Chemical Physics Letters 532 (2012) 22–26

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Chemical Physics Letters

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Nonreactive scattering of the O+ + H2: A time dependent wave packet approach

Jacek Kłos a,⇑, Niyazi Bulut b, Sinan Akpinar b

a Department of Chemistry and Biochemistry, University of Maryland, College Park, MD 20742-2021, USAb Department of Physics, Faculty of Science, Firat University, Elazig 23169, Turkey

a r t i c l e i n f o

Article history:Received 19 January 2012In final form 22 February 2012Available online 1 March 2012

0009-2614/$ - see front matter � 2012 Elsevier B.V. Adoi:10.1016/j.cplett.2012.02.059

⇑ Corresponding author.E-mail addresses: jklos@umd.edu (J. Kłos), nbulut@

a b s t r a c t

Time dependent wave packet calculations have been performed for the O+ + H2 nonreactive scattering onthe recent potential energy surface of Martinez et al. [J. Chem. Phys., 120, 4705, 2004]. Exact total reflec-tion probabilities at the total angular momentum J = 0 and approximate ones for J > 0 have been calcu-lated by using Centrifugal Sudden approximation. Integral cross sections over collision energy range of0.08–0.7 eV were obtained. Time independent quantum calculations have also been performed for a com-parison. Initial state-selected rate constants have been obtained by means of Capture model based on asimple and Uniform J-shifting techniques and they display an Arrhenius behavior.

� 2012 Elsevier B.V. All rights reserved.

1. Introduction

The understanding of state-selected and state-to-state unimo-lecular and bimolecular reaction dynamics is of fundamentalimportance in the quest to control the outcome of chemical reac-tions. Research on ion–molecule reactions has made critical contri-butions to the understanding of state-selected reaction dynamics[1]. Moreover, ion–molecule reactions are important processesthat occur in many interesting situations such as interstellar pro-cesses, electric discharges, plasmas, planetary ionospheres andinterstellar clouds [1–10].

There have been a number of experimental and theoretical stud-ies of the O+ + H2 reactions and its isotopic variants [8–16]. In fact,this reaction, which is important in interstellar chemistry [17] andhas a large interest in the Earth’s ionosphere [11], can be consideredas a prototype of moderately exothermic ion–molecule reactionsinvolving a hydrogen atom transfer and occurs via the groundPotential Energy Surface (PES) (lowest energy quadruplet surface,14A00) in a wide collision energy (relative translational energy)range. Experimentally, Ng [11] measured the reaction cross sectionsand branching ratios for O+ and H2 reaction using the uniquetriple-quadrupole double-octopole (TQDO) photoionization massspectrometer. Burley et al. [12] examined the kinetic energy depen-dences of the reactions of ground-state atomic oxygen ion withmolecular hydrogen and its isotopic variant. Their results are inexcellent agreement with phase space theory calculations and withseveral dynamical models. Moreover, experimental studies haveshown that electronically nonadiabatic processes, such as chargetransfer and dissociative charge transfer, are important only at highcollision energies [11].

ll rights reserved.

firat.edu.tr (N. Bulut).

Theoretical dynamics studies are often quite difficult to be per-formed on O+ + H2 system, because electronically nonadiabaticprocesses are easy to occur, since the ground and excited PES’sare frequently close in energy [8,13–15]. Majority of previousstudies were focused on the reactive part of the O+ + H2 collisionprocess, therefore in this study we would like to focus on thenon-reactive part, which can be as well important in modeling ofthe O+ abundance in the interstellar clouds. In this work we con-sider collisions taking place just on a single potential energy sur-face. An analytical ab initio PES for the ground electronic state ofOþð4SÞ þH2ðX1Pþ

g Þ system was developed by Martinez et al. [8].Employing this PES, Martinez et al. [9] reported a theoreticaldynamics study using the quasiclassical trajectory method (QCT).They calculated the reactive cross sections and isotopic effects asa function of collision energy. In the same work, using close-coupling hyperspherical (CCH) exact quantum method, same groupreported the quantum mechanical results compared with QCT todetermine the importance of quantum effects. They found theCCH integral cross section decreased with ET and, although theQCT results were in general quite similar to the CCH ones, they pre-sented a significant deviation from the CCH data within the0.2–0.6 eV collision energy range, where the QCT method did notcorrectly describe the reaction probability [9]. To the best of ourknowledge, there are no experimental measurements of rate con-stants for O+ + H2 nonreactive scattering to be used for comparisonof theoretical results presented in this Letter. Until now, there wereno three-dimensional studies of O+ + H2 nonreactive scatteringusing time dependent quantum wave packet method to obtaincross sections and the reaction rates.

