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inelastic and charge transfer processes
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Chemical Physics Letters 449 (2007) 358–364 www.elsevier.com/locate/cplett
Ab initio study of H+ + H2 collisions: Elastic/inelastic and charge transfer processes A. Saieswari, Sanjay Kumar
*
Department of Chemistry, Indian Institute of Technology Madras, Chennai 600 036, India Received 11 August 2007; in final form 29 October 2007 Available online 1 November 2007
Abstract An ab initio full configuration interaction study has been undertaken to obtain the global potential energy surfaces for the ground and the first excited electronic state of the H+ + H2 system employing Dunning’s cc-pVQZ basis set. Using the ab initio approach the corresponding quasi-diabatic potential energy surfaces and coupling potentials have been obtained. A time-independent quantum mechanical study has been also undertaken for both the inelastic and charge transfer processes at the experimental collision energy Ec.m. = 20.0 eV and the preliminary results show better agreement with the experimental data as compared to the earlier available theoretical studies. 2007 Elsevier B.V. All rights reserved.
1. Introduction The H+ + H2 collision system is an important and fundamental system. Being a two-electron system, it is also one of the simplest ion–molecule system. Yet, the collision dynamics is quite complicated since it involves the ground electronic and low-lying excited electronic potential energy surfaces (PES). The state-resolved experimental data in terms of the differential cross-section (DCS), transition probability and average vibrational energy transfer have become available at center-of-mass (c.m.) collision energy, Ec.m. = 20.0 eV as early as 1987 [1,2] P for the vibrational þ þ 1 þ inelastic channel (VI), H þ H ðX 2 g ; m ¼ 0Þ ! H þ 1 Pþ 0 H2 ðX g ; m ¼ m Þ, and the vibrational charge transþ 1 Pþ fer channel (VCT), H þ H ðX ; m ¼ 0Þ ! Hð2 SÞþ 2 g þ 2 Pþ 00 H2 ðX g ; m ¼ m Þ. State-resolved experimental data (including rotational excitations) also exist but only for the VI at lower collision energies, Ec.m. = 4.67 eV, 6.0 eV and 10 eV [3]. There have been several ab initio studies on the ground electronic state (GS) PES, subsequently improving its qual*
Corresponding author. Fax: +91 44 2257 4202. E-mail address: [email protected] (S. Kumar).
0009-2614/$ - see front matter 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.cplett.2007.10.098
ity [4–12] with global description [5,8,10,12]. In contrast, there have been only few ab initio studies [4,9,13–15] on the excited state(s) mostly representing cuts of the PES(s) in restricted nuclear geometries. Recently, a better ab initio description [9] of the GS and the first excited states (ES) has become available for 680 different nuclear geometries. In absence of global ab initio PESs including the excited states the semi-empirical diatomics-in-molecules (DIM) PESs [16,17] have been utilized to study the VI and VCT processes. The DIM PESs of Preston and Tully [16] provided a good qualitative description but they were not sufficiently accurate. The DIM PESs of Kamisaka–Bian– Nobusada–Nakamura (KBNN) [18] were obtained using ab initio data of Ichihara and Yokoyama [9]. Very recently, ab initio PESs for the GS and the lowest two ESs have been obtained by Viegas et al. [19] using Dunning’s correlation consistent (cc-pV5Z) basis set [20]. From the spectroscopy point of view, there have been many ab initio studies to characterize the Hþ 3 ion in its ground (and excited) electronic states. For relevant details, the reader is referred to Refs. [21,22] and references therein. The early study for the VI and VCT dynamics was performed by Tully and Preston [23] using their DIM PESs [16] employing the trajectory surface hopping approach
A. Saieswari, S. Kumar / Chemical Physics Letters 449 (2007) 358–364
for collision energies below 10 eV. Baer et al. [24] performed an extensive quantum dynamics study under the formulation of vibrational close-coupling rotational infinite-order-sudden approximation (VCC-RIOSA) for both the processes at Ec.m. = 20.0 eV using the DIM PESs [16] and compared their results with those of experiments [1]. Using the improved DIM PESs [17], quantum dynamical calculations have been performed recently by Ushakov et al. [17] for collinear collisions, followed by three-dimensional quantum dynamics study by Kamisaka et al. [18] for total angular momentum, J = 0. Both the studies were performed in the hyperspherical coordinates for low collision energies, Ec.m. < 3.0 eV. It is important to point out here that there have been many dynamics studies for the VI channel using the various available GS PESs (see Ref. [25] and references therein). The present work has been undertaken in view of two important issues. First, there were some marked discrepancies between quantum dynamics calculations [24] using the early version of diabatic DIM PESs [16] and the stateselected experimental data [1]. This comparison was made some 20 years ago. The calculated rainbow maxima were displaced approximately by 3 towards larger angles, and the calculated DCSs showed quite strong strength for small scattering angles (hc.m.) for the VCT channel, where the experiments show mostly a flat behavior or slightly decreasing trend. Second, to obtain yet another but accurate description of quasi-diabatic PESs purely based on ab initio procedures. The dynamics study is mostly performed on diabatic PESs for computational convenience. Although, the adiabatic representation of the PESs is unique, a unique transformation from the adiabatic to a diabatic representation and vice versa does not exist for a multidimensional collision problem. The (quasi)diabatization procedures, their exactness and their associated advantages for numerical computations have been discussed and documented in the literature in detail. A general discussion on it has been recently published [26] (also see references therein). There have also been procedures to obtain quasi-diabatic PESs based on ab initio computations wherein the ‘mixing angle’ (defined in Section 2.2) needed for the unitary transformation matrix is determined from the configuration interaction (CI) coefficients of the adiabatic electronic wavefunctions [27]. An improvement has been suggested in this scheme by determining the diabatic electronic wavefunctions (and the corresponding CI vectors) which vary as little as possible as a function of nuclear coordinates. This procedure was recently applied by Simah et al. [27] for the photodissociation of H2S on electronically coupled PESs explaining successfully the experimental observation. We realize that Viegas et al. [19] have obtained the diabatic PESs very recently by improving the DIM (3 · 3) potential matrix based on their ab initio data, where each diatomic state was dressed with three body potential term and it would be interesting to examine the dynamics on their diabatic PESs. Yet, the ab initio description of the
