ARTICLE pubs.acs.org/JPCA
Quasi-classical Trajectory Study of the Ne þ H2þ f NeHþ þ H Reaction Based on Global Potential Energy Surface Jing Xiao, Chuan-Lu Yang,* Xiao-Fei Tong, Mei-Shan Wang, and Xiao-Guang Ma School of Physics, Ludong University, Yantai 264025, the People’s Republic of China
bS Supporting Information ABSTRACT:
Using the multireference configuration interaction method with a Davidson correction and a large orbital basis set (aug-cc-pVQZ), we obtain an energy grid that includes 32 038 points for the construction of a new analytical potential energy surface (APES) for the Ne þ H2þ f NeHþ þ H reaction. The APES is represented as a many-body expansion containing 142 parameters, which are fitted from 31 000 ab initio energies using an adaptive nonlinear least-squares algorithm. The geometric characteristics of the reported APES and the one presented here are also compared. On the basis of the APES we obtained, reaction cross sections are computed by means of quasi-classical trajectory (QCT) calculations and compared with the experimental and theoretical data in the literature.
I. INTRODUCTION For complex rare gas Ar þ H2þ and Kr þ H2þ reactions, several potential energy surfaces (PESs) are involved (e.g., electronically nonadiabatic processes are possible), and the charge transfer reaction channel is open.1 In contrast, the Ne þ H2þ f NeHþ þ H reaction has a single PES character, and the number of electrons involved is fewer in ab initio calculations. As such, the Ne þ H2þ system is a popular focus in many experimental and theoretical studies. The dynamics of the reaction has been the subject of several experimental2-6 and theoretical studies.1,7-13,19 Some previous efforts14-22 have been made to obtain a PES for the Ne þ H2þ f NeHþ þ H reaction. Vasudevan,14 using a configuration interaction (CI) method, calculated a number of points for the collinear configuration of Ne to H2þ. Using the self-consistent field (SCF) method calculation, Bolotin et al.15 reported that NeH2þ is stable with respect to its dissociation to NeHþ and H or Ne and H2þ. Hayes et al.16 calculated a number of points for the collinear geometry at the SCF level, but the geometries are limited to the range of RNeH e 4.03 and RHH r 2011 American Chemical Society
e4.0 bohr. They also constructed a diatomics-in-molecules23 (DIM) surface parametrized by adjusting the NeH interaction parameters to fit the SCF points on the surface. Zuhrt17 constructed a three-dimensional (3D) DIM surface for the reaction using parameters from the SCF calculations by Hayes et al.16 Urban, Jaquet, and Staemmler18 calculated a number of points for the system using the coupled electron pair approach (CEPA) method. Urban et al.19 reported an analytical potential energy surface (APES) by fitting 120 points obtained with the CEPA method18 using an extended London-Eyring-PolanyiSato (LEPS) form.24 Ischtwan et al.20 reported the minimum energy pathways and stationary points on the PES using manybody methodologies and an extensive number of basis sets. They had not calculated enough points, however, to construct a full 3D surface. Pendergast et al. (PHHJ)21 employed three different function expressions to construct an APES from the ab initio Received: September 17, 2010 Revised: January 13, 2011 Published: February 15, 2011 1486
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The Journal of Physical Chemistry A energies obtained with the CEPA method. Their third APES,21 known as the PHHJ3 surface, was believed to be the “best” one, and several dynamics studies1,7-13 based on it have been reported. However, only 221 energies were involved in the fitting of the APES. Some dynamic results based on the APES deviate from the experimental data.6 Lv et al. (LZHH)22 presented a new APES that was fitted from 7000 energy points using multireference configuration interaction (MRCI).25,26 The ab initio energies calculated in the new APES are smaller than 2.8 eV relative to the dissociation limit of Ne þ H þ Hþ, therefore the APES is inadequate to investigate the dynamic problem of the collision energy greater than 2.8 eV. In the present paper, we report a new APES for the Ne þ H2þ f NeHþ þ H reaction based on MRCI calculations. In order to construct an APES to describe wider ranges of energy and space, we calculate a large energy grid containing 32 038 points using the MRCI method with a Davidson correction and basis set augcc-pVQZ. A total of 31 000 reasonable energy points, which is more than those used in the PHHJ3 or LZHH surface and can cover extensive space of the PES, are selected to fit the APES. An expression of APES suggested by Aguado and Paniagua27 is used to fit the ab initio energies. In order to evaluate the present APES, we also present some dynamic results based on the quasi-classical trajectory (QCT) method. The article is organized as follows. Section II briefly describes the computational method and theoretical details necessary to build the APES of NeH2þ. Section III shows the analytical potential energy surface. Some dynamics analysis for the present APES is presented in Section IV, and the summary and conclusion are contained in Section V.
