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Accurate three-dimensional quantum scattering calculations for Ne + H+2 → NeH+ + H
Volume 179, number 56
CHEMICAL PHYSICS LETTERS
3May 1991
Accurate three-dimensional quantum scattering calculations for NetHz +NeH++H Joel D. Kress Theoretical Division (T- 12, MS 8268). Los Aiamos National Laboratory, Los Alamos. NM 87545, USA
Received 2 I November 1990; in final form 28 January I99 I
Accurate 3D coupled channel calculations are presented for the reaction Net Hz +NeH* t H computed with the diatomicsin-molecules potential of Hayes and co-workers. Reaction probabilities for total angular momentum J=O and total energies up to I. 1 eV are reported. The hyperspherical formulation of Pack and Parker is used, and the adiabatic basis functions are generated using the discrete variable representation (DVR). The total reaction probabilities show much structure as a function of total energy, structure that is indicative of quantum dynamical resonances. The trend in the results agrees with that observed experimentally: vibrational energy is more effective than an equivalent amount of translational energy for promoting the reaction.
1. Introduction The dynamics of rare gas atoms reacting with hydrogen molecular ions have been studied using various experimental approaches. For the reactions NeSH; -NeH++H and He+H: -+HeH++H, Chupka and Russell [ 1 ] measured reactive cross sections using a threshold photoionization mass spectrometry technique and observed that vibrational energy is more efticient than an equivalent amount of relative translational energy for promoting both of these reactions. Using a photoelectronsecondary-ion-coincidence technique, van Pijkeren et al. [ 2 ] measured reactive cross sections at thermal collision energies as a function of the initial vibrational state v for both reactions. Their results were consistent with the known reaction endoergicities and HZ vibrational energies. Using molecular beam techniques, the observations of Chupka and Russell have been verified [ 31 for both reactions, and angular distributions and flux contour maps have been measured [ 41 for Ne+H: . The quantum dynamics for the He + H2+reaction was recently examined [5] using three different techniques: collinear ( ID); approximate 3D (the bending corrected rotating linear model [ 61, BCRLM); and accurate full 3D. Other theoretical work and experimental work for the He+ Hz reac510
tion was also reviewed there. Relatively less is known theoretically about Ne + H; . Kuntz and Roach constructed [ 7 ] a semi-empirical diatomics-in-molecules [ 81 (DIM) potential energy surface (PES) which was obtained by scaling their DIM PES for the ArSH: reaction. Hayes et al. [ 91 generated a DIM PES (hereafter denoted HSCM ) by representing the diatomic fragments as Morse functions. The H:(‘C,+),H$(‘C,f),andHeH+(‘Z+) curveswere fit to accurate diatomic data and the HeH ( 2E+) were adjusted by fitting the DIM PES to three SCF energies along the collinear reaction path. A contour plot of the resulting DIM PES for collinear geometries was quite similar to that for the SCF PES they [9] also computed. Stroud and Raff [lo] computed quasiclassical collinear reaction probabilities using the HSCM PES and found that translational energy was more effective than vibrational energy for promoting reaction in contradiction with the experimental findings. They postulated that the DIM representation was suspect and generated a spline-fitted PES (which they denoted SAI) fit to a grid of collinear SCF energies similar in accuracy to those of Hayes et al. [ 91. With the SAI PES, Stroud and Raff then found reasonable agreement with experiment in terms of vibrational enhancement for the collinear quasiclassical calculations. Very recently Urban et al. reported [ 111 coupled electron pair approxima-
0009-2614/91/$ 03.50 0 1991 - Elsevier Science Publishers B.V. (North-Holland)
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CHEMICALPHYSICSLETTERS
tion (CEPA, a method which correlates the electrons) calculations for both collinear and non-collinear geometries of Ne+ Hz. Furthermore, employing a fit to their collinear PES, they calculated quantum collinear reaction probabilities using a finite element technique [ 121. For total collision energies up to 1.5 eV, these calculations do not agree with experiment in terms of vibrational enhancement, but beyond 1.5 eV they do. In this Letter, what are believed to be the first calculations of accurate 3D quantum dynamics for the NeSH: reaction are presented. The HSCM PES is employed and the full 3D reactive scattering for total momentum J=O is formulated using the adiabatically adjusting principal-axis hyperspherical ( APH ) method of Pack and Parker [ 131. The sector adiabatic basis functions are generated using the discrete variable representation (DVR) method [ 14 1. This APH-DVR approach has yielded accurate 3D results for F+H2 [ 15,161, H+H2 and D+H2 [ 171, and He+H: [ 51. Total reaction probabilities from all energetically allowed initial vibrational states of Hz show much structure indicative of quantum dynamical resonances and agree with the experimental trend of vibrational enhancement.