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quasi-diabatic PESs in the present study would also be worthwhile and desirable for the dynamics study since the quasi-diabatic PESs obtained using different procedures may lead to topological differences. To this end, we have undertaken ab initio study at the full CI (FCI) level using the Dunning’s correlation consistent (cc-pVQZ) basis set [20] to obtain the adiabatic and quasi-diabatic PESs for the GS and the first ES. The Letter is organized as follows: in Section 2 the details of ab initio adiabatic and quasi-diabatic PESs are given. In Section 3, the computed dynamical attributes like total differential cross-section and integral cross-sections have been compared with the previous theoretical results [24] and with the experimental results [1]. A discussion along with conclusion is given in Section 4. 2. Computation details 2.1. Ab initio adiabatic potential energy surfaces Ab initio calculations have been carried out in the Jacobi coordinates, where R is the distance of H+ from the c.m. of H2, r is the interatomic distance of the diatom H2 and c = cos1(R Æ r). The computations have been performed in the C2v (collinear and perpendicular) and Cs point groups (off-collinear) for the singlet spin symmetry employing the Dunning’s cc-pVQZ basis set [20]. The FCI computations were carried out to obtain three-dimensional PESs as a function of r, R and c on the following set of grid points: R = 0.2–5.0 (0.2)a0, 5.0–7.0 (0.4)a0, 7.0–10.0 (1.0)a0 and 10.0–20.0 (2.0)a0; r = 0.6–5.0 (0.1)a0 and c = 0–90 (15). The numbers in parentheses indicate the increment in the stated intervals. The ab initio data are available on request with the corresponding author. In order to compare the quality of our computations, some of the ab initio global minimum energy values for the GS obtained in recent FCI computations are listed in Table 1. Our value is almost equal to that obtained by Aguado et al. [12], but is higher by 7.37 · 104 a.u., 7.37624 · 104 a.u. and by 7.35 · 104 a.u. in comparison with that of Anderson [28], Cencek et al. [21] and Viegas et al. [19], respectively. Since we are primarily interested in collision dynamics, we feel that the presently obtained global PESs would be reasonably accurate.
Table 1 Comparison of global minimum energy of the GS PES obtained in various recent studies E (a.u.)
Reference
Year
1.343835 1.343835624a 1.343100 1.343833 1.343098
Anderson [28] Cencek et al. [21] Aguado et al. [12] Viegas et al. [19] Present
1992 1998 2000 2007 2007
The global minimum corresponds to the equilateral geometry with bond length, Re = 1.65a0. a Accuracy has been reported at the sub-microhartree level.
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The adiabatic PESs (both the GS and the first ES) for c = 0, 45 and 90 as a function of R and r are shown in Fig. 1a. The avoided crossing between these two surfaces is located at r 2.5a0. In Fig. 1a, the adiabatic GS of Hþ 3 þ 1 Pþ corresponds to the asymptotic channel, H þ H2 ðX g Þ, while the first ES corresponds to the asymptotic channel, 2 Pþ Hð2 SÞ þ Hþ 2 ðX g Þ. To further ensure the quality of our adiabatic PESs we have also compared the energy difference (DE) between the GS and the first ES adiabatic PESs at the position of the avoided crossing (r = 2.5a0) for c = 90 as a function of R in Fig. 2 along with the values of Viegas et al. [19] and Kamisaka et al. [18]. DE shows almost an exponential fall as a function of R. The present ab initio data are almost indistinguishable from those of Viegas et al. [19], clearly indicating the reliability of our ab initio PESs. The DE obtained from the KBNN PESs [18] show a similar behavior but it is slightly higher. 2.2. Diabatization For the two-state coupling, the transformation from an adiabatic representation, with electronic wavefunctions wam ,
m = 1, 2, to a diabatic representation characterized by the electronic wavefunctions wdm , m = 1, 2, is achieved by the unitary transformation ! a w1 cos a sin a wd1 ð1Þ ¼ d wa2 sin a cos a w2 where a is the mixing angle describing the mixing between the two adiabatic electronic states and is a function of R b el in the diaand r. Using Eq. (1), the matrix elements of H batic representation are given by: D E b el jwd ¼ V a cos2 a þ V a sin2 a V d11 ¼ wd1 j H ð2Þ 1 1 2 D E b el jwd ¼ V a sin2 a þ V a cos2 a ð3Þ V d22 ¼ wd2 j H 2 1 2 D E b el jwd ¼ ðV a V a Þ cos a sin a ð4Þ V d12 ¼ wd1 j H 2 1 2 where wd1;2 are the electronic wavefunction of the two coupled states in the diabatic representation and their corresponding potential values are given by V d11 and V d22 . V a1;2 are the potential energy values in the adiabatic representation whose corresponding electronic wavefunctions are give
Fig. 1. (a) Adiabatic and (b) quasi-diabatic PESs for the GS and the first ES for c = 0, 45 and 90.
A. Saieswari, S. Kumar / Chemical Physics Letters 449 (2007) 358–364
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Fig. 2. Energy difference (DE) between the GS and the first ES adiabatic electronic states at the positions of the avoided crossing, r = 2.5a0 for c = 90 as a function of R. The ab initio values of the present work (solid circles) are compared with those of Viegas et al. (Ref. [19], open circles). The dashed–dotted curve corresponds to the KBNN PES (Ref. [18]).
