ... W. David Chandler, Keith E. Johnson, Bradley D. Fahlman, and John L. E. Campbe...
0 downloads 0 Views 2MB Size
Download PDF
J. Phys. Chem. 1993,97, 12516-12522

12516

Dynamics of Photoinduced Reactions in Hydrogen-Bonded Clusters: Classical Studies of the Photodissociation of (HC1)z A. B. McCoy,t***gY. Hurwitz,g and R. B. Gerber’J**vo The Institute for Advanced Studies and The Fritz Haber Research Center for Molecular Dynamics, The Hebrew University of Jerusalem, Jerusalem 91 904, Israel, and Department of Chemistry, University of California-Imine, Irvine, California 9271 7 Received: April 27, 1993’

A theoretical investigation of the photodissociation dynamics of (HC1)2 a t 193 nm is presented. Prior to excitation, the cluster is taken to be in its rotation-vibration ground state. A quantal description of this sixdimensional wave function is computed using diffusion quantum Monte Carlo (DQMC). The photodissociation dynamics are simulated by classical trajectories in which the molecule undergoes vertical excitation to an electronic state that is repulsive along one of the HCI stretch coordinates. The initial conditions for these trajectories are sampled according to the Wigner function which was obtained from the DQMC wave function. In a significant fraction of these trajectories, there is a reactive collision in which the H atom interacts with the H’CI’ molecule to form HCl’. Of the remaining collisions, most are nonreactive, but a small number lead to H2 formation. The trajectories in which an exchange reaction occurs result typically in formation of HCI’ molecules that are rotationally and vibrationally hotter and in H atoms with lower kinetic energies than are found in the nonreactive trajectories. Resonances, in which the H atom undergoes multiple collisions with C1 and H’CI’, are observed in all three classes of trajectories. The above results indicate that this is a rich system for the study of photoinduced chemical reactivity in hydrogen-bonded clusters.

I. Introduction The investigation of the effects of solvation on chemical processes is a very important area of chemical research. As these processes are understood much more fully in the gas phase than in condensed media, a productive approach for pursuing this field is by focusing on chemical processes in van der Waals clusters, as here methods developed for small molecules may be exploited. In recent years, weakly bonded clusters have been used as a framework for studying a variety of photoinduced chemical reaction~.I-~Experimental and theoretical investigations also have probed the effects of solvation by rare gas atoms on photodissociation of 12*s9 and hydrogen halides.l@-ls As chemists, we are interested in processes in which bonds are formed as well as broken. Systems in which such processes occur are the hydrogen halide dimers. These dimers provide a simple, yet realistic, framework for studying the chemical dynamics in hydrogen-bonded systems, a topic of major interest.1-5 The photodissociation dynamics of (HX)2 is of interest on its own right as here the photodissociation dynamics provides a sensitive probe of the transition-state region of the H H X reaction, a process which has been the subject of intense experimental and theoretical scrutiny. In the present study, we have investigated the photodissociation dynamics of (HC1)2. As in any calculation, the differences in computational complexity between triatomic and tetraatomic systems is quite substantial, thereby making a study involving (HC1)2 potentially a much larger computational challenge than studies of atom-diatom clusters. The presence of H atoms means that important quantum effects are anticipated. Our assumption is, however, that the most important quantum effects are likely to be those associated with the specific structure of the initial state, the vibrational ground state of the cluster. The high energy of the system following photodissociation of one of the HCl bonds suggests that, as a first step, classical mechanics should provide

+

To whom correspondence should be addressed. The Institute for Advanced Studies. *The Fritz Haber Research Center for Molecular Dynamics. I University of California-Irvine. *Abstract published in Aduunce ACS Absrrucrs, November 1, 1993. t

a useful tool for exploring the dynamical processes on the excitedstate surface. The above implies that an important consideration is the sampling of initial conditions. This issue is important because improper sampling will lead to incorrect conclusions. Here, both classical and quantal approaches have been proposed. We have chosen to follow the work of Schinke and others (cf. ref 16 and reference therein) and use a quantal distribution to assign relative probabilities to a given set of initial conditions. Our workdeviates from earlier studies in that, instead of basing our probability distribution on the zero-order normal mode ground state wave function, a n-dimensional Gaussian, we have used the groundstate solution of the time-independent SchrMinger equation for (HC1)2 on its ground-state potential. This treatment is essential due to the strongly anharmonic nature of this system even in its rotation-vibration ground state. There are a variety of methods for computing the ground-state wave function for a floppy tetratomic s y ~ t e m . ’ ~ -One ~ ] that is extremely general and can be extended easily to larger systems is diffusion quantum Monte Carlo, developed by Anderson.22.23 This approach has the advantage over traditional basis set calculations in that it scales linearly with the number of atoms. It has thedisadvantage that theresultingenergyand wave function are evaluated statistically and consequently contain statistical noise, the effect of which may be minimized through the employment of appropriate averaging schemes. Once we have selected an ensemble of initial conditions, we propagate the dynamics using classical trajectory simulations. The results obtained here for the photodissociation dynamics of (HC1)zcan becontrasted with thoseof therecent studiesofGarciaVela, Gerber, and ~ o - w o r k e r s ~on ~ J ~the photodissociation dynamics of ArHCl. They found evidence for long-lived resonances in which the H atom chattered between the Ar and C1 atoms. There, as many as six collisions were observed. These resonance trajectories lead to a long tail in the H kinetic energy distribution which is to the red of the peak corresponding to dissociation of an isolated HCl molecule. This tail has been observed by Wittig and co-workers in recent experiments in which ArHBr was excited in the UV.15