Here, we report three dimensional time dependent quantumcalculations of Oþð4SÞ þH2ðm; jÞ ! Oþð4SÞ þ H2ðm0; j0Þ nonreactivescattering with m and j being the vibrational and rotational quan-tum numbers, respectively. Finally, we have also performed forselected values of total angular momentum quantum number fully

J. Kłos et al. / Chemical Physics Letters 532 (2012) 22–26 23

quantum close coupling time independent dynamics (TID) usingthe ABC program of Skouteris et al. [18] to calculate reactive andnon-reactive probabilities and compare with the wave packetand previous CCH results. We used the potential energy surfaceof Martinez et al. [8] for all the calculations.

The Letter is organized as follows. In Section 2 we give a briefdescription of methodology. The results of our calculations are pre-sented in Section 3.

2. Theory

In this work, we employed Jacobi coordinates (R,r,c), which areideally suited for the calculations of the nonreactive state-to-stateand total probabilities. The corresponding Hamiltonian operator isexpressed as:

H ¼ � �h2@2

2lR@R2 ��h2

2lr

@2

@r2 þ�h2

2lRR2 þ�h2

2lrr2

!j2

þ �h2

2lRR2 J2 � 2Jzjz � Jþ j� � J� jþh i

þ VðR; r; cÞ ð1Þ

where r and R, respectively, the diatomic (H–H) and atom–diatom(O+–H2) distances with lR and lr as their reduced masses. c is theangle between R and r � j denotes the diatomic rotational angularmomentum, V(R, r,c) is the potential energy function for atom–mol-ecule reaction. The total angular momentum operator is denoted byJ.

Using the Hamiltonian operator in the form given in Eq. (1)makes it necessary to use a large number of grid points in both Rand r and an imaginary damping potential in the end of each grid.Having added and subtracted V(R =1,r,c = 180) to the Hamilto-nian operator given by Eq. (1), we get:

H ¼ � �h2@2

2lR@R2 þ�h2

2lRR2 J2 þ j2 � 2Jzjz � Jþ j� � J� jþh i

þ UðR; r; cÞ þ HBCðrÞ ð2Þ

where HBC(r) is the Hamiltonian operator for the diatomic moleculeand U(R,r,c) = V(R, r,c) � V(R =1,r,c = 180).

Starting from the initial wave packet at t = 0, the time-depen-dent Schrödinger equation is solved in terms of modified complexChebyshev polynomials [2],

wðR; r; c; tÞ ¼ e�ði=�hÞ DE2 þVmin½ �tPN

n¼0ð2� dn0Þ

� JnDEt2�h

� �UnCnð�iHnormÞwðR; r; c; t ¼ 0Þ ð3Þ

where w(R,r,c, t = 0) is the initial wave function, Cn(x) are complexChebyshev polynomials (CP), Jn(x) are the Bessel functions and DEis the magnitude of the entire energy spread of the spectrum ofthe unnormalized Hamiltonian operator H and Hnorm is,

Hnorm ¼H � I 1

2 DEþ Vmin� �

12 DE

ð4Þ

where I is a unitary operator. The propagation requires the opera-tion of the Cnð�iHnormÞ on w. This is performed by using a three-term recursion relation of the Chebyshev polynomials:

Unþ1 ¼ �2iHnormUn þUn�1 ð5Þ

where the recurrence is started by setting two initial values asU0 ¼ wðR; r; c; t ¼ 0Þ and U1 ¼ �iHnormwðR; r; c; t ¼ 0Þ.

