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Volume 77, number 1 CHEMICAL PHYSICS LETTERS 1 Jsnuary 1

ON THE ISOTROPIC AND LEADING ANISOTROPIC TERMSOF THE H-H, POTEN TiAL ENE RGY SURFACEAnt&no J.C. VARANDASDeparramenro de Quim zca , Unc~ers~dad e de Cormbra , 3000 Cozmbra , Por tuga land

Recclvcd 3 1 \larch 1980, m final form 25 September 1980

SCF and SCT Cl calculattons were performed to obtam the tsotrop~c V,potential surface An analyuc emmate of and leading antsotropx V z terms of the H-HI 0 1sobtauted by addm& to the near Hzuuee-Foch curve ~1 un&on whtch apPro-+mateIY f i ts the correlotson energy at small R and reduces at large R to the muItlpo1~ expanrton of the dlsperslon eThe bchnvtor of V2 IS&O dtscussed

1 IntroductionA fundamental example of chemrcal reactlons withactivation barners 1s the D f H, system, D + Hz + DH

t I+. It also provides an ~portant prototype for study-ing melastrc processes m chenucal reactive systems be-cause the relatlveiy h&-energy barrier to reactionmeans that under many low-ener,T condltlons exchangeis neghglble [l] .

Several ab initio calculations of the H, potential en-er,v surface have been reported [Z] , the most exten-stve and accurate (w&h.m less than 1 kcal mol-L) bemgthat of Sregbahn and Lru (SL) [3,4] . However, hkemost prevrous ab imtro calcuiatrons studymg the ex-change process, the SL surface is restricted to the smallmternuclear separatrons whrch encompass the saddle-pomt geometry; the rate of the exchange process ts,to a large extent, determrned by the topology of thesurface close to thrs point. Analytical potent&s m the(R 2, 3) nterpartlde coordinates which closelyfit the SL ab mitio data and allow a reliable extrapola-tion into regxons of the H3 configuration space un-* Present address Instttuut voor Theorehsche Chemle, Katho-InekeUmverslteit, NlJmegen, The Netherlands.

covered by the ab uutio results have also been repor[S-7] *.Ab initio valence bond calculatrons of Norbeck

[8,9f usmg a comparatrvely modest optumzed (2~1Slater basis set extend to larger H-H, separations (but still dtd not m&de the van der Waals mimmum.Exlstmg mformation about this regron comes prrmarfrom the mversion of transport and cross sectron mesurements_ However, transport data depend only wely on the H-H, rnteractron potentrai as has been destrated by the wade range of potential parameters whave been found to provtde an equaliy good representatton of the experunental data [2). More specific aconsistent mformation IS obtamed from molecular bscattermg data [IO-131. Yet, there IS a discrepancy = 45% in the well depths of the most popular molecbeam potentrals, and of = 6% and = 5%, respectively,in the location of the van der Waals mmunum (R,)the distance of energy zero (u) for these potentAs.

For quantum scattenng calculatrons rt 1s onvemeto express the H-H, interaction potenttal as a Legepolynomial expansron,* In ref [6 1 the coordinatesR and r are hbclled X and x rtnely.

0 009--5614/81/0000--0000/S 02.50 0 North-Holland Publishing Company 1

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Voltlme 77, number 1 CHEMICAL PHYSICS LET-l-IRS 1 January 19

I(R , re, e) = T rp , -J qcos e>. (1)where rc IS one bond drstance fixed at the H-, equlirbrrumgeometry and 8 IS the angle made by the mt&nucleara~s of thrs pair wtll R. the separatron of the remammgatom from the centre of mass of the dtatomrc The e\-pansron potentrals, partrcularly I= 0 and 3,. play an lm-portant role m inelastic and reactive scattermg (see ref[S] and references therem) and they can be obtnmedfrom a complete analytxal potzrlttal IQ t_ R-,, R3) bymtegratron or from ab metro data by a least-squaresfittmg procedure