2. Methods and convergence
studies
The APH theory is used to formulate the exact 3D treatment of the reactive scattering, the details of which are given in ref. [ 131. Here a brief sketch of the approach is given. The total scattering wavefunction is expanded in a basis of sector adiabatic surface functions. The surface functions are bound state eigenfunctions of the surface Hamiltonian H(e,X;Pc)~,(e,X;Pc)=~(Be)~~(e,X;P~)
I
(1)
where 0 and Kare the two APH hyperangles. Eq. ( 1) also depends parametrically on pc, the center of a sector, where the range of APH hyperradius is divided into n sectors, <= 1, .... n. When the total scattering wavefunction is substituted into the full 3D Schriidinger equation for J=O, a set of N coupled channel (CC) equations is obtained, where N is the number of surface functions (t= 1, .... N) in the CC expansion. These exact CC equations are propagated from p, to p,, using the log-derivative method [ 18 1,
3 May 1991
then the boundary conditions are applied as usual [131. The discrete variable representation (DVR) method [ 141 is used to calculate the set of (@,( 8, x; pc)} for each p. (= 1, .... n. The application of the DVR method to the calculation of surface functions is detailed in ref. [ 161; a brief summary of the approach follows. A DVR is used [ 191 for both 0 and x. The DVR in 6’is formed in the basis of (normalized) Legendre polynomials, {p,( cos 20), 110, 1, .... I,,,}. A symmetry adapted [ 19 ] DVR in x is formed in the set of (normalized) basis functions {a,cos2(m-1)x, m=l,2 ,..., m,,,}.TheDVRmatrix representation of eq. ( 1) is then constructed and the surface functions are found using the sequential diagonalization-truncation technique [ 201. The resulting matrix eigenvalue problem is solved using direct techniques. The usual approaches are used to: ( 1) transform [ 161 the surface functions from the DVR to the finite basis representation (FBR) necessary for projecting onto the boundary conditions; ( 2) calculate [ 16 ] the potential coupling matrix elements; and (3 ) calculate [ 5,161 the matrix elements of the overlap between surface functions @,(B,x; pr) and @tt(Q,x~~+I).
The topology of the HSCM PES for NeSH: is quite similar to that of the MTJS PES [21,22] for He+H,+ which was used to calculate accurate 3D quantum scattering results [ 51. The Ne+H: reaction is endoergic by about 0.6 eV, whereas the He + Hz reaction is endoergic by about 0.8 eV, with respect to the rovibrational ground state and both PESs have wells about 0.3 eV deep in the entrance channel. (The zero of energy is measured from the bottom of the asymptotic Hz well.) Also, neither PES has any barrier along the minimum energy path. As is the case for He+ H: ,the reaction path, defined by finding the minimum energy with respect to 0 and x for fixed values of p, has a well. For Net H: ,the well occurs near p= 4.0 a, and is about 0.3 eV deep. ( 1 uo= 1 bohr unit.) To illustrate the effect of electron correlation at the CEPA level, the collinear PES of Urban et al. [ 111 has a deeper well of 0.49 eV with the Ne more tightly bound to Hz as compared to the HSCM PES. The DVR calculation is flexible enough to represent the surface functions over three basis regions: the repulsive wall of the potential at small values of 511
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CHEMICAL PHYSICS LETTERS
pb the bound states in the well for intermediate values of pc; and, the nearly asymptotic rovibrational states which are localized in each arrangement channel. To accommodate the shrinkage problem (as pe increases the surface functions access increasingly limited regions of 19and x on the p
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point), &= 1.067357 eV is converged to nearly 6, 6, and nearly 6 figures with respect to 1,,,,,=40, m ,,,=== 80, and EC,,= 2.6 eV, respectively. Finally, at p<= 11.O a, (near the final value p,), &= 1.074647 eV is converged to 5, 6, and nearly 5 figures with respect to l,,,,,=SO, mmax= 100, and E,,,=2.6 eV, respectively. The overall level of convergence is comparable to that of the surface function calculations [5] for He+H:. As pc is reduced from 6 uo, the potential in general becomes more repulsive and the surface function eigenvalues increase. Therefore, Ecut (and thus V,,, = 1.5&,,) must increase as a function of decreasing pc Convergence of &a”with respect to EC,, atpc=2.0, 2.25, 2.5, 2.75, 3.0, 3.5,4.0, 4.5, 5.0, and 6.0 a0 was used to define a linear interpolation scheme [ 161 for determining Ecu, on the interval [ 2.0,6.0] uo. The interpolation knots corresponding to the above list of pc values were E,,,=24.0, 17.0, 12.2, 10.0, 6.5, 5.0, 3.5, 3.2, 2.9, and 2.6 eV. For pc= [ 6.0 czo,pn], EC,, was fixed at 2.6 eV. As was found [ 5 ] for the HeH: system, the energy for three-body breakup E3Bplaces a practical maximum on E,,,. Z2 representations of continuum states in the DVR ray eigenvector basis rapidly increase the size of the DVR matrix as E,,, approaches and surpasses the value of I&,. For NeH:, E3B=2.793eV. Thus, setting E,,,=2.6 eV for pt= [ 6.0 a,, p,,] satisfies the three-body breakup criterion and provides convergence at the levels stated above. A hyperspherical correlation diagram (plot of t.&(pc), for 1=1-100, as a function of pc) for the HSCM PES is not shown, although it is somewhat similar to that [ 5 ] for He + H: . The influence of the well in the PES near 4.0 a0 is observed in the effective potentials ( $ (pE) curves) as many contain wells of various depths. These wells in turn can support bound states which then lead to dynamical resonances, the signature of which appears in the reaction probabilities presented next. To provide a guide to the energetic; for the Ne+H: reaction, the asymptotic correlation energies of some H$ and NeH+ states are provided in table 1. As pc becomes large, the effective potentials correlate with the rovibronic states of Hz and NeH+. The energy of the maximum state used in the scattering calculations, t = 100, approaches 1.84 eV as pc becomes large. Since
Table I Asymptotic vibrational energies t( v, j=O) for some Ht and NeH+ fragment states. 1is the APH quantum number for the correlation at p=oo Fragment
V
e(u,j=O) (eV)
H:
0 1 2 0 3 1 4 5 2 6
0.1358 0.3972 0.6450 0.7144 0.8794 1.0637 1.1002 1.3075 1.381 I 1.5013
W H: NeH+ H: NeH+ HZ HZ NeH+
H2+
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CHEMICAL PHYSICS LETTERS
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t
I 6 12 16 29 41 45 66
15 90
the endoergic threshold (NeH* (u=O, j=O)) is 0.7144 c-V,the lowest three vibrational channels of I-I: are capable of leading to reaction at threshold.