by wa1;2 . The coupling between the two state in the diabatic representation is given by V d12 and V d12 ¼ V d21 . As mentioned in Section 1, we have used an ab initio approach to calculate a values from the coefficients of CI vectors of the diabatic electronic wavefunctions which are determined with the condition that they vary as little as possible as a function of nuclear geometry. This condition was met by using the invariance of the FCI energies with respect to unitary transformations among the active orbitals so that the geometry dependence of the orbitals is minimized with a reference geometry r = rref at which the adiabatic and diabatic electronic wavefunctions are identical. The diabatic electronic wavefunction is obtained at a nearby geometry by maximizing the overlap for all the pairs of active orbitals at rref with those at a neighborhood geometry r 0 using the Jacobi rotation technique. The relevant details are given in Ref. [27]. We have used MOLPRO [29] software to compute the mixing angle (a) for the 2 · 2 case, involving the 11A 0 (GS) and the 21A 0 (the first ES) states with rref = 1.4a0 (for a fixed value of c) as a function of r and parametrically dependent on R. A two-dimensional view of the mixing angle is given in Fig. 3 for c = 0, 45 and 90 as a function of R and r. It shows a sigmoidal nature along r for r P 2.5a0. a varies from 0 to p/2 in the asymptotic region, indicating the change in character of the electronic wavefunction at the avoided crossing region. For c = 0, one notices a hump in a for closer approach of H+ towards elongated H2, that is, for r > 3.0a0 and R < 4.0a0. This indicates the involvement of the second ES in these regions
Fig. 3. Mixing angle (a) for c = 0, 45 and 90.
which, however, appears to be less significant for other angular approaches. Once the mixing angle is known as a function of r and R, the diabatic potential matrix defined in Eqs. (2)–(4) can be obtained. The diagonal terms V d11 and V d22 are shown in Fig. 1b for c = 0, 45 and 90. The global minimum of diad batic Hþ 3 ðV 11 Þ PES is found at equilateral triangle configuration at Re = 1.65a0 with E = 1.34262197 a.u. This value is 0.30 kcal/mol higher in energy than the global minimum of the adiabatic GS PES (11A1 state) of the present calculation. It is important to note here that the location of the potential well in the quasi-diabatic GS PES remains unaffected. Therefore, one expects that the location of rainbow maxima in the state-to-state DCSs would not be altered in the diabatic calculations. There exists a direct surface crossing around r 2.5a0 in the quasi-diabatic PESs as it can be seen from Fig. 1b. The asymptotic correlation of the quasi-diabatic PES goes as follows:
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V
d 11 PES
A. Saieswari, S. Kumar / Chemical Physics Letters 449 (2007) 358–364
:
asymptotic correlation Hþ !fH þ 3
þ H2 X
1
þ X
!
g
for all r values V d22 PES :
asymptotic correlation 2 Hþ !fH ð SÞ 3
!
þ Hþ X2 2
for all r values
ð5Þ
þ X g
ð6Þ
3. The quantum scattering calculations The reported quantum scattering calculations have been performed for Ec.m. = 20.0 eV using the VCC-RIOSA approach which has been well documented elsewhere [30,31], and it is expected to work rather well at this collision energy. A 2D-spline interpolation of the (r, R) data for each of the considered c values was used to construct the vibronic coupling matrix elements. The calculations involve 12 vibrational states in the ground electronic diabatic state ðV d11 Þ and eight vibrational states in the excited electronic diabatic state ðV d22 Þ. The total number of partial waves considered was 1200. Rotationally-summed differential cross-sections have been calculated for 0 6 hc.m. 6 75 with a step of 0.5.
The total DCS for the VI channel (summed over the vibrational states of H2, m 0 6 12) and VCT channel 00 (summed over the vibrational states of Hþ 2 , m 6 8) are shown in Fig. 4a along with the experimental results [1]. We have reproduced the comparison of earlier theoretical results with the experiments in Fig. 4b from Ref. [24]. Since absolute cross-sections could not be measured in the experiments [1], the experimental data were normalized to the theoretical values. In Ref. [24], the normalization for the VI channel was done at hc.m. = 10.5 (hLab = 7) for the m 0 = 0 (elastic) channel, and for the VCT channel it was achieved by matching the experimental and theoretical DCSs values at their respective rainbow maxima. We also adopt the same normalization procedures and report the comparison in Fig. 4a. It is important to point out here that the present and the early theoretical calculations predict different sizes of the DCSs, and the difference can be seen on the ordinates, where the numbers represent the power of 10. We have reproduced all the available experimental data (with the error bars) from Ref. [1]. The rainbow maximum for the VI channel is seen at hc.m. 10.5 (Fig. 4a, upper part) for the present calculations and at hc.m. 12.5 (Fig. 4b, upper part) for the calculations of Baer et al. [24] using the DIM PESs [16]. The present theoretical results are displaced by about 1
00 Fig. 4. Total differential cross-section for VI channel, H+ + H2 (m = 0) ! H+ + H2 (m = m 0 ), and VCT channel, Hþ þ H2 ðm ¼ 0Þ ! H þ Hþ 2 ðm ¼ m Þ, at Ec.m. = 20.0 eV. The numbers on the ordinate indicate powers of 10. (a) Comparison between present theoretical results (dotted line) and experimental data (solid line with error bars) [1]. (b) Comparison between earlier theoretical results (dotted line) and experimental data (solid line with error bars) (reproduced from Ref. [24]). In both (a) and (b), the experimental data were normalized. For details, see the text.
A. Saieswari, S. Kumar / Chemical Physics Letters 449 (2007) 358–364
towards larger angles (hc.m.(expt) 9.8) whereas those of Baer et al. [24] are displaced by about 2.5 towards larger angles. For the VCT channel, the rainbow maximum is located at hc.m. 11.5 for the present work (Fig. 4a, lower part), which is 1 displaced towards the larger angle (hc.m.(expt) 10.5). In contrast, the rainbow maximum of Baer et al. [24] is located around at hc.m. 13.5 (Fig. 4b, lower part), which is displaced by 3 towards larger angles. The resolution in the scattering angle (DhLab) was not reported in the experiments [1]. The experimental set up for this system was reported to be similar to that used earlier for the H+ + O2 system where DhLab was 0.5 [32]. Assuming that DhLab would be approximately the same in the measurements of the present system, the agreement between theory and experiment is quite good. The discrepancy seen at small hc.m. in the VCT channel is discussed in the next section. The experimental data for the DCS were available for the angular range 0 6 hLab 6 18. Beyond 18 the DCSs fall off very rapidly. Therefore, it was conjectured that their contribution towards the integral cross-section (r) was very small [24]. Since, the experimental r values were obtained from the DCS values in the range 0 6 hLab 6 18 the experimental (incomplete) r values were normalized to the theoretical values. In Fig. 5, the experimental, the present and the earlier theoretical values are normalized at m 0 = 0 and m00 = 0 for the VI and the VCT channels,
Fig. 5. Integral cross-section for the (a) VI channel and (b) VCT channel at Ec.m. = 20.0 eV. The present results (solid line) are compared with the experimental results (dotted line) [1] and with the results of Baer et al. [24] (dashed line). The numbers on the ordinate indicate the powers of 10. See the text.
