Analysis of the magnetic coupling in binuclear complexes. I. Physics of the coupling

Analysis o he magne ic coupling in binuclea complexes. I. Physics o he coupling Ca men J. Calzado Labo a oi e de Physique Quan ique, IRSAMC, Uni e si e ´Paul Saba ie , 118, Rou 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, Rou 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 6 No embe 2001兲 Accu a e es ima es o he magne ic coupling in binuclea complexes can be ob ained om ab ini io con igu a ion in e ac ion 共CI兲calcula ions using he di e ence dedica ed CI echnique. The p esen pape shows ha he same echnique also p o ides a way o analyze he a ious physical con ibu ions o he coupling and pe o ms nume ical analysis o hei espec i e oles on ou binuclea complexes o Cu (d9) ions. The ba e alence-only desc ip ion 共including di ec and kine ic exchange兲does no esul in meaning ul alues. The spin-pola iza ion phenomenon canno be neglec ed, i s sign and ampli ude depend on he sys em. The wo leading dynamical co ela ion e ec s ha e an an i e omagne ic cha ac e . The i s one goes h ough he dynamical pola iza ion o he en i onmen in he ionic alence bond o ms 共i.e., he M⫹ ¯M⫺s uc u es兲. The second one is due o he double exci a ions in ol ing simul aneously single exci a ions be ween he b idging ligand and he magne ic o bi als and single exci a ions o he en i onmen . This dispe si e e ec esul s in an inc ease o he e ec i e hopping in eg al be ween he magne ic o bi als. Mo eo e , i is demons a ed o be esponsible o he p e iously obse ed la ge me al-ligand delocaliza ion occu ing in na u al o bi als wi h espec o he Ha ee–Fock ones. © 2002 Ame ican Ins i u e o Physics. 关DOI: 10.1063/1.1430740兴 I. INTRODUCTION The ield o molecula and solid s a e magne ism a ouses inc easing in e es in o ganic and o ganome allic chemis y as well as in ma e ial science.1–3 These sys ems a e cha ac- e ized by he exis ence o localized unpai ed elec ons, usu- ally loca ed on me allic ions. The p ope ies o he ma e ial a e go e ned by he in e ac ion be ween hei unpai ed elec- ons on neighbo cen e s, which may be iewed as an e ec- i e in e ac ion be ween si e-cen e ed spins, and mapped on o a Heisenbe g–Di ac–Van Vleck Hamil onian,4 H ˆ⫽⫺兺 i,jJijS ˆiS ˆj.共1兲 Fo a bicen ic sys em wi h ms⫽⫾1/2 on each cen e , he coupling cons an J⫽1E⫺3Eis nega i e o a single g ound s a e 共an i e omagne ism兲and posi i e in he opposi e case 共 e omagne ism兲.Jis usually e y small, anging om a ew hund eds o a ew cm⫺1, and i s calcula ion is a chal- lenge o heo e icians. Quan um chemis y may be success- ully applied o such e alua ions. Nume ous s udies employ he densi y unc ional heo y app oach, mos ly wi h he popula B3LYP exchange co ela ion po en ial,5–10 using Noodleman’s app oxima ion,11 in which he magne ic cou- pling is e alua ed om he di e ence be ween he Un e- s ic ed 共spin pola ized兲highes mul iple and he b oken- symme y 共BS兲de e minan . Al hough his p ocedu e may be use ul, in pa icula o p edic o o analyze ends in magne o-s uc u al co ela ions, i su e s om wo weak- nesses: he dependence on he Vxc mul ipa ame ic exchange co ela ion po en ial and he spin con amina ion p oblems, especially se e e o he b oken-symme y solu ion. P ope ly using Noodleman’s o mula, i.e., aking in o ac- coun he e y small o e lap be ween he magne ic o bi als in he BS solu ion, he B3LYP unc ional usually gi es al- ues o Jabou wice he expe imen al ones.7,8,12 Mo eo e , his p ocedu e does no allow o analyze he a ious con i- bu ions o Jand o check he alidi y o he semiempi ical a ionales his o ically p oposed o in e p e he sign and he magni ude o Jand i s s uc u al dependencies ha will be b ie ly ecalled. Ab ini io echniques ha wo k wi h he exac Hamil- onian o e come hese di icul ies and limi s p o ided ha all he meaning ul physical e ec s a e inco po a ed in he ea men , esul ing in a good ag eemen wi h expe imen . JOURNAL OF CHEMICAL PHYSICS VOLUME 116, NUMBER 7 15 FEBRUARY 2002 27280021-9606/2002/116(7)/2728/20/$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 09:45:28 Among he success ul me hods one may i s quo e he non- o hogonal con igu a ion in e ac ion 共NOCI兲,13 which in o- duces a e y limi ed numbe o alence bond con igu a ions, including a ious ligand o me al cha ge ans e s a es.14,15 The lis su e s some a bi a iness. The me hod equi es spe- ci ic op imiza ion o he o bi als o each VB con igu a ion and has no ecei ed nume ous applica ions. Mo e ecen ly, he CASPT2 共Re s. 16 and 17兲me hod has been used in his ield. The second o de expansion gi es a he sa is ac o y esul s p o ided ha he a ia ional comple e ac i e space is la ge enough and p ope ly chosen.18–21 In he ecen pas , an al e na i e con igu a ion in e ac ion 共CI兲 echnique has been designed o a di ec e alua ion o e