Analysis of the magnetic coupling in binuclear complexes. II. Derivation of valence effective Hamiltonians from ab initio Cl and DFT calculations

Analysis o he magne ic coupling in binuclea complexes. II. De i a ion o alence e ec i e Hamil onians om ab ini io CI and DFT calcula ions Ca men J. Calzado Labo a oi e de Physique Quan ique, IRSAMC, Uni e si e ´Paul Saba ie , 118, ou e de Na bonne, 31062 Toulouse, F ance Jesu ´s Cab e o Depa amen de Quı ´mica Fı ´sica i Ino ga `nica and Ins i u d’Es udis A anc¸a s, Uni e si a Ro i a i Vi gili, Pl. Impe ial Ta aco, 1. 43005 Ta agona, Spain Jean Paul Mal ieu Labo a oi e de Physique Quan ique, IRSAMC, Uni e si e ´Paul Saba ie , 118, ou e de Na bonne, 31062 Toulouse, F ance Rosa Caballol Depa amen de Quı ´mica Fı ´sica i Ino ga `nica and Ins i u d’Es udis A anc¸a s, Uni e si a Ro i a i Vi gili, Pl. Impe ial Ta aco, 1. 43005 Ta agona, Spain 共Recei ed 4 Sep embe 2001; accep ed 4 Decembe 2001兲 Mos in e p e a ions o he magne ic coupling Jbe ween wo unpai ed elec ons es upon simple alence models ha in ol e essen ially he e omagne ic di ec exchange con ibu ion, Kab , and he an i e omagne ic e ec o he delocaliza ion esul ing om he in e ac ion be ween neu al and ionic de e minan s, ab , whose ene gy di e ence is U.Ab ini io alence-only calcula ions gi e e y poo es ima es o J, wha e e he de ini ion o he magne ic o bi als, and la ge CI expansions a e equi ed o e alua e i p ope ly. I is, howe e , possible o de ine alence e ec i e Hamil onians om he knowledge o he eigenene gies and he eigen ec o s o hese accu a e CI calcula ions. When applied o ou di e en complexes, his s a egy shows ha spin pola iza ion may change he sign o he di ec exchange in e ac ion, Kab , and ha dynamical co ela ion esul s in a d ama ic educ ion o he e ec i e epulsion U. The p esen a icle also shows how Kab , ab , and Ue ec i e pa ame e s can be ex ac ed om densi y unc ional heo y 共DFT兲calcula ions and ha he ypical o e es ima ion o Jin DFT can be a ibu ed o an excessi e lowe ing o he e ec i e on-si e epulsion. © 2002 Ame ican Ins i u e o Physics. 关DOI: 10.1063/1.1446024兴 I. INTRODUCTION The magne ic p ope ies o molecula bi adicals, in e - molecula complexes, ansi ion me al binuclea , o poly- nuclea a chi ec u es a e he subjec o an in ense esea ch e o . In ma e ial science as well, magne ic la ices ecei e an inc easing a en ion.1–4 The basic cha ac e is ics o hese sys ems, which all in ol e localized unpai ed elec ons, a e he sign and he ampli ude o he coupling Jbe ween he unpai ed elec ons on neighbo si es. This in o ma ion may be in oduced in a Heisenbe g–Di ac–Van Vleck spin-only Hamil onian:5 H ˆ⫽⫺兺 i,jJijS ˆiS ˆj,共1兲 whe e S ˆiand S ˆja e he spin ope a o s on si es iand j. The expe imen al alues o Ja e ob ained by i ing he esul s coming om measu emen s o magne ic suscep ibili y, neu- on sca e ing, o Raman spec oscopy o hose ob ained by assuming he Heisenbe g mic oscopic Hamil onian. The magne ic coupling cons an Jmay be nega i e 共an i e omag- ne ism, AF兲o posi i e 共 e omagne ism, F兲.6This coupling is essen ially local.7–9 The in e ac ion be ween he nea es neighbo s usually p e ails, al hough in some a chi ec u es he second neighbo in e ac ion has been assumed o be simi- la o he nea es one as in CuGeO310 and Li2CuO2.11 F om he e y beginning o his domain, models ha e been p oposed which essen ially es upon a e y limi ed se o elec ons and o bi als. Mos o hem in oke he magne ic o bi als and he unpai ed elec ons only. Hence o a bi- nuclea p oblem wi h ms⫽⫾1 2on each magne ic cen e , only wo unpai ed elec ons in wo local o bi als aand ba e con- side ed. The models may be o mula ed in a nono hogonal alence bond 共VB兲model,12,13 in an o hogonal alence bond desc ip ion14 i aand bha e been o hogonalized, i.e., 具 a 兩 b 典 ⫽0, o in a alence con igu a ion in e ac ion 共VCI兲 pic u e.15 All hese models s ay on a minimal alence de- sc ip ion o he p oblem. They ha e led o some quali a i e conclusions: 共i兲 he di ec exchange Kab be ween he magne ic o bi als is a e omagne ic 共 iple a o ing兲con ibu ion; 共ii兲 he o he con ibu ion is an i e omagne ic and comes om he speci ic elec onic delocaliza ion occu ing in he single , h ough he mixing o he neu al domi- nan single VB con igu a ions (1/&)(ab ¯ ⫹ba ¯ ) wi h JOURNAL OF CHEMICAL PHYSICS VOLUME 116, NUMBER 10 8 MARCH 2002 39850021-9606/2002/116(10)/3985/16/$19.00 © 2002 Ame ican Ins i u e o Physics Reuse o AIP Publishing con en is subjec o he e ms: h ps://publishing.aip.o g/au ho s/ igh s-and-pe missions. Downloaded o IP: 150.214.230.47 On: F i, 28 Oc 2016 10:08:14 he ionic VB de e minan s 兩 aa ¯ 典 and 兩 bb ¯ 典 . I s ampli- ude is go e ned by he e ec i e hopping in eg al ab : 2 ab⫽ 冓 1 &共ab ¯ ⫹ba ¯ 兲 冏 H ˆ 冏 1 &共aa ¯ ⫹bb ¯ 兲 冔 共2兲 and by he ene gy di e ence be ween he neu al and ionic VB s uc u es: U⫽1 2关 具 共aa ¯ ⫹bb ¯ 兲 兩 H ˆ 兩 共aa ¯ ⫹bb ¯ 兲 典 ⫺ 具 共ab ¯ ⫹ba ¯ 兲 兩 H ˆ 兩 共ab ¯ ⫹ba ¯ 兲 典 兴.