0022-3654/93/2097- 12516%04.00/0 0 1993 American Chemical Society

Photoinduced Reactions in H-Bonded Clusters

Figure 1. Equilibrium structure of (HC1)2. The dashed lines show the

geometry of a second equivalent isomer. Here the barrier for isomerization between these two structures is 77 cm-I. In the text we refer to the HCI molecule for which the HClCI’ angle is smaller as the hydrogen-bonding monomer and the other is the free monomer. Primarily, the differences in the photodissociation dynamics of these two systems result from the fact that, in (HC1)2, reactive channels are introduced that were not available in ArHCl. The most important of these channels is the exchange reaction in which H H’Cl’ H’ HCl’. The trajectories in which an exchange reaction occurs, along with those that sample the transition state but do not react, result in a long low-energy tail in the kinetic energy distribution for the product H that is similar to that observed in ArHCl and ArHBr photodissociation. Here, the processes that lead to this distribution are much more complex and will be discussed in detail in the body of this paper. Long high-energy tails in the vibrational and rotational energy distributions of the product HCl molecule are also observed. An additional signatureof the photodissociation of (HCI)2 (compared to that of an isolated HC1 molecule) is the formation of H2, which is predicted to account for less than 1% of the total probability. The outline of the remainder of this paper is as follows. In the following section we discuss the ground-state potential used in the present calculation. This is followed by an outline of the approaches we have used to obtain the probability distribution for initial conditions. In section IV we discuss our excited-state potential, and the method for propagating the dynamics is discussed in section V. Results are presented and discussed in section VI.

+

-

+

11. Ground State Potential for (HC1)2

The dimer of HCl has been the subject of recent theoretical2k27 and experimentaP33 investigations. The equilibrium structure of (HC1)2 displays prototypical behavior of hydrogen-bonded clusters, with the H atom position being characterized as being either hydrogen bonding or free (cf. Figure 1). Experimental and theoretical evidence indicates that, in its equilibrium, the two HCl monomers are separated by 3.8 A, with the hydrogenbonding HClC1’ and free ClCl’H’ angles at approximately loo and 90°, respectively.2s-26.30 Two a b initio studies on this c l u ~ t e rpredict ~ ~ , ~a~barrier between the two equivalent configurations, depicted by solid and dashed lines in Figure 1, of 84 and 60 cm-I, respectively. These values are in accord with the experimentally observed tunneling splittings.28-33 In (HC1)2, the tunneling splittings are significantly larger than in (HF)*, and consequently larger amplitude bending motions are expected to be found in (HC1)2 than in (HF)2. Experimental’oand a b initio26 evidence indicates that the mechanism for this isomerization is through a gear type motion (01 - 02 constant) in which the dimer remains nearly planer. This motion can be seen by considering one-dimensional slices through a in-plane bending contributions to the potential. In the 0- = - coordinate, the potential is simply a symmetric double well. In the symmetric bend coordinate [e+ = 01 + 021, the potential has only a single minimum. In the present study, we are interested in what happens when one of the HCl bonds is broken through laser excitation in the ultraviolet. To properly sample these events, it is imperative that we know the extent of the motion of the atoms when the molecule is in its ground rotation-vibration state on the lowest electron

-

The Journal of Physical Chemistry, Vol. 97,No. 48, 1993 12517 surface. Thus, an appropriate model for the ground-state potential of (HC1)2 is required. The ground-state potential used here is the empirical potential of Votava and A h l r i ~ h smodified ,~~ to improve agreement with more recent ab initio calculations of Karpfen et a1.26 and experimental28-3Ofrequencies and geometry. The ab initio points of Karpfen et al. were also fit to an empirical potential with a rms deviation of 19 cm-1 by Bunker et ~ 1 . 2Because ~ we are interested in only theground-state wave function, we havechosen to use the simpler parametrization described below. Clearly, this potential should not be expected to be as accurate a reproduction of these ab initio points as is the fit of Bunker et al. Our potential contains three contributions, those due to the two HCl monomers and the interaction potential between them:

In this model, the two HCl monomers are described by Morse oscillators vmon(ri) = ~ e ( 1 - exp[-P(ri - re)])’ (2) with De = 0.169 836 6 hartree, re = 2.4086 ao, and B = 0.9881 ao-I, parameters appropriate for an isolated HCl molecule. Following model I of Votava and Ahlrichs,24 the van der Waals contribution is further partitioned into

=

‘vdW

‘clc

+

‘rep

+

(3)

‘disp

Here (4)

where the single-primed summation is over centers (described below) and the double-primed summation is over atoms. Note also that i and j represent centers/atoms in different HC1 monomers. The parameters used here are those of Votava and A h l r i ~ h swith , ~ ~the exception of (YCICI’= 1.5 ao-’, AHCI~ = 12.528 hartrees, ADD.= -1.380 899 hartrees, and ccIcIr= 17 219.286 a09 hartree. The electrostatic and repulsive contributions to the potential, given by eqs 4 and 5 , are defined in terms of a three-center model in which the H, C1, and dummy (D) atoms are given charges so as to reproduce the dipole and quadrapole moments of HCl.z4 The configuration of these three charges is constructed so that in the equilibrium they are collinear with the following bond lengths and charges:24 1.o 2.41 = r*lao -D 0.506

H-C -0.909

0.403

=q/au

As either HCl bond stretches, the dummy atom is shifted so as to keep the DClH collinear and the ratio rDCl/rHC1 constant. The equilibrium properties of this potential are found to be in good agreement with those computed ab initio26 and those extracted from the observed spectroscopy of (HCl)2.28,30 111. Ground-State Vibrational Wave Function

As was stated in the Introduction, when using classical approaches to study photodissociation dynamics, one issue that requires careful consideration is the technique used to sample initial conditions. Unphysical sampling procedures will lead to

McCoy et al.

