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J. Phys. Chem. 1984, 88, 210-214
Stablllzation Calculations and Probability Densities for the Well-Studied Collisional H2, F HD, and F 4- D, Resonances in Collinear F
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Todd C. Thompson and Donald G. Truhlar* Department of Chemistry, University of Minnesota, Minneapolis, Minnesota 55455 (Received: June 30, 1983)
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We have used the stabilization method with a two-dimensional harmonic oscillator basis set to calculate resonance wave functions for the collinear reactions F + H2 FH + H, F + HD FH + D, and F + D2 FD + D on the well-studied Muckerman no. 5 potential energy surface. We present plots of eigenvalue vs. scale parameter and contour maps of probability density I$IZboth showing the relative stability of resonance roots for these systems. In addition, the contour maps provide a clear assignment of quantum numbers and a pictorial illustration of the dynamical interpretation of the resonances.
1. Introduction The role of re-sonances in chemical reactions has been the subject of a great deal of theoretical attention in the past few years.' Resonance oscillations in chemical reaction probabilities as a function of energy were first observed for a realistic potential energy surface in collinear H HZe2Later work showed that these resonances correspond to temporary vibrational excitation in a collision complex3 and to significant time delays in the outgoing reactive flux,4 and it was shown that an observable resonance effect should persist for three-dimensional collision^.^ Resonance oscillations were subsequently observed in many collinear reaction probabilities.' The most interesting case has been F + H2 and isotopic analogues. These reactions appear to show a resonance very close to the dynamical threshold energy:,' and the resonance appears to play a role in the vibrational energy distribution of the product molecules. Since thermal systems are dominated by their behavior at and near threshold, this provides a case where quantal interference effects may have dramatic consequences for observable rate constants into specified product states; these rate constants are especially significant because F Hz and isotopic analogues are chemical laser systems of wide interest. The j,-conserving coupled-channel calculations of Redmon and Wyatt indicate that the F H2 resonance effect also occurs in three dimensions.* In molecular beam experiments, Lee and co-workers have observed the energy dependence of the vibrational-state-resolved differential cross sections for these reactions? and their studies appear to provide the first direct experimental evidence relevant to the role of collisional resonances in chemical reactions. Most of the theoretical dynamical studies of F Hz and isotopic analogues have been based on potential energy surface no. 5 of Muckerman, which, although published by the author only recently,1° has been available for many years. This surface is known to predict too high a threshold energyI0 compared to molecular beam experiments" and too small a thermal rate constant,10.'2
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(1) For a general review see A. Kuppermann in "Potential Energy Surfaces and Dynamics Calculations", D. G Truhlar, Ed., Plenum, New York, 1981, p 375. (2) D. G. Truhlar and A. Kuppermann, J . Chem. Phys., 52,3841 (1970); 56, 2232 (1976). (3) R. D. Levine and S.-F. Wu, Mol. Phys., 22,881 (1971); Chem. Phys. Lett., 11, 557 (1971). (4) G. C. Schatz and A. Kuppermann, J . Chem. Phys., 59, 964 (1973). ( 5 ) G. C. Schatz and A. Kuppermann, Phys. Reu. Lett., 35, 1266 (1975). (6) S. F. Wu, B. R. Johnson, and R. D. Levine, Mol. Phys., 25,839 (1973). (7) G. C. Schatz, J. M. Bowman, and A. Kuppermann, J . Chem. Phys., 58, 4023 (1973); 63, 674, 685 (1975). (8) M. J. Redmon and R. E. Wyatt, Chem. Phys. Lett., 63, 209 (1979); R. E. Wyatt, J. F. McNutt, and M. J. Redmon, Ber. Bunsenges. Phys. Chem., 86, 437 (1982). (9) R. K. Sparks, C. C. Hayden, K. Shobatake, D. M. Neumark, and Y. T. Lee in "Horizons of Quantum Chemistry", K. Fukui and B. Pullman, Eds., Reidel, Boston, 1980, p 91; Y. T. Lee, personal communication. (10) J. T. Muckerman, Theor. Chem. Adu. Perspect., 6A, 1 (1981).
