Exact quantum scattering calculations of transport properties: CH2(X̃[sup 3]B1, ã[sup 1]A1)–helium

Exact quantum scattering calculations of transport properties for the H2O–H systemPaul J. Dagdigian and Millard H. Alexander

Citation: The Journal of Chemical Physics 139, 194309 (2013); doi: 10.1063/1.4829681 View online: http://dx.doi.org/10.1063/1.4829681 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/139/19?ver=pdfcov Published by the AIP Publishing Articles you may be interested in A nine-dimensional ab initio global potential energy surface for the H2O+ + H2 H3O+ + H reaction J. Chem. Phys. 140, 224313 (2014); 10.1063/1.4881943 Reactive scattering dynamics of rotational wavepackets: A case study using the model H+H2 and F+H2reactions with aligned and anti-aligned H2 J. Chem. Phys. 139, 104315 (2013); 10.1063/1.4820881 Properties of the B+-H2 and B+-D2 complexes: A theoretical and spectroscopic study J. Chem. Phys. 137, 124312 (2012); 10.1063/1.4754131 Exact quantum scattering calculation of transport properties for free radicals: OH(X 2)–helium J. Chem. Phys. 137, 094306 (2012); 10.1063/1.4748141 Theoretical investigation of rotationally inelastic collisions of CH2(ã) with helium J. Chem. Phys. 134, 154307 (2011); 10.1063/1.3575200

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THE JOURNAL OF CHEMICAL PHYSICS 139, 194309 (2013)

Exact quantum scattering calculations of transport propertiesfor the H2O–H system

Paul J. Dagdigian1,a) and Millard H. Alexander2,b)

1Department of Chemistry, The Johns Hopkins University, Baltimore, Maryland 21218-2685, USA2Department of Chemistry and Biochemistry and Institute for Physical Science and Technology,University of Maryland, College Park, Maryland 20742-2021, USA

(Received 28 September 2013; accepted 28 October 2013; published online 19 November 2013)

Transport properties for collisions of water with hydrogen atoms are computed by means of exactquantum scattering calculations. For this purpose, a potential energy surface (PES) was computedfor the interaction of rigid H2O, frozen at its equilibrium geometry, with a hydrogen atom, using acoupled-cluster method that includes all singles and doubles excitations, as well as perturbative con-tributions of connected triple excitations. To investigate the importance of the anisotropy of the PESon transport properties, calculations were performed with the full potential and with the spherical av-erage of the PES. We also explored the determination of the spherical average of the PES from radialcuts in six directions parallel and perpendicular to the C2 axis of the molecule. Finally, the com-puted transport properties were compared with those computed with a Lennard-Jones 12-6 potential.© 2013 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4829681]

I. INTRODUCTION

The chemical kinetic modeling of combustion systems,such as flames and combustion engines, requires not onlytemperature-, and for some reactions, pressure-dependent re-action rate constants, but also transport properties because thespecies concentrations will have a spatial dependence in suchsystems. While considerable effort has been made to com-pute transport properties, as discussed in a recent review,1

there is a need for more accurate values of these parameters.The overwhelming majority of transport property calculationshave employed a classical treatment with parameterized, an-alytic potentials, such as the Lennard-Jones (LJ) 12-6 poten-tial. There have been a number of quantum scattering calcu-lations of molecular transport properties employing accurateab initio potentials.2–11 Other quantum scattering transportproperty calculations have employed the spherical average ofsuch potentials.12–15 Improvements in computational methodsand resources permit the calculation of state-of-the-art poten-tial energy surfaces (PESs) for systems involving both stablespecies and reactive intermediates.16, 17

In recent work,10, 11 we have explored the calculation oftransport properties for two atom-molecule systems involvingreactive intermediates [OH(X2�)–He and CH2(X3B1, a1A1)–He] through exact quantum scattering calculations with state-of-the-art anisotropic PES’s. For the former system, onlyslight differences (3%–5%) in the computed transport prop-erties were found between those computed with the full PESand truncated potentials (spherical average of the sum poten-tial Vsum or the difference potential Vdif set to zero).10 Moresignificant differences were found for this system between thecalculations with the full PES and those computed with a LJ12-6 potential.

