Chemical Physics Letters 501 (2011) 252–256

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Chemical Physics Letters

journal homepage: www.elsevier .com/locate /cplet t

Quantum dynamics of H + LiH+ reaction on its electronic ground state

Tanmoy Roy, T. Rajagopala Rao, S. Mahapatra ⇑School of Chemistry, University of Hyderabad, Hyderabad 500 046, India

a r t i c l e i n f o a b s t r a c t

Article history:Received 1 September 2010In final form 23 November 2010Available online 26 November 2010

0009-2614/$ - see front matter � 2010 Elsevier B.V. Adoi:10.1016/j.cplett.2010.11.075

⇑ Corresponding author. Fax: +91 40 23011537/230E-mail address: smsc@uohyd.ernet.in (S. Mahapat

State specific dynamics of the H + LiH+ reaction is theoretically investigated on its electronic groundpotential energy surface employing a time-dependent wave packet approach. Channel specific integralreaction cross-sections and thermal rate constants are reported. Impact of the low-energy collision-induced dissociation channel on the reactive dynamics is discussed.

� 2010 Elsevier B.V. All rights reserved.

1. Introduction

Chemistry of the early universe is an obscure subject till date.Collisional excitation of molecular species by the atomic gas andtheir subsequent emission of photons is proposed to have impor-tant contribution in the efficient cooling of the surrounding cloudsas well as in the origin of cosmic background radiation [1–8]. Themolecule LiH has been a potential target to understand theseobservations because of its high dipole moment and owing to itslow ionization potential, its ionic chemistry has become quite pop-ular in the current literature. The anomalous microwave emissionalong the line of sight of young stars may likely to have contribu-tions from the electric dipole radiation from this molecule [1,2].Low excitation threshold and fast radiative decay of LiH/LiH+ arethe additional qualifications favoring their role as coolants. In par-ticular, a study of collision of lightest nuclei H with LiH/LiH+ in thegas phase has received renewed attention in recent years [3–6].

Several semiempirical/ab initio potential energy surfaces (PESs)of LiH2 and LiHþ2 have appeared in the literature [9–12]. Quasi-clas-sical and quantum dynamical studies were also performed on them[13–21]. In contrast to LiH2, the topography of LiHþ2 PESs seems tobe more complex [22–24]. The two low-lying singlet electronicPESs of LiHþ2 have been reported [12]. The absence of any nonadi-abatic coupling between them have suggested the charge exchangeprocesses forbidden [25]. On the ground electronic PES the colli-sions may lead to the following products:

Hþ LiHþ ! H2 þ Liþ ðR1ÞHþ LiHþ ! LiHþ þH ðR2ÞHþ LiHþ ! Hþ LiHþ ðNRÞHþ LiHþ ! HþHþ Liþ ðCIDÞ

Among these the first two are reactive channels. While the channelR1 is highly exoergic (by �4.4 eV) and follow a steep downhill path,

ll rights reserved.

12460.ra).

the channel R2 appears to be thermoneutral. The minimum energypaths of these reactive channels are shown in Figure 1. It can beseen that both the reactive processes proceed via barrierless paths.The channel NR is nonreactive and the channel CID is the collisioninduced dissociation channel. Due to low binding energy of theLiH+ molecular ion (D0 � 0.112 eV) the CID channel seems to makethe primary contribution in the H + LiH+ dynamics on the groundelectronic state.

The PESs of LiHþ2 developed by Martinazzo et al. [12] are com-puted with a multi-reference valance bond approach. The energeticseparation between the ground and first excited electronic state isfound to be quite large [25]. Therefore, the nuclear dynamics onthe electronic ground state can be treated adiabatically. The steepdownhill path (cf. Figure 1) on the ground state arises from thelarge binding energy difference of LiH+ and H2 molecule. This PEShas a shallow well (�0.286 eV below the Li+ + H2 asymptote) atthe C2v configuration. Quantum dynamical studies for the collineargeometry and three dimensional quasi-classical trajectory (QCT)calculations have been performed recently on the ground elec-tronic PES [20,24]. The findings show that the CID process is amajor competing channel at moderate collision energies.

