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Accurate and model collinear reactive probabilities of the Mg+FH reaction
Volume 168, number 5
CHEMICAL PHYSICS LETTERS
llMay1990
ACCURATE AND MODEL COLLINEAR REACTIVE PROBABILITIES OF THE Mg+FH REACT ION A. LAGANA Dipartimento di Chimica, Universitddi Perugia, 06100 Perugia, Italy
Miguel PANIAGUA Departamento de Quimica Fisica Applicada. UniversidadAutonoma de Madrid, Cantoblanco, 28049 Madrid, Spain and
Josi? M. ALVARIRO Departamento de Quimica Fisica, Facultadde Quimica, Universidaddesalamanca, 37008 Salamanca, Spain Received 25 November 1989; in final form 8 February 1990
Reactive probabilities for the Mg+ FH system have been calculated using an accurate collinear quantum technique. To single out the importance of tunneling contributions these have been compared with quasiclassical results. Product vibrational distributions have been rationalized in terms of an energy-dependent Franck-Condon model.
I. Introduction Theoretical interest in reactions of alkaline earth atoms with hydrogen halides has enormously increased in recent years [ l-51. Systems of this family, in fact, in addition to being of practical importance, can be considered as prototypes of asymmetric A+BC reactions having a structured potential-energy surface (PES). The theoretical investigation of these systems is, at present, mainly based on classical trajectory calculations. However, because the impressive progress of computational techniques is making feasible the accurate three-dimensional quantum study of metal hydrogen halide systems [ 61 we have undertaken an investigation of some alkaline-earth hydrogen-halide reactions using quantum means. Recently, ab initio investigations of the PES of the reaction Mg+FH(u)-+MgF(o’)+H
(1)
have been reported [ 7331. A peculiarity of the PES of this reaction is that although the transition state
has a bent geometry (I% 75”; 0 is the MgFH angle) its overall shape and minimum-energy path vary very little in the range 0= 180” (collinear approach) to 8=75” (transition-state approach). This suggests a first inspection of the reactive dynamics of this system using a collinear R-matrix approach [ 9, lo], The quantum investigation of this reaction was carried out using a vectorized version of the R-matrix program [ 111 restructured to run in parallel mode both the calculation of the sector vibrational basis (in the first section) and the propagation through sectors (second section ) according to the scheme discussed for reduceddimensionality quantum programs in ref. [ 121. For the same reaction, quasiclassical calculations were also performed to single out quantum features. A peculiar feature of atom-diatom collinear quantum reactive results is the structure of the product vibrational distribution (PVD), i.e. its sequence of maxima and minima (modality). To rationalize this structure, a Franck-Condon (FC) mechanism has been proposed. Its application, however, has been mainly confined to the reactions
0009-2614/90/$03.50 0 Elsevier Science Publishers B.V. (North-Holland )
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Volume 168, number 5
H+X,(U)+HX(U’)+X
(2)
with X = Cl, F or isotopic variants [ 13-37 1. However, we have recently shown [ 37,381 that some features previously thought to be typical of FC reaction mechanisms are actually not. Therefore, as a first step towards the generalization of an FC modeling of reduced-dimensionality quantum calculations for generic three-atom (or atomic-aggregate) reactions we have applied the FC treatment to Mg+HF. The paper is organized as follows. In section 2 the potential-energy surface and quantum and quasiclassical collinear reactive probabilities (PJ calculated on it are analysed. In section 3 a rationalization of the shape of the product vibrational distributions in terms of a Franck-Condon model is presented.
2. Quantum and quasiclassical collinear reactivity The shape of the MgFH PES is rather structured. However, as already mentioned, its angular variation for 0 ranging in the interval 180-75” (i.e. from the collinear to the transition-state angle of approach) is small while for further decrease of 6’the barrier rises steeply to very large values. Therefore it can be assumed that most of the reaction outcome is obtained from large 8 attacks. In the process of obtaining a global interpolation of the ab initio potential-energy valtis we have at first fitted its collinear cut. The lit was performed by adopting an expansion of the two- and three-body interaction in bond order (BO) variables [39] as done for other systems of this family [ 39-4 11. A property of the BO fit is the accurate reproduction of the long- and intermediate-range interaction and the confinement of undesirable structures of the fitted PES in the short-range part of the three-body term. These short-range structures usually occur at such a short distance so as either not to affect the regions of the reactive channel experienced by lowenergy collisions (as those of molecular-beam reactive experiments) or to be easily removed by forcing a repulsive behaviour. The PES fitted to the adjusted collinear ab initio values gives a rms deviation slightly smaller than 0.2 eV. The barrier is displaced late in the exit channel (the HF and MgF distances at the saddle are 1.40 442
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CHEMICAL PHYSICS LETTERS
and 1.71 A, respectively) and is 1.9 eV higher than the reactants’ asymptote. A schematic view of the related minimum-energy path is shown in fig. 1 where vibrational levels of reactant and product diatomic molecules are also given. Quantum calculations were performed using a 60-, contracted to 40-, harmonic-oscillator-eigenfunction basis set and dividing the reaction channel in about 300 sectors [ 12,421. Quasiclassical calculations were performed by running batches of one hundred trajectories uniformly distributed in the phase of the reactant oscillator for each pair of reactant vibrational number (u) collision energy (Err ) values. Both types of results are plotted in fig. 2 as a function of the total energy (E) for the four lowest excited reactant vibrational states. The upper limit of the investigated energy interval (E=4 eV) is much larger than the barrier height (compare figs. 1 and 2 ) . This enables us to investigate the effect of varying the vibrational energy E, content of the reactant molecule from well below to definitely above the barrier. The reactive probability calculated at v=O (not shown in fig. 2) is always negligible. On the contrary, the reaction is appreciable when HF is vibrationally excited. At v= 1 the threshold energy is quite large. In this case the quantum reactive probability
MgFH
Mg FH
MgF H -
2
%i L’
I
CI- _
I
Fig. 1. A schematic representation of the collinear minimum-energy path for the Mg+ FH reaction. Reactant (lhs) and product (rhs) diatom vibrational levels are also reported.
