Ab initio ground and the first excited adiabatic and quasidiabatic potential energy surfaces of H+ + CO system

Chemical Physics 373 (2010) 211–218

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Ab initio ground and the first excited adiabatic and quasidiabatic potential energy surfaces of H+ + CO system F. George D.X., Sanjay Kumar * Department of Chemistry, Indian Institute of Technology Madras, Chennai 600 036, India

a r t i c l e

i n f o

Article history: Received 21 January 2010 In final form 12 May 2010 Available online 20 May 2010 Dedicated to Professor H. Köppel on his 60th birthday. Keywords: Ab initio potential energy surfaces Nonadiabatic coupling matrix elements Quasidiabatization Coupling potential

a b s t r a c t Ab initio global adiabatic as well as quasidiabatic potential energy surfaces for the ground and the first excited electronic states of the H+ + CO system have been computed as a function of the Jacobi coordinates (R, r, c) using Dunning’s cc-pVTZ basis set at the internally contracted multi-reference (single and double) configuration interaction level of accuracy. In addition, nonadiabatic coupling matrix elements arising from radial motion, mixing angle and coupling potential have been computed using the ab initio procedure [Simah et al. (1999) [66]] for the purpose of dynamics study. The geometrical properties corresponding to the minimum energy of the bound HCO+ and HOC+ isomers have been obtained and compared with those predicted by previous theoretical and experimental results. The HCO+ has been found to be more stable than the HOC+. The minimum energy pathway in the ground electronic state for the isomerization process, HCO+ HOC+ has also been obtained as a function of c. Ó 2010 Elsevier B.V. All rights reserved.

1. Introduction Proton–molecule interactions are important in several areas of chemical physics and astrophysics. It had been pointed out [1] that the solar flare, which mostly consists of protons, can produce nitric oxide by various interactions in the stratosphere which can deplete the stratospheric ozone up to 15%. These protons enter the Earth’s ionosphere with kinetic energy (KE) in the range 1 keV, and loose most of it by several inelastic collision processes. They reach the stratosphere with KE in the range 0–50 eV and interact with the diatomic and polyatomic molecules present there, thereby disturbing the existing equilibria. In order to understand how their KEs are transformed into the internal modes of the molecules, leading to inelastic vibrational–rotational and charge-transfer excitations several experiments have been performed over past three decades [2–8] based on molecular beams and proton energy-loss spectroscopy. An intriguing observation of selective vibrational excitation [3,4,8] has been made; In similar collision energy range on the collisions of H+ on diatomic molecules, the amount of vibrational excitation in O2 is the largest followed by that observed in H2 and it is very low and almost similar for N2, CO and NO. One also observes mode-selective vibrational excitation patterns in the case of polyatomic molecules. While accurate/approximate information on the global GS and the first ES PESs and their nonadiabatic interactions have become available for the H+ + H2 (for a brief review see Refs. [9–11], and * Corresponding author. Fax.: +91 44 2257 4202. E-mail address: [email protected] (S. Kumar). 0301-0104/$ - see front matter Ó 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.chemphys.2010.05.012

references cited therein), the H+ + O2 [12–16], the H+ + N2 [17,18], the H+ + NO [19] systems they are still lacking for the H+ + CO system, except for some preliminary available ab initio data which are barely helpful for the dynamics study. Therefore, the present study focuses on the ab initio computations of the GS and the first ES PESs (and their nonadiabatic interactions) of the H+ + CO system. It is worth pointing out here that the H+ + CO is also an important astrophysical system ever since it was proposed [20] that the bound molecular HCO+ and HOC+ ions could be the likely source of an unidentified microwave line observed [21] from the interstellar space. The experimental observation of the rotational spectrum of the HCO+ in the laboratory using the microwave technique [22] confirmed its presence in the interstellar media, and perhaps it was the first polyatomic ion to be detected in the outer space. Since then, a number of ab initio structural calculations have been performed near the interaction well [23–29] of the GS of the H+ + CO system to characterize the equilibrium structure of the [HCO]+ ion. Interestingly, early ab initio study [30] on proton affinity of CO had predicted the existence of [HCO]+ ion in the collinear geometry, which also got confirmed in subsequent refined ab initio studies [31,32]. There have been a few early ab initio calculations [32] at the self-consistent field (SCF) and restricted configuration interaction (CI) level of accuracies to construct the potential energy curves (PEC) for the system for the GS as well as the low-lying ESs as a function of proton approaches for the collinear and a few non-collinear geometries with CO fixed at its equilibrium geometry, req. Subsequently, treating the CO molecule as a rigid rotor, the GS PES was computed at the SCF level [33].