363
respectively. Both the theoretical results agree with the experimental results quite well showing a rapid decrease in cross-sections for higher vibrational states in both the channels. 4. Discussion and conclusion New global ab initio FCI PESs for the GS and the first ES of the H+ + H2 system have been obtained in the Jacobi coordinates. The corresponding quasi-diabatic PESs have also been obtained using the ab initio approach. A timeindependent quantum mechanical study within the VCCRIOSA framework has been performed at Ec.m. = 20.0 eV using the presently obtained quasi-diabatic PESs. For both the VI and VCT channels the present study shows better results against the earlier theoretical study and an improved agreement with experiments reproducing the results almost within the experimental error bars in terms of: (i) the locations of the rainbow maximum, and (ii) the overall behavior of DCS. The improvement is achieved mainly because of two reasons: (i) the ab initio computations yield quite accurate PESs and the location of the global minimum remains unaltered in the quasi-diabatic PESs, and (ii) the long range interaction potential is properly accounted in terms of charge–polarizability and charge– quadrupole interactions. While the former is related with the location of the rainbow maximum, the latter is mostly responsible for the behavior of DCS at small scattering angles. The present calculations show marked improvement in the DCS for the VCT channel, almost reproducing the experimental results for 3 6 hc.m. 6 20. Yet, the discrepancy remains for hc.m. < 3 where the experiments shows a decreasing trend in the DCS. In contrast, the theoretical predictions for the VI channel for small scattering angles agree very well with the experiments. One may like to count on various possibilities to improve the theoretical predictions against this particular discrepancy in the VCT channel. In comparison with the recently obtained ab initio PESs of Viegas et al. [19] (see Fig. 2 and Table 1) the present ab initio PESs can be considered to be quite accurate for the dynamics. Therefore, we believe that the accuracy of the present PESs and the applicability of the VCC-RIOSA for quantum dynamics may not be the case for this discrepancy since the collision attributes are well reproduced in other scattering regions. The other possibility to improve the theoretical prediction is to further include the second ES in the dynamics calculations. Presumably, it will not have significant role to play since it lies high in energy. Also, it would be interesting to include and examine the role of dissociation channels in the dynamics. Yet, another possibility is to include the reactive channel which, however, has a very low probability at these collision energies and can be completely neglected [1,24]. An elaborate quantum dynamics study on the newly obtained diabatic PESs comparing state-to-state DCSs and other collision attributes with those of experiments for both the channels is being reported elsewhere [33].
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However, in view of low detection efficiency of H-atom and a fairly small magnitude of VCT cross-section (approximately three orders of magnitude less as compared to that of the VI channel and smaller than the background signal which peaked at hLab = 0) it was noted in the experiments [1] that the H-atom angular distribution could be somewhat uncertain below hLab = 2 (hc.m. = 3). A new experimental study would be also worthwhile and desirable in this regard. Acknowledgements This study was supported by a grant from Department of Science and Technology, New Delhi under the fast track scheme. The financial assistance by IIT Madras in procuring MOLPRO software is also gratefully acknowledged. References [1] G. Niedner, M. Noll, J.P. Toennies, Ch. Schlier, J. Chem. Phys. 87 (1987) 2685. [2] G. Niedner-Schatteburg, J.P. Toennies, Adv. Chem. Phys. LXXXII (1992) 553. [3] H. Schmidt, V. Hermann, F. Linder, J. Chem. Phys. 69 (1978) 2734. [4] C.W. Bauschlicher Jr., S.V. O’Neil, R.K. Preston, H.F. Schaefer III., C.F. Bender, J. Chem. Phys. 59 (1973) 1286. [5] C.F. Giese, W.R. Gentry, Phys. Rev. A 10 (1974) 2156. [6] I.G. Csizmadia, R.E. Kari, J.C. Polanyi, A.C. Roach, M.A. Robb, J. Chem. Phys. 52 (1970) 6205. [7] G.D. Carney, R.N. Porter, J. Chem. Phys. 60 (1974) 4251. [8] R. Schinke, M. Dupuis, W.A. Lester Jr., J. Chem. Phys. 72 (1980) 3909. [9] A. Ichihara, K. Yokoyama, J. Chem. Phys. 103 (1995) 2109.
