Buenker, R.

Kimura, M.

Li, Y.

Liebermann, H.

Pichl, L.

Schneider, I.

English

MPG.PuRe

EPJ manuscript No.(will be inserted by the editor)Steric effect in H2O+ and H3O+ collision complexesCharge transfer in O+/H2 and H+/H2O molecular ion systemsMineo Kimura1, Lukáš Pichl2a, Yan Li3, Robert J. Buenker3, and Ioan F. Schneider41 Graduate School of Sciences, Kyushu University, Fukuoka 812-8581, Japan2 Max Planck Institute for the Physics of Complex Systems, Nöthnitzer Str. 38, D-01187 Dresden, Germany3 Fachbereich C-Mathematik und Naturwissenschaften, Bergische Universita̋t Wuppertal, D-42119 Wuppertal, Germany4 UFR Sciences et Techniques, Universite du Havre, 25, rue Philippe Lebon, BP 540, 76058 Le Havre Cedex, France.Received: July 30th, 2005 / Revised version:Abstract. The positive water and hydronium ions are of interest in a variety of chemical and biologicalapplications. Here we study the steric effect in charge transfer collisions, i.e. the spatial dependence ofsingle electron capture, in collisions mediated by these ions. In particular, the steric effect is demonstratedin the O+(2D)/H2 and H+/H2O charge transfer collisions in the energy range of 100 eV/u to 10 keV/u.PACS. 34.70.+e charge transfer – 34.50.Gb electronic excitation and ionization of molecules; intermediatemolecular states – 31.15.Ar ab initio calculations1 IntroductionIn charge-transfer collisions of atomic ions with molecules,the relative orientation of target and projectile as well asthe details of potential energy surfaces are known to havea limited effect on the integral cross sections, at leastin the keV/u energy range. Nevertheless, a considerablesteric effect is evident even at those high energies whenthe differential cross sections or transition probabilitiesalong particular collision trajectories are studied. Thesephenomena have been somewhat neglected in the past,due to both the experimental difficulties in orienting neu-tral molecular targets and theoretical complexity inherentin polyatomic collisional systems, which typically involvesrovibrational motion of the colliding system in a numberof coupled electronic states.Owing to large collision velocities in the keV/u energyrange, the period of (ro)vibrational motion of the targetmolecule is much larger than the collision time, and hencethe nuclear geometry of the target can be kept fixed. Inaddition, the presence of long-range Coulomb interactionin ion-ion channels may dominate the details of local elec-tronic structures, imposing thus a partial spherical sym-metry on the system. That is often the reason why the in-tegral charge transfer cross sections can be well estimatedby using a few, if not a single one, orientation of the col-lisional molecular ion system. Much theoretical work hasbeen devoted to the development and accuracy assessmentof such approximations [1,2].Send offprint requests to: lukas@pichl.jpa On summer leave from: International Christian University,Osawa 3-10-2, Mitaka, Tokyo 181-8585, JapanHowever, the electron capture process does depend onthe orientation of the projectile and details of the poten-tial energy surface, if either the collision trajectory or tar-get orientation is resolved. It is the purpose of this paperto demonstrate the steric effect in the H2O+ and H3O+systems. To that aim, we first study a central collisionof O+(2D) with H2 molecule, depending on the angle be-tween projectile velocity and molecular axis. Next we pro-ceed to the collision of H+ with H2O using a proton beamoriented perpendicularly to the plane of water molecule,and screen the asymptotic electron capture probabilitiesalong a two-dimensional set of impact parameter vectors.The paper is organized as follows. In Section 2, the abinitio calculations and the electronic state close-couplingformalism are summarized along with references to previ-ous work [3,4]. Section 3 then resolves the steric effecton electron capture probabilities and cross sections forthe H2O+ and H3O+ collision intermediates and discussesthe accuracy of the integral charge transfer cross sections.Concluding remarks close the paper in Section 4. Atomicunits are used throughout the paper.2 Theoretical ModelThe potential energy curves (PEC) for H2O+ and H3O+systems were obtained ab initio for the most general molec-ular symmetries by using the multireference single- anddouble-excitation configuration interaction (MRD-CI) method[5–8].A gaussian (6s3p2d1f) basis contracted to [4s3p2d1f]was used for the hydrogen atom, while a (12s6p3d2f) basis2 Mineo Kimura et al.: Steric effect in H2O+ and H3O+ collision complexesFig. 