共3兲 Up o he second o de , he an i e omagne ic con ibu ion14 o Jis ⫺4 ab 2/Uand he quan i y 2 ab can be ela ed o he ene gy di e ence be ween he symme y-adap ed gand u molecula o bi als:15 2 ab⫽␧g⫺␧u. These models ha e been widely used no only as a pos- e io i a ionaliza ion o he expe imen , o ins ance o in- e p e he s uc u al dependence o Jin a se ies o complexes,16–18 bu also as an in ellec ual guide, o in- s ance, o build e o- o e imagne ic la ices.19 Howe e , ab ini io calcula ions appa en ly ail o suppo he alidi y o hese elemen a y pic u es. I is ac ually pos- sible o de ine accu a e magne ic o bi als om sel -consis en ield 共SCF兲calcula ions on he uppe o lowe mul iple s o a binuclea sys em. These calcula ions minimize he ene gy o a simple desc ip ion o hese s a es 共in e ms o wo de e mi- nan s o a wo elec on in wo o bi al p oblem兲in a mean ield app oxima ion. Na u al magne ic o bi als may as well be de ined om e y accu a e desc ip ions, in ol ing ex en- si e con igu a ion in e ac ion 共CI兲expansions. Bu in bo h cases he alence-only desc ip ion, i.e., he in e ac ion be- ween 兩 ab ¯ 典 , 兩 ba ¯ 典 , 兩 aa ¯ 典 , and 兩 bb ¯ 典 de e minan s, gi es e y poo esul s, he alues o Jbeing equen ly o inco ec sign and, when no , one o de o magni ude oo small.20–25 Accu a e alues o Jcan be ob ained by ab ini io CI echniques.8,9,22–30 A pe u ba i e second-o de analysis was pe o med by Mal ieu e al.31–33 showing he impo ance o p ocesses which in ol e o he o bi als and elec ons. Since he pe u ba ion expansion is no e y eliable due o con e - gence p oblems,32 a selec ed CI scheme has been de ined om pe u ba i e a gumen s. The ene gies and wa e unc- ions o he desi ed s a es a e ob ained by diagonalizing di - e en selec ed spaces. The dynamical co ela ion e ec is ob ained h ough exci a ions in ol ing ei he occupied 共holes, h兲o i ual 共pa icles, p兲inac i e o bi als. Up o he second o de in a pe u ba i e expansion, he exci a ion can conce n a mos wo holes and wo pa icles. These (2h ⫹2p) exci a ions do no con ibu e o he ene gy di e ence and can be omi ed in he CI expansion, leading o he a ia- ional so-called di e ence dedica ed CI me hod.34 This ap- p oach has led o e y accu a e alues o he magne ic cou- pling in a wide se ies o sys ems. A p eceding pape 35 has shown ha i is possible o analyze he ole o he a ious ypes o p ocesses which go beyond he alence-only de- sc ip ion 共spin-pola iza ion, dynamical epola iza ion o ionic VB s uc u es, e c.兲using di e en CI spaces. The i s aim o his wo k is o analyze whe he i is possible o e u n om his complex pic u e o a simple a- lence space desc ip ion, in which he in e ac ions a e no longe he ba e ones imposed by he di ec ac ion o he Hamil onian, bu e ec i e in e ac ions inco po a ing he e - ec s o ex e nal co ela ion. The heo y o e ec i e Hamil- onians is a ool o a igo ous concen a ion o he in o ma ion.36,37 I s p inciple is ecalled in Sec. III. When applied o a wo-elec on in wo-o bi al p oblem i gi es a d essed alence-only Hamil onian. The compa ison be ween he ba e and he d essed alence-only Hamil onians, ob- ained o a se ies o ou binuclea complexes o Cu 共II兲ions 共desc ibed in Sec. II兲, shows he ac ion o he dynamical co ela ion, i.e., he modi ica ion o he Kab , ab , and U pa ame e s. Th ee s a egies a e employed o his concen a- ion o in o ma ion, namely, 共i兲 he o iginal heo y o Bloch,38 which uses in o ma ion coming om ou exac eigens a es, when a ailable; 共ii兲an al e na i e o malism, which de ines a He mi ian e ec i e Hamil onian, by using he G am–Schmid o hogonaliza ion;39 and 共iii兲 he heo y o in e media e e ec i e Hamil onians, p oposed by Mal ieu e al.,40 which handles he wo lowes eigens a es only. All hese me hods lead o a consis en conclusion conce ning a d ama ic educ ion o he e ec i e on-si e epulsion Uas an e ec o he dynamical pola iza ion. The second p ospec o he p esen wo k conce ns DFT calcula ions on magne ic complexes 共Sec. IV兲. I is shown ha i is possible o de i e DFT e alua ions o he in eg als Kab , ab, and U, om a ious solu ions o he Kohn–Sham equa ions 共closed shell single , es ic ed open shell iple , and b oken-symme y single solu ions兲. These calcula ions lead o an o e es ima ion o he 兩 ab /U 兩 a io, compa ed o he bes ab ini io 兩 ab e /Ue 兩 alue, esul ing in a oo la ge ionic VB componen in he b oken-symme y solu ion. This ex- plains he gene ally obse ed o e es ima ion o Jin hese app oaches. The key ole o he exchange po en ial will be illus a ed and discussed. Finally, Sec. V con ains he conclu- sions o bo h pape