Time-dependent wave packet dynamics of the H+HLi reactive scatteringR. Padmanaban and S. Mahapatra Citation: The Journal of Chemical Physics 117, 6469 (2002); doi: 10.1063/1.1504702 View online: http://dx.doi.org/10.1063/1.1504702 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/117/14?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Time-dependent quantum wave packet dynamics of the C + OH reaction on the excited electronic state J. Chem. Phys. 138, 094318 (2013); 10.1063/1.4793395 An eight-degree-of-freedom, time-dependent quantum dynamics study for the H 2 + C 2 H reaction on a newmodified potential energy surface J. Chem. Phys. 127, 154304 (2007); 10.1063/1.2794757 Resonances in three-dimensional H + HLi scattering: A time-dependent wave packet dynamical study J. Chem. Phys. 120, 1746 (2004); 10.1063/1.1634559 Time-dependent quantum wave packet study of the H+DCN→HD+CN reaction J. Chem. Phys. 117, 5642 (2002); 10.1063/1.1501888 Time-dependent quantum wave packet study of H + HCN → H 2 + CN reaction J. Chem. Phys. 117, 172 (2002); 10.1063/1.1481385

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Time-dependent wave packet dynamics of the H ¿HLi reactive scatteringR. Padmanaban and S. Mahapatraa)

School of Chemistry, University of Hyderabad, Hyderabad, 500 046, India

~Received 23 May 2002; accepted 15 July 2002!

We report the initial-state-selected and energy-resolved total reaction probabilities for the H1HLisystem calculated with the aid of a time-dependent wave packet approach. Theab initio potentialenergy surface~PES! of Dunneet al. @Chem. Phys. Lett.336, 1 ~2001!# is employed for the purpose.The reaction probabilities are reported for both the collinear and the three-dimensional arrangementsof the reacting system. In the collinear arrangement the exothermic reaction path H1HLi→H21Liis investigated only, whereas in the three-dimensional arrangement both competing reaction pathsare investigated and the channel specific reaction probabilities are reported. The hydrogen exchangechannel, in general, is found to be more favored over the LiH depletion channel. Both the collinearand the three-dimensional reaction probabilities reveal that the scattering occurs via resonanceformation at low energies and the dynamics follows a more direct path at high energies. The overalldynamical characteristics of the system are consistent with the absence of any barrier in theunderlying PES. The effect of the rotationally and vibrationally excited reactant LiH molecule onthe dynamics is discussed. The importance of the noncollinear configuration of the reacting systemon the LiH depletion dynamics is also delineated. ©2002 American Institute of Physics.@DOI: 10.1063/1.1504702#

I. INTRODUCTION

The cooling of the primordial matter and its collapse iscontrolled by the presence of a trace amount of gas mol-ecules. Way back in 1984 the preliminary study of Lepp andShull1 indicated that a large fraction of primordial lithiumexists as LiH. Subsequently, the effects of presence of thisspecies in the ‘‘early universe’’ have been discussed by vari-ous researchers2,3 and the formation and depletion of LiHand its ionic counterparts were believed to play rather impor-tant role in the steller evolution and galactic lithium produc-tion. Furthermore, the presence of such molecular speciesmay cause significant anisotropy of the cosmic backgroundradiation by undergoing Thompson scattering with the emit-ted photon.4 Recently, Stancilet al.5 have carried out a de-tailed study and pointed out possible pathways for the for-mation and depletion of LiH via collisions with H1 and H, inthe interstellar cloud.

We focus here on the dynamics of H1HLi collisionswhich can proceed via the following two reactive paths:

H1HLi→H21Li @reaction 1 ~R1!#

→Li1H @reaction 2 ~R2!#.

While the path~R1! is highly exothermic and considered tobe one of the event contributing to LiH depletion, the hydro-gen exchange path~R2! is mostly of thermoneutral type,leading to the retention of LiH. In addition, there is anotherpath leading to the retention of LiH via nonreactive colli-sions with H~NR1!. The possibility of the reverse endother-mic process of the reaction~R1!, contributing to the LiHformation, has also been pointed out5 and studied in the

literature.6 The rates of the reaction~R1! in the temperaturerange of interest have been estimated qualitatively by Stancilet al.5 To date, however, very little is known~both theoreti-cally and experimentally! about the H1 HLi collision dy-namics. In this paper we undertake this exercise and attemptto study it theoretically by a time-dependent quantum me-chanical approach.

