Ab initio group model potentials including electron correlation effects

J. Chem. Phys. 113, 6082 (2000); h ps://doi.o g/10.1063/1.1290003 113, 6082 © 2000 Ame ican Ins i u e o Physics. Ab ini io g oup model po en ials including elec on co ela ion e ec s Ci e as: J. Chem. Phys. 113, 6082 (2000); h ps://doi.o g/10.1063/1.1290003 Submi ed: 03 Feb ua y 2000 . Accep ed: 07 July 2000 . Published Online: 02 Oc obe 2000 No ge C uz He nández, and Ja ie Fdez. Sanz Ab ini io g oup model po en ials including elec on co ela ion e ec s No ge C uz He na ´ndez and Ja ie Fdez. Sanza) Depa amen o de Quı ´mica Fı ´sica, Facul ad de Quı ´mica, E-41012, Se illa, Spain 共Recei ed 3 Feb ua y 2000; accep ed 7 July 2000兲 A me hod o de e mina ion o ab ini io g oup model po en ials, wi h he in en ion o desc ibing he e ec s o a whole molecule o a chemical g oup wi hin a densi y unc ional heo y amewo k, is epo ed. The one-elec on pa o he Kohn–Sham equa ions is modi ied by inco po a ion o a Coulomb ope a o , which accoun s o he classical elec on in e ac ion a ising om he g oup. Exchange and co ela ion e ec s a e in oduced by a sui able modi ica ion o he exchange-co ela ion unc ionals. The s ong o hogonali y condi ion, usually equi ed by he heo y o sepa abili y o many elec on sys ems, is w i en in e ms o i s o de educed densi y ma ices. In o de o check he me hod a g oup model po en ial o H2O共en i onmen 兲was ob ained and employed in he calcula ion o HF¯H2O and H2O¯H2O complexes using se e al unc ionals. Equilib ium in e g oup dis ances and binding ene gies a e compa ed wi h all-elec on calcula ions. ©2000 Ame ican Ins i u e o Physics. 关S0021-9606共00兲30337-3兴 I. INTRODUCTION A winning app oxima ion in quan um chemis y, espe- cially help ul in educing he huge compu a ional e o usu- ally in ol ed, is he classi ica ion o elec ons o a gi en sys em as ac i e o inac i e.1–4 The inac i e elec ons may be assumed o beha e as spec a o s and he e o e can be ozen. This idea has been success ully exploi ed o desc ibe he beha io o a oms o ions o model en i onmen e ec s in clus e embedded calcula ions,5,6,7 as well as o he de e - mina ion o e ec i e po en ials o agmen s ha could be employed as sa u a o g oups. In his sense, e ec i e po en- ials o NH3,8SiH3,9COOH,10 and cyclopen adienyl ligand11 ha e been epo ed. On he o he hand, some e o s ha e been de o ed o eplace a comple e molecule, o chemical g oup, in calcula- ions in which he en i onmen is desc ibed h ough an em- bedding ype echnique. Along his line, in 1993 Mejı ´as and Sanz12 epo ed on he de e mina ion o ab ini io g oup model po en ials 共GMPs兲 o he HF molecule, oge he wi h some applica ions. La e , F ank e al.13 epo ed on he ea- sibili y o ep esen ing spec a o molecules h ough e ec i e po en ials gene a ed by a semiempi ical NDDO calcula ion. Qui e ecen ly, Day e al.14 came up wi h a me hod o ob ain wha hey called an ‘‘e ec i e agmen po en ial’’ which makes use o a dis ibu ed mul ipole