1. Collision of H+ with H2O: beam of trajectories per-pendicular to the water molecule plane. The beam intersectswith the plane along two lines: (1) O-S line, where S is locatedin the middle of H-H line (φ = 0), and (2) O-P line φ = π/2),perpendicular to O-S. Four values of impact parameters, 0.9,1.8 and 3.6 a.u. are considered, resulting in 10 straight-linetrajectory rays.set contracted to [5s4p3d2f] was employed for the oxygenatom. All basis sets are of the cc-pVQZ type [9]. In addi-tion, a diffuse [2s2p2d] basis centered at oxygen atom isemployed to describe the Rydberg electronic states of thewater molecule.The selection threshold and the number of configu-rations are 2.0 × 10−7 a.u. and 500, 000 for H2O+, and8.0×10−7 a.u. and 200, 000 for H3O+, respectively. The ef-fect of all unselected configurations is checked by using theperturbative energy extrapolation procedure [8]. The re-sulting highly correlated wavefunctions are used to obtainthe nonadiabatic coupling matrix elements by a numericaldifferentiation method [10]. The origin of electronic coor-dinates is placed at the molecular center of mass for allgeometries.In the case of H2O+, the internuclear distance of hy-drogen atoms was fixed at 1.4 and 1.6, to estimate theeffect of hydrogen molecular vibrations. The potential en-ergy surfaces for the doublet states (asymptotically cor-responding to the metastable O+(2D,2 P ) ions were pro-jected along three O+ lines passing through the center ofthe H-H line at the angles γ = 0, π/4 and π/2.In case of H3O+, the target water molecule is fixed atthe equilibrium geometry. The trajectories of impactingproton are perpendicular to the plane of water molecule[x, y], which they intersect at: (1) the oxygen atom (z-direction approach), (2) on the OS line, where S is thecenter of H-H line, or (3) the OP line, where OP andOS are perpendicular in-plane lines. Nuclear geometry isinvariant with respect to an inversion along the OS line.Potential energy curves and nonadiabatic couplings wereobtained for three magnitudes of the impact parameterrelative to the oxygen atom, b = 0.9, 1.8, and 3.6. Thegeometry of the collision is shown in Fig. 1.The collision dynamics at the keV energy range is givenby coupled equations [1,3] in the above basis of electronic-state amplitudes fn on potential energy surfaces Γ la-belled by the impact parameters b,[iddz− En(R)− E0v]fbn (z;Γ ) =∑n′′cnn′fbn′(z; Γ ). (1)Here v is the velocity, z = vt is the coordinate along pro-jectile’s trajectory, R = (b2 + z2)1/2, cnn′ is the couplingmatrix and E0 is the collision energy. The equations aresolved for the initial condition fbn′(−∞) = δnn′ . The crosssection formula for electron capture in the general case isobtained by integration over the plane of impact param-eters and averaging over the molecular orientations Γ asappropriate,σnn′ =∫dbxdby〈|fbn′(∞; Γ )|2〉Γ. (2)Details of the above formulation for ion-atom collisions(fb → f b, single PEC Γ ) can be found in Refs. [1,11–14];the ion-diatomic collision case is dealt with e.g. in Refs.[2,15–17] (PEC are parameterized by one angle γ). In thepresent ion-triatomic case, H+/H2O collision, the molecu-lar plane is aligned perpendicular to the proton beam, andhence each value of the impact parameter corresponds toits potential energy surface Γ . Introducing polar coordi-nates for the impact parameter vector, (b, φ), the crosssection formula then readsσ⊥ = 2∫ ∞0db∫ π0dφ b p(b, φ; R(b, z →∞)). (3)3 Steric Effect in the Electron CaptureBased on the MRD-CI potential energy curves and cou-plings, we solved the coupled equations (1) along the col-lision trajectories and for target orientations defined inSection 2. The next two subsections discuss the steric ef-fect seen in the resulting charge transfer probabilities.3.1 Collision of O+(2D) with H2Figure 2 (upper panel) displays the manifold of doubletH2O+ potential energy curves, which asymptotically cor-respond to O+(2D,2 P ) states, or O states in the elec-tron capture channel. Excited states of H2/H+2 are alsoincluded. The PEC passing through the target center areshown for two angles between