3. Results, discussions, and conclusions Converged 3D reaction probabilities for J=O are presented in this section. PUiJi(L+,jf) is the distinguishable particle reaction probability from the initial Hz rovibrational state (v,, ji) into both NeH+ product channels of final state (v,, j,). The reaction probability into a vf vibrational manifold summed over open j, states is pu,,j,(uf) +
1
if
p"i,ji(~f,jf)
.
(2)
The total reactivity Pvwi (total) is defined as the sum of Pvi,,,( vf) over open vf states. In table 2, a convergence study of the J=O reaction probabilities is presented. The probabilities are certain to two significant figures (unless otherwise specified), although three figures are reported for comparative purposes. The probabilities computed
Table 2 A convergence study ofNe+ H$ reaction probabilities for J= 0 on the Hayes et al. PES [ 9 1.p. = 10.86 a0 unless otherwise specified Total energy
N”
Pv,,ji=0( total)
E (W 0.8
0.9
1.0
1.1
1.2
vi=0
Q=l
I?,=2
vi=3
80
0.187
0.227
0.180
90 100 100b’
0.187 0.187 0.187
0.226 0.226 0.227
0.182 0.182
0.0 0.0 0.0
0.180
0.0
80 90
0.872x 10-l 0.863x IO-’
0.154 0.155
0.248 0.248
0.383 0.381
100 100 b,
0.867x 10-l 0.867x 10-l
0.156 0.156
0.248 0.249
0.382 0.381
80
0.126
0.146
0.272
0.258
90 100 100b’
0.125 0.126 0.126
0.146 0.147 0.147
0.271 0.272 0.272
0.258 0.257 0.256
80 90 95 100 IOOb’
0.144 0.162 0.157 0.157 0.157
0.160 0.147 0.150 0.150 0.150
0.356 0.327 0.325 0.323 0.323
0.333 0.384 0.382 0.382 0.379
80
0.238
0.193
90 95 100 100b)
0.237 0.236 0.234 0.234
0.189 0.187 0.188 0.188
0.300 0.329 0.380 0.339 0.339
0.380 0.381 0.428 0.401 0.402
‘) N is the number of channels (surface functions). b’ pn= 10.97 a&
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3 May 1991
with N= 100 and pn= 10.86 a0 for E=0.8, 0.9 and
1.OeV are converged to 1% or better with respect to N, for E= 1.1 and 1.2 eV, the convergence with respect to N is 2% or better, and 5Ohor better, respectively. The slight degradation in convergence between 1.0 and 1.1 eV is understood in terms of the support in the surface function basis. The N= 100 basis contains surface functions which correlate asymptotically with the v=O, 1 and 2 manifolds of NeH+ . It has been observed previously [ 5,16 ] that excellent convergence can be obtained if the basis correlates asymptotically with two closed vibration manifolds. Below the threshold energy of NeH+ (v= 1,&O) = 1.0637 eV this criteria is fulfilled. For energies greater than this, there is only one closed vibrational manifold. The closed vibrational manifolds are necessary to provide a complete enough basis when expanding the surface functions of the new sector p<+1in terms of those from the previous sector Pt:
The spacing between successive p
0.4 0.2 1, 0.0
1
....... .
.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Vi=2
e
. .
Total Energy (ev) Fig. 1.Total reaction probabilities for Ne + HZ+on the Hayes et al. PES [ 91 plotted versus total energy I?. The solid lines are the present J=O quantum 3D results (P,,,,,,(total)) for u,=O-3. The solid circles are the quasiclassical collinear results [ lo] for U,=O-2.
for He+H:. As was analyzed in detail for the He+H: case, the structure indicates the presence of dynamical resonances due to bound states in the effective potentials and due to rovibrational product channels opening. Vibrational enhancement is observed in fig. 1 as Vi= 1 and 2 appear to be most reactive below 0.88 eV, and vi= 2 and 3 appear to be most reactive above the (v= 3, j=O) threshold of 0.88 eV. This trend of vibrational enhancement is in agreement with the trend in reaction cross sections measured experimentally [ 1,3]. However, in contrast to that observed [ 51 for He t HZ where P,,o(total) was never greater than 0.0 1, the Vi= 0 reaction probabilities in fig. 1 are not negligible and at a few energies display strong resonance behavior. The behavior of the reaction probabilities at total reaction threshold is also in contrast to that calculated [5] for He+H:; the results at threshold are nearly zero in fig. 1 whereas those for He +H: were large and finite and appeared to be in the middle of a resonance. Also for Net H: in fig. 1, the reaction probability for the first
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CHEMICAL PHYSICS LETTERS
initial vibrational channel which opens after the total reaction threshold, P3,0(total), enters with a sharp energy threshold; it increases from zero to nearly 0.3 within an interval of 0.001 eV. In contrast for He+H: , the reaction probability for the first initial vibrational channel which opens after the total reaction threshold, P&total), exhibits a delayed classical threshold of 0.0 10 eV. The difference in the bchavior, between the HeH: and NeH: systems, at both types of thresholds discussed above could be due to either differences in the topologies of the PESs or due to the difference in mass, although neither possibility can be confirmed with the present results. Also shown in fig. 1 are the quasiclassical (QC) collinear reaction probabilities [ lo]. Since the same PES was utilized, the discrepancy between the quantum 3D and QC collinear results is due to either the reduced