expansion, and which allowed hem o eplace wa e molecules by an e ec i e po- en ial, wi h excellen esul s.15 Mo e ecen ly we ha e de- eloped polycen e compac model po en ials in which he sho - ange con ibu ion was expanded as a spec al ep e- sen a ion. Also, we epo ed an algo i hm o sol e he p ob- lem associa ed wi h o a ion o he model po en ial.16 A s aigh o wa d way o imp o e g oup model po en- ials would in ol e he inco po a ion o he elec on co ela- ion e ec s be ween ac i e and ozen elec ons. On a gen- e al basis, his con ibu ion could in p inciple be inco po a ed in o he densi y unc ional heo y 共DFT兲共Re s. 17, 18兲 amewo k h ough a modi ica ion o he exchange- co ela ion e m. This opic has been conside ed o a omic co es in se e al wo ks19,20,21 and, in his con ex , one o he mos widely pseudopo en ials used in solid-s a e calcula ions is he so-called ‘‘ul aso pseudopo en ial’’ 共UPP兲o Vande bil .22 As o he ep esen a ion beyond a omic co es, Aba en- ko e al.23 epo ed a me hod in which he wa e unc ion o a small clus e o Li2Mg was compu ed by inco po a ing he exchange-co ela ion in e ac ion wi h a neighbo ing g oup in an i e a i e way. Mo e ecen ly Ca e e al.24 p esen ed a new embedding echnique ha combines pe iodic-DFT me hods and explici elec on-co ela ion p ocedu es. G ound-s a e p ope ies o chemical g oups ha e also been s udied by means o he ‘‘ eeze-and- haw’’ cycle o Kohn–Sham equa ions wi h cons ained elec on densi y 共KSCED兲as p oposed by Wesolowski and Webe .25 In his app oach, he elec on densi y o each agmen is kep o- zen while he in e ac ion ene gy is calcula ed using e ms de i ed om DFT, and, in addi ion o he exchange co ela- ion unc ional, he analy ical o m o he app oxima e ki- ne ic ene gy unc ional is needed. Examples like he wa e dime , HF–HF, HCl–HCl o HCN–HF may be ound as applica ions o he s udy o he hyd ogen bond.26 The main goal o his pape is o model he e ec s o a chemical g oup, o he en i onmen in clus e embedded cal- cula ion, bu now wi hin he one-pa icle Kohn–Sham sel - consis en ield 共SCF-KS兲 amewo k. The s a ing poin is he elec onic sepa abili y p inciple p oposed by McWeeny1 and Huzinaga2–4 bu in ol ing, in his case, he i s -o de educed densi y ma ices o he sepa able sys ems. Using he Kohn–Sham o bi als such condi ions lead o an ope a o simila o he Vande bil 22 and Phillips–Kleinman27 pseudo- po en ials. The Coulomb in e ac ion be ween elec ons o a gi en g oup was handled by i ing he elec onic densi y o a se o auxilia y Gaussian unc ions,28 acco ding o one o he a兲Au ho o whom co espondence should be add essed. Elec onic mail: [email p o ec ed] JOURNAL OF CHEMICAL PHYSICS VOLUME 113, NUMBER 15 15 OCTOBER 2000 60820021-9606/2000/113(15)/6082/6/$17.00 © 2000 Ame ican Ins i u e o Physics s a egies o Dunlap e al.29 The a icle is a anged as ol- lows. The heo e ical ame is b ie ly o e iewed in Sec. II. A new sepa abili y condi ion in e ms o g oup densi y ma- ices is p oposed and he way o inco po a e he exchange- co ela ion e ec coming om he en i onmen is p esen ed. In Sec. III some es calcula ions conce ning he H2O model po en ial a e epo ed. Finally he main conclusions a e sum- ma ized in Sec. IV. II. THEORETICAL ASPECTS In he densi y unc ional heo y, he SCF Kohn–Sham 共SCF-KS兲equa ions17 ha e o be sol ed, 兵 ⫺1 2ⵜ2⫹Vc共 兲⫹ ␮ xc共 ␳ 兲 其 ␺ i KS共 兲⫽␧i ␺ I KS共 兲.