[10] R. Prosmiti, O.L. Polyansky, J. Tennyson, Chem. Phys. Lett. 273 (1997) 107. [11] O.L. Polyansky, R. Prosmiti, W. Klopper, J. Tennyson, Mol. Phys. 98 (2000) 261. [12] A. Aguado, O. Roncero, C. Tablero, C. Sanz, M. Paniagua, J. Chem. Phys. 112 (2000) 1240. [13] D. Kawaoka, R.F. Borkman, J. Chem. Phys. 54 (1971) 4234. [14] L.J. Schaad, W.V. Hicks, J. Chem. Phys. 61 (1974) 1934. [15] D. Talbi, R.P. Saxon, J. Chem. Phys. 89 (1988) 2235. [16] R.K. Preston, J.C. Tully, J. Chem. Phys. 54 (1971) 4297. [17] V.G. Ushakov, K. Nobusada, V.I. Osherov, Phys. Chem. Chem. Phys. 3 (2001) 63. [18] H. Kamisaka, W. Bian, K. Nobusada, H. Nakamura, J. Chem. Phys. 116 (2002) 654. [19] L.P. Viegas, A. Alijah, A.J.C. Varandas, J. Chem. Phys. 126 (2007) 074309. [20] T.H. Dunning Jr., J. Chem. Phys. 90 (1989) 1007. [21] W. Cencek, J. Rychlewski, R. Jaquet, W. Kutzelnigg, J. Chem. Phys. 108 (1998) 2831. [22] W. Kutzelnigg, R. Jaquet, Philos. Trans. Roy. Soc. Lond. Ser. A 364 (2006) 2855. [23] J.C. Tully, R.K. Preston, J. Chem. Phys. 55 (1971) 562. [24] M. Baer, G. Niedner-Schatteburg, J.P. Toennies, J. Chem. Phys. 91 (1989) 4169. [25] F.A. Gianturco, S. Kumar, Chem. Phys. 196 (1995) 2109. [26] Faraday Discuss. 127 (2004) 81, general discussion. [27] D. Simah, B. Hartke, H.-J. Werner, J. Chem. Phys. 111 (1999) 4523, and references therein. [28] J.B. Anderson, J. Chem. Phys. 96 (1992) 3702, and related references. [29] MOLPRO is a package of ab initio programs written by H.-J. Werner et al. [30] R. Schinke, P. McGuire, Chem. Phys. 31 (1978) 391. [31] F.A. Gianturco, A. Palma, E. Semprini, F. Stefani, M. Baer, Phys. Rev. A 42 (1990) 3926. [32] M. Noll, J.P. Toennies, J. Chem. Phys. 85 (1986) 3313. [33] S. Amaran, S. Kumar, J. Chem. Phys. 127 (2007), in press.
Chemical Physics Letters 449 (2007) 358–364 www.elsevier.com/locate/cplett
Ab initio study of H+ + H2 collisions: Elastic/inelastic and charge transfer processes A. Saieswari, Sanjay Kumar
*
Department of Chemistry, Indian Institute of Technology Madras, Chennai 600 036, India Received 11 August 2007; in final form 29 October 2007 Available online 1 November 2007
Abstract An ab initio full configuration interaction study has been undertaken to obtain the global potential energy surfaces for the ground and the first excited electronic state of the H+ + H2 system employing Dunning’s cc-pVQZ basis set. Using the ab initio approach the corresponding quasi-diabatic potential energy surfaces and coupling potentials have been obtained. A time-independent quantum mechanical study has been also undertaken for both the inelastic and charge transfer processes at the experimental collision energy Ec.m. = 20.0 eV and the preliminary results show better agreement with the experimental data as compared to the earlier available theoretical studies. 2007 Elsevier B.V. All rights reserved.
1. Introduction The H+ + H2 collision system is an important and fundamental system. Being a two-electron system, it is also one of the simplest ion–molecule system. Yet, the collision dynamics is quite complicated since it involves the ground electronic and low-lying excited electronic potential energy surfaces (PES). The state-resolved experimental data in terms of the differential cross-section (DCS), transition probability and average vibrational energy transfer have become available at center-of-mass (c.m.) collision energy, Ec.m. = 20.0 eV as early as 1987 [1,2] P for the vibrational þ þ 1 þ inelastic channel (VI), H þ H ðX 2 g ; m ¼ 0Þ ! H þ 1 Pþ 0 H2 ðX g ; m ¼ m Þ, and the vibrational charge transþ 1 Pþ fer channel (VCT), H þ H ðX ; m ¼ 0Þ ! Hð2 SÞþ 2 g þ 2 Pþ 00 H2 ðX g ; m ¼ m Þ. State-resolved experimental data (including rotational excitations) also exist but only for the VI at lower collision energies, Ec.m. = 4.67 eV, 6.0 eV and 10 eV [3]. There have been several ab initio studies on the ground electronic state (GS) PES, subsequently improving its qual*
Corresponding author. Fax: +91 44 2257 4202. E-mail address: [email protected] (S. Kumar).
0009-2614/$ - see front matter 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.cplett.2007.10.098
ity [4–12] with global description [5,8,10,12]. In contrast, there have been only few ab initio studies [4,9,13–15] on the excited state(s) mostly representing cuts of the PES(s) in restricted nuclear geometries. Recently, a better ab initio description [9] of the GS and the first excited states (ES) has become available for 680 different nuclear geometries. In absence of global ab initio PESs including the excited states the semi-empirical diatomics-in-molecules (DIM) PESs [16,17] have been utilized to study the VI and VCT processes. The DIM PESs of Preston and Tully [16] provided a good qualitative description but they were not sufficiently accurate. The DIM PESs of Kamisaka–Bian– Nobusada–Nakamura (KBNN) [18] were obtained using ab initio data of Ichihara and Yokoyama [9]. Very recently, ab initio PESs for the GS and the lowest two ESs have been obtained by Viegas et al. [19] using Dunning’s correlation consistent (cc-pV5Z) basis set [20]. From the spectroscopy point of view, there have been many ab initio studies to characterize the Hþ 3 ion in its ground (and excited) electronic states. For relevant details, the reader is referred to Refs. [21,22] and references therein. The early study for the VI and VCT dynamics was performed by Tully and Preston [23] using their DIM PESs [16] employing the trajectory surface hopping approach
A. Saieswari, S. Kumar / Chemical Physics Letters 449 (2007) 358–364
for collision energies below 10 eV. Baer et al. [24] performed an extensive quantum dynamics study under the formulation of vibrational close-coupling rotational infinite-order-sudden approximation (VCC-RIOSA) for both the processes at Ec.m. = 20.0 eV using the DIM PESs [16] and compared their results with those of experiments [1]. Using the improved DIM PESs [17], quantum dynamical calculations have been performed recently by Ushakov et al. [17] for collinear collisions, followed by three-dimensional quantum dynamics study by Kamisaka et al. [18] for total angular momentum, J = 0. Both the studies were performed in the hyperspherical coordinates for low collision energies, Ec.m. < 3.0 eV. It is important to point out here that there have been many dynamics studies for the VI channel using the various available GS PESs (see Ref. [25] and references therein). The present work has been undertaken in view of two important issues. First, there were some marked discrepancies between quantum dynamics calculations [24] using the early version of diabatic DIM PESs [16] and the stateselected experimental data [1]. This comparison was made some 20 years ago. The calculated rainbow maxima were displaced approximately by 