85
J. phys. Chem. 1982, 86, 85-91
Theoretical Analysis of the Reaction F 4- H2(Y = 0)
-+
HF( Y = 0, 1, 2, 3)
+H
Edward F. Hayes Divlsbn of Chemistry. Natbnal Science Foundatbn. Washington, D.C. 20550
and Robert B. Walker' Theoretical Divlsbn, Los Alamos National Lakatoty, Los Alaros, New M x b o 87545 (Received: April 28, 1981; In Flnal Form: August 3, 198 1)
A simple model is presented for the low-energy resonance in the PO2 reactive transmission probability for collinear dynamics. Detailed quantum-dynamicalcalculations on the Muckerman-5 surface are reported. These dynamical results suggest that the model adequately captures the physics underlyingthe low-energy resonance. In addition, an analysis of the vibrationally adiabatic potential-energy curves for the exit channel provides an explanation for the decay of the HF(u = 3) threshold obtained by using both quantum and quasi-classicalreverse dynamical calculationson this potential-energysurface. The resonance model and the analysis of the vibrationally adiabatic potential-energy curves provide some insight into features of the potential-energy surface that need to be determined accurately before full a priori theoretical predictions can be expected to provide quantitatively meaningful results.
-
Introduction Recently, the reaction F + Hz HF + H has been the subject of extensive theoretical' and experimental2 investigation, and it seems that this reaction will become one of the key testing grounds for chemical theory and experimental efforts with quantum state-tostateresolution. The system is small enough that accurate potential-energy-surface calculations are within reach. In addition, the atoms are light enough that full dynamical calculations, including a sufficient number of rotational and vibrational states to achieve convergence, will be possible in the near future. On the experimental side, Sparks, Hayden, Shobatake, Neumark, and Lee2 have already reported stateresolved, high-resolution crossed molecular beam studies of this reaction, and more detailed studies are underway. Our interest in this reaction is aimed toward a better understanding of quantum-dynamical resonances in reactive scattering. The first indication that there might be a quantum resonance feature in the F + H2 reaction came from collinear quantum-mechanical ~ t u d i e s . The ~ collinear results indicated a peak (resonance feature) in the reactive transmission probability, PO2, at low scattering energies (0.01-eV relative translational energy). Recently, Redmon and Wyatt' have reported three-dimensional theoretical calculations which predict that the same resonance feature, though broadened, is still present in the total cross section. Subsequently, Sparks et aL2 found changes in the center-of-mass angular distributions for HF(u = 2) formation as the total energy was increased; these observations are consistent with the theoretically predicted resonance phenomenon. The apparent agreement between theory and experiment for this reaction suggests that it might be instructive to reanalyze the theoretical calculations in order to pinpoint those features of the potential-energy surface re(1) (a) M. J. Redmon and R. E. Wyatt, Chem. Phys. Lett., 63, 209 (1979); (b) R. E. Wyatt, "Quantum Mechanical Study of Chemical Re-
action Dynamics",proceedings of the International Quantum Chemistry Congress, Kyoto, 1979. (2) R. K. Sparks, C. C. Hayden, K. Shobat.de, D. M. Neumark, and Y. T. Lee, "Molecular Beam Studies of Reaction Dynamics F + H2", Proceedings of the International Quantum Chemistry Congress, Kyoto, 1979. (3) (a) S. F. Wu, B. R.Johnson, and R.D. Levine, Mol. Phys., 25,839 (1973); (b) G.C. Schatz, J. M. Bowman, and A. Kuppermann, J. Chem. Phys., 63, 674 (1975); 58, 4023 (1973). 0022-365418212086-0085$0 1.2510
sponsible for the resonance feature. In the treatment of the F + H2 reaction and its isotopic analogues, the potential surface that has received the most attention is an extended LEPS surface reported by Muckerman (the M-5 surface)! This surface was regarded as the best of several LEPS surfaces because it provided agreement with the (then accepted) transition-state-theory activation energy of 1.71 kcal/m01,~and it optimized agreement between experimental data and three-dimensional trajectory results on the average vibrational energy of the product HF molecule? Considering the essential involvement of classical mechanics with the history of the M-5 potential surface, we are led to ask the obvious question: can it be more than a fortunate accident that such a surface would correctly predict the existence of so quantum-mechanical a feature as a resonance? To answer this question, it will be necessary to characterize the resonance feature more carefully to determine what features of the potential surface influence the position, the width, and the amplitude of the resonance. With these questions in mind, we have attempted to develop a simple, yet meaningful, explanation of this low-energy resonance feature. As we proceeded, it became natural to explore other quantum effects, including tunneling and local zero-point energy. Although the main focus of this paper is to characterize the low-energy F H2 resonance, we will address other issues pertinent to a complete description of this reaction and to the design of future theoretical treatments. In the hope that the reader will be better able to follow the discussion, we state our conclusion about the resonance at the outset. The resonance feature is a pre-classicalthreshold resonance, dependent upon the shape of the potential-energy surface (and especially the reaction barrier) in the entrance valley. To substantiate this claim, we present scattering results for a few simple models to
+
(4) J. T. Muckennan, J. Chem. Phys., 64,1155 (1971);56,2997 (1972); 57, 3388 (1972). (5) P. D. Mercer and H. 0. Pritchard, J. Phys. Chem., 63,1468 (1959). A lower experimental activation energy, 1.6 kcal/mol, was reported by
K. H. Homann, W. C. Solomon, J. Warnatz, H. Gg. Wagner, and C. Zetzsch, Ber. Bumenges. Phys. Chem., 74, 585 (1970). (6) K. G. Anlauf, P. E. Charters,D. S. Horne, R. G.MacDonald, D. H. Maylotte, J. C. Polanyi, W. J. Slrrlos, D. C. Tardy, and K. B. Woodall, J. Chem. Phys., 63,409 (1970); J. C. Polanyi and D. C. Tardy, ibid., 51, 5717 (1969).