dimensionality or the classical mechanics. Since the present data cannot sort out this issue, quantum collinear and approximate quantum 3D (BCRLM) calculations employing the same PES are planned [24]. The quantum collinear results of Urban et al. [ 1 I ] are not useful in this context as a different PES was used. Quasiclassical 3D and quantum 3D dynamics have been compared in detail [ 25,261 for the H + H2 and D+H2 systems. For H+H2, Zhao et al. [25] found for vibrationally resolved partial reaction cross sections and for vibrational branching ratios that the QC results agreed with the quantum results to within a factor of 1.5 to 2. For DS Hz, Blais et al. [ 26 ] found for vibrationally resolved partial reaction cross sections that the QC results were systematically larger than the quantum results by approximately lOoh to 30%. Such detailed comparisons to the present quantum 3D results for NeSH: are not currently possible because QC 3D results do not exist, to my knowledge, for the HSCM PES. However, I do not anticipate the comparison for Ne+Hf to be as good as that discussed above, as H + Hz and D + Hz do not exhibit the rich structure of dynamical resonances observed presently in fig. 1; it is not possible to describe quantum resonances with classical mechanics. In conclusion, accurate 3D quantum dynamics for the Ne+H; reaction are presented. The total reaction probabilities for J= 0 and different initial vibrational states of Hz all show much structure indicative of quantum dynamical resonances. These
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resonances will be examined in more detail in future work, where quantities such as time delays [ 271 and eigenvalues of the collision lifetime matrix [ 271 will be calculated using efficient direct techniques [28] to compute the energy derivatives of the scattering matrix. The trend in the total reaction probabilities agrees with that for the experimental reaction cross sections [ 1,3]: vibrational energy is more efficient than translational energy. This is, perhaps, not unexpected as vibrational enhancement for the related endoergic reaction of He + Hz has been predicted by both quantum-mechanical statistical phase-space theory [ 291 and recent quantum 3D studies [ 5,301. Finally, the trend in vibrational enhancement for Ne+H: predicted by the present results is in contradiction with the trend in the collinear classical results [lo] computed with the same PES.
Acknowledgement It is a pleasure to acknowledge Ed Hayes, Russ Pack, and Bob Walker for critically reading the manuscript and participating in useful discussions and Greg Parker for providing upgraded versions of the APH scattering codes. This work was performed under the auspices of the US Department of Energy.
References [ 1] W.A. Chupka and M.E Russell, J.
Chem. Phys. 49 (1968) 5426. [ 2 ] D. van Pijkeren, J. van Eck and A. Niehaus, Chem. Phys. Letters 96 (1983) 20; D. van Pijkeren, E. Boltjes, J. van Eck and A. Niehaus, Chem. Phys. 91 ( 1984) 293. [ 31 Z. Herman and I. Koyano, J. Chem. Sot. Faraday Trans. II 83 (1987) 127. [4] R.M. Bilotta and J.M. Farrar, J. Chem. Phys. 75 (1981) 1776. [ 51J.D. Kress, R.B. Walker and E.F. Hayes, J. Chem. Phys. 93 (1990) 8085. [ 61 R.B. Walker and E.F. Hayes, J. Phys. Chem. 87 (1983) 1255; 88 (1984) 1194. [ 71 P.J. Kuntz and A.C. Roach, J. Chem. Sot. Faraday Trans. II 68 ( 1972) 259. [8] F.O. Ellison, 1. Am. Chem. Sot. 85 ( 1963) 3540. [9] E.F. Hayes,A.K.Q. Siu,F.M.Chapman Jr.andRL. Matcha, J. Chem. Phys. 65 (1976) 1901. [ IO] C. Stroud and L.M. Ra,ff, Chem. Phys. 46 (1980) 3 13.
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[ 111J. Urban, R. Jaquet and V. Staemmler, Intern. J. Quantum Chem. 38 (1990) 339. [ 121 R. Jaquet, Comput. Phys. Commun. 58 (1990) 257. [ 131 R.T Pack andG.A. Parker, J. Chem. Phys. 87 ( 1987) 3888. [ 141J.C. Light, I.P. Hamilton and J.V. Lill, J. Chem. Phys. 82 (1985) 1400. [ 151J.D. Kress, 2. BaEiC,G.A. Parker and R.T Pack, Chem. Phys. Letters 157 (1989) 484. [ 161Z. BaEiC,J.D. Kress, GA. Parker and R.T Pack, J. Chem. Phys. 92 (1990) 2344. [ 17 ] J.D. Kress, Z. BaEiC,G.A. Parker and R.T Pack, J. Phys. Chem. 94 (1990) 8055. [ 181 B.R. Johnson, J. Chem. Phys. 67 (1977) 4086; 69 (1978) 4678. [ 191 R.M. Whitnell and J.C. Ligbt, J. Chem. Phys. 89 (1988) 3674. [20] Z. BaEiCand J.C. Light, J. Chem. Phys. 85 ( 1986) 4594. Ann. Rev. Phys. Chem. 40 (1989) 469.
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[ 211 D.R. McLaughlin and D.L. Thompson, J. Chem. Phys. 70 ( 1979) 2748. [22] T. Joseph and N. Sathyamurthy, J. Chem. Phys. 86 (1987) 704. [ 23 ] R.T Pack, private communication. [24] E.F. Hayes, J.D. Kress and R.B. Walker, in progress. [25] M. Zhao, M. Mladenovic, D.G. Truhlar, D.W. Schwenke, Y. Sun, D.J. Kouri and N.C. Blais, J. Am. Chem. Sot. 1 I 1 ( 1989) 852. [26] NC. Blais, M. Zhao, M. Mladenovic, D.G. Truhlar, D.W. Schwenke, Y. Sun and D.J. Kouri, J. Chem. Phys. 91 (1989) 1038. [27] ET. Smith, Phys. Rev. 118 (1960) 349. [28] Z. Darakjian and E.F. Hayes, J. Chem. Phys. 93 (1990) 8793. [29] D.G. Truhlar, J. Chem. Phys. 56 (1972) 1481. [30] J.Z.H. Zhang, D.L. Yeager and W.H. Miller, Chem. Phys. Letters 173 ( 1990) 489.