s. II. DESCRIPTION OF THE SYSTEMS AND COMPUTATIONAL DETAILS Fou binuclea sys ems in ol ing wo d9Cu共II兲cen e s i.e., wo ac i e elec ons in wo magne ic o bi als ha e been conside ed. The i s one is a agmen o he CuO2squa e la ice o he La2CuO4pe o ski e, which p esen s supe con- duc ing beha io a e hole doping, while i is a wo- dimensional an i e omagne ic la ice wi h J⬃⫺1000 cm⫺1 be o e doping (Jexp⫽⫺1032⫾48 cm⫺1,⫺1081⫾40 cm⫺1).41,42 ACu 2O7clus e p ope ly embedded in a se o pseudopo en ials and poin cha ges 关Fig. 1共a兲兴 has been used o ep esen his sys em. The emaining h ee examples come om chemis y, namely, 共i兲 he 关Cu2Cl6兴2⫺complex in a plana geome y, Fig. 1共b兲, wi h a weak an i e omagne ic cha ac e (Jexp ⫽0,⫺40 cm⫺1);43 3986 J. Chem. Phys., Vol. 116, No. 10, 8 Ma ch 2002 Calzado e al. Reuse o AIP Publishing con en is subjec o he e ms: h ps://publishing.aip.o g/au ho s/ igh s-and-pe missions. Downloaded o IP: 150.214.230.47 On: F i, 28 Oc 2016 10:08:14 共ii兲关Cu2(N3)2(NH3)6兴2⫹wi h end- o-end b idging-azido g oups, Fig. 1共c兲, leading o a e y s ong an i e o- magne ic beha io (Jexp ⬍⫺800 cm⫺1);44 共iii兲Cu2(CH3COO)4(H2O)2wi h ou ace a o b idges, Fig. 1共d兲, p esen ing an in e media e an i e omag- ne ism (Jexp ⫽⫺286,⫺294⫾4cm ⫺1).45,46 The ela i e o ien a ion o he magne ic o bi als in hese ou sys ems is p esen ed in he Scheme: o 共a兲 he Cu2O7clus e , 共b兲 he 关Cu2Cl6兴2⫺complex, 共c兲 he 关Cu2(N3)2(NH3)6兴2⫹complex, and 共d兲 he Cu2(CH3COO)4(H2O)2molecule, espec i ely. A mo e de ailed desc ip ion o hese sys ems has been e- po ed in pape I,35 oge he wi h he compu a ional de ails o he CI calcula ions. The MOLCAS 4.1 package47 has been used o ob ain he ROHF molecula o bi als. The CASDI p og am48 has been used in he CI calcula ions and he NATURAL p og am49 in he de e mina ion o he na u al MOs. All DFT calcula ions ha e been pe o med by means o he GAUSSIAN 98 code.50 The B3LYP51 pa ame iza ion has been used in he DFT calcula ions, wi h he ollowing basis se s: o he 关Cu2Cl6兴2⫺and 关Cu2( ␮ -N3)2(NH3)6兴2⫹com- plexes, he Hay and Wad co e po en ial and basis unc ions ha e been used o coppe a oms.52 All elec on basis se s ha e been used o he emaining a oms, he 6-311G basis se o chlo ine53 and he 6-31G one o hyd ogen and ni ogen a oms.54 Fo he Cu2( ␮ -CH3COO)4(H2O)2molecule, he e ec i e co e pseudopo en ial and basis unc ions p oposed by S e ens, Basch, and K auss55 ha e been employed o he Cu, O, and C a oms. Fo he cup a e, he same basis unc- ions as in he HF–CI calcula ions ha e been used o Cu and O a oms. In o de o analyze a ious aspec s o he dynamical co - ela ion, ou di e en CI spaces a e used, namely, 共i兲 he ba e alence CAS 共CASCI兲; 共ii兲 he DDCI1 space con aining all he con igu a ions eached by single exci a ions on he op o he CAS. Th ee ypes o exci a ions can be dis inguished: •1h, whe e an elec on mo es om an inac i e occupied o bi al o an ac i e o bi al, wi h a possible simul aneous single exci a ion inside he ac i e space; •1p, whe e an ac i e elec on is mo ed o a i ual o - bi al, and as in he p eceding case, his mo emen can be coupled wi h single exci a ions inside he ac i e space; •1h⫹1p, whe e an elec on mo es om an occupied o a i ual o bi al, wi h possible simul aneous single ex- ci a ions inside he ac i e space. The spin pola iza ion con ibu ions, i.e., a simul aneous exci a ion o a co e elec on o he ac i e space and o an ac i e elec on o di e en spin o a i ual o bi al, a e included. 共iii兲 he DDCI2 space, which adds o he p e ious space he 2hde e minan s, i.e., wo co e elec ons mo ing o he ac i e o bi als, and 2pexci a ions, i.e., wo elec ons om he ac i e o bi als o he i ual ones; 共i 兲 he DDCI space, con aining also he 2h⫹1pde e mi- nan s 共 wo co e elec ons mo ing o an ac i e o bi al and o a i ual one, espec i ely兲and 1h⫹2pde e - minan s 共one co e and one ac i e elec ons mo ing o wo i ual o bi als兲. Rega ding he DFT calcula ions, Noodleman’s b oken- symme y app oach56 has been used o es ablish he alue o he magne ic coupling. Since he o e lap be ween he mag- ne ic o bi als is a he small in all he s udied sys ems 共see Sec. IVB兲, he limi o s ong o hogonali y has been consid- e ed, Jbeing calcula ed om J⫽2共EBS⫺ET兲,共4兲 whe e EBS and ETa e he un es ic ed b oken symme y de- e minen and iple s a e ene gies, espec i ely. III. STRICT DETERMINATION OF VALENCE EFFECTIVE HAMILTONIANS FROM ACCURATE AB INITIO CI CALCULATIONS A. The ba e alence-only Hamil onian Le us ecall b ie ly he na u e o he model space, buil om wo o hogonal magne ic local o bi als, aand b.I is composed o ou de e minan s, wo neu al, 兩 ab ¯ 典 and 兩 ba ¯ 典 , and wo ionic ones, 兩 aa ¯ 典 and 兩 bb ¯ 典 . The Hamil onian ex- p essed in his educed comple e ac i e space akes he o m FIG. 1. Schema ic ep esen a ion o he ou models conside ed: 共a兲 he Cu2O7clus e ; 共b兲 he 关Cu2Cl6兴2⫺complex, in a plana geome y; 共c兲 he 关Cu2( ␮ -N3)2(NH3)6兴2⫹complex wi h end- o-end b idging azido ligands; and 共d兲 he Cu2( ␮ -CH3COO)4(H2O)2molecule. 