共1兲 The e m ⫺1 2ⵜ2is he kine ic ene gy ope a o , Vc( ) ep esen s he elec os a ic 共o Coulomb兲po en ial, and ␮ xc( ␳ ) in oduces he exchange and co ela ion con ibu ion, which, in gene al, may be w i en as ␮ xc共 ␳ 兲⫽ ⳵ Exc共 ␳ 兲 ⳵␳ ,共2兲 whe e Exc( ␳ ) is he so-called exchange and co ela ion unc- ional. In local densi y app oxima ion 共LDA兲o in he gen- e alized g adien app oxima ion 共GGA兲, i is gene ally w i - en as Exc共 ␳ 兲⫽ 冕 xc共 ␳ ,ⵜ ␳ 兲d3 .共3兲 The o al elec on densi y ␳ in Eq. 共1兲is ela ed o he i s -o de densi y ma ix o an Npa icles sys em in he o m ␳ 共 兲⫽ 冕 ⌫共x,x⬘兲d ␴ ,共4兲 whe e he spin–o bi al no a ion x⫽( ␴ ) is used and he i s -o de educed densi y ma ix is de ined as30 ⌫共x,x⬘兲⫽N 冕冕 ¯ 冕 ⌽*共x,x1,x2,...,xN⫺1兲 ⫻⌽共x⬘,x1,x2,...,xN⫺1兲dx1dx2¯dxN⫺1, 共5兲 ⌽(x1,x2,..,xN) being he elec onic wa e unc ion. The necessa y and su icien condi ion o he educed densi y ma ix o be N- ep esen able18 can be exp essed as ⌫ ˆ⫽兺 ini 兩 ␸ i 典具 ␸ i 兩 共6兲 wi h ␸ ibeing he spin–o bi als and 0⭐ni⭐1. Choosing ni ⫽0,1 in Eq. 共5兲, he o al elec on densi y may be spanned on a se o spinless single-pa icle o bi als, each one occupied wi h wo elec ons, ␳ 共 兲⫽兺 occ 兩 ␺ i共 兲 兩 2.共7兲 Le us now assume a sys em cons i u ed by wo mol- ecules o chemical g oups, Aand B, weakly in e ac ing, wi h ⌽Aand ⌽Bbeing he wa e unc ions. Acco ding o he building block p inciple o McWeeny and Huzinaga, he o- al elec onic wa e unc ion ⌽can be w i en as ⌽⫽MA ˆ共⌽A⌽B兲,共8兲 whe e Mis a no maliza ion ac o and A ˆis he in e g oup an isyme ize ope a o . Fo ma hema ical con enience in he de i a ion o ab ini io model po en ials, he g oup wa e unc ions a e selec ed so as o be s ongly o hogonal,1i.e., 冕 ⌽A *共x,i,j,...兲⌽B共x,k,l,...兲dx⫽0. 共9兲 In o de o keep he same condi ion in e ms o elec on densi y. Eq. 共9兲is mul iplied by ⌽A共2,i,j,...兲⌽B *共3,k,l,...兲 and in eg a ed wi h espec o i,j,..,k,l,.... Bea ing in mind he o m o he educed densi y ma ix, Eq. 共9兲becomes 冕 dx⌫B共3,x兲⌫A共x,2兲⫽0. 共10兲 Since in Eq. 共9兲, i is possible o w i e Ain place o B and Bplace o A, hen he s ong o hogonali y condi ion 共9兲 in e ms o educed densi y ma ices may be w i en as ⌫ ˆA⌫ ˆB⫽⌫ ˆB⌫ ˆA⫽0. 共11兲 I his condi ion is ul illed, he p oblem o inding he local p ope ies o he g oup o in e es 共clus e , A o B兲,in he p esence o a spec a o g oup 共en i onmen , B o A兲can be se up using he DFT o malism. Acco ding o he de ini ion o he densi y ma ix gi en in Eq. 共6兲, he s ong o hogonali y condi ion 共11兲is sa is ied i 具 ␺ i,clus KS 兩 ␺ j,en KS 典 ⫽0. 