ical ene gy di e ences, he di e ence dedica ed con igu- a ion in e ac ion 共DDCI兲,22 which has been success ully em- ployed in a wide se ies o s udies, conce ning molecula 23–26 as well as solid s a e magne ic ma e ials.27–30 An e o smalle han 10 cm⫺1is ypical o a alue o J⬃100 cm⫺1. The de ini ion o he DDCI space is based on second o de quasidegene a e pe u ba ion heo y a gumen s as ex- plained in ou ea ly pape s in his domain.22,23 A second- o de pe u ba i e app oach has he ad an age o p o iding a pa i ion o he inal J alue in o a sum o di e en physical con ibu ions31 such as di ec exchange, Ande son’s an i e - omagne ic 共o kine ic exchange兲con ibu ion, spin pola iza- ion, e c. Howe e he second o de pe u ba i e expansion is no nume ically eliable because o con e gence p oblems and a bi a iness in he choice o he ze o o de Hamil onian. The DDCI me hod goes h ough an exac diagonaliza ion and hence gains in accu acy, bu he analysis o he a ious physical mechanisms in ol ed in he coupling is no s aigh - o wa d any mo e. Howe e , he use ulness o such a igo ous analysis is indubi able since i would allow o check he alidi y o he quali a i e models, equen ly used in an a pos e io i a ionaliza ion and in he molecula design o new ma e ials.1 The a ge o he p esen pape is o show ha i is possible o combine nume ical accu acy 共going h ough la ge basis se s and long CI expansions兲and analysis in e ms o physical con ibu ions. Fou di e en an i e omagne ic sys ems ha e been conside ed in he p esen wo k, all o hem in ol ing wo elec ons on wo Cu (d9) si es: he 关Cu2Cl6兴2⫺, he 关Cu2( ␮ -N3)2(NH3)6兴2⫹, and he Cu2( ␮ -CH3COO)4(H2O)2complexes and he Cu2O7clus- e , a model o he La2CuO4pe o ski e. Sec ion II desc ibes he compu a ional de ails. A e ecalling he alence only ea men s in Sec. III, he heo e ical de elopmen and he s a egy o analysis a e p esen ed in Sec. IV and applied on hese ou binuclea sys ems. Finally, Sec. V eanalyzes he ole o he di e en con ibu ions when using na u al o bi als ins ead o he iple Ha ee–Fock o bi als used in Secs. III and IV. II. DESCRIPTION OF THE MODELS AND COMPUTATIONAL DETAILS The ou sys ems conside ed he e, ep esen ed in Fig. 1, con ain wo Cu(d9) cen e s, b idged by di e en ligands and wi h di e en ela i e posi ions. Figu e 1共a兲depic s ou i s model, he 关Cu2Cl6兴2⫺, which magne os uc u al depen- dence has been ex ensi ely s udied in he ecen pas .9,23,32,33 We ha e concen a ed on he plana s uc u e, whe e each Cu a om has a squa e plana coo dina ion and bea s i s unpai ed elec on in a dxy- ype o bi al 关Fig. 2共a兲兴. The Cu–Cu dis ance in he eal complex is 3.44 Å. Expe imen ally, he plana s uc u e o his complex belongs o he Cipoin g oup bu only e y sligh ly dis o ed om D2h, conside ed in he p esen wo k. A simila si ua ion is ound in he 关Cu2( ␮ -N3)2(NH3)6兴2⫹complex 关Fig. 1共b兲兴 wi h a squa e py amidal coo dina ion o he Cu cen e s 关Fig. 2共b兲兴. The Cu–Cu dis ance is 5.19 Å, la ge han in he 关Cu2Cl6兴2⫺ complex, bu he magne ic coupling cons an in he o me is one o de o magni ude la ge han in he la e , showing ha he end- o-end b idged (N3)⫺g oups play a c ucial ole in he exchange phenomenon.34,35 As in he p eceden case, he expe imen al Cigeome y has been symme ized o C2h. Rega ding he Cu2( ␮ -CH3COO)4(H2O)2molecule 关Fig. 1共c兲兴, he Cu a oms ha e also a squa e py amidal coo - dina ion, bu a e placed in pa allel planes, b idged by ou biden a e ace a o-ligands and sepa a ed by 2.64 Å only. Each Cu a om bea s an unpai ed elec on in a dx2–y2o bi al 关Fig. 2共c兲兴. The expe imen al Cigeome y36 has been used in he calcula ions. The na u e o he magne ic in e ac ion in his sys em was a ma e o con o e sy du ing h ee decades: Do he wo dx2–y2unpai ed elec ons in e ac di ec ly ia a ␦ -bond o indi ec ly h ough he ace a o-ligands?37 The ex- pe imen al wo ks o Gu ¨del e al.38 and Ewald and Sinn39 oge he wi h he ab ini io calcula ion o de Lo h e al.31 uled ou he ␦ -bond hypo hesis and showed he ole o he ace a o-ligand in he Cu–Cu exchange. The las conside ed sys em is a model o he an i e o- magne ic pe o ski e La2CuO4. I consis s o a Cu2O7clus e 关Fig. 1共d兲兴, embedded in a se o poin cha ges o mimic he Madelung ield o he La2CuO4la ice. The unpai ed elec on is placed in a dx2–y2o bi al, bu unlike he p eceden models, he Cu a oms a e b idged by jus one oxygen ligand 关Fig. 2共d兲兴. The magne ic coupling in his compound is qui e la ge FIG. 1. Schema ic ep esen a ion o he ou models conside ed: 共a兲 he 关Cu2Cl6兴2⫺complex, in a plana geome y; 共b兲 he 关Cu2( ␮ -N3)2(NH3)6兴2⫹ complex wi h end- o-end b idging azido ligands; 共c兲 he Cu2( ␮ -CH3COO)4(H2O)2complex; 共d兲 he Cu2O7clus e . 