3987J. Chem. Phys., Vol. 116, No. 10, 8 Ma ch 2002 Magne ic coupling Reuse o AIP Publishing con en is subjec o he e ms: h ps://publishing.aip.o g/au ho s/ igh s-and-pe missions. Downloaded o IP: 150.214.230.47 On: F i, 28 Oc 2016 10:08:14 兩 ab ¯ 典 兩 ba ¯ 典 兩 aa ¯ 典 兩 bb ¯ 典 冋 HNN Kab ab ab Kab HNN ab ab ab ab HNN⫹UK ab ab ab Kab HNN⫹U 册 ,共5兲 whe e HNN is he ene gy o he neu al de e minan s. This space gene a es 共i兲a pu ely neu al iple s a e 兩 Tu 典 ⫽(1/&)( 兩 ab ¯ 典 ⫺ 兩 ba ¯ 典 ), 共ii兲a pu ely ionic single s a e o usymme y 兩 Su 典 ⫽(1/&)( 兩 aa ¯ 典 ⫺ 兩 bb ¯ 典 ), 共iii兲 wo single s a es o gsymme y, 兩 Sg 1 典 and 兩 Sg 2 典 , ha may be exp essed as linea combina ions o a pu ely neu al single 兩 Sg N 典 ⫽(1/&)( 兩 ab ¯ 典 ⫹ 兩 ba ¯ 典 ), also de- no ed Sab in pape I,35 and a pu ely ionic single 兩 Sg I 典 ⫽(1/&)( 兩 aa ¯ 典 ⫹ 兩 bb ¯ 典 ). The Hamil onian can be exp essed in he basis o S ˆ2eigen- ec o con igu a ions. Taking he ene gy o he iple 3Eu⫽HNN⫺Kab 共6兲 as he ene gy o igin, he Hamil onian can be w i en 兩 Sg N 典 兩 Sg I 典 兩 Tu 典 兩 Su 典 冋 2Kab 2 ab 00 2 ab 2Kab⫹U00 0000 000U 册 .共7兲 The 兩 Su 典 s a e lies a he ene gy Uabo e he iple s a e. In he gsymme y he lowes single s a e 兩 Sg 1 典 ⫽cN 兩 Sg N 典 ⫹cI 兩 Sg I 典 共wi h cN⬎cI⬎0兲is essen ially neu al. I s ene gy is 1Eg 1⫽Kab⫹U⫺ 冑 U2⫹16 ab 2 2.共8兲 The second oo 兩 Sg 2 典 ⫽⫺cI 兩 Sg N 典 ⫹cN 兩 Sg I 典 is essen ially ionic and much highe in ene gy: 1Eg 2⫽Kab⫹U⫹ 冑 U2⫹16 ab 2 2,共9兲 lying close o Uwhen 兩 ab 兩 ⰆU. Pape I o his se ies35 has shown how o ob ain he alues o he Kab , ab , and Uin eg als om he CASCI solu ion. A second-o de pe u ba i e expansion leads o he well-known exp ession o he magne ic coupling:14 J⫽2Kab⫺4 ab 2 U.共10兲 Using he ela ions 2 ab⫽␧g⫺␧uand U⫽EI⫺EN⫽Jaa ⫺Jab 共Jaa and Jab being he one and wo-cen e Coulomb epulsions, espec i ely兲, Eq. 共10兲can be w i en as15 J⫽2Kab⫺共␧g⫺␧u兲2 Jaa⫺Jab ,共11兲 exploi ed in mos o he quali a i e a ionaliza ions o he magne ic coupling and o i s s uc u al dependence.16–18 B. The e ec i e Hamil onian app oach Wha e e he de ini ion o he alence space 共Ha ee Fock o na u al o bi als兲, he physics o he magne ic cou- pling canno be con ained in he wo elemen a y ea u es, namely di ec exchange and kine ic exchange. I would be impossible as well o educe i o an enla ged alence space including one o a ew occupied MOs o he b idging ligand, as sugges ed by he wo-band model, popula in solid s a e physics, since dynamical co ela ion phenomena in ol ing i ual MOs appea o be c ucial in he de e mina ion o he ampli ude o he magne ic coupling. Bu i is possible o p ojec he exac in o ma ion ob ained om he la ge CI cal- cula ions in o he alence space, conside ed as a model space, using he igo ous heo y o e ec i e Hamil onians. This is he scope o he p esen sec ion. 1. Quasi-degene a e pe u ba ion heo y A way o p oduce an e ec i e Hamil onian, spanned by a gi en model space S, consis s in using he well-known quasidegene a e pe u ba ion heo y.38,57–59 This is a leas a concep ual guide, as i was in he p eceding pape ,35 o which we shall e e o he iden i ica ion o he a ious e ec s. The second-o de exp ession o he ma ix elemen s o H ˆe a e gi en by 具 I 兩 H ˆe 共2兲 兩 L 典 ⫽ 具 I 兩 H ˆ 兩 L 典 ⫹兺 兩 ␣ 典 苸S 具 I 兩 H ˆ 兩 ␣ 典具 ␣ 兩 H ˆ 兩 L 典 EL 共0兲⫺E ␣ 共0兲, ᭙I,L苸S,共12兲 whe e Sis he model space and he denomina o is he ze o h-o de ene gy di e ence be ween he igh componen 兩L典o he ma ix elemen and he ou e space de e minan 兩 ␣ 典. F om his o mula ion i is possible o iden i y which ma ix elemen s a e a ec ed by he a ious pe u be s 兩 ␣ 典. I should be poin ed ou ha he o malism is no He mi ian, as shown by Eq. 共12兲.I 兩I典and 兩L典ha e di e en ze o h-o de ene - gies, as occu s in magne ic sys ems when one o hem is a neu al de e minan and he o he an ionic de e minan , hen 具 I 兩 H ˆe (2) 兩 L 典 ⫽ 具 L 兩 H ˆe (2) 兩 I 典 . In pa icula , he consequence o be ex- pec ed is he e ec o he pe u be s o be la ge o he 具 neu al 兩 H ˆe (2) 兩 ionic 典 han o he 具 ionic 兩 H ˆe (2) 兩 neu al 典 e ec i e hopping, since he ene gy denomina o s a e la ge in absolu e alue o he la e . Re u ning o he de elopmen s o pape I,35 some p edic- ions can be made: 共i兲The spin pola iza ion co ec ion di ec ly a ec s Kab since i couples 兩 I 典 ⫽ 兩 co e ab ¯ 典 and 兩 L 典 ⫽ 兩 co e ba ¯ 典 h ough he spin pola ized aa ¯ ⫹ah ¯ ap ⫹aa 兩 co e ab ¯ 典 de e minan . 