II. COMPUTATIONAL METHODS A. Ab Initio Calculations. The MRCI method25,26 is used to
obtain the energy grids for the ground state of NeH2þ. The reference wave functions of the MRCI include all single and double excitations and are built on the complete active space involving nine electrons and six active orbitals. The 1s orbital of the Ne atom is kept inactive in the calculations. Nevertheless, some spurious energies are found when the orbitals are only produced from the complete active space self-consistent field (CASSCF) of the ground state. In order to solve this problem, we use a two-state averaged CASSCF calculation to determine the molecular orbitals for the MRCI calculations. This is an effective approach because the calculation results show that all the spurious energies are eliminated, and the PES becomes smooth. Dunning’s correlation-consistent basis set, aug-cc-pVQZ,28,29 is used in the calculations, after considering both the accuracy and CPU time. All calculations are performed using the MOLPRO set of programs.30 The geometries used to calculate the energies to fit the APES are selected according to the internal coordinate system in Figure 1a. The symbols R1 and R2 represent the two Ne-H distances, which range from 0.60 to 10.0 Å. The symbols θ and β represent the angles, where θ varies from 0° to 180°. To emphasize some important regions of the reaction, such as regions near the entrance or exit valleys, we also use the Jacobi coordinate system (represented by RR, R3, and σ in Figure 1b) to select some geometries. The values of R3 range from 0.60 to 10.0 Å, while RR ranges from 0.0 to 10.0 Å. The geometries are discarded when one or more distances between two atoms are less than 0.5 Å because these geometries having very high
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Figure 1. Coordinate systems for selecting geometries to build the APES of NeH2þ. (a) Internal coordinate; (b) Jacobi coordinate.
energies are difficult to reach in the reaction and small interatomic distances often result in convergence problems in the MRCI calculations. B. Fitting of the PES. The APES can be represented by the many-body expansion proposed by Murrell and Carter:31 3 X ð1Þ X Vtot ðR1 , R2 , R3 Þ ¼ Vi þ Vnð2Þ ðRn Þ n¼1
i
þV
ð3Þ
ðR1 , R2 , R3 Þ
ð1Þ þ
where Vtot is the potential energy of the NeH2 reaction system, Vi(1) (i = 1,2,3) is the atomic energy terms, Vn(2) (n = 1,2,3) represents the two-body terms, and V(3) is a three-body term, which becomes zero at all dissociation limits. In the present case, the values for Vi(1) are set to zero because the two atoms are in the ground state in the dissociation limits. The extended-Rydberg (ER) function32 is used to represent the two-body terms, which are fitted from the potential energy curves of the NeHþ product and the H2þ reactant molecules. The function is also the same as the one for the two-body terms in ref 21 of fits 1 and 2. The ER function has the following form: n X ai Fi Þ expð - a1 FÞ ð2Þ V ðFÞ ¼ - De ð1 þ i¼1
where F = R-Re. Here, Re is the equilibrium distance, and R is the distance between two atoms. In the present work, we truncate the polynomial to a degree of 9, which will result in 11 parameters (De, Re, a1, a2.......a9) to be determined. The AP function suggested by Aguado and Paniagua,27 which was used in PHHJ3, is used to represent the three-body term. It can be written as: 8 M > X j > > cjkl λ1 λk2 λl3 ðk ¼ lÞ > > < j, k, l ¼ 0 V ð3Þ ðR1 , R2 , R3 Þ ¼ M X > > j > c λ ðλk λl þ λl2 λk3 Þ ðk 6¼ lÞ > > : j, k, l ¼ 0 jkl 1 2 3 ð3Þ where cjkl are the linear coefficients. The constraints j þ k þ l 6¼ j 6¼ k 6¼ l, j = k þ l e M, and k e l are used to ensure that the 1487