2729J. Chem. Phys., Vol. 116, No. 7, 15 Feb ua y 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 09:45:28 共a ound ⫺1000 cm⫺1兲, well es ablished om Raman40 and neu on di ac ion41 expe imen s. Di e en heo e ical ap- p oaches ha e been employed o e alua e J关 o ins ance, ini e8and pe iodic42 DFT calcula ions, nono hogonal CI 共Re . 14兲and DDCI 共Re s. 27, 28兲calcula ions兴 he CI p o- cedu es gi ing he mos co ec esul s. Since he ou models p esen ed he e ha e been s udied in p e ious independen wo ks, di e en basis se s ha e been used in he HF–CI calcula ions. In he 关Cu2Cl6兴2⫺complex A, he azido-b idged complex B, and coppe ace a e C, an e ec i e co e po en ial o Ba andia a ´n and Seijo has been employed o he Cu a oms, whe e he alence elec ons a e desc ibed by a (9s6p6d)/关3s3p4d兴basis se .43 ANO- ype unc ions44 ha e been used o Cl, N, and H, wi h he ol- lowing con ac ions: ANO-s.4s3p1d o Cl, ANO-s.3s2p1d o he b idging N a oms, ANO-s.3s2p o he N a oms o he ex e nal NH3g oups, and ANO-s.2s o he H a oms. Fo he coppe ace a e C, due o he size o he sys em, he e ec i e co e po en ials p oposed by Huzinaga45 ha e been used o he C and O a oms, wi h basis se s (5s6p1d)/关2s2p1d兴 o oxygen a oms and (5s5p1d)/ 关2s2p1d兴 o he ca bon a oms. In he CH3g oups and H2O molecules, minimal basis se s ha e been used, (6s3p)/关2s1p兴 o O and C, and (3s)/关1s兴 o H.46 Finally, in he Cu2O7clus e D, he Hay–Wad pseudopo en ial and basis unc ions ha e been used o Cu a oms.47 Fo he oxy- gen a oms an all elec on basis se (10s5p) con ac ed o 关3s2p兴has been employed,48 enla ged wi h a dpola iza ion unc ion ( ␣ ⫽0.85) in he b idging-oxygen a om. The ROHF molecula o bi als ha e been ob ained by using he MOLCAS 4.1 package.49 The CASDI 共Re . 50兲p o- g am has been used in he CI calcula ions and he NATURAL 共Re . 51兲p og am in he de e mina ion o he na u al MOs. III. VALENCE-ONLY DESCRIPTION A. Theo y 1. One-band model F om he ea ly se en ies, se e al quali a i e models we e p oposed52–54 o in e p e he physical ac o s go e n- ing he magne ic coupling. Ea ly analysis poin ed ou he exis ence o wo an agonis con ibu ions, J⫽JF⫹JAF , whe e Fand AF indica e e omagne ic and an i e omag- ne ic con ibu ions, espec i ely. JFis a ibu ed o he di ec exchange be ween he magne ic o bi als 共always e omag- ne ic兲, and JAF is in e p e ed h ough he delocaliza ion e ec ha can only occu in he single s a e 共hence an i e omag- ne ic兲and is some imes called kine ic exchange. These a io- naliza ions may ollow o hogonal alence bond 共VB兲,52 nono hogonal VB 共Re . 53兲o alence con igu a ion in e - ac ion 共VCI兲共Re . 54兲a gumen s, bu in gene al hey handle wo elec ons in wo magne ic 共local o symme y-adap ed兲 o bi als. Le us conside he VCI and he o hogonal VB ap- p oaches o he simples symme ical A•–B•sys em wi h wo elec ons in wo o bi als. The o bi als a e supposed o be p e iously de e mined, gi ing o hogonal co e o bi als 共closed shell兲and magne ic o bi als. These magne ic o bi als may be ei he symme y-adap ed, gand u, o hei equi alen localized ans o ms, aand b, g⫽a⫹b &and u⫽a⫺b & o 共2兲 a⫽g⫹u &and b⫽g⫺u &. Wi h wo ac i e elec ons in he magne ic o bi als, he CI space is limi ed o ou de e minan s. In he o hogonal VB app oach, o he Ms⫽0 componen , he e a e wo neu al de e minan s, 兩 ab ¯ 典 ⫽ 兩 co e ab ¯ 典 and 兩 ba ¯ 典 ⫽ 兩 co e ba ¯ 典 ,共3兲 and wo ionic de e minan s, 兩 aa ¯ 典 ⫽ 兩 co e aa ¯ 典 and 兩 bb ¯ 典 ⫽ 兩 co e bb ¯ 典 .共4兲 The CI ma ix akes he o m 兩 ab ¯ 典 兩 ba ¯ 典 兩 aa ¯ 典 兩 bb ¯ 典 冋 0Kab ab ab Kab 0 ab ab ab ab UK ab ab ab Kab U 册 ,共5兲 whe e he ene gy o he neu al VB de e minan s is consid- e ed as he ene gy o igin. The di ec exchange, Kab ⫽ 具 ab ¯ 兩 1/ 12 兩 ba ¯ 典 , is necessa ily posi i e since i is he sel - epulsion o he ab o e lap elec onic dis ibu ion, as well as U, he ene gy di e ence be ween he ionic and he neu al o ms. The hopping in eg al ab⫽ 具 ab ¯ 兩 H ˆ 兩 aa ¯ 典 ⫽ 具 a 兩 F ˆ 兩 b 典 , whe e F ˆis he Fock ope a o , is he coupling be ween he neu al and he ionic VB o ms. As shown by Hay e al.54 his in eg al can also be exp essed by he diagonal Fock el- emen s o he symme y adap ed o bi als, ab⫽1 2(Fgg ⫺Fuu). The ou solu ions a e: 共i兲in he usymme y: FIG. 2. Rela i e o ien a ion o he magne ic o bi als in 共a兲 he 关Cu2Cl6兴2⫺ complex; 共b兲 he 关Cu2( ␮ -N3)2(NH3)6兴2⫹complex; 共c兲 he Cu2( ␮ -CH3COO)4(H2O)2molecule; and 共d兲 he Cu2O7clus e . 2730 J. Chem. Phys., Vol. 116, No. 7, 15 Feb ua y 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 09:45:28 共a兲a pu ely neu al iple s a e, 兩 Tu 典 ⫽1 &共 兩 ab ¯ 典 ⫺ 兩 ba ¯ 典 )共6兲 o ene gy 3Eu⫽⫺Kab .共7兲 We also e e o his Ms⫽0 componen o he iple con igu a ion as Tab 0⫽1 &共ab ¯ ⫺ba ¯ 兲;共8兲 共b兲a pu ely ionic single s a e, 兩 Su 典 ⫽1 &共 兩 aa ¯ 典 ⫺ 兩 bb ¯ 典 )共9兲 o ene gy 1Eu⫽U⫺Kab ;共10兲 共ii兲in he gsymme y, wo single s a es which a e mix- u es o he neu al and he ionic componen s: 共a兲 he lowes one is essen ially neu al, 兩 Sg 典 ⫽ ␦ 共 兩 ab ¯ 典 ⫹ 兩 ba ¯ 典 )⫹ ␥ 共 兩 aa ¯ 典 ⫹ 兩 bb ¯ 典 ), 共 ␦ ⬎ ␥ ⬎0兲.