Hartree Fock Wave Functions with a Modified GTO Basis for Atoms

Transcription

1 Hrtree Fok Wve Funtions with Moifie GTO Bsis for Atoms E. BUENDIA, F. J. GALVEZ, A. SARSA Deprtmento e Fısi Moern, Fult e Cienis, Universi e Grn, E-807 Grn, Spin Reeive Mrh 996; revise 7 Mrh 997; epte 0 Mrh 997 ABSTRACT: We hve solve the tomi Hrtree Fok equtions y using the lgeri pproh, expning the single-prtile ril wve funtion in terms of moifie Gussin type oritls Ž GTOs. sis. Severl tomi properties suh s Kto s usp onition for the eletron ensity or the orret symptoti ehvior of the eletron momentum ensity istriution re urtely verifie. Aitionlly the energy of the tomi groun stte n e otine y using smller numer of sis funtions thn in stnr GTO expnsions. This stuy hs een performe for severl toms of the first three rows. 997 John Wiley & Sons, In. Int J Qunt Chem 65: 59 64, 997 Key wors: Atomi struture; Hrtree Fok; moifie Gussin oritls; momentum ensity; single-prtile ensity Introution H rtree Fok Ž HF. wve funtions for toms my e otine numerilly or y mens of lgeri pproximtions, in whih the ril wve funtion n e expne either in terms of Slter-type oritls Ž STOs. Žthe t of Clementi n Roetti re the most use within this type of sis. or in terms of Gussin-type oritls Ž GTOs. 4, 5. Reently the Hrtree Fok t of Clementi n Roetti hve een improve y reoptimiztion Corresponene to: F. J. Glvez. Contrt grnt sponsor: Spnish Direion Generl e Investigion Cientıfi y Teni Ž DGICYT.. Contrt grnt numer: PB95--A. of the exponents of the STOs 6 n y vrying the omposition of the STO sis for eh tom 7, 8. It shoul e note tht fewer STOs thn GTOs in tomi lultion re neessry to otin the energy with given ury. However, this lst type is use minly in moleulr lultions euse ll the multienter integrls whih pper n e nlytilly evlute 9. But there exist two importnt limittions of working with GTOs euse ny finite liner omintion of Gussins n hve neither usp t the nuleus in the ril eletron ensity nor orret symptoti ehvior of the momentum ensity istriution. The Gussin-type oritls re efine in spheril polr oorintes s l r Nr e Y Ž,. Ž. lm lm Interntionl Journl of Quntum Chemistry, Vol. 65, John Wiley & Sons, In. CCC / 97 /

2 BUENDIA, GALVEZ, AND SARSA eing N onstnt of normliztion. Note tht the power of the r vrile oinies with the ngulr momentum of the tomi oritl. Mny lultions hve een performe y using this sis; perhps one of the most reent n etile stuies for ifferent toms re those of Prtrige 5. We lso mention the work of Primor n Kovevi ˇ 0 where they introue the Hermite Gussin sis funtions Ž. l H r Nr H Ž r. e r Y Ž,. nlm n lm n lm, Ž. eing H n the Hermite polynomil of n egree. This expression inlues even powers on r Žprt the term r l. in the GTO sis to esrie the ril wve funtion of the oritls for the Helium tom n H moleulr ion. Both types of GTO sis given respetively y Eqs. Ž. n Ž. provie wve funtions tht re le to reproue quite well the est energy of the toms otine either numerilly or y mens of STO prmeteriztion, ut none of them is le to reproue the Kto s eletron nuleus usp onition, : Ž 0. Ž 0. Z 0 eing Ž r. the spherilly verge single-prtile ensity, nor the long p rnge ehvior of the momentum ensity istriution. As it is known 6, preise esription of oth the singleprtile ensity t the nuleus n the momentum ensity istriution is neessry to reh n greement with the experimentl t. A previous work tht tkes into ount the usp onition y using Gussins ws one y E. Steiner 7, who employe GTO sis n n extr usp funtion of the form t Ž. Ž. r r if r 0 if r, where n t re vritionl prmeters. The prie to e pi for using this sis set is tht some integrls must e numerilly performe. Finlly, we woul like to omment tht reful stuy of ifferent types of GTOs hs een me y W. Klopper n W. Kutzelnigg 8 y testing them in the hyrogen tom. The min onlusion of tht work is tht one must use sis set tht stisfies the eletron nuleus usp in orer to get the fstest onvergene in the lultion of, for exmple, the energy of the system or the vlue of the eletron ensity t the nuleus. Our purpose is to show how GTO sis moifie in suh wy tht it inlues o power on r provies not only similr results for the energy to those provie y nother sis of Gussin type ut lso goo esription of the single-prtile ensity roun r 0, n the orret symptoti ehvior of the orresponing momentum ensity istriution. Moifie GTO Bsis n Results The elementl funtion of the moifie GTO sis use in this work will e enote y Ž. nlm r n hs the following nlytil expression: Ž. n r r Ž. r nlm lm n r l Nr e r Y Ž,. lm, n,,,... Ž. where we hve expliitly seprte the power r n from the usul GTO. We only nee oritls with l 0 to esrie the ril wve funtion of the helium, lithium, n eryllium toms. These elementl oritls will e lle s,s,s,... for n,,,..., respetively. In oing so we re using similr nottion to the stnr one use in STO expnsion. To esrie the other toms, we nee l, oritls, whih will e enote Ž n. p n Ž n. with n,,... respetively. n The inlusion of the term r Ž n,,... to esrie the oritl Ž. nlm r llows one to get the groun-stte energy, with given ury, with smller numer of sis funtions thn tht neee in other previous GTO sis 5, 0, lthough this numer is still greter thn the one neee with STO expnsion 8. This n e seen in Tle I where we show the energy of the helium, eryllium, n neon toms in terms of the numers of elements in the sis, n we ompre these results with those otine y Prtrige 5 with those from STO lultion 8, n with others from numeril lultion 9. The sis set use in the esription of oritls with ngulr momentum l 0 will e omintion of n oritls of s type, n oritls of s type, et. tht will e enote herefter y Ž n, n,.... For the neon tom we hve to give the sis set use to esrie the single-prtile wve funtion with l 60 VOL. 65, NO.