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Table 1. Spectroscopic Constants of the Two-Body Terms for Reactant H2þ and Product NeHþ Re/Å
De/ cm-1
we/cm-1
H2þ(present APES)
1.057
22512.81
2332.00
H2þ(present ab initio)
1.057
22513.60
2319.70
experiment34
1.052
22535.12
2323.6
H2þ(LZHH APES)22
1.056
22527.06
2331.24
H2þ(LZHH ab initio)22
1.057
22615.78
2323.4
NeHþ(present APES)
0.990
18811.04
2924.86
NeHþ(present ab initio)
0.987
18813.08
2942.26
experiment NeHþ(LZHH APES)22
0.99135 0.989
18486.22
291736 2955.53
NeHþ(LZHH ab initio)22
0.990
18550.74
2927.76
three-body term becomes zero at all dissociation limits. The two expressions of eq 3 are used to satisfy the permutation symmetry of the two H atoms. M equals 10 in the present work, which results in 140 linear coefficients to be determined in the expression. The λi term is expressed as the following: λi ¼ Ri expð - ωi Ri Þ
III. ANALYTICAL PES A. The Fitting of Two-Body Terms. Using the ER function described in Section II and the nonlinear least-squares33 fitting method, we obtain the APESs for the two-body terms of the NeHþ product and the H2þ reactant from ab intio points. A total of 209 points for NeHþ and 200 points for H2þ are used to obtain the APESs. All the fitted parameters are presented in the Supporting Information. The root-mean-square (RMS) error can be used to quantitatively determine the quality of the fitting process. The present RMSs are 1.41 10-4 and 3.84 10-4 eV for NeHþ and H2þ respectively, which are smaller than those reported for the PHHJ3 surface21 and close to those for the LZHH surface.22 This implies that the fitting process is of high quality, and that the ER function is suitable for reproducing the potential energy curves of the two molecules. To evaluate the APESs more widely, we calculate the spectroscopic constants of
Be/cm-1
Re/cm-1
68.31
29.91
1.54
66.2
30.2
1.68
114.07
17.91
1.09
18.0034
Table 2. RMSs and Maximum Energy Deviations of the APES PHHJ321
present APES
θ (deg)
ð4Þ
where ωi is the nonlinear parameter and Ri is the distance between two atoms. There are three nonlinear parameters in expression 4, but the two ωNeHþ values should be the same because of the commutative symmetry of the two H atoms. Therefore, only two nonlinear parameters (i.e., ωNeHþ, ωHHþ) need to be determined in the present case. The nonlinear least-squares fitting method is used to determine the parameters from the ab initio results. The parameters of the two-body terms, which are first fitted from the ab initio energy points of NeHþ and H2þ, are fixed in the fitting process for the parameters of the three-body term. Therefore, 142 parameters (140 linear plus 2 nonlinear) are available in the last fitting process. The fitting calculations are performed using an adaptive nonlinear least-squares algorithm.33 As there are a large number of both fitting parameters and energies, the fitting will be difficult to converge if the initial parameters are unsuitable. However, suitable initial parameters are difficult to determine. In order to obtain convergent fitting results, we have not used all the ab initio energy points to fit at the beginning of the fitting process. Instead, we start from 10 000 energy points and increase the points step by step. Finally, 31 000 energies are used to determine the 142 parameters.