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Fig. 1. Jacobi coordinates.

Recently, a three dimensional global ab initio PES for the GS (in the Jacobi coordinates, R, r, c, see Fig. 1) of the system was obtained in our group and quantum dynamics was performed [34] only for the inelastic vibrational excitations (IVE) of CO. Very recently, studies of the charge transfer (CT) dynamics was carried out in our group [35] in the restricted (r fixed at its equilibrium values, req = 2.13a) geometry involving the GS and the three lowest ES PECs as a function of R for c = 180° and including only the GS and the first ES PECs as a function of R for c = 0°, 90° and 180° at the collision energy, Ecoll = 28.96 eV. The CT dynamics (electron capture) has also been studied in the system at very high collision energies at Ecoll = 10 keV/u [36] and Ecoll = 0.5–1.0 keV/u [37] using the PECs for the GS and the first ES with CO fixed at req for the collinear and the perpendicular approaches of H+. The focus of the present study is to compute the global ab initio adiabatic PESs in the Jacobi scattering coordinates for the GS and the first ES, and to construct the coupled PESs relevant for the dynamics to rationalize the experimental findings [3,4,8] of low amount of IVE and CT processes in the system. Therefore, we have computed the nonadiabatic couplings arising from the radial motions since at the experimental collision energy Ecoll = 28.96 eV, it is expected that collisions would be dominated mostly by the translational–vibrational couplings in view of relatively much shorter collision time scale (almost two orders of magnitude less) as compared to that of rotation of the diatom. The construction and computation of the corresponding quasidiabatic PESs are also desirable for the dynamics studies. A preliminary study towards this goal is also reported by us very recently [38]. The present paper is organized as follows: In Section 2, we describe the details of ab initio computations. In Section 3, the characteristics of ab initio adiabatic PESs are given. In Section 4, we present the form of asymptotic potential in terms of multipolar components. The radial nonadiabatic coupling matrix elements, the mixing angle and the corresponding quasidiabatic PESs and their characteristics are presented in Section 5, followed by a summary and conclusions in Section 6. 2. Computational methodology Ab initio calculations have been performed in the Jacobi coordinates (Fig. 1) where R is the H+ distance from the center  of mass  of !

CO, r is the internuclear distance of CO and c ¼ cos1 ~ r  R . The

calculations have been carried out at the internally contracted multi-reference configuration interaction (MRCI) level of accuracy [39–41] employing Dunning’s correlation consistent polarized valence triple zeta (cc-pVTZ) basis set [42] using the MOLPRO 2002.6 suite of programs [43]. At the Hartree–Fock (HF) level, the chosen basis set produced 74 molecular orbitals (MO) from contracted gaussian atomic orbitals and they are listed as [33a1, 17b1, 17b2, 7a2] and [50a0 , 24a00 ] in the C2v and the Cs point groups, respectively. The calculations were done for the singlet spin symmetry in the C2v (collinear) and the Cs (off-collinear) point