共12兲 The o al educed densi y ma ix in a sepa able sys em may be w i en as ⌫ ˆall⫽⌫ ˆclus⫹⌫ ˆen ,共13兲 and he elec on densi y, ␳ 共 兲⫽ ␳ clus共 兲⫹ ␳ en 共 兲,共14兲 whe e he elec on densi ies ␳ clus( ) and ␳ en ( ) a e no mal- ized o he numbe o elec ons o he clus e (Nclus) and he en i onmen (Nen ), espec i ely. Unde hese condi ions, he clus e ene gy in he p es- ence o he en i onmen 共 he e ec i e ene gy Eclus e )is Eclus e 共 ␳ clus兲 ⫽E0共 ␳ clus ,nclus兲 ⫹Ein 共 ␳ clus⇔ ␳ en , ␳ clus⇔nen ,nclus⇔ ␳ en ,nclus⇔nen 兲. 共15兲 The i s e m on he igh -hand side o Eq. 共15兲is he iso- la ed clus e ene gy, excluding all componen s depending on he en i onmen . The second e m is he in e ac ion o elec- on densi y in clus e ␳ clus , wi h elec ons and nuclei o he en i onmen ␳ en and nen , espec i ely. This las e m also includes he in e ac ion be ween he clus e nuclei, nclus , and he en i onmen nega i e and posi i e cha ges, which can be 6083J. Chem. Phys., Vol. 113, No. 15, 15 Oc obe 2000 G oup model po en ials added a he end o he elec onic ene gy calcula ion. No ice ha ollowing Eq. 共7兲 he clus e elec on densi y ␳ clus can be w i en as ␳ clus共 兲⫽兺 occ 兩 ⌿i,clus KS 共 兲 兩 2.共16兲 Applica ion o he a ia ional p inciple o Eq. 共15兲, o- ge he wi h he condi ion 共12兲leads o he se o SCF-KS- like equa ions 兵 ⫺1 2ⵜ2⫹Vc clus⫹Ve 共 兲⫹ ␮ xc e 共 ␳ clus , ␳ en 兲 其 ␺ ,clus KS 共 兲 ⫽␧i ␺ i,clus KS 共 兲.共17兲 In his equa ion he hi d and ou h e ms ep esen he in- e ac ion be ween he clus e and he en i onmen , ep e- sen ed h ough a g oup model po en ial 共KS-GMP兲. Analyz- ing hese con ibu ions, one o he cen al poin s in he p esen wo k, he hi d e m expands as Ve 共 兲⫽⫺ 兺 i nucl. en . zi 兩 ⫺Ri 兩 ⫹ 冕 ␳ en 共 ⬘兲 兩 ⫺ ⬘ 兩 d3 ⬘⫺P ˆen ,共18兲 whe e he summa ion ep esen s he in e ac ion be ween he clus e elec ons and he en i onmen nuclei, while he in e- g al accoun s o he elec on–elec on in e ac ion be ween he wo g oups. In he p esen wo k, such an in e ac ion has been simula ed by i ing he cha ge densi y o he spec a o g oup o a se o coe icien and Gaussian unc ions. The algo i hm used has been one o hose p oposed by Dunlap e al.29 whose main poin is o ind he se o coe icien s ha minimizes he unc ional D⫽ 冕 ⌬ ␳ 共1兲⌬ ␳ 共2兲 12 d 1d 2共19a兲 wi h ⌬ ␳ 共 兲⫽ ␳ en exac 共 兲⫺ ␳ en 共 兲.共19b兲 The se o Gaussian unc ions used he e was aken om Re . 28. Finally P ˆen is a weigh ed p ojec ion ope a o o e he occupied o bi als o he spec a o g oup ha allows o e- s ic ion o he a ia ional space o he clus e space e lec - ing he condi ion gi en by Eq. 共11兲, P ˆen ⫽ ␣ 兺 occ 兩 ␺ occ,en KS 典 ␧occ,en KS 具 ␺ occ,en KS 兩 ,共20兲 whe e he sum uns o e he occupied molecula o bi als o he en i onmen wi h eigen alues ␧occ,en KS . The p ojec o ac- o ␣ is se o 2.31 Conce ning he ou h e m o Eq. 