CHEMICAL PHYSICS LETTERS
Volume 168, number 5
11 May 1990
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E / eV Fig. 2. Quantum (dots connected by solid lines) and quasiclassical (circles) reactive probabilities for the Mg+FH reaction reported as a function of the total energy for v= 1 (Ihs lower panel), u=2 (rhs lower panel), v=3(lhs upper panel) and v=4 (rhs upper panel). Connecting lines have been drawn for greater clarity.
(dots connected by solid lines in the lower left panel of fig. 2) increases smoothly at threshold (,I$= 1.7 eV) to reach a maximum around E{, = 2.5 eV. At higher collision energy, the reactive probability decreases quite rapidly to zero. At v=2 (lower right panel) the quantum reactivity increases more rap idly than for v= 1 at threshold (Efr= 1 eV). At this v, the maximum is reached after an energy interval of about 0.5 eV. Then the reactive probability has a plateau for an energy interval of about 1 eV. For ZI values larger than 2 the probability increase at threshold is almost stepwise (the rise to a plateau occurs within 0.3 and 0.02 eV respectively for P3and P4).
These four cases well illustrate the role played by vibrational energy in enhancing reactivity for endoergic late-barrier reactions. When the reactant vibrational energy is smaller than the height of the barrier an increase of reactant vibrational excitation has the effect of lowering the threshold to reaction (along the total energy scale) because it acts upon the molecular motion favouring the bond breaking. On the contrary, when the amount of vibrational energy is larger than the saddle height a further supplying of energy as diatomic vibration subtracts it to a more effective use for reactivity. Accordingly, the threshold moves to larger total energy values and the reactive probability suddenly switches from zero to a
maximum at threshold. This seems to suggest that the only role vibrational energy can play for these reactions is to lower the threshold. On the contrary, fig. 2 shows that there is a second less apparent way through which initial vibrational excitation can contribute to increasing reactive probability. This is by rising the height of the reactive probability plateau. The height of the plateau, in fact, rises to about 0.6, 0.8, 0.9 and 1.Ofor P,,Pz, P3and P., respectively. This can be rationalized in terms of a transition-state picture as a better overlap between the phase space covered by the reactant oscillator and the window to reaction at the saddle and has been confirmed by a detailed analysis of classical trajectories. Deviations of quasiclassical probabilities (reported as circles in the same figure) from quantum values emphasize some similarities of the reactive behaviour of this reaction with that of the Li+HF system [ 421. The figure indicates that (especially in the vicinity of the threshold) quasiclassical reactive probabilities are larger than the related quantum values when E,is clearly lower than the height of the barrier to reaction (UC 3 ). The opposite IStrue when the initial vibrational energy is larger than the barrier (to 3). As discussed in ref. [ 421, this parallels the effect of increasing collision energy for early-bar443
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CHEMICAL PHYSICSLETTERS
tier reactions and has been rationalized in terms of different tunneling components.