groups. The [1a1, 2a1] MOs in the C2v and [1a0 , 2a0 ] MOs in the Cs were treated as core orbitals. At the HF level, the ground state electronic configurations were [5a1, 1b1, 1b2] and [6a0 , 1a00 ] for the C2v and the Cs geometries, respectively. All the MOs were doubly occupied accounting for 14 electrons for the triatomic [HCO]+ system. For the MRCI calculations [41,39,40] the two doubly occupied core orbitals were kept frozen, that is, they were excluded from the excitations. Thus, only the 10 valence electrons were considered in the single and the double excitations. The number of active orbitals for the C2v and the Cs point groups were [3–7a1, 2b1, 1–2b2] and [3–9a0 , 1–2a00 ], respectively. Complete active space self-consistent field (CASSCF) calculations [44,45] were done to optimize the MOs. In the subsequent internally contracted MRCI calculations, the CASSCF wavefunctions were used as reference to compute the first two roots of 1R+(1A0 ) symmetry. Typically, the MRCI wavefunction for the 1A0 states consisted of 13,019,888 uncontracted configuration state functions (CSFs) which were internally contracted to 448,260 configurations. For the 1R+ states these numbers were 4,405,136 and 214,088, respectively. The threshold value of the CSFs selection was kept at 0.32  106 a.u. The PESs were obtained on the following grid points: c = 0– 180°(15°); r = 1.5–3.2(0.1); R = 1.4–2(0.2), 2.1–20(0.2, 0.1, 1.0). r and R are in the units of Bohr and the numbers in the parenthesis indicate the interval. Typically, for R the initial values differed for different angular approaches and the stated intervals also varied for different ranges of R. The adopted ab initio procedure to compute the corresponding quasidiabatic PES and the coupling potentials is described in Section 5. The ab initio data set of the adiabatic as well as quasidiabatic PESs are available on request with the corresponding author.

3. Ab initio adiabatic PESs Before we present and discuss the results it would be worthwhile to compare the computed parameters for the diatoms CO (CO+) obtained using the similar internally contracted MRCI/ccpVTZ computations in the present calculations with those available from the experiments as well as theoretical calculations. The comparisons are reported for the CO and the CO+ molecules in Table 1. The various symbols used in the table have been explained there. All the computed data in the present calculations in Table 1 agree fairly well with the available experimental and theoretical data. It is important to point out here that accurate theoretical prediction of the dipole moment of CO has been a long-standing difficult task since it appears to be quite sensitive to the amount of electron correlation included in the computation. A large number of calculations have been performed employing different levels of theory and basis sets (see Ref. [46, and references cited therein]). The most reliable prediction came from Scuseria et al. [46] who predicted a value of 0.0492 a.u. at req as compared to the experimental values, 0.0481 a.u. [47] and 0.0484 a.u. [48]. Maroulis [49,50] performed extensive ab initio computations using different levels of theory and variety of larger and larger basis sets confirming the absolute need of accounting for the electron correlation for the prediction of dipole moment of CO. For example, his calculations predicted the following values at different levels of computations using BS1 basis set [6s, 4p, 4d, 2f] at r = req, l = 0.1052 a.u. and l = 0.0570 a.u. The first and the second values correspond to the calculations at the self-consistent field and the coupled cluster single and double excitations with perturbative correction for the triple excitations levels. Our computed values is comparatively large but fairs well particularly in view of the sensitiveness on accounting for the electron correlation. In order to further test the quality of the generated PESs, we compare the collinear equilibrium geometry properties of the

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Table 1 Comparison of computed molecular properties of CO and CO+ with experimental data. req is equilibrium bond distance. D0 is the dissociation energy corresponding to the GS. I.P. is the ionization potential. DE0!1 is the energy of first rotational excitation, j = 0 ? j = 1 expressed in milli eV in the GS. DE0!1 v ib is the energy of first vibrational excitation, rot v = 0 ? v = 1 in the GS. DE0!1 is the energy of first electronic excitation from v = 0 of the GS to v0 = 0 of the first ES and D is the dipole moment for the GS expressed in atomic units elec and for req = 2.147a.

CO (1R+) Present study Experiment [73] Experiment [74] Experiment [75] Experiment [76] Experiment [77] Experiment [47] Experiment [48] Theory [37] Theory [46] Theory [49,50]

req (Bohr)

D0 (eV)

I.P. (eV)

DE0!1 ðmeVÞ rot

DE0!1 v ib ðeVÞ

DE0!1 elec ðeVÞ

D (a.u.)

2.147 2.1357

10.825 11.09

13.7

0.39

0.264

6.046

0.0725

14.0 14.1 14.014 14.0

0.48

0.266

5.898

0.48

0.269

9.6 2.1325

11.108

0.0481 0.044 13.8 0.0492 0.0514

CO+(2R+) Present study Experiment [77] Experiment [78]

2.123 2.1077

8.17 9.9 8.33

27.19 27.90

Table 2 Equilibrium geometry (collinear) data for the bound HCO+ and HOC+ ions. The bond distances are in Å.

