Comment on symmetry of the interaction between an asymmetric rigid rotor and alinear rigid rotorSheldon Green

Citation: The Journal of Chemical Physics 103, 1035 (1995); doi: 10.1063/1.469813 View online: http://dx.doi.org/10.1063/1.469813 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/103/3?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Semiclassical wave packet treatment of the rigid asymmetric rotor J. Chem. Phys. 87, 1116 (1987); 10.1063/1.453344 Accurate semiclassical determination of rigid asymmetric rotor eigenvalues J. Chem. Phys. 83, 2363 (1985); 10.1063/1.449277 A semiclassical determination of the energy levels of a rigid asymmetric rotor J. Chem. Phys. 68, 745 (1978); 10.1063/1.435747 The Stark Effect for a Rigid Asymmetric Rotor J. Chem. Phys. 16, 669 (1948); 10.1063/1.1746974 An Asymptotic Expression for the Energy Levels of the Rigid Asymmetric Rotor J. Chem. Phys. 16, 78 (1948); 10.1063/1.1746662

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Comment on symmetry of the interaction between an asymmetric rigidrotor and a linear rigid rotor

Sheldon GreenNASA Goddard Space Flight Center, Institute for Space Studies, New York, New York 10025

~Received 14 March 1995; accepted 14 April 1995!

In fitting an ab initio potential for H2O–H2, Phillips et al. @J. Chem. Phys.101, 5824 ~1994!#excluded certain terms in the angular expansion they believed to vanish because of ‘‘the requirementthat the potential is invariant to inversion of all coordinates through the origin.’’ However, there hasbeen some question in the literature as to whether these terms must, in fact, vanish owing to spatialinversion symmetry. By providing counterexamples, it is demonstrated here that this isnot requiredby fundamental spatial symmetry. However, these terms do appear to vanish for realistic molecularinteractions and this symmetry may arise from the two-body nature of the electrostatic Hamiltonian.

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I. INTRODUCTION

An accurate method for calculating energy transfernonreactive molecular collisions is the close couplingproach which solves the time independent Schro¨dinger equa-tion by expanding the total system wave function in a baset of rotation~and vibration! functions of the asymptoticmolecular species and a partial wave expansion for thelision coordinate. This leads to coupled second-order difential equations for radial functions which are labeled byasymptotic vibration-rotation states and the partial wavebital angular momentum; the coupling comes from the oentation dependence of the interaction potential. Calculaof the required matrix elements of the interaction potenover the expansion basis functions is facilitated by expaing the orientation dependence of the potential in an apppriate complete set of orthonormal functions so that thegular part of the matrix elements can be obtained frangular momentum coupling theory.

It is always advantageous to use any symmetry inhein the collision system to decouple the problem into smalnoninteracting sets of states. To give the simplest examfor the case of a diatomic molecule colliding with a strutureless atom1 it is convenient to combine the diatomic rottional angular momentum,j , with the orbital angular mo-mentum of the collision, l , to form a total angularmomentumJ. Because of conservation of both total angumomentum~owing to rotational invariance! and parity~ow-ing to invariance on spatial inversion!, the problem can beseparated into uncoupled subsets labeled byJ and by parity,P5~21!l1 j . For this case appropriate expansion functiofor the interaction potential are Legendre polynomiaPl~cosu!, whereu is the angle between the diatomic bonand the collision coordinate. Molecular symmetries mlimit the required angular expansion terms. For example,homonuclear diatomic molecules only terms with evenl canbe nonzero. Because the matrix elements for evenPl termsvanish unless the rotational levels,j and j 8 have the sameparity, one immediately obtains the collisional selection rthat D j must be even, and calculations can be done serately for the even and the odd rotational levels. Similarless obvious, collisional selection or ‘‘propensity’’ rules habeen obtained for more complex systems.2

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Of particular relevance to the present discussion issymmetry of the angular terms used to describe the intertion of two linear rigid rotors.3 These are labeled by threeindices;l 1 and l 2 which describe the tensor order of the dependence on orientation of molecules 1 and 2, respectivand l which is the vector sum ofl 1 and l 2. Because theinteraction potential must be invariant to spatial inversionthe whole system and because the expansion terms withl 11 l 21 l change sign under this symmetry operation, theterms must vanish.