γ = π/4 (dashed line, A′and A′′ symmetry) and γ = π/2 (full line, A1, A2, B1,B2 symmetry). Although both groups of PEC are similarfor |HH| ∼2.5 (polarization interaction), substantial dif-ferences arise at lower collision distances. Owing in partto symmetry reasons, the PEC for γ = π/4 have higherenergies than those for γ = π/2. The lower panel of Fig.2 shows the cross section for electron capture from theO+(2D) projectile in fixed-γ approximation [4,15,16]. Thesteric effect can clearly be resolved in Fig. 2. On the otherhand, since γ = π/4 is the middle value interpolating theextreme cases γ = π/2 and γ = 0, the range of the stericMineo Kimura et al.: Steric effect in H2O+ and H3O+ collision complexes 3Fig. 2. Electron capture cross section for collisions of O+(2D)with the H2 molecule using fixed-γ potential energy surfacesfor γ = π/2 and γ = π/4.effect is only about 30% of the total cross section value forkeV energies. This is a very reasonable accuracy for theO+(2D) charge transfer cross sections that vary in theexperiments by a factor of 2 [18–20].A detailed discussion of the integral cross sections forelectron capture on H2 by O+(2D, 2P, 4S) can be found inRef. [3,4] along with a comparison to recent measurements[21,22].3.2 Collision of H+ with H2OThe potential energy curves for the proton beam intersect-ing water molecular plane on the OS line (φ = 0) and theOP line φ = π/2 are shown in the upper and lower panel ofFig. 3, respectively. The labels correspond to the verticalordering of PEC for the four different impact parametervalues and the initial (dashed line) and final (full line)collision channels. As the impact parameter increases, theenergy difference diminishes, converging to the asymp-totic energy gain value E(H2O) − E(H2O+)+0.5, whichis approximately 0.0467. The absolute values of the non-adiabatic coupling between the initial and final state isshown in Fig. 4 for all values of impact parameters andφ = 0, π/2.Solving the coupled equations (1) along the H+ beam,we obtained the electron capture probability for each straight-Fig. 3. Potential energy curves of the initial and final chargetransfer state (1A1 symmetry) for φ = 0 (upper panel) andφ = π/2 (lower panel) for four impact parameter values (R >b).line trajectory. These are plotted in Figs. 5 and 6, respec-tively, demonstrating a marked steric effect along eachtrajectory. Except for b = 0, the angle between RO−H+(connecting the oxygen atom and the impacting proton)and the plane of water molecule varies in the full rangefrom −π/2 through 0 to π/2, scanning the potential en-ergy surface in a complex way. The probabilities of elec-tron capture for φ = 0 and φ = π/2 exhibit oscillationsas a function of energy, the frequency of which dimin-ishes when b increases. Although for any fixed value of en-ergy the capture probability substantially differs betweenφ = 0 and φ = π/2, the overall behavior (frequency of os-cillations, relative heights of the peaks) is similar, exceptfor the highest value of the impact parameter consideredhere, b = 3.6. This similarity between the two cases canbe explained analytically e.g. in the 2-state exponentialmodel, since the asymptotic energy separation is smalland the coupling vanishes quite slowly both for φ = 0 andφ = π/2.Finally, let us estimate the variation of the unresolvedelectron capture cross section in the case of perpendicularcollisions, based on Eq. (3). To perform the integral overthe entire plane of impact parameters with high accuracy,the present 10-point grid may still be quite inaccurate.However, we can reasonably assess the variation of the4 Mineo Kimura et al.: Steric effect in H2O+ and H3O+ collision complexes2 4 6 8-1.5-1.0-0.50.00.51.01.5 b=0.0, φ=0 b=0.9, φ=0 b=1.8, φ=0 b=3.6, φ=0- |<1|d/dz|2>| b=0.0, φ=π/2 b=0.9, φ=π/2 b=1.8, φ=π/2 b=3.6, φ=π/2 |<1|d/dz|2>| (a.u.)RH+-O (a.u.)Fig. 4. Magnitude of the nonadiabatic coupling between thetwo states in Fig. 1 (φ = 0; positive sign) and Fig. 2 (φ = π/2,negative sign). Four impact parameters are considered in eachcase.1021031040.010.10.51 b=0.0 b=0.9 b=1.8 b=3.6Capture probabilityEnergy (eV)φ=0Fig. 5. Probability of electron capture along the four trajec-tories with φ = 0.cross section integral (2) by altering the PEC projectionsΓ and studying the difference. This can be formalized asfollows.Let us denote the fixed potential