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Volume 61. number 3 CHEMICAL PHYSICS LETTERS 1 March 1979

CHERCHER LE CROISEMENTAJ .C. VARAN?AS *, J . TEN NYSON and 1-N. MURRELLS&ml o f Molecu lcrr Scien ces , Un irem fy o f Su ssex. Bright on B;VZ FQJ, UKReceived 2 November 1978Revised manuscript received 20 November 1978

An intersection of the two lo%*est doublet potential energy surfaces of LiNaK has been established b; ab-initi o CalcuIa-tions.

An interesting controversy in recent years has beenon whether potential-energy surfaces of poiyatomicmolecules cross in regions of space where the twowavefunctions belong to the same symmetry species[t -61. Such crossings are forbidden for diatomic po-tential energy curves hut, as was first pointed out byTeller [l] , and later ampiified by l-lerzberg andLonguet-Eggins [Z] , crossings are allowed in (Ar - 2)-dimensional space for a polyatomic with N internalcoordinates. This conclusion has recently b een chal-lenged [3,4] but, in turn, the basis for this challengehas been refuted [S] .The existence of a crossing point can be establishedby the following theorem [2,5,6]. If S is a surface innuclear configuration space bounded by a closed loopL and if the electronic wavefunction changes signwhen transported adiabatically once around L thenthere must be at least one point on S at which thewavefunction is a discontinuous function of the inter-nal coordinates. Longuet-Higgins [5] showed that forthe lowest energy doublet state of three S atomsthere was such a change in sign provided that a loopcould be found around which certain bamiltonian ma-_ trk elements had an assumed parametric form. It wasargued that such a loop was qualitatively j ustified.Stone [6] has likewise shown by qualitative argu-ments the existence of an intersection for the surfaceof a d1 transition metal with six nonequivalent li-* Permanent address: Iaboratorio Quimico, Universidade deCoitnbta, Coimbra, Portugal-

gands. Our objective in this paper is to establish by ab-initio calculations the existence of a loop satisfyingthe sign reversal criterion for the system L.iNaKThe system was examined using a minimal ST OSGexpansion of the molecular orbital with no p orbitalsin the valence shell _ Although this basis will not, ofcourse, give accurate wavefunctions, there bar been nosuggestion that approximate potential energy surfacesshould have different characteristics to the true Bom-Oppenheimer surface provided that there are no hid-den symmetries within the approximationPaths in the potential ener,v surface for a triatomicmolecule are convenientIy expressed in the coordi-nates (SI , p, 6) where$7 3-w

I r3-W 3-w 11R, Ry_is, =rO 2 -m -2 -e

1I h -R; i 01($p _& /2 &A Rx -&j 11

andp = (Sf +sp, tan 0 = SrfS3-R, are the three internuclear distances and RF are ref-erence bond lengths. ?Ve took these to be all equal tothe arithmetic mean of the equilibrium bond lengthsof the three homonuclear diatomic molecules, thus

Calculations were all made along a path for which S,= 0, that is, the perimerer of the triangle is fried andequal to the sum of the three h o m o n u c i e ~ diatomic431


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VoIume 61 , mem ber 3 Ct iEMfCAL PHYSICS LF. lTERS 1 Mar ch i919

Na

F i $ l_ C o o rd i i r e s d e f%in g th e p a th s a l c n g \ i h i c h u l c u h t io r sfL lre been mzde . The c i rc le s ha ve ra d i i p = I [2 ,& an d 4 .8 /, /6a n d th e in t e rm e d ia t e a rc 1 r a d iu s 3 /J 6 . T h e o r ig in c o r re sp o n d sto an equ ik tem i t r im@e c f pe r ime t e r 9 .6 A.

d i s t a n c e s_ We a r e t h u s se e k in g a point of discontinuityin the wavefunction and not the line of such pointsthat can exist in the complete three-dimensiona sur-face.Wavefunctions were first obtained for the re-stricted Hartree-Fock approximation along the circu-

lar path shown in fig. 1. The wavefirnction is a singledetetrninant with a set of doubly fil led orbitals repre-senting inner shells, one doubly occupied valence or-bital and one singly occupied orbitaf; there i s also onevirtual valence orbital. The Iatter three are ah of Asymmetry in the C, group.It is clear from f ig. 2 that the RHF wavefunctionchanges sign on passing once around the loop. Thiswavefunction is not a variational lower bound for theMom-Oppenhe~er hamiltonian within the adoptedbasis set because it does not allow for ful l correlationof the electrons. We can, however, obtain such a boundby configuration interaction within the RPlF orbitalsto give a wavefunction.