At short atom-molecule separatrons, I/u has an ap-proLunate mverse exponenttJ dependence on R thatIS reliably gven by the SL ab metro results Conversely,the weakly attractive long-range regions of 10 are con-vemently described by the asymptotic 1/R drspersronseries expnnsion,- Id,sp(R) = C$- 6 + C,R+ + C,,R-o -i- (2)whrch IS normally truncated after the fust few termsYet thrs expansion (2) 1s based on a second-order per-turbation treatment which takes no account of overlapand exchange effects and it dtvergcs at small R wherethese effects are srgmfrcant. The drvergence csn beavoided by dampurg the expansion An example of thusIS the umvcrsJ potentral function for closed-shell mter-actions prcposed by Ahlrrchs et al [14],V(R) = ~$,@) + vdlsp(R) D(R), (3)D(R) = eup[-_P(l 28 R,R- - I)], R < 1.28 R,,

(4)D(R) = 1, R> 1.25 Rm.

(9where fl IS a scalmg parameter origmally assumed to beumty, R, 1s the locatron of the van der Waals rmmmumwhrch IS self-consistently determmed, and T/HF IS aHartree-Fock repulsron curve. THIS functiona! formhas recently been used by ToreUo and Dondt 1131 whofitted H-HZ differential cross section data by trral anderror admstment of the parameters m a modified Born-Meyer (MBM) potentral,vttF(R) = A exp [-R(bo + b, R)] . (6)

Another approach to thrs problem IS to use accurate152

ab untro CL data from small R and extrapolate out-wards [5,6] . A thud approach, whrch 1s also possiblem the case of the H-H, system, IS to mterpolate be-tween the accurate ab rmtro data obtarncd for smalland the dispersron expansron, vahd at large R, by chof a suitable dampmg function For thrs rt 1s necessaryto have an accurate ab mitro estmlate of VHt- at largR. However the only theoretical mformation at largR comes from early perturba tlon calculattons [ 15.16Although perturbation theory could in prmclple affoan accurate means to determine the small mteractionenergies mvolved, the approxunatlons made in evalu-atmg the molecular mtegrals and the srmphcrty of thmolecular wavefuncttons used cast doubt on the rehabrhty of these calculatrons.

In an attempt to elucrdate the drscrepancy betweeevlstmg H-H? interactron potentrals, we have perforSCF and SCFCI calculations for a range of values gobeyond the van der Waals muumunt and obtamed esmates of V. and rr) over thrs range. We also tested vahdrty of truncatmg eq (1) at small R by performmgSCF calculations at several values of 0 _

2. SCF calculationsSCF calculahons were performed for a H atom approachmg a H, molecule with fiied bond length (re

1.4 ~~ *) as a functron of R and 0 We consrdered vaof R between 1 6 a0 and 9.0 c10 w t h 0 = 0 (C,, symmetry) and 90 (C2, symmetry). A few SCF calcula-tions were also performed for 6 = 30 5556O, 54.7356and 70.1243 (all C, symmetry) to allow the Importaat small R, of the I > 2 terms m the Legendre expan-sion (1) to be mvestrgnted and, as a result, a more reliabesrunate of Vo and Vl m Uns reson. AU calculations eployed the (4s3pld) gaussran basrs set recommendedby Sregbahn and Ltu [4]. Although the Siegbahn-Ltu basis set may not contatn sufficiently diffuse funtions to describe the long-range drspersron forces III CI calculatron, we b&eve that rt is sufficrent to obtama potenhal of near Hartree-Fock quakty. The resultsobtamed are grven m table 1, and from these we haveestunated the harmonrc components of the H-H, inaction potentral gtven 111 able 2. Basis-set superposttion 1 bohr (U(J) = 1 au of bond length = 5 29177 X 16 m, hartree (Eh) = 1 au of energy = 4.3598 aJ

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Volum e 77, nu mber 1 CHEMICAL PHYSICS LETTERS 1 J anu ar > 19