matrix as V (b0, φ0, z).Then we define a numerical estimate on the range of crosssection in Eq. (3), s⊥(b0, φ0) ≡2π∫ ∞0db b p(b0, φ0; R(√b2 + b20, z →∞)). (4)The most natural choice is s⊥(0, ·), hereafter referred toas the central collision. In mathematical terms, the dia-batic matrix V (b, φ, z) is approximated by V (0, ·, z) for0 ≤ b < Rt, where Rt is the size of the transition region.Variation in the values of s⊥(b0, φ0) (for b0’s falling in thetransition region and arbitrary φ0’s) then indicates theaccuracy of the central collision approximation. We alsonote that the value s⊥(0, ·) corresponds to the fixed-γ ap-proximation, which has been used extensively [23–25]. The1021031040.010.10.51φ=π/2 b=0.0 b=0.9 b=1.8 b=3.6Capture probabilityEnergy (eV)Fig. 6. Probability of electron capture along the four trajec-tories with φ = π/2.1021031041.0x10-152.0x10-153.0x10-15H++H2O(X) H(1s) +H2O+(X) Calculation with b=0 Calculation b=1.8, φ=0 B. Lindsay et al. Phys. Rev. A 55 (1997) 3945.R. Dagnac et al. J. Phys. B 3 (1970) 1239. Cross section (cm2)Energy (eV)Fig. 7. Electron capture cross section: experimental data andstereodynamical projections. The calculated curves correspondto an infinite beam of perpendicular trajectories (0 ≤ b <∞) using a frozen potential energy surface with parametersindicated in the figure; the spread between the curves a usefulaccuracy measure of the central collision model.resulting cross section estimates, s⊥(0, ·) and s⊥(1.8, 0.0)are shown in Fig. 7 along with experimental values for sin-gle electron capture by the proton on the water molecule.The experimental cross sections can be well reproducedby s⊥(0, ·) in accordance with our experience and previouswork [4,23–25]; the size of the steric effect on the electroncapture cross section is also much smaller as compared tothe transition probabilities along particular collision tra-jectories.4 Concluding RemarksIn this paper, we have first discussed the steric effect inthe charge transfer collisions of the metastable O+(2D) ionwith H2. The steric effect in the keV range of collision ener-gies, defined as the cross section ratio for the H2 moleculealigned at a fixed angle toward the O+(2D) beam, variesMineo Kimura et al.: Steric effect in H2O+ and H3O+ collision complexes 5in the range of 30% of the orientation-unresolved crosssection, an effect which falls within the error bars of mostelectron capture experiments. We further observed a markedsteric effect in the electron capture probabilities for indi-vidual rays in an H+ beam oriented perpendicular to theplane of the H2O target molecule. The steric characteris-tics of charge transfer probabilities are pronounced in theentire energy range studied, i.e. 100 eV/u to 10 keV/u. Anumerical criterion was developed to assess the variationof the cross section integral based on a sample of projectiletrajectories from the beam. Using this theoretical criterionand the experimental data available, it was demonstratedthat the central collision approximation yields a very rea-sonable value of the capture cross section.AcknowledgementsThis work was supported in part by the Japan Societyfor the Promotion of Science (JSPS), the Japanese Min-istry of Science, Sports, Culture and Education (MEXT),and by the grant BU 450/7-3 of the Deutsche Forschungs-gemeinschaft and the Fonds der Chemischen Industrie. L.Pichl acknowledges support by a JSPS Grant-in-Aid, MaxPlanck Society, and the Academic Frontier Program byMEXT. Special thanks are due to Prof. T. Kasai.References1. M. Kimura and N. F. Lane, Adv. At. Mol. Opt. Phys. 26,79 (1989).2. V. Sidis, Adv. At. Mol. Phys. 26, (1989) 161.3. L. Pichl, Y. Li, H.-P. Liebermann, R.J. Buenker, M.Kimura, J. Chem. Phys. 118, 4872 (2003).4. L. Pichl, Y. Li, H.-P. Liebermann, R. J. Buenker, and M.Kimura, Phys. Rev. A 69, (2004) 062715.5. R. J. Buenker, and S. D. Peyerimhoff, Theo. Chim. 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Steric effect in O+/H2 and H+/H2O collisions - Charge transfer in H2O+ and H3O+ collision intermediate systems

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Steric effect in O<sup>+</sup>/H<sub>2</sub> and H<sup>+</sup>/H<sub>2</sub>O collisions - Charge transfer in H<sub>2</sub>O<sup>+</sup> and H<sub>3</sub>O<sup>+</sup> collision intermediate systems

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