Earlier theoretical studies on the LiH2 system have fo-cused on various aspects of its electronic structure. Boldyrevand Simons7 have reported the vertical and adiabatic ioniza-tion potentials of LiH2

2 at the MP2 level of theory withtriple-zeta plus polarization and 6-31111G* diffuse bases.Their study revealed a stableD`h structure (1sg

21su2) of

LiH22 and the detachment of the extra electron from the high-

est occupied molecular orbital~HOMO! (1su) leads to theformation of LiH2 in the 2Su

1 state with a vertical ionizationpotential of;3.06 eV.7 The latter is in close agreement withthat reported by Senekowitsch and Rosmus8 calculated at theCEPA level and of Boldeyrev and Von Niessen9 calculated atthe Green’s function level. LiH2 in its 2Su

1 state is reportedto possess an imaginary bending frequency of 277i cm21,indicating the absence of a local minimum.7 On the otherhand, a local minimum along the bending mode for the bent(C2v) geometry of LiH2 correlating to a2B2 species hasbeen discovered.7 This structure is reported to be more stablewith respect to H1HLi dissociation but unstable with respectto Li1H2 dissociation. At the optimal geometry of this2B2

state a low-lying 2A1 state correlating with the Li(2S)1H2(1Sg

1) asymptote is also reported to exist.7 This 2A1

state can undergo a nonadiabatic crossing with the2B2 stateand therefore can form a conical intersection.6 Such an inter-section with a minimum lying only;2.4 kcal mol21 abovethe minimum of the2B2 state has been found, which causesan indirect dissociation of the2B2 state into Li1H2 througha!Electronic mail: smsc@uohyd.ernet.in

JOURNAL OF CHEMICAL PHYSICS VOLUME 117, NUMBER 14 8 OCTOBER 2002

64690021-9606/2002/117(14)/6469/9/$19.00 © 2002 American Institute of Physics

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the nonadiabatic interactions with the2A1 state. This revealsthe importance of the noncollinear configuration of LiH2 inthe H1HLi collision dynamics. The detachment of an elec-tron from the 1sg molecular orbital of LiH2

2 leads to theformation of LiH2 in the 2Sg

1 state. The latter supports alocal minimum at theD`h configuration. The correspondingvertical and adiabatic ionization potentials were found tohave nearly the same value of;3.35 eV.7

In a latter theoretical study Clarkeet al.10 have reportedthe reactive potential energy surface of~R1! for the collineararrangements of the three nuclei obtained through extensivevalence bond calculations using nonorthogonal, spin-coupledwave functions with optimized orbitals. These authors havealso calculated10 the initial-state-selected reaction probabili-ties for the forward and the reverse processes of~R1! for thecollinear configuration by a quasiclassical trajectory as wellas a time-dependent wave packet~WP! approach. Their find-ings revealed a strong exothermicity of~R1! with an energygain of ;2.0238 eV including the zero-point energy~ZPE!correction, and a small early barrier of;0.036 eV.10 Thedynamical results mostly indicated a direct nature of the re-action proceeding without any transition state. A sharp reso-nance feature, very sensitive to the initial most probabletranslational energy, has been observed in the threshold re-gion of the reaction probability curve. Furthermore, it is re-ported that low translational energy of the reagent fragmentsfavors the depletion process and vibrational excitation ofLiH does not have any noticeable effects on it.10 These find-ings therefore indicate that ‘‘cold’’ primordial LiH moleculescan be depleted efficiently by reactive encounters with Hatoms even at a very low value of translational energy. Theirstudy also indicated that much of the reagent translationalenergy emerge as the product vibrational energy in the exo-thermic LiH depletion path.10

Very recently a global three-dimensional analytical po-tential energy surface~PES! for the H1HLi system is re-ported by Dunne, Murell, and Jemmer~DMJ PES!.11 Theseauthors have derived this PES by fitting a many-body poten-tial to energies obtained by performing high qualityab initioconfiguration interaction calculations using an augmentedcorrelation consistent valence double zeta basis set. The re-sulting PES did not reveal the existence of any barrier to thereaction.11 Apart from this minor difference, the collinearsurface of Clarkeet al.10 is well reproduced by the globalanalytical DMJ PES.11 The latter authors have also studiedthe temperature dependence of the rate of the depletion re-action ~R1! and provided with a satisfactory analytical fit tothis rate.11

We in this paper use the DMJ PES~Ref. 11! and reportthe initial-state-selected and energy-resolved channel-specific reaction probabilities of H1HLi collisions with theaid of a time-dependent WP approach. In addition to report-ing the three-dimensional reaction probabilities, we alsoshow results obtained from the calculations for the collinearconfiguration of~R1! in order to check the consistencies ofthe present results with those of Clarkeet al.10 Furthermore,the collinear calculations enable us to optimize the grid pa-rameters more carefully for this reaction which proceeds on asteep downhill path. Apart from these, it can be seen that the

collinear results have no real significance when comparedwith the three-dimensional results. The sharp resonance fea-tures seen at the onset of the collinear reaction survives inthe three-dimensional results. The channel-specific probabili-ties show a preference of the reactive path~R2! over ~R1! inthe H1HLi collision dynamics for any given energy.