12518 The Journal of Physical Chemistry, Vol. 97,No. 48, 1993

incorrect results. The distribution of the positions of the atoms that comprises theground-state wave function is highly unclassical, asisexpected for a systemcontainingtwo hydrogen atoms. Indeed, in (HC1)2 the zero-point energy in each of the HCl monomers exceeds the stabilization energy of the dimer, which is approximately 600 cm-l . The challenge in sampling the phase space probability distribution that corresponds to (HC1)Z in itsquantum ground state is that even a t this low energy the Hamiltonian is substantially anharmonic with strong coupling between the lowfrequency modes. Therefore, we must use a quantum mechanical method appropriate for calculating anharmonic ground states. A. Diffusion Quantum Monte Carlo (DQMC). An approach developed by A n d e r ~ o n for ~ ~ solving , ~ ~ for ground-state wave functions and energies is diffusion quantum Monte Carlo (DQMC). This method has been applied to studies of vibrations in clusters.19,~0,34,35A closely related approach, Green’s function Monte Carlo, has been used recently by BaEi6 et al.36and by Whaley et al.37in studies of quantum clusters. While DQMC is primarily a method for calculating nodeless ground-state wave functions, extensions have been proposed for dealing with states with nodes by geometrically constraining these nodes23J5J8 or by orthogonalization p r ~ e d u r e s . 3Anderson ~ gives a clear description of the general approache~.~2J~ Here, we offer a summary. The DQMC approach is basedon the idea that, for an N-particle system, the time-dependent Schriidinger equation in imaginary time (7 = it/h)

The rate of diffusion of the realizations through the barrier in the 8- coordinate is slow compared to the time scale of our calculation. Therefore, if we start with all of the psips in one of the two equivalent minima in the 8- coordinate, the psips will not tunnel to the other minimum during the time of the calculation. To take into account this symmetry, which must be present in the wave function, we symmetrized the bending wave function by equating 90(81,82) and 9 0 ( 8 r x , 8 1 + r ) . Then we performa change of variables of 8+ and e-, defined in section 11. This change of coordinates in important because the algorithm used to obtain the Wigner distribution (discussed below) is much more stable numerically in the symmetrized bend coordinates. We investigated the sensitivity of the energy and wave function to the number of psips N(7=0) = 2000-6000 and the number of time steps used in the binning procedure, and we found that the the energy and wave function are both stable with respect to these two parameters. The results presented here use those obtained with N(s-0) = 6000 and binned over 800 au in time. C. Wigner Distributions. Having obtained an internal coordinate wave function, we need a way to convert this function to a phase space probability distribution. A quantity which accomplishes this is the n-dimensional Wigner function

For convenience, we take advantage of the physically based approximation discussed above that 90is nearly separable in the internal coordinates, giving

(7) closely resembles a diffusion equation. By analogy to concentration in a diffusion equation, 9 is represented by an ensemble of diffusing particles that correspond to realizations of configurations of the molecule, expressed as a function of their coordinates. These realizations are allowed to diffuse through the 3N-dimensional configuration space with a diffusion constant in the ith dimension, Di = h2/2mi. The potential, Vin eq 7,gives rise to probabilities for creation and destruction of each realization as a function of its coordinates. The general solution to eq 7 is

where the 9,are the solutions to the time-independent Schriidinger equation with eigenvalues E,. In the limit of large T , it can be shown that the above expression is dominated by the n = 0 contribution. In other words, the probability distribution of realizations converges to the unnormalized ground-state wave function. Theground-state energy is simply the potential averaged over the final ensembleof realizations.22 It should be emphasized at this point that DQMC gives the ground-state wave function 90and not the probability density, Po2. B. Results of DQMC. We ran DQMC on (HC1)z confined to a plane. To obtain the steady-state solution, for the first 200 time steps AT = 10 au, for the next 200 time steps AT = 1 au, and AT = 0.1 au for the remainder of the run. We first ran DQMC until T = 2260 au, at which point the solution to eq 7 is already completely dominated, for all practical purposes, by the n = 0 contribution to eq 8. After this point we began to bin the psips (the diffusion particles that are realizations of possible configurations of the dimer) to construct 90.Analysis of the full five-dimensional internal coordinate wave function demonstrates that the two HCl stretching motions are nearly separable from the other three modes (as is expected from the large disparity in the harmonic frequencies). The ClCl stretching mode is also nearly separable, and we average the five-dimensional DQMC wave function over the three stretch coordinates to form a twodimensional bending wave function.

Note that in the ground state the bending modes are strongly coupled. Therefore, even to a first approximation, they cannot be approximated by separate one-dimensional contributions. We calculate $1, J.2, and cp by solving the one-dimensional Schriidinger equation, all other coordinates constrained to their equilibrium values, in a distributed Gaussian basis?

where the centers of the basis functions are given by r i and the widths by Ai. The c j are the expansion coefficients, and NGare the number of Gaussians in the basis. Here we have taken Ai = 133 ao-2 and ri+I - rJ = 0.08 a0 for the two HCl stretches (i = 1,2) andA3 = 38ao-2andr~+1-r3i=0.14aofortheClClstretch. The advantage of this form for the basis is that the corresponding exact one-dimension Wigner function is obtained analytically by evaluating

ex,[ -2Ai( ri -

$4 + r:))’]

(1 3)

The out-of-plane motion is modeled by a Gaussian with A = 4.69 ao-2 centered at the origin. This coordinate corresponds to the distance of the H’ atom from the plane containing the H, C1, and C1’ atoms. Finally, the Wigner distribution for the in-plane bending motions is calculated numerically using fast Fourier transforms.40 One of the potential problems in treating W(i,@)as a quantal phase space probability density is that it may become negative in classically forbidden regions of phase space. This is not a

Photoinduced Reactions in H-Bonded Clusters

The Journal of Physical Chemistry, Vol. 97, No. 48, 1993 12519

concern hereas theseareregionsin which weexpect the probability to be very small, so for W I0 we assume zero probability. A second issue is that the possible configurations do not necessarily correspond to zero total angular momentum. Instead, our sampling corresponds to an average total angular momentum in the center-of-mass frame of 10 h . This need not be considered as a drawback of using the Wigner distribution to sample the initial conditions, as, to lowest order, overall rotation of the system is decoupled from the dynamics occurring within the system. Also, a finite angular momentum is closer to the experimental situation than J = 0.