共ii兲The 1h⫹1pde e minan s gi ing he ap ⫹ahaa ¯ ⫹ab ¯ 兩 co e ab ¯ 典 in e media e s a es change he ac i e pa o he wa e unc ion and lead o • a small second-o de modi ica ion o Kab due o he in e ac ion be ween he wo neu al VB de e minan s, • a modi ica ion o ab due o he in e ac ion be ween he neu al and ionic de e minan s, and • a mo e impo an modi ica ion o Udue o he pola iza- ion o he ionic o ms, as shown in he Diag am: 3988 J. Chem. Phys., Vol. 116, No. 10, 8 Ma ch 2002 Calzado e al. Reuse o AIP Publishing con en is subjec o he e ms: h ps://publishing.aip.o g/au ho s/ igh s-and-pe missions. Downloaded o IP: 150.214.230.47 On: F i, 28 Oc 2016 10:08:14 which may be seen as he in e nal pa o Diag am 11 in pape I.35 I gi es Ue ⫽U⫹⌬U⫽U⫺兺 h兺 p 具 h 兩 J ˜ a⫺J ˜ b 兩 p 典 2 U⫹⌬Eh→p,共13兲 whe e J ˜ a⫽Ja⫺Ka/2 and ⌬Eh→pis he exci a ion ene gy o he h→pp omo ion. The co ec ed kine ic exchange is ob ained acco dingly: ⫺ ab 2 Ue ⫽⫺ ab 2 U 冉 1⫹兺 h兺 p 具 h 兩 J ˜ a⫺J ˜ b 兩 p 典 2 U共U⫹⌬Eh→p兲⫹highe o de s 冊 . 共14兲 The ou h-o de e m was de i ed in he p e ious pape 关Eq. 共56兲o Re . 35兴. The highe o de s in oduced by he change om U o Ue a e physical con ibu ions o J h ough in ini e summa ions o he diag ams. 共iii兲The 2hand 2pexci a ions essen ially consis in modi ica ions o he exchange in eg al h ough he in e me- dia e s a es aa ¯ ⫹ah ¯ ⬘ab ⫹ah 兩 co eab ¯ 典 and ap ¯ ⬘ ⫹ab ¯ ap ⫹aa 兩 co eab ¯ 典 ; 共i 兲 he 2h⫹1pand 1h⫹2pde e minan s essen ially ouch he hopping in eg al up h ough he in e media e s a es ap ⫹ah⬘aa ¯ ⫹ah ¯ 兩 co e ab ¯ 典 and ap⬘ ⫹ahap ¯ ⫹ab ¯ 兩 co e ab ¯ 典 . To summa ize his sec ion, he expec ed esul s a e 共a兲 he spin pola iza ion should modi y he Kab alue, he sign o he co ec ion being sys em dependen ; and 共b兲 he 1h⫹1pdynamical pola iza ion o he ionic com- ponen s should educe he ene gy di e ence be ween he neu al and he ionic pa s, Ue . As shown in he Appendix, i may be p edic ed ha a he hi d o de he same pe u be s should lowe he 兩 具 aa ¯ 兩 ⌬H ˆ(3) 兩 ab ¯ 典 兩 , i.e., he 兩 具 ionic 兩 ⌬H ˆ(3) 兩 neu al 典 兩 ⫽ 兩 IN 兩 alue, while he 兩 具 ab ¯ 兩 ⌬H ˆ(3) 兩 aa ¯ 典 兩 elemen , ha is he 兩 具 neu al 兩 ⌬H ˆ(3) 兩 ionic 典 兩 ⫽ 兩 NI 兩 coun e pa , should be le p ac ically unchanged, con ibu ing o he non- He mi ian cha ac e o he e ec i e Hamil onian. The 2h⫹1pand 1h⫹2pde e minan s should a ec essen- ially he hopping in eg als, and pape I35 has shown ha he e ec is an inc ease o he ab magni ude. O cou se he p esen discussion is based on low-o de con- side a ions and he cons uc ion o H ˆe om he a ia ional CI calcula ions may somewha di e om second-o de de- elopmen s. Mo eo e , he QDPT expansion has no chance o con e ge when wo king wi h his ou -dimensional space. Ac ually, some ligand o me al cha ge ans e 共LMCT兲 s a es lie below he ionic M⫹M⫺ 兩 Su 典 and 兩 Sg 2 典 s a es. These LMCT s a es ac as in ude s, esul ing in small posi i e en- e gy denomina o s and inducing he di e gence o he se ies. Hence he QDPT a gumen s a e pu ely quali a i e.60 We now go on o nonpe u ba i e app oaches using ou a ia ional calcula ions in o de o build e ec i e Hamil onians. 2. E ec i e Hamil onian om he exac spec um The heo y de eloped by Bloch38 es ablishes a p ocedu e o de ine e ec i e Hamil onians, expanded in a low- dimensional model space, om he knowledge o he eigenene gies and eigens a es o he exac Hamil onian. Le us conside a model space, S, o small dimension, i.e., spanned by n ec o s 兩 ⌽I 典 . I s p ojec o is P ˆS⫽兺 I苸S 兩 ⌽I 典具 ⌽I 兩 .共15兲 He e we a e in e es ed in a alence CAS, spanned by he ou VB de e minan s S⫽ 兵 兩 ab ¯ 典 , 兩 ba ¯ 典 , 兩 aa ¯ 典 , 兩 bb ¯ 典 其 , o hei ou combina ions: S⫽ 兵 兩 Sg N 典 , 兩 Sg I 典 , 兩 Su 典 , 兩 Tu 典 其 . Suppose we know a la ge numbe o eigensolu ions o he exac Hamil onian: H ˆ 兩 ⌿k 典 ⫽Ek 兩 ⌿k 典 .共16兲 Conside now he neigen ec o s o H ˆha ing he la ges componen s 共o p ojec ions兲in he model space S. These eigen ec o s de ine a a ge space, ST, isodimensional o S: P ˆST⫽兺 k苸ST 兩 ⌿k 典具 ⌿k 兩 .共17兲 The wa e ope a o ⍀ ˆsends S o ST,P ˆST⫽⍀ ˆP ˆS. We would like o de ine an e ec i e Hamil onian in S, he eigensolu- ions o which a e he mos exac and in o ma i e. This means ha we wan an e ec i e Hamil onian: H ˆe ⫽P ˆSH ˆe P ˆS,共18兲 such ha i s neigen alues a e exac , and ha i s eigen ec o s a e p ojec ions o he co esponding exac eigen ec o s in he model space: H ˆe Bloch 兩 P ˆS⌿k 典 ⫽Ek 兩 P ˆS⌿k 典 .共19兲 This is he de ini ion o he Bloch e ec i e Hamil onian.38 Since he p ojec ions o he 共necessa ily o hogonal兲eigen- ec o s may be nono hogonal, Skl⫽ 具 P ˆS⌿k 兩 P ˆS⌿l 典 ⫽0 o k⫽l,共20兲 he Bloch e ec i e Hamil onian may be non-He mi ian.61 I is mo e con enien o o hogonalize he 兩 P ˆS⌿k 典 ec o s by a p ocedu e ha modi ies hem as li le as possible. The S⫺1/2 ans o ma ion, 3989J. Chem. Phys., Vol. 116, No. 10, 8 Ma ch 2002 Magne ic coupling Reuse o AIP Publishing con en is subjec o he e ms: h ps://publishing.aip.o g/au ho s/ igh s-and-pe missions. Downloaded o IP: 150.214.230.47 On: F i, 28 Oc 2016 10:08:14 兩 ⌿k ⬘ 典 ⫽S⫺1/2 兩 P ˆS⌿k 典 .