The paper is organized in the following way: In Sec. IIwe describe the theoretical framework of calculating the re-action probabilities. In Sec. III we discuss the features of theDMJ PES and present the reaction probabilities obtained forthe collinear and the three-dimensional arrangements and in-terpret them. The paper is closed by summarizing the find-ings in Sec. IV.

II. THEORETICAL FRAMEWORK: WAVE PACKETDYNAMICAL TREATMENT OF REACTIVESCATTERING

The time-dependent WP approach is well established totreat bimolecular reactive collision processes and has beenvery successfully applied to numerous atom-diatom, diatom-diatom, and atom-polyatomic gas phase chemical reactionsin recent years~for example, see Refs. 12 and 13 and refer-ences therein!. Since the details of such methods are welldocumented in the literature, we will be concerned only withthe essentials here.

The three-dimensional~3D! interaction Hamiltonian forthe H1HLi system in mass-scaled reactant channel Jacobicoordinates (R,r ,g) and for the total angular momentumJ50 in the body-fixed frame is given by14

H51

2m@PR

21Pr2#1

j2

2I1V~R,r ,g!

52\2

2m F ]2

]R2 1]2

]r 2G2\2

2I

1

sing

]

]g S sing]

]g D1V~R,r ,g!, ~1!

wherePR andPr are the momentum operators correspondingto the two mass-scaled Jacobi distancesR ~distance of the Hatom from the center of mass of the HLi molecule! and r~HLi internuclear distance!, respectively, andj is the rota-tional angular momentum operator of the HLi molecule as-sociated with the Jacobi angleg ~angle betweenRW and rW).The quantitym, m5AmH

2mLi /(2mH1mLi) ~wheremH andmLi are the masses of the nuclei H and Li, respectively!, isthe three-body uniform~for both channelsR and r ) reducedmass, andI, I 5mR2r 2/(R21r 2), is the three-body momentof inertia. The body-fixedz axis is defined to be parallel tothe Jacobi vectorRW and HLi lies in the (x,z) plane. Theanalyticalab initio DMJ PES~Ref. 11! for the ground elec-tronic state of LiH2 is employed forV(R,r ,g).

The dynamics of the system is followed by numericallysolving the time-dependent Schro¨dinger equation~TDSE! ona grid in (R,r ,g) space. For an explicitly time-independentHamiltonian, the solution reads

uC~ t !&5expF2 iH t

\G uC~ t50!&. ~2!

6470 J. Chem. Phys., Vol. 117, No. 14, 8 October 2002 R. Padmanaban and S. Mahapatra

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Here uC(t50)& is the initial wave function pertinent to theH1HLi reactants anduC(t)& is the wave function of thesystem at timet, depending on the set of Jacobi coordinatesR, r andg. The initial wave functionuC(t50)& is preparedin the asymptotic reactant channel (R→`) where there is noinfluence of the interaction potential. In such a situation itcan be described as a product of the translational wave func-tion F(R) for motion alongR and the rovibrational wavefunction fv j (r ) of the HLi molecule. In the present case forJ50, it is given by

uC~ t50!&5F~R!fv j~r !A2 j 11

2Pj~cosg!. ~3!

We choose a minimum uncertainty Gaussian wave packet~GWP! for F(R):

F~R!5S 1

2pd2D 1/4

expF2~R2R0!2

4d2 2 ik0~R2R0!G . ~4!

The quantityd is the width parameter of the GWP andR0

and k0 correspond to the location of its maximum in thecoordinate and momentum space, respectively. The functionsfv j (r ) along with the normalized Legendre polynomials@Pj (cosg)# represent the rovibrational eigenfunction corre-sponding to a vibrationalv and rotationalj state of the HLimolecule. The functionfv j (r ) are obtained by solving theeigenvalue equation of the free HLi molecule:

F2\2

2m8

d2

dr821V~r 8!1

j ~ j 11!\2

2m8r 82Gfv j~r 8!5ev jfv j~r 8!.

~5!

Herem8 is the reduced mass of HLi molecule,ev j the energyeigenvalue,r 85r (m/m8)1/2 is the unscaled internuclear dis-tance, andV(r 8) is the potential energy of the HLi moleculeobtained from the DMJ PES Ref. 11 by settingR→`. Weused the sine-DVR approach of Colbert and Miller15 to solvethe above eigenvalue equation.