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so it cannot be quantitatively correct.' Quasiclassical trajectory calculations1° based on this surface do, however, predict the first moment of the product vibrational distribution at 300 K in good agreement with experiment. Most important, however, is the fact that this surface has been used for so many theoretical studies that it has become a testing ground for new theoretical methods of studying resonances. Each new theoretical method that is applied to the resonances associated with the Muckerman 5 surface has given new insights into the subtleties of reactive resonances. There is also, of course, the hope that these studies might illuminate the dynamics of the real F + Hz reaction, but it may be more important at this stage to learn more about reactive resonances in general and about the usefulness of various techniques for studying them. It is well-known in scattering theory that resonance effects are more sensitive to potential energy functions than nonresonant scattering is, so we should be very cautious about comparing results obtained with model potential energy surfaces to experiments. The stabilization method13 is a general method for calculating resonance energies, wave functions, and widths, and it has been most widely used for electron-atom and electron-molecule scattering. We have recently applied this method to collinear reactive resonances in H H2, H FH, and isotopic analogues using approximate and model potential energy surfaces.l"l6 In this article we apply it to the nonsymmetric collinear reactions F H2, F HD, and F D2 on the Muckerman 5 potential energy surface.
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11. Calculations The basic idea of the calculations is that we diagonalize the Hamiltonian in a basis set containing a scale factor a. The value of a determines the range of coordinates spanned by the basis, and hence, effectively, a box size to which the basis localizes the wave functions. Since the Hamiltonian supports no three-body bound states, all eigenvalues are energies in the continuum. For large a,corresponding to sufficient variational freedom in the strong-interaction region, the eigenvalues corresponding to non(11) T. P. Schafer, P. E. Siska, J. M.Parson, F. P. Tully, Y. C. Wong, and Y. T. Lee, J . Chem. Phys., 53, 3385 (1970). (12) B. C. Garrett, D. G. Truhlar, R. S. Grev, and A. W. Magnuson, J . Phys. Chem., 84, 1730 (1980); 87,4554 (E) (1983). (13) E. Holoien and J. Midtal, J . Chem. Phys., 45, 2209 (1966); H. S. Taylor, J. K. Williams, and I. Eliezer, ibid., 47, 2165 (1967); H. S . Taylor, Ado. Chem. Phys., 18, 91 (1970); H. S. Taylor and A. U.Hazi, Phys. Reo. A , 14, 2071 (1976). Additional references for the stabilization method are given in ref 14-16. For other related work see D. G. Truhlar, Chem. Phys. Lett., 15, 483 (1972); 26, 377 (1974); N. Moiseyev, P. R. Certain, and F. Weinhold, Mol. Phys., 36, 1613 (1978); P. Kaijser and J. Simons, Phys. Reu. A , 21, 1093 (1980); B. C. Garrett and D. G. Truhlar, Chem. Phys. Lett., 92, 64 (1982); D. Farrelly, R. M. Hedges, Jr., and W. P. Reinhardt, ibid., 96, 599 (1983). ((4) T.'C. Thompson and D. G. Truhlar, J . Chem. Phys., 76, 1790 (1982); 77, 3777 (E) (1982). (15) T. C. Thompson and D. G. Truhlar, Chem. Phys. Lett., 92,71 (1982). (16) T. C. Thompson and D. G. Truhlar, Chem. Phys. Lett., 101, 235 (1983).
0022-3654/84/2088-0210$01.50/0 0 1984 American Chemical Society
Resonance Calculations on F
+ H2 Reactions
The Journal of Physical Chemistry, Vol. 88, No. 2, 1984 211
TABLE I: Constants Determining Coordinate Origins and Reactant Zero Point Energies
A
B
C
R,, a,
I; F F
H H D
H
4.4450 5.5997 5.9918
D D
,RG, R'AB, a, R'Bc, a, kcal/mol 1.792 1.830 1.797
2.469 2.184 2.417
6.2 5.1 4.4
resonant wave functions will decrease as a decreases, corresponding to a more diffuse basis and hence a box with lower eigenvalues. For smaller a the eigenvalues eventually increase as a decreases because of the loss of variational freedom in the strong-interaction region. The corresponding wave functions approximate the nonresonant scattering states a t somewhat arbitrary energies, depending on a,and the same nonresonance wave function occurs for two different values of a only by accident. In contrast, resonant wave functions are localized by dynamical effects to smaller regions than the basisset "box"; these wave functions and their associated eigenvalues tend to recur almost unchanged for every value of a. For the present calculations we used a two-dimensional harmonic-oscillator basis set given by
X
Figure 1. Coordinate system.