3 towards larger angles, and the calculated DCSs showed quite strong strength for small scattering angles (hc.m.) for the VCT channel, where the experiments show mostly a flat behavior or slightly decreasing trend. Second, to obtain yet another but accurate description of quasi-diabatic PESs purely based on ab initio procedures. The dynamics study is mostly performed on diabatic PESs for computational convenience. Although, the adiabatic representation of the PESs is unique, a unique transformation from the adiabatic to a diabatic representation and vice versa does not exist for a multidimensional collision problem. The (quasi)diabatization procedures, their exactness and their associated advantages for numerical computations have been discussed and documented in the literature in detail. A general discussion on it has been recently published [26] (also see references therein). There have also been procedures to obtain quasi-diabatic PESs based on ab initio computations wherein the ‘mixing angle’ (defined in Section 2.2) needed for the unitary transformation matrix is determined from the configuration interaction (CI) coefficients of the adiabatic electronic wavefunctions [27]. An improvement has been suggested in this scheme by determining the diabatic electronic wavefunctions (and the corresponding CI vectors) which vary as little as possible as a function of nuclear coordinates. This procedure was recently applied by Simah et al. [27] for the photodissociation of H2S on electronically coupled PESs explaining successfully the experimental observation. We realize that Viegas et al. [19] have obtained the diabatic PESs very recently by improving the DIM (3 · 3) potential matrix based on their ab initio data, where each diatomic state was dressed with three body potential term and it would be interesting to examine the dynamics on their diabatic PESs. Yet, the ab initio description of the
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quasi-diabatic PESs in the present study would also be worthwhile and desirable for the dynamics study since the quasi-diabatic PESs obtained using different procedures may lead to topological differences. To this end, we have undertaken ab initio study at the full CI (FCI) level using the Dunning’s correlation consistent (cc-pVQZ) basis set [20] to obtain the adiabatic and quasi-diabatic PESs for the GS and the first ES. The Letter is organized as follows: in Section 2 the details of ab initio adiabatic and quasi-diabatic PESs are given. In Section 3, the computed dynamical attributes like total differential cross-section and integral cross-sections have been compared with the previous theoretical results [24] and with the experimental results [1]. A discussion along with conclusion is given in Section 4. 2. Computation details 2.1. Ab initio adiabatic potential energy surfaces Ab initio calculations have been carried out in the Jacobi coordinates, where R is the distance of H+ from the c.m. of H2, r is the interatomic distance of the diatom H2 and c = cos1(R Æ r). The computations have been performed in the C2v (collinear and perpendicular) and Cs point groups (off-collinear) for the singlet spin symmetry employing the Dunning’s cc-pVQZ basis set [20]. The FCI computations were carried out to obtain three-dimensional PESs as a function of r, R and c on the following set of grid points: R = 0.2–5.0 (0.2)a0, 5.0–7.0 (0.4)a0, 7.0–10.0 (1.0)a0 and 10.0–20.0 (2.0)a0; r = 0.6–5.0 (0.1)a0 and c = 0–90 (15). The numbers in parentheses indicate the increment in the stated intervals. The ab initio data are available on request with the corresponding author. In order to compare the quality of our computations, some of the ab initio global minimum energy values for the GS obtained in recent FCI computations are listed in Table 1. Our value is almost equal to that obtained by Aguado et al. [12], but is higher by 7.37 · 104 a.u., 7.37624 · 104 a.u. and by 7.35 · 104 a.u. in comparison with that of Anderson [28], Cencek et al. [21] and Viegas et al. [19], respectively. Since we are primarily interested in collision dynamics, we feel that the presently obtained global PESs would be reasonably accurate.
Table 1 Comparison of global minimum energy of the GS PES obtained in various recent studies E (a.u.)
Reference
Year
1.343835 1.343835624a 1.343100 1.343833 1.343098
Anderson [28] Cencek et al. [21] Aguado et al. [12] Viegas et al. [19] Present
1992 1998 2000 2007 2007
The global minimum corresponds to the equilateral geometry with bond length, Re = 1.65a0. a Accuracy has been reported at the sub-microhartree level.
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The adiabatic PESs (both the GS and the first ES) for c = 0, 45 and 90 as a function of R and r are shown in Fig. 1a. The avoided crossing between these two surfaces is located at r 2.5a0. In Fig. 1a, the adiabatic GS of Hþ 3 þ 1 Pþ corresponds to the asymptotic channel, H þ H2 ðX g Þ, while the first ES corresponds to the asymptotic channel, 2 Pþ Hð2 SÞ þ Hþ 2 ðX g Þ. To further ensure the quality of our adiabatic PESs we have also compared the energy difference (DE) between the GS and the first ES adiabatic PESs at the position of the avoided crossing (r = 2.5a0) for c = 90 as a function of R in Fig. 2 along with the values of Viegas et al. [19] and Kamisaka et al. [18]. DE shows almost an exponential fall as a function of R. The present ab initio data are almost indistinguishable from those of Viegas et al. [19], clearly indicating the reliability of our ab initio PESs. The DE obtained from the KBNN PESs [18] show a similar behavior but it is slightly higher. 2.2. Diabatization For the two-state coupling, the transformation from an adiabatic representation, with electronic wavefunctions wam ,
m = 1, 2, to a diabatic representation characterized by the electronic wavefunctions wdm , m = 1, 2, is achieved by the unitary transformation ! a w1 cos a sin a wd1 ð1Þ ¼ d wa2 sin a cos a w2 where a is the mixing angle describing the mixing between the two adiabatic electronic states and is a function of R b el in the diaand r. Using Eq. (1), the matrix elements of H batic representation are given by: D E b el jwd ¼ V a cos2 a þ V a sin2 a V d11 ¼ wd1 j H ð2Þ 1 1 2 D E b el jwd ¼ V a sin2 a þ V a cos2 a ð3Þ V d22 ¼ wd2 j H 2 1 2 D E b el jwd ¼ ðV a V a Þ cos a sin a ð4Þ V d12 ¼ wd1 j H 2 1 2 where wd1;2 are the electronic wavefunction of the two coupled states in the diabatic representation and their corresponding potential values are given by V d11 and V d22 . V a1;2 are the potential energy values in the adiabatic representation whose corresponding electronic wavefunctions are give
Fig. 1. (a) Adiabatic and (b) quasi-diabatic PESs for the GS and the first ES for c = 0, 45 and 90.
