J. Phys. Chem. 1988, 92, 3195-3201
3195
a “figure-of-eight” trajectory as discussed elsewhere6b). If so, then the effective barrier to reaction on the sf-POLCI surface is better thought of as located at the collinear stationary point of Table 11, which is 0.02 eV higher in energy than the LEPS saddle point. Further work (trajectories) will be needed to establish the validity of this argument. One additional topic, which will be deferred to a future paper, concerns the comparison of our CSDW rate coefficients with values obtained from VTST and from reduced dimensionality quantum scattering theories. The observed differences between the sf-POLCI and LEPS rate coefficients bring to question whether vibrationally adiabatic theories that do not impose angular momentum constraints can properly describe threshold energies. We will examine this point in future work.
activation energy on the sf-POLCI surface.
V. Discussion Perhaps the most important finding in this paper concerns the comparison of the LEPS and sf-POLCI results. This comparison indicates that the biggest difference occurs in the energy dependence of the cross sections, with the LEPS results showing lower threshold energies than sf-POLCI, despite the fact that LEPS has the higher ground-state adiabatic barrier. This leads to a noticeable difference (0.03 eV) in the activation energies, even though the overall rate coefficients both agree with experiment within the experimental uncertainties. Other differences between LEPS and sf-POLCI are more subtle, such as the slightly higher product rotational excitation on sf-POLCI. Some quantities, such as angular distributions, show almost no differences at all. The fact that rotation plays an important role in the C1 HCl reaction is a significant result, and although this is true for both potential surfaces, it may also contribute to the differences in the threshold energies for the sf-POLCI and LEPS surfaces. In particular, we note that for highly excited rotational states (j = 10, say) the rotational period (-0.16 ps), while still long compared to the vibrational period (-0.01 ps), is comparable to the collision duration. As a result, it is best to consider reaction from the perspective of Figure 1 wherein the reacting hydrogen atom follows an angular path over the saddle-point region (executing part of
+
Acknowledgment. This research was supported by the Office of Basic Energy Science, Division of Chemical Sciences, U S . Department of Energy, under Contract W-3 1- 109-ENG-38, by N S F Grant CHE-8416026, and by the UK Science and Engineering Research Council. The computations were carried out on the Cray XMP and Cray 2 computers at the National Magnetic Fusion Energy Computer Center, and on the CDC 7600, CDC Cyber 176, and CDC Cyber 205 computers at the University of Manchester Regional Computer Centre. Registry No. C1, 22537-15-1; HC1, 7647-01-0.
Rotational State Distributions following Direct Photodissociation of Triatomic Molecules: Test of Classical Modek Reinhard Schinke Max-Planck-Institut fur Stromungsforschung, 0-3400 Gottingen, FRG (Received: April 15, 1987; In Final Form: June 11, 1987)
We test the applicability of classical mechanics to calculate rotational state distributions following the direct photodissociation of triatomic molecules under conditions when rotational excitation is str6ng. The agreement with exact quantal calculations for a model system is generally very good. However, we also find that in some cases the agreement is poor. Discrepancies can be traced back to peculiar, nonmonotonic trajectories; We also examine the reliability of classical calculations employing a classical distribution function for the initial state.
I. Introduction Photodissociation of polyatomic molecules in the UV range is an active field, both e~perimentallyl-~ and t h e ~ r e t i c a l l y . ~ - ~ Traditionally, the total absorption cross section as a function of photon wavelength has been measured for almost every molec ~ l e . ~ -It’ is~ related to the energetical ordering of the excited states involved in the photoexcitation process, and its overall shape might give some clues about the nature of the dissociation process. For example, a broad and structureless spectrum is expected for a fast (direct) dissociation mechanism. In contrast, sharp structures indicate an indirect (predissociating) decay mechanism. (1) Leone, S. R . Adu. Chem. Phys. 1982, 50, 255. (2) Simons, J. P.J . Phys. Chem. 1984>88, 1287. (3) Bersohn, R. J . Phys. Chem. 1984, 88, 5145. (4) Jackson, W. M.;Okabe, H . In Advances in Photochemistry; Volman, D. H., Gollnick, K., Hammond, G. S., Eds.; Wiley: New York, 1986: Vol. 13, p 28. (5) Freed, K. F.;Band, Y. B. In Excited States; Lim, E. C.: Ed.; Academic: New York, 1978; Vol. 3. (6)Shapiro, M.; Bersohn, R . Annu. Rev. Phys. Chem. 1982, 33, 409. (7) Shapiro, M.; Baht-Kurti, G. G. In Photodissociation and Photoionization; Lawley, K. P., Ed.; Wiley: New York, 1985; p 403. (8) Schinke, R . J . Phys. Chem. 1986, 90, 1742. (9) Okabe, H.Photochemistry in Small Molecules; Wiley: New York, 1978. (10) Robin, M.B. Higher Excited States of Polyatomic Molecules; Academic: New York, 1974.