In this Letter we attempt to carry out a three dimensional quan-tum dynamical study within the centrifugal sudden (or coupledstates) (CS) approximation of the H + LiH+ reaction on the elec-tronic ground state. Efforts have been made to minimize the con-tribution of dissociative flux into the reactive ones. The channelspecific integral reaction cross-section and thermal rate constantsare reported. While the theoretical and computational methodolo-gies are briefly outlined in Section 2 below, the results are pre-sented and discussed in Section 3. The summarizing remarks aregiven in Section 4.

2. Theoretical and computational details

A time-dependent wave packet propagation approach is em-ployed in the present study. The adiabatic Hamiltonian to describeH + LiH+ collisions can be written as

-10 -5 0 5 10 15 20-5

-4

-3

-2

-1

0

-10 0 10-0.08

-0.04

0

LiH+(v=0,j=0)

H2(v=0,j=0 )

H+HLi+

Li+ +H2

ΔE~4.36 eV

ΔE~0.0281 eV

r1 - r 2 [a0]

Ene

rgy

[eV

]

H---Li+ ---H

Li+ ---H---H

r1 r2

r2r1

E~0.0632 eVΔ

r1 - r 2 [a0]

Figure 1. Minimum energy path (occurring at collinear geometry) of the Hþ LiHþ ! H2 þ Liþ (depletion) reaction. The reaction exothermicity is 4.36 eV. The minimumenergy path of the Hþ LiHþ ! HLiþ þ H (exchange) reaction is shown in the insert.

Table 1Numerical grid parameters and properties of the initial wave function used in thepresent study.

Parameter Value Description

NR=Nr=Nc 256/128/48 Number of grid pointsRmin=Rmax ða0Þ 1.0/26.50 Extension of the grid along Rrmin=rmax ða0Þ 1.0/13.70 Extension of the grid along rDR=Dr ða0Þ 0.10/0.10 Grid spacings along R and rrd ða0Þ 10.50 Location of the dividing surface in the

product channelRmask=rmask ða0Þ 19.30/11.00 Starting point of the masking functionR0 ða0Þ 18.10 Initial location of the center of the GWP in

the coordinate spaceEtrans ðeVÞ 0.5 Initial translational kinetic energyd ða0Þ 0.25 Initial width parameter of the GWPDt ðfsÞ 0.135 Length of the time step used in the WP

propagationT ðfsÞ 540 Total propagation time

T. Roy et al. / Chemical Physics Letters 501 (2011) 252–256 253

bH ¼ � �h2

2l@2

@R2 þ@2

@r2

" #þ

^j2

2lr2 þ^l2

2lR2 þ VðR; r; cÞ: ð1Þ

Here R (H to the center-of-mass of LiH+ distance), r (LiH+ internu-

clear distance) and c (angle between~R and~r) define the set of Jacobicoordinates in the body-fixed frame of the reactant channel. The

quantity l ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffim2

HmLiþ=ð2mH þmLiþ Þq

, is the three body reduced

mass and j is the rotational angular momentum operator of LiH+.The orbital angular momentum operator of the collisional system

is denoted as l and is given by

^l2 � ðbJ2 � j2Þ ¼ ^J2 þ ^j2 � 2Jz jz � Jþ j� � J� jþ: ð2Þ

In the above equation bJ is the total angular momentum operator, bJz

and jz are the body-fixed z-components and JþðbJ�Þ and jþðj�Þ areraising (lowering) operators of bJ and j, respectively. Within the CSapproximation, the last two terms of Eq. (2) are neglected and thequantum number X (the projection of j and also of J on the body-fixed z-axis) is treated as good quantum number. The ground elec-tronic PES of LiHþ2 developed by Martinazzo et al. [12] is used forV(R,r,c).

