J. Phys. Chem. 1996, 100, 2759-2761
2759
Resonances in Collinear H- + H2 f H2 + H- Reaction: Energy-Resolved Reaction Probabilities by the Time-Dependent Wave Packet Approach Susanta Mahapatra and N. Sathyamurthy*,† Department of Chemistry, Indian Institute of Technology, Kanpur 208 016 India ReceiVed: August 2, 1995; In Final Form: NoVember 21, 1995X
We report here the results of a time-dependent quantal calculation for the collinear reaction H- + H2(V)0) f H2 + H-, which reveals a large number of resonances, in contrast to a relatively smaller number for the neutral analogue, despite the similarities in their potential energy surfaces.
In this letter we show the existence of a large number of resonances in the collinear reaction
H- + H2(V)0) f H2 + H-
(R1)
by computing the energy-resolved reaction probability, PR(E), values for the system using the time-dependent wave packet (TDWP) approach.1 Dynamics of collisions involving negative ions have received considerable attention in recent years because of their importance in interstellar media. In addition, photodetachment of negative ions has opened up an interesting avenue for investigating transition state species of their neutral analogues.2 In case of H3-, this becomes of fundamental interest. There have been a number of experimental studies of reaction R1.3 The availability of a recent ab initio potential energy surface (PES)4 for the reaction makes it possible to investigate its dynamics in detail. Gianturco and Kumar5 have undertaken a detailed dynamical study of vibrational and rotational inelasticity in the system and compared the results with those of (H+, H2 ) collisions. The system has an interesting feature: it has an electron detachment channel with an energetic threshold of EEdth ∼ 1.45 eV competing with the charge-transfer rearrangement channel (R1) for which the cross section becomes appreciable at ∼0.45 eV. Qualitatively, the PES of (R1) looks similar to that of the collinear reaction6
H + H2 f H2 + H
(R2)
Quantitatively, the transition state for (R1) occurs at a slightly extended configuration rH--H ) rH-H ) 1.997 au when compared to that with rH-H ) rH-H ) 1.757 au for (R2). The barrier height (Eb) for H3-, 0.4648 eV,4 is only slightly larger than that for H3 (Eb ) 0.4249 eV).6 Yet, we find a large number of transition-state resonances (TSR) for the collinear reaction (R1) when compared to a relatively smaller number for (R2). Belyaev et al.7 had computed the state-resolved reaction probabilities for (R1) and its isotopic analogue (H- + D2 ) in collinear geometry by the S-matrix Kohn variational method on a diatomics-in-molecules (DIM) PES. Their results did not give any indication of resonances in the system. In contrast, our studies, which involved the time evolution of a wave packet (WP) placed initially in the interaction region, revealed a large number of resonances.8 Our studies also revealed unusual features in the eigenfunctions corresponding to the quasibound states. † Honorary Professor, S. N. Bose National Center for Basic Sciences, Calcutta, India. X Abstract published in AdVance ACS Abstracts, February 1, 1996.
0022-3654/96/20100-2759$12.00/0
To compute PR(E) explicitly and to verify that a large number of reactive scattering resonances indeed exist for the system, we have adapted the strategy outlined by Neuhauser and Baer9 and others,10-12 in which the configuration space is divided into reagent, interaction, and product regions and PR(E) values are calculated from the energy-resolved flux across a dividing line chosen in the product region. The method was initially applied by Neuhauser and Baer9 to (R2) and then to the collinear F + H2 reaction. It was adapted by Zhang and Zhang10,11 to the H + O2 and OH + H2 reactions and by Balakrishnan and Sathyamurthy12 to the He + H2+ reaction. The time-dependent Schro¨dinger equation for the system is solved numerically on a grid of size 256 × 256 in mass-scaled Jacobi coordinates (R,r). The Hamiltonian for the collinear configuration is
H)-
[
]
p2 ∂2 ∂2 + + V(R,r) 2µ ∂R2 ∂r2
(1)
with µ ) [mambmc/(ma + mb + mc)]1/2. The grid spacings are taken to be 0.05 au in both coordinates. The Sta¨rck-Meyer (SM) PES4 is used for V(R,r). The initial wave function Φ(R,r,t)0) is chosen as a product of a minimum uncertainty Gaussian wave packet (GWP), F(R), corresponding to H-, H2 relative translation, and a Morse oscillator wave function, φV(r), corresponding to the vibrational state of H2 with vibrational quantum number V:
