Article pubs.acs.org/JPCA
Time-Dependent Wave Packet Quantum and Quasi-Classical Trajectory Study of He + H2+, D2+ → HeH+ + H, HeD+ + D Reaction on an Accurate FCI Potential Energy Surface Juan Zhao*,†,‡ and Yi Luo*,† †
State Key Laboratory of Fine Chemicals, School of Pharmaceutical Science and Technology, Dalian University of Technology, Dalian 116024, China ‡ State Key Laboratory of Molecular Reaction Dynamics, Dalian Institute of Chemical Physics, Chinese Academy of Sciences, Dalian 116023, China ABSTRACT: The quantum scattering dynamics and quasi-classical trajectory (QCT) calculations have been carried out for the title reaction on an accurate potential energy surface (PES) computed using the full configuration interaction (FCI). On the basis of the PES, the integral cross-sections of He + H2+(v = 0−3, j = 1) → HeH+ + H reaction have been calculated, and the results are generally agreed with the experimental cross-sections obtained by Tang et al. [J. Chem. Phys. 2005, 122, 164301] after taking into account the experimental uncertainties, which proves the reliability of implementing dynamics calculations on the FCI PES. The reaction probability of He + D2+(v = 0−2, j = 0) → HeD+ + D reactions for total angular momentum J = 0 and the integral cross-section (ICS) have been calculated. The significant quantum effect has been explored by the comparison between the QCT reaction probabilities (or ICS) and the quantum mechanical (QM) reaction probabilities (or ICS), which may be attributed to the deep well in the PES of this light atoms system. Furthermore, the role of Coriolis coupling (CC) effects has also been found not important by the comparison between the CC calculation and the centrifugal sudden (CS) approximation calculation, except that the CC total cross-sections for the v = 1 and 2 states show the collision energy-dependent behaviors in the low-energy area, which are different from those based on the CS calculation.
1. INTRODUCTION Over several decades, the ion−molecule reactions of He + H2+ → HeH+ + H and its isotopic variants involving HD+ and D2+ have received much attention due to their importance in the fundamental understanding of ion−molecule reaction dynamics.1−32 Especially after year 2000 when Palmieri et al.33,34 published an accurate potential energy surface (PES), labeled PPA PES, of the ground state of the HeH2+ reactive system, many studies35−41 have been carried out on this PES. In 2005, the new state-selected integral reaction cross-sections determined with the guided-ion beam (GIB) technique in conjunction with high-resolution vacuum ultraviolet (VUV) reactant ion preparation and a pulse-field ionization photoelectronsecondary ion coincidence (PFI-PESICO) approach are reported for the H2+(v) + He proton transfer reaction by Tang et al.,35 and the experimental cross-sections reasonably agreed with the QCT results. Later, Chu et al.36 used the Palmieri surface to conduct time-dependent wave packet quantum scattering calculations including Coriolis coupling (TWQS-CC) to determine absolute integral H2+(v = 0−2,4,6, j = 1) + He proton-transfer cross-sections over a translational energy range, ET = 0−2.4 eV. Their results demonstrated that the inclusion of the Coriolis coupling (CC) is essential for obtaining satisfactory agreement between theoretical and experimental cross-sections for HeH+ formation from the H2+ © 2012 American Chemical Society
+ He reaction, and the TWQS-CC results compared well to the state-selected experiments by Tang et al.35 In 2007, Tang et al.37 examined the absolute integral cross-sections for the formation of HeH+ and HeD+ from the collisions of HD+(v = 0−15, j = 1) + He using the same VUV and PFI-PESICO method and carried out QCT calculation proving the agreement with the experimental results. Later, they also presented the 3D TWQC-CC calculations for HD+(v = 0−3, j = 1) + He collisions in the center-of-mass collision energy range of 0.0−2.0 eV using the PPA PES,38 and there was excellent agreement between the experimental and theoretical results above the threshold. Both of the TWQS-CC results of the H2+(v = 0−2,4,6, j = 1) + He and HD+(v = 0−3, j = 1) + He reactions have been found having very rich quantum resonances. Lately, Liu and his co-workers did many theoretical analyses in the scalar properties and vector properties of the reactions of He + H2+, its isotopic variants, and its reverse using the QCT method.39−41 In 2008, Xu et al. computed a new PES for the ground state of the He + H2+ → HeH+ + H reaction using an ab initio multireference configuration interaction method (MRCI) with Davidson correction and a large orbital Received: December 18, 2011 Revised: February 18, 2012 Published: February 21, 2012 2388
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Figure 1. Comparison among the QCT-computed integral cross-sections in this work, the previous QCT results, and the experimental results of Tang et al. for He +H2+(v = 0−3, j = 1) → HeH+ + H reactions.