a)Electronic mail: [email protected])Electronic mail: [email protected]

The methylene CH2–He system offered an opportunityto investigate in more detail the effect of the PES on trans-port properties since the anisotropies of the CH2(X3B1)–Heand CH2(a1A1)–He systems are dramatically different. ThePES for CH2(a)–He is very anisotropic because of the stronginteraction of the electrons on the helium atom with the unoc-cupied CH2 orbital perpendicular to the molecular plane.18 Bycontrast, the anisotropy of the PES for CH2(X)–He is muchsmaller because this orbital is singly occupied in this elec-tronic state.19 These differences were found to yield transportproperties of different magnitudes for the two CH2 electronicstates, ranging from 3% to 15% higher for the a state at thelowest and highest temperatures considered (200 to 1500 K,respectively).11 More significant differences were observedfor the transport properties computed with the full PES’s andthose computed with LJ potentials whose parameters were de-rived by combination rules.20

Here, we carry out exact quantum scattering calculationsfor collisions of the water molecule with the hydrogen atom.Both species have significant concentrations in combustionmedia. Moreover, the transport of hydrogen is expected to berapid because of its small mass. The PES for the H2O–H inter-action was computed more than 20 years ago by Zhang et al.21

using several different methods [the unrestricted open-shellHartree-Fock method, second-order Møller-Plesset perturba-tion theory (MP2), and fourth-order Møller-Plesset perturba-tion theory (MP4)]. These authors determined a sphericallyaveraged, isotropic potential from radial cuts in six directionsparallel and perpendicular to the C2 axis of the molecule. Inour approach, we have computed the interaction energy formany more angular orientations of the H atom and have de-termined a full PES, using a coupled-cluster theory with areasonably large basis set. Note that our study focusses on therepulsive ground electronic state of the H3O system, not thebound excited state in which one of the electrons is promotedinto a Rydberg orbital on the O atom.22

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194309-2 P. J. Dagdigian and M. H. Alexander J. Chem. Phys. 139, 194309 (2013)

At sufficiently high collision energy (or, equivalently,temperature) or with high vibrational excitation of themolecule, H + H2O can react to form H2 + OH products.The activation energy for this reaction is high, and the rateconstant is only approximately 3 × 10−15 cm3 molecule−1

s−1 at 1000 K.23 Two different types of experimental studiesof the reaction dynamics have been carried out on this system,namely reaction of H2O in its ground vibrational level withphotolytically generated hyperthermal H/D atoms24–26 and re-action of thermal H/D atoms with vibrationally excited watermolecules.27–29 The theoretical work has involved calculationof a potential energy surface30, 31 for the reaction and quasi-classical trajectory and time-dependent quantum mechanicalwave packet calculations of cross sections.26, 32, 33

II. CALCULATION OF TRANSPORT PROPERTIES

In this section, a brief review of the calculation of trans-port properties of molecule immersed dilutely in a bath ofa collision partner is presented. Transport properties can becomputed from collision integrals �(n, s)(T), which can be ob-tained by taking an appropriate integral over the collision en-ergy and carrying out a Boltzmann state average.4, 20, 34, 35 Re-versing the usual order of carrying out this integration andstate averaging, we can express the collision integral as aBoltzmann state average over state-dependent collision inte-grals �

(n,s)ji

(T ):

�(n,s)(T ) = 1

qR

∑ji

(2ji + 1) exp(−εi/kBT )�(n,s)ji

(T ). (1)

In Eq. (1), εi is the energy of the ith rotational level, qR isthe rotational partition function, and kB is the Boltzmann con-stant. The state-dependent collision integrals are the integralsof state-dependent effective cross sections Q

(n)ji

(E) over thecollision energy E:

�(n,s)ji

= 1

2

(kBT

2πμ

)1/2 1

(kBT )s+2

×∫ ∞

0Es+1 exp(−E/kBT )Q(n)

ji(E)dE. (2)