Theory [25] Theory [28] Theory [29] Theory [34] Present study Experiment [79]

HCO+ (c = 180°)

HOC+ (c = 0°)

rHC

rCO

rHO

rCO

1.0930 1.0935 1.0912 1.0898 1.1018 1.0972

1.1080 1.1086 1.1066 1.1056 1.1082 1.1047

0.9900 0.9904 0.9891 0.9889 0.9882 0.9750

1.1579 1.1579 1.1553 1.1563 1.1532 1.1570

bound molecular ions, HCO+ and HOC+, in terms of bond distances at their respective minima in the global GS PES in Table 2 along with the earlier theoretical and experimental values. The highest level of computations involved CCSD (T) calculations employing cc-pVQZ basis set [25,28]. The latter authors generated the semi-global GS PES characterizing the potential wells in the Jacobi coordinates corresponding to stable HCO+ and HOC+ isomers. More recent calculations have been done at CCSD (T)/aug-cc-pVQZ and MRDCI/cc-pVTZ levels [29,34]. It is gratifying to note that the present calculations predict the bond distances of these collinear isomers in good agreement with the existing theoretical and experimental data (data in the present work printed in bold in Table 2) thus lending confidence in the present ab initio computations. We further compare our results with those of the early theoretical results obtained by Bruna et al. [32] in Table 3 in terms of the location and the well depths of the bound collinear HOC+ and HCO+ ions. Their ab initio study on the bound HCO+ and HOC+ ions was done at the SCF level of accuracy with the energy values (in a.u.): Table 3 Calculated minimum energy geometries for H+-CO. rm is the distance between the atoms of the diatom, Rm is the distance between the center of mass of the diatom and the proton, c is the angle between rm and Rm,  is the well depth. c = 0°:HOC+; c = 180°:HCO+.

a

rm (Å)

Rm (Å)

c (°)

 (eV)

Theory [32]

1.17 1.11

1.47 1.56

0 180

5.46 6.26

Present study

1.153 1.108

1.482 1.735

0 180

4.87 6.56

Theory [37]a

1.128 1.128

1.45 1.73

0 180

4.73 6.55

Calculations for req = 2.1377a (see the text).

0.485 0.49

0.2677 0.2745

2.574

1.029

E(HCO+) = 112.9044, E(COH+) = 112.8751, E(CO) = 112.6745. The corresponding values from our calculations are: 113.3820, 113.3176 and 113.1418. The derived well depths representing the HOC+ and HCO+ configurations with respect to the asymptotic energy, H+ + CO is listed in Table 3. For the sake of comparison from the published ab initio data of Kimura et al. as PECs as a function of R with r = req = 2.13a, which is close to the optimized r-value of the ions, for collinear geometries configurations we derived the well depths of these ions with respect to the asymptotic value at R = 15a. The values are reported there along with our optimized data with the asymptotic, H+ + CO value. As expected, we observe that our values are close to those of Kimura et al. In the asymptotic regions (R > 10a), the two PECs run almost parallel to each other, and in the asymptotic limit, the experimental [8] endoergicity value of the CT channel, (H (2S) + CO+(X2R+)) with that of IVE channel, (H+ + CO (X1R+)) is estimated to be 0.40 eV. This value is derived from the experimental value of the ionization potential (IP) of CO (14.0 eV) and H (13.6 eV). Our predicted value of IP of CO is 13.7 eV while that of Kimura et al. [37] is 13.8 eV and both predictions are within the accuracy of 2% against the experimental value. We now present the adiabatic PECs for the GS (red1) and the first ES (blue1) for three molecular orientations c = 0°, 90°, 180° in Fig. 2 (left) as a function of R at req = 2.13a. Fig. 2 (right) shows the diabatic PECs which we will discuss in the following section. The symmetry designations of the GS and ES PECs are 11R+ and 21R+ for the collinear orientations (C1v) and 11A0 and 21A0 for off-collinear orientations (Cs), respectively. Kimura et al. [37] had also computed the GS and the first ES PECs of R or A0 symmetries only for these three molecular orientations for req using Dunning’s cc-pVTZ [42] basis set and MRDCI (single and double excitations with Davidson correction) method [37]. We compare our results with theirs in Fig. 2 (left) and observe that both the results agree rather closely except a few noticeable deviations as can be seen. The reasons for this difference could be attributed mainly due to the difference in the optimization of orbitals in their computations since the selection threshold for the configuration was kept at the same value, 0.32  106 a.u. Their PECs for the first ES also show the sign of avoided crossing (more so for c = 180°) with the second ES as a function of R at r = req. There is a hump which is more pronounced in our calculations at R = 4.4a for c = 180° which, 1 For interpretation of the references to color in Fig. 2, the reader is referred to the web version of this paper.

