www.elsevier.com/locate/cplett

Chemical Physics Letters 428 (2006) 227–230

Rotational energy transfer in H2 + H2

Renat A. Sultanov *, Dennis Guster

Business Computing Research Laboratory, St. Cloud State University, 31 Centennial Hall, 720 Fourth Avenue South, St. Cloud,

MN 56301-4498, United States

Received 30 May 2006; in final form 8 July 2006Available online 14 July 2006

Abstract

Quantum-mechanical close-coupling calculations for state-to-state cross-sections and thermal rate coefficients are reported forH2 + H2 collisions. Two recently developed potential energy surfaces (PESs) for the H2–H2 system are applied, namely, the global poten-tial surface from the work [A.I. Boothroyd et al., J. Chem. Phys., 116 (2000) 666], and a restricted, model surface from the work [P. Diepand J.K. Johnson, J. Chem. Phys. 113 (2000) 3480; P. Diep, J.K. Johnson, J. Chem. Phys. 112 (2000) 4465]. The low temperature limit isinvestigated. We found significant differences in cross-sections and corresponding thermal rate coefficients calculated with these twoPESs.� 2006 Elsevier B.V. All rights reserved.

1. Introduction

The investigation of elastic and inelastic collisionsbetween molecules and atoms can provide valuable infor-mation about interactions, chemical properties and energytransfer dynamics [1–15]. The hydrogen molecule is thesimplest and most abundant molecule in the universe’smolecular clouds and plays an important role in manyareas of astrophysics. For example, knowledge of the ro-vibrational excitation and de-excitation rate constants inmolecular hydrogen collisions is of fundamental impor-tance for understanding and modeling the energy balancein the interstellar medium. The energy transfer processesinvolving H2 molecules control the evolution of shockfronts and photodissociation regions (PDRs) in the inter-stellar medium. Additionally, the energy transfer betweenH2 molecules and between H2 and other atoms/moleculesis important for cooling of primordial gas and shockwave-induced heating in the interstellar media. However,to accurately model the thermal balance and kinetics ofsuch important systems one needs accurate state-to-staterate constants kvjv0j0 ðT Þ.

0009-2614/$ - see front matter � 2006 Elsevier B.V. All rights reserved.

doi:10.1016/j.cplett.2006.07.019

* Corresponding author.E-mail addresses: sultanov@bcrl.stcloudstate.edu, r.sultanov2@yahoo.

com (R.A. Sultanov), dcguster@stcloudstate.edu (D. Guster).

Experimental measurement of quantum state-resolvedcross-sections and rate coefficients is a very difficult techni-cal problem. On the other hand, accurate theoretical datarequire precise potential energy surfaces and reliabledynamical treatment of the collision processes. The firstattempt to construct a realistic full-dimensional ab initioPES for the H2–H2 system was done in works [3], andthe potential was widely used in a variety of methods andcomputation techniques.

Recently, the H2–H2 system has been reinvestigated andan accurate interaction potential from the first principleshas been developed in work [12]. However, in this workthe Diep and Johnson potential energy surface (DJ PES)was extrapolated only for the rigid rotor model of H2–H2. An extensive study of the H2–H2 PES has been recentlyreported by Boothroyd et al. [13], where the potential ener-gies have been represented at 48180 geometries, respec-tively, with a large basis set at the multireferenceconfiguration interaction level.

In this work, we provide a test of sensitivity for the newPESs for collisions of rotationally excited H2 molecules

H2ðj1Þ þH2ðj2Þ ! H2ðj01Þ þH2ðj02Þ: ð1ÞWe apply only the new surfaces mentioned above.

The scattering cross-sections and their correspondingrate coefficients are calculated using a non reactive quan-

228 R.A. Sultanov, D. Guster / Chemical Physics Letters 428 (2006) 227–230

tum-mechanical close-coupling approach. In the followingsection, we will briefly outline the method. Our results anddiscussion are presented in Section 3.