The time-evolution operator exp@2iHt/\# in Eq. ~2! isevaluated by dividing the time axis intoN segments of lengthDt. The exponential operator at each time step is then ap-proximated by the split-operator scheme,16

expF2 iHDt

\G5expF2 iVDt

2\ GexpF2 i j2Dt

4I\ G3expF2 iTDt

\ GexpF2 i j2Dt

4I\ G3expF2 iVDt

2\ G1O@~Dt !3#, ~6!

where T represents the total radial kinetic energy operatoralong the Jacobi coordinatesR andr. Equation~6! is used inconjunction with the fast Fourier transform~FFT! method17

to evaluate the action of the exponential containing the radialkinetic energy operator and the discrete variable representa-tion ~DVR! method18–20 to evaluate the action of the expo-nential containing the rotational kinetic energy operator( j2/2I ) on the wave function. The coordinate grid consists ofequally spaced pointsRl andr m along the Jacobi distancesR

and r, respectively. The grid along the Jacobi angleg ischosen as the nodes of ann-point Gauss-Legendre quadra-ture ~GLQ!.21 The initial wave function at each node(Rl ,r m ,gn) of this grid is given by

uC~Rl ,r m ,gn ,t50!&

5uC lmn&5AwnF~Rl !fv j~r m!A2 j 11

2Pj~cosgn!, ~7!

wherewn is the weight of the GLQ associated with the gridpointn. The action of the rotational kinetic energy operator isthen carried out by transforming this DVR wave function@Eq. ~7!# to the angular momentum space@finite basis repre-sentation~FBR!#, multiplying it by the diagonal value of theoperator (e2 i j ( j 11)Dt\/4I), and transforming it back to theDVR representation. Numerically this is accomplished in asingle step:20

expF2 i j2Dt

4I\ G uC lmn8&

5(n

H(j

Tn8, j† e2 i j ( j 11)Dt\/4ITj ,nJ uC lmn&, ~8!

where j is the rotational quantum number of the HLi mol-ecule. The coefficientsTj ,n are the elements of the DVR-FBR transformation matrix, constructed in terms of Leg-endre polynomials~eigenfunctions of thej2 operator forJ50) ~Refs. 18–20!,

Tj ,n5Aw~n!A2 j 11

2Pj~cosgn!, ~9!

andTn, j† are the elements of the inverse transformation ma-

trix, the Hermitian conjugate toTj ,n .Once the TDSE@Eq. ~2!# is solved, the initial statei (v j

state of reactant HLi molecule! selected and energy resolvedfrom the reaction probability is calculated from the expecta-tion value of the flux operatorF in the basis of the energynormalized time-independent reactive scattering wave func-tion evaluated at a dividing surface along the reaction coor-dinate~herer ) ~Refs. 22 and 23!:

PiR~E!5(

fuSf i

Ru25^F~R,r ,g,E!uFuF~R,r ,g,E!&ur 5r d,

~10!

where Sf iR is the reactive scattering matrix from an initial

statei of the reactant to a final statef of the product. The fluxoperatorF is defined in terms of the dividing surface~at r5r d) and is given by24

F52i\

2m F ]

]rd~r 2r d!1d~r 2r d!

]

]r G . ~11!

In terms of Eq.~11!, Eq. ~10! can be written as

PiR~E!5

\

mImF K F~R,r d ,g,E!U]F~R,r d ,g,E!

]r L G . ~12!

The quantity on the right-hand side of the above equation isintegrated over the entire range ofR andg. The quantityErepresents the total energy~relative translational1rovibrational! of the collisional system. The energy nor-

6471J. Chem. Phys., Vol. 117, No. 14, 8 October 2002 Wave packet dynamics of H1HLi

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malized time-independent reactive scattering wave functionis calculated along the dividing surface atr 5r d as

uF~R,r d ,g,E!&5uc~R,r d ,g,E!&/kE . ~13!

The functionc(R,r d ,g,E) is obtained by Fourier transform-ing the time-evolved WPC(R,r ,g,t) along the dividing sur-face:

c~R,r d ,g,E!51

A2pE

2`

1`

C~R,r ,g,t !eiEt/\dtur 5r d.

~14!

The quantitykE in Eq. ~13! is the weight of the translationalenergy component contained in the initial WP for a giventotal energyE:

kE5S m

2p\kD 1/2E2`

1`

F~R!eikRdR, ~15!

where k5A2m(E2ev j )/\, with ev j being the initial rovi-brational energy of the HLi molecule.

The reactive flux flowing into a specific reaction channel@~R1! or ~R2!# is estimated by comparing the internucleardistances of the product H2 and LiH molecules. The reactiveflux in which the H2 distance is smaller than the LiH~prod-uct! distance is considered to represent the~R1! channel andthe rest is considered to represent the~R2! channel.25 Finally,the channel-specific probabilities are obtained by integratingthis flux in the respective channel.