11 May 1990
16
t
3. Product vibrationaldistributions A distinctive feature of collinear atom-diatom reactions is the modality (the structure of maxima) of the product vibrational distribution. The most accurate characterization of these PVDs has been carried out for reactions occurring via a Franck-Condon mechanism. The first detailed analysis of collinear quantum probabilities carried out using such a model treatment has been performed for the H-tCIZ(~)~HCl+C1 reaction [ 131. Such a treatment approximates the reactive S-matrix elements as overlap integrals between shifted reactant and product diatomic vibrational wavefunctions. Since then extensive studies of FC approaches have been reported in the literature using either quantum or semiclassical arguments [ 20-381. Reactions occurring via an FC mechanism are usually believed to lead to PVDs having v+ 1 peaks (u t 1 modality). In a recent investigation of the H + Cl, reaction and isotopic variants 137,381 we found that such a requisite is not mandatory and that an FC approach is also able to reproduce less structured PVDs. The Mg+HF reaction differs in many respects from the H t Cl1 reactions. Its potential-energy surface is endo- rather than exo-ergic and its mass combination is not of the light+ heavy-heavy type. Nonetheless, reactive probabilities calculated for significantly vibrationally excited reactants, seem to fulfill reasonably well the basic requirement for a reactive FC mechanism, i.e. the process is (almost) fully reactive (see P3 and P4 of fig. 2). In fact, a reactive probability close to one guarantees that back reflection is negligible. Accordingly, as for the H+ CIZ reaction we have approximated the S-matrix elements as overlap integrals between properly shifted wavefunctions of reactant and product diatoms and attempted a rationalization of the PVDs in terms of a model FC treatment. Curves interpolating the location of maxima of the quantum collinear PVD calculated for v= 3 and u= 4 are plotted in fig. 3 as a function of the total energy. The inclusion of the v=3 case is motivated by the fact that, as for H +CIZ [ 37,381, we assume that the
444
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----_:-;: ISi
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0
3
E /
3.5
4
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Fig. 3. Curves interpolating the locations of the PVD maxima of the Mg+FH reaction reported as a function of the total energy for ~3 (lower panel) and ~4 (upper panel).
FC treatment can be applied when the reactive probability is larger than 0.8. As clearly apparent from the figure, both PVDs show for the whole energy interval a bimodal structure whose peaks move regularly to higher u’ values when energy increases. Both features, however, are perfectly compatible with a Franck-Condon reaction mechanism. To compare model with accurate dynamical results we have calculated the overlap integrals between the wavefunctions of the two Morse potentials fitted to reactant and product asymptotic diatomic potentials of the MgFH BO PES. Locations of the maxima of the corresponding PVDs are shown in the upper panels of fig. 4 as a function of d (the relative displacement of the product from the reactant wavefunction). The two lowest peaks of the plots are bracketed by dashed lines to evidence the interval of d (root interval ) for which the lowest portion of the model PVD coincides with that obtained from the accurate dynamical calculations. As discussed in ref. [ 381 this is relevant to the factoring of the reactive probability and to its parameterizing for artificial intelligence purposes [ 43 1. The figure shows also that the evolution with energy of such a structure is re-
Volume 168, number 5
CHEMICAL PHYSICS LETTERS
11May 1990
totic to the saddle. In this approach the S, element of the S-matrix is given as the overlap of the wavefunctions: &j=NR,iNPj<$R,i~h’,j~,
\ ‘,
\
-
-
cl/
w
Fig. 4. Location of the maxima of the Franck-Condon PVD for the Mg+HF reaction (upper panels) and the H+C12 reaction (lower panels) plotted as a function of the displacement d. The initial vibrational numbers arc 3 (lhs panel) and 4 (rhs panel)
for the Mg+HFreactionand 2 (Ihs panel),and 3 (rhs panel) for the H + Cl, reaction.
produced if d is allowed to vary with energy. For comparison, PVD maxima determined using the FC model treatment for the H t Cl2 reaction at v= 2 and 3 are shown in the lower panels of the same figure. As discussed in ref. [ 381, the structure of the PVD for the H t CIZ reaction derived from the accurate quantum collinear calculation is well reproduced by model calculations for the energy interval investigated (as evidenced by the dashed line in this case too). A first clear differenceis the dependence of the root interval upon vibrational energy. For the Mg t HF reaction, d moves to the right with energy while for H t CIZ it moves to the left. This can be easily rationalized in terms of the characteristics of our FC model treatment: both reactant < &I and product I $+diatom wavefunctions are assumed to propagate with little distortion from the related asymp
(3)
where the N coefficients are the related reactant and product normalization factors. In our simplified approach the distortion of the diatomic wavefunctions can be either neglected (by setting d equal to the difference of the diatomic equilibrium distances) or approximately accounted for by adjusting the value of d. At small vibrational energies the distortion of the diatomic wavefunctions is negligible. Accordingly, the displacement parameter d has almost the same value as the difference between the equilibrium distances (because of our definition of d, the difference is positive when the reactant diatom equilibrium distance is smaller than that of the product and negative in the opposite case). Obviously, larger distortions have to be considered at higher energies because a larger region of the potential is explored during the propagation. In our approach, this is taken into account simply by shifting the root interval with v and v’ (i.e by imposing a larger displacement to the wavefunctions’ origin). A more realistic formulation of the pdtential at the Franck-Condon transition region and therefore a more appropriate representation of the local vibrational wavefunctions would certainly reduce the need for large displacements and allow a better FC parametrization. Another important difference between the H + CIZ and the Mg + HF model results is that for the latter the FC treatment populates more excited vibrational states than the accurate treatment. This means that some excited product vibrational levels are dynamically closed even when energetically accessible and allowed by FC mechanisms. Therefore, a generalization of the FC model needs not only to improve the description of the potential at the FC transition point but also to include additional constraints. Since our model is based on a simplistic approach, we rendered this additional constraint as a limit to the maximum amount of energy to be disposed of as product vibration. Such a type of cut-off is not surprising owing to the fact that heavy+ heavy-light latebarrier reactions can be considered as the inverse process of a light t heavy-heavy early-barrier reaction. Because of that, any restriction on the collision 445
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CHEMICAL PHYSICS LETTERS
energy for light+ heavy-heavy reactions should be paralleled by a restriction on the product vibrational energy of heavy + heavy-light ones.