It is important that angular terms in the expansion of tinteraction which must vanish owing to symmetry not bincluded inadvertently. This may seem like a trivial poinbut it is quite possible to obtain nonzero values for suterms, e.g., when fitting a limited set ofab initio points. Theproblem is easily avoided if these terms are simply excludfrom the fit. For most systems studied to date the approprsymmetries have been fairly obvious. However, as computional abilities have increased, close coupling studies habeen attempted for more complex collision systems whthe symmetries may be less readily apparent. Recent whas focused on collisions of symmetric top~NH3! ~Refs.4–7! and asymmetric top~H2O! ~Refs. 8–10! rigid rotorscolliding with a linear rigid rotor, H2, and, unfortunately, aquestion is raised in the literature concerning whether certterms in the angular expansions appropriate to these systmust vanish owing to symmetry.7,11 The terms in questionare analogous to the oddl 11 l 21 l terms discussed above fotwo linear rotors. In the case of NH3 these terms affect col-lisional propensity rules and might explain some discrepacies between experiment and the most recent theoretvalues.11

The present work addresses the question of symmetrthe interaction between a~n! ~a!symmetric rotor and a linearrotor. The next section presents the formalism and derisome of the accepted symmetry relations. Section III psents numerical examples which indicate that the symmein question isnot a consequence of any fundamental spatsymmetry. On the other hand itdoesappear to be generallyvalid for empirical functions that are expected to be typic

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1036 Sheldon Green: Interaction between an asymmetric linear rotor

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of intermolecular forces. Also, attempts at fitting thH2O–H2 ab initio potential in the course of work leading tRef. 9 which did include these terms, found them to bemerically small. While the symmetry relation in questionnot a general property of the symmetry of space to rotatioreflections, and inversion through the origin, it may, nonetless, result from underlying symmetry of the electronic strture Hamiltonian as suggested by Ristet al.7 although theproof presented there is valid only for long-range intertions.

II. COORDINATES AND ANGULAR EXPANSIONFUNCTIONS

Slightly different conventions for describing the coordnates for the collision of a~n! ~a!symmetric rigid rotor with alinear rigid rotor have been used. The present work follothe convention adopted in this laboratory; the connectwith other choices was discussed in Ref. 9. The overall stem is described in space-fixed coordinates with origin atcollisional center of mass. The collision coordinate,R, thevector from the~a!symmetric top center of mass to the linerotor center of mass, is then conveniently described byusual spherical polar coordinates~R,Q,F!, whereR is theradial distance,Q is measured from thez axis, andF ismeasured from thexz plane. The orientation of the lineamolecule can also be described by space-fixed polar an~Q8,F8!. The orientation of the~a!symmetric top requiresthree angles, traditionally the Euler angles~a,b,g! which ro-tate the space-fixed axes to an axis system fixed in the frof the molecule. Following earlier work we do not considhere the most general case, but assume that the top hplane of symmetry; this is always true for symmetric topsuch as NH3, and is also true for H2O.

9 We choose themolecule-fixed axes such that thexz plane is a plane of symmetry. If the molecule has a twofold or higher axis of symmetry~always true for a symmetric top! we chose this axis athe molecule-fixedz axis; otherwise~e.g., a planar moleculesuch as deuterated water! we choose one of the two principaaxes of inertia which are in the molecular plane~the xz-plane! as the molecule-fixedz axis. The molecular orientation a,b,g50,0,0 then corresponds to alignment of tmolecule-fixed and space-fixed axis systems. A generalentationa,b,g is produced by rotating the molecule-fixeaxis system, beginning at 0,0,0, by an anglea about themolecule-fixedz axis, followed by a rotationb about themolecule-fixedy axis, and then another rotationg about themolecule-fixedz axis.

We choose angular expansion functions as rotationinvariant contractions of spherical harmonics forQ,F andQ8,F8 and rotation matrices fora,b,g,

V~R,Q,F,Q8,F8,a,b,g!

5( v l1m1l2l~R!Tl1m1l2l

~Q,F,Q8,F8,a,b,g!, ~1!

where

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Tl1m1l2l~Q,F,Q8,F8,a,b,g!

5~11dm10!21( S l 1 l 2 l

r 1 r 2 r DYl2r2~Q8F8!Ylr ~QF!

3@Dm1r1

l1 ~a,b,g!1~21! l11m11 l21 lD2m1r1

l1 ~a,b,g!#.

~2!

Here ~:::! is a Wigner 3-j symbol,Dmrl ~a,b,g! is a Wigner

rotation matrix, andYlm~QF! is a spherical harmonic, all asdefined by Silver,12 and the sum is overr 1, r 2, andr . Equa-tion ~1! is a generalization of the expansion terms usedtwo linear rotors.3 Because the first molecule is no longecylindrically symmetric it requires a rotation matrix labeleby l 1 ,m1 in place of a spherical harmonic; otherwisel 1, l 2,and l serve analogous roles.

Because of rotational invariance, the intermolecular ptential, and other physically meaningful quantities, can dpend on only a smaller number ofrelativeangles~sometimescalled body-fixed angles, not to be confused with the robody-fixed coordinate system discussed above!. In general,one can choose three angles in an arbitrary way, correspoing to rotation of the collision system as a whole. A convnient choice does this by fixing the symmetric or asymmetrotor orientation ata,b,g50,0,0 and definingu,w andu8,w8as the collision direction and the linear molecule orientatirelative to the rotor molecule body-fixed axis system.9 Interms of these relative coordinates Eqs.~1!–~2! can be writ-ten as

V~R,u,w,u8,w8!5( v l1m1l2l~R!t l1m1l2l

~u,w,u8,w8!,

~3!

where

t l1m1l2l~u,w,u8,w8!