Here, μ is the atom-molecule reduced mass.The state-dependent effective cross section in Eq. (2) is

a sum over final levels jf of state-to-state effective cross sec-tions:

Q(n)ji

(E) =∑jf

Q(n)ji→jf

(E). (3)

The state-to-state effective cross sections on the right-hand side of Eq. (3) are weighted angle averages of theji → jf differential cross section:

Q(n)ji→jf

(E) =∫ (

dσ

d�

)ji→jf

�n(E)dR, (4)

where R = (θ, φ) is the orientation of the Jacobi vector R.The weighting factors in Eq. (4) for n = 1 and 2 are4, 11, 34

�1(E) = 1 − (E′/E)1/2P1(cos θ ), (5)

�2(E) = 56 − 1

6 (E′/E)2 − 23 (E′/E)P2(cos θ ), (6)

where E′ is the relative translational energy after scatteringinto level jf. It should be noted that for integral cross sectionsthe weighting factor in Eq. (4) is �0(E) = 1.

Details about the evaluation of the Legendre moments inthe effective cross sections in Eq. (4) can be found in theliterature.36, 37 The effective cross sections in Eq. (4) can bewritten as the sum of several low-order Legendre moments ofthe differential cross sections:

Q(1)ji−jf

= π (−1)ji−jf

(2ji + 1)k2i

[A0 − 1

3 (E′/E)1/2A1], (7)

Q(2)ji−jf

= π (−1)ji−jf

(2ji + 1)k2i

[{56 − 1

6 (E′/E)2}A0 − 2

15 (E′/E)A2].

(8)In Eqs. (7) and (8), ki is the initial wave vector and Aλ equals

Aλ =∑

JJ ′l1l2l′1l′2

Z(l1J l2J′; jiλ)Z(l′1J l′2J

′; jf λ)

× T Jji l1,jf l′1

(T J ′

ji l2,jf l′2

)∗. (9)

In Eq. (9), T designates a T-matrix element, expressed in thespace-fixed frame, and the Z coefficients36, 38 are given by

Z(lJ l′J ′; jλ) = (−1)12 (λ−l+l′)+J+J ′ (

[λ][l][l′][J ][J ′])1/2

×(

l l′ λ

0 0 0

){l J j

J ′ l′ λ

}, (10)

where [x] = 2x + 1,( · · ·· · ·

)is a 3j symbol, and

{ · · ·· · ·}

is a 6j

symbol.39

We employed Eqs. (3) and (7)–(10) to compute the state-dependent effective cross sections in the scattering calcula-tions described in Sec. IV. These cross sections were used inEq. (2) to derive state-dependent collision integrals. A Boltz-mann state average of these integrals was finally carried out[Eq. (1)] in order to compute collision integrals.

We can relate the collision integrals to two transportproperties, namely the binary diffusion coefficient Dab and thequantity ηab that determines the curvature of the mixture vis-cosity as a function of concentration. The relations are20, 34, 35

Dab = 3kBT

16Nμ�(1,1), (11)

ηab = 5kBT

8�(2,2), (12)

where N is the total number density of the gas.

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194309-3 P. J. Dagdigian and M. H. Alexander J. Chem. Phys. 139, 194309 (2013)

III. POTENTIAL ENERGY SURFACE

The coordinates used to describe the H2O–H PES are thesame as those used previously in our study of the interactionof the He atom with both H2O and CH2.18, 19, 40 As depicted inFig. 1 of Ref. 18, the body-frame z axis is chosen to lie alongthe a inertial axis of the molecule, and the molecule lies in thexz plane. The water molecule was taken to be rigid, with bondlength and bond angle (r = 1.8361 bohr and θ = 104.69◦,respectively) corresponding to the vibrationally averaged ge-ometry reported by Mas and Szalewicz.41

The PES for the H2O–H system was computed with theMOLPRO 2010.1 suite of computer codes,42 by means of re-stricted coupled cluster calculations with inclusion of singleand double excitations, as well as perturbative contributionsof connected triple excitations [RCCSD(T)].43, 44 An atom-centered avqz atomic-orbital basis was used,45 with the ad-dition of a set of bond functions located midway between theatom and the molecule.46, 47 A counterpoise correction wasused at all geometries to correct for basis set superposition er-ror (BSSE).48 No hint of access to the reactive channel lead-ing to OH + H2 products was found in the calculations; thisis due to the fact that the H2O geometry was held fixed. Sincewe are interested in collisions at thermal energies and onlyslightly above, it is reasonable to neglect the reactive channel.