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Fig. 2. (Left) Comparison between present computed adiabatic PECs for the GS and the first ES for c = 0°, 90° and 180° as a function of R at r = req = 2.13a with those of Kimura et al. [37]. (Right) Quasidiabatic PECs as a function R at req = 2.13a for c = 0°, 90° and 180° orientations for the GS and the first ES superimposed with the corresponding adiabatic PECs.

however, is missing in other two angles. In the asymptotic regions, the GS correlates with the H+ + CO (X1R+) while the first ES correlates with the H (2S) + CO+(X2R+). Therefore, the IVE and the CT channels for the system are: 0

Hþ þ COðX1 Rþ ; v ¼ 0; j ¼ 0Þ ! Hþ þ COðX1 Rþ ; v 0 ; j Þ Hþ þ COðX1 Rþ ; v ¼ 0; j ¼ 0Þ ! Hð2 SÞ þ COþ ðX2 Rþ ; v 00 ; j Þ 00

where v and j stand for vibrational and rotational levels, respectively. In our earlier study [35] of PECs as a function of R with r = req for the collinear and perpendicular approaches we had observed two low-lying ESs, 31R+ and 11P in the collinear geometries and 31A0 and 11A00 in the off-collinear geometries, which asymptotically correlate to H (2S) + CO+(12R+) and H (2S) + CO+(12P), respectively. No low-lying triplet state was observed in this energy range. The PECs for the first ES (21R+) showed clear avoided crossing with

the second ES (31R+) only for c = 180°. In the present study, the hump in the first ES appears for c = 180° at smaller R values (3– 5a). Therefore, we have plotted the PECs for the GS and the first ES as a function of r for c = 0°, 90° and 180° in Fig. 3 with R fixed at 3.0a and 5a, respectively, to highlight the nonadiabatic interactions which are seen gain for c = 0° and c = 180° but the humps are located at slightly extended r values from req. A natural question arises at this point whether or not the nonadiabatic interactions between the first and the second ESs would influence the energy transfer dynamics of the IVE and the CT channels. Also whether there would be effective coupling between the R and P states considering the fact that the experiments [8] were carried out at Ecoll = 28.96 eV, for which the collision time expected to be much shorter than that of rotation, one expects that the nonadiabatic collisions would be occurring through mostly by the radial motions. Therefore, the involvement of 11P-state is largely ruled out since there would be no radial coupling between R and P states. The

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as a function of R and r for c = 0°, 90° and 180°. The GS PES shows a deep interaction well for all the angular approaches. A minimum energy path computed by locating the local minimum energy positions as a function of c, is plotted in Fig. 5, showing the possible [HOC+] [HCO+] interconversion. From Fig. 5 it is clear that between the two, HCO+ is more stable. The transition state isomer of this process lies at c = 97.2° (Eh = 113.261), 3.29 eV higher relative to the most stable isomer, HCO+ and the energy difference between the two stable isomers in the collinear configurations is estimated to be 1.75 eV. 4. Asymptotic interaction potential The long-range interaction potential Vas for the H+ + CO system is modelled and obtained in terms of multipolar expansion terms which is given below.