IV. Excited-State Potential for (HC1)z The excited-state potential for (HC1)Z is constructed from the sum of two three-body and one two-body contributions: The repulsive surface for the I l l state of HCl was obtained by fitting the ab initio points of Hirst and Guest“ to an exponential: where A = 7.9699 hartrees and a = 1.7331 ao-l. For the HCl’H’ interactions we use the GWS LEPS potential of Stern et modified by Schwenke et to agree with ab initio points in the transition region. The VC~H~CII term was obtained by multiplying the eiiltermsin the H6(3) ArHCl potential of Hutson4 by 1.4, the ratio of the well depths of the Ar245and ArC112.46 ground-state potentials. Before discussing the results, a few comments on this choice of potential are in order. First, our choice to use an ArHCl potential to model the ClH’Cl’ interactions means that in the present calculations we have eliminated the channel for C12 formation. This reactive channel requires an energetic collision of C1 with H’Cl’, but the amplitude of the bending motions in (HC1)z makes the probability of sampling an initial geometry that could lead to such a collision infinitesimally small. As a result, these trajectories are only expected to sample the longrange part of the Cl H’Cl’ interaction potential. This part of the potential is anticipated to be approximated well by VCIWCI.. Incontrast, theH + H’Cl’interactionshouldbeveryimportant in thedynamics studied here; when breaking the hydrogen bonding HCl bond, the H atom is accelerated in the direction of the other HCl molecule. The potential used to model this interaction has been used previously to model H HCI4’ and H + D C P collisions and is found to give qualitative agreement with experiment at the collision energies considered here. Further, we believe that experiments on the photodissociation dynamics of this cluster should contain valuable information regarding the transition state of the H HCl reaction that could be used as part of a detemination of the potential for this reaction.

+

+

+

V. Propagation on the Excited Surface As was discussed in the preceding section, we use the Wigner distribution to obtain probabilities for sampling initial conditions for the trajectories, These initial conditions are selected at random, and we propagateonly those for which W(i&)> 0.001% of the maximum value of the Wigner distribution. Because the Wigner distribution is a function of the internal coordinates and momenta, and the trajectories are propagated in the Cartesian coordinates, we must perform the appropriatechanges ofvariables. This is achieved by first evaluating the velocity associated with the ith internal coordinate ui = pi/meff,i,where mcff,iis the reduced mass associated with mode i, which is simply the reciprocal of the diagonal C-matrix elements given by Wilson et al.49 As the out-of-plane coordinate is ill-defined when either 81 or 02 = n r , melt., is multiplied by the volume element, sin el sin 02. The rest of the transformation is an exercise in trigonometry. Given that 90is chosen to be separable in all but the 8+ and 8- coordinates,

the only additional assumption introduced here is that the kinetic coupling between these two coordinates can be neglected. This is simply a statement that on average Gee = Grr. Once a set of initial conditions is chosen, the trajectory is run by propagating Hamilton’s equations, xi = aH/apt andbi = 4 H / a x , using the Gear method. We assume that the energy on the excited state is high enough to justify the use of classical dynamics, at least for the purpose of this first study on (HC1)2. We are encouraged by previous studies on the photodissociation dynamics of ArHCl. For this system, Garcia-Vela and co-workers demonstrated that the main dynamical effects are classical, although detailed investigation of the spectroscopy requires a quantal treatment.13.14 VI. Results and Discussion To properly sample the photodissociation dynamics of (HC1)2, we ran 2800 classical trajectories in which the hydrogen bonding and 2600 in which the free HC1 bond was severed ( c - Figure 1 for definitionsof the two HCl molecules in thedimer). We assume that the excitation is a vertical Franck-Condon promotion from thegroundelectronicstate toanexcited state in which the potential in one of the two HCl stretch coordinates is repulsive. While classical mechanics treats the two HCl configurations as distinct in the ground state of the system, the large-amplitude bending motion means that the cross sections for electronic excitation of the monomers are essentially identical. Further, quantum mechanically there will be interference effects between the electronic excitations of the two types of HCI molecules due to the fact that there are couplings between these zero-order representations of the excited electronic surfaces for (HC1)z. We assume here that this effect can be neglected. A. Reactivity. The most interesting feature of the photodissociation dynamics in (HC1)2, that was not present in ArHCl, is the possibility to observe photoinduced chemical reactions. In the discussion that follows, we use the terms reactive/nonreactive to refer to what happens after the HCl bond is broken through laser photolysis in the UV, reactive trajectories are those in which either the H or C1 atom undergoes a chemical reaction with the H’Cl’ molecule. There are a several possible outcomes for both reactive and nonreactive trajectories. In the nonreactive channel, the H atom may exit without interacting with the H’Cl’ molecule. This is the typical situation when the free HCl bond is broken. It may also undergo an inelastic collision with the H’CI’ molecule, as is observed when the hydrogen-bonded HCl bond is broken. If a trajectory is reactive, there are four possible outcomes:

H + C1+ H’CI’

-

+ H’CI’ (recombination) H + CICl’ + H’ (Cl’ abstraction) C1+ HH’ + CY (H’ abstraction) C1+ HC1’ + H’ (exchange)

HCI

Of these channels, the first two are closed by the energetics of the system and consequently by our choice of excited-state potential surface. The H + Cl contribution is solely repulsive, and as was discussed above, the C1+ H’C1’interaction is modeled after an Ar + HCl potential. Therefore, it does not contain a minimum for C12 formation. The third channel, H2 extraction, is observed with < l % probability. These reactions occur via two mechanisms: direct and orbiting collision of the two hydrogen atoms. In the orbiting collisions the hydrogen encircles the H’Cl’ molecule prior to collision. The resulting H2 molecules are observed with angular momentum as high as 10h and vibrational energies up to 9000 cm-1.