2. Method

In this section, we provide a brief outline of the quan-tum-mechanical close-coupling approach used in our calcu-lations. All relevant details have been developed in work[1]. The hydrogen molecules are treated as linear rigidrotors. The model has been applied in few previous works,see for example, [1,4]. For the considered range of kineticenergies of astrophysical interest the rotor model is consid-ered to be adequate [14].

The log-derivative matrix is propagated to large R-inter-molecular distances, since all experimentally observablequantum information about the collision is contained inthe asymptotic behaviour of functions UJM

a ðR!1Þ. Aset of coupled second order differential equations for theunknown radial functions UJM

a ðRÞ is solved

d2

dR2� LðLþ 1Þ

R2þ k2

a

� �UJM

a ðRÞ

¼ 2M12

Xa0

Zh/JM

a ðr1; r2;~RÞjV ð~r1;~r2;~RÞj/JMa0

� ðr1; r2;~RÞiUJMa0 ðRÞdr1 dr2 dbR; ð2Þ

where a ” (j1j2j12L), j1 + j2 = j12 and j12 + L = J. We applythe hybrid modified log-derivative-Airy propagator in thegeneral purpose scattering program MOLSCAT [15] to solvethe coupled radial Eq. (2). Different propagator schemesincluded in MOLSCAT have been tested. Our calculationsshowed that other propagators can also provide quite stableresults.

The numerical results are matched to the known asymp-totic solution to derive the physical scattering S-matrix

UJa �R!þ1

daa0e�iðkaaR�ðlp=2ÞÞ � kaa

kaa0

� �12

� SJaa0e

�iðkaa0R�ðl0p=2ÞÞ: ð3Þ

The method was used for each partial wave until a con-verged cross-section was obtained. It was verified that theresults are converged with respect to the number of partialwaves as well as the matching radius, Rmax, for all channelsincluded in our calculations.

The cross-sections for rotational excitation and relaxa-tion phenomena can be obtained directly from the S-matrix. In particular, the cross-sections for excitation fromj1j2 ! j01j02 summed over the final m01m02 and averaged overthe initial m1m2 corresponding projections of the H2 mole-cules angular momenta j1 and j2 are given by

rðj01; j02; j1j2; �Þ ¼p

ð2j1 þ 1Þð2j2 þ 1Þkaa0

�X

Jj12j012

LL0ð2J þ 1Þjdaa0 � SJ

aa0 ðEÞj2: ð4Þ

The kinetic energy is � = E � B1j1(j1 + 1) � B2j2(j2 + 1).Here E is the total energy in the system,B1(2) = 60.8 cm�1 are the rotation constants of the collidingH2 molecules, J is total angular momenta of the 4-atomicsystem, a ” (j1j2j12L), kaa0 ¼ 2M12ðE þ Ea � Ea0 Þ1=2 is thechannel wavenumber and Ea(a 0) are rotational channelenergies.

The relationship between the rate coefficient kj1j2!j01j02ðT Þ

and the corresponding cross-section rj1j2!j01j02ðEkinÞ can be

obtained through the following weighted average:

kj1j2!j01j02ðT Þ ¼ 8kBT

pl1

ðkBT Þ2Z 1

�s

�d�rj1j2!j01j02ð�Þe��=kBT ; ð5Þ

where � ¼ E � Ej1� Ej2

is precollisional translational en-ergy at the translational temperature T, kB is Boltzmannconstant, l is reduced mass of the molecule–molecule sys-tem and �s is the minimum kinetic energy for the levels j1and j2 to become accessible.

3. Results

As we mentioned in Section 1, in this work we apply thenew PESs from works [12, 13]. The DJ PES [12] is con-structed for the vibrationally averaged rigid rotor modelof the H2–H2 system to the complete basis set limit usingcoupled-cluster theory with single, double and triple excita-tions. A four term spherical harmonics expansion modelwas chosen to fit the surface. It was demonstrated, thatthe calculated PES can reproduce the quadrupole momentto within 0.58% and the experimental well depth to within1%.