V as ðR; r; cÞ 

lðrÞ R2 

P 1 ðcos cÞ þ

a2 ðrÞ 2R4

QðrÞ R3

P2 ðcos cÞ 

a0 ðrÞ 2R4

P 2 ðcos cÞ

ð1Þ

where Vas is the asymptotic potential, l(r) is the dipole moment of CO, Q(r) is the quadrupole moment of CO, a0(r) and a2(r) are the polarizability components of CO, P1, P2 are the Legendre polynomials. Maroulis [49,50] had performed extensive high quality ab initio computations for l, Q, a0 and a2 as a function of r of CO at different levels of theory using variety of basis sets. In order to model the long-range asymptotic interactions we have chosen his ab initio data obtained using his BS1 basis set at CCSD (T) level of accuracy. We obtained a polynomial fit of his data by expanding around req as

f ðrÞ ¼ a0 þ a1 ðr  req Þ þ a2 ðr  req Þ2 þ a3 ðr  r eq Þ3 þ a4 ðr  req Þ4 ð2Þ where req = 2.132a is the equilibrium bond distance of CO. The obtained coefficients a0, a1, a2, a3, a4 are tabulated in Table 4. For the CT process, H (2S) + CO+(12R+), only the polarizability component of the H atom (a0 = 2.73729 a.u.) was used considering CO+ as a point charge to generate asymptotic potentials. These are then matched with the respective ab initio electronic energies to get the longrange potential. 5. Ab initio quasidiabatic PESs Fig. 3. The GS and the first ES adiabatic PECs for c = 0°, 90° and 180° as a function of r at R fixed at 3a and 5a.

experiments also did not hint at any observation of the CT channels, H (2S) + CO+(12R+) and H (2S) + CO+(12P). Yet, the nonadiabatic interactions between the first and second ESs of the collision system may influence the energy transfer dynamics. In view of the fact that these interactions are located at slightly extended r values from req and expected to become effective mostly for the collinear and angular approaches, in the range c = 150– 180°, and that the experiments show very low amount of vibrational excitations in the system, we believe that a two-state nonadiabatic interaction dynamics would largely capture the energy transfer dynamics. It should be noted that the initial state of the collision in the system is prepared with CO (v = 0). Therefore, in the following section, we focus our attention on the description of two-state quasidiabatic PESs and the nonadiabatic interactions arising from the radial couplings. To further illustrate the characteristics of the PESs, the contour diagrams of the GS and the first ES adiabatic PES are shown in Fig. 4

An adiabatic-to-diabatic transformation has been made for the sake of computational convenience for the dynamics study. In b becomes the diabatic representation, the nuclear KE operator T b becomes nondiagonal diagonal and potential energy operator V which are nondiagonal and diagonal, respectively, in the adiabatic representation. The diabatization procedures, their exactness and their associated advantages for numerical computations have been discussed and documented in the literature in detail [51–62]. A general discussion on it has been recently published [63]. The nonadiabatic coupling matrix elements (NACME) in the KE operator   b are of the form T

*

  +  ol   a / oql  2

 /a1 

ð3Þ

where l = 1 (first-order NACME) or 2 (second-order NACME). The kets j/a1 j and j/a2 j represent the real electronic wavefunctions of the two involved adiabatic electronic states and q stands for R, r in the present calculations. The terms with l = 2 are mostly smaller in magnitude and are generally ignored in the dynamical calculations. In the present calculations, they have been ignored. The NACME values have been computed between the GS and the first ES by

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Fig. 4. Contour diagrams of the GS (left) and ES (right) adiabatic PESs of the H+ + CO system as a function of R and r for c = 0°, 90° and 180°.

    d

1 a wa1  wa2 ¼ w ðR þ DRÞjwa2 ðR  DRÞ dR 2 DR 1 R

Fig. 5. Minimum energy pathway on the adiabatic GS for the interconversion process, HOC+ HCO+.