basis set (aug-cc-pv5z);42 the new potential energy surface has a root-mean-square (rms) error of 0.095 kcal/mol much smaller than the rms error in PPA PES, where it is 0.75 kcal/mol. They also calculated the integral cross-sections of He + H2+(v = 0−3, j = 1) → HeH+ + H reactions on their new PES using the QCT method. After comparing their results with the previous QCT results based on the PPA surface, they found that the new QCT cross-sections also agreed well with the experimental crosssections, especially for the collision energy ET < 0.7 eV in the v = 3 initial vibrational state case. However, the quantitative discrepancies between experimental and theoretical results still exist, and the MRCI surface of Xu et al. lead to unsatisfactory agreement with the vibrational state-to-state integral crosssection. In 2009, in order to both validate and possibly improve the previous calculated points, Ramachandran et al. constructed a new PES computed using the full configuration interaction (FCI) method and the cc-pVQZ basis set with the MOLPRO suite of programs.43 Up to now, there has been seldom calculation done on the FCI PES. To date, the dynamics of He + H2+ and He + HD+ reactions have been well studied experimentally and theoretically, while the He + D2+ reaction has neither the time-dependent wave packet quantum scattering calculations nor the experimental results. In the present work, we will report a comparative dynamics studies of the He + D2+(v = 0−2, j = 0) → HeD+ + D reactions by performing a time-dependent wave packet (TDWP) calculation with a split-operator scheme and a QCT calculation on the new FCI PES computed by Ramachandran et al. First, by comparing the present QCT cross-sections and the previous QCT and experimental results of the He + H2+ collision, the accuracy of the FCI PES is proven. In order to estimate the quantum effect, we have calculated the reaction probabilities of the title reaction for J = 0
using the TDWP and QCT methods. In addition, to investigate the influence of the CC on the reaction system, we have also performed CS approximation calculations in the collision energy range of 0.0−2.5 eV for comparison with the CC results. This article is structured as follows: section 2 presents the parameters employed in the quantum mechanical (QM) TDWP and the QCT methods; section 3 presents the results and discussion; and finally, section 4 briefly concludes the material.
2. THEORETICAL METHODS 2.1. Time-Dependent Wave Packet Method. The general theory of the time-dependent wave packet method has been very perfect and standard;44 the reader is referred to the relevant articles and to the references therein. The following parameters were used to get the converged results of the He + D2+ reaction: 160 translational basis functions for the R coordinate in the range of 0.5−18.5 au (200 translational basis functions for the high collision range of 1.6−2.5 eV), 130 vibrational basis functions for the r coordinate of 1.0−13.06 a0, jmax = 90 for rotational basis functions, and a propagation time of 60 000 au. In order to get the converged results, many test calculations on each parameter have been done. The number of K used in the CC calculations is up to 5, where K is the projection of J on the body-fixed z axis. 2.2. QCT Calculations. The calculation method of the QCT is the same as the one used in the literature.45,46 The classical Hamilton’s equations are numerically integrated in three dimensions. In the present work, 50 000 trajectories are run for each reaction; the initial distance from He atom to the center of mass of H2+ is 10.0 Å, and the integration step size in the trajectories is chosen to be 0.1 fs, which can guarantee the 2389
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mediate HeH2+ without saddle point. The endothermicity and the lightness of the atoms involved make possible a fully converged quantum mechanical treatment of the dynamics.34 This implies that the QCT method may not be very suitable for studying the dynamics of the HeH2+ system and that the quantum calculation is required; the corresponding results of He + D2+ reactions have been presented in the following text, where the number of K used in the CC calculations is up to 5. As mentioned above, although there is a little discrepancy between the present results and the experimental data, the FCI PES computed by Ramachandran et al. is accurate enough to be used in dynamic calculation. On the basis of this PES, the dynamics of He + D2+(v = 0−2, j = 0) → HeD+ + D reactions have been studied by QCT and quantum mechanical timedependent wave packet methods. Figure 2a−c shows the comparison between the timedependent wave packet result of the reaction probability and
conservation of the total energy and total angular momentum. The impact parameter, bmax, is computed by calculating 50 000 trajectories at a fixed value of the impact parameter b and systematically increasing the value of b until no reactive trajectories are obtained. The reaction cross-section is defined as σr = πbmax2(Nr/N) (Nr, the number of reactive trajectories; N, the total number of trajectories). The sampling error is defined as ε = ((N − Nr)/NNr)1/2 × 100%, and the reaction cross-section error in our calculation is defined as σrε = εσr. The same parameters are used in QCT calculation of the He + D2+ reactions.