0022-3654/88/2092-3195$01 S O / O
Obviously, more information about the photodissociation process is obtained if the quantum states of the products are also resolved. Due to the availability of sophisticated probing techniques (LIF; REMPI, etc.), the past 10 years or so have witnessed an enormous growth of final state distributions for many dissociation products (OH, NO, CO, e t ~ . ) . l - ~ J Product l state distributions are determined by the preparation of the parent molecule prior to-the electronic excitation and by the dynamics on the excited state potential energy surface. The so-called “final state interaction” is nothing else than the coupling (forces) between the various degrees of freedom induced by the corresponding multidimensional potential energy surface V X .The product state distributions somehow ”reflect” these forces, and thus certain information about Pxmay be deduced which would be difficult to obtain otherwise. Quantum mechanically, the photoabsorption cross section is usually calculated from Fermi’s Golden Rule6-7~12 where Qtj is the nuclear wave function in the ground electronic state and rotational-vibrational level (i), is the nuclear wave function in the excited electronic state asymptotically dissociating (1 1) Buelow, S.;Noble, M.:Radhakrisiinar, G.;Reisler, H.; Wittig, C.; Hancock, G. J . Phys. Chem. 1986, 90, 1015. (12) Weissbluth, M.Atoms and Molecules; Academic: New York, 1978.
0 1988 American Chemical Society
3196 The Journal of Physical Chemistry, Vol. 92, No. 11, 1988 into final product state (f), and p is the transition dipole function. Equation 1 is based on first-order perturbation theory concerning the light-matter interaction.12 It is considered to be an excellent approximation if the photon intensities are small as it is usually the case in most phgtodissociation experiments. The wave functions in eq 1 are exact solutions of Schrodinger’s equation for the nuclear motion within the ground and the excited state, respectively. The quantum mechanical formulation is rigorously defined and does not need any further discussion. The only problem is its practical implementation. Several methods have been suggested in the literature which basically avoid the direct calculation of the wave function^.'^-'^ They have been applied to a number of collinear, collinear-like, and even three-dimensional problems.16-22 In contrast to what was frequently stated in the literature, the direct calculation of eq 1 is also possible even if the dissociation wave function is highly o ~ c i l l a t o r y . ~With ~ “direct” we mean the separate calculation of both wave functions and the subsequent evaluation of the transition matrix elements. This method also has been applied to several problem^.*^-^^ Within the time-independe?t formulation the total dissociation wave functions are usually expanded in terms of product eigenfunctions which leads to the well-known set of coupled radial equations. The solution of these equations has been amply studied, and many efficient computer codes are now a~ailable.~’Such close-coupling calculations are only limited by the number of channels which very rapidly exceeds practical feasibility. The n u m k r of coupled equations sometimes can be drastically reduced by invoking further approximations, for example the famous 10.5 a p p r o x i m a t i ~ n . ~‘However, ~-~~ even then rigorous quantal calculations are very often impractical for polyatomic molecules with several degrees of freedom which are certainly the most interesting from the physicochemical point of. view. Thus, other, completely different approaches are desired. Most appealing is certainly the use of ordinary classical mechanics. Classical trajectory calculations have been very successfully applied in gas-phase scattering over the past two decades or so. They can be implemented in a straightforward way even for larger systems; see, for example, ref 34,. Since direct photodissociation can be interpreted as t.he second step of a full scattering process (half-collision), it seems plausible that classical mechanics is also applicable here. Classically, a photodissociation process consists of three steps: (i) A vertical Franck-Condon (FC) transition in which the
(13) Shapiro, M. J . Chem. Phys. 1972,56, 2582. (14) Kylander, K. C.; Light, J. C. J . Chem. Phys. 1980, 73, 4337. (15) Kulander, K. C.; Heller, E. J. J . Chem. Phys. 1978, 69, 2439 (16) Lee, S. Y.; Heller, E. J. J . Chem. Phys. 1982, 76, 3035. (17) Shapiro, M.; Becsohn, R. J . Chem. Phys. 1980, 73, 3810. (18) Clary,_D.C. J . Chem. Phys. 1986, 84, 4288. (191 Kulander. K. C. Chem. Phvs. Lett. 1984. 103. 373. (20) von Veen,’G. N. A.; B a l k : T.; de Vries, A.E.’;Stiapiro, M. Chem. Phys. 1985, 93, 277. (21) Heather, R. W.; Light, J: C. J . Chem. Phys. 1983, 78, 5513; 1983, 79, 147. (22) Segev, E.; Shapiro, M. J . Chem. Phys. 1980,73,2001;1982,77,5604. (23) Engel, V. Max-Planck-Institut fur StrGmungsforschung, Bericht 14/ 1986. (24) Hennig, S.; Engel, V.; Schinke, R. J . Chem. Phys. 1986, 84, 5444. (25) Engel, V.; Schinke, R. Chem. Phys. Lett. 1985, 122, 103. (26) Engel, V.; Schinke, R.; Staemmler, V. Chem. Phys. Lett. 1986, 130, 413; J . Chem. Phys., to be published. (27) Thomas, L. D.; Alexander, M. H.; Johnson, B. R.; Lpster, Jr., W. A,; Mchnithan, K. D.; Parker, G. A.; Redmond, M. J.; Schmalz, T.G.;Secrest, D.; Walker, R. B. J . Comput. Phys. 1981, 41, 407. (28) Beswick, J. A.; Delgado-Barrio, G. J . Chem. Phys. 1980, 73, 3653. (29) Atabek, 0.;Beswick, J . A.; Delgado-Barrio, G. J . Chem. Phys. 1985, 83, 2954. (30) Segev, E.; Shapiro, M. J . Chem. Phys. 1983, 78, 4969. (31) Schinke, R; Engel, V.; Staemmler, V. J. Chem. Phys. 1985,83, 4522. (32) Kulander, D. C.; Light, 3. C. J . Chem. Phys. 1986,.85, 1938. (33) Clary, D. C.; Henshaw, J. P. In The Theory of Chemical Reaction Dynamics; Clary, D. C., Ed.;Reidel: Dordrecht, 1986. Henshaw, J. P.; Clary, D . C . J . Chem. Phys., to be published. (34) Schatz, G. C. In Molecular Collision Dynamics; Bowman, J. M., Ed.; Springer: Heidelberg, 1983; Chapter 3 .