CHEMICAL PHYSICS LETTERS
3May 1991
Accurate three-dimensional quantum scattering calculations for NetHz +NeH++H Joel D. Kress Theoretical Division (T- 12, MS 8268). Los Aiamos National Laboratory, Los Alamos. NM 87545, USA
Received 2 I November 1990; in final form 28 January I99 I
Accurate 3D coupled channel calculations are presented for the reaction Net Hz +NeH* t H computed with the diatomicsin-molecules potential of Hayes and co-workers. Reaction probabilities for total angular momentum J=O and total energies up to I. 1 eV are reported. The hyperspherical formulation of Pack and Parker is used, and the adiabatic basis functions are generated using the discrete variable representation (DVR). The total reaction probabilities show much structure as a function of total energy, structure that is indicative of quantum dynamical resonances. The trend in the results agrees with that observed experimentally: vibrational energy is more effective than an equivalent amount of translational energy for promoting the reaction.
1. Introduction The dynamics of rare gas atoms reacting with hydrogen molecular ions have been studied using various experimental approaches. For the reactions NeSH; -NeH++H and He+H: -+HeH++H, Chupka and Russell [ 1 ] measured reactive cross sections using a threshold photoionization mass spectrometry technique and observed that vibrational energy is more efticient than an equivalent amount of relative translational energy for promoting both of these reactions. Using a photoelectronsecondary-ion-coincidence technique, van Pijkeren et al. [ 2 ] measured reactive cross sections at thermal collision energies as a function of the initial vibrational state v for both reactions. Their results were consistent with the known reaction endoergicities and HZ vibrational energies. Using molecular beam techniques, the observations of Chupka and Russell have been verified [ 31 for both reactions, and angular distributions and flux contour maps have been measured [ 41 for Ne+H: . The quantum dynamics for the He + H2+reaction was recently examined [5] using three different techniques: collinear ( ID); approximate 3D (the bending corrected rotating linear model [ 61, BCRLM); and accurate full 3D. Other theoretical work and experimental work for the He+ Hz reac510
tion was also reviewed there. Relatively less is known theoretically about Ne + H; . Kuntz and Roach constructed [ 7 ] a semi-empirical diatomics-in-molecules [ 81 (DIM) potential energy surface (PES) which was obtained by scaling their DIM PES for the ArSH: reaction. Hayes et al. [ 91 generated a DIM PES (hereafter denoted HSCM ) by representing the diatomic fragments as Morse functions. The H:(‘C,+),H$(‘C,f),andHeH+(‘Z+) curveswere fit to accurate diatomic data and the HeH ( 2E+) were adjusted by fitting the DIM PES to three SCF energies along the collinear reaction path. A contour plot of the resulting DIM PES for collinear geometries was quite similar to that for the SCF PES they [9] also computed. Stroud and Raff [lo] computed quasiclassical collinear reaction probabilities using the HSCM PES and found that translational energy was more effective than vibrational energy for promoting reaction in contradiction with the experimental findings. They postulated that the DIM representation was suspect and generated a spline-fitted PES (which they denoted SAI) fit to a grid of collinear SCF energies similar in accuracy to those of Hayes et al. [ 91. With the SAI PES, Stroud and Raff then found reasonable agreement with experiment in terms of vibrational enhancement for the collinear quasiclassical calculations. Very recently Urban et al. reported [ 111 coupled electron pair approxima-
0009-2614/91/$ 03.50 0 1991 - Elsevier Science Publishers B.V. (North-Holland)
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CHEMICALPHYSICSLETTERS
tion (CEPA, a method which correlates the electrons) calculations for both collinear and non-collinear geometries of Ne+ Hz. Furthermore, employing a fit to their collinear PES, they calculated quantum collinear reaction probabilities using a finite element technique [ 121. For total collision energies up to 1.5 eV, these calculations do not agree with experiment in terms of vibrational enhancement, but beyond 1.5 eV they do. In this Letter, what are believed to be the first calculations of accurate 3D quantum dynamics for the NeSH: reaction are presented. The HSCM PES is employed and the full 3D reactive scattering for total momentum J=O is formulated using the adiabatically adjusting principal-axis hyperspherical ( APH ) method of Pack and Parker [ 131. The sector adiabatic basis functions are generated using the discrete variable representation (DVR) method [ 14 1. This APH-DVR approach has yielded accurate 3D results for F+H2 [ 15,161, H+H2 and D+H2 [ 171, and He+H: [ 51. Total reaction probabilities from all energetically allowed initial vibrational states of Hz show much structure indicative of quantum dynamical resonances and agree with the experimental trend of vibrational enhancement.