The PES was determined on a grid of 20 values of theatom-molecule separation R ranging from 3.0 to 10 bohr insteps of 0.5, with additional points at 11, 12, 13, 15, and 20bohr. By exploiting symmetry we can limit the calculationsto the angular range 0◦ ≤ θ ≤ 90◦ and 0◦ ≤ φ ≤ 180◦, bothvaried in steps of 10◦, for a total of 190 orientations. The totalnumber of nuclear geometries at which the interaction energywas computed was 3800.

The computed potential as a function of R and the orien-tation was fit to the following expansion in spherical harmon-ics:

V (R, θ, φ) =∑

λ,μ≥0

Vλμ(R)(1 + δμ0)−1

× [Yλμ(θ, φ) + (−1)μYλ,−μ(θ, φ)]. (13)

Symmetry considerations restrict the sums in Eq. (13) toterms with λ + μ even.18 To get an acceptable fit of the PES,we needed to include all terms with λ ≤ 10 and μ ≤ 8, for atotal of 33 anisotropic terms.

Figure 1 presents a contour plot of the dependence of thepotential energy upon the orientation of the H atom collisionpartner at an atom-molecule separation R = 6.50 bohr, theseparation at the global minimum. The anisotropy of the PESat this value of R is seen to be small. This is true for mostvalues of R, except at small atom-molecule separations forwhich there is significant repulsion between the H atom colli-sion partner and the H atoms on the water molecule.

The global minimum of the PES has an energy of 61.0cm−1 below the H2O + H asymptote, at a geometry of R =6.50 bohr, θ = 30◦, φ = 180◦. The hydrogen atom collisionpartner thus lies within the molecular plane at the global min-imum and close to one of the H atoms of the water molecule.Because of the small anisotropy of the PES, the angular ori-entation for the minimum interaction energy varies somewhat

−60

−60

−56

−56

−56 −56

−56

−56

−53

−53

−53

−53

−53

−53

−53

−53

−50

−50

−50

−50

−50

−50

−50

−50

−47

−47 −47−47

−47

−47 −47

−47

−44 −44

−44

−44

−44−44

−44

−44

−41−41

−41

−41

θ / degrees

φ / d

egre

es

0 30 60 90 120 150 1800

60

120

180

240

300

360

x

y

z

R

φ=180o

θ=30o

H

O

H

H

FIG. 1. (Top panel) Dependence of the potential energy (in cm−1) on theorientation of the hydrogen atom collision partner with respect to the H2Omolecule for an atom-molecule separation R = 6.5 bohr. (Bottom panel) Ar-rangement of the atoms at the H2O–H equilibrium geometry [R = 6.50 bohr,θ = 30◦, φ = 180◦].

as a function of R. The binding energy for H2O–H is some-what larger than for the related H2O–He system, for whichthe binding energy is computed to be 35 cm−1, while theatom-molecule separation R for the latter system is slightlysmaller (5.92 bohr).49 In their calculations on H2O–H, Zhanget al.21 considered that their best estimate for the well depthof this system (53 ± 6 cm−1) was intermediate between thevalues computed in MP4 calculations without and with thecounterpoise correction (69.9 and 36.8 cm−1, respectively).The presently computed well depth of 61 cm−1 for H2O–H isslightly larger than the estimate of Zhang et al.21

Figure 2 presents a plot of the larger expansion coeffi-cients Vλμ as a function of the atom-molecule separation R inthe region of the van der Waals well. The largest coefficientin the region of the well is the isotropic V00 term. We see thatfor values of R smaller than that of the global minimum someof the anisotropic coefficients become compatible to or largerthan the V00 term.