共11兲 I s ene gy is 1Eg⫽Kab⫹U⫺ 冑 U2⫹16 ab 2 2.共12兲 I is also use ul o de ine he single con igu a ion, Sab⫽1 &共ab ¯ ⫹ba ¯ 兲;共13兲 共b兲 he second one is essen ially ionic, 兩 Sg ⬘ 典 ⫽⫺ ␥ 共 兩 ab ¯ 典 ⫹ 兩 ba ¯ 典 )⫹ ␦ 共 兩 aa ¯ 典 ⫹ 兩 bb ¯ 典 ), 共14兲 and much highe in ene gy, 1Eg ⬘⫽Kab⫹U⫹ 冑 U2⫹16 ab 2 2.共15兲 The single – iple ene gy sepa a ion is gi en by J⫽⌬EST⫽1Eg⫺3Eu⫽2Kab⫹U⫺ 冑 U2⫹16 ab 2 2.共16兲 When UⰇ 兩 ab 兩 , a powe expansion, equi alen o a pe u - ba ion o he neu al single by he ionic one, is possible and he coupling cons an ends o52 J⫽2Kab⫺4 ab 2 U,共17兲 whe e he opposi e signs o bo h con ibu ions become e i- den . I may be wo h o in oducing a diag amma ic pic u e o he coupling. The wo neu al VB de e minan s may be de- pic ed as in Diag am 1: Diag am 1 whe e he double a ows indica e he magne ic o bi als. He ea e , simple le - o- igh a ows →symbolize holes, i.e., inac i e doubly occupied o bi als, om which an elec on is exci ed, and simple igh - o-le a ows ←symbolize pa - icles, i.e., i ual o bi als, in o which elec ons a e exci ed. The di ec exchange appea s as a i s o de in e ac ion, ep esen ed in Diag am 2: Diag am 2 while he kine ic exchange is a second o de e ec , as shown in Diag am 3: Diag am 3 The same esul is ob ained when wo king wi h symme y- adap ed MOs, 兩 Sg 典 ⫽␭ 兩 gg ¯ 典 ⫺ ␮ 兩 uu ¯ 典 , 兩 Tu 典 ⫽1 &共 兩 gu ¯ 典 ⫺ 兩 ug ¯ 典 ), 共18兲 兩 Sg ⬘ 典 ⫽ ␮ 兩 gg ¯ 典 ⫹␭ 兩 uu ¯ 典 , 兩 Su 典 ⫽1 &共 兩 gu ¯ 典 ⫹ 兩 ug ¯ 典 ). The coe icien s in bo h app oaches a e ela ed by 2 ␦ ⫽␭⫹ ␮ and 2 ␥ ⫽␭⫺ ␮ .共19兲 When 兩 /U 兩 is small, ␭and ␮ a e bo h close o 1/&and ␥ is small. The ela ions ha gi e he alues o Kab , ab , and U om he solu ions o he (2e⫺/2MO) CI a e easily es ab- lished, Kab⫽ 1Eg⫹1Eg ⬘⫺3Eu⫺1Eu 4,共20兲 2731J. Chem. Phys., Vol. 116, No. 7, 15 Feb ua y 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 09:45:28 U⫽1Eu⫺3Eu,共21兲 ab⫽⫺ 冑 共1Eg ⬘⫺1Eg兲2⫺共1Eu⫺3Eu兲2 4.共22兲 Hence his a ia ional CI calcula ion enables us o e alua e he di ec exchange, Kab , as well as he hopping in eg al, ab , and he on-si e epulsion, U. Analogously, e ec i e Kab , ab , and U alues ha include he e ec s o he dy- namic co ela ion can be ex ac ed in an exac way om la ge CI expansions, as will be shown in pa II o his wo k.58 2. Two-band model Analogous elemen a y pic u es may explici ly in oduce ligand o bi als, when he exchange be ween he me al cen e s is supposed o p oceed h ough a b idging-ligand in a A•–L–B•s uc u e. Fo simplici y, he ligand o bi als will be limi ed o a unique doubly occupied o bi al, l. The wo neu- al VB de e minan s may be depic ed as in Scheme 1: Scheme 1 In addi ion o he p e iously discussed di ec coupling, a ou h-o de indi ec coupling be ween hem is possible ha p oceeds h ough a ligand o me al cha ge ans e 共LMCT兲 p ocess, acco ding o Scheme 2: Scheme 2 whe e ⌬ECT is he exci a ion ene gy 共hence, posi i e兲 o he ligand o me al cha ge ans e s a es (A⫺L⫹B•). Ob iously, he e is an equi alen con ibu ion in ol ing (A•L⫹B⫺) in- e media e. The diag amma ic ep esen a ion o one o hem is shown in Diag am 4: Diag am 4 The co esponding con ibu ion o Jis an i e omagne ic, J←⫺4 la 2 lb 2 ⌬ECT 2U共23兲 and is he usual in e p e a ion o supe exchange. The con i- bu ion o he consecu i e cha ge ans e p ocesses h ough he ligand may be included in an e ec i e hopping in eg al, ab e ⫽ ab⫹ la lb ⌬ECT ,共24兲 and when ab is small, J⫽2Kab⫺4共 ab e 兲2 U,共25兲 which e u ns o he one-band model, Eq. 共17兲. In p inciple, o he mechanisms p oceeding h ough he double cha ge ans e (A⫺L⫹⫹B⫺) in e media es 共o ela i e ene gy ⌬E2CT兲may also be conside ed gi ing a con ibu ion, J←⫺8 la 2 lb 2 ⌬ECT 2⌬E2CT.共26兲 A ypical ep esen a ion is Diag am 5: Diag am 5 This double ligand o me al ans e is known in solid s a e physics as he Goodenough mechanism.55,56 Since he ela- i e ene gy o he doubly ionic s uc u e ⌬E2CT is ce ainly la ge han U his mechanism may be supposed o b ing e y small con ibu ions, al hough i has been in oked some- imes.29,56 B. Nume ical esul s The p esen sec ion epo s calcula ions o he alence- only CASCI 共2e⫺in 2MOs兲 o he ou model p oblems. I 2732 J. Chem. Phys., Vol. 116, No. 7, 15 Feb ua y 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 09:45:28 has been shown57 ha he o bi als o he ROHF iple and o he (2e/2MO) CASSCF single a e almos iden ical. The e- sul s p esen ed in Table I a e ob ained om he iple s a e MOs and show ha : 共1兲The Kab e omagne ic di ec exchange is small, wi hou a clea co ela ion wi h he Cu–Cu dis ance, since he o ien a ion o he magne ic o bi als 共Fig. 2兲is no he same and he ails on he b idging-ligands depend on hei chemical na u e. 