2. Methods and convergence
studies
The APH theory is used to formulate the exact 3D treatment of the reactive scattering, the details of which are given in ref. [ 131. Here a brief sketch of the approach is given. The total scattering wavefunction is expanded in a basis of sector adiabatic surface functions. The surface functions are bound state eigenfunctions of the surface Hamiltonian H(e,X;Pc)~,(e,X;Pc)=~(Be)~~(e,X;P~)
I
(1)
where 0 and Kare the two APH hyperangles. Eq. ( 1) also depends parametrically on pc, the center of a sector, where the range of APH hyperradius is divided into n sectors, <= 1, .... n. When the total scattering wavefunction is substituted into the full 3D Schriidinger equation for J=O, a set of N coupled channel (CC) equations is obtained, where N is the number of surface functions (t= 1, .... N) in the CC expansion. These exact CC equations are propagated from p, to p,, using the log-derivative method [ 18 1,
3 May 1991
then the boundary conditions are applied as usual [131. The discrete variable representation (DVR) method [ 141 is used to calculate the set of (@,( 8, x; pc)} for each p. (= 1, .... n. The application of the DVR method to the calculation of surface functions is detailed in ref. [ 161; a brief summary of the approach follows. A DVR is used [ 191 for both 0 and x. The DVR in 6’is formed in the basis of (normalized) Legendre polynomials, {p,( cos 20), 110, 1, .... I,,,}. A symmetry adapted [ 19 ] DVR in x is formed in the set of (normalized) basis functions {a,cos2(m-1)x, m=l,2 ,..., m,,,}.TheDVRmatrix representation of eq. ( 1) is then constructed and the surface functions are found using the sequential diagonalization-truncation technique [ 201. The resulting matrix eigenvalue problem is solved using direct techniques. The usual approaches are used to: ( 1) transform [ 161 the surface functions from the DVR to the finite basis representation (FBR) necessary for projecting onto the boundary conditions; ( 2) calculate [ 16 ] the potential coupling matrix elements; and (3 ) calculate [ 5,161 the matrix elements of the overlap between surface functions @,(B,x; pr) and @tt(Q,x~~+I).
The topology of the HSCM PES for NeSH: is quite similar to that of the MTJS PES [21,22] for He+H,+ which was used to calculate accurate 3D quantum scattering results [ 51. The Ne+H: reaction is endoergic by about 0.6 eV, whereas the He + Hz reaction is endoergic by about 0.8 eV, with respect to the rovibrational ground state and both PESs have wells about 0.3 eV deep in the entrance channel. (The zero of energy is measured from the bottom of the asymptotic Hz well.) Also, neither PES has any barrier along the minimum energy path. As is the case for He+ H: ,the reaction path, defined by finding the minimum energy with respect to 0 and x for fixed values of p, has a well. For Net H: ,the well occurs near p= 4.0 a, and is about 0.3 eV deep. ( 1 uo= 1 bohr unit.) To illustrate the effect of electron correlation at the CEPA level, the collinear PES of Urban et al. [ 111 has a deeper well of 0.49 eV with the Ne more tightly bound to Hz as compared to the HSCM PES. The DVR calculation is flexible enough to represent the surface functions over three basis regions: the repulsive wall of the potential at small values of 511
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pb the bound states in the well for intermediate values of pc; and, the nearly asymptotic rovibrational states which are localized in each arrangement channel. To accommodate the shrinkage problem (as pe increases the surface functions access increasingly limited regions of 19and x on the p
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point), &= 1.067357 eV is converged to nearly 6, 6, and nearly 6 figures with respect to 1,,,,,=40, m ,,,=== 80, and EC,,= 2.6 eV, respectively. Finally, at p<= 11.O a, (near the final value p,), &= 1.074647 eV is converged to 5, 6, and nearly 5 figures with respect to l,,,,,=SO, mmax= 100, and E,,,=2.6 eV, respectively. The overall level of convergence is comparable to that of the surface function calculations [5] for He+H:. As pc is reduced from 6 uo, the potential in general becomes more repulsive and the surface function eigenvalues increase. Therefore, Ecut (and thus V,,, = 1.5&,,) must increase as a function of decreasing pc Convergence of &a”with respect to EC,, atpc=2.0, 2.25, 2.5, 2.75, 3.0, 3.5,4.0, 4.5, 5.0, and 6.0 a0 was used to define a linear interpolation scheme [ 161 for determining Ecu, on the interval [ 2.0,6.0] uo. The interpolation knots corresponding to the above list of pc values were E,,,=24.0, 17.0, 12.2, 10.0, 6.5, 5.0, 3.5, 3.2, 2.9, and 2.6 eV. For pc= [ 6.0 czo,pn], EC,, was fixed at 2.6 eV. As was found [ 5 ] for the HeH: system, the energy for three-body breakup E3Bplaces a practical maximum on E,,,. Z2 representations of continuum states in the DVR ray eigenvector basis rapidly increase the size of the DVR matrix as E,,, approaches and surpasses the value of I&,. For NeH:, E3B=2.793eV. Thus, setting E,,,=2.6 eV for pt= [ 6.0 a,, p,,] satisfies the three-body breakup criterion and provides convergence at the levels stated above. A hyperspherical correlation diagram (plot of t.&(pc), for 1=1-100, as a function of pc) for the HSCM PES is not shown, although it is somewhat similar to that [ 5 ] for He + H: . The influence of the well in the PES near 4.0 a0 is observed in the effective potentials ( $ (pE) curves) as many contain wells of various depths. These wells in turn can support bound states which then lead to dynamical resonances, the signature of which appears in the reaction probabilities presented next. To provide a guide to the energetic; for the Ne+H: reaction, the asymptotic correlation energies of some H$ and NeH+ states are provided in table 1. As pc becomes large, the effective potentials correlate with the rovibronic states of Hz and NeH+. The energy of the maximum state used in the scattering calculations, t = 100, approaches 1.84 eV as pc becomes large. Since
Table I Asymptotic vibrational energies t( v, j=O) for some Ht and NeH+ fragment states. 1is the APH quantum number for the correlation at p=oo Fragment
V
e(u,j=O) (eV)
H:
0 1 2 0 3 1 4 5 2 6
0.1358 0.3972 0.6450 0.7144 0.8794 1.0637 1.1002 1.3075 1.381 I 1.5013
W H: NeH+ H: NeH+ HZ HZ NeH+
H2+
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t
I 6 12 16 29 41 45 66
15 90
the endoergic threshold (NeH* (u=O, j=O)) is 0.7144 c-V,the lowest three vibrational channels of I-I: are capable of leading to reaction at threshold.