wexe/cm-1
Npts
ΔVrms
ΔVmax
θ
(eV)
(eV)
(deg)
Npts
ΔVrms
ΔVmax
(eV)
(eV) 0.138
0-30
6326
0.0501
0.4339
30
33
0.0392
40-60
4255
0.0261
0.5151
60
28
0.0224
0.089
70-90
4188
0.0245
0.1109
90
39
0.0103
0.027
100-120 130-150
4152 4144
0.0338 0.0275
0.1483 0.1314
120 150
32 33
0.0134 0.0085
0.042 0.024
4131
0.0239
0.1004
180
56
0.0120
0.030
31000
0.0391
0.5151
All
221
0.0193
0.138
160-180 all
the two diatomic molecules using the APESs and list them in Table 1. The ab initio calculation results, experimental data, and the results calculated by LZHH22 are also presented. The spectroscopic constants based on present APESs are closer to both the ab initio results and the experimental data,34 which implies that the present APESs can describe the interactions between atoms and be safely used to build an APES for the triatomic system. B. The Fitting of the Three-Body Term. We have calculated a total of 32 038 structures according to the theoretical level and method to select geometries of the NeH2þ system described in Section II. Considering the typical energy of the incident atom in reaction dynamics experiments, we have not used geometries with energies larger than 6.0 eV in the fitting process. A total of 31 000 geometries are used to fit the APES. It was difficult to get the 142 parameters converged, due to both the degree of the polynomial in eq 3 and the large number of points. The final fitted parameters are presented in the Supporting Information. The RMSs and maximum energy deviations are reported in Table 2. The present RMS errors and maximum energy deviations are larger than those of PHHJ3.21 However, it should be noted that the 31 000 points used in the present fitting are significantly larger in number than Pendergast et al.’s21 221 points from the previous study. For each section, our points span several angles (e.g., 0°, 10°, 20°, and 30°), but the points in PHHJ3 use only one angle (cf. Table 221), which enlarges the correspondent RMSs in each section. To satisfy the dynamic calculations, we construct analytical energy derivatives for our APES in the internal coordinate system. The FORTRAN subroutine, including the energy and the derivatives, is provided in the Supporting Information. C. Characteristics of the APES. To compare the characteristics of the present APES with those in the literature, we provide the contour plot of the collinear approach of Ne to H2þ in 1488
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Figure 2. Contour plots of the APES for NeH2þ (β = 180°). Energies are in electron volts, and distances are in angstroms.
Table 3. Location, Depth, and Harmonic Vibrational Frequencies for the Collinear Minimuma RNeHþ
RHHþ
depth
frequencies
present APES
1.221
1.110
0.580
677.18, 677.18
ab initiob
1.211
1.099
0.557
674.48, 674.48 986.20, 1903.10
PHHJ3
1.197
1.101
0.510
683.39, 683.39
LZHH
1.212
1.090
0.536
669.90, 669.90
ab initio22
1.209
1.101
0.534
976.19, 1906.41
930.75, 1868.07 997.21, 1871.10 656.50, 656.55 977.35, 1890.13 a b
The distances are in angstroms, and the energies are in electron volts. UCCSD(T)/aug-cc-pVQZ.
Figure 2. On the basis of the figure, it can be seen that there is a well about 0.58 eV deep with respect to the reaction asymptote Ne þ H2þ. We present the location and depth of the well for the present APES, the ab initio calculations, and the PHHJ3 and LZHH surfaces in Table 3. The harmonic vibrational frequencies for the present APES, the ab initio calculations, and the PHHJ3 and LZHH surfaces are also shown in Table 3. It can be seen in the table that the location and depth are in accordance with those by found ab initio calculations, and better than those of PHHJ3. As mentioned in the previous study,21,22 along the minimum energy path and Ne-H-H approaches not very far from collinearity, the potential energy of this system has no barriers at the entrance and exit channels, so the reaction can easily occur for these geometries, if there is enough energy to overcome the reaction endoergicity (0.55 eV, including the zero point energy). For 90°e β e 180°, the minimum energy pathways of our APES and the PHHJ3 are plotted in Figure 3. Because the minimum energy paths for the LZHH surface and for our APES almost overlap, we omit those of the LZHH surface in Figure 3. We can see from the figure that the barrier of our APES for β = 90° is lower, and the wells in the other angles are deeper than their counterparts in PHHJ3. In the figure, the results of our APES and the PHHJ3 surface show that the minimum energy pathways for β = 180° has the lowest energy well, implying that
Figure 3. Minimum energy pathways for our APES and the PHHJ3 surface. The abscissa is in units of length. The βs are 90°, 120°, 150°, and 180°, respectively.