3. RESULTS AND DISCUSSION Figure 1a−d shows the comparison of the present calculated cross-sections of He + H2+(v = 0−3, j = 1) → HeH+ + H reactions with the previous QCT-computed results based on PPA and Xu surfaces and the experimental cross-sections obtained by Tang et al.35 In ref 35, Tang et al. have also fitted their cross-section curve according to the line-of-center (LOC) collision model, and the LOC fit after the deconvolution of the experimental kinetic-energy function is also shown in Figure 1. In Figure 1, it is clear that the deconvoluted cross-sections provide sharp threshold behavior except for v = 0. It can also be seen that the vibrational energy of the reactants is much more effective than translational energy to enhance the reaction cross-section, and the theoretical threshold energy decrease obviously with the increasing of the vibrational quantum number. For the four v cases, the present measurements are more similar to the ones calculated by Xu et al.42 In the v = 0 case, our results are in agreement very well with Xu’s QCT results, which are somewhat lower than the data from PPA PES and experiment. In the v = 1 case, the three QCT results show good agreement with the experimental results at collision energy lower than 0.6 eV and higher than 1.0 eV and are larger than the experimental results at collision energies between 0.6 and 1.0 eV. For v = 2, the trends of three calculated results are consistent, but they are all larger than the experimental results. Our results are slightly higher than PPA QCT and experimental results near 0.75 eV, but they are in line with the experimental data and are better than Xu QCT results at the higher collision energy. For v = 3, the present results are closest to the LOC fit cross-sections in the whole collision energy range except for E ≈ 0.3 eV. Through the description above, the discrepancy among the theoretical results may be attributed to the different interaction region and the depth of the potential well in the different PESs, the parameters of three PESs are shown in Table 1. As shown in Table 1, the well in FCI PES is deepest. Maiti and Sathyamurthy have pointed out that the well depth for various PESs can affect the number of bound state of the systems and hence the dynamics observed.28,43 In addition, this endothermic reaction proceeds through the stable linear triatomic inter-
Figure 2. Comparison between the QCT-computed reaction probability and the QM results for He + D2+(v = 0−2, j = 0) → HeD+ + D(J = 0) reactions.
the present QCT results for He + D2+(v = 0−2, j = 0) → HeD+ + D(J = 0) reactions. As shown in this figure, the trends from QCT data are consistent with QM results for the different vibration levels, i.e., the reaction probabilities increase as the collision energy increases. However, the differences between the two results are also obvious. The threshold calculated by QCT is lower than the one by QM. This happens because the product states below the zero point energy are accessible in the QCT calculations. In detail, for v = 0, the QCT result has an obvious salient point near 1.7 eV, while the QM result is a smooth curve, and at the collision energy larger than 2.2 eV, QCT values are smaller than QM ones; For v = 1, QCT results are slightly bigger than QM results near the threshold energy and are consistent with QM results, and then the results are obviously smaller than QM results with a further increase (about E ≥ 1.7 eV) in collision energy. The same phenomenon appears in the v = 2 case except that QCT results are obviously smaller than QM results at E ≥ 1.3 eV. For the same system and PES, the reason for the discrepancy of two results is the different method used in the calculations. Compared with the QM method, QCT method has a serious drawback: its inability
Table 1. Parameters of the Three PESsa equilibrium distances of HeH+ equilibrium distances of H2+ well depth a c
PPAb
Xu et al.b
FCIc
rHeH = 1.934
rHeH = 1.932
rHeH = 1.933
rHH = 2.075 7.78
rHH = 2.075 7.75
rHH = 2.075 7.84
The well depth in kcal/mol, distances in Bohr. Reference 43.
b
Reference 41.