McCoy et al.

12520 The Journal of Physical Chemistry, Vol. 97, No. 48, 1993

We believe that the low probability of H2 found here is due to the restriction placed on the geometry of H + H’C1’ by our choices of initial conditions. This is a very important consequence of carrying out chemical reactions in clusters as compared to traditional gas-phase collisions. The connection between the transition-state geometry and the outcome of reactions is demonstrated by the work of Barclay et on the H + DCl reaction. Based on these results and those of Aker and Valentini on the H H’Cl’rea~tion~~ the geometriessampled by the cluster in its ground state are expected to favor the H atom exchange reaction over the H2 abstraction reaction, as we have found. The exchange channel accounts for about 10% of the total probability. It proceeds through two mechanisms. In the first, the HClCl’complex is nearly linear in the transition state. These trajectories will be referred to as direct collisions in the discussion that follows. The second mechanism, which we shall call the insertion mechanism, precedes through a transition state in which the HClCl’angle is 190’. Of these two mechanisms, the insertion dominates over the direct mechanism by a factor of 4: 1. Some care should be taken in giving quantitative significance to the relative reaction probabilities for the different channels described above. These quantities are extremely sensitive to the HH’Cl’ potential. For example, if we do not add the barrier correctionterm of Schwenke et ~ 1to the . LEPS ~ ~ surface of Stern et al.,42 we find a significantly higher total reaction probability. We also find a larger probability of H2 formation. The comparison of classical trajectory studies to the results of experiments for the H + DCl reaction indicates that the corrected HH’Cl’ potential used in the present study predicts cross sections for exchange reactions that are larger than experiment. This discrepancy is not surprising as the potential of Stern et was fit to kinetic data for the C1+ H2 reaction. These findings show how useful future experiments on (HC1)2 photolysis could prove in the elucidation of the H + H’Cl’ reactive potential surface. B. Resonances. Before discussing the spectroscopic implications of these different mechanisms, we consider the role of resonances observed in the trajectories. In a series of studies, Garcia-Vela et al. investigated the classical photodissociation dynamics of ArHC1.l23l3 One of their findings was that this triatomic system exhibited behavior that corresponds to the cage effect. This effect, which is known to exist in bulk systems, results from the fact that the H atom can undergo single or multiple collisions with the heavy atoms before leaving the interaction region. This contrasts with the photodissociation dynamics of an isolated HC1 molecule in which there can be no collisions of the photofragments. In ArHCl, Garcia-Vela et al. observed that trajectories for which the initial HClAr angle 8 was smaller than 22’ the hydrogen underwent as many as six collisions before exiting the interactionregion. The number of collisions was shown to correlatewith 8, where smaller angles resulted in more collisions. By analogy, for HClC1’ angles in this range, we observe nonreactive trajectories in which the H atom undergoes two collisions, one with the H’Cl’ molecule and one with the C1 atom, before exiting the interaction region. In contrast with the results for ArHC1, we have not found nonreactive trajectories with more than two collisions of this type. This result is a reflection of a fundamental difference between these two systems-the H atom can undergo reactive collisions with H’Cl’ whereas it cannot with Ar. The region of 8 for which reactions occur is also 8 I22O, thereby accounting for the decrease in the observed number of nonreactive resonances relative to the results for the ArHCl system. The other types of resonance trajectories occur when the H atom samples the H + H’C1’ transition state. These resonances result from collisions of the H atom with either the C1 or the H end of the H’C1’ molecule or from long-lived transition states in which H orbits around the H’Cl’ molecule before exiting the reaction region. In some of these trajectories a correlated motion of the two H atoms lasting several femtoseconds is observed. As we shall show in the following subsection, these trajectories lead

0 E 0.6 Y

$a 0.4

1

0.2 0.0

+

0.6 a g 0.4

a

0.2 0.0

0

10

20

30

40

50

Angular momentum of the HCI [in a.u.1

Figure 2. Angular momentum distribution for the classical trajectories discussed in the text. In (a) the probability distribution as a function of angular momentum (J) in units of h is plotted for events in which the hydrogen-bonding HCl bond is broken, and in (b) the analogous plot is given for the photodissociation of the free HCl bond. We distinguish the probability that arises from reactive and nonreactive trajectoriesby shading them in black and gray, respectively.