3. Results, discussions, and conclusions Converged 3D reaction probabilities for J=O are presented in this section. PUiJi(L+,jf) is the distinguishable particle reaction probability from the initial Hz rovibrational state (v,, ji) into both NeH+ product channels of final state (v,, j,). The reaction probability into a vf vibrational manifold summed over open j, states is pu,,j,(uf) +
1
if
p"i,ji(~f,jf)
.
(2)
The total reactivity Pvwi (total) is defined as the sum of Pvi,,,( vf) over open vf states. In table 2, a convergence study of the J=O reaction probabilities is presented. The probabilities are certain to two significant figures (unless otherwise specified), although three figures are reported for comparative purposes. The probabilities computed
Table 2 A convergence study ofNe+ H$ reaction probabilities for J= 0 on the Hayes et al. PES [ 9 1.p. = 10.86 a0 unless otherwise specified Total energy
N”
Pv,,ji=0( total)
E (W 0.8
0.9
1.0
1.1
1.2
vi=0
Q=l
I?,=2
vi=3
80
0.187
0.227
0.180
90 100 100b’
0.187 0.187 0.187
0.226 0.226 0.227
0.182 0.182
0.0 0.0 0.0
0.180
0.0
80 90
0.872x 10-l 0.863x IO-’
0.154 0.155
0.248 0.248
0.383 0.381
100 100 b,
0.867x 10-l 0.867x 10-l
0.156 0.156
0.248 0.249
0.382 0.381
80
0.126
0.146
0.272
0.258
90 100 100b’
0.125 0.126 0.126
0.146 0.147 0.147
0.271 0.272 0.272
0.258 0.257 0.256
80 90 95 100 IOOb’
0.144 0.162 0.157 0.157 0.157
0.160 0.147 0.150 0.150 0.150
0.356 0.327 0.325 0.323 0.323
0.333 0.384 0.382 0.382 0.379
80
0.238
0.193
90 95 100 100b)
0.237 0.236 0.234 0.234
0.189 0.187 0.188 0.188
0.300 0.329 0.380 0.339 0.339
0.380 0.381 0.428 0.401 0.402
‘) N is the number of channels (surface functions). b’ pn= 10.97 a&
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with N= 100 and pn= 10.86 a0 for E=0.8, 0.9 and
1.OeV are converged to 1% or better with respect to N, for E= 1.1 and 1.2 eV, the convergence with respect to N is 2% or better, and 5Ohor better, respectively. The slight degradation in convergence between 1.0 and 1.1 eV is understood in terms of the support in the surface function basis. The N= 100 basis contains surface functions which correlate asymptotically with the v=O, 1 and 2 manifolds of NeH+ . It has been observed previously [ 5,16 ] that excellent convergence can be obtained if the basis correlates asymptotically with two closed vibration manifolds. Below the threshold energy of NeH+ (v= 1,&O) = 1.0637 eV this criteria is fulfilled. For energies greater than this, there is only one closed vibrational manifold. The closed vibrational manifolds are necessary to provide a complete enough basis when expanding the surface functions of the new sector p<+1in terms of those from the previous sector Pt:
The spacing between successive p
0.4 0.2 1, 0.0
1
....... .
.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Vi=2
e
. .
Total Energy (ev) Fig. 1.Total reaction probabilities for Ne + HZ+on the Hayes et al. PES [ 91 plotted versus total energy I?. The solid lines are the present J=O quantum 3D results (P,,,,,,(total)) for u,=O-3. The solid circles are the quasiclassical collinear results [ lo] for U,=O-2.
for He+H:. As was analyzed in detail for the He+H: case, the structure indicates the presence of dynamical resonances due to bound states in the effective potentials and due to rovibrational product channels opening. Vibrational enhancement is observed in fig. 1 as Vi= 1 and 2 appear to be most reactive below 0.88 eV, and vi= 2 and 3 appear to be most reactive above the (v= 3, j=O) threshold of 0.88 eV. This trend of vibrational enhancement is in agreement with the trend in reaction cross sections measured experimentally [ 1,3]. However, in contrast to that observed [ 51 for He t HZ where P,,o(total) was never greater than 0.0 1, the Vi= 0 reaction probabilities in fig. 1 are not negligible and at a few energies display strong resonance behavior. The behavior of the reaction probabilities at total reaction threshold is also in contrast to that calculated [5] for He+H:; the results at threshold are nearly zero in fig. 1 whereas those for He +H: were large and finite and appeared to be in the middle of a resonance. Also for Net H: in fig. 1, the reaction probability for the first
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initial vibrational channel which opens after the total reaction threshold, P3,0(total), enters with a sharp energy threshold; it increases from zero to nearly 0.3 within an interval of 0.001 eV. In contrast for He+H: , the reaction probability for the first initial vibrational channel which opens after the total reaction threshold, P&total), exhibits a delayed classical threshold of 0.0 10 eV. The difference in the bchavior, between the HeH: and NeH: systems, at both types of thresholds discussed above could be due to either differences in the topologies of the PESs or due to the difference in mass, although neither possibility can be confirmed with the present results. Also shown in fig. 1 are the quasiclassical (QC) collinear reaction probabilities [ lo]. Since the same PES was utilized, the discrepancy between the quantum 3D and QC collinear results is due to either the reduced dimensionality or the classical mechanics. Since the present data cannot sort out this issue, quantum collinear and approximate quantum 3D (BCRLM) calculations employing the same PES are planned [24]. The quantum collinear results of Urban et al. [ 1 I ] are not useful in this context as a different PES was used. Quasiclassical 3D and quantum 3D dynamics have been compared in detail [ 25,261 for the H + H2 and D+H2 systems. For H+H2, Zhao et al. [25] found for vibrationally resolved partial reaction cross sections and for vibrational branching ratios that the QC results agreed with the quantum results to within a factor of 1.5 to 2. For DS Hz, Blais et al. [ 26 ] found for vibrationally resolved partial reaction cross sections that the QC results were systematically larger than the quantum results by approximately lOoh to 30%. Such detailed comparisons to the present quantum 3D results for NeSH: are not currently possible because QC 3D results do not exist, to my knowledge, for the HSCM PES. However, I do not anticipate the comparison for Ne+Hf to be as good as that discussed above, as H + Hz and D + Hz do not exhibit the rich structure of dynamical resonances observed presently in fig. 1; it is not possible to describe quantum resonances with classical mechanics. In conclusion, accurate 3D quantum dynamics for the Ne+H; reaction are presented. The total reaction probabilities for J= 0 and different initial vibrational states of Hz all show much structure indicative of quantum dynamical resonances. These
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resonances will be examined in more detail in future work, where quantities such as time delays [ 271 and eigenvalues of the collision lifetime matrix [ 271 will be calculated using efficient direct techniques [28] to compute the energy derivatives of the scattering matrix. The trend in the total reaction probabilities agrees with that for the experimental reaction cross sections [ 1,3]: vibrational energy is more efficient than translational energy. This is, perhaps, not unexpected as vibrational enhancement for the related endoergic reaction of He + Hz has been predicted by both quantum-mechanical statistical phase-space theory [ 291 and recent quantum 3D studies [ 5,301. Finally, the trend in vibrational enhancement for Ne+H: predicted by the present results is in contradiction with the trend in the collinear classical results [lo] computed with the same PES.