Φ(R,r,t)0) ) F(R) φV(r)
(2)
where
[
F(R) ) (2πδ ) exp 2 1/4
(R - R0)2 4δ2
- ik0R
]
(3)
Here δ is the width parameter of the GWP, and R0 and k0 are its locations in the coordinate and momentum spaces, respectively. The kinetic energy part of the Hamiltonian is evaluated by the fast Fourier transform (FFT) method, and the time evolution of the wave packet is carried out by the split-operator method.13 The width parameter of the GWP is chosen (0.25 au) sufficiently narrow in coordinate space so that in momentum space it is broad enough to contain most of the energy components in it. This also exploits the power of the TDWP approach in that a single WP propagation gives rise to reaction probabilities at various energies, as long as those energy components are adequately represented in the initial WP. The upper limit of the potential on the grid is fixed at 4 eV which © 1996 American Chemical Society
2760 J. Phys. Chem., Vol. 100, No. 8, 1996
Letters
allows a time step of 0.1616 fs for a stable propagation of WP by the split-operator scheme. The reaction probability, PR(E), is computed by12
PVR(E) ) (p/µ)Im[〈Ψ(R,rI,E |∂Ψ(R,rI,E)/∂r〉]
(4)
where the quantity within the angular brackets is the energyresolved flux of the WP, which is integrated over the entire range of R. Ψ(R,rI,E) is the energy-resolved wave function normalized with respect to initial translational energy distribution of the WP, i.e.
Ψ(R,rI,E) ) Φ(R,rI,E)/aE
(5)
Φ(R,rI,E) is obtained by Fourier transforming the time evolved WP, Φ(R,r,t), along the dividing line at r ) rI:
Φ(R,rI,E) ) (1/x2π)∫-∞Φ(R,r,t)exp(iEt/p) dt|rdrI (6) ∞
where E is the total energy of the WP. aE in eq 5 is the weight of the initial energy component contained in the WP and is evaluated by ∞ 1 F(R)exp(ikR) dR aE ) (µ/pk)1/2 ∫ -∞ x2π
(7)
) (µ/pk)1/2F(k)
(8)
where F(k) is the momentum representation of F(R). The factor (µ/pk)1/2 ensures that Ψ(R,rI,E) is energy normalized. The final expression for the reaction probability can be written as
PVR(E) )
(
) [〈
|
〉]
∂Φ(R,rI,E) p2k Im Φ(R,r ,E) I ∂r µ2|F(k)|2
(9)
with k ) x2µ(E-EV)/p, and EV is the initial vibrational energy of the diatomic molecule. In the present calculation the WP is initially taken in the asymptotic reagent channel at R0 ) 10.0 au, which is sufficiently far away from the interaction region so that initially the separation of variable condition (eq 2) is met. The dividing line in the product channel is taken at rI ) 8.5 au. The WP is time evolved for 662 fs which is sufficient to remove essentially all the WP from the interaction region. Operationally, the WP and its derivative with respect to r are computed at each time step along r ) rI and Fourier transformed to the energy domain. The values of the Fourier transformed quantities at each time step are added coherently to the respective values at the previous step, which reduces the storage requirement of the computer to the size of the R grid times the number of energy values at which PR(E) is sought. Once the contributions from each time step are added to the values from the previous step, the WP is absorbed by a negative imaginary potential (NIP) placed near the edge of the grid in both reactant and product channels. A linear type of NIP as suggested by Neuhauser and Baer14 is used in the present calculation. We have checked for convergence in PR(E) by varying the height and the width of the NIP and the results have been further verified by obtaining them using a sin2 masking function in place of the NIP. The PR(E) values for (R1) for V ) 0 state of H2 thus obtained are plotted in Figure 1 (as a solid line). Clearly, there is a large number of oscillations in PR(E), far in excess of what one would have expected, based on the results of Belyaev et al.7 on a DIM PES and also based on the well-known results9a,15 for collinear
Figure 1. Energy-resolved reaction probability, PR, as a function of energy, E, for collinear H- + H2(V)0) reaction on SM PES (solid line). The PR(E) values of collinear H + H2(V)0) reaction on LSTH PES are shown by a dashed line to highlight the dynamical differences between the two systems. The arrow along the abscissa indicates the threshold for the electron detachment channel.