The initial wave packet is located in the asymptotic region ofthe entrance channel and propagated on the potential energy sur-face toward the strong interaction region. We wish to compute

total nonreactive scattering probabilities and we have to followthe development of the wave packet being reflected from the inter-action region. The flux that goes into reactive channel is absorbedand not analyzed. In order to extract the cross section and otherobservable quantities from the wave packet dynamics, the wavepacket is analyzed at each time step by taking cuts through at afixed value of the scattering coordinate R = R1.

Cv 0j0 ðtÞ ¼Z 1

r¼0

Xk

wðR1; r; ck; tÞPj0ðckÞwk

!/v0j0ðrÞdr ð6Þ

where wðR1; r; ck; tÞ is initial wave function,Pj0 ðckÞ is an angularwave function for a rotational state j0;/v 0 j0 ðrÞ is vibrational wavefunction of H2 molecule, wk are the weights in Gauss quadratureformula. The transition probabilities for the production of specificfinal vibrational–rotational states from a specified initial reactantlevel are given by [19],

PJ¼0vj;v 0 j0 ðEÞ ¼

�h2

lRlrkvjkv 0j0

Av 0 j0 ðEÞf ð�jkv jÞ

��������2

ð7Þ

where Av0 j0(E) is the Fourier transform of time dependent coeffi-cients (Cm0 j0(t)). kvj and kv0j0 are the wave vectors for initial and finalchannels. f ð�jkv jÞ is the Fourier transform of initial Gaussian wavepacket.

When using a time dependent quantum method for scatteringproblems one is always faced with numerical difficulties associatedwith the reflection of the wave function from the end of the grid.Therefore, in order to avoid such a reflection, an imaginary poten-tial is used to damp the wave packet at the edges of the grid. Theabsorbing potential parameters are optimized as instructed by Vi-bok and Balint-Kurti [20]. At present calculations, a negative com-plex damping potential with a quadratic form has been used atboth edges of the grid.

Usually, it is not enough to obtain the integral cross sectionsand the thermal rate constants by nonreactive probabilities calcu-lated only for J = 0. All important contributions from different par-tial waves must be used to calculate converged cross sections, andfor high values of J each calculation becomes computationallyexpensive. This problem is often handled approximately by the J-shifting method based on Capture method [21] or Centrifugal Sud-den (CS) approximation.

Time dependent wave packet close coupling (TDWP–CC) calcu-lations for J > 0 are computationally very expensive so TDWP–CScalculations have been performed at selected total angularmomentum values in step of 10 and compared with QM–CCcalculations.

The calculation of total cross sections requires having the tran-sition probabilities for all available J values:

rv 0 j0 ðEcolÞ ¼p

k2v 0 j0

X1J¼0

ð2J þ 1ÞPJvjðEÞ ð8Þ

where Ecol = E � evj is the collision energy and evj is the initialrovibrational energy of the diatomic molecule, PJ

vjðEÞ is the en-ergy-dependent total reaction probability for a given initial state.

The initial-state resolved rate constant is calculated by Boltz-mann averaging of the integral cross section over the collisionenergy,

kvjðTÞ ¼8

plRk3BT3

!1=2 Z 1

0dEcolEcole�Ecol=kBTrvjðEcolÞ ð9Þ

where kB is the Boltzmann constant, and T is temperature.Another very appealing method for evaluating nonreactive rate

constants is the Uniform J-Shifting approach developed by Zhangand Zhang [22]. In this approach, the optimized value of rotational

Table 1Parameters of the wave packet calculations (Values are given in a.u.).