共2兲The ab hopping in eg al con ibu ion is qui e di e en om one sys em o ano he . Compa ing he chlo ide and he azido complexes, bo h wi h dxy magne ic o bi als 共Fig. 2兲, makes e iden he ole o he delocaliza ion ails on he b idging ligand since ab is la ge in he second one despi e he la ge Cu–Cu dis ance. The ene gy di - e ence be ween he neu al and he ionic VB compo- nen s, U, is e y la ge 共a ound 2⫻105cm⫺1,⬃24–25 eV兲, much la ge han usually assumed in model Hamil- onians. One may no ice ha his quan i y is almos in- a ian o he ligand and o he me al-me al dis ance. This nea -in a iance is in be e ag eemen wi h he on- si e epulsion o he simple Hubba d model,52 han wi h mo e elabo a ed me hods54 in which U⫽Jaa⫺Jab ,共27兲 Jaa and Jab being he one-cen e and wo-cen e s Cou- lomb epulsions, espec i ely. 共3兲The o e all J alue is e y a om he expe imen al one. I is e en o he w ong sign o he chlo ide com- plex. I shows ha a lo o physics is lacking a his s age which ne e heless con ains he wo leading ac o s usu- ally in ol ed o in e p e a ion. I is clea ha he an i- e omagne ic con ibu ion is g ossly unde es ima ed. IV. BEYOND THE VALENCE-ONLY DESCRIPTION: DYNAMICAL CORRELATION EFFECTS A. The B illouin’s heo em and i s consequences A his s age i is impo an o discuss he e ec s o he o bi als used in he calcula ions. In Secs. III and IV, sel - consis en o bi als ob ained om a a ia ional es ic ed open-shell Ha ee–Fock calcula ion o he iple s a e a e used. These o bi als sa is y he B illouin’s heo em, which consequences can be summa ized as ollows. When ac ing on he Tu iple s a e, he single exci a ions ai ⫹aj共whe e a and a⫹a e annhila ion and c ea ion ope a o s, espec i ely兲 lead o ai ⫹ajTude e minan s ha do no in e ac wi h Tu, 具 ai ⫹ajTu 兩 H ˆ 兩 Tu 典 ⫽0. 共28兲 Th ee ypes o single exci a ions a e o be dis inguished: 共1兲The 1hde e minan s co espond o he single exci a- ions ha send one elec on om an occupied ho bi al o he amagne ic o bi al, aa ⫹aho aa ¯ ⫹ah ¯ , 兩 hh ¯ (ab ¯ ⫺ba ¯ ) 典 ⇒ 兩 ha ¯ ab ¯ ⫺ah ¯ ba ¯ 典 . The B illouin’s heo em imposes 具 a 兩 h ˆc⫹J ˆa⫹J ˆb 兩 h 典 ⫽0, 共29兲 whe e h ˆc ep esen s he Fock ope a o o he closed shells o he sys em, h ˆc⫽h ˆ⫹兺 k苸co e 2J ˆk⫺K ˆk,共30兲 J ˆand K ˆbeing he usual Coulomb and exchange ope a- o s. 共2兲The 1pde e minan se con ains all he single exci a- ions om he magne ic o bi als o he i ual se , ap ⫹aao ap ¯ ⫹aa ¯ , 兩 hh ¯ (ab ¯ ⫺ba ¯ ) 典 ⇒ 兩 hh ¯ pb ¯ ⫺hh ¯ bp ¯ 典 . The B illouin’s heo em imposes in his case, 具 p 兩 h ˆc⫹J ˆa⫺K ˆa⫹J ˆb⫺K ˆb 兩 a 典 ⫽0. 共31兲 共3兲Fo he 1h⫹1pexci a ions, ap ⫹ah, he B illouin’s heo- em gi es 具 p 兩 h ˆc⫹J ˆa⫺K ˆa 2⫹J ˆb⫺K ˆb 2 兩 h 典 ⫽0. 共32兲 This condi ion implies ha he s a e ob ained unde he ac- ion o he single ap ⫹ah⫹ap ¯ ⫹ah ¯ exci a ions, Shp⫽1/&(hp ¯ ⫹ph ¯ ) on he iple s a e, 兩 Shp•Tu 典 ⫽ 兩 Shp•Tab 0 典 ⫽1 2 兩 共hp ¯ ⫹ph ¯ 兲共ab ¯ ⫺ba ¯ 兲 典 共33兲 does no in e ac wi h i , 具 Shp•Tu 兩 H ˆ 兩 Tu 典 ⫽0. 共34兲 TABLE I. Valence-only desc ip ion wi h ROHF molecula o bi als. Di ec exchange, Kab , hopping in eg al, ab , on-si e epulsion, U, coupling cons an , J. All esul s in cm⫺1. 关Cu2Cl6兴2⫺关Cu2(N3)2(NH3)6兴2⫹Cu2(CH3COO)4(H2O)2Cu2O7 dCu–Cu(Å) 3.44 5.19 2.64 3.78 2Kab 27 12 4 67 2 ab ⫺1756 ⫺4436 ⫺2085 ⫺7917 U(⫻10⫺4) 19.0 20.8 19.1 19.4 J⫹11 ⫺82 ⫺19 ⫺255 Jexp 0,⫺40a⬍⫺800b⫺286c,⫺294⫾4d⫺1032⫾48e ⫺1081⫾40 aRe e ence 62. dRe e ence 38. bRe e ence 63. eRe e ence 40. cRe e ence 64. Re e ence 41. 2733J. Chem. Phys., Vol. 116, No. 7, 15 Feb ua y 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 09:45:28 I is easy o show ha he same exci a ion ac ing on he single s a e, Shp•Sab⫽1 2 兩 共hp ¯ ⫹ph ¯ 兲共ab ¯ ⫹ba ¯ 兲 典 ,共35兲 sa is ies a simila equa ion, 具 Shp•Sab 兩 H ˆ 兩 Sab 典 ⫽0, 共36兲 whe e Sab⫽1/&(ab ¯ ⫹ba ¯ ) was de ined in Eq. 共13兲. In summa y, single exci a ions om he co e o he i - ual space do no in e ac wi h any o bo h s a es. Single exci a ions om he co e o he ac i e o bi als, Eq. 共29兲,o om he ac i e o bi als o he i ual ones, Eq. 共31兲, do no in e ac wi h he SCF iple s a e bu hey can gi e small in e ac ions wi h he single ha ha e o be ca e ully ana- lyzed. As a consequence, he ligand- o-me al cha ge ans e exci a ions esponsible o he delocaliza ion ails on he ligand do no in e ac wi h he iple s a e. I is shown in Sec. IVB1a ha hei in e ac ion wi h he single is also ex emely small. The o -diagonal elemen s la and lb o he Fock ope a o a e null and consequen ly he supe exchange mechanisms, Eq. 共23兲and Eq. 