The initial (for time t = 0) wave packet (WP) of the reagentH + LiH+ is prepared in the asymptotic reactant channel ðR!1Þand expressed as a direct product of a minimum uncertaintygaussian wave packet (GWP) to describe the translational motionalong R, the ro-vibrational eigenfunction of LiH+ for a given vibra-tional (v) and rotational (j) level (for the motion along r) andassociated Legendre polynomials (for the motion along c). ThisWP is represented on a grid in the (R,r,c) space and the time-dependent Schrödinger equation is solved with the aid of a sec-ond-order split operator integrator. While the kinetic energyassociated with R and r in the Hamiltonian (cf. Eq. (1)) are eval-uated by the fast Fourier transform method, the rotational kineticenergy term associated with c is evaluated by a Gauss–Legendrediscrete variable representation method. The unphysical reflec-tion or wraparound of the time evolved WP at the grid bound-aries is minimized by activating a sine-type damping functionalong R and r [26]. The grid parameters employed in the presentstudy are given in Table 1.

The reaction probabilities are calculated by collecting flux of theWP in the asymptotic product channel along a dividing surface atr = rd. The expectation value of the flux operator in the basis of en-ergy normalized reactive scattering wave function defines the totalreaction probability

PJXi ðEÞ ¼

Xf

SJXfi

��� ��� ¼ �hl

Im /ðR; rd; c; EÞj@/ðR; rd; c; EÞ

@r

� �� �; ð3Þ

where SJXfi is the reactive scattering matrix from an initial state i of

the reactant to a final state f of the product.The initial state selected and energy resolved integral reaction

cross section is calculated from these probabilities and is given by

rvjðEÞ ¼pk2

Xj

X¼0

gX

2jþ 1

XJmax

JPX

ð2J þ 1ÞPJXvj ðEÞ; ð4Þ

where gX is the degeneracy factor; gX = 1 for X = 0 and gX = 2 forX – 0 and k ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2lRðE� �i

pÞ=�h, with �i representing the internal

(rovibrational) energy of LiH+. The thermal rate constants areobtained from the integral reaction cross sections as

0

0.2

0.4

0.6R1R2Total (R1+R2)

0 0.5 10

0.2

0.4

0.6

0 0.5 1 0 0.5 1Collisional Energy [eV]

Rea

ctio

n P

roba

bilit

y

J=0 J=10 J=20

J=90J=60J=40

Figure 2. The probability of H + LiH+ (v = 0, j = 0, X = 0) reaction as a function of collision energy. The probabilities of R1 and R2 channel are shown by the solid and dottedlines, respectively. The dashed line indicate the total probability of the H + LiH+ reaction.

254 T. Roy et al. / Chemical Physics Letters 501 (2011) 252–256

kðTÞ ¼

ffiffiffiffiffiffiffiffiffiffiffi8kBTpl

s1

ðkBTÞ2Z 1

0ErvjðEÞeð�E=kBTÞdE; ð5Þ

where kB is the Boltzmann constant.The channel specific reaction probability is calculated by com-

paring the internuclear distance of the product diatom. The H2 for-mation channel R1 is populated when dH2 < dHLiþ (where d is theinternuclear distance), otherwise the H exchange channel R2 ispopulated. Since the CID channel is a major competing channelin the dynamics and since both the reactive and dissociative fluxproceed through the same dividing surface, we minimize the con-tribution of CID flux into the reactive one by discarding the WP fluxhaving dH2 > 4:2 a0 from R1 and dHLiþ > 9:1 a0 from R2. These arethe distances which represent approximately the dissociationthreshold of the respective diatomic potential energy function esti-mated from the employed PES in this study. Admittedly, this is theonly strategy than can be adopted in the Jacobi coordinate frame-work. This may however appears crude, but nevertheless this strat-egy does discards considerably the CID flux flowing into both thereactive channels.