Schinke positions and momenta of the parent molecule in the ground electronic state at time t = 0 are preserved and projected onto the excited state potential surface. (ii) The decay of the “excited complex” following the classical equations of motion with the coordinates and momenta at t = 0 as initial conditions. (iii) The resolution of the individual trajectories into appropriate “quantum boxes” after the dissociating forces have decreased to zero. The second step is well-defined and does not need any further explanation. The third step also does not cause severe problems, especially if many quantum states are finally populated as it is often the case for the rotational degree of freedom. The most crucial point is the first step. Immediate questions concern the weighting of the positions and momenta within the parent molecule and the applicability of the FC assumption. In ordinary scattering the initial conditions are defined for the infinitely separated molecular subsystems, and a variation of them is expected to have only little influence on the final cross sections and distributions. Not so in dissociation where the initial conditions for the trajectory are defined in the inner region of the potential where both subsystems are close together. If the decay is fast, which implies that strong forces are immediately acting, we expect significant changes in the final product distributions if the initial conditions are somewhat varied. There are several classical studies of photodissociation reported in the l i t e r a t ~ r e . ~ In ~ -most ~ ~ cases the weighting of the initial conditions is performed according to the quantal distribution of positions and momenta of the parent molecule. This seems to be a generally accepted recipe by now, although also classical distributions have been utilized.49 Since accurate excited state potentials are not known for almost all dissociation systems, a rigorous test of classical models cannot be based on a comparison with experimental data but must be based on comparison with exact quantal calculations using the same (model) potential. As far as we know, such tests have been made in only three cases, concerning vibrational distributions in collinear-type processes. Pattengill@compared classical calculations with supposedly exact quantal calculations for the model N20S0and CH311’ systems. Since it was later noted that these quantal calculations are wrong due to a simple programming error,51this early comparison is meaningless. Henriksen4’ used the semiclassical Wigner method again for the model CH31 problem and obtained very good agreement with the exact quantal calculation of Lee and Heller.I6 However, since final state interaction (Le., translational-vibrational coupling) is very weak for this system, this study is of limited value for a general test of classical models. Recently, Henriksen et al.48 tested the same method for the dissociation of symmetric triatomic molecules, namely H 2 0 and C 0 2 , and reasonably good agreement was obtained with the corresponding exact quantal calculations. As far as we know, the applicability of classical mechanics has not been rigorously tested for problems involving the rotational degree of freedom. Since usually many (up to 60 0; more) rotational states are excited in a direct dissociation process, classical
(35) Holdy, K. E.; Klotz, L. C.; Wilson, K. R. J . Chem. Phys. 1970, 52, 4588. (36) Goursaud, S.; Sizun, U.; Fiquet-Fayard, F.J . Chem. Phys. 1976, 65. 5453. (37) Heller, E. J. J . Chem. Phys. 1978, 68, 2066. (38) Brown, R. C.; Heller, E. J. J . Chem. Phys. 1981, 75, 186. (39) Waite, B. A.; Helvajian, H.; Dunlap, B. I.; Baronavski, A. P. Chem. Phys.-Lett. 1984, 111, 544. (40) Pattengill, M. D. Chem. Phys. 1982, 68, 7 3 . (41) Pattengill, M. D. Chem. Phys. 1983, 78, 229. (42) Pattengill, M. D. Chem. Phys. Lett. 1984, 104, 462. (43) Pattengill, M. D. Chem. Phys. Lett. 1984, 105, 651. (44) Pattengill, M. D. Chem. Phys. 1984, 87, 419. (45) Goldfield, E. M.; Houston, P. L.; Ezra, G. S. J . Chem. Phys. 1986, 84, 3120. (46) Bersohn, R.; Shapiro, M. J . Chem. Phys. 1986, 85, 1396. (47) Henriksen, N. E. Chem. Phys. Lett. 1985, 121, 139. (48) Henriksen, N. E.; Engel, V.; Schinke, R. J . Chem. Phys. 1987, 86, 6862. (49) Waite, B. A,; Dunlap, B. I. J . Chem. Phys. 1986, 84, 1391. (50)Shapiro, M. Chem. Phys. Lett. 1977, 46, 442. (51) Shapiro, M. J . Phys. Chem. 1986, 90, 3644.