to significant transfer of H kinetic energy to internal energy in the H’Cl’ molecule. A second resonance occurs in reactive trajectories when 8 0. Here, the HC1’ molecule that is formed comes out with with very low angular momentum, and trajectories have been observed in which the product remains in a nearly collinear ClHCl’ geometry for more than 100fs. These trajectories account for approximately 20% of the total reactive probability. Needless to say, these trajectories are very different from the multiplenonreactive elastic collisions observed by Garcia Vela et al. C. Product-State Distributions. There are several ways to partition the probability distribution and, from it, analyze the trajectories and make connections to experiment. The partitionings we consider here are the angular momentum and vibrational energy distributions of the HC1 molecule and the angular and velocity distributions of the outgoing H atom. For clarity, analysis of the HCl molecule and H atom will be discussed separately. Due to the low probability of H2 forming trajectories, we have not obtained statistics required to make quantitative statements on the product distributions of these reactions. 1 . Rotational and Vibrational Distributions HCl Molecule. The total energy of the HCl molecule can be separated into three contributions: the kinetic, vibrational, and rotational energy. In the true system, quantum mechanics dictates that these latter two quantities be quantized. Because the trajectories are run classically, we will not achieve this quantization, but we can still make quantitative predictions of the range of energies that are expected to be observed. In Figure 2 we plot the angular momentum distribution for the HC1 product. This distribution, and all distributions presented below, are constructed by weighting the results of each trajectory according to the value of the Wigner distribution at the initial conditions. The total probability for the reactive plus nonreactive trajectories is then normalized to unity. In Figure 2b we display the results of the trajectories in which we selectively break the free HCl bond. Because here the H atom interacts only very weakly with the H’Cl’ molecule, this angular momentum distribution provides a snapshot oft he angular momentum distribution of the H’Cl’ molecule immediately following the photodissociation event. Figure 2a shows the results when the hydrogen-bonded HCl bond is broken. The histogram that is shaded in black gives the results of the reactive trajectories, and the part shaded in grey gives the results for nonreactive trajectories. Whereas when the free HCl bond was broken the probability falls off rapidly with increasing J, here the probability falls off much more slowly and

Photoinduced Reactions in H-Bonded Clusters

The Journal of Physical Chemistry, Vol. 97, No. 48, 1993 12521

1

g 0.2

a

0.1

0.0

L

0

trajectories with J > 30h are observed. This higher J results come from the 25-30% of the probability that comes from trajectories that sample tne H + H’C1’ transition state. Energy redistribution is very efficient in those trajectories that reach the transition state, regardless of whether or not a reaction follows. In Figure 3 we make a similar comparison, but here we plot the vibrational energies of the HC1 molecule using a binning schemethat corresponds roughly to half of the harmonic frequency of HCl. Here the two probability distributions are much more similar than were the two angular momentum distributions. Because the HCl vibrational frequency is quite high, transfer of energy from translational energy of the H atom to vibrations in the molecule through an inelastic collision is less efficient than transfer to rotational energy. The nature of the rotational and vibrationalenergy distributions for the reactive trajectories is better understood when oneconsiders the two basic types of trajectories that lead to exchange reactions. First consider the case of a direct exchange event. In the transition state, the ClHCl’ intermediate is nearly linear, and thejj~c1’and ?HC]’ for the reactive H atom are approximately perpendicular. This means that the resulting HC1’ molecule will pick up substantial rotational and vibrational energy, and the largest J and Euib found in the distributions in Figures 2 and 3 result mainly from trajectories that sample this portion of the HCl’H’ transition state. The insertion mechanism is expected to transfer primarily vibrational energy as this mechanism corresponds to a situation in which the HClCl’ angle is small. In these trajectories, the translational energy of the H atom goes primarily into vibrational energy in the newly formed HCl’ molecule. In the limit of the reactive resonance trajectories discussed above, 3 ~ ~ and 1 ’ ?HCI’ are nearly collinear and the product HC1’ molecule comes out with very low angular momentum. 2. H Atom. The outgoing H atom is characterized by its angle (relative to a reference axis) and its kinetic energy. If we photodissociate an isolated HCl molecule at 193 nm, then we would expect to find a very narrow velocity distribution for the H atom, the width of which results from the width of the FranckCondon window for the transition. This is exactly the type of distribution that is displayed in Figure 4b for the photodissociation of the free HC1 bond. When the hydrogen-bonded HC1 bond was broken, the resulting distribution is broadened and has a low-energy tail. This tail results primarily from reactive trajectories and nonreactive trajectories that sample the HCl’H’ transition state. While the mechanism is quite different, the result is similar to that predicted for ArHCl by Garcia-Vela et a1.12J3and observed by Wittig and c o - w o r k e r ~for ~ ~ArHBr. Figure 5 shows the probability as a function of the outgoing angle 8 of the hydrogen atom relative to the ClCl’ axis at t = 0. Here, 8 = 0 corresponds to forward scattering and 8 = 180° to backscattering. In the case of photodissociation of the free HC1

5000 10000 15000 20000

Kinetic Energy of H atom [in wavenumbers]

Vibrational energy of the HCI [in wavenumbers]

Figure 3. As in Figure 2, but for the vibrational energy distribution.

-

AL

Figure 4. As in Figure 2, but for the distribution of the kinetic energies of the hydrogen atom. The spectrum that results from breaking the free HC1 bond (b) closely resembles that for an isolated HCl molecule. The part of the spectrum in (a) that is to the red of the peak in (b) results from inelastic or reactive collisionsof the H atom with the H’Cl’molecule.

g 0.4 Q

0.2 0.0

i

180

0

30

60

90

120 150

Angular distribution of the H atom [in degrses]

Figure 5. As in Figure 2, but for the angular distribution of the hydrogen. We have defined the angle fl relative to the ClCl’ axis at the time of photoexcitation. The angular distribution for photodissociation of the free HCl bond (b) resembles closely the projection of \ko2 onto this mode. Thevery different shapeof the angular distributions in (a) and (b) reflects the fact that when the hydrogen-bonding HCl bond is broken in a large fraction of the trajectories, the H atoms undergo inelastic or reactive collisions with the H’Cl’ molecule.

molecule (Figure 5b), this distribution is approximately the projection of the square of the ground-state wave function onto the HClC1’ angle. The angular distribution for trajectories in which the hydrogen-bonded HC1 bond is broken is much more interesting. Examination of Figure 5a shows that the distribution decreases from 8 = 30’ to 180’. That the probability is negligible for 8 < 30* reflects the geometry of the cluster; for small Cl’HCl angles, H must collide with H’C1’ before exiting the interaction region. In summary, the velocity and angular distributions of the H are of considerable interest. The very long tail of low-energy H, observed when the hydrogen-bonded HC1 bond is broken, is a direct reflection of those events in which there is an exchange reaction and a slow H atom is released. They also reflect the energy transfer that occurs when the nonreactive H + H’C1’ collision samples the transition state. Deuteration of the H’C1’ molecule may throw further light on the roles of the different processes in the context of future experimental studies. VII. Conclusion We have presented a general approach for quantal sampling of initial conditions for classical trajectory simulations of photodissociation of clusters in the UV. This type of approach