In their investigation of the interaction of H2O with H,Zhang et al.21 derived a spherically averaged, isotropic poten-tial by taking the average of radial cuts along six orientations

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194309-4 P. J. Dagdigian and M. H. Alexander J. Chem. Phys. 139, 194309 (2013)

4 5 6 7 8 9 10−200

−150

−100

−50

0

50

100

150

200

R / bohr

ener

gy /

cm−

1

V00

V20

V11

V31

V22

FIG. 2. Dependence of the larger expansion coefficients Vλμ [defined inEq. (13)] on the atom-molecule separation R.

parallel and perpendicular to the C2v axis of the molecule.With the availability of our full PES for this system, we cancheck the accuracy of this method to generate a sphericallyaveraged H2O–H potential. Figure 3 presents a plot of thespherical average of our full PES [which equals (4π )−1/2V00].A plot of the average of the radial cuts of our PES along the±x, ±y, and ±z directions is indistinguishable from the spher-ical average on the scale of this figure. At the minimum of thepotential, the two curves differ by 0.5 cm−1, while the welldepth of the former equals 48.6 cm−1. At 4 bohr, the poten-tials differ by 31 cm−1, as compared to an interaction energyof 1240 cm−1. As an indication of the small anisotropy forthis system, we note that the well depth for the sphericallyaveraged potential is only slightly smaller than the well depthfor the full PES.

This comparison of the two ways to obtain a sphericalaverage of the potential suggests that it may be reasonable tocompute an isotropic potential for the interaction of an atom

4 5 6 7 8 9 10−50

0

50

100

R / bohr

ener

gy /

cm−

1

present workZhang et al.

FIG. 3. Spherically averaged H2O–H potentials determined by averaging ra-dial cuts along the ±x, ±y, and ±z directions. Red: present work; blue: MP4calculation with counterpoise correction by Zhang et al. (Ref. 21).

with a small nonlinear polyatomic molecule by averaging cutsalong six orthogonal approach directions of the perturber tothe molecule, at least for the H2O–H system. A similar ap-proach has been employed by Stallcop et al.12, 14 for severalatom-molecule systems. Jasper and Miller50 have consideredmethods to spherically average potentials involving larger,floppy molecules.

Also plotted in Fig. 3 is the spherical average obtainedby Zhang et al.21 by taking the average of MP4 calculationswith a BSSE correction along the ±x, ±y, and ±z directions.This potential energy curve has a slightly shallower well andlonger equilibrium atom-molecule separation than the presentcalculation.

IV. SCATTERING CALCULATIONS

Close-coupling calculations were carried out with the HI-BRIDON suite of programs,51 which has recently been ex-tended to allow the determination of the Q1 and Q2 effec-tive cross sections [Eqs. (7) and (8)]. The cross sections werechecked for convergence with respect to the inclusion of asufficient number of partial waves and energetically closedchannels. At the highest energies for which calculations werecarried out (4000 cm−1), the rotational basis included all lev-els whose energy was less than 4600 cm−1, and the scatteringcalculations included all total angular momenta J ≤ 133 ¯.In previous work,11 we checked that the computed effectivecross sections agreed with values obtained by numerical in-tegration of the differential cross sections weighted by the�1(E) and �2(E) factors [Eqs. (5) and (6)].

The high-energy tails of the integrals over collision en-ergy [Eq. (2))] were evaluated by exponential extrapolationof the integrand. The sum over rotational levels ji in Eq. (1)included all levels whose energy was less than 3000 cm−1.For levels whose energy is between 1200 and 3000 cm−1, thetotal effective cross section [Eq. (4)] was taken to be equal tothat of the highest rotational level of the same ka projectionnumber with energy less than 1200 cm−1.

V. RESULTS

The rotational levels of an asymmetric top like water arelabeled jka kc

, where j is the rotational angular momentum, ka

is its projection along the principal axis in the prolate limit,and kc is its projection along the principal axis in the oblatelimit.52 Each rotational level exists for only one of the nu-clear spin modifications (ortho and para). The two nuclearspin modifications cannot be easily interconverted in nonre-active collisions.