共21兲 p esen s such a p ope y and leads o he des Cloizeaux e - ec i e Hamil onian:57 H ˆe dC 兩 ⌿k ⬘ 典 ⫽Ek 兩 ⌿k ⬘ 典 .共22兲 O he o hogonaliza ions a e possible, o ins ance he G am–Schmid one39 ha is used he ea e . O cou se he p e ious de elopmen s a e applicable when using an app oxima e spec um o H ˆ, esul ing, o ins ance, om unca ed CI calcula ions. Le us speci y his echnique in ou wo-elec on/ wo-o bi al p oblem. The model space being spli in o h ee subspaces o di e en spin and space symme y, as shown p e iously, he e ec i e Bloch Hamil onian can only ake he ollowing o m, aking as ze o o ene gy he iple s a e one: 兩 Sg N 典 兩 Sg I 典 兩 Tu 典 兩 Su 典 冏 2Kab B2 NI B00 2 IN BUB⫹2Kab B00 000 0 000UB⫹2共Kab B⫺Kab ⬘B兲 冏 ⫽H ˆe Bloch , 共23兲 whe e Kab B⫽ 具 ab ¯ 兩 H ˆe Bloch 兩 ba ¯ 典 ,共24兲 Kab ⬘B⫽ 具 aa ¯ 兩 H ˆe Bloch 兩 bb ¯ 典 ,共25兲 NI B⫽ 具 ab ¯ 兩 H ˆe Bloch 兩 aa ¯ 典 ,共26兲 IN B⫽ 具 aa ¯ 兩 H ˆe Bloch 兩 ab ¯ 典 ,共27兲 and UB⫽ 具 aa ¯ 兩 H ˆe Bloch 兩 aa ¯ 典 ⫺ 具 ab ¯ 兩 H ˆe Bloch 兩 ab ¯ 典 ⫹Kab ⬘B⫺Kab B. 共28兲 This Hamil onian is non-He mi ian, NI B⫽ IN B,共29兲 and in oduces i e in eg als: NI B, IN B,UB,Kab B, and Kab ⬘B. The p ojec ions o he eigen ec o s 兩 3⌿u 典 and 兩 1⌿u 典 on o he model space a e ixed by symme y. I is no he case o he single s a es 兩 1⌿g 1 典 and 兩 1⌿g 2 典 , whose p ojec ions in he model space ha e a deg ee o eedom, namely he a io o he coe icien s on 兩 Sg N 典 and 兩 Sg I 典 : 兩 P ˆS1⌿g 1 典 ⫽cN 兩 Sg N 典 ⫹cI 兩 Sg I 典 ,cN⬎cI⬎0, 共30兲 兩 P ˆS1⌿g 2 典 ⫽⫺cN ⬘ 兩 Sg N 典 ⫹cI ⬘ 兩 Sg ⬘ 典 ,cI ⬘⬎cN ⬘⬎0. Hence he knowledge o he ene gy o he ou s a es, 3Eu, 1Eu,1Eg 1, and 1Eg 2, and o he wo cI/cNand cI ⬘/cN ⬘ a ios ixes uni ocally he alues o he i e pa ame e s o he Bloch e ec i e Hamil onian. Le us de ine he o e lap be ween he p ojec ions o he wo single s a es as s⫽ 具 P ˆS1⌿g 1 兩 P ˆS1⌿g 2 典 ⫽⫺cNcN ⬘⫹cIcI ⬘.共31兲 The o e lap ma ix akes he o m S⫽ 冉 1s s1 冊 .共32兲 The bio hogonal ec o s a e de ined by 兩 P ˆS⌿k † 典 ⫽S⫺1 兩 P ˆS⌿k 典 ,共33兲 which o he case o he p ojec ions 兩 P ˆS1⌿g 1 典 and 兩 P ˆS1⌿g 2 典 become 兩 P ˆS1⌿g 1† 典 ⫽1 1⫺s2共 兩 P ˆS1⌿g 1 典 ⫺s 兩 P ˆS1⌿g 2 典 ), 共34兲 兩 P ˆS1⌿g 2† 典 ⫽1 1⫺s2共⫺s 兩 P ˆS1⌿g 1 典 ⫹ 兩 P ˆS1⌿g 2 典 ). Using he spec al ep esen a ion o he e ec i e Hamil- onian, He Bloch⫽兺 i⫽1,4 兩 P ˆS⌿i 典 Ei 具 P ˆS⌿i † 兩 ,共35兲 i is possible o ex ac he exp ession o he i e e ec i e pa ame e s NI B, IN B,UB,Kab B, and Kab ⬘B: 2 NI B⫽1 1⫺s2关1Eg 1共cNcI⫺scNcI ⬘兲⫹1Eg 2共scN ⬘cI⫺cN ⬘cI ⬘兲兴, 2 IN B⫽1 1⫺s2关1Eg 1共cNcI⫹scN ⬘cI兲⫺1Eg 2共scNcI ⬘⫹cN ⬘cI ⬘兲兴, UB⫽1 1⫺s2关1Eg 1„cI 2⫺cN 2⫺s共cI ⬘cI⫹cNcN ⬘兲… ⫹1Eg 2„cI ⬘2⫺cN ⬘2⫺s共cI ⬘cI⫹cNcN ⬘兲…兴,共36兲 2Kab B⫽1 1⫺s2关1Eg 1共cN 2⫹scNcN ⬘兲⫹1Eg 2共cN ⬘2⫹scN ⬘cN兲兴⫺3Eu, 2Kab ⬘B⫽1 1⫺s2关1Eg 1共cI 2⫺scIcI ⬘兲⫹1Eg 2共cI ⬘2⫺scI ⬘cI兲兴⫺1Eu. The exchange Kab ⬘Bis di e en om Kab B, bu o simplici y, i s alue will no be epo ed no discussed he ea e . Le us conside now he p ocedu e o ob ain a He mi ian e ec i e Hamil onian. The S⫺1/2 o hogonaliza ion o he eigen ec o s p ojec o on o he model space, leading o he des Cloizeaux e ec i e Hamil onian, symme ically a ec s he essen ially neu al s a e 兩 1⌿g 1 典 , and he mos ly ionic s a e 兩 1⌿g 2 典 . Since we a e mainly in e es ed in he lowes 共neu al兲single , i is p e e able o lea e 兩 P ˆS1⌿g 1 典 un- changed, and o o hogonalize 兩 P ˆS1⌿g 2 典 共 he p ojec ions o he ionic eigens a e兲 o 兩 P ˆS1⌿g 1 典 . This is a G am–Schmid o hogonaliza ion. The second eigen ec o has o sa is y 具 P ˆS1⌿g 2† 兩 P ˆS1⌿g 1 典 ⫽0, and hus cN ⬘⫽cI,cI ⬘⫽cN, 共37兲 兩 P ˆS1⌿g 2† 典 ⫽⫺cI 兩 Sg N 典 ⫹cN 兩 Sg I 典 3990 J. Chem. Phys., Vol. 116, No. 10, 8 Ma ch 2002 Calzado e al. Reuse o AIP Publishing con en is subjec o he e ms: h ps://publishing.aip.o g/au ho s/ igh s-and-pe missions. Downloaded o IP: 150.214.230.47 On: F i, 28 Oc 2016 10:08:14 is he second eigen ec o o he G am–Schmid e ec i e Hamil onian H ˆe GS , wi h 1Eg 2as eigenene gy. This e ec i e Hamil onian akes he o m 兩 Sg N 典 兩 Sg I 典 兩 Tu 典 兩 Su 典 冋 2Kab GS 2 ab GS 00 2 ab GS UGS⫹2Kab GS 00 000 0 000UGS⫹共2Kab GS⫺2Kab GS兲 册 ⫽H ˆe GS .共38兲 This He mi ic Hamil onian only in oduces ou pa ame e s. The knowledge o he h ee ene gy di e ences and o he cI/cN a io is su icien o de e mine he alues o hese pa- ame e s. Since in his case he o e lap be ween 兩 P ˆS1⌿g 1 典 and 兩 P ˆS1⌿g 2† 典 is ze o, as ollows om Eqs. 共30兲and 共37兲, we ob ain he ela ions 2 ab GS⫽cIcN共1Eg 1⫺1Eg 2兲, UGS⫽共cI 2⫺cN 2兲共1Eg 1⫺1Eg 2兲,共39兲 2Kab GS⫽cN 21Eg 1⫹cI 21Eg 2⫺3Eu. In p ac ice i is mo e con enien o w i e he p ojec ion o he g ound single s a e 兩 P ˆS1⌿g 1 典 in e ms o he 兩 gg ¯ 典 and 兩 uu ¯ 典 symme y-adap ed alence de e minan s: 兩 P ˆS1⌿g 1 典 ⫽␭ 兩 gg ¯ 典 ⫺ ␮ 兩 uu ¯ 典 ,␭⬎ ␮ ⬎0, 共40兲 whe e cN⫽␭⫹ ␮ &and cI⫽␭⫺ ␮ &.