In dynamical studies involving scattering systems, as theWP moves forward in time, its fast moving components ap-proach the grid boundaries and are no longer relevant for therest of the dynamics.22 Therefore, to avoid unphysical reflec-tions or wrap arounds of these components from the bound-aries of a finite-sized grid, the WP at each time step is mul-tiplied by a damping function26

f ~Xi !5sinFp2 ~Xmask1DXmask2Xi !

DXmaskG , Xi>Xmask,

~16!

which is activated outside the dividing line in the productchannel and also in the asymptotic reactant channel.Xmask isthe point at which the damping function is initiated andDXmask (5Xmax2Xmask) is the width ofX over which thefunction decays from 1 to 0, withXmax being the maximum

value of X in that direction, in a particular channel. Theproperties of the initial WP and the grid parameters used forthe numerical calculations are listed in Table I.

III. RESULTS AND DISCUSSION

A. Potential energy surface

In the following we use the analytical DMJ PES~Ref.11! to calculate the reaction probability of~R1! and ~R2!.Dunneet al.11 have derived this PES by fitting theab initio~including configuration interaction, employing a double-zeta basis set! calculated potential energy values to a many-body expansion function in the following way. The two-bodyterms appearing in this expansion were fitted to the extendedRydberg function with parameters derived from the spectro-scopic data on H2 and LiH, in order to impose the correctdissociation limits of these diatomic species and to reproducethe experimental exothermicity of the reaction~R1!. Thethree-body energies were obtained by subtracting the atomicterms and the two-body potential energies from the calcu-latedab initio energies and were fitted to an analytical equa-tion of the form11

V~R1 ,R2 ,R3!5V0F11(i jk

Ci jk~S1! i~S2! j~S3!kG3@12tanh~g1S1/21g3S3/2!#, ~17!

where, S15R11R222RLiH0 ; S25R12R2, and S35R3

2RHH0 ~with R1 andR2 representing two LiH distances and

R3 the HH distance!. The parameters in the above functionwere obtained by a least-squares fit with 21 coefficients(Ci jk) to represent the PES in the entire configuration rangesatisfactorily.

In order to demonstrate the microscopic details of thedynamics of~R1! and ~R2! we briefly discuss on the salientfeatures of the above PES. In Fig. 1 we show the minimumenergy path for the depletion reaction~R1! for the collinearconfiguration. The zero of the energy scale corresponds tothe minimum of the LiH potential occurring forR→` in thereactant channel. The zero-point vibrational level of the re-actant LiH as well as the product H2 is also shown in thefigure. The highly exothermic nature of the depletion reac-tion leading to Li1H2 formation is clearly revealed by Fig.

TABLE I. Numerical grid parameters and properties of the initial wave function used in the calculations ofreaction probabilities.

Parameter Value Description

NR /Nr /Ng 128/64/64 Number of grid pointsRmin /Rmax (a0) 1.0/18.78 Extension of the grid alongRr min /r max (a0) 1.0/8.56 Extension of the grid alongrDR/Dr (a0) 0.14/0.12 Grid spacings alongR and r, respectivelyr I (a0) 5.44 Location of the dividing surface in the product channelRmask/r mask (a0) 14.72/5.8 Starting point of the masking functionR0 (a0) 12.0 Initial location of the center of the GWP in the coordinate

spaced (a0) 0.3 Initial width parameter of the GWPDt ~fs! 0.1347 Length of the time step used in the WP propagationT ~fs! 413.76 Total propagation time

6472 J. Chem. Phys., Vol. 117, No. 14, 8 October 2002 R. Padmanaban and S. Mahapatra

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1. The reaction exothermicity is;2.086 56 eV including theZPE corrections. The height of the diffuse barrierlike struc-ture apparently occurring for nearly equal HH and LiH dis-tances is;20.028 eV.

In Fig. 2 we plot the contour level diagram of the abovePES in the (R,r ) plane and for various values of the Jacobiangleg indicated in each panel. The steep decrease in thepotential energy while going from the reagent to the productchannel revealing the attractive nature of the surface isclearly observed from the contour line diagrams of Fig. 2.For g<90° the equilibrium LiH ~reactant! distance issmaller in the small-R region than in the large-R region,which indicates a weak excitation of LiH vibration in the

interaction region and hence low reaction probability to both~R1! and ~R2! channels. The topology of the PES forg>90°, however, indicates a relatively large vibrational exci-tation of the reactant LiH in the interaction region and hencehigh probability for the reactions to take place. Therefore, itis expected that the noncollinear collisions may make signifi-cant contribution to the reaction dynamics of H1HLi.