3 HARTREE FOCK WAVE FUNCTIONS TABLE I Energy of the helium, eryllium, n neon toms for severl onfigurtions of the sis set s ompre with oth STO expnsion n numeril lultion. Helium Beryllium Neon Bsis Energy Bsis Energy Bsis Energy ( ) , ( 6) ( 5) (( ), ( 8)) ( 4,, ) , (( 7, ), ( 6)) ( 5, 5) , (( 8, 5 ), ( 8)) , (( 9, 5 ), ( 9)) STO STO STO Num Num Num Ref. [ 5 ]. This work. Ref. [ 9 ]., n this is one y inluing seon prentheses with the numer n p of p funtions use in suh lultion. For exmple, ŽŽ 9, 5., Ž 9.. mens tht we hve expne the oritls with ngulr momentum l 0 y using nine s type funtions n five s funtions, n the oritl with ngulr momentum l y using nine p funtions. The sum n n n p gives the totl numer of oritls use in the lultion. The results for the energy for severl toms re presente in Tles II, III, n IV for the three kins of prmeteriztions stuie in this work. We lso give the oeffiient V efine s the quotient etween the potentil n the kineti energy, whih is equl to for the ext solution of the Hrtree Fok equtions. The viril rtio from the numeril Hrtree Fok lultion oes not iffer from the ext one more 0 thn 0 9. Besies inresing the onvergene, the most importnt ontriutions of the o power in the GTO sis is on oth the spherilly verge single-prtile position, Ž r., n momentum, Ž p., ensities. The former hs the goo ehvior t the origin with this new prmeteriztion of the ril wve funtion, wht it is impossile to get with the stnr GTO prmeteriztion or with TABLE II Vlues for the totl energy n for the viril onition ( V ) for ifferent expnsions for the helium, lithium, n eryllium toms. Atom Configurtion E V He S ( 5, 5) STO Num Li( S) ( 6, 5) STO Num Be( S) ( 4) ( 7, 7) STO Num Ref. [ 9 ]. TABLE III Vlues for the totl energy n for the viril onition ( V ) for ifferent expnsions for the nitrogen, neon, n silion toms. Atom Configurtion E V 4 ( ) N S, (( 8, 4 ), ( 0)) STO Num Ne( S) (( 4 ), ( 9)) (( 9, 5 ), ( 9)) STO Num Si( P) (( 5 ), ( 0)) (( 0, 5 ), ( 0)) STO Num Ref. [ 9 ]. INTERNATIONAL JOURNAL OF QUANTUM CHEMISTRY 6