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Figure 3. Comparison between the CC and CS probabilities in the collision energy range of 0.0−2.5 eV for initial quantum numbers v = 2, j = 0, and k0 = 0, and for total angular momenta (a) J = 10; (b) J = 20; (c) J = 30; (d) J = 40; (e) J = 50; and (f) J = 60.
calculations is the appearance of a very rich pattern of narrow quantum resonances, which is similar to the quantum resonance structure in He + H2+ and He + HD+ collisions. For the latter two reactions, large efforts have been dedicated to the characterization of these resonances both theoretically and experimentally,25,32,33,36,38,47−53 These are essentially Feschbach resonances originating from the deep potential well in the triatomic interaction region and are indicative of large quantum effects contributing to the mechanism of the reaction. For the reaction considered in the present work, the marked jagged resonance is as well found in the CC (CS) probability, and when J gets larger, the resonance in CS probabilities sharply increases. The similar situation is presented for the initial quantum numbers j = 0, v = 1, and v = 2, and the results have not been present here. Figure 4a−c shows the comparison of QCT, CS, and CC integral cross-sections (ICSs) for He + D2+(v = 0−2, j = 0) → HeD+ + D reactions in the collision energy range of 0.0−2.0 eV. As seen in this figure, the vibration excitation of the reactants also obviously enhances the reaction cross-section and decreases the theoretical threshold energy. In more specific terms, in Figure 4a, the differences between the CC and CS cross-sections are rather small, which means that the influence of Coriolis coupling on the ICS for v = 0, j = 0 can be negligible, while the big differences were found when we compared the CC (CS) results with the QCT results. The values from QCT calculation are much smaller than the ones calculated by the QM method. The large gap between them further illustrates the strong quantum effect. In Figure 4b,c, many differences among the three results have been found. Seen in Figure 4b, the CC results show a sharp increasing trend
to properly treat the quantum mechanical zero-point energy, which can lead to the inaccuracy of the results near the threshold; in addition, the character of the ground state PES may be the main reason. For the HeH2+ system, the PES has a well depth of about 7.84 kcal/mol in the entrance channel at a collinear geometry rHeH = 1.9330 Bohr and rHH = 2.0756 Bohr. For the isotopic HeD2+ system, the potential well in the reaction path also exists; this deep potential well and relatively light weight of the atoms together cause the significant quantum effect in the He+D2+ collision, the higher the collision energy, the more significant the quantum effect, which conforms to the general quantum theory. The vibrational excitation of reactants can also increase the quantum effect. Figure 3a−f shows the CC and CS probabilities of the He + D2+(v = 2, j = 0) → HeD+ + D reaction for the initial quantum numbers v = 2, j = 0, and k0 = 0 (k0 is the projection of the initial rotational angular momentum j on the body-fixed z axis), and the total angular momentum J = 10, 20, 30, 40, 50, and 60, respectively, in the collision energy range of 0.0−2.5 eV. Seen in Figure 3, the values and the pronounced extent of the resonance peaks in the calculated probabilities are quite different for the two sets of calculations. For low J values, such as J = 10 and 20, the differences between the CC and CS probabilities are rather small; but as J becomes larger, for J = 30−60, remarkable differences are observed in that the CC probabilities are larger than the CS results near the threshold and that the resonance exhibited in the CC probabilities are also quite different from those in the CS probabilities all over the collision energy range. It can also be seen in Figure 3 that the important feature of this reaction in the energy range covered by the quantum 2391
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molecule rotating in various directions because the vibrational modes excited through CC might provide more possible chances to break the bond of the complex and thus facilitating the formation of the products;54 but the CS approximation artificially hindered the collision-induced orientation changes in the molecular angular momentum vector. Therefore, it works well with sudden collision; however, for the indirect reaction in which the resonances last long enough, collisions do not occur suddenly and hence lead to the failure of the CS approximation in describing the dynamical behaviors.49 As can be seen in Figure 4b,c, the trends of the QCT crosssection dependence of the collision energy are similar to that of CC, but the threshold energy and the magnitude of ICSs calculated by the two methods are very different. For He + D2+(v = 1,2, j = 0) → HeD+ + D reactions, the QCT method decreases the threshold, and QCT results are larger than the CC results in the low collision energy region but smaller than the CC results in the high collision energy region. In view of the accuracy of the QM method and the intrinsic shortage of the QCT method and considering the nature of the present system, the great differences between CC and QCT ICSs may be ascribed to the unfeasibility of the QCT method in describing the dynamical behaviors of the light−atom system with a deep potential well. Unfortunately, there have been no experimental data about the title reactions to verify the reasonability of our prediction. We hope that this calculation may provide experimentalists with some help when they interpret their experimental data in the future.