The bond length was fixed at 1.449 a.u. or 0.7668 A. DJPES is defined by the center-of-mass intermolecular dis-tance, R, and three angles: h1 and h2 are the plane anglesand /12 is the relative torsional angle. The angular incre-ment for each of the three angles defining the relative ori-entation of the dimers was chosen to be 30�.

The BMKP PES [13] is a global six-dimensional poten-tial energy surface for two hydrogen molecules. It was espe-cially constructed to represent the whole interaction regionof the chemical reaction dynamics of the four-atomic sys-tem and to provide an accurate as possible van der Waalswell. In the six-dimensional conformation space of the fouratomic system the conical intersection forms a complicatedthree-dimensional hypersurface. Because the BMKP PESuses cartesian coordinates to compute distances betweenfour atoms, we have devised some fortran code, which con-verts spherical coordinates used in Section 2 to the corre-sponding cartesian coordinates and computes thedistances between the four atoms. In all our calculationswith this potential the bond length was fixed at 1.449 a.u.or 0.7668 A as in the DJ PES.

A large number of test calculations have also been doneto secure the convergence of the results with respect to allparameters that enter into the propagation of the Schro-dinger equation [11]. This includes the intermolecular dis-

0

T(K)

1e-16

1e-15

1e-14

1e-13

1e-12

1e-11

1e-10

k 00-2

0(T)

[cm

3 s-1]

E[cm-1

]

1e-05

0.0001

0.001

0.01

0.1

1

σ00

−02

[10-1

6 cm-2

]

Flower et al.

Flower, 1998

This work, DJ PES

DJBMKP

0 2000 4000 6000 8000500 1000 1500 2000

Fig. 2. Temperature dependence of the state-resolved thermal rateconstant (left panel) and corresponding cross-sections (right panel) forthe transition j1 ¼ j2 ¼ 0! j01 ¼ 2; j02 ¼ 0. Squares and circles are the datafrom [4, 5], respectively. Our results with the DJ PES are depicted withsolid lines, the open triangles up represent our results with the BMKPsurface.

R.A. Sultanov, D. Guster / Chemical Physics Letters 428 (2006) 227–230 229

tance R, the total angular momentum J of the four atomicsystem, Nlvl the number of rotational levels to be includedin the close-coupling expansion and others (see the MOLS-

CAT manual [15]). We reached convergence for the integralcross-sections, r(Ekin), in all considered collisions. In thecase of the DJ PES the propagation has been done from2 to 10 A, since this potential is defined only for those spe-cific distances. For the BMKP PES we used rmin = 1 A tormax = 30 A. We also applied a few different propagatorsincluded in the MOLSCAT program.

Now we present our results for different rotational tran-sitions in collisions between para/para- and ortho-/ortho-hydrogen molecules (1). Our main goal in this work is first:to carry out complete quantum-mechanical calculationswith new potentials for different transitions in p-H2+p-H2

and o-H2+o-H2 collisions and, second, to provide a com-parative study and check the sensitivity of the two newestsurfaces for the very important and fundamental H2–H2

system.The energy dependence of the elastic integral cross-sec-

tions on the total energy rel(Etot) in the H2 + H2 systemis represented in Fig. 1 (upper plots) together with thestate-resolved integral cross-sections rj1j2!j0

1j02ðEtotÞ for the

j1 ¼ j2 ¼ 0! j01 ¼ 2; j02 ¼ 2 and j1 ¼ j2 ¼ 1! j01 ¼ 1; j02 ¼3 rotational transitions (lower plots) for both the BMKPand the DJ PESs, respectively. As can be seen both PESsprovide the same type of the behaviour in the cross-section.These cross-sections are in basic agreement with recenttime-dependent quantum-mechanical calculations in work[7]. However, our results show, that the DJ PES generatesabout 30% higher values for the cross-sections relatively tothe BMKP PES.