11 May 1990
and CICYT (PB 85-0316) is acknowledged.
References 4. Conclusions
[ I] H.‘Schor, S. Chapman, S. Green and R.N. Zare, J. Phys. Chem. 83 (1979) 920.
[ 21S.Chapman, J. Chem. Phys. 8 1 ( 1984) 262.
Collinear calculations performed for the Mg+FH reaction, with the purpose of carrying out a preliminary investigation of this system on a reasonable potential-energy surface, have evidenced typical features of its dynamical behaviour. The calculated energy dependence of the reactive probability well illustrates the role played by vibrational energy in enhancing reactivity. The rationalization of the value of the plateau of the quantum reactivity and its rise at the threshold (smooth for vibrational energies lower than the barrier, stepwisc for those higher than the barrier) has led to the conclusions that the relative importance of collisional and vibrational energy is similar to that found for the alkali-atom hydrogen-fluoride reactions. Differences between quantum and quasiclassical calculations have also been rationalized as for the alkali-atom hydrogenfluoride reactions in terms of the bias of dominant tunneling contributions. Product vibrational distributions have been rationalized by applying (for the first time to a heavy -I-heavy-light reaction ) a Franck-Condon modeling linking overlaps between shifted reactant and product diatomic vibrational wavefunctions to the reaction detailed probabilities. The FC model treatment has been able to reproduce the location of the two lower peaks as well as their evolution with collision energy. As for the H+Clz reaction the reproduction of the energy dependence of the location of the maxima is made possible by making the shift between asymptotic vibrational functions vary with collision energy. In the case of the MgSFH reaction, however, an upper limit to the amount of energy disposed of as product vibration more restrictive than the total energy conservation had to be applied.
Acknowledgement Financial support from CNR (within the Progetto Finalizzato Sistemi Informatici e Calcolo Parallelo) 446
[3] J.M. Alvariiio, A. Laganzl and M. Paniagua. Studia Chem. 11 (1985) 419. [4] R.L. Jaffe,M.D. Pattengill, F.G. MascarelloandR.N.Zare, J. Chem. Phys. 86 ( 1987) 6150. [5] J.M. Alvariiio and A. IaganP, Chem. Phys. Letters 144 (1988) 558. [6] A. Lagan.%,R.T Pack and G.A. Parker, Faraday Discussion Chem. Sot. 84 (1987); G.A. Parker, R.T Pack, A. Lagana, B.J. Archer, J.D. Kress and Z. BaEic, in: Supercomputer algorithms for reactivity, dynamics and kinetics of small molecules, ed. A. Lagana (Kluwer, Dordrecht, 1989) p. 105; D.W. Schwenke, M. Mladenovic, M. Zhao, D.G. Truhlar, Y. Sun and D.J. Kouri, in: Supercomputer algorithms for reactivity, dynamics and kinetics of small molecules, ed. A. Lagana (Kluwer, Dordrccht, 1989) p. 131. [7] M. Paniagua, J.M. Garcia de La Vega, J.R. Alvarez CoJlado, J.M. Alvariiio andA. Lagana, Chem. Phys. 101 (1985) 55. [S] P.G. JasieuandC.E. Dykstra, J.Am. Chem. Sot. 107 (1985) 1891; P. Chaquin, J. Phys. Chem. 91 (1987) 1440. [9] J.C.Light andR.B. Walker, J. Chem. Phys. 65 (1976) 1598. [lo] J.C. Light and R.B. Walker, J. Chem. Phys. 65 (1976) 4272, [ 111 L. Ciccarelli, E. Garcia and A. Lagana, Chem. Phys. Letters 120 (1985) 75. [ 121 A. LaganP, E. Garcia and 0. Gervasi, in: Supercomputer algorithms for reactivity, dynamics and kinetics of small molecules, ed. A. Lagana (Kluwer, Dordrecht, 1989) p. 271; A. Lagana, 0. Gervasi, R. Baraglia and D. Laforenza, in: High performance computing, eds. J.L. Delhaye and E. Gelembc (North-Holland, Amsterdam, 1989) p. 287. [ 131 M. Baer, J. Chem. Phys. 62 ( 1975) 4545; J. Phys. Chem. 85 (1981) 3974. [ 141 G.L. Hofacker and R.D. Levine, Chem. Phys. Letters 9 (1971) 617; 15 (1972) 165. [ 151 M.D. Pattengill and J.C. Polanyi, Chem. Phys. 3 (1974) I. [ 161 M.J. Berry, Chem. Phys. Letters 27 (1974) 73. [17]B.H.ChoiandK.T. Tang, J.Chem.Phys.61 (1974) 5147; 65 (1977) 5161. [ 181 U. HalaveeandM. Shapiro, J. Chem. Phys. 64 (1976) 2826. [ 19) B.C. Eu, Mol. Phys. 31 (1976) 1261. [ 201 G.C. Schatz and J. Ross, J. Chem. Phys. 66 (1977) 1201, 1037,2943. [ 211 S.Mukamel and J. Ross, J. Chem. Phys. 66 (1977) 3759. [22] R.J. Bienniek,Phys. Rev. A 15 (1977) 1513. [ 231 S. Fischer and G. Venzl, J. Chem. Phys. 67 (1977) 1335; S. Fischer, G. Venzl, J. Robin and M.A. Ratner, Chem. Phys. 27 (1978) 251; G. Venzl and S. Fischer, Chem. Phys. 33 (1978) 305.