12522 The Journal of Physical Chemistry, Vol. 97, No. 48, 1993

is particularly important for systems in which large-amplitude motions and zero-point effects are important in the spectroscopy of the initial state. We used diffusion quantum Monte Carlo, developed by Anderson?Q3 to construct the n-dimensional groundstate wave function and from that the 2n-dimensional Wigner distribution. As a simplifying approximation,we assumed partial separation of vibrational modes. The prescription outlined above is extremely general and can be applied to the ground state of a wide variety of molecular systems. As it scales linearly with the number of atoms, it can be used to compute the ground-state wave functions for fairly large molecular systems. We have applied this methodology in a study of the photodissociation dynamicsof (HC1)2. Here the importanceof sampling of initial conditions is borne out in our results. The outcome of a single trajectory is highly sensitive to the initial value for the HClCI’ angle. Improper sampling of this coordinatecould alter the results significantly. In the preceding section,we presented results for the photolysis of each of the two HCl bonds in (HCI)2. Clearly, the fact that both HCl molecules in the dimer absorb a 193-nm photon with nearly identical cross sections makes bond selective photolysis in this cluster impossible in a one-color experiment. But, theobserved tails, resulting from those trajectories that sample the transition state of the H H’Cl’reaction and appearing in all four probability distributions described above, should have experimental consequences. By using a two-photon approach, one could selectively photodissociate the hydrogen-bonded HCl bond and not the free HCl bond. A clear signature of the photodissociation of (HC1)z would be the detection of H2. Given the potentials used in the present study, we predict that this channel accounts for less than 1% of the total probability. This is due in part to the restrictions on the geometry of the collision complex imposed by the geometry of (HC1)Zinitsgroundstate. Inexperimentson theH+ HIreaction, inwhich theHatomsareobtained byUVphotolysisofH1, Buntine et a1.50 observed Hz products with velocities that are too small to have resulted from an H H collision. One proposed mechanism for this result was they were photodissociating(HI)2. This mechanism is consistent with our calculated photodissociation dynamics of (HC1)2. Whilemuchofthedynamicsofthephotodissociationof (HC1)2 can be treated well by classical methods, the final vibrational energy and angular momentum distributions are quantized. By treating this system by a quantal or mixed quantum/semiclassical TDSCF approach,11J4we hope to explore these distributions more rigorously in work that is now in progress.

+

+

Acknowledgment. We thank Professors J. Valentini and C. Wittig for helpful discussions and Dr. A. Garcia-Vela for his comments on the manuscript. This research is supported by a grant from the Ministry of Science, Culture and Arts, State of Neidersachsen, Germany (to R.B.G.). We thank Silicon Graphics Inc. for the loan of INDIGO workstations to the Institute for Advanced Studies at the Hebrew University. The Fritz Haber Research Center is supported by the Minerva Gesellschaft fiir die Forschung, mbH, Munich, Germany. References and Notes (1) Wittig, C.;Sharpe, S.;Bcaudet, R. A. Acc. Chem. Res. 1988,II.341. Wittig, C.; Engel, Y. M.; Levine, R. D. Chem. Phys. Lett. 1988, 153, 411. Buelow, S.; Noble, M.; Radhakrishnan, G.;Reisler, H.; Wittig, C.; Hancock, G.J . Phys. Chem. 1986,90, 1015. (2) Hoffman, G.; Oh, D.; Iams, H.; Wittig, C. Chem. Phys. Lett. 1989, 155,356. Bdhmer, E.; Shin, S.K.; Chen, Y.; Wittig, C. J . Chem. Phys. 1992, 97, 2536. (3) Ionov, S. I.; Brucker, G. A.; Jaques, C.; Valachovic, L.; Wittig, C. J . Chem. Phys. 1992,97,9486. (4) Schercr, N. F.; Khundkar, L. R.; Bernstein, R. R.; Zewail, A. H. J. Chem. Phys. 1987, 87, 1451. Schcrer, N. F.; Sipes, C.; Bernrtein, R. R.; Zewail, A. H. J. Chem. Phys. 1990, 92, 5239.