Table 4 Values of coefficients (a.u.) used in Eq. (2) for the prediction of l, Q, a0 and a2 as a function of r(a). f

a0

a1

a2

l

0.057 1.46 13.19 3.66

0.6636 1.03 5.52 8.28

0.0008 0.40 1.74 3.32

Q

a0 a2

a3 0.1337 0.35 0.24

a4 0.0054 0.33 0.42

the numerical differentiation using the finite difference method [64]

ð4Þ

where DR is the small increment. We have used MOLPRO to compute the NACME values with DR = 0.0002a. Our computed NACME values were found to be in good agreement with those obtained from Kimura et al. [37] as a function of R for fixed c(= 0°, 90° and 180°) and r(= req). Other approximate methods for diabatization have also been suggested which avoid direct computation of NACME and where the mixing angle (a) and coupling potential are obtained from the CI coefficients of the electronic wavefunctions. They have been employed recently in studying the photodissociation dynamics on the coupled PESs [65]. An improvement in this scheme has been suggested by Werner and co-worker [66] by determining the (quasi) diabatic wavefunctions (and the corresponding CI vectors) so that they vary as little as possible as a function of geometry. This condition is met by using the invariance of the MRCI energies with respect to the unitary transformation among the active orbitals so that the geometry dependence of the orbitals is minimized. This is accomplished by maximizing the overlap for all the pairs of active orbitals at Rref with those at neighbourhood geometry R0 using the Jacobi rotation technique. Rref is identified at which both the adiabatic and diabatic states become identical. This procedure was applied to study the photodissociation of H2S on electronically coupled PES by computing the quasidiabatic PESs and the coupling potentials and the quantum dynamics using the quasidiabatic potential matrix was able to explain the experimental observation [66]. It is also worth pointing out here that Balint-Kurti et al. (see in General Discussion [63]) have also recently computed the quasidiabatic PECs corresponding to the five lowest ESs in the O3 system using the same procedure. Recently, a new set of quasidiabatic potential matrix (2  2) involving the GS and the first ES for the H+ + H2 system was also obtained using this procedure in our

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group and quantum dynamics yielded results in near qualitative agreement with experiments settling some long-standing discrepancies between theory and experiment [9–11]. In the present study, we have computed the 2  2 quasidiabatic potential matrix involving the GS and the first ES by determining the quasidiabatic electronic wavefunctions using the ab initio procedure suggested by Werner and co-workers [66] which has been incorporated in the MOLPRO (version 2002.6) suite of programs [43]. In our calculations we choose the reference geometry, Rref at R = 16a, in the asymptotic limit, around which both adiabatic and the diabatic states become identical. The computations were carried out in similar manner as described in Section 2, that is CASSCF reference space was used for the subsequent internally contracted MRCI calculations keeping other parameters like selection threshold for CSFs the same. The computation was carried out as a function  of R fora fixed value of r and c. The obtained coupling potential V d12 ¼ V d21 is shown as a function of R and r for c = 0°, 90° and 180° in Fig. 6. One can see that the coupling potentials become vanishingly small for R > 8a indicating that both the adiabatic and the diabatic states become identical in the asymptotic limit. The diagonal elements of the coupling matrix V d11 and V d22 represent the quasidiabatic PESs corresponding to the GS and the

Fig. 6. Coupling potential between the GS and the first ES as a function R and r for c = 0°, 90° and 180° orientations.