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[24] M. Zvijac and J. Ross, J. Chem. Phys. 68 (1978) 4458. [ 25 ] K.H. Fung and K.F. Freed, Chem. Phys. 30 ( 1978) 249. [26] MS. Child, Mol. Phys. 35 (1978) 759. [27] P.M. Hunt and M.S. Child, Chem. Phys. Letters 58 (1978) 202. [ZS] H. Kruger, Theoret, Chem. Acta 51 (1979) 311. [ 291 M.S. Child and R.B. Gerber, Mol. Phys. 38 (1979) 421. [ 301 MS. Child and K.B. Whaley, Faraday Discussions Chem. Sot. 67 (1979) 57. [ 311 J.K.C. Wongand P. Bmmer, Chem. Phys. Letters 68 (1979) 517. [ 321 J.N.L. Connor, A. Lagan& J.C. Whitehead, W. Jalcubetz and J. Manz, Chem. Phys. Letters 62 (1979) 479. [33] J.N.L. Connor, J. Chem. Phys. 74 (1981) 1047. [ 341 J.N.L. Connor, J.C. Whitehead, W. Jakubetz and A Lag&, Nuovo Cimento 63 ( 1981) 116.
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[ 351 N. Agmon, J. Chem. Phys. 76 ( 1982) 1759. [36] C. Nyeland, B.C. Eu and G.D. Billing, J. Phys. Chem. 87 (1983) 488. [37] A. Lagan& J. Chem. Phys. 86 (1987) 5523. [ 381 A. Laga@ E. Garcia and J.M. Alvarifio, Nuovo Cimento (1989), in press. [39] E. Garciaand A. Lagan& Mol. Phys. 56 (1985) 621,629. [40] P. Palmieri, E. Garcia and A. Lagan&,J. Chem. Phys. 88 (1988) 181; A. Laganb, 0. Gervasi and E. Garcia, Chem. Phys. Letters 143 (1988) 174. [41] A. LaganP, M. Dini and M. Paniagua, in preparation. [42] A. Lagand, E. Garcia, M.L. Hemandez and J.M. Alvaririo, Nuovo Cimento 5 D (1985) 541. [43] A. Lagan8 and S. Rivoira, in: Artificial intelligence in numerical and symbolic simulation, ed. E. Pavk (1989)
447
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llMay1990
ACCURATE AND MODEL COLLINEAR REACTIVE PROBABILITIES OF THE Mg+FH REACT ION A. LAGANA Dipartimento di Chimica, Universitddi Perugia, 06100 Perugia, Italy
Miguel PANIAGUA Departamento de Quimica Fisica Applicada. UniversidadAutonoma de Madrid, Cantoblanco, 28049 Madrid, Spain and
Josi? M. ALVARIRO Departamento de Quimica Fisica, Facultadde Quimica, Universidaddesalamanca, 37008 Salamanca, Spain Received 25 November 1989; in final form 8 February 1990
Reactive probabilities for the Mg+ FH system have been calculated using an accurate collinear quantum technique. To single out the importance of tunneling contributions these have been compared with quasiclassical results. Product vibrational distributions have been rationalized in terms of an energy-dependent Franck-Condon model.