共41兲 Equa ion 共39兲can be w i en as 2 ab GS⫽1 2共␭2⫺ ␮ 2兲共1Eg 1⫺1Eg 2兲, UGS⫽⫺2␭ ␮ 共1Eg 1⫺1Eg 2兲,共42兲 2Kab GS⫽␭ ␮ 共1Eg 1⫺1Eg 2兲⫹1 2共1Eg 1⫹1Eg 2兲⫺3Eu. 3. Nume ical esul s a. The Bloch e ec i e Hamil onian. Table I con ains he Bloch e ec i e Hamil onian ob ained om ei he ROHF o - bi als 共ROHF MOs兲o na u al o bi als 共NOs兲using he DDCI wa e unc ions o he Cu2O7 agmen o he pe o - ski e la ice. Figu e 2 shows he e ec i e pa ame e s NI B, IN B,UB, and Kab Band he cI/cN a io ob ained o his sys- em by using inc easingly co ela ed CI wa e unc ions. When using ROHF MOs, se e al ea u es a e ele an : 共i兲The ba e Kab B alue is small and impo an changes appea a he DDCI le el. 共ii兲The non-He mi ici y is impo an : ( NI B⫺ IN B)/( NI B ⫹ IN B)⬇20%–30% and 兩 NI B 兩 ⬎ 兩 IN B 兩 , as expec ed, a all pos -CASCI le el. Figu e 2 shows ha 兩 IN B 兩 begins o dec ease due o hi d-o de e ec s ela ed wi h he 1h⫹1pexci a ions 共DDCI1兲as explained in he Ap- pendix. As shown om QDPT 共al eady in pape I35兲 he 2h⫹1ppe u be s appea ing a he DDCI le el enhance he e ec i e 兩 ab B 兩 alue. 共iii兲The dominan e ec is he d as ic dec ease o UB, educed o 30% o i s ba e alue by he e ec o he dynamical pola iza ion 共1h⫹1p, DDCI1兲. Table I and Fig. 2 共bo om兲show ha NOs signi ican ly change he ze o h-o de alues 共la ge Kab B, la ge 兩 ab B 兩 , smalle UBa he CASCI le el, due o la ge delocaliza ion ails in he magne ic o bi als62兲, bu he ends a e simila and he inal DDCI e ec i e in e ac ions a e e y close 共same alue o UB, same alue o he p oduc NI B• IN B兲. b. The G am–Schmid e ec i e Hamil onian. In o de o a oid he uncom o able non-He mi ici y p oblem, he G am–Schmid e ec i e Hamil onians a e gi en o h ee sys ems: he cup a e clus e 共Table I, Fig. 2兲, he chlo ide complex 共Table II, Fig. 3兲, and he azido complex 共Table III, Fig. 4兲. The case o coppe ace a e is discussed a he end o he sec ion. As in Table I, Tables II and III con ain he inal DDCI esul s o he cho ide and azido sys ems, while as in Fig. 2, Figs. 3 and 4 also epo alues ob ained om sho e CI expansions. The conclusions a e qui e simila o he p e- ceding ones: 共i兲The Kab GS e ec i e exchange may become nega i e. This change o sign may be co ela ed wi h he sign o he spin pola iza ion con ibu ion, which is an i e o- magne ic in he cup a e and he azido complex 共c . pape I35兲, bu he changes in Kab GS a e la ge han he spin pola iza ion con ibu ion o J. Hence, he spin pola iza ion is only a pa o he e ec s con ibu ing o Kab . 共ii兲When s a ing om ROHF MOs 共 op o igu es兲, he 兩 ab GS 兩 alue dec eases unde he e ec o he 1h⫹1p de e minan s 共compa e CASCI and DDCI1 in Figs. 2–4兲and aises close o he o iginal CASCI alue a TABLE I. Magne ic coupling 共J兲, ionic/neu al a io o he g ound s a e (cI/cN), and e ec i e pa ame e s 共 ab ,Kab , and U兲 o he Cu2O7 ag- men o he La2CuO4la ice om CI and B3LYP calcula ions. All he alues in cm⫺1, excep Uwhich is in eV. Fo he CI calcula ions, he CASCI and he DDCI lis s o de e minan s ha e been employed, wi h wo se s o o bi - als 共ROHF and na u al o bi als兲. The CI pa ame e s ha e been ob ained by using he h ee kinds o e ec i e Hamil onians: Bloch, G am–Schmid 共GS兲, and in e media e 共In 兲. MOs Le el JacI/cNHe U2Kab ab ROHF CASCI ⫺255 0.041 24.12 67 ⫺3960 DDCI ⫺1077 0.118 Bloch 7.24 581 IN ⫺3541 NI ⫺7041 GS 7.44 ⫺229 ⫺3589 In 8.21 ⫺139 ⫺3960 Na u al CASCI ⫺451 0.070 19.65 334 ⫺5589 DDCI ⫺1136 0.136 Bloch 7.21 339 IN ⫺4057 NI ⫺5412 GS 7.31 ⫺22 ⫺4089 In 9.98 386 ⫺5589 B3LYP ⫺1689 0.230 4.20 108 ⫺3903 aJexp :⫺1032⫾48 cm⫺1,⫺1081⫾40 cm⫺1, om Re s. 41 and 42. 3991J. Chem. Phys., Vol. 116, No. 10, 8 Ma ch 2002 Magne ic coupling Reuse o AIP Publishing con en is subjec o he e ms: h ps://publishing.aip.o g/au ho s/ igh s-and-pe missions. Downloaded o IP: 150.214.230.47 On: F i, 28 Oc 2016 10:08:14 he DDCI le el unde he e ec o he 2h⫹1pand 1h⫹2pde e minan s. When s a ing om NOs 共bo - om o igu es兲, he CASCI alue o 兩 ab GS 兩 is much la ge . I is again educed by he 1h⫹1pe ec s and sligh ly aises unde he e ec s o he 2h⫹1pand 1h⫹2pexci a ions. The DDCI alue wi h NOs is in- e media e be ween he ze o h-o de alues 共CASCI兲 o 兩 ab 兩 om ROHF MOs and NOs. 共iii兲F om bo h se s o o bi als, he main e ec is he d a- ma ic dec ease o UGS o5eVin关Cu2Cl6兴2⫺and 7.5 eV in he cup a e. This dec ease is essen ially due o he 1h⫹1pexci a ions 共dynamical epola iza ion o ionic VB s uc u es兲. Fo he azido complex he esul s ha e o be conside ed wi h cau ion since he iden i ica ion o he essen ially ionic a- lence 兩 1⌿g 2 典 s a e was qui e di icul and ambiguous. Fo in- s ance, a he DDCI2 le el wi h NOs, wo s a es, which a e he 14 h and 24 h o hei symme y, ha e p ac ically equal weigh s on he 兩 Sg I 典 con igu a ion. A he DDCI le el, he iden i ica ion o he pe inen oo s was impossible when us- ing ROHF MOs. The a he a bi a y choice be ween hem leads o e y di e en e ec i e in eg