共26兲, do no con ibu e. The a ia ional op imiza ion o he o bi als in he SCF p ocedu e de ines an op imal delocaliza ion be ween he ligand and he magne ic cen e s in such a way ha hese e ec s a e inco - po a ed in he alence-only CI. B. Taking only he neu al de e minan s as model space 1. Theo y Following he logics o a p e ious wo k,31 he quaside- gene a ed pe u ba ion heo y may be used as a guideline o unde s and he e ec i e coupling be ween he neu al o ms, 兩 ab ¯ 典 and 兩 ba ¯ 典 . The pe u ba i e analysis is essen ially con- cep ual, since he nume ical analysis p esen ed he e is based on a ia ional calcula ions. These wo de e minan s may be conside ed as de ining a physically meaning ul wo- dimensional model space, upon which quasidegene a e pe - u ba ion heo y allows o build an e ec i e Hamil onian, H ˆe . I s o -diagonal ma ix elemen gi es J ⫽2 具 ab ¯ 兩 H ˆe 兩 ba ¯ 典 . Up o he second o de , 具 ab ¯ 兩 H ˆe 共2兲 兩 ba ¯ 典 ⫽Kab⫹兺 兩 ␣ 典 ⫽ 兩 ab ¯ 典 , 兩 ba ¯ 典 具 ab ¯ 兩 H ˆ 兩 ␣ 典具 ␣ 兩 H ˆ 兩 ba ¯ 典 E0 共0兲⫺E ␣ 共0兲, 共37兲 whe e E0 (0)⫽E 兩 ab ¯ 典 . The second o de con ibu ions may be schema ized as in Diag am 6: Diag am 6 In he p eceding sec ion, he ole o 兩 ␣ 典 ⫽ 兩 aa ¯ 典 , 兩 bb ¯ 典 has been analyzed. O he ou e -space de e minan s can be ob- ained by exci a ions in ol ing non alence 共inac i e兲o bi - als, ei he co e-o bi als o i ual o bi als. The lis o de e - minan s con ibu ing up o he second o de as well as he co esponding physical con ibu ions ha e been discussed by de Lo h e al.31 The de e minan s o his lis will be labeled he ea e by he numbe no he exci ed elec ons om he co e o holes, nh, and he numbe mo p omo ed elec ons o he i ual space o pa icles, mp. Up o he second o de , 兩 ␣ 典in ol e i e ypes o de e minan s: 1h,1p,1h⫹1p,2h, and 2p,as ep esen ed in Fig. 3. a. The 1h and 1p de e minan s. Among he p e iously de ined 1hde e minan s, he mos impo an a e he ligand o me al (L→M) cha ge ans e 共LMCT兲s a es, which do no in e ac wi h he iple due o B illouin’s heo em, bu lead o a small in e ac ion wi h he single s a e, 冓 1 &共hb ¯ ⫹bh ¯ 兲aa ¯ 冏 H ˆ 冏 1 &hh ¯ 共ab ¯ ⫹ba ¯ 兲 冔 ⫽2 具 h 兩 K ˆb 兩 a 典 ⫽2共hb,ab兲,共38兲 whe e he con en ional no a ion o bielec onic in eg als is used. The in eg al, 具 i共1兲j共2兲 兩 1 l2 兩 k共1兲l共2兲 典 ⫽ 具 ij 兩 kl 典 ⫽共ik,jl兲共39兲 ep esen s he in e ac ion be ween he ik and jl o e lap dis- ibu ions. Since he Aand Bmagne ic cen e s a e usually well sepa a ed, he ab dis ibu ion is e y small. This esul s in a small in eg al and consequen ly small coe icien s o he LMCT s a es in he single wa e unc ion, ypically o he o de o 10⫺2. FIG. 3. Schema ic ep esen a ion o he a ious exci a ion ope a o s leading o he ele an CI spaces. 2734 J. Chem. Phys., Vol. 116, No. 7, 15 Feb ua y 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 09:45:28 Simila ly, he 1pde e minan s, which in ol e me al o ligand cha ge ans e exci a ions, do no in e ac wi h he iple s a e, bu hei in e ac ion wi h he single s a e is 冓 1 &hh ¯ 共ap ¯ ⫹pa ¯ 兲 冏 H ˆ 冏 1 &hh ¯ 共ab ¯ ⫹ba ¯ 兲 冔 ⫽2 具 p 兩 K ˆa 兩 b 典 ⫽2共pa,ab兲.共40兲 As in he p eceden case, o wo dis an magne ic cen e s he magni ude o his in eg al is small and he e o e he coe i- cien s o he 1pde e minan s on he single wa e unc ion a e small as well. b. The 1h ⫹ 1p de e minan s. The 1h⫹1pde e minan s belong o h ee di e en classes. 共i兲The pu e single exci a ions in he inac i e pa , Shp , ep esen ed in Scheme 3: Scheme 3 ac on he Sab and Tab 0model space s a es gi ing he single , Shp•Sab⫽1 2 兩 (hp ¯ ⫹ph ¯ )(ab ¯ ⫹ba ¯ ) 典 , and he iple , Shp•Tab 0 ⫽1 2 兩 (hp ¯ ⫹ph ¯ )(ab ¯ ⫺ba ¯ ) 典 , de ined in Eqs. 共35兲and 共33兲, e- spec i ely. The in e ac ion o hese single-exci ed s a es wi h he single and iple s a es does no b ing any con ibu ion o he S–Tene gy di e ence, due o he B illouin’s heo em, as shown in Sec. IVA. 共ii兲The iple exci a ions in he inac i e pa , as shown in Scheme 4: Scheme 4 in oduce he spin pola iza ion e ec . The inac i e pa be- comes Thp 0⫽1 &共hp ¯ ⫺ph ¯ 兲,Thp ⫹⫽hp,Thp ⫺⫽h ¯ p ¯ ,共41兲 which ac ing on he model space gi es 共1兲 wo iple s a es: Thp 0•Sab⫽1 2 兩 共hp ¯ ⫺ph ¯ 兲共ab ¯ ⫹ba ¯ 兲 典 , 1 &共Thp ⫹ •Tab ⫺⫺Thp ⫺ •Tab ⫹兲⫽1 & 兩 共hpa ¯ b ¯ ⫺h ¯ p ¯ ab兲 典 ;共42兲 one single s a e: 1 )共Thp ⫹ •Tab ⫺⫹Thp ⫺ •Tab ⫹⫺Thp 0•Tab 0兲.共43兲 The in e ac ion o he iple s a es wi h Tab 0b ings he ol- lowing con ibu ions: 具 Thp 0•Sab 兩 H ˆ 兩 Tab 0 典 ⫽1 & 具 p 兩 K ˆa⫺K ˆb 兩 h 典 , 共44兲 1 & 具 Thp ⫹ •Tab ⫺⫺Thp ⫺ •Tab ⫹ 兩 H ˆ 兩 Tab 0 典 ⫽ 具 p 兩 K ˆa⫹K ˆb 兩 h 典 . Fo he single s a e, 1 ) 具 Thp ⫹ •Tab ⫺⫹Thp ⫺ •Tab ⫹⫺Thp 0•Tab 0 兩 H ˆ 兩 Sab 典 ⫽⫺ 3 & 具 p 兩 K ˆa⫺K ˆb 兩 h 典 .共45兲 The local K ˆaand K ˆbexchange ope a o s in oduce spin po- la iza ion o he inac i e o bi als 共o closed shells兲. The o al spin pola iza ion ene gy in he iple s a e is 3Ehp 共2兲⫽⫺ 1/2 具 p 兩 K ˆa⫺K ˆb 兩 h 典 2⫹ 具 p 兩 K ˆa⫺K ˆb 兩 h 典 2 ⌬Eh→p,共46兲 whe e ⌬Eh→pis a posi i e quan i y ep esen ing he exci a- ion ene gy o he p omo ed inac i e elec ons, E 兩 hp ¯ ab ¯ 典 ⫺E 兩 hh ¯ ab ¯ 典 . The o al spin pola iza ion ene gy o he single s a e is 1Ehp 共2兲⫽⫺ 3/2 具 p 兩 K ˆa⫺K ˆb 兩 h 典 2 ⌬Eh→p.共47兲 The con ibu ions o bo h s a es a e signi ican . The e ms 具 p 兩 K ˆa 兩 h 典 2/⌬Eh→p共 esp. b兲, which in oduce he spin pola - iza ion o he elec ons a ound each magne ic cen e , con ib- u e equally o he single and he iple s a es and cancel in he ansi ion. Hence, he con ibu ion o he S–Tene gy spli ing comes only om he c ossed e ms, J←⫹4 具 h 兩 K ˆa 兩 p 典具 p 兩 K ˆb 兩 h 典 ⌬Eh→p.