共17兲, he ope a o ␮ xc e ( ␳ clus , ␳ en ) a ises om he exchange-co ela ion compo- nen o he in e ac ion be ween he wo g oups. I can be w i en as ␮ xc e ⫽ ⳵ Exc,clus e 共 ␳ clus兲 ⳵␳ clus ⫽ ␮ xc共 ␳ clus⫹ ␳ en 兲,共21兲 whe e Exc,clus e ( ␳ clus) is he e ec i e exchange-co ela ion en- e gy o he clus e Exc,clus e 共 ␳ clus兲⫽Exc共 ␳ clus⫹ ␳ en 兲⫺Exc共 ␳ en 兲共22兲 I a ac ion o he exac Ha ee–Fock exchange is used, o in a simple Ha ee–Fock calcula ion, he exchange ope a- o can be simula ed using a nondiagonal spec al ep esen a ion.5 Equa ions 共21兲and 共22兲in oduce he exchange and elec on co ela ion con ibu ion be ween clus e and he su - ounding sys em in a DFT amewo k. These wo equa ions a e simila o hose p oposed by Vande bil 22 in he UPP o mula ion in o de o include he co ela ion be ween co e and alence elec ons, al hough in ou case he de e mina ion o he Ve ( ) does no need he use o a cu o adius. To build Ve ( ) we only need o know an app oxima e densi y and he occupied o bi als o he su ounding sys em, wha - e e he unc ional used in he clus e g oup is. Mo eo e , in con as wi h he KSCED,25,26 in ou me hod i is no neces- sa y o use an app oxima ed kine ic ene gy unc ional, and he inclusion o he weigh ed p ojec ion ope a o P ˆen con- e s a nonlocal componen o Ve ( ) as in he UPP app oach. The e alua ion o Eq. 共22兲and he pa o he SCF-KS equa ion ela ed o Eq. 共21兲was sol ed using he g id o poin s gene a ed by he clus e sys em as i i we e isola ed.32 These algo i hms we e implemen ed in he HONDO p og am.33 The in eg als in he a omic basis we e e alua ed using he King, Dupuis, and Rys34,35 and Gauss–He mi e quad a u es. Fo in eg als conce ning Eqs. 共19兲and 共20兲, he ecu si e o mulas o Oba a and Saika we e used.36 III. NUMERICAL EXAMPLES We epo in his sec ion a ew nume ical applica ions o he p ocedu e de eloped abo e. Among o he aspec s, he sui abili y o he KS-GMPs o simula e a wa e molecule in he compu a ion o he 共H2O兲2dime and he HF–H2O com- plex is explo ed. Mo e ex ensi e examples in which he me hod is es ed in su ace chemis y calcula ions will be epo ed in a o hcoming wo k. Fo he p esen we ha e cen e ed ou a en ion on he use o a wide a ie y o unc- ionals commonly used in DFT calcula ions. Fo he ex- change we ha e selec ed ha o Sla e ,37 共SLT兲as well as ha p oposed by Becke,38 共B88兲. Fo he co ela ion pa we employed he unc ional o Lee–Yang–Pa 39 共LYP兲, which added o SLT o B88 gi es wo di e en exchange- co ela ion unc ionals: SLYP and BLYP. The widely used hyb id h ee pa ame e unc ional B3LYP has also been conside ed.40 To se up he p ocedu e, a i s single calcula ion o he isola ed wa e molecule was ca ied ou using each o hese unc ionals and om he KS molecula o bi als he KS-GMP was de e mined. These KS-GMPs we e hen inco po a ed in o he DFT calcula ions o he ac i e g oup (H2Oo HF兲 and he in e ac ion ene gy was es ima ed a se e al dis ances. The esul ing p o iles we e compa ed wi h hose a ising om e e ence DFT all elec on 共AE兲calcula ions ca ied ou on he supe molecules HF¯H2O and H2O¯H2O. All he cal- cula ions we e pe o med using he s anda d TZP basis se . Resul s o he HF¯H2O and H2O¯H2O complexes a e epo ed in Tables I and II, espec i ely. S a ing wi h he HF¯H2O sys em we can see ha he equilib ium in e mo- lecula dis ance es ima ed wi h he KS-GMP ag ees