where
A(QJ
= NiffLyQJ exp(-y2Q?/2)
(2)
In eq 2, Ni is the normalization constant, Hi is a Hermite polynomial, and y is a constant. In eq 1, a is the scale factor. Since we will vary a in both directions from 1.0, we can choose the coordinates q1 and q2 and the constant y somewhat arbitrarily. Indeed, it is one of the advantages of the method that a stable root corresponding to a resonance should not depend sensitively on coordinate systems or the basis set. We choose our coordinate axes so that one of them is parallel to the line that bisects the reactant- and product-valley directions in the usual mass-scaled coordinate system, the other is perpendicular to this, the reduced mass is the same in both directions, and the origin is on the minimum-energy point of the bisector. In mass-scaled coordinates for an A BC AB C reaction, the angle between the reactant and product valleys of the potential is given by"
+
-
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= arccos
(-)
(3)
in terms of the masses. Then the coordinate choice specified in words above may be specified mathematically by q1 = f cos y2/3
+ J sin y2p - Ro
1.2
'
I
1.0
"
'
0.8
'
1
0.6
a
mAmC
~ A B ~ B C
-25"
(4)
where Rotranslates the origin and f and J are defined in terms of nearest-neighbor distances by
Figure 2. Eigenvalues vs. scale factors for FHH.
The constant reduced mass M~ is chosen as 1 amu, which is the usual choice in the vibrational spectroscopy literature. The constant Roand the nearest-neighbor distances at the origin, ROAB and RoBC, are given in Table I. The coordinates are illustrated in Figure 1. The constant y in eq 2 can be any convenient value and for the present work it was chosen to correspond when a = 1 to a vibrational quantum of 3816 cm-I; this is the local vibrational frequency for the F H2 reaction at the point where the minimum-energy path crosses the q2 = 0 axis. This yields y = 0.17761 ao-l. Finally, we took Nl = N2 = 12 for all calculations. The zero of energy for all calculations is A BC infinitely separated and at classical equilibrium. The zero point energies eRG of H,, HD, and D2 are given in Table I; relative translational energies in the entrance channel are obtained by subtracting these values from the total energies.
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where
and
111. Results and Discussion Figure 2 shows the calculated eigenvalue spectrum as a function of scale factor ("aplot") for F + H2. The lowest three resonance, at energies of about -1 1, 0, and 8 kcal/mol, are clearly visible. The first two resonances would be observable in nonreactive FH H collisions, and the third is the well-studied reactive resonance. The reactive resonance is clearly more stable on the right side of the plot, where a is smaller and the basis is more diffuse; this indicates that basis sets with Nl = N2 = 12 and a 2 1 do not span a wide enough region of coordinate space to fully represent the resonant part of the wave function. Figure 3 shows contour maps
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(17) See,e.g., D. G. Truhlar, A. D. Isaacson, and B. C. Garrett in "Theory of Chemical Reaction Dynamics", M. Baer, Ed.,CRC Press, Boca Raton, FL, in
press.
212 The Journal of Physical Chemistry, Vol. 88, No. 2, 1984
Thompson and Truhlar
18
Figure 3. Contours of normalized 1+12 for roots 18-22 of FHH at (Y = 0.6. The horizontal axis is q, in the range -1.2 to +1.6 ao; the vertical axis is q2 in the range -1.6 to +l.2 a,,. The contour values start at 0.25 ao-2and are spaced by 0.50 ao-2. The reactant channel is in the lower right, and the product channel is in the upper right; the plus sign denotes the saddle point location. The number in the upper left is the number of the root, and the dashed curves are potential energy contour values corresponding to the energy of the root.