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Fig. 2. Energy difference (DE) between the GS and the first ES adiabatic electronic states at the positions of the avoided crossing, r = 2.5a0 for c = 90 as a function of R. The ab initio values of the present work (solid circles) are compared with those of Viegas et al. (Ref. [19], open circles). The dashed–dotted curve corresponds to the KBNN PES (Ref. [18]).
by wa1;2 . The coupling between the two state in the diabatic representation is given by V d12 and V d12 ¼ V d21 . As mentioned in Section 1, we have used an ab initio approach to calculate a values from the coefficients of CI vectors of the diabatic electronic wavefunctions which are determined with the condition that they vary as little as possible as a function of nuclear geometry. This condition was met by using the invariance of the FCI energies with respect to unitary transformations among the active orbitals so that the geometry dependence of the orbitals is minimized with a reference geometry r = rref at which the adiabatic and diabatic electronic wavefunctions are identical. The diabatic electronic wavefunction is obtained at a nearby geometry by maximizing the overlap for all the pairs of active orbitals at rref with those at a neighborhood geometry r 0 using the Jacobi rotation technique. The relevant details are given in Ref. [27]. We have used MOLPRO [29] software to compute the mixing angle (a) for the 2 · 2 case, involving the 11A 0 (GS) and the 21A 0 (the first ES) states with rref = 1.4a0 (for a fixed value of c) as a function of r and parametrically dependent on R. A two-dimensional view of the mixing angle is given in Fig. 3 for c = 0, 45 and 90 as a function of R and r. It shows a sigmoidal nature along r for r P 2.5a0. a varies from 0 to p/2 in the asymptotic region, indicating the change in character of the electronic wavefunction at the avoided crossing region. For c = 0, one notices a hump in a for closer approach of H+ towards elongated H2, that is, for r > 3.0a0 and R < 4.0a0. This indicates the involvement of the second ES in these regions
Fig. 3. Mixing angle (a) for c = 0, 45 and 90.
which, however, appears to be less significant for other angular approaches. Once the mixing angle is known as a function of r and R, the diabatic potential matrix defined in Eqs. (2)–(4) can be obtained. The diagonal terms V d11 and V d22 are shown in Fig. 1b for c = 0, 45 and 90. The global minimum of diad batic Hþ 3 ðV 11 Þ PES is found at equilateral triangle configuration at Re = 1.65a0 with E = 1.34262197 a.u. This value is 0.30 kcal/mol higher in energy than the global minimum of the adiabatic GS PES (11A1 state) of the present calculation. It is important to note here that the location of the potential well in the quasi-diabatic GS PES remains unaffected. Therefore, one expects that the location of rainbow maxima in the state-to-state DCSs would not be altered in the diabatic calculations. There exists a direct surface crossing around r 2.5a0 in the quasi-diabatic PESs as it can be seen from Fig. 1b. The asymptotic correlation of the quasi-diabatic PES goes as follows:
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V
d 11 PES
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:
asymptotic correlation Hþ !fH þ 3
þ H2 X
1
þ X
!
g
for all r values V d22 PES :
asymptotic correlation 2 Hþ !fH ð SÞ 3
!
þ Hþ X2 2
for all r values
ð5Þ
þ X g
ð6Þ
3. The quantum scattering calculations The reported quantum scattering calculations have been performed for Ec.m. = 20.0 eV using the VCC-RIOSA approach which has been well documented elsewhere [30,31], and it is expected to work rather well at this collision energy. A 2D-spline interpolation of the (r, R) data for each of the considered c values was used to construct the vibronic coupling matrix elements. The calculations involve 12 vibrational states in the ground electronic diabatic state ðV d11 Þ and eight vibrational states in the excited electronic diabatic state ðV d22 Þ. The total number of partial waves considered was 1200. Rotationally-summed differential cross-sections have been calculated for 0 6 hc.m. 6 75 with a step of 0.5.
The total DCS for the VI channel (summed over the vibrational states of H2, m 0 6 12) and VCT channel 00 (summed over the vibrational states of Hþ 2 , m 6 8) are shown in Fig. 4a along with the experimental results [1]. We have reproduced the comparison of earlier theoretical results with the experiments in Fig. 4b from Ref. [24]. Since absolute cross-sections could not be measured in the experiments [1], the experimental data were normalized to the theoretical values. In Ref. [24], the normalization for the VI channel was done at hc.m. = 10.5 (hLab = 7) for the m 0 = 0 (elastic) channel, and for the VCT channel it was achieved by matching the experimental and theoretical DCSs values at their respective rainbow maxima. We also adopt the same normalization procedures and report the comparison in Fig. 4a. It is important to point out here that the present and the early theoretical calculations predict different sizes of the DCSs, and the difference can be seen on the ordinates, where the numbers represent the power of 10. We have reproduced all the available experimental data (with the error bars) from Ref. [1]. The rainbow maximum for the VI channel is seen at hc.m. 10.5 (Fig. 4a, upper part) for the present calculations and at hc.m. 12.5 (Fig. 4b, upper part) for the calculations of Baer et al. [24] using the DIM PESs [16]. The present theoretical results are displaced by about 1
00 Fig. 4. Total differential cross-section for VI channel, H+ + H2 (m = 0) ! H+ + H2 (m = m 0 ), and VCT channel, Hþ þ H2 ðm ¼ 0Þ ! H þ Hþ 2 ðm ¼ m Þ, at Ec.m. = 20.0 eV. The numbers on the ordinate indicate powers of 10. (a) Comparison between present theoretical results (dotted line) and experimental data (solid line with error bars) [1]. (b) Comparison between earlier theoretical results (dotted line) and experimental data (solid line with error bars) (reproduced from Ref. [24]). In both (a) and (b), the experimental data were normalized. For details, see the text.