Figure 4 displays plots of the collision energy depen-dence of the state-dependent effective cross sections Q

(1)ji

and

Q(2)ji

[Eq. (3)] for collisions of para-H2O in its lowest and ahigher rotational level [000 and 551, respectively]. The rota-tional energy of the latter is 758 cm−1. The elastic contribu-tion to these effective cross sections is also plotted. As weobserved in OH(X)–He and CH2(X, a)–He collisions,10, 11 theQ

(1)ji

effective cross sections decrease monotonically with in-creasing collision energy.

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194309-5 P. J. Dagdigian and M. H. Alexander J. Chem. Phys. 139, 194309 (2013)

elastic

collision energy / cm–1

elastic

(c) 551

(a) 000

totalV00

totalV00Q

(1) e

ffect

ive

cros

s se

ctio

n / Å

2j i

(b) 000

(d) 551

totalV00

totalV00

elastic

elastic

Q(2

) effe

ctiv

e cr

oss

sect

ion

/ Å2

j i

FIG. 4. Dependence on collision energy of the state-dependent effectivecross sections Q

(1)ji

[panels (a) and (c)] and Q(2)ji

[panels (b) and (d)] for col-lisions of H2O in the 000 and 551 initial levels with hydrogen atoms. Theelastic contribution to the effective cross section is indicated in each panel. Inaddition, the effective cross section computed with the spherical average ofthe potential is displayed for comparison.

At low collision energies, elastic collisions make by farthe dominant contribution to the Q

(1)ji

effective cross section,while at higher energies inelastic scattering plays a muchlarger role. As we have noted previously,10, 11 this contrastswith the dominant role of elastic scattering in total integralcross sections. We showed in our study of CH2–He transportproperties11 that the differing importance of elastic scatteringin the total integral and transport effective cross section resultsfrom the angular weighting of the differential cross section inthe latter [see Eq. (4)]. Because of the weighting factor �n(E)[Eqs. (5) and (6)], transport cross sections, particularly forelastic transitions, are much smaller than the correspondingintegral cross sections, as discussed previously.11

Also included in the plots in Fig. 4 are the effectivecross sections computed with the spherical average of thePES [(4π )−1/2V00]. Of course, only elastic collisions can takeplace with this potential. We see that the Q

(1)ji

effective crosssections for this isotropic potential and the full PES are verysimilar in magnitude for all collision energies. We considerbelow the effect of the anisotropy of the PES through exami-nation of state-dependent collision integrals [Eq. (1)].

It can be seen in Fig. 4 that the Q(2)ji

effective cross sec-tion for the 000 level displays a similar collision energy de-pendence as for Q

(1)ji

for this level, namely a monotonicallydecreasing magnitude vs. collision energy. By contrast, theQ

(2)ji

effective cross section for the 551 level becomes verylarge and negative at low collision energies. As noted pre-viously for Q

(2)ji

effective cross sections for higher rotationallevels,10, 11 this is a consequence of large values for the factorE′/E in the �2(E) weighting factor in Eq. (6) for superelas-tic collisions (E′ > E) at small collision energies. As we havediscussed previously,10, 11 this singularity does not cause prob-lems in the evaluation of the Q

(2)ji

effective cross sections forthe higher rotational levels, and hence also the determination

0 2 4 6 8 10

11

12

13

14

15

V00

term only

rotational angular momentum j

(a) 300 K

j i Ω

(1,1

) / cm

3 10–1

1 s–1

0 2 4 6 8 1014

15

16

17

18

19

20

V00

term only

rotational angular momentum j

(b) 1500 K

j i Ω

(1,1

) / cm

3 10–1

1 s–1

FIG. 5. State-dependent collision integrals �(1,1)ji

as a function of the rota-tional angular momentum j for H2O–H at temperatures of (a) 300 K and (b)1500 K. Red and blue symbols denote ortho and para levels, respectively.Levels with different values of the body-frame projection quantum numberare denoted by the following symbols: ka = 0, circles; ka = 1, plus signs;ka = 2, diamonds; ka = 3, upward-pointing triangles; ka = 4, downward-pointing triangles; ka = 5, left-pointing triangles; and ka = 6, right-pointingtriangles. In each panel, the value of the collision integral computed with thespherical average of the potential is indicated with a dashed line.

of the �(2,2) collision integral. As with the Q(1)ji

effective crosssections, elastic collisions make a decreasing contribution toQ

(2)ji

effective cross sections with increasing collision energy.