共48兲 This con ibu ion, labeled ␧DSP 2in de Lo h e al.,31 is non- negligible only when he hp dis ibu ion is impo an nea A and B, i.e., o hand pbeing b idging-ligand o bi als. I s sign canno be p edic ed since i depends on he signs o he hp dis ibu ion nea he magne ic cen e s, which depends on he ligand. The co esponding second-o de diag ams o he cou- pling a e o Diag am 7 ype. The con ibu ion o he spin pola iza ion o Jis easily iden i ied by ex ending he CI space, spanned by he alence CAS, wi h he de e minan s o he Thp ⫹ •Tab ⫺and Thp ⫺ •Tab ⫹ ype, which a e esponsible o he di e en ial e ec , 2735J. Chem. Phys., Vol. 116, No. 7, 15 Feb ua y 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 09:45:28 Diag am 7 共iii兲The inac i e single exci a ions wi h a spa ial change in he ac i e pa , shown in Scheme 5, namely, he single exci a ions on he ionic VB o ms 共single by na u e兲gi e Shp•aa ¯ and Shp•bb ¯ single s and Thp 0•aa ¯ and Thp 0•bb ¯ ip- le s:Scheme 5 The in e ac ion o Shp•aa ¯ wi h he neu al single is 具 Shp•aa ¯ 兩 H ˆ 兩 Sab 典 ⫽2共ph,ab兲⫺共pb,ah兲,共49兲 while he in e ac ion o Thp 0•bb ¯ wi h he iple s a e is 具 Thp 0•bb ¯ 兩 H ˆ 兩 Tab 0 典 ⫽⫺共pb,ah兲.共50兲 The inal second-o de con ibu ion o he single – iple spli ing is J←⫺8共ph,ab兲2⫺4共ph,ab兲关共pa,bh兲⫹共pb,ah兲兴 共U⫹⌬Eh→p兲. 共51兲 The ab dis ibu ion has weak ampli ude e e ywhe e and he e o e his second o de con ibu ion, labeled ␧SE⫹P 2in he wo k o de Lo h e al.,31 is expec ed o be small. The co e- sponding p ocesses a e depic ed in Diag am 8: Diag am 8 Howe e hese ou e space 1h⫹1pde e minan s ha e la ge in e ac ions wi h he ionic VB de e minan s 兩 aa ¯ 典 and 兩 bb ¯ 典 which a e pa s o he alence CI single s a e, 兩 Sg 典 关Eq. 共11兲兴. Ac ually, 1 & 具 共hp ¯ ⫹ph ¯ 兲aa ¯ 兩 H ˆ 兩 hh ¯ aa ¯ 典 ⫽& 具 p 兩 h ˆc⫹2J ˆa⫺K ˆa 兩 h 典 . 共52兲 The B illouin’s heo em implies ha 关Eq. 共32兲兴, 具 p 兩 h ˆc⫹J ˆa⫹J ˆb⫺K ˆa 2⫺K ˆb 2 兩 h 典 ⫽0, 共53兲 which gi es 1 & 具 共hp ¯ ⫹ph ¯ 兲aa ¯ 兩 H ˆ 兩 hh ¯ aa ¯ 典 ⫽& 具 p 兩 J ˆa⫺K ˆa 2⫺J ˆb⫹K ˆb 2 兩 h 典 . 共54兲 The 具 p 兩 J ˆa⫺J ˆb 兩 h 典 in eg al is la ge since i ep esen s he cou- pling be ween he hp ansi ion dis ibu ion wi h he A⫺ ¯A⫹ alence dipole, esul ing om he modi ica ion o he ield in he ionic VB s uc u e wi h espec o he mean ield. The co esponding ene gy co ec ion ep esen s he e - ec o he dynamical pola iza ion o he ligands unde he e ec o he alence cha ge luc ua ion in he single s a e. The ma ix elemen s may be isualized as in Diag am 9: Diag am 9 2736 J. Chem. Phys., Vol. 116, No. 7, 15 Feb ua y 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 09:45:28 o m as Eq. 共63兲, such as ep esen ed in Diag ams 21 and 22, espec i ely: Diag am 21 Diag am 22 These con ibu ions o 1h⫹2pand 2h⫹1pde e minan s a e nonaddi i e con ibu ions since mixed ou h-o de e ms also exis s, such as ep esen ed in Diag am 23: Diag am 23 Adding he 2h⫹1pand 1h⫹2pde e minan s o he DDCI2 lis leads o he ull DDCI space and has been shown in he ecen pas 26–28,60,61 o p o ide a sys ema ic ag eemen wi h expe imen , in all he sys ems s udied. The e ec o he 1h ⫹2pde e minan s goes h ough a modi ica ion o ab in ol - ing p oduc s o wo bielec onic in eg als, each o hem co - esponding o he in e ac ion o wo ansi ion dipoles. The bielec onic in eg al (ap⬘,hp)关 esp. (bp⬘,hp)兴is in ol ed in he dispe sion ene gy be ween he elec ons in a共 esp. b兲 and in h, which is p opo ional o (ap⬘,hp)2关 esp. (bp⬘,hp)2兴. The con ibu ion o he e ec i e hopping in e- g als no longe in ol es he quad a ic e ms, bu a c ossed p oduc (ap⬘,hp)(bp⬘,hp). This con ibu ion may he e- o e be conside ed o dispe si e o igin. A simila physical con en can be a ibu ed o he 2h⫹1pp ocesses, now in- ol ing al ansi ion dipoles. The e ec o 2h⫹1pand 1h⫹2pcan he e o e be con- side ed as a dispe si e con ibu ion o he kine ic exchange. In o he e ms, he pola izabili y o he spec a o elec ons o he ligand helps o ans e he ac i e elec ons om one magne ic si e o he o he . I is expec ed ha he mo e pola - izable he b idging-ligand, he la ge his e ec will be. 2. Nume ical esul s When compa ing DDCI2 and DDCI esul s in Table II i appea s ha he esul s a e signi ican ly imp o ed o e DDCI2. The 关Cu2Cl6兴2⫺complex is now weakly an i e o- magne ic, in acco dance wi h some expe imen al da a.62 Mo eo e , in all cases a easonable quan i a i e ag eemen wi h expe imen is ob ained now. When looking a he iso- la ed con ibu ion o 2h⫹1pand 1h⫹2pde e minan s, e- po ed in Table III, se e al ea u es can be commen ed. Table III shows ha he o e all e ec o hese de e mi- nan s in he sys ems s udied is sys ema ically an i e omag- ne ic. Addi ional calcula ions adding only he 2h⫹1po he 1h⫹2p o DDCI2 ha e been pe o med on he CuO2la ice agmen and on he coppe ace a e. They show ha : 共i兲 he 2h⫹1pcon ibu ion is la ge and an i e omag- ne