Figure 4. Comparison among the QCT-computed integral crosssections, CC results, and CS results in the collision energy range of 0.0−2.0 eV for He + D2+(v = 0−2, j = 0) → HeD+ + D reactions.
near the threshold region, followed by a slight decline and a slow increase with a further increase in the investigated energy range, while only an increasing trend exists in the CS results; in addition, there is a good agreement between the CS and CC calculations for collision energies below about 0.7 eV, but as the collision energy increases, the agreement between the CS and CC calculations worsen, and the CC integral cross-section is always larger than the CC result with a maximum difference of ∼62% at the collision energy of 0.83 eV, but at the collision energies over about 1.6 eV, only a slight difference exists. These QCT results together with the ones in Figure 4c will be discussed later. The results for v = 2, j = 0 are present in Figure 4c. From this figure, we can see the similar situations. The CC results rise rapidly from the threshold energy of 0.47 eV and reach a maximum at collision energy of ∼0.7 eV, then decrease with further increasing collision energy and finally level off at high collision energies, while the CS results only increase and also finally level off at high collision energies. Except for a well agreement near the threshold, the CC integral cross-section is also larger than the CC result with a maximum difference of ∼64% at the collision energy of 0.64 eV, but they are closer to each other at the larger collision energies. As the reason mentioned in some other articles,36,44,54 due to the J-averaging effect, the pronounced resonances observed in the probability curves of Figure 3 are greatly suppressed in the CC and CS cross-sections shown in Figure 4c. However, the CS cross-section curves reveal more fine resonance features than the CC cross-section curves, which can be observed in Figure 4a−c, particularly in Figure 4c. The reason for this fact that there are almost no resonance features in the CC total cross-section may be the further washing out effects on the resonance probabilities by including different K states in the CC calculation. By the comparison between CC and CS calculated results in Figure 4b,c, we conclude that the effect of including CC here is to enlarge the calculated integral cross-sections. As analyzed by Lv et al.54 and also Chu et al.,36 because the title reaction is dominated by the formation of a complex, numbers of resonances in the calculated reaction probabilities are shown in Figure 3. The complex-forming mechanism usually favors the
4. CONCLUSIONS We have carried out a quasi-classical trajectory calculation for the state-selected reactions He + H2+(v = 0−3, j = 1) → HeH+ + H in the collision energy range of 0.0−2.0 eV, by employing the new PES of Ramachandran et al. On the whole, the calculated integral cross-sections are in agreement with the experimental cross-sections obtained by Tang et al., which proves the reliability of implementing dynamics calculations on the new FCI PES. Using this PES, we also performed the threedimensional time-dependent wave packet scattering and QCT calculations for He + D2+(v = 0−2, j = 0) → HeD+ + D reactions in the collision energy range of 0.0−2.5 eV. Comparison between the collision energy dependence of the QCT reaction probabilities and that of the CS reaction probabilities for J = 0 reveals a significant quantum effect, which may be attributed to the deep well in the PES of this light− atom system. The number of resonances in the reaction probability is an important feature of the title reaction; these are essentially Feschbach resonances originating from the deep potential well in the triatomic interaction region and are indicative of large quantum effects contributing to the mechanism of the reaction. The huge mismatch between QCT ICSs and CC ICSs further proves this significant quantum effect, which implies that QCT calculations will not be reliable because of its inability to properly treat the quantum mechanical zero-point energy and quantum tunneling in the present system. Comparison between the collision energy dependence of the CC cross-sections and that of the CS crosssections for v = 1 and 2 reveals that neglecting the CC can significantly influence the trend and the value of the calculated cross-sections. The above conclusions are only obtained from theoretical calculation, and we hope that there will be corresponding experimental data to compare with the theoretical results in the future. 2392
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AUTHOR INFORMATION
Corresponding Author
*E-mail:
[email protected] (J.Z.);
[email protected] (Y.L.). Notes
The authors declare no competing financial interest.
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