This view is further supported by investigating the cutsof the above PES along the Jacobi angleg for fixed values ofR and keeping the reactant LiH molecule at its equilibriumconfiguration. Such cuts are shown in Fig. 3. The meaning ofdifferent line types is indicated in the panel. It can be seenfrom Fig. 3 that at smaller values of the Jacobi distanceR thepotential exhibits an asymmetric double-well shape. Thisasymmetry results from the asymmetric nature of the reactantLiH molecule. The collisions at the bent geometry~nearC2v) of LiH2 seem to be more favorable at small values ofR.Therefore, an insertion type of mechanism appears to bemore likely at such geometries. With increase in the ap-proach distanceR the well at the bent geometry transformsinto a small barrier and the collisions at near collinear geom-etry appear to be more favorable. To this end we draw atten-tion of the readers to the polar plot presented in Ref. 11 inFig. 2, which clearly reveals a narrow ‘‘approach cone’’ forthe H abstraction from LiH by the incoming H atom, as aresult of the steric hindrance caused by the relatively bulkyLi atom.27

B. Collinear reaction dynamics

The approach of the incoming H atom to the H end ofthe reactant LiH molecule is considered in the collinearmodel and the dynamical calculations are mathematicallyconstrained to the (R,r ) plane. In this configuration thedepletion path~R1! is the only reactive channel and thedynamics takes place on the steep downhill PES~cf. Figs. 1and 2!.

The total reaction probability for H1HLi( v50)→H2 (Sv8)1Li, for an initial most probable collision en-ergy of 0.55 eV as a function of the total energyE ~H-HLi

FIG. 1. The minimum energy path for the H1HLi→H21Li depletion reac-tion for the collinear approach. Energy is measured relative to the minimumof HLi potential. The zero-point vibrational energies of the reactant HLi andthe product H2 molecules are indicated in the figure. The magnitude of thereaction exothermicity is;2.09 eV. The height of the diffuse barrierlikestructure occurring for nearly equal LiH and HH distances is;20.028 eV.

FIG. 2. Contour line diagrams of the DMJ PES plotted in the (R,r ) planeand for various values ofg indicated in the panel. The spacing between thesuccessive contour lines is 0.5 eV and the zero of the energy corresponds toR→`. The minimum of the contour occurs at22.281,21.363,20.991,20.866, 20.0136, and20.275 eV forg50°, 30°, 60°, 90°, 120°, and180°, respectively.

FIG. 3. Cuts of the DMJ PES along the H-HLi approach angleg and forvarious values ofR, indicated in the panel. The value ofr is fixed at 3.015a0, corresponding to the equilibrium internuclear distance of the LiH mol-ecule.

6473J. Chem. Phys., Vol. 117, No. 14, 8 October 2002 Wave packet dynamics of H1HLi

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translational1HLi vibrational! is shown in Fig. 4. The en-ergy distribution of the initial GWP is shown in the inset.The properties of the initial WP and other grid parametersare the same as those listed in Table I. The total energyE ismeasured relative to the minimum of the HLi potential. TheWP is time evolved for a total of 413.76 fs, and the norm(^CuC&) of the WP goes down to 9.1731023 at the end ofthe time evolution. The zero-point vibrational energy of theHLi molecule is 0.086 56 eV. From Fig. 4 it can be seen thatthe onset of the reaction occurs nearly at the ZPE of HLimolecule indicating a zero reaction threshold and hence theabsence of any barrier in the underlying PES. The pro-nounced oscillations in the reaction probability curve at lowenergies reveal the formation ofresonancesduring thecourse of the reaction at these energies. These resonancescorrespond to the metastable excited vibrational levels ofLiH2 . The resonance structures are less pronounced in thehigh-energy part (.0.9 eV! of the reaction probability curve.Therefore, at high energies the collisions become more di-rect, leading to the formation of the product. The above re-sults are in fairly good agreement with the classical andquantum mechanical findings of Clarkeet al.10 The remain-ing quantitative differences between the two may be attrib-uted to the differences in the potential energy surfaces usedin the two calculations.

In order to further check on the sensitivity of the aboveresults, we have carried out calculations for different initialmost probable collision energies and different width param-eters of the GWP and also for different choices of the loca-tion of the dividing line at the product channel. We obtainedconsistent reaction probabilities calculated for initial mostprobable translational energies of 0.35 eV, 0.55 eV, 0.62 eV,and 0.7 eV. Also the variation of the width parameter of theGWP in the range~0.25–0.35! a0 is considered. We findsome minor variation of the reaction probabilities for differ-ent choices of the dividing line at the product channel. Forinstance, the first sharp peak in the reaction probability curve

disappears when this line is moved closer to the interactionregion of the PES.