Rotational State Distributions of Triatomic Molecules mechanics is expected to be highly accurate. The lack of meaningful comparison between classical and quantal calculations for such cases is certainly due to the lack of accurate close-coupling calculations when many states have to be considered. Recently, we performed elaborate quantal calculations for a simple model s y ~ t e m in~ order ~ , ~ ~to understand some general experimental findings: In many cases the final rotational product state distributions are highly inverted, narrow, and of Gaussian shape (see Figure 1 of ref 52, for example). They can qualitatively be explained by the rotational reflection principle, an effect which was found by us in earlier approximate quantal calculations em~ ~ a brief discussion see section ploying the 10s a p p r ~ x i m a t i o n(for IV below). In order to interpret the quantal results, we also performed extremely simple classical calculations. Although the general trends were nicely reproduced, some differences were observed which certainly are caused by the simplicity of that classical model. In the present work we extend the classical calculations and thereby provide a rigorous test of classical mechanics for cases with strong rotational excitation. The article is organized in the following way: The model is briefly summarized in section 11, and the corresponding classical calculations are defined in section 111. The results and the comparison with the quantal calculations will be reported in section IV. In section V we also discuss some other classical calculations. The main conclusions are summarized in section VI.
The Journal of Physical Chemistry, Vol. 92, No. 11, 1988 3197 If dPX/dyis equal to zero for all distances R , the initial angular momentum j , = j(t=O) will not be changed (no final state interaction). The excited state interaction potential is represented in the simple form
-R
P x ( R , y )= A exp(-a[R f(y) =
€
+ f l y ) -f(y=O)])
(4)
cos2 y
where t controls the coupling between the dissociation and the rotational degree of freedom. For e = 0 the potential is isotopic. In the present work we consider only negative values of t appropriate for bent excited state molecules. The Hamiltonian for the ground electronic state is identical with eq 2 with Pxreplaced by P , the corresponding interaction potential. For simplicity and following Goldfield et al.,45we choose Prto be separable in R and y,i.e. p r ( R * y )=
y2kR(R
- R e ) 2 + l/zky(y - ye)2
(5)
where Re and ye define the equilibrium geometry of the parent molecule. For small displacements from equilibrium the ground state Hamiltonian can be written as f f ( R , y ) = HR(R) + ffy(y)
(6a)
11. The Model We consider the photodissociation of a triatomic molecule, Le., hw A BCU), ignoring the vibrational degree of ABC f r d o m of BC. For simplicity it is assumed that the total angular momentum is zero in both the ground and the excited state. The classical two-dimensional Hamiltonian in the excited state is given by55
+
-
+
where m is the reduced mass for the A-BC system, B is the rotational constant of BC, and Pxis the interaction potential. The coordinates are R;the distance of A with respect to the center of mass of BC, and y,the orientation angle of R with respect to the BC internuclear vector r, Le., cos y = R.r/Rr. The corresponding momenta are the linear momentum P and the molecular rotational momentumj. Please note that y and j are conjugate variables. This is not generally true but follows here from the assumption that the total angular momentum is ~ e r o . ~ ~In, e~q' 2 we explicitly used that 1 = -j with 1 being the orbital angular momentum. The advantage of using action-angle variables becomes immediately apparent from the corresponding equations of motion: dR -= dt
dY = 2 j ( B +
P m
A)
In this study we will consider only linear parent molecules, i.e., ye = 0. The (unnormalized) quantal wave function for the vibrational ground state with no quantum of excitation in the stretch or the bending motion is given by \kgr(R,y) =
4 g 1 , R ( ~ 4) g r , y ( y )
+gr.R(R) = ex~[-aR(R - Re)*] @ g r , y ( ~ )=
ex~[-ay(y -
(7a) (7b) (7c)
The exponents aRand ay are related to the potential parameters by (YR
= Y2(mk~)'/~
ay = & ( f i k , ) ' f 2
@a) (8b)
The parent wave function eq 7 is identical with-that used in our previous quantal studies (ye= O).5233There we gave explicit values for the full widths at half-maximum (fwhm), AR,and A y , instead of values for aR and ay (or k R and k y ) . They are related by
dt
(9)
The change o f j along the trajectory is directly determined by the anisotropy, i.e., the angle dependence of the interaction potential. ~~
(52) Schinke, R. J . Chem. Phys. 1986, 85,5049. (53) Schinke, R.; Engel, V. Faraday Discuss. Chem. SOC.1986, 82, 1 1 1. (54) Schinke, R.; Engel, V. J . Chem. Phys. 1985, 83,5068. (55) McCurdy, C. W.; Miller, W. H. J . Chem. Phys. 1977, 67, 463. (56) Miller W. H. J . Chem. Phys. 1970, 53, 1949. (57) Smith, N. J . Chem. Phys. 1986, 85, 1987.