of llc/l2 for roots 18-22 at a = 0.6. One of the wave functions, corresponding to root 20, is much more compact than the others. Examination of 1$l2 plots for other a values shows that this wave function is reasonably stable, but those corresponding to other roots are not. The stability of the eigenvalue and the compactness and stability of the wave function are all consistent with the resonant interpretation for root 20 a t a = 0.6. Figure 4 compares the contours of I$I2 for the resonant root of the F Hzsystem at two other values of a. Consistent with the discussion above we see that, at a = 1, the resonant wave function does not extend as far into the saddle point region and is apparently not well converged. The resonant wave functions at a = 0.665 and a = 0.6, in contrast, are very similar despite being on opposite sides of an avoided crossing. The a plot for the F Dz system is shown in Figure 5. The lowest-energy resonances are at about -15, -7, 1, and 6 kcal/mol, with the fourth resonance being the well-studied reactive resonance. The a plot for the F H D system is similar to that for F Hz and it shows a reactive resonance a t about 6 kcal/mol. The resonance identifications are confirmed by analysis of the resonant and nonresonant wave functions at various a values, as described above for the F Hz system. Figure 6 shows contours of for the resonant roots of the F + Dz and F + H D system.
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IV. Discussion Examination of the resonance roots in Figures 3,4, and 6 leads immediately to the assignment of quantum numbers (0,3) for FHz and FHD and (0,4) for FD2. The resonances correspond to an almost rectilinear motion between the saddle point region and a repulsive wall in the exit channel. The orientation of the pseudo-one-dimensional motion would not have been obvious beforehand, and it is a major advantage of the method used here that no assumptions of separability or adiabaticity in any set of coordinates are necessary. It is encouraging though that the same
set of quantum numbers obtained here, by quantal stabilization calculations with no dynamical assumptions, were obtained earlier for FHz by Launay and LeDo~rneuf'~ by means of an adiabaticity assumption in hyperspherical coordinates. The potentials along the pseudo-one-dimensional motions of the resonant states are very anharmonic and the resonance wave functions exhibit the familiar accumulation of probability density at the effective turning point with the least steep local potential, which is the region near the saddle point. Another coordinate system that has been widely used to discuss resonances is the natural-collision-coordinate system consisting of a curvilinear reaction coordinate and a locally transverse vibrational coordinate.z0 For more symmetric systems, adiabatic potential curves as functions of the reaction coordinate have been used for quantitative predictions of resonance energies and widths.21 For F Hz and isotopic analogues the resonances are apparently vibrationally nonadiabatic. Hayes and WalkerZZhave explained the F Hz resonance as nonadiabatic trapping between an adiabatic barrier with vibrational quantum number n = 0 in the entrance channel and another with n = 3 in the exit channel. The present contour plots for F + H, show that the resonance state corresponds to no vibrational excitation in the reactant channel and to 3 quanta of vibrational excitation in the product channel. This is consistent with the relevant adiabatic barrier in
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(18) R. D. Levine and R. B. Bernstein, "Molecular Reaction Dynamics", Oxford University Press, New York, 1974, pp 98-9. (19) J. M. Launay and M. LeDourneuf, J. Phys. B, 15, L455 (1982). (20) R. A. Marcus, J. Chem. Phys., 45,4493 (1966). (21) B. C. Garrett and D. G. Truhlar, J . Phys. Chem., 83, 1079 (1979); 84,682 (E) (1980); 87, 4553 (E) (1983); B. C. Garrett, D. G. Truhlar, R. S. Grev, G. C. Schatz, and R. B. Walker, ibid.,85, 3806 (1981); B. C. Garrett and D. G. Truhlar, ibid.,86, 1136 (1982); 87,4554 (E)(1983); R. T. Skodje, D. G. Truhlar, and B. C. Garrett, ibid.,to be published in the Crawford issue. (22) E. F. Hayes and R. B. Walker, J. Phys. Chem., 86, 85 (1981).
Resonance Calculations on F
+ H2 Reactions
The Journal of Physical Chemistry, Vol. 88, No. 2, 1984 213
0.665
Figure 4. Same as Figure 3 except root 19 at a = 0.665 and root 16 at a = 1.0.