I. Introduction Theoretical interest in reactions of alkaline earth atoms with hydrogen halides has enormously increased in recent years [ l-51. Systems of this family, in fact, in addition to being of practical importance, can be considered as prototypes of asymmetric A+BC reactions having a structured potential-energy surface (PES). The theoretical investigation of these systems is, at present, mainly based on classical trajectory calculations. However, because the impressive progress of computational techniques is making feasible the accurate three-dimensional quantum study of metal hydrogen halide systems [ 61 we have undertaken an investigation of some alkaline-earth hydrogen-halide reactions using quantum means. Recently, ab initio investigations of the PES of the reaction Mg+FH(u)-+MgF(o’)+H
(1)
have been reported [ 7331. A peculiarity of the PES of this reaction is that although the transition state
has a bent geometry (I% 75”; 0 is the MgFH angle) its overall shape and minimum-energy path vary very little in the range 0= 180” (collinear approach) to 8=75” (transition-state approach). This suggests a first inspection of the reactive dynamics of this system using a collinear R-matrix approach [ 9, lo], The quantum investigation of this reaction was carried out using a vectorized version of the R-matrix program [ 111 restructured to run in parallel mode both the calculation of the sector vibrational basis (in the first section) and the propagation through sectors (second section ) according to the scheme discussed for reduceddimensionality quantum programs in ref. [ 121. For the same reaction, quasiclassical calculations were also performed to single out quantum features. A peculiar feature of atom-diatom collinear quantum reactive results is the structure of the product vibrational distribution (PVD), i.e. its sequence of maxima and minima (modality). To rationalize this structure, a Franck-Condon (FC) mechanism has been proposed. Its application, however, has been mainly confined to the reactions
0009-2614/90/$03.50 0 Elsevier Science Publishers B.V. (North-Holland )
441
Volume 168, number 5
H+X,(U)+HX(U’)+X
(2)
with X = Cl, F or isotopic variants [ 13-37 1. However, we have recently shown [ 37,381 that some features previously thought to be typical of FC reaction mechanisms are actually not. Therefore, as a first step towards the generalization of an FC modeling of reduced-dimensionality quantum calculations for generic three-atom (or atomic-aggregate) reactions we have applied the FC treatment to Mg+HF. The paper is organized as follows. In section 2 the potential-energy surface and quantum and quasiclassical collinear reactive probabilities (PJ calculated on it are analysed. In section 3 a rationalization of the shape of the product vibrational distributions in terms of a Franck-Condon model is presented.
2. Quantum and quasiclassical collinear reactivity The shape of the MgFH PES is rather structured. However, as already mentioned, its angular variation for 0 ranging in the interval 180-75” (i.e. from the collinear to the transition-state angle of approach) is small while for further decrease of 6’the barrier rises steeply to very large values. Therefore it can be assumed that most of the reaction outcome is obtained from large 8 attacks. In the process of obtaining a global interpolation of the ab initio potential-energy valtis we have at first fitted its collinear cut. The lit was performed by adopting an expansion of the two- and three-body interaction in bond order (BO) variables [39] as done for other systems of this family [ 39-4 11. A property of the BO fit is the accurate reproduction of the long- and intermediate-range interaction and the confinement of undesirable structures of the fitted PES in the short-range part of the three-body term. These short-range structures usually occur at such a short distance so as either not to affect the regions of the reactive channel experienced by lowenergy collisions (as those of molecular-beam reactive experiments) or to be easily removed by forcing a repulsive behaviour. The PES fitted to the adjusted collinear ab initio values gives a rms deviation slightly smaller than 0.2 eV. The barrier is displaced late in the exit channel (the HF and MgF distances at the saddle are 1.40 442
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and 1.71 A, respectively) and is 1.9 eV higher than the reactants’ asymptote. A schematic view of the related minimum-energy path is shown in fig. 1 where vibrational levels of reactant and product diatomic molecules are also given. Quantum calculations were performed using a 60-, contracted to 40-, harmonic-oscillator-eigenfunction basis set and dividing the reaction channel in about 300 sectors [ 12,421. Quasiclassical calculations were performed by running batches of one hundred trajectories uniformly distributed in the phase of the reactant oscillator for each pair of reactant vibrational number (u) collision energy (Err ) values. Both types of results are plotted in fig. 2 as a function of the total energy (E) for the four lowest excited reactant vibrational states. The upper limit of the investigated energy interval (E=4 eV) is much larger than the barrier height (compare figs. 1 and 2 ) . This enables us to investigate the effect of varying the vibrational energy E, content of the reactant molecule from well below to definitely above the barrier. The reactive probability calculated at v=O (not shown in fig. 2) is always negligible. On the contrary, the reaction is appreciable when HF is vibrationally excited. At v= 1 the threshold energy is quite large. In this case the quantum reactive probability
MgFH
Mg FH
MgF H -
2
%i L’
I
CI- _
I
Fig. 1. A schematic representation of the collinear minimum-energy path for the Mg+ FH reaction. Reactant (lhs) and product (rhs) diatom vibrational levels are also reported.
CHEMICAL PHYSICS LETTERS
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E / eV Fig. 2. Quantum (dots connected by solid lines) and quasiclassical (circles) reactive probabilities for the Mg+FH reaction reported as a function of the total energy for v= 1 (Ihs lower panel), u=2 (rhs lower panel), v=3(lhs upper panel) and v=4 (rhs upper panel). Connecting lines have been drawn for greater clarity.
(dots connected by solid lines in the lower left panel of fig. 2) increases smoothly at threshold (,I$= 1.7 eV) to reach a maximum around E{, = 2.5 eV. At higher collision energy, the reactive probability decreases quite rapidly to zero. At v=2 (lower right panel) the quantum reactivity increases more rap idly than for v= 1 at threshold (Efr= 1 eV). At this v, the maximum is reached after an energy interval of about 0.5 eV. Then the reactive probability has a plateau for an energy interval of about 1 eV. For ZI values larger than 2 the probability increase at threshold is almost stepwise (the rise to a plateau occurs within 0.3 and 0.02 eV respectively for P3and P4).