3. Results and discussion

The initial state selected and energy resolved reaction probabil-ities, integral reaction cross sections and thermal rate constants ofthe H + LiH+ reaction are presented and discussed in this section.The reaction probabilities are calculated up to a collision energyof 1.0 eV. Partial wave contributions up to J = 95 and J = 50 werefound to be necessary to obtain converged reaction cross sectionfor the R1 and R2 channels, respectively. The convergence of the re-sult is checked with respect to the numerical grid parameters listedin Table 1. It is found that the reaction dynamics is quite sensitiveto the choice of the location of the dividing surface in the asymp-totic product channel. We tried with several choices and finallyfixed the dividing surface at r = 10.5a0 which yields the best con-verged results along with the other numerical parameters givenin Table 1. Despite the best efforts made to prevent the flow ofthe dissociative WP flux into the reactive channels, there may still

remain some minor contribution as the CID seems to be a majorchannel in this reaction.

The channel specific reaction probability as a function of colli-sion energy is plotted in Figure 2 for a few representative valuesof the total angular momentum J = 0, 10, 20, 40, 60 and 90. Theprobabilities of the LiH+ depletion channel (R1) and the hydrogenexchange channel are shown by the solid and dotted lines, respec-tively. Since the reaction is intrinsically barrierless it proceedswithout any threshold as can be clearly seen from the J = 0 reactionprobability curves. For higher J values the reaction develops athreshold originating from an increase in the height of the centrif-ugal barrier. As found earlier for the H + LiH reaction [13,15], sharpresonance structures at low energies for both R1 and R2 channelscan be seen from Figure 2. The resonances are less prominent athigher energies, revealing a more direct nature of the collisiondynamics. In contrast to the findings for the neutral H + LiH reac-tion on the Dunne, Murrell, and Jemmer (DMJ) PES [9], the LiH+

depletion channel (as can be seen from Figure 2) is preferred overthe hydrogen exchange channel for the H + LiH+ reaction. Such adifference in the reaction probabilities computed on the DMJ vs.a recent ab initio PES for the H + LiH reaction is also reported re-cently [10].

Reaction probability of both R1 and R2 channels decrease withincreasing J. In the given collision energy range of Figure 2, theexchange channel makes negligible contribution beyond J = 40.Therefore, for higher angular momentum the reactivity of theH + LiH+ system is governed by the depletion path. Resonances alsobecome less prominent for higher J values.

The initial state selected and channel specific cross section as afunction of collision energy of the H + LiH+ reaction are shown inFigure 3a and b. These are integral cross sections represent aweighted sum of the reaction probabilities over different partialwave contributions [�minðj; JÞ 6 X 6 þminðj; JÞ]. The reactioncross sections are calculated for v = 0 and j = 0–4 of the reagentLiH+ and are shown by different line types indicated in panel a.The reaction cross section are converged for the lowest collisionenergy of � 0.02 eV. Analogous to the observations made for theneutral H + LiH reaction [15], the cross sections for the H + LiH+

reaction for both the channels generally decrease with increasing

0 0.2 0.4 0.6 0.8 1

10

20

30

j=0j=1j=2j=3j=4

0 0.2 0.4 0.6 0.8 10

0.5

1

1.5

2

2.5

3

3.5

4

0.01 0.1 1

1

10

100 our resultref. 20

0.01 0.1 1

1

10

100

our resultref. 20

(a) LiH+ depletion channel (b) H exchange channel

Collision Energy [eV]

Cros

s Se

ctio

n [A

2]

Cros

s Se

ctio

n /A

2

Collision Energy /eV

ο

ο

Collision Energy /eV

Cros

s Se

ctio

n /A

2ο

Figure 3. Channel specific integral cross-section as a function of collision energy for the H + LiH+ (v = 0, j) reaction. The QCT cross sections for the v = 0, j = 0 reaction(dotted line) are extracted from Ref. [20] and compared with our results (solid lines) in the inset of both the panel a and b.