(4)where qi are other configurations. As the individualRHF orbit& vary smoothly around the loop so mustthe qi. Providing that configuration inferaction issmall (Xi = 0) it is reasonable to assume that hi is con-tinuous and hence, if JrRRF changes sign so must~~:ixi\k,and hence 8.The variation of wavefunction was next examinedalong the diagonals S, = 0 and S3 = 0 of the circularpath already described. For the Sr = 0 diagonal thewavefunction was continuous and we established thatthe non-analytic point was in the sector - $r < 0


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Vo iu m e 6 1 , n u m b e r 3 CHEMICAL PHYSICS LETTERS I March 1979

F ig _ 3 . Va r i a t io n in t h e c o e ffk re n t s o f t h e v a l e n c e b a s i s fu n c t io n s fo r t h e R H F ( fu l l l i n e ) a n d S E R I IF (d a sh e d l in e ) s in g l : o c c u p ie do rb i t a l . T h e p a th fol lo lxe d i s t h e p e r im e t e r o f t h e sh a d e d r e g io n sh o w n in f ig. 1 . S y m b o l s a s i n f i _r .2_

suggested that the crossing point lies near this line_ Awedge-shaped path bounded by the radial paths 0=&r and 0 =$n and the circular paths p = 0.5 and p= 3-O was then studied and as can be seen from fig_ 3the RHF coefficients are smoothly varying but changesign on going around this path.The RHF hamiltonian allows for Coulomb repul-sion and exchange between the electrons and is there-fore a function of its eigenvector and the orbital popu-lation. For incompletely filled shells the hamiltoniandoes not necessarily have the symmetry of the Bom-Oppenheimer hamiltonian. For example.. a standardRHF ca!culation on H3 in D;, configurations will notgive degenerate E orbit& because only one of theseorbitals will be occupied and this does not give an RHFhamiltonian with Dsh symmetry. One can correct thisby a symmetry equivalencing (SE) procedure which ef-fectively puts equal fractional populations of electronsinto symmetry equivalent orbitals [7]. Moreover, ifthis procedure is adopted when there is no symme-try requirement (e.g. for H, in C, symmetries) thenthe convergence of open shell calculations is improvedand the eigenfunctions merge smoothly into those ap-&propriate to the high symmetry structures.SE RXF calculations on LiIWS were made for anelectron configuration which has equal (notionallyhalf-electron) occupation of the two highest molecularorbitals. The resulting orbitals are very little differentfrom those obtained by the standard RHF procedurefor the path shown in fig. 2 and this is because thepath is far away from the surface crossing point. Close

to the crossing point the orbitak do show significantdifferences as can be seen from fig. 3 which showsSE RHF and RHF coefficients for the same path. Abrief examination of the wavefunctions within thisbounded area shows that degeneracy of the two equiv-alent orbitals occurs approximately at the point p= 1_2Oag,O =o.We have finally to show that the coefficients of tlzeCI expansion are analytic around the closed loop. CIcalculations, using dil contigurations that can be gener-ated from the three valence orbitals, have been carriedout around the closed loop defined in fig. 3. The varia-tion in t!le coeffklent of the SE RHF component issho\\n in fig 4 and, wivithin he interval chosen, ap-pears to be continuous. If one coefficient is continu-ous then all are. Moreover, we can argue on perturba-tion grounds that the inclusion of configurations inwhich the inner shell electrons are promoted wouldnot change this conclusion.

Fig_ 4_ Coeffic ien t o f th e SE RHF roo t func t ion in th e CI ex-p a n s io n fo r t h e p a th r e fe r r e d to i n f i g _ 3 _ T h e p o in t s c & u -l a t e d w e r e in d ic a t e d th u s * .

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Vo lu m e 6 5 , n u m b e r 3 CHEMICAL PHYSICS LETTERS 1 Mar ch 1979

It might be argued that no fmite set of computa-tions can disprove the existence of additions signchanges in the intern& between computed points. Weca n appeal only to the experienced eye, Are the inter-vais shown in figs. 2, 3 and 4 sufficiently convincing?To us they are and we can only suggest that those whodoubt fXl in a few more points,All calculations reported in this paper were carriedout on the ATMOLj3 and SPLICE sys tem of programsdocumented by the -4tlas Computing IX-vision,Rutherford Laboratoly.

AMT_C_V_hanks the British Council for f&an&lsupport during the period nwhich this work was com-pleted.

[I] E . Teller,3. P h y s , them. 4l(t937) 109.f2] G -H. Herzberg nd H.C. Longaet-~~~ DiscussionsFzr ad ay Sot. 3s (15163) 77s

f31 K Raz i Naqu i , Chem . Ph ys- Letters 1 .5 (1972) 634.[4f G.J. Hoytink,Chem. Phys. Letters 34 (1975) 414.fs] H-C?. ong ue Z-H@ins , P roc R oy. Sot. A344 Q.975) 147.[6 ] A-J. Ston e, P roc . Roy. Sot A351 (1976) 141.171 N-E Gue st an d V-R. Sa un der s, Mot P hys . 28 (1974) 819..

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