and similarly for ay. In order to make contact with the exact quantal calculations of ref 52 and 5 3 , the parameters for the ground and the excited states have the following values: Re = 2 A, ye = 0, AR = 0.2 A, and A y = 20' for the ground state; I? = 2 A, a = 2 k', and A = 1.3 eV for the excited state. The reduced mass m is that of OC-S, and the rotational constant B is that of CO. The strength of the anisotropy of Pxwill be varied as well as E , the total available energy in the excited state. 111. Classical Theory A . Full Classical Theory. We define the classical cross section for absorption of a photon with energy h w and producing BC in a particular rotational state j ( = 0, 1, 2, ...) by
3198 The Journal of Physical Chemistry, Vol. 92, No. 11, 1988
where Ro,etc., represent the start values of the trajectory at time t = 0. p ( R , y )is the transition dipole function, which is in principle coordinate-dependent. Since it is not known in most cases, it is usually assumed to be constant and therefore ignored. The energy E = E,, + h w is the total available energy in the excited state which can be partitioned into translation and rotation. The first .delta function in eq 10 assures that E in the excited state is well-defined (energy conservation). The second delta function selects only those trajectories (Ro,Po,yo,jo)which lead to the particular final state j . Here J(Ro,Po,yoJo)is the so-called excitation function J = j(t-m), Le., the final rotational angular momentum after the collision.s Since J is a continuous fmction, IIJI 5 j I / > and zero it is understood that S(J-j) = 1 if j otherwise. This means that the usual “boxing” procedure is applied. Utilizing energy conservation, we can reduce the four-dimensional phase space integral to a three-dimensional one. We choose Po,yo.and j o as independent initial variables and fix the distance as the turning point R,(Po,yojolE)according to Hex(R,,Po,yoJo) = E . Then the classical cross section becomes [ p ( R , y )= ljs8.s9
+
.,10’lE)
- wJ-dPoJ-dy0
Schinke appropriate can only be decided by comparison with exact quantal calculations. The full classical calculations are performed in the following way: First, an arbitrary set of initial values Po, yo, and j o is randomly selected from an uniform distribution. The appropriate starting value R,(Po,yoJolE)is then determined from energy conservation. Second, Hamilton’s equations (3) are solved numerically until j ( t ) and P ( t ) become constant, defining the excitation function J(Po,yoJolE).Finally, the cross section is calculated as
where the summation is over all trajectories u which lead to the particular rotational state j . In all cases 60 000 trajectories are calculated for a given energy E. The calculations are so fast that an improvement of convergency by importance sampling was not necessary. B. Simplified Classical Theory. A drastic simplification is possible if the two momenta, Po and j o ,are set to zero initially. We expect this to be a good approximation if the forces at the turning point are strong. Then the cross section formula (1 1) reduces to the one-dimensional integral
sin Y o j d j o ~ ( R J o 3 Y o J o )x
The sin yo term in eq 10 and 11 simply reflects the fact that y is a polar angle. It is very important as we will demonstrate below, especially for linear triatomic molecules. So far the definition of the classical cross section is straightforward and does not need any further discussion. It has been used in this or slightly different ways in all classical studies. The most important question concerns the distribution function W(Ro,Po,yojo) which is introduced to incorporate the vibrational motion of the parent m o l y l e prior to dissociation. It is widely accepted now that a quantum mechanical distribution is most appropriate, although also classical distributions have been sampled49(see also section V). In most cases the so-called Wigner function is e m p l ~ y e d . ~However, ~ - ~ ~ ~ all ~ ~choices are nothing else than more or less reasonable recipes. There is no rigorous derivation of which initial distribution is superior. Another ”natural choice” would be where the first term is simply the nuclear wave function and the second term is the corresponding wave function in momentum space, Le., the corresponding Fourier transform.. If we insert the harmonic ground state wave functions of eq 7 , the quantum mechanical distribution function in eq 12 becomes
In this particular case the distribution function definedjn eq 12 and the corresponding Wigner f ~ n c t i o n ~are ~ , ~identical. ’ This is, of course, not always the case. Which choice for W is most ( 5 8 ) Messiah, A. Quuntum Mechanics; North-Holland: Amsterdam, 1972; Vol. I, p 469. (59) We became aware of the normalization term lafPx/aR\only after the present calculations were completed. It is therefore not included in all of the present calculations. However, since the excited state potential is very steep and since the transition region in R is very narrow, we believe that it is negligible, especially for the rotational state distributions which are the primary focus of this study. (60) Hay, P. J.; Pack, R. T.; Walker, R. B.; Heller, E. J. J . Phys. Chem. 1982,86, 862. Sheppard, M. G.; Walker. R. B . J . Chem. Phys. 1983, 78, 7191.