These four cases well illustrate the role played by vibrational energy in enhancing reactivity for endoergic late-barrier reactions. When the reactant vibrational energy is smaller than the height of the barrier an increase of reactant vibrational excitation has the effect of lowering the threshold to reaction (along the total energy scale) because it acts upon the molecular motion favouring the bond breaking. On the contrary, when the amount of vibrational energy is larger than the saddle height a further supplying of energy as diatomic vibration subtracts it to a more effective use for reactivity. Accordingly, the threshold moves to larger total energy values and the reactive probability suddenly switches from zero to a
maximum at threshold. This seems to suggest that the only role vibrational energy can play for these reactions is to lower the threshold. On the contrary, fig. 2 shows that there is a second less apparent way through which initial vibrational excitation can contribute to increasing reactive probability. This is by rising the height of the reactive probability plateau. The height of the plateau, in fact, rises to about 0.6, 0.8, 0.9 and 1.Ofor P,,Pz, P3and P., respectively. This can be rationalized in terms of a transition-state picture as a better overlap between the phase space covered by the reactant oscillator and the window to reaction at the saddle and has been confirmed by a detailed analysis of classical trajectories. Deviations of quasiclassical probabilities (reported as circles in the same figure) from quantum values emphasize some similarities of the reactive behaviour of this reaction with that of the Li+HF system [ 421. The figure indicates that (especially in the vicinity of the threshold) quasiclassical reactive probabilities are larger than the related quantum values when E,is clearly lower than the height of the barrier to reaction (UC 3 ). The opposite IStrue when the initial vibrational energy is larger than the barrier (to 3). As discussed in ref. [ 421, this parallels the effect of increasing collision energy for early-bar443
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tier reactions and has been rationalized in terms of different tunneling components.
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16
t
3. Product vibrationaldistributions A distinctive feature of collinear atom-diatom reactions is the modality (the structure of maxima) of the product vibrational distribution. The most accurate characterization of these PVDs has been carried out for reactions occurring via a Franck-Condon mechanism. The first detailed analysis of collinear quantum probabilities carried out using such a model treatment has been performed for the H-tCIZ(~)~HCl+C1 reaction [ 131. Such a treatment approximates the reactive S-matrix elements as overlap integrals between shifted reactant and product diatomic vibrational wavefunctions. Since then extensive studies of FC approaches have been reported in the literature using either quantum or semiclassical arguments [ 20-381. Reactions occurring via an FC mechanism are usually believed to lead to PVDs having v+ 1 peaks (u t 1 modality). In a recent investigation of the H + Cl, reaction and isotopic variants 137,381 we found that such a requisite is not mandatory and that an FC approach is also able to reproduce less structured PVDs. The Mg+HF reaction differs in many respects from the H t Cl1 reactions. Its potential-energy surface is endo- rather than exo-ergic and its mass combination is not of the light+ heavy-heavy type. Nonetheless, reactive probabilities calculated for significantly vibrationally excited reactants, seem to fulfill reasonably well the basic requirement for a reactive FC mechanism, i.e. the process is (almost) fully reactive (see P3 and P4 of fig. 2). In fact, a reactive probability close to one guarantees that back reflection is negligible. Accordingly, as for the H+ CIZ reaction we have approximated the S-matrix elements as overlap integrals between properly shifted wavefunctions of reactant and product diatoms and attempted a rationalization of the PVDs in terms of a model FC treatment. Curves interpolating the location of maxima of the quantum collinear PVD calculated for v= 3 and u= 4 are plotted in fig. 3 as a function of the total energy. The inclusion of the v=3 case is motivated by the fact that, as for H +CIZ [ 37,381, we assume that the
444
;:-
----_:-;: ISi
a-g
0
3
E /
3.5
4
eV
Fig. 3. Curves interpolating the locations of the PVD maxima of the Mg+FH reaction reported as a function of the total energy for ~3 (lower panel) and ~4 (upper panel).
FC treatment can be applied when the reactive probability is larger than 0.8. As clearly apparent from the figure, both PVDs show for the whole energy interval a bimodal structure whose peaks move regularly to higher u’ values when energy increases. Both features, however, are perfectly compatible with a Franck-Condon reaction mechanism. To compare model with accurate dynamical results we have calculated the overlap integrals between the wavefunctions of the two Morse potentials fitted to reactant and product asymptotic diatomic potentials of the MgFH BO PES. Locations of the maxima of the corresponding PVDs are shown in the upper panels of fig. 4 as a function of d (the relative displacement of the product from the reactant wavefunction). The two lowest peaks of the plots are bracketed by dashed lines to evidence the interval of d (root interval ) for which the lowest portion of the model PVD coincides with that obtained from the accurate dynamical calculations. As discussed in ref. [ 381 this is relevant to the factoring of the reactive probability and to its parameterizing for artificial intelligence purposes [ 43 1. The figure shows also that the evolution with energy of such a structure is re-
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11May 1990
totic to the saddle. In this approach the S, element of the S-matrix is given as the overlap of the wavefunctions: &j=NR,iNPj<$R,i~h’,j~,
\ ‘,
\
-
-
cl/
w
Fig. 4. Location of the maxima of the Franck-Condon PVD for the Mg+HF reaction (upper panels) and the H+C12 reaction (lower panels) plotted as a function of the displacement d. The initial vibrational numbers arc 3 (lhs panel) and 4 (rhs panel)
for the Mg+HFreactionand 2 (Ihs panel),and 3 (rhs panel) for the H + Cl, reaction.