0 2000 4000 6000 8000 100002e-10

4e-10

6e-10

8e-10

1e-09 Our resultFitted line

0 2000 4000 6000 8000 100000

2e-11

4e-11

6e-11

8e-11

1e-10

Temperature (K)

Rat

e C

onst

ant [

cm3m

olec

ule-1

s-1

]

(a) LiH + depletion channel (b) H exchange channel

Figure 4. Boltzmann averaged (over j = 0, 4) thermal rate constants for the H + LiH+ reaction. The rate constant for the depletion channel is fitted to the empirical formulaproposed by Stancil et al. [3] and shown as dotted line in panel a (see text for details).

T. Roy et al. / Chemical Physics Letters 501 (2011) 252–256 255

collision energy. This is a typical feature of reactions which pro-ceed on a PES that has no barrier. For hydrogen exchange channel(cf. Figure 3b) the cross section show some increase for the inter-mediate collision energies (�0.2–0.4 eV). With reagent rotationalexcitation the cross sections decreases for both the channels. Theresonance oscillations seen in the reaction probability curves of

Figure 2 average out with different partial wave contributions inthe reaction cross sections. The reaction cross sections of the LiH+

depletion channel are much larger than the hydrogen exchangechannel. This is in contrast to the findings on the H + LiH reactionon DMJ PES [15]. The reaction cross sections obtained by a three-dimensional QCT calculations are extracted from Ref. [20] and

256 T. Roy et al. / Chemical Physics Letters 501 (2011) 252–256

compared with our v = 0, j = 0 results in the inset of Figures 3a andb. The QCT and the present quantum mechanical results are shownby the dashed and solid lines, respectively. It can be seen that theoverall trend of variation of cross section with energy is similar inboth cases, although their magnitude differs.

Thermal rate constants for the H + LiH+ (v = 0,j = 0–4) arecalculated up to 10000 K within the CS approximation. The stateselected thermal rate constant, kvj(T), shows a decrease (notshown here for brevity) with increasing j for both the R1 andR2 channels. This is in contrast to the H + LiH reaction, wherethe rate constant for the LiH depletion channel was found to in-crease with increasing j [15]. The Boltzmann averaged (overj = 0–4) thermal rate constants of the depletion and exchangechannel are shown in Figure 4. It can be seen that the depletionrates are higher than the exchange rates for any giventemperature.

The depletion rate constants for the reagent LiH+(cf. Figure 4a)

are fitted to the functional form, kðtÞ ¼ a1T

300

� a2 exp � Ta3

�as

proposed by Stancil et al. [3] in their kinetic model of primordiallithium reaction. Rate constant values obtained from the fit areincluded in panel a of Figure 4 and shown as dotted lines. Weobtained values a1 ¼ 4:16106� 10�10 cm3 s�1, a2 = 0.397502,a3 = 6953.43 K as compared with the corresponding valuea1 ¼ 3:0� 10�10 cm3 s�1 estimated by Stancil et al. [3].

4. Summary

Quantum dynamics of H + LiH+ reaction is studied employingthe ab initio PES developed by Martinazzo et al. [12]. A time depen-dent wave packet approach within the CS approximation is under-taken and state selected reaction probability, integral reactioncross section and thermal rate constant are reported. Effort is madeto minimize the contribution of low energy CID channel into thereactive dynamics. The LiH+ depletion reaction is found to be morefavoured over the H exchange path.

Acknowledgments

This study is supported in part by a research grant from theDepartment of Science and Technology, New Delhi, India (GrantNo. DST/SF-04/2006). T.R. and T.R.R. thank the Council of Scientificand Industrial Research (CSIR), New Delhi, for a doctoral fellow-ship. Thanks to an anonymous referee for his/her useful commentsto revise the manuscript.

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