Using the properties of the delta f u n c t i ~ nwe , ~can ~ ~rewrite ~ ~ eq 15 as
where the summation is over all trajectories yo,+defined by the simple relation
The excitation function J ( y o )is the only function, besides Px and W,which enters the cross section expression. Since J ( y o ) is usually a simple and smooth function, it is sufficient to calculate only very few (-20) trajectories and fit J ( y o ) to some analytical form (cubic spline, for example). Then the cross section can be calculated analytically. The required computational time is obviously negligible, compared to the quantal or full classical calculations. This simple classical theory was used in our previous s t ~ d i e s , and ~ ~ ,astonishingly ~~ good agreement with the exact quantal results has been obtained. Moreover, it allows a very simple explanation of the rotational state distributions, the rotational reflection principle. The present full classical calculations are performed to test the influence of nonzero initial momenta and thereby to hopefully improve the agreement between the classical and the quantal approach.
IV. Results In Figures 1 and 2 we present the main results of this article, rotational state distributions for fixed energy E and different anisotropies t (Figure 1) and for various energies, but the same excited state potential (Figure 2). Shown are the exact quantal and the simple classical distributions of ref 52 in comparison with the full classical calculations of this work. The form of the distributions has beenamply described in terms of the ground state weighting function Wand the excitation function J ( y o ) which follows solely from the dynamics in the excited state.s2ss3The excitation funtion J(yolPo=jo=O)for one of the cases is plotted vs. yo in Figure 3. It is zero at yo = 0, rises monotonically, and approaches a maximum at around yo 30°. In this region eq 18 has exactly one solution yoG) for each j = 0, 1, 2, .... There are otker solutions at larger angles, but they do not contribute since W ( y o )sin yo, which is also shown in Figure 3, is negligibly
-
Rotational State Distributions of Triatomic Molecules E = 10eV
r o t a t i o n a l state Figure 1. Rotational state distributions for various anisotropy parameters c (in A) and E = 1 eV. Comparison between exact quantal (-), full classical ( O ) , and simplified classical (- - -) calculations. The arrows
indicate the highest available state for this energy.
0
20
LO
60
80
rotational state Figure 2. Same as Figure 1 but for c = -1 .O A and various energies E .
small. Therefore, eq 18 constitutes a one-to-one correspondence between the initial angle variable yoand the final action variable j . According to eq 17 the classical cross section is primarily a reflection of @(yo)onto the quantum number axis, mediated by the excitation function J ( y o ) , as indicated in Figure. 3. This justifies the term rotational reflection principle. Although the agreement of the extremely simple classical calculations with the exact ones is astonishingly good, there are distinct deviations. First we note that the classical distributions are generally narrower, especially at the high energy side. Second, eq 17 breaks down in the vicinity of the maximum of J(yo),where the normalization factor dJ/dyo approaches zero and Zcl becomes singular. This effect is called a “rotational rainbown6’ in normal (61) Schinke, R.; Bowman, J. M. In Molecular Collision Dynamics; Bowman, J. M., Ed.; Springer: Heidelberg, 1983; Chapter 4.
Figure 3. (a) Classical excitation function J(yo,Po=OjolE) vs. initial orientation angle yofir various initial angular momenta j o as indicated. p ( y o ) is the weighting function, eq 10, including sin yo. (b) Simple classical rotational state distribution vs. j for j o = 0.
scattering, and the same effect obviously exists also in a direct photodissociation. The variation of the initial momenta Po and especiallyjo leads naturally to a broadening which in most cases brings the full classical calculations in almost perfect agreement with the quantal distributions. The improvement is most strikingly seen for e = -2.0 in Figure 1 and E = 0.9 eV in Figure 2. In both cases the rainbow effect is dominant. However, in so.me cases (e = -0.25 an&-0.5 in Figure 1 and E = 1.5 eV in Figure 2) the agreement is not so good. Here, the probabilities for the low rotational states are strongly overestimated. The influence of the variation ofjo is easily understood in terms of the corresponding excitation functions J(Po=O,yojo),shown in Figure 3 forjo = 0, f3, and f6. The variation of Po is of minor importance and not considered here. The various curves have all the same qualitative behavior but are vertically shifted, depending on the sign of j o . For negative j o there is even a small angular interval, for which the final momentum is also negative. The corresponding trajectories are very peculiar in the sense that y ( t ) as well asj(t) is a nonmonotonous function. The outcome of such a trajectory depends very sensitively on the interplay between the initially negative angular momentum and the initially positive force -dPx/dy, which tends to increase j ( t ) . The corresponding simple classical distributions, eq 17, would be similar to thejo = 0 cross section, however, shifted along the j axis. For example, thej, = 6 cross section would start at j 15 with a maximum at j 27. The), = -6 cross section would start a t j = -15 (if we would distinguish between positive and negative final angular momenta) 4. The averaging over all curves, with a maximum around j including the appropriate weighting, leads to the desired broadening, but also to the significant probability around j = 0, which is inconsistent with the quantum calculation (see Figure 1). The artifact around j = 0 is obviously not present in the simple classical theory with j o = 0. The “untypical” trajectories, which lead to this artifact, all originate from negative initial angular momenta and positive initial angles yo (or the other way around). Their significance depends very much on the special case. In Figures 1 and 2 it is strongest for relatively weak anisotropies or quite broad distributions. We note once again that the Wigner distribution function would give the same results in this case. In section VI we will speculate
-
-
-
3200
Schinke
The Journal of Physical Chemistry, Vol. 92, No. 11, 1988 1.0 I
-*
I
E = 0.9eV a
0 4
a *
/ 0 5. /. * e a
-
1
0
O
0
*
20
60
LO
Figure 4. Rotational state distributions for two anisotropy parameters c (in A) and E = 1 eV. Comparison between exact quantal calculations (-), full classical calculations without sin y o in eq 11 (0),and full classical calculations employing the classical distribution function for the initial state, eq 19 ( 0 ) .