Also, it is important to point out here, that for compar-ison purposes we do not include the compensating factor of2 mentioned in [2]. However, in Fig. 2 (left plot) and in our

070

80

90

100

110

120

el[1

0-16 cm

2 ]

80

90

100

110

E[cm-1

]

0.01

0.1

1

tr[1

0-16 cm

2 ]

E[cm-1

]

0.01

0.1

1

p-/p-H2 Elastic o-/o-H2 Elastic

31>-1122>-00

BMKP

DJ

BMKP

DJ

BMKPDJ DJBMKP

2000 4000 6000 8000 0 2000 4000 6000 8000

0 2000 4000 6000 8000 0 2000 4000 6000 8000

Fig. 1. Rotational state-resolved integral cross-sections for elasticscattering in the case of para-/para- and ortho-/ortho-hydrogen andtransitions, when j1 ¼ j2 ¼ 0! j01 ¼ 2; j02 ¼ 2 and j1 ¼ j2 ¼ 1! j01 ¼ 1;j02 ¼ 3. Calculations are done with the DJ (bold lines) and BMKP (opentriangles up) PESs. The compensating factor of 2 is included only in theelastic cross-sections.

subsequent calculations of the thermal rate coefficients,kjj0 ðT Þ, the factor is included.

Significant differences in the cross-sections of the twopotentials are reflected in the state-resolved transitionstates j1 ¼ 0; j2 ¼ 0! j01 ¼ 2; j02 ¼ 0, as shown in Fig. 2(right panel). That is why it seems that the DJ PES can pro-vide much better results, as seen in the same figure in theleft panel. Specifically, when we present the results forthe corresponding thermal rate coefficients k00�20(T) calcu-lated with the DJ potential together with results of othertheoretical calculations [4, 5] the agreement is almost per-fect. Next, Fig. 3 provides the same results for the cross-sections, but includes the details at low energies, togetherwith the corresponding experimental data from [9]. Nowone can better see the considerable differences in these

350

Energy [cm-1]

1e-05

0.0001

0.001

0.01

0.1

Cro

ss s

ecti

on σ

00−2

0 [Α2 ]

Experiment: B. Mate, et al., JCP 122, 064313 (2005)

Theory: this work (DJ PES)

Theory: this work (BMKP PES)

o

400 450 500 550 600 650 700 750 800

Fig. 3. Cross-sections for the 00! 20 rotational transition calculatedwith the DJ and BMKP PESs for the H2 + H2 collision. The circles aresome experimental data from the work [9], triangles up and triangles downare the results of this work using the DJ and BMKP PESs, respectively.

0

T[K]

1e-22

1e-20

1e-18

1e-16

1e-14

1e-12

1e-10

k 00-2

2(T)

[cm

-3s-1

]

T[K]

1e-16

1e-15

1e-14

1e-13

1e-12

1e-11

1e-10

k 02-2

2(T)[

cm-1

s-1]

T[K]

1e-13

1e-12

1e-11

k 20-0

0(T)[

cm-3

s-1]

T[K]

1e-18

1e-16

1e-14

1e-12

1e-10k 11

-13[c

m-3

s-1]

This work, DJ PES

Flower, 1998

DJ PES

Flower, 1998

DJ PES

Danby et al., 1987

DJ PES

Flower, 1998

500 1000 1500 2000 0 500 1000 1500 2000

0 500 1000 1500 20000 500 1000 1500 2000

Fig. 4. Thermal rate coefficients calculated with the DJ PES for the00! 22, 02! 22, 20! 00 and 11! 13 transitions and other results [2,4].

Table 1Thermal rate coefficients k00!20(T) and k02!22(T) (m3s�1) at low and verylow temperatures calculated with the DJ surface in comparison withavailable experimental* and theoretical data from [9]

T(K) k00!20 k02!22 P

DJ [9]* [9] DJ [9]

100 1.64 2.2(4) 1.92 2.63 3.23 10�20

60 3.81 6.0(7) 4.75 5.62 7.97 10�22

30 0.54 1.1(1) 0.80 0.65 1.34 10�25

20 0.87 2.3(3) 1.63 0.93 2.71 10�29

10 0.38 2.7(7) 1.65 0.33 2.62 10�40

230 R.A. Sultanov, D. Guster / Chemical Physics Letters 428 (2006) 227–230

two cross-sections. However, the DJ PES is able to providevery good agreement with the experimental data [9]. Thuswe conclude, that DJ PES is much better suited for theH2–H2 system. Moreover, in Fig. 4 we provide thermal ratecoefficients for different transition states calculated withonly the DJ PES and in comparison with other theoreticaldata obtained within different dynamical methods andPESs. Again the agreement is very good.