reaction (R2) in the same energy range. To ensure that we have not obtained any spurious results because of the choice of (R,r) grid or the computational scheme, we computed the PR(E) values for reaction R2 under identical conditions on the LiuSiegbahn-Truhlar-Horowitz (LSTH) PES6 and found them (included as dashed lines in Figure 1) to be in agreement with the values published in the literature.9a,15 Thus it becomes clear that indeed there is a large number of collinear reactive scattering resonances for (H-, H2) in contrast to a much smaller number for (H, H2) despite comparable Eb values for the two reactions. We must add that for both reactions PR(E) becomes appreciable at ∼0.45 eV, rises to near unity, and eventually becomes nearly zero around 3.0 eV. The most prominent resonance observed for (R2) is at 0.882 eV, near the threshold of the V′ ) 1 channel of H2 (0.782 eV). The first substantial dip in PR(E) occurs at a higher energy (1.1 eV) in the case of (R1) and is less sharp. Our earlier calculations8 using the autocorrelation function approach had predicted the resonance at 1.12 eV. Preliminary investigations indicate that this is a Feshbach (compound state) resonance arising from a quasibound state supported by the vibrationally adiabatic potential (VAP) for V ) 2 in hyperspherical coordinates.16 Figure 1 also reveals that at higher energies, near each (vibrational) channel threshold, there is a resonance as anticipated by Friedman and Truhlar.17 Some of the congestion near each such threshold resonance could be attributed to the Feshbach resonances arising from quasibound states supported by the higher VAPs. Still, there are some more oscillations in PR(E) remaining to be accounted for, and their origin is presently under investigation. The Sta¨rck-Meyer PES has a shallow well of depth of ∼0.05 eV at (R, r) ) (5.75, 1.52) au. But this is unlikely to be the source of the large number of resonances found for (R1). As mentioned earlier, there is an electron detachment channel opening up at ∼1.45 eV, and one can anticipate that some of the rearrangement channel resonances would be quenched at higher energies. We have tried to include the detachment channel by using a NIP, along the detachment seam in the PES. The magnitude of PR(E) is highly sensitive to the choice of NIP and a more careful investigation is necessary. The barrier for the exchange reaction is the lowest in the collinear approach for (R1) as well as (R2). Therefore, the
Letters dynamical differences found here for the collinear geometry could be indicative of the differences in the dynamics in three dimensions for reaction (R1). A three-dimensional study of reaction (R1) is presently in progress. To conclude, we have shown the existence of a large number of resonances for collinear reaction (R1) on SM PES4 in contrast to their absence on the DIM PES7 and for reaction (R2) on the LSTH PES.6 Acknowledgment. This study was supported in part by a grant from the Commission of European Communities. References and Notes (1) (a) Mohan, V.; Sathyamurthy, N. Comput. Phys. Rep. 1988, 7, 213. (b) Kosloff, R. J. Phys. Chem. 1988, 92, 2087. (c) Thematic issue on: TimeDependent Methods for Quantum Dynamics; Kulander, K. C., Ed. Comput. Phys. Commun. 1991, 63, 1. (d) Kosloff, R. Annu. ReV. Phys. Chem. 1994, 45, 145. (e) Balakrishnan, N.; Kalyanaraman, C.; Sathyamurthy, N. Phys. Rep., submitted. (2) Metz, R. B.; Bradforth, S. E.; Neumark, D. M. AdV. Chem. Phys. 1992, 81, 1. (3) (a) Michels, H. H.; Paulson, J. F. Potential energy surfaces and dynamics calculations; Truhlar, D. G., Ed.; Plenum Press: New York, 1981; p 535. (b) Haq, M. S.; Doverspike, L. D.; Champion, R. L. Phys. ReV. 1983, A27, 2831. (c) Zimmer, M.; Linder, F. Chem. Phys. Lett. 1992, 195, 153. (d) Reinig, P.; Zimmer, M.; Linder, F. Nucl. Fusion Suppl. 1992, 2, 95. (e) Zimmer, M.; Linder, F. J. Phys. B, in press.
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