0 1982 American Chemical Society
88
Hayes and Walker
The Journal of Physical Chemistry, Vol. 86, No. 1, 1982 I
I
1
000
04
I
I
1
08
12
. . .
A -
.-
1
Energy (eV) Flgure 1. Transmission probabiilty for the onedimensional-model problem (model 1) descrlbed by eq 1 (solid curve). Transmission probability for onedimensional-model problem described by eq 2 (dashed curve).
provide insight into the physics underlying the low-energy resonance. We will relate these model calculations to quantum collinear reactive scattering calculations on the M-5 surface. The quantum collinear reactive scattering calculations were performed by using the RXNlD ~ r o g r a m ,with ~ - ~a set of parameters chosen to provide sufticient accuracy (i-e., convergence) to reproduce the resonance features with confidence. We have worked with a primitive harmonic oscillator basis of modest size (20 functions), and a smaller set of coupled equations based on the local internal eigenstates. The number of coupled equations was reduced to 14 in all cases. We have found this level to be sufficiently accurate to produce all of the resonance features not only for this system but also for the reaction He + H2+ HeH+ H.g Errors in amplitudes due to lack of convergence in most cases are less than 10% of the calculated amplitude. The plan in the remainder of this paper is as follows. In section 11,we present resulta of several model scattering systems. In addition to their pedagogical virtue, these model problems exhibit resonance features which we feel qualitatively mimic the resonance feature observed in the F H2reaction. Then, in section 111,we discuss the M-5 surface in some detail, in order to point out several of ita qualitative features that are related to the dynamical calculations presented in section JY.Section V concludes with a discussion of results.
-
+
+
11. Simple Models
In this section, we would like to consider two simple classea of onedimensional models. The first model, model 1, is the simple two-barrier problem with the following potential parameters: V = 0.0 eV x C 0,0.8 C x C 1.5, and x > 2.3 V = 0.046 eV 0.0 Ix I0.8 and 1.5 Ix I2.3 (1) where the distances are given in b o k radii. The scattering problem for this potential can be solved exactly in the usual way by dividing the Schriidinger equation into several regions and matching wave functions and derivatives between regions. For a particle with a mass of 1 amu, a very distinct resonance is obtained at a total energy of 0.028 eV. These results are presented as a function of scattering energy in Figure 1(solid curve). The resonance feature is basically due to the formation of a standing wave between the two barriers at particular scattering energies. For this reason, the position of the resonance is very sensitive to the separation between the barriers. Moreover, the amplitude and the width of the resonance depend on the height and the width of the barriers. In addition to (7) Robert B. Walker, QCPE Program No. 352. (8)J. C. Light and R. B. Walker, J. Chem. Phys., 66, 4272 (1976). (9) J. T. Adams, Chen. Phys. Lett., 33, 275 (1975).
I -
1
.o
I
0.0
1 .o
Re(S) Flgure 2. Argand diagram for transmission matrix S for onedimensbnaknodel problem (model 1) described by eq 1. The second panel corresponds to the singlabarrier model described by eq 2. Dots indicate unit circle. Arrow shows the direction of increasing energy. Energy range is the same as in Figwe 1. Tick marks locate energies € = 0.02, 0.04, ..., 0.14.
this symmetrical-barrier problem, it is also possible to defiie scattering problems with unequal barriers that will exhibit similar resonance features. The dashed curve in Figure 1 corresponds to the single-barrier problem obtained by bringing the two barriers of eq 1 together, i.e. V = 0.0 eV x C 0 and x > 1.6 (2) 0 Ix I1.6 V = 0.046 eV The resonance feature for the two-barrier potential is a prethreshold resonance, since the classical dynamical threshold is 0.046 eV. Figure 2 shows an Argand diagram for the two-barrier problem for energies near the resonance. The rapid change in the phase of the scattering S-matrix element with energy, apparent in the figure, is characteristic of such a resonance. Next we consider a second class of models in order to illustrate that prethreshold resonances can be obtained with a variety of model potentials. The first member of this class is a simple barrier (model 2a). The second member (model 2b) extends the first by including a potential well at some distance from the barrier. The third member (model 2c) resembles the second, except that two channels are coupled in the region of the second barrier. The potentials for each of these models are defined in Table I. The first two model problems (models 2a and 2b) can be solved exactly in the usual way be dividing the Schrodinger equation into several regions and matching solutions. The third model problem (model 2c) is solved similarly once the coupling potential is diagonalized in
Analysis of F
-
+ HAv = 0)
HF(v = 0, 1, 2, 3)
+H
TABLE I: Potential Definition for Model Problems 1.0
potential
-
l-'
. . .
-
. . .
model 2c 1-2 model model channel channel coup1 2 ling region coordinate 2a 2b
1
0
Oa
2 3 4
--