Table 1Calculated SCF and SCT CI cnergles of H(R,O) Hz@ = 14 UO), m atomx unrt s, for the Slegbahn-Llu [41 basts setR (a01 SCF

e = oSCF Cl--

0 = 30 5556O e = 54 7356O e = 70.1243O e = 9o e = o 0 = 9o-

16 -1436907 - 1495332 -1 528953 -1 529647 -1 52567120 -1 556533 -1 564464 -1 567019 --I 565620 -1 564402 -1 6 15087 -1 617779a 2.5 -1 579838 -1 581624as -1 5931172.75 -1 60299830 -1 6108323s -162160840 -1 6275964s -163069250 -1 63220855 -1 63292160 -1 63324565 -1.63338870 -1 63345080 -1 63348790 -1 633493m -1 633494

-1594159 -1 594882 -1 594992 -1 595002-1 605286

-1 611538 -1 612454 -1 612871 -1 613079-1 623173- 1 62784 1 -1 6787_09 -1 618396 - 1628492-I 631152-1 632430-1633022-1 633290-1 633407-1 633458-1 633488-1 633493-1 633494 -1 633494 -1633494 -1 633491

-1650016 - 1 644034-1 660317 -1 658213-1 669729 -1 669818-1 672648 -1 672667-1 673062 -1 673047- 1 673221 -1 6731?5-1 673270 -1 673247-1 673278 -1.673261-1 673267 -1 673258-1 673255 -1 673251-1 673241 -1.673241

Table 2Ha rm omc components , m kcal mol- of the H-H? (r = 1 4 no) SCF an d SCF Cl lntera ctlon potcntxxlsR (~0) SCF scr Cl

vo 3) r, b) v, a) v2 b) vj b) if6 b, rro 3) I, C) rr , a) v2 c)

16 86 23 72 95 37 13 24 95 21 73 3 692 6160 264820 45 00 4287 3 292 1 209 3.470 0.740 35 36 3167 1 126 -0.9222 25 32 92 0 74825 24 55 24 34 0 788 0 59s 0 335 0 0630 17 08 16 32 -2 502 -2 5972 75 18.18 0 95730 13 28 13 23 0 940 0 898 0.08 11 0 0100 8 990 8 699 -0 880 -091135 6 804 0 655 4 477 0.17040 3 326 3 320 0 355 0 370 0 0102 0 0001 Z-167 2 100 0 0371 -0.04145 1 566 0 193 0 838 -0 04450 0 714 0 0926 0 364 0 266 0 0079 -0 OOG55 0 317 0 0424 0 119 -0 00636.0 0.138 0.0187 0 0234 -0 010665 0 0587 0 0080 -0 0085 -0 00987.0 0.0246 0 0033 -0 0160 -0 007480 0 0041 0 0005 -0 0125 -0.003690 0 0006 0 0001 -0.0071 -0 0017

a) Values obtained from th e ener gies of table 1 for e = O and 90 nssu xmng a two-term Legendr e eupa nslon (1).) Values obtam ed from a least-squa res tit t o th e dat a of table 1 for e = O, 30 56O, 54.74, 70.12 and 90 usm g a four-ter m

Legendr e expans ion (I).) Ref. 151 , values obtam ed from the complete leastsquares potenual surface by mtesat ion

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Volume 77, number 1105

CHLMIC4L PtlYSICS LETTCRS 1 January 19

118 1 SphencAI~ .tvcrJped ki-112 (r = 1 4~2~) cpulswe potentrnk first-order perturbatron tbcor) results n ret [ 151. a ret [ 1SCF c.kxlattom + ret [91 o d11r \\orh nlod0icd Born-Xleqer analytic potentials - - - ref [ 131. - t l l lS \ t arkerrorswereneglrgible iO-~~~and7Xt0-6FI,atR=65a0 and R = 2 5~ respectixelf Our SCF results for V. canbe well fitted by A mod&ed Born-Meyer potent& (6)wtthA=403 43kcA moP1.bG=O 97919aoand bl =0 05775ag2, root-mean-square de\mrton 5s 0 31 kcalmot-l and ryp~cal errors are =3% They agree surpn-SU& weltwtth the Torelio-Dondi emptrrcai Hartree-Fock curve tn the range 3.0