The reaction probability for the collisions of H atomwith the vibrationally excited reactant HLi molecule isshown in Fig. 5. The initial vibrational quantum number ofthe HLi molecule is indicated in each panel. It can be seenfrom Fig. 5 that the onset of the reaction progressively shiftsto the expected higher total energy. The exact location of thisonset on the energy axis in each panel corresponds to thevibrational energy of the reactant HLi molecule. This there-fore again supports the fact that the reaction has no thresholdand the underlying PES is of early attractive nature. Thesharp resonance structures at low energies also survive incollision with the vibrationally excited reactant molecule. Itcan also be seen that the reaction probability decreases withthe vibrational excitation of the reactant HLi molecule,which is also indicated by the results of Clarkeet al.10

C. The three-dimensional reaction dynamics

In three-dimensional collisions both the reactive chan-nels~R1! and~R2! are open. In order to calculate the reactionprobability for ~R1! and ~R2! it is necessary to separatelyintegrate the reactive flux flowing into these channels. Asmentioned above, we distinguish these two channels by com-paring the internuclear distances of the product H2 and LiHmolecules. The calculations are carried out on the same(R,r ) grid as employed for the collinear dynamics and a64-point GLQ is used along the angleg. The properties ofthe initial WP and other grid parameters are the same asthose listed in Table I. The reaction probabilities are calcu-lated at an energy interval of 0.005 eV. The convergence ofthe results is checked with respect to the grid parameters.

The reaction probabilities for the H1HLi ( v50, j50) collisions as a function of the total energyE are plottedin Fig. 6. The energy distribution of the initial GWP is thesame as that shown in Fig. 4. The norm of the WP at the endof the time evolution goes down to 2.4731023 in this par-ticular case. The probabilities for the H21Li ~R1! andLiH1H ~R2! reactive channels are shown in the figure, and

FIG. 4. Total reaction probability as a function of total energyE ~H, LiHtranslational1HLi vibrational! for the H1HLi( v50)→H2 (Sv8)1Lidepletion reaction in collinear configuration and for a initial most probablecollision energy of 0.55 eV. The energyE is measured relative to the mini-mum of the HLi potential. The energy distribution of the initial translationalGWP is shown in the inset.

FIG. 5. Same as in Fig. 1, obtained for the vibrationally excited reactant HLi~indicated in the panel! showing the effect of reagent vibration on the dy-namics of the depletion reaction in collinear configuration.

6474 J. Chem. Phys., Vol. 117, No. 14, 8 October 2002 R. Padmanaban and S. Mahapatra

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the corresponding channels are indicated on the respectivecurves. These are the total reaction probabilities for the par-ticular channel which resulted from summation over all openvibrational (v8) and rotational (j 8) levels of the correspond-ing product molecule at a given energy. The overall reactionprobability considering both the channels together, obtainedfrom a separate calculation, is included in the figure and isshown by the solid curve marked as R11R2. To confirm theidentity of the results the sum of the probabilities for R1 andR2 channels on a coarse energy grid is superimposed on itand is shown by the dots. It can be seen that the sum of thechannel specific probabilities exactly converges to the over-all reaction probabilities. The results of Fig. 6 reveal that the

hydrogen exchange probability is larger than the LiH deple-tion probability. Therefore, at any given energy the majorportion of the reactive flux flows into the hydrogen exchange~R2! channel in three-dimensional collisions. This observa-tion is consistent with the small value of the rate constantreported for the LiH depletion path by Stancil and Dalgarno5

and also by Dunneet al.11 We note that, due to a relativelybulky nature of the Li atom, only a narrow cone available tothe incoming H atom to approach to the H end of LiH whichin part reduces the probability of the LiH depletion.11,27 Itcan be seen from Fig. 6 that the onset of both reactions~R1!and ~R2! occurs at the zero-point vibrational energy of LiH,indicating that these reactions proceed without any barrier.The oscillations in reaction probability curves indicate theexistence of short-lived resonances at low energies. On theother hand, a relatively smooth variation of the reactionprobability reveals a direct nature of the collision dynamicsat high energies. It can be seen by comparing the abovereaction probability for the LiH depletion channel~R1! to thesame obtained for the collinear configuration~cf. Fig. 4! that,despite an overall qualitative agreement the collinear modeldoes not provide a realistic description of the dynamics ofthis system. Mechanistically, both the collinear and three-dimensional reaction probabilities reveal the existence ofresonances and therefore an indirect nature of the scatteringat low energies and a more direct nature of the same at highenergies. A quantitative comparison of the two reaction prob-ability curves clearly indicates the importance of the noncol-linear collisions in the dynamics which is also revealed bythe topological characteristics of the PES~cf. Figs. 2 and 3!.