Finally, we note that the values of Q(2)ji

computed with thespherical average of the potential are very similar to the val-ues computed with the full PES, except at the lowest collisionenergies for the higher rotational levels.

From the state-dependent effective cross sections we canuse Eq. (1) to compute the �(1,1) and �(2,2) collision integrals.We plot in Fig. 5 the variation with rotational level of the state-dependent �

(1,1)ji

collision integrals at two temperatures. The

�(2,2)ji

collision integrals display a similar dependence uponthe rotational quantum numbers and hence are not plottedhere. The rotational levels for which �

(1,1)ji

is plotted are thosefor which collision-energy dependent effective cross sectionshave been explicitly computed (see Sec. IV). The most prob-able value of j is ∼3 at 300 K and ∼7 at 1500 K. We see thatat 300 K the dependence of �

(1,1)ji

on the rotational quantumnumbers is weak. This dependence is stronger at 1500 K. Inparticular, the �

(1,1)ji

values are seen in Fig. 5(b) to show a de-crease with increasing j. Presumably, this reflects the greateranisotropy of the PES at higher energies, because of the repul-sion of the hydrogen atom collision partner by the hydrogenatoms on the water molecule. We also plot in Fig. 5 the effec-tive cross sections under the assumption that the interaction ispurely spherical (only the V00 term); in this case there is nodependence on rotational level.

As shown in Fig. 5, it is easy to use the spherically av-eraged potential to predict collision integrals. Figure 6 dis-plays the temperature dependence of these quantities. We seethat the collision integrals computed with the spherically av-eraged potential are slightly larger than those computed withthe full PES. Specifically, the collision integrals computedwith the former are ∼1.4% larger at 300 K and 6% largerat 1500 K. These differences are smaller than what we foundfor the CH2(X, a)–He systems, likely a reflection of the largeranisotropies for the PESs for these systems.

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194309-6 P. J. Dagdigian and M. H. Alexander J. Chem. Phys. 139, 194309 (2013)

full PESV00 only

full PESV00 only

FIG. 6. The collision integrals �(1,1) and �(2,2) as a function of temperaturefor the H2O–H system. These quantities have been computed for the full PESand for the spherically averaged potential.

VI. DISCUSSION

We have employed quantum scattering calculations tocompute collision integrals for the H2O–H system, fromwhich transport properties can readily be predicted. As an ex-ample we will consider the H2O–H diffusion coefficient. Wecan also compare these values with a conventional calcula-tion using a parameterized LJ 12-6 potential. The parametersspecifying this potential were estimated through conventionalcombining rules: To estimate the well depth ε and the lengthparameter σ for H2O–H, we took the geometric and arith-metic means, respectively, of the corresponding parametersfor the like systems [ε = 572.4 K and σ = 2.605 for H2O–H2O and ε = 145 K and σ = 2.050 for H–H].53 Since H2Opossesses a dipole moment, there is an induction contributionto the long-range attractive interaction. Using the dipole mo-ment of water (1.844 D)53 and the polarizability of the H atom(0.663 3),54 we follow the procedure of Hirschfelder et al.20

to correct the LJ parameters; this correction is quite small(<4%) for this system. The final LJ parameters for H2O–H areε = 204 cm−1 and σ = 4.47 bohr.