217

first ESs, respectively. For the sake of comparison and clarity, the obtained diabatic PECs of the GS and the first ES are displayed in Fig. 2 (right) as a function of R with r = req = 2.13a for c = 0°, 90° and 180° along with the corresponding adiabatic PECs. The magnitudes of coupling potential between the GS and the first ES for the system are relatively smaller as compared to those computed for the H+ + O2 [16] and the H+ + NO [19] systems. Therefore, one expects that the CT processes would be weaker in strength in comparison with the IVE processes. In fact, the experiments [8] carried out at Elab = 30 eV also reveal that the CT is almost two orders of magnitude less than the IVE in the system. In Figs. 2 and 3, we noticed the regions of avoided crossings with the first ES with that of second ES at closer approaches of H+, and at slightly extended r from req. This suggests to a Landau–Zener type of nonadiabatic coupling [67] where the corresponding diabatic PECs would cross leading to a cross over of electronic character of the two states in the regions of avoided crossings. Such couplings are quite effective for the nonadiabatic transitions and would be dominantly coupled for the first and the second CT channels. However, as discussed above, in the present study we have ignored them because of their existence in the localized regions of collision interactions and focussed our attention on the other channel for the nonadiabatic transitions below. In Fig. 2 (right), we observe that both the adiabatic and the diabatic PECs run almost parallel to each other very closely with small energy difference (DE) in the asymptotic regions for R > 10a. Interestingly, the diabatic PECs do not show any crossing. Because of (relatively) weak (potential) coupling between the two states the corresponding diabatic PECs may not exhibit crossing unlike the general case. This has been observed in model systems [60]. In contrast to the curve crossing case, the two diabatic potentials exhibit a very weak R-dependence such that their energy difference can be assumed to be constant. However, their coupling potential shows a strong (or exponential) R-dependence as shown in Fig. 6. In these systems, the nonadiabatic transition does not become so effective generally as compared to the curve crossing case unless the DE is quite small. Yet, they are important especially when the two PECs are in near resonance asymptotically. The experiments of NiednerSchatteburg and Toennies [8] show that the magnitude of total differential cross section (DCS) of the CT channel is an order of magnitude less as compared to that of IVE channel at collision energy of Ecoll = 28.96 eV. This mode of nonadiabatic transition has been studied in the literature through model coupling potentials providing useful insights and it is often referred to occur via Rosen–Zener–Demkovtype of coupling. Although in their original work, Rosen and Zener [68] developed the time-dependent theory of the double Stern– Gerlach experiment, their mathematical basis corresponds to the problem of nonadiabatic transition; the corresponding DE and coupling potential were constant and sec-hyperbolic function of time, respectively. Later, Demkov [69] obtained the same formula for the transition probability as that of Rosen and Zener by assuming a constant DE and exponential function for the coupling for the study of near resonant charge-transfer processes in the time-dependent formalism. Some more elaborations of Demkov-type of coupling can also be found elsewhere [70,71,67]. In this model, in the absence of apparently no avoided crossing, the nonadiabatic transition occurs quite locally at Rx ¼ RðR Þ, where the adiabatic potentials just start to diverge and R* is the complex crossing point closest to the real axis. As there is no crossing between the PECs, it implies that there is no switch over of the character of the electronic states. Therefore, the nonadiabatic transition does not approach to unity but rather one-half in the limit of high energy. For the present system, in Fig. 6 the computed coupling potentials start exhibiting their strengths for approximately R < 8a from where the adiabatic PECs also start diverging. However, a full quantum dynamics study

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would perhaps reveal the dynamical intricacies in this realistic system yielding quantitative estimates of the relevant cross sections. Quantum dynamics involving this coupling potential matrix is currently been carried out to study the IVE and the CT processes and it would be reported soon [72] comparing the results with those of experiments [4,8]. 6. Summary and conclusions In the present study, we have generated new global ab initio adiabatic PESs for the GS and the first ES of the H+ + CO system using Dunning’s cc-pVTZ basis set at the internally contracted MRCI (with the single and the double excitations) level of accuracy. Comparisons of various computed diatomic data for CO and CO+ and the equilibrium structure data of the bound HCO+ and HOC+ ions in the GS with those of experiments and high level ab initio calculations available for these ions lend credence to the accuracy of the present ab initio calculations. The various characteristics and topologies of the adiabatic PESs have been analysed in detail. The corresponding quasidiabatic PESs and the coupling between them are also obtained using the ab initio procedure at the same level of accuracy. The nonadiabatic interactions have been analysed in details in terms of NACME and the coupling potentials. We believe that perhaps for the first time the ab initio global quasidiabatic PESs for the GS and the first ESs have become available for the system. The present set of quasidiabatic PESs have been obtained taking into account only the nonadiabatic couplings arising through radial motion, that is, vibrational and translational motions, in view of the experimental data available for collision energies Ecoll = 9.5– 30 eV. In these range of collision energies, the collision time is almost two order of magnitude shorter than that of molecular rotations. Therefore, effective nonadiabatic interactions are expected to arise mostly from the radial couplings. Acknowledgements This study was supported by a grant from the Department of Science and Technology, New Delhi. F. George D.X. acknowledges the CSIR, New Delhi, for the Senior Research Fellowship. The financial assistance by IIT Madras in procuring the MOLPRO code through inter-disciplinary research grant in the chemical physics program is also gratefully acknowledged. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15]

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