produced if d is allowed to vary with energy. For comparison, PVD maxima determined using the FC model treatment for the H t Cl2 reaction at v= 2 and 3 are shown in the lower panels of the same figure. As discussed in ref. [ 381, the structure of the PVD for the H t CIZ reaction derived from the accurate quantum collinear calculation is well reproduced by model calculations for the energy interval investigated (as evidenced by the dashed line in this case too). A first clear differenceis the dependence of the root interval upon vibrational energy. For the Mg t HF reaction, d moves to the right with energy while for H t CIZ it moves to the left. This can be easily rationalized in terms of the characteristics of our FC model treatment: both reactant < &I and product I $+diatom wavefunctions are assumed to propagate with little distortion from the related asymp
(3)
where the N coefficients are the related reactant and product normalization factors. In our simplified approach the distortion of the diatomic wavefunctions can be either neglected (by setting d equal to the difference of the diatomic equilibrium distances) or approximately accounted for by adjusting the value of d. At small vibrational energies the distortion of the diatomic wavefunctions is negligible. Accordingly, the displacement parameter d has almost the same value as the difference between the equilibrium distances (because of our definition of d, the difference is positive when the reactant diatom equilibrium distance is smaller than that of the product and negative in the opposite case). Obviously, larger distortions have to be considered at higher energies because a larger region of the potential is explored during the propagation. In our approach, this is taken into account simply by shifting the root interval with v and v’ (i.e by imposing a larger displacement to the wavefunctions’ origin). A more realistic formulation of the pdtential at the Franck-Condon transition region and therefore a more appropriate representation of the local vibrational wavefunctions would certainly reduce the need for large displacements and allow a better FC parametrization. Another important difference between the H + CIZ and the Mg + HF model results is that for the latter the FC treatment populates more excited vibrational states than the accurate treatment. This means that some excited product vibrational levels are dynamically closed even when energetically accessible and allowed by FC mechanisms. Therefore, a generalization of the FC model needs not only to improve the description of the potential at the FC transition point but also to include additional constraints. Since our model is based on a simplistic approach, we rendered this additional constraint as a limit to the maximum amount of energy to be disposed of as product vibration. Such a type of cut-off is not surprising owing to the fact that heavy+ heavy-light latebarrier reactions can be considered as the inverse process of a light t heavy-heavy early-barrier reaction. Because of that, any restriction on the collision 445
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energy for light+ heavy-heavy reactions should be paralleled by a restriction on the product vibrational energy of heavy + heavy-light ones.
11 May 1990
and CICYT (PB 85-0316) is acknowledged.
References 4. Conclusions
[ I] H.‘Schor, S. Chapman, S. Green and R.N. Zare, J. Phys. Chem. 83 (1979) 920.
[ 21S.Chapman, J. Chem. Phys. 8 1 ( 1984) 262.
Collinear calculations performed for the Mg+FH reaction, with the purpose of carrying out a preliminary investigation of this system on a reasonable potential-energy surface, have evidenced typical features of its dynamical behaviour. The calculated energy dependence of the reactive probability well illustrates the role played by vibrational energy in enhancing reactivity. The rationalization of the value of the plateau of the quantum reactivity and its rise at the threshold (smooth for vibrational energies lower than the barrier, stepwisc for those higher than the barrier) has led to the conclusions that the relative importance of collisional and vibrational energy is similar to that found for the alkali-atom hydrogen-fluoride reactions. Differences between quantum and quasiclassical calculations have also been rationalized as for the alkali-atom hydrogenfluoride reactions in terms of the bias of dominant tunneling contributions. Product vibrational distributions have been rationalized by applying (for the first time to a heavy -I-heavy-light reaction ) a Franck-Condon modeling linking overlaps between shifted reactant and product diatomic vibrational wavefunctions to the reaction detailed probabilities. The FC model treatment has been able to reproduce the location of the two lower peaks as well as their evolution with collision energy. As for the H+Clz reaction the reproduction of the energy dependence of the location of the maxima is made possible by making the shift between asymptotic vibrational functions vary with collision energy. In the case of the MgSFH reaction, however, an upper limit to the amount of energy disposed of as product vibration more restrictive than the total energy conservation had to be applied.
Acknowledgement Financial support from CNR (within the Progetto Finalizzato Sistemi Informatici e Calcolo Parallelo) 446
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