Figure 6. Same as Figure 3 except root 29 for FDD at a = 0.58 and root 16 for FHD at a = 0.76.
for n = 2, in terms of an n = 3 adiabatic barrier in the exit channel. Similar considerations apply to the other isotopes, and, for the F D2reaction, the relevant exit channel barrier would be slightly above the total energy of 4.22 kcal/mol of the n = 4 product. An interesting explanation of how this vibrationally nonadiabatic trapping may m u r classically was provided by Pollak and Child?4 They showed the existence of periodic classical trajectories oscillating between n = 0 states in the entrance channel and n = 3 states in the exit channel. Quantizing these resonance periodic orbits gave estimates of the resonance energies. Comparison of the resonant periodic trajectories published by Child and Pollak to the contours of 1$12 presented here shows that the trajectories are located in approximately the same regions of space as the larger probability densities of the quantal calculations; this confirms the basic correctness of the Pollak-Child interpretation of the resonances. It is interesting to compare the probability densities of the present calculation to a contour map of the scattering wave function probability density at the resonance energy as presented by Latham et al.25 Their contour map (Figure 12 of ref 25) is very similar to Figure 3 (root 20) or Figure 4 (a = 0.665) of the present study. However, the probability density of the scattering wave function is harder to interpret because it includes even larger peaks in the entrance channel due to interference of incoming reactive flux and outgoing elastically scattered flux. Latham et al. did not interpret the nodal structure in the interaction region or use it to assign quantum numbers to the resonance.
+
10
0
0
Y
Y
%
gr -10
W
-20c 1.2
1.0
0.8
a Figure 5. Eigenvalues vs. scale factor for FDD.
0.6
the entrance channel being the n = 0 one, and the relevant adiabatic barrier in the exit channel being an n = 3 one at an energy slightly above the total energy of 6.63 kcal/mol of the n = 3 product. This is consistent with the analyses of Hayes and Walker. In related work, P01lak~~ has explained the delayed threshold for producing the n = 3 state of HF, as compared to the threshold (23) E.Pollak, J. Chem. Phys., 74,5586 (1981); 75,4435 (1981). See also S. Ron, M. Baer, and E.Pollak, J . Chem. Phys., 78, 4414 (1983).
V. Conclusions We have shown that the well-studied resonances in F H, and isotopic analogues on the Muckerman 5 potential energy surface, although broad, can be studied by the stabilization method.
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(24) E. Pollak and M. S . Child, Chem. Phys., 60, 23 (1981). (25) S. L. Latham, J. F. McNutt, R. E. Wyatt, and M. J. Redmon, J . Chem. Phys., 69, 3746 (1978).
214
J. Phys. Chem. 1984,88, 214-221
Contour maps of 1$12, where $ is the resonance wave function for the collinear reaction, show that the resonance wave functions have a very simple structure and allow for unequivocal assignment of quantum numbers (0,3)for F H2 and F H D and (0,4) for F + D2. The assignment is consistent with a previous adiabatic treatment of FH2 in hyperspherical coordinates. The structure of 1$12 is also consistent with interpretations based on adiabatic barriers in natural collision coordinates and on classical resonant periodic orbits. The contour maps directly illustrate the dynamical localization of the resonance wave function to a compact region
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of the potential energy surface in the strong-interaction region between the potential energy barrier in the entrance channel and an effective dynamical barrier in the exit channel.
Acknowledgment. Helpful collaboration with Bruce C. Garrett in related studies of F + H2 and isotopic analogues is gratefully acknowledged. This work was supported in part by the National Science Foundation through Grant No. CHE80-25232. Registry No. Hydrogen, 1333-74-0;deuterium, 7782-39-0;hydrogen deuteride, 13983-20-5; atomic fluorine, 14762-94-8.
A Quasiclasslcal Trajectory Study of Colllsional Excitation in H
+ CO
Lynn C. Geiger and George C. Schatz* Department of Chemistry, Northwestern University, Evanston, Illinois 60201 (Received: July 13, 1983) The results of a quasiclassical trajectory calculation of cross sections for collisional excitation in H (D) + CO at 1-4-eV translational energy are presented and used to interpret recent laser photolysis measurements. A realistic potential energy surface was used in these calculations based on Dunning’s recent ab initio studies. Overall agreement of the calculated results with experiment is generally good, and this enables us to assess in detail what features of the potential energy surface the measured results are sensitive to. For the rotationally summed vibrational distributions, we find that the high u tail of these distributions is almost exclusively due to collisions which form a COH complex. The average COH lifetime is found to be about 3 OH vibrational periods, and the cross sections for complex formation are found to be very sensitive to the H + CO COH barrier. Based on comparisons with experiment we revise this barrier from Dunning’s 1.72-eV value down to 1.52 eV. Many other features of the measured results, such as the average vibrational energy transfer, are found to be primarily sensitive to impulsive collisions of H with either the C or 0 atom. Although the HCO portion of the potential surface was sampled with significant probability, none of the HCO complexes formed had a lifetime of more than one inner turning point. The rotational distributions were found to be composed of a high j component (for which ( j ) increases with increasing u ) arising from impulsive collisions of H with the C atom, and a lower j component (for which ( j ) is constant or decreases with increasing v) arising mostly from collisions with the 0 atom.