Finally, Table 1 depicts the thermal rate coefficientsk00!20(T) and k02!22(T) at lower kinetic temperatures,down to 10 K. Numbers in each column should be multi-plied by corresponding factor P. As can be seen our resultsare in very good agreement with recent experimental andtheoretical data at higher temperatures. However, at T [

30 K we could not reproduce effectively the results of [9].It may be because of the intermolecular H2–H2 distancerestrictions in the DJ surface in which the potential rangesfrom 2 A to only 10 A. It seems plausible, that for such lowenergies one needs to carry out calculations even beyondthe 10 A limit.

We provide close-coupling quantum-mechanical calcu-lations of the state-resolved rotational excitation and de-excitation cross-sections and rate coefficients for molecularhydrogen collisions. A test of convergence and the resultsfor cross-sections and rate coefficients using two differentpotential energy surfaces for the H2–H2 system have beenobtained for a wide range of kinetic energies [11].

Our calculations revealed, that both PESs can providethe same type of behaviour in regard to cross-sectionsand rate coefficients for different transition states. How-

ever, significant differences in the cross-section of the00! 20 transition have been found. Also, it was alreadyindicated, that at even larger kinetic energies the DJ poten-tial overestimates relative to the BMKP surface the resultsby about 20–40% [11].

In conclusion, the results of these calculations show,that additional work is needed to further improve theBMKP PES, particularly the part of the surface, which isresponsible for the 00! 20 transition. On the other hand,as it was shown in [11] and in this work, the DJ PES is ableto provide very good results for different transition states atrelatively higher temperatures. However we also found, ascan be verified from Table 1, it is difficult to use the DJ PESand carry out reliable calculations at low temperatures:T [ 30 K.

References

[1] S. Green, J. Chem. Phys. 62 (1975) 2271.[2] G. Danby, D.R. Flower, T.S. Monteiro, Mon. Not. R. Astr. Soc. 226

(1987) 739.[3] D.W. Schwenke, J. Chem. Phys. 89 (1988) 2076.[4] D.R. Flower, Mon. Not. R. Astron. Soc. 297 (1998) 334.[5] D.R. Flower, E. Roueff, J. Phys. B 31 (1998) 2935.[6] S.K. Pogrebnya, D.C. Clary, Chem. Phys. Lett. 363 (2002) 523.[7] S.Y. Lin, H. Guo, Chem. Phys. 289 (2003) 191.[8] M. Bartolomei, M.I. Hernandez, J. Campos-Martinez, J. Chem. Phys.

122 (2005) 064305.[9] B. Mate, F. Thibault, G. Tejeda, J.M. Fernandez, S. Montero, J.

Chem. Phys. 122 (2005) 064313.[10] F. Gatti, F. Otto, S. Sukiasyan, H.-D. Meyer, J. Chem. Phys. 123

(2005) 174311.[11] R.A. Sultanov, D. Guster, Chem. Phys. 326 (2006) 641.[12] P. Diep, J.K. Johnson, J. Chem. Phys. 112 (2000) 4465.[13] A.I. Boothroyd, P.G. Martin, W.J. Keogh, M.J. Peterson, J. Chem.

Phys. 116 (2002) 666.[14] J. Le Bourlot, G. Pineau des Forets, D.R. Flower, Mon. Not. R.

Astron. Soc. 305 (1999) 802.[15] J.M. Hutson, S. Green, MOLSCAT ver. 14, 1994 (Distrib. by Collabor.

Comp. Proj. 6, Daresbury Lab., UK, Eng. Phys. Sci. Res. Council).