5~11dm10!21( S l 1 l 2 l

r 1 r 2 r DYl2r2~u8w8!Ylr ~uw!

3@dm1r11~21! l11m11 l21 ld2m1r1

#, ~4!

di j is a Kronecker delta, equal to one ifi5 j and to zerootherwise, and the sum is again overr 1 ,r 2 ,r .

Another body-fixed system has been used previously5,7,8

in which the orientation of the linear molecule is measurnot with respect to the water-fixed axes but with respectthe intermolecular vector,R; i.e., the body-fixedz axis isaligned withR, the body-fixedx axis is in the plane definedby the water-fixedz axis andR, and the body-fixedy axis isperpendicular to bothR and the water-fixedz axis, The lin-ear molecule orientation is then described by polar coornatesu9,w9 in this local, body-fixed coordinate system. Equtions ~1! and ~2! are then written as

V~R,u,w,u9,w9!5( v l1m1l2l~R!t l1m1l2l

~u,w,u9,w9!,

~5!

where

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1037Sheldon Green: Interaction between an asymmetric linear rotor

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t l1m1l2l~u,w,u9,w9!

5~11dm10!21@~2l11!/4p#1/2(

s1S l 1 l 2 l

s1 2s1 0D3~21!s1Yl2s1

~u9w9!@Ds1m1

l1 ~w,u,0!

1~21! l11m11 l21 lDs12m1

l1 ~w,u,0!#. ~6!

The orthonormality integrals of the expansion functionEqs. ~2!, ~4!, and ~6!, are useful for expanding the angdependence of various quantities, such as the interactiontential, and for calculating matrix elements,

E sin QdQ dF sin Q8dQ8 dF8 da sin b db dg

3Tl1m1l2l~Q,F,Q8,F8,a,b,g!*

3Tl18m18 l28 l 8

~Q,F,Q8,F8,a,b,g!

516p2~11dm10!22~2l 111!21d l1l18dm1m18

d l2l28d l l 8 , ~7!

and

E sin u du dw sin u8 du8 dw8

3t l1m1l2l~u,w,u8,w8!* t l

18m18 l28 l 8~u,w,u8,w8!

5E sin u du dw sin u9 du9 dw9

3t l1m1l2l~u,w,u9,w9!* t l

18m18 l28 l 8~u,w,u9,w9!

52~11dm10!22~2l 111!21d l1l18dm1m18

d l2l28d l l 8 . ~8!

Symmetry properties of these angular functions habeen discussed already in the literature, although details hnot been given for the body-fixed coordinates, Eq.~4!, whichwe prefer to use. Consider first the complex conjugatet l1m1l2l

(u,w,u8,w8). Using the relation for the complex conjugate of a spherical harmonic,

Yl ,m~u,w!*5~2 !mYl ,2m~u,w!, ~9!

the fact that the three projection values in a 32j symbolmust sum to zero, and the fact that

S a b c

2d 2e 2 f D 5~2 !a1b1cS a b cd e fD , ~10!

it is readily shown that t l1m1l2l(u,w,u8,w8)*

5 t l1m1l2l(u,w,u8,w8), i.e., that these angular expansion c

efficients are real. Since the potential is real, the radial coficients are likewise real quantities.

Consider next reflection in the rotorxz plane, which wehave taken to be a plane of symmetry. This has the effecchangingw→2w andw8→2w8. Using the fact that

Yl ,m~u,2w!5~2 !mYl ,2m~u,w!, ~11!

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changing signs of the projection values in the 32j symbolsusing Eq.~10!, which is allowed because these are dummsummation indices, and using the fact that the three projetion values sum to zero, Eq.~4! becomes

t l1m1l2l~u,2w,u8,2w8!

5~11dm10!21( ~2 ! l11 l21 l1r1S l 1 l 2 l

r 1 r 2 r D3Yl2r2

~u8w8!Ylr ~uw!@dm1 ,2r1

1~21! l11m11 l21 ld2m1 ,2r1#. ~12!

Because of the delta symbols, (2) r15(2)m1. It is also ob-vious thatd i , j5d2 i ,2 j , giving finally

t l1m1l2l~u,2w,u8,2w8!5t l1m1l2l

~u,w,u8,w8!. ~13!

Thus, any function expanded as in Eq.~3! is automaticallysymmetric on reflection in the body-fixedxz plane. Note thatthe two delta functions, which derive from the two rotationmatrices in the space-fixed coordinates, Eq.~2!, have beenswitched to accomplish this. In fact, the phased sum overm1in Eqs. ~2!, ~4!, and ~6! was chosen in the expansion func-tions precisely to account for reflection symmetry in thexzplane.