Translational energy center of the initial WP 0.4 eVR center and width of the initial WP 13.32 and 5R range and no. of grid points 1.18–29.5 and 512R range and no. of grid points 0.56–8.5 and 64No. of Legendre polynomials and of c points 40R and r absorption start at 22 and 6.52R and r absorption strength 0.01Analysis at R 17

24 J. Kłos et al. / Chemical Physics Letters 532 (2012) 22–26

constant (B) at a given temperature (T) for a range (Ji and Ji+1) of Jvalue, which are obtained by using TDWP–CS approximation, is ex-tracted from these accurate probability functions:

BiðTÞ ¼kBT

Jiþ1ðJiþ1 þ 1Þ � JiðJi þ 1Þ lnQJi

Q Jiþ1

!ði ¼ 1;2Þ ð10Þ

where QJi and QJiþ1 are partition like functions for Ji and Ji+1 refer-ence angular momentum and can be written in a simple form:

Q Ji ðTÞ ¼Z

PJi ðEcolÞe�Ecol=kBT dEcol ð11Þ

0.0

0.2

0.4

0.6

0.8

1.0

J=0

pro

babi

lity

WP inelastic; this work CCH Reactive ; J. Chem. Phys.,

124, 144301(2006) ABC reactive; this work ABC inelastic; this work

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.70.0

0.2

0.4

0.6

0.8

1.0

J=20

prob

abilit

y

collision energy (eV)

Figure 1. Nonreactive WP transition probabilities for O+ + H2 (m = 0, j = 0) ? O+ + H2 (m0 ,different J values. � CCH (close-coupling hyperspherical) results were taken from Ref. [9].method using the ABC program [18] are plotted for comparison with WP and Ref. [9] re

and

QJiþ1 ðTÞ ¼Z

PJiþ1 ðEcolÞe�Ecol=kBT dEcol ð12Þ

In result, the rate constant is given by,

kðTÞ ¼ 2pl3k3

BT3

!1=2

Q 0ðTÞX

J

ð2J þ 1Þe�BiðTÞJðJþ1Þ=kBT ð13Þ

3. Results and discussions

We have calculated exact nonreactive probabilities at totalangular momentum J = 0 for initial v = 0, j = 0 state by propagatingof the initial wave packet in reactant Jacobi coordinates R, r, and c.The properties of initial wave packet and the grid parameters aregiven in Table 1 which were converged for J = 0 and then usedfor the rest of J0s calculated. The calculation required 50000 itera-tions steps to converge.

In addition we also calculated quantum close coupling (QM–CC)reactive and non-reactive transition probabilities using ABC

J=10

WP CS inelastic CCH Reactive ; J. Chem. Phys.,

124, 144301(2006) ABC k

max=4 inelastic

ABC kmax

=4 reactive

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

J=30

collision energy (eV)

j0) summed over all final ro-vibrational states as a function of collision energy forReactive and nonreactive transition probabilities from fully quantum close-couplingsults.

0.1 0.2 0.3 0.4 0.5 0.6 0.70

20

40

60

80

100

WP inelastic; this work CCH reactive; J. Chem. Phys.,

124, 144301(2006)

σ

collision energy/eV

v=0,j=0

Figure 2. Initial state selected integral cross sections for O+ + H2 (v = 0, j = 0)nonreactive scattering as a function of the collision energy. � CCH (close-couplinghyperspherical) results were taken from Ref. [9].

1 2 3 4 5 6 7 8 9 101E-13

1E-12

1E-11

1E-10

1E-9

k(T)

(cm

3 s-1/m

olec

ule)

1000/T(K)

Uniform J-shifting Capture

Figure 3. TDWP–CS nonreactive rate constants depend on the initial quantumnumbers for O+ + H2 (v = 0, j = 0) nonreactive scattering as a function oftemperature.

J. Kłos et al. / Chemical Physics Letters 532 (2012) 22–26 25

scattering program [18] from the initial (v = 0, j = 0) state of H2 andsummed over all final product quantum numbers. These calcula-tions were performed at four selected values of J = 0, 10, 20 and30 of the total angular momentum with a maximum projectionquantum number kmax = 4. We compared the results with kmax = 6and confirmed that kmax = 4 is enough to converge probabilities forJ = 10 for collision energies up to 1 eV. The basis in QM–CC calcula-tions was composed of reactants and products channel limited bymaximum of channel energy of 3 eV. The propagation was ex-tended up to 25 bohr using 400 sectors.