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Volume 77. number I CHEMICAL PHYSKS LETTERS I Jwn~ary 19

10

00

ii ?z06

0.4

02

00I-IS 2 Sphervzally averaged H-H, (r = 1 4 go) dlspersron damprng factorsD(R) = (VW - I)&$?* -i-C@- -+ C& 2-o) - -ref [131,--- this ~orh. rq (8), 0 calculated aiues from t ie SCF energxes of the present =orb and the Ci dara of ref [S] Shom the insert ss the approumate llnenr dependence of t he calculated pomts when plotted as Inf I - D(R)f /R2 versus R

4033 2 au respectively. Thus suggests that a darnpmgfunction of the formD(R) = 1 - exp[--R2(dl f d,R)I (8)ml&t provide a simple means to interpolate for R >2 n o between long- and short-range potentrals. Tlusfunction has the advantage that rt IS analytic (contin-uous m all derlvatlves) for au R and thus It allows short-and long-range potentials to merge smoothly_

The Linear constants dI and ~2, were determmedfrom a least-squares Et to points& the repon I A 5.0 a@ behveen our Hartree-Fock curve and that ehpertmentally determrned byToreilo and Dondl [13] _

Investigation ot the leadmg term rn the anisotroplcpotential at the SCF level suggests that a two-term trun-cation of the Legendre expansion IS vahd except atsmall R, R < 2 5 ao_ Th u s, one may expect that a rea-sonably accurate estunnte of Vz can also be obtamedfrom the 0 = 0 and 90 mteraction potent& m a waysmdar to rhat we used to get VO. for such an approachto the He-H, and M--H7 systems, see ref 1171 How-ever. one requzes for thzskcurate estimates of theamsotropies of the C6. C, and Cl0 coefficients

Llmlted calculattons at the SCF CI level suggest thatthere s stdl work to be done on the anisotroplc poten-t1a1 at large I?.

AcknowledgementThe authors gratefully acknowledge helpful drscus-

slons with Professor J N. Murrell. The calculationsreported m this paper were performed on the ATMOL

3 and SPLICE programme suttes documented by AComputmg Dlvlsion, Rutherford Laboratory TIushas been carried out with the help of a NATO grant

References[ 11. D G Truhlar and R E W>att, Ann Rev. Phys Chem (1976) 1[Z] DC Truhlnr and R E. Watt. Advw Chem Phys 36(1977) 141131 6 LIU, J Chcm Phys SS (1973) 1925[+%I S~egbabn and B Lm. J Chem Phss 68 (1975) 245[S] DC Truhlrtrand C J Horo1tttz.J Chem Phys 68 (1

2466161 A J C Varnndas J Chem Phys 70 (1979) 3786[71 A A Wu, hlol Phys 38 (1979) 843[S] J XI Norbeck, P R Certam and K T Tang, J Chem 63 (197.5) 590f9] J Bf Norbeck and P R Certam, J Chem Phys 63 (19-1127[ 101 R Gengenbach Ch Hahn and J.P Toenntes, J. Chem

Phys 62 (1975) 3620[I 11 W We% Dwertntlon, Unwers~ry of Bonn (1976)1121 N Hlshmuma, J Phys Sot Japan 41 (1976) 1733.[ 131 r Torelio and XI G Dondr, J Chem. Phfs 70 (1979)1565(141 R Ahtnchs, R Pence nnd G Stoles, Chem ~11~s 19(1977) 119[Is] H hlargenau,Ph}s Rev 66 (1944) 303[161 R A Mason and J 0 Hrrschfelder, J Chem Phys 26(1957) 756[I71 K T Tang and J-P Toennies, J Chem Phys 68 :197S)5501

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