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towards larger angles (hc.m.(expt) 9.8) whereas those of Baer et al. [24] are displaced by about 2.5 towards larger angles. For the VCT channel, the rainbow maximum is located at hc.m. 11.5 for the present work (Fig. 4a, lower part), which is 1 displaced towards the larger angle (hc.m.(expt) 10.5). In contrast, the rainbow maximum of Baer et al. [24] is located around at hc.m. 13.5 (Fig. 4b, lower part), which is displaced by 3 towards larger angles. The resolution in the scattering angle (DhLab) was not reported in the experiments [1]. The experimental set up for this system was reported to be similar to that used earlier for the H+ + O2 system where DhLab was 0.5 [32]. Assuming that DhLab would be approximately the same in the measurements of the present system, the agreement between theory and experiment is quite good. The discrepancy seen at small hc.m. in the VCT channel is discussed in the next section. The experimental data for the DCS were available for the angular range 0 6 hLab 6 18. Beyond 18 the DCSs fall off very rapidly. Therefore, it was conjectured that their contribution towards the integral cross-section (r) was very small [24]. Since, the experimental r values were obtained from the DCS values in the range 0 6 hLab 6 18 the experimental (incomplete) r values were normalized to the theoretical values. In Fig. 5, the experimental, the present and the earlier theoretical values are normalized at m 0 = 0 and m00 = 0 for the VI and the VCT channels,
Fig. 5. Integral cross-section for the (a) VI channel and (b) VCT channel at Ec.m. = 20.0 eV. The present results (solid line) are compared with the experimental results (dotted line) [1] and with the results of Baer et al. [24] (dashed line). The numbers on the ordinate indicate the powers of 10. See the text.
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respectively. Both the theoretical results agree with the experimental results quite well showing a rapid decrease in cross-sections for higher vibrational states in both the channels. 4. Discussion and conclusion New global ab initio FCI PESs for the GS and the first ES of the H+ + H2 system have been obtained in the Jacobi coordinates. The corresponding quasi-diabatic PESs have also been obtained using the ab initio approach. A timeindependent quantum mechanical study within the VCCRIOSA framework has been performed at Ec.m. = 20.0 eV using the presently obtained quasi-diabatic PESs. For both the VI and VCT channels the present study shows better results against the earlier theoretical study and an improved agreement with experiments reproducing the results almost within the experimental error bars in terms of: (i) the locations of the rainbow maximum, and (ii) the overall behavior of DCS. The improvement is achieved mainly because of two reasons: (i) the ab initio computations yield quite accurate PESs and the location of the global minimum remains unaltered in the quasi-diabatic PESs, and (ii) the long range interaction potential is properly accounted in terms of charge–polarizability and charge– quadrupole interactions. While the former is related with the location of the rainbow maximum, the latter is mostly responsible for the behavior of DCS at small scattering angles. The present calculations show marked improvement in the DCS for the VCT channel, almost reproducing the experimental results for 3 6 hc.m. 6 20. Yet, the discrepancy remains for hc.m. < 3 where the experiments shows a decreasing trend in the DCS. In contrast, the theoretical predictions for the VI channel for small scattering angles agree very well with the experiments. One may like to count on various possibilities to improve the theoretical predictions against this particular discrepancy in the VCT channel. In comparison with the recently obtained ab initio PESs of Viegas et al. [19] (see Fig. 2 and Table 1) the present ab initio PESs can be considered to be quite accurate for the dynamics. Therefore, we believe that the accuracy of the present PESs and the applicability of the VCC-RIOSA for quantum dynamics may not be the case for this discrepancy since the collision attributes are well reproduced in other scattering regions. The other possibility to improve the theoretical prediction is to further include the second ES in the dynamics calculations. Presumably, it will not have significant role to play since it lies high in energy. Also, it would be interesting to include and examine the role of dissociation channels in the dynamics. Yet, another possibility is to include the reactive channel which, however, has a very low probability at these collision energies and can be completely neglected [1,24]. An elaborate quantum dynamics study on the newly obtained diabatic PESs comparing state-to-state DCSs and other collision attributes with those of experiments for both the channels is being reported elsewhere [33].
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However, in view of low detection efficiency of H-atom and a fairly small magnitude of VCT cross-section (approximately three orders of magnitude less as compared to that of the VI channel and smaller than the background signal which peaked at hLab = 0) it was noted in the experiments [1] that the H-atom angular distribution could be somewhat uncertain below hLab = 2 (hc.m. = 3). A new experimental study would be also worthwhile and desirable in this regard. Acknowledgements This study was supported by a grant from Department of Science and Technology, New Delhi under the fast track scheme. The financial assistance by IIT Madras in procuring MOLPRO software is also gratefully acknowledged. References [1] G. Niedner, M. Noll, J.P. Toennies, Ch. Schlier, J. Chem. Phys. 87 (1987) 2685. [2] G. Niedner-Schatteburg, J.P. Toennies, Adv. Chem. Phys. LXXXII (1992) 553. [3] H. Schmidt, V. Hermann, F. Linder, J. Chem. Phys. 69 (1978) 2734. [4] C.W. Bauschlicher Jr., S.V. O’Neil, R.K. Preston, H.F. Schaefer III., C.F. Bender, J. Chem. Phys. 59 (1973) 1286. [5] C.F. Giese, W.R. Gentry, Phys. Rev. A 10 (1974) 2156. [6] I.G. Csizmadia, R.E. Kari, J.C. Polanyi, A.C. Roach, M.A. Robb, J. Chem. Phys. 52 (1970) 6205. [7] G.D. Carney, R.N. Porter, J. Chem. Phys. 60 (1974) 4251. [8] R. Schinke, M. Dupuis, W.A. Lester Jr., J. Chem. Phys. 72 (1980) 3909. [9] A. Ichihara, K. Yokoyama, J. Chem. Phys. 103 (1995) 2109.
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