4 BUENDIA, GALVEZ, AND SARSA TABLE IV Vlues for the totl energy n for the viril onition ( V ) for ifferent expnsions for the lium, snium, n titnium toms. Atom Configurtion E V ( ) C S 5, (( 0, 5 ), ( 0)) STO Num S( D) (( 4 )()()), 9, (( 9,5, ) ( 9, ) ( 8)) STO Num Ti( F ) (( 4 )()()), 9, (( 9,5, ) ( 9, ) ( 8)) STO Num Ref. [ 9 ]. the Hermite Gussin sis. In prtiulr the Kto usp onition, is stisfie quite well. In Tles V, VI, n VII we show, for severl toms n for ifferent sis sets, the vlue of the ensity t the origin, Ž. 0, n the vlue of the prmeter given y en Ž 0., Ž 4. en Z Ž 0. whih must e equl one ue to the Kto s usp onition. These results re ompre with those otine from STO expnsion 8 Žrell tht with the stnr GTO expnsion this oeffien TABLE VI Vlues of the single prtile eletron ensity t the origin, ( 0 ), n of the Kto usp ( ) en for ifferent expnsions for the nitrogen, neon, n silion toms. Atom Configurtion 0 n 4 ( ) N S, (( 8, 4 ), ( 0)) STO Ne( S) (( 4 ), ( 9)) (( 9, 5 ), ( 9)) STO Si( P) (( 5 ), ( 0)) (( 0, 5 ), ( 0)) STO [ ] Ref. 8. ient is lwys zero.. One n notie how Ž 0. n en here otine re very similr to the vlues lulte with the STO prmeteriztion of the ril wve funtion. It is known tht the ltter, Ž p., ehves for smll vlues of p s 4, 0 : Ž. Ž 4. Ž. p A A p O p. 5 0 This expression n e reproue y using STO or GTO sis. It is lso known tht the lrge p ehvior of the eletron momentum ensity is governe y the short-rnge ehvior of the ril oritl 4,. So one n see tht STO expnsion gives ensity Ž p. whih symptotilly TABLE V Vlues of the single prtile eletron ensity t the origin, ( 0 ), n of the Kto usp ( ) en for ifferent expnsions for the helium, lithium, n eryllium toms. Atom Configurtion 0 n He S ( 5, 5) STO Li( S).64 0 ( 6, 5) STO Be( S) ( 4) ( 7, 7) STO [ ] Ref. 8. TABLE VII Vlues of the single prtile eletron ensity t the origin, ( 0 ), n of the Kto usp ( ) en for ifferent expnsions for the lium, snium, n titnium toms. Atom Configurtion 0 n ( ) C S 5, (( 0, 5 ), ( 0)) STO S( D) (( 4 )()()), 9, (( 9, 5 )()()), 9, STO Ti( F ) (( 4 )()()), 9, (( 9, 5 )()()), 9, STO [ ] Ref VOL. 65, NO.

5 HARTREE FOCK WAVE FUNCTIONS eys s 0 B8 B0 Ž. Ž p O p.. Ž p p A liner omintion of Gussins les to momentum ensity istriution whih eys symptotilly s Gussin, ut the inlusion of o powers in the GTO sis to esrie the ril wve funtion of the sttes with l 0 mkes tht the momentum ensity istriution hs the orret symptoti ehvior given y Eq. Ž. 6, eing the reson the following: while the Fourier trnsform of Gussin is nother Gussin, the Fourier trnsform of re r is funtion tht is proportionl to p 4 for lrge vlues of p. This n e otine, for exmple, y using some results of MKinnon. To see the qulity of the momentum ensity istriution here otine we show in Tles VIII, IX, n X the vlues of the prmeters A0 n B8 Z 5, where Z is the nuler hrge, for severl toms n for ifferent GTO n MGTO expnsions of the ril wve funtion. These results re ompre with those oth otine from STO 0 n from numeril 9 lultion. We notie how the vlues here otine ompre quite well with those of Refs. 9, 0. We lso n hek the improvement tht we otin in the vlue of the spherilly verge momentum ensity t p 0 with respet to stnr GTO lultion. TABLE VIII Stuy of the symptoti ehvior of the eletron momentum ensity istriution, oth t smll n lrge vlues of p for ifferent onfigurtions for the helium, lithium, n eryllium toms. Atom Configurtion A B /Z He S ( 5, 5) STO Num Li( S) ( 6, 5) STO Num Be( S) ( 4) ( 7, 7) STO Num Ref. [ 0 ]. Ref. [ 9 ]. TABLE IX Stuy of the symptoti ehvior of the eletron momentum ensity istriution, oth t smll n lrge vlues of p for ifferent onfigurtions for the nitrogen, neon, n silion toms. Atom Configurtion A B /Z ( ) N S, (( 8, 4 ), ( 0)) STO Num Ne( S) (( 4 ), ( 9)) 0.46 (( 9, 5 ), ( 9)) STO Num Si( P) (( 5 ), ( 0)).8944 (( 0, 5 ), ( 0)) STO Num Ref. [ 0 ]. Ref. [ 9 ]. As we hve mentione in the introution n urte esription of the wve funtion is essentil for n greement with the experimentl results. So one oul sk for the role of the orreltions in the quntities here lulte within the Hrtree Fok frmework. It is known tht orreltions re very importnt to reproue the energy of the tomi systems n to esrie some tomi properties like the intereletroni ensity, hž. s, n TABLE X Stuy of the symptoti ehvior of the eletron momentum ensity istriution, oth t smll n lrge vlues of p for ifferent onfigurtions for the lium, snium, n titnium toms. Atom Configurtion A B /Z ( ) C S 5, (( 0, 5 ), ( 0)) STO Num S( D) (( 4 )()()), 9, (( 9, 5 ), ( 9 ), ( 8)) STO Num Ti( F ) (( 4 )()()), 9, (( 9, 5 ), ( 9 ), ( 8)) STO Num Ref. [ 0 ]. Ref. [ 9 ]. INTERNATIONAL JOURNAL OF QUANTUM CHEMISTRY 6