As discussed in some detail by Rist, Valiron, andAlexander7 them1 index is closely related to rotation aboutthe ~a!symmetric top axis; in general, if the rotor has aCnvaxis of symmetry only values ofm1 which are integral mul-tiples ofn are allowed. For a molecule like H2O which has aC2v axis, this is equivalent to theyz plane as well as thexzplane being a plane of symmetry. Rather than consider rotion about thez axis, which was discussed by Rist, Valiron,and Alexander, we consider here reflection in theyz planewhich has the effect of takingw→p2w andw8→p2w8. Us-ing the fact that

Yl ,m~u,p2w!5Yl ,2m~u,w!, ~14!

changing the signs of the projection indices in the 32j sym-bol, and swapping the role of the two delta functions aabove, it is straightforward to show that

t l1m1l2l~u,p2w,u8,p2w8!5~2 !m1t l1m1l2l

~u,w,u8,w8!.~15!

Because the potential must be invariant to reflection in theyzplane, V(R,u,2w,u8,2w8)5V(R,u,w,u8,w8), which im-plies that

( v l1m1l2l~R!t l1m1l2l

~u,2w,u8,2w8!

5( v l1m1l2l~R!~2 !m1t l1m1l2l

~u,w,u8,w8!

5( v l1m1l2l~R!t l1m1l2l

~u,w,u8,w8!. ~16!

Because the expansion functions are orthonormal the equity holds separately for each angular term so that

~2 !m1v l1m1l2l~R!5v l1m1l2l

~R!, ~17!

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1038 Sheldon Green: Interaction between an asymmetric linear rotor

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and potential expansion coefficients with oddm1 must van-ish.

We consider finally the symmetry of inverting the sytem through the origin, the symmetry which defines the pity of a system. Although it is agreed that this symmetmust leave the potential energy unaffected, it is the effectthis symmetry on the angular expansion functions whichsomewhat controversial. Ristet al.7 recently proved thatterms with (2 ) l11 l21 l odd must vanish owing to this symmetry for the long-range electrostatic part of the interactioThey further noted that these terms did not appear torequired in a numerical fit to the short-range part of anabinitio NH3–H2 interaction. These terms were also found tovery small in numerical fits to anab initio H2O–H2potential,9 and they were excluded in the final fit.

It is easiest to consider this symmetry in the full spacfixed coordinate system of Eqs.~1! and~2!. Space inversiontakes a vector into its negative, i.e.,R→2R which is equiva-lent to Q→p2Q, andF→F1p. The spherical harmonicstransform as

Yl ,m~p2Q,F1p!5~2 ! lYl ,m~Q,F!. ~18!

There is no comparable transformation for the rotation mtrices; in general, space inversion converts a right-hancoordinate system into a left-handed one and there is notation which will interconvert these. Because we have poslated a plane of symmetry for our system, however, andcause space inversion followed by reflection is equivalena regular rotation there is a simple transformation which dscribes this combination;a→a1p, b→p2b, g→p2g.13

The effect of this transformation on the rotation matricesgiven by

Dm,rl ~a1p,p2b,p2g!

5eim~p2g!dm,rl ~p2b!eir ~a1p!

5~21! l1mD2m,rl ~a,b,g!, ~19!

where Eqs.~4.2.4! and~4.2.6! of Edmunds14 have been usedCombining Eqs.~18! and ~19! with Eq. ~2! it is readilyshown that

Tl1m1l2l~p2Q,F1p,p2Q8,F82p,a1p,p2b,p2g!

5Tl1m1l2l~Q,F,Q8,F8,a,b,g!. ~20!

Thus, each term is separately invariant to this symmetryplaces no restrictions on the allowed terms.

It should be noted that angular terms withm150 havecylindrical symmetry about the~a!symmetric top rotor axisand thus have the same symmetry as a linear rotor; the rtion matrices in Eq.~2! reduce to a spherical harmonic fowhich the effect of spatial inversion can be described by E~18!. In this case it is straightforward to show that terms wiodd l 11 l 21 l do, in fact, vanish owing the spatial inversiosymmetry.

III. EMPIRICAL POTENTIALS

Studies of NH3–H2 ~Refs. 4–7! and H2O–H2 ~Refs.8–10! to date have been based on fits toab initio points on

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the interaction potential energy surface. It is importantsuch fits to impose the appropriate symmetries from the oset; otherwise it is easy to find spurious contributions frofits to limited numbers of points. If a particular angular termXl~V!, vanishes owing to some symmetry,S , which trans-forms the angles according toS V5V8, then it is importantto include bothV andV8 in the fit ~for all V! to ensure thatXl does, in fact, go to zero in the fit. Of course, evensymmetrically related points are not specifically includefits should converge to zero for such symmetry forbiddeterms if enough angular points are available, but this convgence can be quite slow. For a~n! ~a!symmetric top and alinear rotor, there are four angular degrees of freedom, whmakes it very difficult to obtain adequate numbers of pointo ensure such convergence.