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I. Introduction have developed a new and promising Recently, two technique for studying collisional excitation and chemical reactions in small molecules using fast hydrogen and deuterium atoms. In this technique, excimer laser photodissociation of H2S, D2S, HCl, and other species is used to produce translationally monochromatic H or D atoms with center-of-mass energies between 0.95 and 4.0 eV, and the collisions of these atoms with small molecules are monitored on a microsecond timescale by infrared chemiluminescence. Among the first applications of this technique have been studies of translational to vibrational energy transfer in H CO and D + C 0 3 and translational to rotational energy transfer in H The H (D) CO system is especially interesting from the point of view of these experiments because metastable intermediates corresponding to the formyl radical isomers H C O and COH are energetically accessible. This means that the vibrational and/or rotational distributions might contain information that pertains to regions of the potential surface corresponding to these species. The kinetics of H C O H C O is, of course, familiar from its importance in comb~stion,~ and HCO is known to be stable (by 0.81 eV6) relative to H + CO, with only a small (-0.11 eV7>*)barrier to formation. COH, on the other hand, has never been observed experimentally, but accurate ab initio calculationss indicate that a metastable minimum corresponding to COH does exist at 1.04 eV above H CO, with a barrier of 0.68 eV to dissociation. Adding the last two numbers together yields an estimated barrier to formation of COH from H + CO of 1.72 eV which is nicely within the range of energies spanned by the photolysis experiments. In this paper, we present the results of a detailed quasiclassical trajectory study of H (D) + C O collisional excitation using a
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*Alfred P. Sloan Fellow; Camille and Henry Dreyfus Teacher-Scholar.
0022-3654/84/2088-0214$01.50/0
realistic potential energy surface. These results will be used to interpret CO vibrational distributions measured by Wight and Leone3 and rotational distributions reported by Wood, Flynn, and west or^.^ Evidence will be given which suggests that certain features of the vibrational and rotational distributions are sensitive to the presence of metastable COH (COD), and that the barrier to formation of this metastable can, at least crudely, be inferred from the measured results. To summarize the rest of this paper, the section 11, we describe our potential energy surface as obtained by fitting a combination of experimental and ab initio data, and we briefly summarize the trajectory calculations. Section I11 presents the resulting state resolved cross sections, including a detailed analysis of how the region of the potential surface sampled influences the final state distributions. This section also contains an analysis of the experimental data. A summary of our conclusions is presented in Section IV. 11. Theory A . The HCO Potential Energy Surface. Although small
portions of the ground (zA’) electronic potential energy surface (1) Magnotta, F.; Nesbitt D. J.; Leone, S. R. Chem. Phys. Lett. 1981,83, 21. (2) Quick, Jr., C. R.; Weston, Jr., R. E.; Flynn, G. W. Chem. Phys. Lett. 1981, 83, 15. (3) Wight, C. A.; Leone, S.R. J. Chem. Phys. 1983, 78,4875. (4) Wood, C. F.; Flynn, G. W.; Weston, Jr., R. E. J. Chem. Phys. 1982, 77, 4776. (5) Ahumada, J. J.; Michael, J. V.; Osborne, D. T. J . Chem. Phys. 1972, 57, 3736, and references therein. (6) Warneck, P. Z . Naturforsch. A 1974, 29, 350. (7) This is the highest of a range of estimates made in ref 8 based on the 0.087-eV activation energy measured by Wang, H. Y . ;Eyre, J. A,; Dorfman, L. M. J . Chem. Phys. 1973, 59, 5199. (8) Dunning, T. H. J . Chem. Phys. 1980, 73, 2304.
0 1984 American Chemical Society