6 BUENDIA, GALVEZ, AND SARSA they re essentil to fit severl experimentl results relte to inelsti proess, 4. However, they o not ply n importnt tsk in the esription of one-oy quntities s one noties y ompring the result here otine for Ž r. n Ž p. of the helium tom with those otine inluing orreltions 5. In ition it is known tht oth Hrtree Fok n onfigurtion intertion Ž CI. lultions provie prtilly the sme results for the momentum istriution of the highest oupie oritls of the nole gses n some oritls of severl moleules 5, 6. Conlusions We woul like to emphsize the following points reltive to the effet of the inlusion of o powers in GTO sis to prmeterize the ril Hrtree Fok wve funtion. The first one is tht it oes not require ny extr omputtionl prolem to solve the Hrtree Fok equtions for toms, lthough this is not the se for moleulr systems for whih the numeril iffiulties re not voie with this type of sis. The seon point is the improvement of the onvergene of this sis with respet to the stnr GTO prmeteriztion or to the Hermite Gussin sis. Besies the energy hypersurfe is not so flt s the one in pure GTO lultion n the minimum in the energy is rehe fster. However, the most importnt fts tke ple in the single-prtile ensity funtion n in the momentum ensity istriution where the o powers in the sis provie the orret symptoti ehvior ner the origin n t lrge vlue of p, respetively, otining similr vlues to those otine with STO prmeteriztion. These lst results re not reprouile with the stnr GTO sis. ACKNOWLEDGMENTS This work hs een prtilly supporte from the Spnish Direion Generl e Investigion Cientıfi y Teni Ž DGICYT. uner ontrt PB95--A n from the Junt e Anluı. Referenes. Ch. Froese Fisher, The Hrtree Fok Metho for Atoms ŽWi- ley, New York, C. C. J. Roothn, Rev. Mo. Phys., 69 Ž 95.;, 79 Ž E. Clementi n C. Roetti, At. Dt Nul. Dt Tles 4, 77 Ž S. Chkrvorty n E. Clementi, Phys. Rev. A 9, 90 Ž H. Prtrige, J. Chem. Phys. 87, 664 Ž 987.; 90, 04 Ž T. Kog, H. Ttewki, n A. J. Thkkr, Phys. Rev. A46, 450 Ž C. F. Bunge, J. A. Brrientos, A. V. Bunge, n J. A. Cogorn, Phys. Rev. A46, 69 Ž T. Kog, S. Wtne, K. Knym, R. Ysu, n A. J. Thkkr, J. Chem. Phys. 0, 000 Ž F. E. Hrris, Rev. Mo. Phys. 5, 558 Ž M. Primor n K. Kovevi, ˇ Phys. Rev. A 46, 540 Ž T. Kto, Commun. Pure Appl. Mth. 0, 5 Ž E. Steiner, J. Chem. Phys. 9, 65 Ž T. Aerg, Phys. Rev. A, 76 Ž A. J. Thkkr, J. Chem. Phys. 86, 5060 Ž P. Duffy, M. E. Csi, C. E. Biron, n D. P. Chong, Chem. Phys. 59, 47 Ž P. Duffy, S. A. C. Clrk, C. E. Biron, M. E. Csi, n D. P. Chong, Chem. Phys. 65, 8 Ž E. Steiner, Mol. Phys., 669 Ž 97., n referenes therein. 8. W. Klopper n W. Kutzelnigg, J. Mol. Strut. ŽTheohem 8. 5, 9 Ž T. Kog n A. J. Thkkr, J. Phys. B: At. Mol. Opt. Phys. 9, 97 Ž A. J. Thkkr, A. L. Wonfor, n W. A. Peersen, J. Chem. Phys. 87, Ž J. C. Kimll, J. Phys. A: Mth. Gen. 8, 5 Ž R. F. MKinnon, Mth. Comput. 6, 55 Ž J. Wng, R. O. Esquivel, n V. H. Smith, Jr., Phys. Rev. A 5, 8 Ž S. Gupt n M. K. Srivstv, Phys. Rev. A5, 08 Ž F. Aris e Sver, E. Buenı, n F. J. Glvez, Z. Physik D8, 5 Ž VOL. 65, NO.