On the other hand, if an interaction potential can breadily obtained~e.g., analytically! for arbitrary orientations,it is straightforward to obtain the angular expansion termusing orthogonality relationships, Eq.~8!, and four-dimensional numerical quadrature. In particular, the bodfixed integrals overd~cosu! and d~cosu8! are done withstandard Gauss–Legendre quadratures and integrals ovewandw8 are done with even spaced~Gauss–Mehler! quadra-tures. The present work has analyzed a number of ‘‘empical’’ analytic potential energy functions into angular expansion terms in the hope of shedding some light on symmeproperties. Most of the calculations were done for H2O–H2,but NH3–H2 and HDO–H2 which have different symmetrywere also considered. The geometries describing eachthese systems are listed in Table I; note that the molecuare placed with their center of mass at the origin, with thsymmetry axis along thez-axis, and with thexz-plane aplane of symmetry, as required for the internal coordinatspecified by Eq.~4!. The interaction potentials were assumeto be functions only of interatomic distances, which arreadily calculated from the atomic coordinates in Table I anthe H2 coordinates. The latter are easily calculated for arbtrary collision coordinates by starting at the center of the H2molecule which is located at polar coordinates,R, u, w, andadding or subtracting Cartesian coordinates correspondingthe polar coordinatesRHH/2, u8, w8, where the hydrogenbond length has been taken asRHH51.4022 a0 ~1 a05Bohr radius55.29231029 cm!. Since the interaction de-pends only on interatomic distancesit must be invariant toall spatial symmetries; overall rotation, reflection, or spaceinversion. All the empirical potentials were further designeto reflect exchange of identical particles. If the H2 hydrogensare designated as Ha and Hb and the H2O hydrogens as H1and H2, atom–atom potential terms,Vx(R1a), Vx(R2a),Vx(R1b), Vx(R2b), where the notation for atom–atom distances should be evident, were always included symmecally.

Several calculations were done using pairwise additiatom–atom interactions; this is a very common model finteraction potentials and is thought to provide a reasonaaccurate description of short-range forces, which are of mconcern. The atom–atom forms were either a Lennard-Jon

VLJ~Rm ,e,R!5e@~Rm /R!1222~Rm /R!6#, ~21!

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1039Sheldon Green: Interaction between an asymmetric linear rotor

This article is copyrighte

TABLE I. Geometries and potential parameters for H2O, NH3, and HDO with H2.

H2O coordinates (x,y,z), a0O 0 0 0.123 9H1 21.430 3 0 20.983 23H2 1.430 3 0 20.983 23potential parameters, distances ina0, energies in cm21

H–H Rm54.5 e540 A530 000 a53.5O–H Rm55.5 e580 A550 000 a52.5

NH3 coordinates (x,y,z), a0N 0 0 0.127H1 1.771 0 0 20.592 8H2 20.855 5 1.533 7 20.592 8H3 20.885 5 21.533 7 20.592 8potential parameters, distances ina0, energies in cm21

H–H A53 000 a53.0O–H A54 000 a53.5

HDO coordinates (x,y,z), a0O 0.135 91 0 0.133 26H 21.412 55 0 1.068 17D 20.380 93 0 21.600 13potential parameters, distances ina0, energies in cm21

H–H A510 000 a53.5O–H A550 000 a52.5

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or a simple exponential repulsion

Vexp~A,a,R!5A exp~2R/a!. ~22!

The potential parameters were chosen rather arbitrarilyare included in Table I.

Convergence of selected expansion terms for H2O–H2using the pairwise additive Lennard-Jones potential, E~21!, at a collision distance ofR54 a0 is shown as a functionof the number of integration points~the same number wasused for each of the four dimensions! in Table II. It turns outthat this interaction potential is highly anisotropic. Nonethless these terms with relatively low indices show reasonaconvergenceexcept for terms with l11 l 21 l odd. The latterare found to approach zero, but require relatively large nu

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bers of integration points for convergence. Interestingly, covergence of these terms is much slower using even numbof points.~It should, perhaps, be noted that the smallest number of integration points reported in Table II corresponds ttwo orders of magnitude more orientations than have beavailable in even the largestab initio studies.! Results usinga pairwise additive exponential interaction, Eq.~22!, wereentirely analogous and are not shown; this calculation like athe others reported here was done for a collision distanR54 a0.

Results for NH3–H2 using a pairwise additive exponen-tial function are shown in Table III. They too are seen to bentirely analogous to results for H2O–H2 in Table II. Similar

TABLE II. Convergence of selected angular expansion terms for H2O–H2 with the Lennard-Jones interaction asa function of number of Gauss integration points. Terms with oddl 11 l 21 l are indicated with an asterisk.Number in parentheses is power of ten.