als. The e a e a la ge numbe o ligand o me al o me al o ligand cha ge ans e s a es ha can ha e lowe ene gy han he me al o me al cha ge ans e , i.e., he ionic alence-bond s a e, which ex- plains he high ank o he 兩 1⌿g 2 典 s a e. The same is ue o he 兩 1⌿u 典 ionic s a e. A s ong mixing be ween LMCT and ionic s a es can occu , which esul s in he di icul assign- men o he p edominan ly alence ionic s a es. Fo he ac- e a o complex he iden i ica ion o hese s a es happened o FIG. 2. E ec i e pa ame e s ab ,Kab , and U共in eV兲and ionic/neu al a io (cI/cN) in he g ound s a e wa e unc ion ob ained o he Cu2O7clus e om inc easingly co ela ed CI wa e unc ion. Th ee ypes o e ec i e Hamil onians: Bloch, G am–Schmid 共GS兲, and in e media e 共In 兲ha e been used and wo di e en se s o molecula o bi als, ROHF 共on he op兲and na u al o bi als 共on he bo om兲, ha e been conside ed. TABLE II. Magne ic coupling 共J兲, ionic/neu al a io o he g ound s a e (cI/cN), and e ec i e pa ame e s 共 ab ,Kab , and U兲 o he 关Cu2Cl6兴⫺2 complex. Fo he CI calcula ions, he CASCI and he DDCI lis s o de e - minan s ha e been employed, wi h wo se s o o bi als 共ROHF and na u al o bi als兲. The CI pa ame e s ha e been ob ained by using he G am– Schmid 共GS兲and he in e media e 共In 兲e ec i e Hamil onians. All alues in cm⫺1, excep o U, which is in eV. MOs Le el JacI/cNHe U2Kab ab ROHF CASCI 11 0.009 23.61 27 ⫺878 DDCI ⫺22 0.042 GS 5.07 49 ⫺853 In 5.22 52 ⫺878 Na u al CASCI 78 0.024 18.32 162 ⫺1764 DDCI ⫺15 0.056 GS 5.07 113 ⫺1145 In 7.81 182 ⫺1764 B3LYP ⫺99 0.122 2.15 161 ⫺1062 aJexp :0,⫺40 cm⫺1, om Re . 43. 3992 J. Chem. Phys., Vol. 116, No. 10, 8 Ma ch 2002 Calzado e al. Reuse o AIP Publishing con en is subjec o he e ms: h ps://publishing.aip.o g/au ho s/ igh s-and-pe missions. Downloaded o IP: 150.214.230.47 On: F i, 28 Oc 2016 10:08:14 be echnically di icul , he ionic s a es being ou o he i s 25 lowes oo s o he co esponding symme y. These ema ks illus a e he concep ual limi o he s ic e ec i e Hamil onian app oach when he model space gen- e a es a se o eigens a es wi h a b oad ene gy spec um. In such a case in ude s a es appea . Thei impac is no simply he di e gen beha io o he QDPT, as usually belie ed. The in ude s a es esul in an impossible o a bi a y de ini ion o he a ge space, i.e., o he se o he exac eigen ec o s which a e supposed o be gene a ed om he model space. This commen suppo s he idea ha one has o de ine less ambi ious e ec i e Hamil onians, which, in his case, no longe y o gene a e he ionic exci ed s a es and concen a e on he wo low-lying magne ic s a es. Such e ec i e Hamil- onians may be de ined using he concep and heo y o he in e media e e ec i e Hamil onians. C. Valence in e media e e ec i e Hamil onian 1. Theo y The in e media e Hamil onian,40 buil on an n-dimen- sional model space, is only asked o ep oduce m(m⬍n) exac eigen alues and he p ojec ions o he co esponding m eigens a es on o he model space: H ˆin 兩 P ˆS⌿k 典 ⫽Ek 兩 P ˆS⌿k 典 ,k⫽1, m⬍n.共43兲 This imposes m(m⫺1) condi ions and he e is an in insic lexibili y in he de ini ion o H ˆin . When only he ks a e is looked o , i is possible o impose H ˆin ⫽P ˆSH ˆP ˆS⫹P ˆS⌬ ˆkP ˆS,共44兲 whe e ⌬ ˆkis a s a e speci ic diagonal ope a o . Fo he pa - icula de e minan 兩I典o he model space, he eigenequa ions o he exac Hamil onian lead o FIG. 3. E ec i e pa ame e s ab ,Kab , and U共in eV兲and ionic/neu al a io (cI/cN) in he g ound s a e wa e unc ion ob ained o he 关Cu2Cl6兴2⫺complex om inc easingly co ela ed CI wa e unc ion. The G am–Schmid and in e media e e ec i e Hamil onians ha e been used and wo di e en se s o molecula o bi als, ROHF 共on he op兲and na u al o bi als 共on he bo om兲, ha e been conside ed. TABLE III. Magne ic coupling 共J兲, ionic/neu al a io o he g ound s a e (cI/cN), and e ec i e pa ame e s 共 ab ,Kab , and U兲 o he 关Cu2( ␮ -N3)2(NH3)6兴2⫹complex. Fo he CI calcula ions, he CASCI and DDCI lis s o de e minan s ha e been employed, wi h wo se s o o bi als 共ROHF and na u al o bi als兲. The CI pa ame e s ha e been ob ained by using he G am–Schmid 共GS兲and he in e media e 共In 兲e ec i e Hamil o- nians. All alues in cm⫺1, excep o U, which is in eV. MOs Le el JacI/cNHe U2Kab ab ROHF CASCI ⫺82 0.021 25.77 12 ⫺2218 DDCI ⫺802 0.095 In 5.73 ⫺380 ⫺2218 Na u al CASCI ⫺253 0.080 18.85 720 ⫺6100 DDCI ⫺1103 0.159 GS 4.57 ⫺147 ⫺3007 In 9.27 837 ⫺6100 B3LYP ⫺3026 0.341 3.26 ⫺144 ⫺4482 aJexp :⬍⫺800 cm⫺1, om Re . 44. 3993J. Chem. Phys., Vol. 116, No. 10, 8 Ma ch 2002 Magne ic coupling Reuse o AIP Publishing con en is subjec o he e ms: h ps://publishing.aip.o g/au ho s/ igh s-and-pe missions. Downloaded o IP: 150.214.230.47 On: F i, 28 Oc 2016 10:08:14 la ions in Exchange Coupled Sys ems, edi ed by R. D. 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