Figure 7 presents a comparison of diffusion coefficientscomputed through quantum scattering calculations using thefull PES and the spherically averaged potential, as well as aclassical calculation with the LJ 12-6 potential using the pa-

200 400 600 800 1000 1200 14000

0.5

1

1.5

2

2.5

temperature / K

Dab

/ 10

4 Tor

r−1 c

m2 s

−1

LJ 12–6

full PES

V00 only

FIG. 7. Diffusion coefficients for the H2O–H system. This quantity has beencomputed for the full PES, for the spherically averaged potential, and the LJ12-6 potential described in the text.

rameters described above. As we found for OH(X)–He andCH2–He,10, 11 we see that the LJ 12-6 potential predicts a dif-fusion coefficient smaller than that computed with either thefull ab initio PES or its spherical average. The disagreementis relatively independent of temperature, in this case. Specif-ically, the LJ diffusion coefficient is 12% and 15% smaller at300 and 1500 K, respectively, than that computed with the fullPES.

To investigate further the difference in the diffusion co-efficient calculated using the ab initio PES and the LJ poten-tial, it is useful to compare the spherically averaged ab initioand the LJ 12-6 potentials; both of these potentials are dis-played in Fig. 8. The well depth of the LJ potential is signif-icantly deeper than that of the spherically averaged potential,and the equilibrium atom-molecule separation is smaller. Thissuggests that the use of combining rules to estimate isotropicpotentials for unlike collision partners may be suspect, at leastfor this system.

Additionally, the repulsive wall of the LJ potential ismuch steeper than that of the ab initio potential. In previouswork on the OH(X)–He system,10 we showed that the LJ 9-6

4 5 6 7 8 9

0

500

1000

1500

2000

2500

3000

R / bohr

ener

gy /

cm−

1

LJ 12-6 potential

ab initio isotropic V00 potential

FIG. 8. Comparison of the spherically averaged ab initio H2O–H potentialand the LJ 12-6 potential system.

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194309-7 P. J. Dagdigian and M. H. Alexander J. Chem. Phys. 139, 194309 (2013)

potential yielded a diffusion coefficient that agreed much bet-ter with that computed with the ab initio potential, since therepulsive wall for this LJ potential has a slope more in linewith that of the ab initio potential. This would presumablyalso be the case here, but the comparison with the LJ 12-6 po-tential is more relevant since this is the functional form thatis currently widely used to estimate transport properties. Pauland Warnatz55 had previously suggested that an exponentialrepulsive part may be more appropriate than that of the LJ 12-6 potential for the calculation of transport properties at hightemperatures.

For the three atom-molecule systems for which we havecomputed transport properties, the spherical average of theab initio PES has been found to possess a less steep repul-sive wall than the commonly employed LJ 12-6 potential. Asa result, the LJ potential predicts a smaller diffusion coeffi-cient than does the ab initio PES, particularly at combustiontemperatures. This observation may also apply to other atom-molecule systems.

One goal of our recent quantum scattering calculationsof transport properties for atom-molecule systems [OH–He(Ref. 10), CH2(X, a)–He (Ref. 11), and H2O–H (presentwork)], is to assess how the anisotropy of the PES can affectthe calculated transport properties. For most of the systems in-vestigated, transport properties computed using the full PESdiffer by only a few percent from those computed using justthe spherical average of the PES. The one exception was theCH2(a)–He system. The large PES anisotropy in this systemoccurs because of the strong interaction of the electrons onthe He atom with the low-lying lowest unoccupied molecu-lar orbital (LUMO) on CH2, which lies perpendicular to themolecular plane.

We might anticipate that PESs involving molecules hav-ing a low-lying LUMO would also exhibit a large anisotropy.An example would be electron deficient molecules such asBH3. In preliminary work, we have computed several radialcuts of the BH3–He PES, along the C3 axis perpendicular tothe molecular plane and in the molecular plane along a B–Hbond and bisecting two B–H bonds. Indeed, the potential issignificantly more attractive for approach of He along the C3

axis, where the electrons on helium can interact strongly withthe unoccupied boron 2p orbital. In general, then the impor-tance of the PES anisotropy should be taken into account incalculating transport properties involving molecules possess-ing a low-lying LUMO.

ACKNOWLEDGMENTS

This work was supported by the Chemical, Geosciencesand Biosciences Division, Office of Basic Energy Sciences,Office of Science, U.S. Department of Energy, under GrantNo. DESC0002323.

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