l 1 m1 l 2 l

Number of Gauss points

12 15 18 21 24 32 48

0 0 0 0 3.0~6! 3.1~6! 3.1~6! 3.1~6! 3.06~6! 3.06~6! 3.06~6!1 0 0 1 4.2~6! 4.3~6! 4.3~6! 4.3~6! 4.26~6! 4.26~6! 4.26~6!1 0 2 1 22.5~6! 22.6~6! 22.6~6! 22.6~6! 22.62~6! 22.62~6! 22.62~6!1 0 2 3 3.9~6! 4.0~6! 4.0~6! 4.0~6! 3.95~6! 3.95~6! 3.95~6!2 2 0 2 2.3~6! 2.3~6! 2.3~6! 2.3~6! 2.35~6! 2.35~6! 2.35~6!2 2 2 0 7.9~5! 9.0~5! 9.0~5! 9.0~5! 9.00~5! 9.01~5! 9.01~5!2 2 2 1* 2.6~4! 5.0~0! 1.0~3! 24.3~25! 2.30~1! 8.47~22! 4.46~27!2 2 2 2 21.1~6! 21.2~6! 21.2~6! 21.2~6! 21.22~6! 21.22~6! 21.22~6!2 2 2 3* 23.1~4! 25.1~0! 21.1~3! 21.3~3! 22.32~1! 28.03~22! 23.76~27!2 2 2 4 2.1~6! 2.2~6! 2.2~6! 2.2~6! 2.19~6! 2.19~6! 2.19~6!4 4 2 2 2.6~5! 6.4~5! 6.3~5! 6.4~5! 6.44~5! 6.44~5! 6.44~5!4 4 2 3* 1.1~5! 1.1~1! 5.9~3! 6.4~23! 1.52~2! 6.30~21! 3.67~26!4 4 2 4 25.8~5! 28.1~5! 28.0~5! 28.1~5! 28.08~5! 28.09~5! 28.09~5!4 4 2 5* 29.9~4! 21.0~1! 24.3~3! 23.0~23! 29.82~1! 23.66~21! 21.88~26!4 4 2 6 1.5~6! 1.7~6! 1.7~6! 1.7~6! 1.67~6! 1.67~6! 1.67~6!

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1040 Sheldon Green: Interaction between an asymmetric linear rotor

This article is copyrighted

TABLE III. Same as Table II, but using a pairwise additive exponential interaction potential for NH3–H2.

l 1 m1 l 2 l

Number of Gauss points

8 11 14 20

0 0 0 0 3.155~5! 3.155~5! 3.155~5! 3.155~5!0 0 2 2 1.303~3! 1.303~3! 1.303~3! 1.303~3!3 0 0 3 23.141~3! 23.141~3! 23.141~3! 23.141~3!3 0 2 1 1.403~1! 1.389~1! 1.389~1! 1.389~1!3 0 2 3 28.602~1! 28.595~1! 28.595~1! 28.595~1!3 3 0 3 1.890~3! 1.890~3! 1.890~3! 1.890~23!3 3 2 1 28.114~0! 28.356~0! 28.356~0! 28.356~0!3 3 2 2* 26.948~22! 1.496~26! 7.008~26! 8.641~28!3 3 2 3 5.164~1! 5.171~2! 5.171~1! 5.171~1!3 3 2 4* 3.077~22! 3.108~25! 23.375~26! 23.969~28!3 3 2 5 2.160~2! 2.160~2! 2.160~2! 2.160~2!

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calculations with a pairwise additive exponential interactfor HDO–H2, one which set the H–H and H–D interactionequal~as expected from the Born–Oppenheimer approximtion! and another which used a different interactions~H–Dstronger than H–H by a factor of 2! also showed the sambehavior, and results are not presented here. Expansion twhich are expected to vanish owing to other symmetriesthese systems, for example, terms with oddl 2 or, forH2O–H2, terms with oddl 1 or m1, were also examined anfound to converge quickly to zero; for most of these tsymmetrically related points are included automaticallythe integration scheme and so cancel even for very smnumbers of integration points.

It appears that terms with oddl 11 l 21 l generally vanishfor these pairwise additive atom–atom potentials, althourelatively large numbers of integration points are neededobtain numerically small values. This may be a general prerty of potentials which can be written as pairwise additatom–atom interactions as suggested by Briels,15 althoughhis derivation was limited specifically to pairwise inverpowers and/or exponentials and also to collision distanlarge enough that atoms of the two species do not interpetrate.

It therefore seemed reasonable to consider more genforms for the interaction potential, for example, three-boterms. As noted by Cooper and Hutson,16 an important three-body term, especially at long-range, is the triple dipole intaction. A simplified functional form based on the triple dpole term was therefore considered,

Vddd5a~3 cosu1 cosu2 cosu311!r 123r 2

23r 323, ~23!

where the angles refer to the interior angles of the trianformed by three atoms,r i are the distances which form thsides of the triangle, anda is a proportionality constant. Acalculation was done for H2O–H2 including this terms forthe H2 hydrogens with each of the H2O atoms, and usinga53000 for the interaction with oxygen,a51000 for theinteraction with the hydrogens, distances in atomic units,numbers of integration points,N58, 15, and 21. The oddl 11 l 21 l terms, however, converged smoothly to zerothis interaction in a manner analogous to that seen in TaII and III.