about the basic physical origin of this artifact and possible improvements.
V. Comparison with Other Calculations The full classical calculations, described in this work, are almost identical with those of Goldfield et al.45to model the dissociation of ICN. However, there is one important difference: They did not include the sin yo term in the definition of the classical cross section eq 10. This factor is especially important for a linear ground state, because it shifts the net weighting function away from yo = 0 to some finite value. If the anisotropy is large in this range (and that is the case for ICN and the model system of this work), the final rotational distribution can be significantly in error. Since the sin yo term is so important, we test its influence for the present model system and compare in Figures 4 and 5 the results omitting it with the exact quantal distributions. The corresponding classical results with sin yoincluded are given in Figures 1 and 2. The differences are clearly visible and need no further discussion. The sin yo term will obviously become less influential for bent ground states, ye # 0. Recently, Waite and D ~ n l a studied p ~ ~ the photodissociation of ClCN, using classical mechanics. This work was a great step forward, because the dynamical calculations were performed on an a b initio potential for the excited state, not including any adjustable parameter, Most studies before were done either without any final state interaction or with simple modef potentials. The calculated C N rotational state distribution manifests clearly the rotational reflection principle for this system. It agrees reasonably well with the measurement of Halpern and Jackson.62 Despite the great achievement, we must criticize the definition of the classical cross section in ref 49. The dynamical calculations are identical with ours, but the weighting of the trajectories is done quite differently. Instead of giving explicitly the distribution function, they select the initial conditions by importance sampling in the following way (see eq 2.13 of ref 49) Po = sin t1 Ro = Re + ( w R / k R ) l I 2cos
(19a)
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( 6 2 ) Halpern, J . B.; Jackson, W . M. J . Phys. Chem. 198.2, 86, 3528.
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60 rotational state Figure 5. Same as Figure 4 but for t = -1.0 8, and two energies E . 0
rotational state
oo
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where wR and w7 are the frequencies for the stretch and the bending mode, respectively, and (, and t2are "random numbers, selected from a uniform distribution between 0 and 2a". Since Cl-CN is linear, ye = 0. The resulting distributions are the classical distributions for a harmonic oscillator with the quantum mechanical zero-point energy 0/2. For example, the distributions for the coordinates peak sharply at the classical turning points and are zero outside the classical region. It is thus completely different from the quantal distribution which is maximal a t the equilibrium and tunnels also into the nonclassical region. As qualitatively discussed by Heller,37 the classical distribution function for the parent molecule will lead to absorption cross sections which are drastically different from the corresponding quantal spectrum. Another point which we find important to mention is that the choice of initial conditions employed by Waite and Dunlap covers a wide range of energies in the excited state. For example, they average over as much as 5000 cm-' in their study of C1-CN d i s s ~ c i a t i o n .This ~ ~ averaging may not be important if the energy variation is insignificant. However, this is not necessarily the case and in particular not known beforehand. In Figures 4 and 5 we show results for our model system, obtained from the classical distribution eq 19, however, with one modification: In order to specify the total energy, only Po,jo,and yo are selected independently, while Rois chosen to be the corresponding turning point. The agreement with the corresponding quantal cross sections is not good in these cases and certainly much worse than in Figures 1 and 2, where the quantal distribution for the ground state has been employed. Agreement is best for the very high rotational states beyond the maximum. These probabilities are primarily determined by the shape of the excitation function (which is the same in both classical calculations) and not so much by the initial weighting. The CNiotational distribution of Waite and Dunlap admittedly agrees reasonably well with the m e a s ~ r e m e n t .It~ ~is also in good accord with the simple classical calculations (Po = j o = 0) which we performed for the same system using their ab initio potential.53 Therefore, it seems as if for C1-CN the quantal and the classical distributions for the initial conditions give roughly the same results which is in contrast to the present model system. In order to unravel this contradiction, we reexamine the C1-CN photodissociation and perform classical calculations with four different distribution functions. In Figure 6b we show the classical rotational state distributions as obtained with the quantal distribution, eq 13, and the classical distribution, eq 19; however, incorporating a fixed total energy E . Also shown are the experimental results of ref 6 2 . The parameters are the same as in ref 53: E = 2.083 eV, R, = 2.255
Rotational State Distributions of Triatomic Molecules
The Journal of Physical Chemistry, Vol. 92, No. 11, 1988 3201 the theory of Waite and Dunlap would predict a rotational distribution with two peaks, one originating from each turning point (see also the discussion of Heller3’). In the case of a linear molecule the two peaks naturally coincide.