the well is a global minimum on the PES. For β = 30° and 60°, the contour plots of our APES, PHHJ3, and LZHH are presented in Figure 4 as a function of R1 and R3. We can see from Figure 4a that a well is located at RNeH = 2.2 Å and RHH = 1.05 Å, which is coincident with RNeH = 2.2 Å and RHH = 1.06 Å of PHHJ3 in Figure 4b and RNeH = 2.18 Å and RHH = 1.05 Å of LZHH in Figure 4c. The present well depth of 0.39 eV is a little deeper than that reported in the previous literature. We also find a barrier located at RNeH = 1.55 Å and RHH = 2.65 Å. The height of the barrier is about 2.16 eV, which is slightly lower than that obtained with PHHJ3 (2.21 eV)21 but a little higher than that of LZHH (2.12 eV). Moreover, we found that the barrier is close to that reported by Urban et al. (2.17 eV)18 for β = 30°. We can clearly see in Figure 4d,e,f that similar barriers exist for our APES and the PHHJ3 and LZHH surfaces. The barrier of our APES is located at RNeH = 1.38 Å and RHH = 1.83 Å, and the height is about 1.48 eV. That of PHHJ3 is located at RNeH = 1.38 Å and RHH = 1.89 Å, with a height of about 1.53 eV, and that of LZHH is located at RNeH = 1.40 Å and RHH = 1.85 Å, with a height of about 1.49 eV. The others contour plots for the present APES and LZHH are presented in the Supporting Information. The barrier for β = 30° and 60° is rather high, so the reactive system may find difficulty in passing through a barrier with these configurations when the collision energy is 0.7 eV, although the degree of difficulty will depend on the internal energy content of H2þ. This will be discussed in the following section. In order to show the difference of the present APES and the PHHJ3 and LZHH surfaces more clearly, we give the geometries and energies of the stationary points (fixed angle) on the PESs in the Supporting Information. In conclusion, the characteristics of the present APES are in accordance with those of PHHJ321 and very similar to those of LZHH surface,22 although quantitatively different. We have noted that the well of our APES is somewhat deeper, and the barrier of our APES is a little lower than those of the PHHJ3 and LZHH surfaces, which could cause the reaction cross sections based on our APES to become closer to the experimental data in many cases.
IV. QUASI-CLASSICAL TRAJECTORY CALCULATIONS In order to evaluate the APES, we calculate the cross sections of the reaction using the QCT calculations. The QCT program is a modified version of Muckerman’s program, CLASTR,37 which 1489
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Figure 4. Contour plots of the present APES, PHHJ3, and LZHH for NeH2þ (β = 30° and 60°). (a) The present APES for β = 30°; (b) PHHJ3 for β = 30°; (c) LZHH for β = 30°; (d) the present APES for β = 60°; (e) PHHJ3 surface for β = 60°; (f) LZHH for β = 60°. Energies are in electron volts, and distances are in angstroms.
is available as Program No. 229 of the Quantum Chemistry Program Exchange (QCPE). In order to compare our results with experimental data and some theoretical reports for the reaction in the literature, the collision energies used are 0.7, 1.7, and 4.5 eV. The present calculations involve an initial rotational number j = 1 and initial vibrational number v = 0-16 for 0.7 eV, v = 0-14 for 1.7 eV, and v = 0-12 for 4.5 eV. For every reaction condition, a batch of 100 000 trajectories is run, and the integration step size in the trajectories is selected to be 0.1 fs, which guarantees conservation of the total energy and total angular momentum. The trajectories are started at an initial distance of 12.0 Å between the Ne atom and the center of mass of H2þ. Zero point energy correction is realized using the scheme proposed by Gillbert et al.,9 which is the same as that used in previous QCT calculations6 for the reaction. The values of impact parameter bmax, which are tested in each run, range from 1.1 to 5.2 Å. In order to accurately compare our results with those of the previous APESs, we also recalculate the cross sections of the reaction using PHHJ3 and LZHH surfaces. However, frequent unexpected interruptions happen in the calculations for collision energy 4.5 eV using the LZHH surface. It is believed that there are some spurious points in the range of the APES because it is constructed using ab initio energies under 2.8 eV. We avoid the spurious points by varying the bmax repeatedly and obtain the cross sections at last. However, we know that bmax is not changed freely, therefore the LZHH surface is inadequate to investigate the dynamic problem of the collision energy greater than 2.8 eV. The calculation results for collision energies 0.7, 1.7, and 4.5 eV are plotted in Figure 5. We also present the experimental data6 and the previous reaction cross sections calculated using