Figure 1 display the exact (calculated using QM–CC) andapproximate (TDWP–CS) total O+ + H2 (v = 0, j = 0) nonreactivetransition probabilities as a function of the collision energy forJ = 0 and J > 0. Transition probabilities show no threshold andshows narrow peaks or resonances. These quantum resonancesare associated with quasi-bound states of the well depth of the

potential energy surface in the reactant channel. The reactionprobabilities calculated by Martinez et al. [8], using a close-coupling hyperspherical (CCH) method, are also plotted in Figure1 to confirm the accuracy of the TDWP–CS and QM-CC inelastictransition probabilities. That is, the sum of the total reactive andnonreactive probabilities in Figure 1 would have been close toone. The nonreactive probability decreases slowly in the low en-ergy interval considered while the reactive probabilities obtainedby CCH method are increasing in the low collision energy range.

All the transition probabilities show the same structure withmany resonances located in the whole energy region. The generaltrend of the nonreactive probabilities is to increase with increasingtotal angular momentum. The reason for this increases in the tran-sition probabilities with increasing J0s are that the centrifugal bar-rier to the reactive channel. The agreement between TDWP andQM nonreactive probabilities are excellent at J = 0 total angularmomentum but TDWP-CS calculations slightly overestimate prob-abilities for larger J0s as can be seen in Figure 1.

Figure 2 shows TDWP–CS nonreactive integral cross section(ICS) as a function of collision energy for the title reaction. TheICS decrease as collision energy increases without showing anythreshold. This behavior is a characteristic of a barrierless exother-mic reaction. The CCH reactive excitation function is considerablelarger (approx. 10 times) than the WP inelastic one as one can beexpected from the results displayed in Figure 1.

The initial state-selected rate constant for the O+ + H2 (v = 0,j = 0) rovibrational state has been calculated up to 1000 K usingthe TDWP–CS inelastic cross section and it is shown in Figure 3.In the figure it is compared with Uniform J-Shifting method, whichis using as well the TDWP–CS probabilities. The rate constant issensitive to temperature and it shows threshold. This behaviormay be attributed to the well in the entrance channel. As can beseen, Capture approximation yields rate constants in good agree-ment with the Uniform J-Shifting method. The rate constants forboth methods increase monotonically with temperature anddisplay an Arrhenius behavior as expected for many of the atom–diatom reactions. A linear fit of ln [k(T)] = A⁄(1000/T) + B yieldsA = �0.94858 and �0.92809 and B = �21.544 and �21.642 for theCapture model and Uniform J-Shifting methods, respectively. Weshould stress that due to the fact that the rate constants arederived from the TDWP–CS integral cross sections and as notedthese slightly differ from the TIQM calculations, we can expect thatthe current rate is slightly overestimated.

4. Conclusions

In this Letter, we presented a time-dependent quantum wavepacket and time-independent close-coupling calculations for theO+ + H2 nonreactive scattering and gave the dynamics informationof nonreactive probability, cross section and rate constant. Thenonreactive cross section has been obtained by summing up thenonreactive probabilities. The probability for J = 0 was calculatedand for some selected higher J0s transition probabilities obtainedby using TDWP–CS and QM–CC approximations. The intermediatevalues of J0s were estimated by means of two different J-Shiftingapproximations. The calculations showed that the transition crosssection does not exhibit a threshold behavior and the initialstate selected rate constant is significantly dependent on thetemperature.

Acknowledgments

We would like to thank M. González for his providing of potentialenergy function. Partial financial support from the Firat UniversityScientific Research Projects Unit (FUBAP) (Project No: FUBAP-1775) is gratefully acknowledged. The numerical calculations

26 J. Kłos et al. / Chemical Physics Letters 532 (2012) 22–26

reported in this Letter were performed at TUBITAK ULAKBIM, HighPerformance and Grid Computing Center (TR-Grid) and this workwas supported by the Turkish Scientific and Technological ResearchCouncil of TURKEY (TUBITAK) (Project No. 109T447). J.K. acknowl-edges support by the Chemical, Geosciences and Biosciences Divi-sion, Office of Basic Energy Sciences, Office of Science, USDepartment of Energy, under Grant No. DESC0002323 to MillardH. Alexander.

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