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Although it was beginning to appear that restriction toeven l 11 l 21 l might be a general property, some other, es-sentially ad hoc functional forms were also tried, a few ofwhich did, in fact, provide counterexamples. These were alldone for H2O–H2; to describe these it will be convenient tolabel the atom–atom distances asR1a, R1b, R2a, R2b fordistances between hydrogens in the H2 ~labeleda andb! andthose in H2O ~labeled 1 and 2! andROa andROb for distancesbetween the oxygen and the two atoms in H2. The first func-tion which was found to require oddl 11 l 21 l terms multi-plied the pairwise additive exponential interaction discussedabove by the factor,

factor5exp~2R1aR1b /10!* exp~2R2aR2b /10!, ~24!

where distances are in atomic units. Selected angular expan-sion terms for this interaction are shown in Table IV. Al-

TABLE IV. Convergence as a function of number of Gauss integrationpoints for selected angular expansion terms for the H2O–H2 pairwise addi-tive exponential interaction multiplied by the factor given in Eq.~24!. Termswith odd l 11 l 21 l are indicated with an asterisk. Number in parentheses ispower of ten.

l 1 m1 l 2 l

Number of Gauss points

9 12 15 24

2 2 0 2 8.3983~2! 8.3981~2! 8.3983~2! 8.3983~2!2 2 2 0 9.3548~1! 9.3490~1! 9.3548~1! 9.3548~1!2 2 2 1* 21.2276~2! 21.2274~2! 21.2276~2! 21.2276~2!2 2 2 2 1.5307~2! 1.5310~2! 1.5307~2! 1.5307~2!2 2 2 3* 21.4298~2! 21.4300~2! 21.4298~2! 21.4298~2!2 2 2 4 2.7853~2! 2.7852~2! 2.7853~2! 2.7853~2!3 2 0 3 8.3521~2! 8.3519~2! 8.3521~2! 8.3521~2!3 2 2 1 4.1774~1! 4.1719~1! 4.1770~1! 4.1770~1!3 2 2 2* 25.8239~1! 25.8273~1! 25.8239~1! 25.8239~1!3 2 2 3 1.0600~2! 1.0605~2! 1.0601~2! 1.0601~2!3 2 2 4* 21.0364~2! 21.0363~2! 21.0364~2! 21.0364~2!3 2 2 5 3.2128~2! 3.2127~2! 3.2128~2! 3.2128~2!4 4 0 4 4.6665~1! 4.6421~1! 4.6665~1! 4.6665~1!4 4 2 2 3.4711~0! 2.8429~0! 3.4710~0! 3.4710~0!4 4 2 3* 25.3440~0! 25.1005~0! 25.3449~0! 25.3449~0!4 4 2 4 1.1656~1! 1.1791~1! 1.1656~1! 1.1656~1!4 4 2 5* 21.2677~1! 21.2758~1! 21.2676~1! 21.2676~1!4 4 2 6 4.0559~1! 4.0548~1! 4.0559~1! 4.0559~1!

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1041Sheldon Green: Interaction between an asymmetric linear rotor

This

though the pattern of convergence on using more integrapoints was similar to that found for other interactions, itclear that oddl 11 l 21 l terms donot vanish.

Another function for which the oddl 11 l 21 l terms didnot vanish was obtained by multiplying the pairwise additiexponential interaction by the factor,

factor5431021~R1a1R1b1R2a1R2b!212

3~RO a1RO b!212. ~25!

Expansion terms converged with respect to number of ingration points are shown in Table V. Another counterexaple which does not include a pairwise additive factor is

V553104 exp@2~R1a1R1b1R2a1R2b!/3.5#

3exp@2~ROa1ROb!/2.5#. ~26!

Expansion terms for this function are also shown in TableA handful of other functional forms—of varying plausibilityas representative of interaction potentials—were tried; macould be expanded using only evenl 11 l 21 l terms but atleast one more was found which required odd terms.

IV. DISCUSSION

Several simple models for the interaction potential btween a~n! ~a!symmetric top rotor and a linear rotor werexpanded in terms of the angular functions used in quantmolecular scattering calculations. These angular expansfunctions are labeled by indicesl 1 andl 2, which describe the

TABLE V. Angular expansion terms for the functions described by Eq~25! and ~26! in the text. Terms with oddl 11 l 21 l are indicated with anasterisk.N is the number of Gauss points; comparison with resultssmallerN indicate that these values are converged. Number is parenthespower of ten.

l 1 m1 m2 lEq. ~25!N527

Eq. ~26!N524

0 0 0 0 3.077~2! 4.286~2!0 0 2 2 3.794~1! 1.393~1!1 0 0 1 5.728~2! 3.913~2!1 0 2 1 24.113~1! 25.556~0!1 0 2 3 5.958~1! 1.313~1!2 0 0 2 4.222~2! 1.163~2!2 0 2 0 1.978~1! 6.160~21!2 0 2 2 22.646~1! 21.577~0!2 0 2 4 4.263~1! 4.107~0!2 2 0 2 3.843~1! 2.309~1!2 2 2 0 2.520~0! 3.678~21!2 2 2 1* 21.075~0! 21.970~21!2 2 2 2 21.091~0! 3.590~22!2 2 2 3* 21.522~0! 22.607~21!2 2 2 4 6.092~0! 1.687~0!3 0 0 3 1.879~2! 8.512~0!3 0 2 1 9.500~0! 21.159~21!3 0 2 3 21.172~1! 23.400~21!3 0 2 5 1.658~1! 28.264~21!3 2 0 3 5.157~1! 1.677~1!3 2 2 1 3.055~0! 2.033~21!3 2 2 2* 26.142~21! 28.099~22!3 2 2 3 22.228~0! 1.272~22!3 2 2 4* 21.548~0! 21.721~21!3 2 2 5 8.666~0! 1.669~0!