the QCT and the real wave packet (RWP) methods1 based on PHHJ3. In order to see the deviations clearly, we provide the calculation results and experimental data in the Supporting Information. In the case of Ecol = 0.7 eV, Figure 5a shows that the reaction cross sections of our APES are within the ranges of the previous experimental deviations for v = 0-10. The whole tendency of QCT results based on the present APES are in accordance with that of the experiment data. We also found in Figure 5a that the reaction cross sections of our APES are almost the same as those of the LZHH surface. In contrast, the present QCT results using PHHJ3 obviously deviate from the experimental data when v > 5 as well as the calculation results in ref 6. It is should be noted that the calculation results using PHHJ3 are close to those in ref 6 if we fix the bmax at 3.0 Å. However, it is inadequate because the bmax values are much different for each vibrational state and the maximum one reaches 5.7 Å in the test calculations. The obvious difference between the cross sections of the present calculations and those in ref 6 may be brought about by the different bmax. However, we cannot be sure that it is the only cause because ref 6 did not provide the parameters for the QCT calculations. As for Ecol = 1.7 eV, Figure 5b shows that our results are good agreement with the results based on the LZHH surface. Although the results based on our APES for v = 0, 1, 4, and 9-14 are slightly smaller than the experimental data and those for v = 2, 3, and 5-8 are larger, they remain close to the experimental data and are better than the results based on PHHJ3, except for v = 9 and 10. We also find from Figure 5, there are peaks on the curves of the cross sections. For example, an obvious peak at v = 7 occurs in Figure 5b. The peaks are associated with the different values of the barrier with a different 1490
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on our APES, PHHJ3, and LZHH are at least a factor of 2 smaller than the experimental data, except for v = 0, where it is slightly higher. According to the experiment,6 there is another reaction channel Ne þ H2þ f NeH þ Hþ for Ecol = 4.5 eV. Therefore, only one PES cannot give an accurate result for the Ne þ H2þ reaction because two reaction channels occur at the same time in the large collisional energy range.
V. CONCLUSIONS With the two-state averaged CASSCF orbitals overcoming the spurious energies on the PES, we successfully used the MRCI/ aug-cc-pVQZ method to obtain a large energy grid for the construction of an APES for the Ne þ H2þ f NeHþ þ H reaction. The 142 parameters of the APES are fitted using 31 000 ab initio energy points selected from the calculated grid (32 038 energies). It is found that the main geometric characteristics of the present APES are similar to the ones reported in the literature. The QCT results of LZHH and the present APES are very similar to each other. Moreover, QCT calculations show that the reaction cross sections calculated using the present APES are in rather good agreement (with the exception of what happens at Ecol = 4.5 eV) with experimental values and are much better than the theoretical data based on PHHJ3. It is expected that the new APES can provide an effective basis for future investigations of the dynamics of the Ne þ H2þ f NeHþ þ H reaction. ’ ASSOCIATED CONTENT
bS
Supporting Information. FORTRAN subroutine of APES and first derivatives with respect to the internuclear distances for the Ne þ H2þ f NeHþ þ H reaction. More tables for the parameters and cross sections. This material is available free of charge via the Internet at http://pubs.acs.org.
’ AUTHOR INFORMATION Corresponding Author
*Tel: þ86 535 6672870; fax: þ86 535 6672870; e-mail address:
[email protected].
’ ACKNOWLEDGMENT This work is supported by the National Science Foundation of China under Grant Nos. NSFC-10974078 and NSFC-10674114. Figure 5. Comparison of the theoretical and experimental results. (a) Ecol = 0.7 eV; (b) Ecol = 1.7 eV; (c) Ecol = 4.5 eV. The reaction cross sections are in Å2.
H-H-Ne angle as shown in Figure 2 and Table 1 in the Supporting Information. For v = 7, the total energy 3.3 eV, which includes the collision energy 1.7 eV and the vibrational energy 1.6 eV, is close to the energy of the highest barrier or so for the 30° angle of H-H-Ne. The cross section reaches the maximum value because the highest barrier has exerted effect on the collision. When the total energy further increases along with the increase of the vibrational number, more and more trajectories will directly overreach the highest barrier not react. Therefore, the cross section decreases. One can see from Figure 5c that for Ecol = 4.5 eV all of the theoretical results based
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dx.doi.org/10.1021/jp108922c |J. Phys. Chem. A 2011, 115, 1486–1492