eason- 6084 J. Chem. Phys., Vol. 113, No. 15, 15 Oc obe 2000 N. C. He na ´ndez and J. F. Sanz ably wi h hose ob ained om AE calcula ions. Wi h he ex- cep ion o SLT based unc ionals, he KS-GMP dis ances a e ound o be la ge han he AE ones, he e o s anging be- ween 0.9% and 6.9%. The KS-GMP binding ene gies com- pu ed a he op imized in e molecula dis ances appea o be signi ican ly lowe han he AE ones; howe e , i should be no ed ha he AE binding ene gies a e o e es ima ed be- cause o he basis se supe posi ion e o 共BSSE兲. Since he KS-GMP a e ee o BSSE, we co ec ed he AE ene gies using he well-known coun e poise me hod o Boys and Be naldi.41 These co ec ed ene gies a e also epo ed in he able unde Eco en y. As can be seen, he ag eemen is clea ly imp o ed, wi h de ia ions be ween 3% and 20%. The obse ed gene al end is an unde es ima ion o he binding ene gies e lec ing he ac ha he model po en ial is ozen and, he e o e, no epola iza ion o he elec on dis ibu ion is allowed. Also, because no basis se is p esen in he GMP, he elec on densi y and, he e o e, he in e g oup egion a e no p ope ly desc ibed. An addi ional app oxima ion in o- duced in he KS-GMP calcula ions a ises om he way o compu ing he ene gy and in eg als ela ed o Eq. 共21, 22兲, whe e only he g id gene a ed by he clus e is used. Finally, in Fig. 1, he p o ile ene gies ob ained om KS-GMP and AE calcula ions a e plo ed. I shows ha beyond he nume i- cal easonable ag eemen , he shape o hese cu es appea s o be qui e sa is ac o y, wi h smoo h beha io and a sui able desc ip ion o he equilib ium egion. Resul s ob ained o he H2O¯H2O dime a e epo ed in Table II and Fig. 2. Bo h in e g oup and binding ene gies a e ound o ollow he ends al eady men ioned abo e. The equilib ium dis ances appea o be o e es ima ed 共10%– 14%兲again wi h he excep ion o he SLT unc ionals, while he binding ene gies a e unde es ima ed wi h espec o he BSSE co ec ed AE ones. The ene gy p o iles also show a sui able beha io , and as in he HF¯H2O case, he cu a- u es a e ound o be lowe acco ding o he unde es ima ion o he g oup–g oup in e ac ion. Finally, we will commen on one o he di e ences be- ween ou KS-GMP and he KSCED me hod epo ed in Re . 25. This di e ence conce ns he use o he p ojec ion ope a- o de ined in Eq. 共20兲, which is no p esen in he KSCED o malism. In o de o show he impo ance o his e m, a se o calcula ions o he HF¯H2O sys em was ca ied ou bu now se ing he p ojec ion ac o a ␣ ⫽0 in Eq. 共20兲. The p o ile ene gies hus ob ained a e epo ed in Fig. 3. As can be seen, he absence o he p ojec o gi es ise o a com- ple ely a ac i e sys em wha e e he unc ional is. This e- sul shows he necessi y o he p esence o he p ojec ion ope a o in ou ep esen a ion. The p esence o such a p o- jec o comes di ec ly om he s ong o hogonali y condi ion as exp essed in Eq. 共11兲in e ms o he i s -o de densi y ma ices, and aking in o accoun he N- ep esen abili y con- di ion leads o Eq. 