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tensor order of the dependence of the interaction on oriention of molecules 1 and 2, respectively, andl which is thevector sum of l 1 and l 2. The chosen functions could bereadily evaluated analytically for any specified collision orientation, so the angular expansion terms could be determinby converged four-dimensional numerical quadratures. Thehas been some controversy in the literature whether termwith odd l 11 l 21 l must vanish owing to spatial symmetry.For some of the functions considered here these terms wfound to be nonzero. Because the potential energy functioconsidered here depend only on interatomic distances abecause allowed spatial symmetry operations must presethese distances, the functions considered here must be invant to these symmetries. The fact that some functions wefound which required expansion terms with oddl 11 l 21 ldemonstrates thatsuch terms cannot be excluded on thgrounds of the underlying symmetry of space itself, i.e., ro-tations, reflections, or inversion.

On the other hand, it is possible that the electronic struture Hamiltonian has additional symmetries which excludsuch terms from realistic potential energy functions. ThiHamiltonian consists of two-body interactions, specificallsimple inverse powers of distances, and such terms are offound to lead to only evenl 11 l 21 l terms,7,15,17although aproof for the general case does not yet seem to be availabBecause of its importance for fitting interaction potentials tuse in molecular scattering calculations further work tprove the generality~or falsehood! of this conjecture wouldbe most useful.

ACKNOWLEDGMENTS

This work was supported in part by NASA Headquarters, Office of Space Science and Applications, AstrophysiDivision, Infrared and Radio Astrophysics Program. I amgrateful to David Flower, Bruce Haas, Jeremy Hutson, TimPhillips, Claire Rist, and Pierre Valiron for informative dis-cussions on this subject.

1A. M. Arthurs and A. Dalgarno, Proc. R. Soc. London Ser. A256, 540~1960!.

2A small sampling of work in this area includes M. H. Alexander, J. ChemPhys.77, 1855 ~1982!; M. H. Alexander and D. C. Clary, Chem. Phys.Lett. 97, 319~1983!; M. H. Alexander and P. J. Dagdigian, J. Chem. Phys83, 2191~1985!; M. H. Alexander and G. C. Corey,ibid. 84, 100 ~1986!.

3K. Takayanagi, Adv. At. Mol. Phys.1, 149 ~1965!; S. Green, J. Chem.Phys.67, 715 ~1977!.

4G. D. Billing and G. H. F. Diercksen, Chem. Phys.118, 161 ~1987!; 124,77 ~1988!.

5A. Offer and D. Flower, J. Phys. B22, L439 ~1989!; A. Offer and D.Flower, J. Chem. Soc. Faraday Trans.86, 1659~1990!.

6J. Schleipen, J. J. ter Meulen, and A. Offer, Chem. Phys.171, 347~1993!.7C. Rist, M. H. Alexander, and P. Valiron, J. Chem. Phys.98, 4662~1993!.8V. Balasubramanian, G. G. Balint-Kurti, and J. H. van Lenthe, J. ChemSoc. Faraday Trans.89, 2239~1993!.

9T. R. Phillips, S. Maluendes, A. D. McLean, and S. Green, J. Chem. Phy101, 5824~1994!.

10T. R. Phillips, S. Maluendes, and S. Green, J. Chem. Phys.102, 6024~1995!.

11D. R. Flower and A. Offer, inMolecules and Grains in Space, edited by I.Nenner~AIP, New York, 1994!, p. 477.

12B. L. Silver, Irreducible Tensor Methods: An Introduction for Chemists~Academic, New York, 1975!.

13E. P. Wigner,Group Theory, translated by J. J. Griffin~Academic, NewYork, 1959!, p. 217. Note that Wigner specifies the transformation whictakes a dipod oriented ata,b,g into the dipod obtained by space inversion

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1042 Sheldon Green: Interaction between an asymmetric linear rotor

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asa→a6p, b→p2b, g→p2g, suggesting that this transformation mabe more general than the inversion-rotation discussed here.

14A. R. Edmunds,Angular Momentum in Quantum Mechanics~PrincetonUniversity, Princeton, 1957!.

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15W. J. Briels, J. Chem. Phys.73, 1860~1980!; see especially the commentfollowing Eq. ~3.8!.

16A. R. Cooper and J. M. Hutson, J. Chem. Phys.98, 5337~1993!.17R. A. Sack, J. Math. Phys.5, 245, 253~1964!.

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