Acknowledgement It is a pleasure to acknowledge Ed Hayes, Russ Pack, and Bob Walker for critically reading the manuscript and participating in useful discussions and Greg Parker for providing upgraded versions of the APH scattering codes. This work was performed under the auspices of the US Department of Energy.
References [ 1] W.A. Chupka and M.E Russell, J.
Chem. Phys. 49 (1968) 5426. [ 2 ] D. van Pijkeren, J. van Eck and A. Niehaus, Chem. Phys. Letters 96 (1983) 20; D. van Pijkeren, E. Boltjes, J. van Eck and A. Niehaus, Chem. Phys. 91 ( 1984) 293. [ 31 Z. Herman and I. Koyano, J. Chem. Sot. Faraday Trans. II 83 (1987) 127. [4] R.M. Bilotta and J.M. Farrar, J. Chem. Phys. 75 (1981) 1776. [ 51J.D. Kress, R.B. Walker and E.F. Hayes, J. Chem. Phys. 93 (1990) 8085. [ 61 R.B. Walker and E.F. Hayes, J. Phys. Chem. 87 (1983) 1255; 88 (1984) 1194. [ 71 P.J. Kuntz and A.C. Roach, J. Chem. Sot. Faraday Trans. II 68 ( 1972) 259. [8] F.O. Ellison, 1. Am. Chem. Sot. 85 ( 1963) 3540. [9] E.F. Hayes,A.K.Q. Siu,F.M.Chapman Jr.andRL. Matcha, J. Chem. Phys. 65 (1976) 1901. [ IO] C. Stroud and L.M. Ra,ff, Chem. Phys. 46 (1980) 3 13.
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[ 111J. Urban, R. Jaquet and V. Staemmler, Intern. J. Quantum Chem. 38 (1990) 339. [ 121 R. Jaquet, Comput. Phys. Commun. 58 (1990) 257. [ 131 R.T Pack andG.A. Parker, J. Chem. Phys. 87 ( 1987) 3888. [ 141J.C. Light, I.P. Hamilton and J.V. Lill, J. Chem. Phys. 82 (1985) 1400. [ 151J.D. Kress, 2. BaEiC,G.A. Parker and R.T Pack, Chem. Phys. Letters 157 (1989) 484. [ 161Z. BaEiC,J.D. Kress, GA. Parker and R.T Pack, J. Chem. Phys. 92 (1990) 2344. [ 17 ] J.D. Kress, Z. BaEiC,G.A. Parker and R.T Pack, J. Phys. Chem. 94 (1990) 8055. [ 181 B.R. Johnson, J. Chem. Phys. 67 (1977) 4086; 69 (1978) 4678. [ 191 R.M. Whitnell and J.C. Ligbt, J. Chem. Phys. 89 (1988) 3674. [20] Z. BaEiCand J.C. Light, J. Chem. Phys. 85 ( 1986) 4594. Ann. Rev. Phys. Chem. 40 (1989) 469.
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[ 211 D.R. McLaughlin and D.L. Thompson, J. Chem. Phys. 70 ( 1979) 2748. [22] T. Joseph and N. Sathyamurthy, J. Chem. Phys. 86 (1987) 704. [ 23 ] R.T Pack, private communication. [24] E.F. Hayes, J.D. Kress and R.B. Walker, in progress. [25] M. Zhao, M. Mladenovic, D.G. Truhlar, D.W. Schwenke, Y. Sun, D.J. Kouri and N.C. Blais, J. Am. Chem. Sot. 1 I 1 ( 1989) 852. [26] NC. Blais, M. Zhao, M. Mladenovic, D.G. Truhlar, D.W. Schwenke, Y. Sun and D.J. Kouri, J. Chem. Phys. 91 (1989) 1038. [27] ET. Smith, Phys. Rev. 118 (1960) 349. [28] Z. Darakjian and E.F. Hayes, J. Chem. Phys. 93 (1990) 8793. [29] D.G. Truhlar, J. Chem. Phys. 56 (1972) 1481. [30] J.Z.H. Zhang, D.L. Yeager and W.H. Miller, Chem. Phys. Letters 173 ( 1990) 489.