In order to study the effect of the reagent vibration androtation on the dynamics, in Fig. 7 we show the reactionprobability curves obtained with the vibrationally and rota-tionally excited reactant LiH molecule. The initial vibrationaland rotational (v, j ) quantum state of the latter is indicated in

FIG. 6. Total reaction probability as a function of the total energyE for theH1HLi ( v50, j 50) collision in three dimensions and for total angularmomentumJ50. The total reaction probabilities for the~R1! and~R2! chan-nels are shown separately, and the corresponding curves are marked as R1and R2, respectively. The overall reaction probability~considering bothchannels together! is shown by the solid curve marked as R11R2, and thesum of the probabilities for~R1! and~R2! is superimposed on it on a coarseenergy grid and is shown by the dots.

FIG. 7. Same as in Fig. 1, obtainedwith different vibrational and rota-tional (v, j ) states of the reactant LiH,illustrating the effect of reagent vibra-tion and rotation on the H1HLi colli-sion dynamics. In each panel the solidcurve marked as R11R2 representsthe sum of the two probability curvesR1 and R2.

6475J. Chem. Phys., Vol. 117, No. 14, 8 October 2002 Wave packet dynamics of H1HLi

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each panel. The overall trend of variation of reaction prob-abilities in each panel is similar to that observed for thereaction with LiH (v50, j 50) ~which is again included inthe figure for clarity!. The onset of the reaction with thevibrationally and rotationally excited reactant LiH occurs atthe internal energy of the corresponding (v, j ) state, indicat-ing the absence of any barrier to the reaction. The sharpresonance structures near the onset of the reaction also showup in all probability curves, which confirms an indirect na-ture of the dynamics at low energies. The resonance struc-tures disappear at high energies, and the dynamics is moredirect at those energies. It can also be seen that for any (v, j )state the LiH depletion probability is lower than the H ex-change probability for any given energy.

Unlike in the collinear case for the reaction~R1! thethree-dimensional reaction probability results do not showany systematic variation with the vibrational excitation of thereactant LiH. However, the reaction probabilities for~R2!show a slight decrease with the vibrational excitation of thereactant LiH. The rotational excitation of the reactant LiH,on the other hand, do not cause any systematic variation ofthe reaction probabilities of either~R1! or ~R2! and appearsto play only a minor role in the corresponding dynamics.

IV. SUMMARY AND OUTLOOK

A detailed theoretical account of the dynamics ofH1HLi collisions in three dimensions is presented. The ini-tial state-selected and channel-specific energy-resolved reac-tion probabilities are reported. The reaction probabilities areobtained with the aid of a time-dependent WP approach andusing the analytical DMJ PES~Ref. 11! for the system.

The reaction probability curves for both the LiH deple-tion channel~R1! as well as the hydrogen exchange channel~R2! exhibit sharp resonance structures at low energies. Thistherefore indicates an indirect nature of the scattering atthose energies. The resonances are less pronounced at furtherhigh energies and the scattering becomes more direct. Anexamination of the probability results at the onset of thereactions indicates a zero threshold energy, in agreementwith the absence of any barrier in the underlying PES. Thechannel-specific reaction probability values indicate thatH1HLi collision dynamics predominately follow the hydro-gen exchange path~R2! at any given energy which is con-sistent with the very small rate constant reported by Stanciland Dalgarno5 and also by Dunneet al.11 for LiH depletion.This finding can be partly attributed to the steric effects as-sociated with the bulky Li atom, which hinders the incomingH atom from approaching to the H end of the LiH moleculeto react via the path~R1!.

A careful examination of the reaction~R1! for the col-linear configuration revealed a decrease of probability withthe vibrational excitation of the reactant LiH molecule. Sucha general trend, however, is not revealed by the correspond-ing three-dimensional reaction probabilities. The probabilityfor the exchange path~R2! in three dimensions, on the otherhand, decreases slightly with the vibrational excitation of thereactant LiH. The rotational excitation of the reactant LiHmolecule, however, does not show any such systematicvariation of reaction probabilities either for~R1! or ~R2!.

A comparison of the reaction probabilities for~R1! forthe collinear and the three-dimensional configuration clearlyrevealed the importance of the noncollinear collisions in thecorresponding dynamics. In the three-dimensional collisionsa substantial amount of reactive flux is pushed away from thecollinear path. The dynamics at the bent geometry of LiH2 ismost likely to follow an insertion type of mechanism. Thisanticipation is further strengthened by the observation of aconical intersection in the bent geometry of LiH2.7 A furtherdetailed analysis of the energy disposal mechanism and theimportance of the nonadiabatic effects on the H1HLi colli-sion dynamics is presently underway by our group and willbe considered in a forthcoming publication.

ACKNOWLEDGMENTS

Thanks are due to Dr. T. P. Radhakrishnan for his interestin this work and support. This study is in part supported by agrant from the Department of Science and Technology, NewDelhi under the fast track scheme.

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6477J. Chem. Phys., Vol. 117, No. 14, 8 October 2002 Wave packet dynamics of H1HLi

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