McCoy et al. (5) Gruebele, M.;Sims, I. R.;Potter, E. D.; Zewail, A. H. J . Chem.Phys. 1991, 95, 7763. Sims, I. R.; Gruebele, M.; Potter, E. D.; Zewail, A. H. J . Chem. Phys. 1992, 97,4127. (6) Jouvet, C.; Boivinear, M.; Duval, M. C.; Soep, B. J . Phys. Chem. 1987, 91, 5416. Breckenridge, W. H.; Jouvet, C.; Soep, B. J . Chem. Phys. 1986.84, 1443. (7) Socp, B.; Ab%, S.; Keller, A.; Visticot, J. P. J. Chem. Phys. 1992, 96, 440. (8) Valentini, J. J.; Cross, J. B. J. Chem. Phys. 1982, 77, 572. (9) Burke, M. L.; Klemperer, W. J . Chem. Phys. 1993, 98, 1797. (10) Alimi, R.; Gerber, R. B. Phys. Reo. Lett. 1990, 64, 1453. (1 1) Alimi, R.; Gerber, R. B.; Hammerich, A. D.; Kosloff, R.;Ratner, M. A. J. Chem. Phys. 1990, 93,6484. (12) Garcla-Vela, A.; Gerber, R. B.; Valentini, J. J. Chem. Phys. Lett. 1991, 186, 223. (13) Garcia-Vela, A.;Gerber, R. B.; Valentini, J. J. J . Chem. Phys. 1992, 97, 3297. (14) Garcia-Vela, A.; Gerber, R. B.; Imre, D. G.J . Chem. Phys. 1992, 97, 7242. Garcia-Vela, A.; Gerber, It. B. J. Chem. Phys. 1993, 98, 427. Garcla-Vela, A.; Gerber, R. B.; Imre, D. G.; Valentini, J. J. Chem. Phys. Lett. 1993, 202, 473. (15) Segall, J.; Yen, Y.; Singer, R.; Wittig, C.; Garcia-Vela, A.; Gerber, R. B. Chem. Phys. Lett., in press. (16) Schinke, R. J . Phys. Chem. 1988, 92, 3195. (17) Tennyson, J.; van der Avoird, A. J . Chem. Phys. 1982, 77, 5665. (18) Hutson, J. M.; &wick, J. A.; Halberstadt, N. J . Chem. Phys. 1989, 90, 1337. (19) Sun, H.; Watts, R. 0. J . Chem. Phys. 1990, 92, 603. (20) Quack, M.; Suhm, M. A. J. Chem. Phys. 1991, 95, 28. (21) Bunker, P. R.; Epa, V. C.; Jensen, P.; Karphen,A. J. Mol.Spectrosc. 1991, 146, 200. (22) Anderson, J. B. J. Chem. Phys. 1975,63, 1499. (23) Anderson, J. B. J. Chem. Phys. 1976,65,4121. (24) Votava, C.; Ahlrichs, R. In Pullman, B., Ed. Intermolecular Forces; D. Rcidcl Publishing: Dordrecht, Holland, 1981. Votava, C.; Ahlrichs, R.; Geiger, A. J. Chem. Phys. 1983, 78, 6841. (25) Latajka, 2.;Scheiner, S.Chem. Phys. 1988, 122,413. (26) Karpfen, A.; Lischka, H.; Bunker, P. R. In Halberstadt, N., Janda,

K. C., Eds. Dynamics of Polyatomic Van der Waals Complexes; Plenum Press: New York, 1990. Karpfen. A.; Bunker, P. R.; Jensen, P. Chem. Phys. 1991, 149, 299. (27) Augspurger, J. D.; Dykstra. C. E. Chem. Phys. Lett. 1992,189,303. (28) Ohashi, N.; Pine, A. S . J. Chem. Phys. 1984, 81, 73. (29) Furlan, A.; Wllfert, S.; Leutwyler, S . Chem. Phys. Lett. 1988,153, 291. (30) Blake, G.A.; Busarow, K. L.; Cohen, R. C.; Laughlin, K. B.; Lee, Y. T.; Saykally, R. J. J . Chem. Phys. 1988,89, 6577. (31) Moazzen-Ahmadi, M.; McKellar, A. R. W.; Johns, J. W. C. Chem. Phys. Lett. 1988, 151, 318. Moazzen-Ahmadi, M.; McKellar, A. R. W.; Johns, J. W.C. J. Mol. Spectrosc. 1989, 138, 282. (32) Blake, G.A.; Bumgarner, R. E. J . Chem. Phys. 1989, 91, 7300. (33) Schuder, M. D.; Lovejoy, C. M.; Nelson, D. D.; Nesbitt, D. J. J. Chem. Phys. 1989, 91,4418. (34) Coker, D. F.; Miller, R.E.; Watts, R. 0. J. Chem. Phys. 1985,82, 3554. (35) Buch, V. J . Chem. Phys. 1992, 97, 726. (36) BaEiC, Z.; Kennedy-Mandzuik, M.; Moskowitz. J. W.; Schmidt, K. E. J . Chem. Phys. 1992, 97,6472. (37) Rama Krishna, M. V.; Whaley, K. B. 2.Phys. D 1991, 20, 223. Barnett, R. N.; Whaley, K. B. Phys. Reo. A 1993, 47, 4082. (38) Coker, D. F.; Watts, R. 0. Mol. Phys. 1986, 58, 1113. (39) Hamilton, I. P.; Light, J. C. J. Chem. Phys. 1986, 84, 306. (40)Coo1ey.J. W.;Tukey, J. W. Math. Comput. 1%5,19,297. Temperton, C. J. Compur. Phys. 1983, 52, 1. (41) Hirst, D. M.; Guest, M. F. Mol. Phys. 1980, 41, 1483. (42) Stern, M. J.; Penky, A.; Klein, F. S.J . Chem. Phys. 1973,58,5697. (43) Schwenke, D. W.; Tucker, S. C.; Steckler, R.; Brown, F. B.; Lynch, G.L.; Truhlar, D. G.;Garrett, B. C. J . Chem. Phys. 1989, 90, 3110. (44)Hutson, J. M. J. Chem. Phys. 1988,89,4550. (45) Aziz, R. A,; Slaman, M. J. Mol. Phys. 1986, 58, 679. (46) Aquilanti, V.; Candori, R.;Cappelletti, D.; Lorant, V.; Luzzatti, E.; Piranr, F. J. Phys. Chem. 1993, 97, 2063. (47) Aker, P. M.; Valentini, J. J. Isr. J . Chem. 1990, 30, 157. (48) Barclay, V. J.; Collings, B. A.; Polanyi, J. C.; Wang, J. H. J. Phys. Chem. 1991, 95, 2921. (49) Wilson, E. B.; Decius, J. C.; Cross, P. C. Molecular Vibrations; McGraw-Hill: New York, 1955. (50) Buntine, M. A.; Baldwin, D. P.; a r e , R. N.; Chandler, D. W. J . Chem. Phys. 1991, 94,4672.