VI. Summary and Discussion The main goal of this study is the test of classical mechanics in direct photodissociation processes. We performed full classical trajectory calculations for a model ABC A BC system and compared the final rotational state distributions with previously published exact quantal results. A quantum mechanical distribution was used to weight the starting values in the four-dimensional phase space. In most cases the agreement with the exact calculations is very satisfactory and generally better than as obtained previously with the initial momenta set to zero. However, there are cases where agreement is not so good, especially at low rotational states. Here, the classical calculation artificially overestimates the population. Analyzing the individual trajectories, we find that this artifact originates from very peculiar trajectories. For all these trajectories the angular velocity dy/dt and the torque -aIFx/ay at the twrning point have different directions. This leads to a nonmonotonous behavior in time and finally to the artifact at low j ’ s . The extent of this failure will sensitively depend on the parameters of the system and, last but not least, on the interaction potential. It is tempting to speculate whether semiclassical theories, in which the phase associated with each trajectory is also incorporated, might be more However, because of the double-ended boundary conditions such calculations are technically not simple (multidimensional root searching) in classical S matrix t h e ~ r y . ~The ~ . ~time-dependent ~ semiclassical formulation of Heller65 avoids this problem and may be more appropriate. Another, more fundamental reason for this artifact might be connected with the Franck-Condon (FC) assumption. In the quantal first-order perturbation theory the ground and the excited state wave functions are completely independent of each other. They are “coupled” only by the matrix element (1). In the classical theory the motion in the excited state depends, through the initial conditions, directly on the motion in the ground state. At the present time we cannot suggest any better “recipe” to perform the classical calculations. However, since for larger, real polyatomic systems classical mechanics might be the only feasible method, we find it worthwhile to further investigate these fundamental questions. In addition, we tested in this article the possibility of using the classical distribution function to sample the initial conditions. On the basis of the-comparison with exact quantal calculations for our model system, we must conclude that the classical distribution function is not appropriate and certainly cannot be recommended. Under certain conditions and because of averaging over several degrees of freedom, the final cross sections might be in reasonable accord with calculations employing a quantal distribution function. However, that agreement must be considered to be accidental.
-
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Figure 6. (a) Classical excitation function J(y0,Po=j,=0) and weighting function m(y0)of eq 16 including sin yo for the photodissociation of CI-CN. All parameters are defined in the text. Also shown is the classical distribution function (dashed line) as obtained from eq 19d, and ymalis the classical “turning point” of the bending motion. (b) Comparison of rotational state distributions as obtained from the following: and simplified (-) classical calculations employing the quantal full (0) distribution function for the initial state; full (0)and simplified (- - -) calculations employing the classical distribution function for the initial state. ( 0 )Experimental results of ref 6 2 .
-
A, AI? = 0.131 A, and Ay = 19O.-Both results agree surprisingly well and peak a t j 60 in excellent accord with the experiment. This result is unexpected in view of the quite different representations of the initial state. The differences become much more pronounced on the basis of the simple classical theory, where Po and j o are set to zero initially. The result for the quantal distribution function is only slightly different from the corresponding full calculation. It is only slightly narrower but has the same shape. The differences are of the same order as observed in Figures 1 and 2. The rotational distribution can be readily understood in terms of the (Po =, j o = 0) excitation function b(yo) and the weighting function W ( y o )which are shown in Figure 6a (including the sin yot_erm). 60 arises from the peak of W(yo)’ The pronounced peak at j at around y So. This is the rotafional reflection principle. The differences between the full and the simple classical calculations are much more pronounced, if.the classical distribution function is employed. Here, the Po = j o= 0 cross section peaks strongly at j,,, 63 and is a direct reflection of the classical distribution function in yowhich is also shown in Figure 6a. As discussed above, it peaks at some maximum angle ymax 8 O (i.e., the turning point) and is zero at larger angles. Figure 6 clearly indicates to us that the averaging effect, due to nonzero initial momenta, is much stronger if the classibal distribution function is used. W e consider the close agreement of the two full classical calculations to be only accidental. Incidentally, we note that the original calculation of Waiteand Dunlap agrees very well with our own calculation using their sampling procedure, but incorporating energy conservation. We went through this rather detailed analysis in order to demonstrate that under certain circumstances basically different methods can yield similar results, although the underlying physical pictures are quite different. If the parent molecule would be bent,
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-
-
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+
Acknowledgment. The author acknowledges several stimulating discussions with Drs. E. M. Goldfield and N. E. Henriksen. (63) Miller, W. H. Adu. Chem. Phys. 1974, 25, 36; 1975, 30, 11. (64) Gray, S. K.; Child, M. S. Mol. Phys. 1984, 51, 189. (65) Heller, E. J. J . Chem. Phys. 1975, 62, 1544.