共12兲. IV. CONCLUSIONS In his wo k a me hod o ob ain polycen e g oup model po en ials wi hin a DFT amewo k is epo ed. The main idea unde lying his app oach is o ca y ou he calcula ion o an ac i e subsys em 共clus e 兲 aking in o accoun he in- luence o he su ounding g oups 共en i onmen 兲 h ough an ab ini io model po en ial, including elec on co ela ion e - ec s in he clus e –en i onmen in e ac ion. Wi h his pu - FIG. 1. Binding ene gy p o iles o he H2O–HF complex om all-elec on and KS-GMP calcula ions. TABLE I. Op imized in e g oup dis ances 共Å兲and binding ene gies 共kcal/ mol兲ob ained o he H2O–HF sys em om all elec on and 共KS-GMP兲 calcula ions using SLT, B88, SLYP, BLYP, and B3LYP unc ionals. De ia- ions in pe cen a e be ween pa en heses. DFT unc ional dBinding ene gy All elec on KS-GMP All elec on KS-GMP Eco SLT 1.690 1.573 共6.9兲⫺13.2 ⫺11.2 共3.4兲⫺11.6 B88 1.936 1.954 共0.9兲⫺6.6 ⫺4.7 共14.5兲⫺5.5 SLYP 1.594 1.540 共3.4兲⫺17.5 ⫺15.1 共3.8兲⫺15.7 BLYP 1.786 1.822 共2兲⫺9.6 ⫺6.8 共18兲⫺8.3 B3LYP 1.763 1.849 共4.9兲⫺10.1 ⫺7.2 共20兲⫺9.0 TABLE II. Op imized in e g oup dis ances 共Å兲and binding ene gies 共kcal/mol兲ob ained o he H2O–H2O sys em om all elec on and 共KS-GMP兲calcula ions using SLT, B88, SLYP, BLYP, and B3LYP unc- ionals. De ia ions in pe cen a e be ween pa en heses. DFT unc ional dBinding ene gy All elec on KS-GMP All elec on KS-GMP Eco SLT 1.839 1.785 共2.9兲⫺8.4 ⫺6.5 共6.1兲⫺6.9 B88 2.200 2.576 共14.6兲⫺3.2 ⫺1.9 共24.9兲⫺2.5 SLYP 1.719 1.708 共0.6兲⫺11.8 ⫺8.9 共15.3兲⫺10.5 BLYP 1.976 2.192 共10.9兲⫺5.3 ⫺3.5 共22.2兲⫺4.5 B3LYP 1.953 2.182 共11.7兲⫺5.7 ⫺3.8 共22.4兲⫺4.9 6085J. Chem. Phys., Vol. 113, No. 15, 15 Oc obe 2000 G oup model po en ials pose in mind, wo main modi ica ions a e in oduced in o he KS equa ions. The i s one accoun s o he classical elec on–elec on in e ac ion be ween he clus e and he en- i onmen . Such an in e ac ion in ol es a new e m in he one-elec on pa and is ep esen ed h ough a Coulomb op- e a o buil up by i ing he elec on densi y o he en i on- men g oup o a se o Gaussian unc ions. The second modi- ica ion in ol es an adequa e co ec ion o he exchange- co ela ion ope a o , depending on he unc ional used. The exchange-co ela ion e ec s a e in oduced in o he KS equa ions by means o a g id o poin s de ined in he clus e sys em. In o de o p ese e he s ong o hogonali y condi- ion be ween g oup wa e unc ions, as equi ed by he build- ing block p inciple, an equi alen es ic ion is p oposed us- ing he densi y ma ix o malism. This allows o he de ini ion o a p ojec ion ope a o simila o ha used in he g oup model po en ial implemen a ion a he Ha ee–Fock le el.16 The whole p ocedu e was implemen ed in an all-pu pose p og am 共HONDO兲,33 pe mi ing us o compu e he clus e elec on densi y wi h se e al exchange-co ela ion unc ion- als 共SLT, B88, SLYP, BLYP, B3LYP兲wi hin he DFT amewo k. In o de o es he me hod a g oup model po en- ial o H2O was de e mined and used in he calcula ions o HF¯H2O and H2O¯H2O complexes. 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