ic 共leading o an exagge a ed alue o J⫽⫺1532 cm⫺1in he cup a e and ⫺284 cm⫺1in he ace a e兲; 共ii兲 he 1h⫹2pcon ibu ion is e omagne ic 共 educing J o ⫺563 cm⫺1in he cup a e and o ⫺62 cm⫺1in he ace a e兲; 共iii兲 hey in e ac a ou h o de , as p e iously no iced 共c . Diag am 23兲since he inal con ibu ion is no he sum o he sepa a e con ibu ions. The ole o he b idging-ligand pola izabili y is mani es . The ela i e e ec o he dispe si e con ibu ion o he o al alue o Jis much la ge o he azido and he ace a o ligands 共whe e hey mul iply Jby a ac o close o 3兲 han o he oxo b idge 共30% inc ease o J兲. The chlo ide case is excep- ional since he e ec changes he sign o an o e all e y small J. V. USE OF NATURAL ORBITALS Na u al o bi als ob ained om diagonaliza ion o he one-elec on densi y ma ix, calcula ed om he DDCI mos 2743J. Chem. Phys., Vol. 116, No. 7, 15 Feb ua y 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 09:45:28 exac wa e unc ion can be used o eanalyze he p oblem. To apply he di e ence dedica ed echnique, a common se o MOs is o be used o bo h s a es, and he e o e mean na u al o bi als ha e been calcula ed om he a e age o he single and iple s a es densi y ma ices. As a ionalized in Sec. IV o he p esen wo k, he delocaliza ion ails be ween he ligand and he me allic cen e s become la ge in he na u- al o bi als han in he ROHF ones, and he e o e he ab o e lap dis ibu ion is expec ed o ha e la ge ampli udes, especially on he b idging-ligand. The emaining pa ame e s should be modi ied in consequence: he di ec exchange, Kab , is expec ed o be enla ged as well as he hopping in e- g al, 兩 ab 兩 , and he on-cen e e ec i e epulsion, U, should be sligh ly educed. The esul s in Table V by compa ing wi h Table I show ha 共i兲Kab is d as ically inc eased, by a ac o 5 in 关Cu2Cl6兴2⫺and Cu2O7, by 20 in he ace a e, by 60 in he azido complex; 共ii兲 兩 ab 兩 is la ge , mul iplied by 2 in 关Cu2Cl6兴2⫺and in he ace a e, by 1.4 in Cu2O7, and by 3 in he azido complex; 共iii兲Uis educed by 20%–25% in all he sys ems. The esul ing alence-only CI emains un eliable. I is w ong in 关Cu2Cl6兴2⫺,⫹78 cm⫺1共inco ec sign兲, and weakly im- p o ed in Cu2O7,⫺452 cm⫺1共ins ead ⫺255 cm⫺1兲, mo e signi ican ly in he azido complex, ⫺253 ins ead o ⫺82 cm⫺1, and ⫺33 ins ead o ⫺19 cm⫺1in he ace a e. The e is no eason o hope ha he complex dynamical co ela ion e ec s can be kep h ough a me e e ision o he o bi als. I is in e es ing o epea he DDCI1, DDCI2, and DDCI calcu- la ions wi h hese o bi als. Compa ing Tables II and VI one sees ha he DDCI1 and DDCI2 esul s a e signi ican ly im- p o ed, i.e., close o he inal alue. The inal 共DDCI兲 e- sul s a e qui e compa able o hose ob ained om ROHF o bi als, bu in sligh ly be e ag eemen wi h expe imen . The 2h⫹1pand 1p⫹2hin oduced in DDCI ha e now a much weake e ec , as shown in Table VII 共⫹6cm ⫺1 ins ead ⫺30 cm⫺1 o ROHF o bi als in 关Cu2Cl6兴2⫺,⫺185 cm⫺1ins ead o ⫺333 cm⫺1in he Cu2O7 agmen , ⫺310 cm⫺1ins ead o ⫺427 cm⫺1in he azido complex and p ac- ically in a ian in he ace a e兲. In he azido complex hese e ec s now inc ease Jby 38% ins ead o 114% when using ROHF o bi als. Thei e ec s ill doubles he alue o J o he ace a e, while i mul iplies i by a ac o 3 when using ROHF o bi als. These wo las cases show ha i is no pos- sible o ely on he use o quasina u al o bi als o es on he DDCI2 le el o calcula ion, omi ing he 2h⫹1pand 1h ⫹2pexci a ions. One may ac ually obse e ha he coe i- cien s o he ligand o me al cha ge ans e con igu a ions a e no longe ze o a he 1h⫹1po he DDCI2 le el, c . Table IV, since he B illouin’s heo em is no sa is ied any- mo e. Howe e hese coe icien s a e dec eased when he 2h⫹1pand 1h⫹2pcon igu a ions a e in ol ed. This is op- posi e o he phenomenon obse ed when s a ing om ROHF o bi als. VI. CONCLUSIONS Recen wo ks ha e shown he possibili y o ob ain accu- a e alues o he magne ic coupling cons an h ough ab ini io CI calcula ions, especially when using he di e ence TABLE V. Con ibu ions o he coupling cons an , Ja he CASCI le el, using quasina u al o bi als. Di ec exchange, Kab , hopping in eg al, ab , on-si e epulsion, U. All esul s in cm⫺1. 关Cu2Cl6兴2⫺关Cu2(N3)2(NH3)6兴2⫹Cu2(CH3COO)4(H2O)2Cu2O7 2Kab 162 720 66 334 2 ab ⫺3528 ⫺12200 ⫺3992 ⫺11179 U(⫻10⫺4) 14.8 15.2 16.0 15.8 J⫹78 ⫺253 ⫺33 ⫺452 Jexp 0,⫺40a⬍⫺800b⫺286c,⫺294⫾4d⫺1032⫾48e ⫺1081⫾40 aRe e ence 62. dRe e ence 38. bRe e ence 63. eRe e ence 40. cRe e ence 64. Re e ence 41. TABLE VI. Coupling cons an , J共in cm⫺1兲, a a ious CI le els using quasina u al o bi als. 关Cu2Cl6兴2⫺关Cu2(N3)2(NH3)6兴2⫹Cu2(CH3COO)4(H2O)2Cu2O7 CASCI ⫹78 ⫺253 ⫺33 ⫺452 DDCI1 ⫺9⫺745 ⫺114 ⫺903 DDCI2 ⫺21 ⫺815 ⫺120 ⫺952 DDCI ⫺15 ⫺1125 ⫺238 ⫺1137 Jexp 0,⫺40a⬍⫺800b⫺286c,⫺294⫾4d⫺1032⫾48e ⫺1081⫾40 aRe e ence 62. dRe e ence 38. bRe e ence 63. eRe e ence 40. cRe e ence 64. Re e ence 41. 2744 J. Chem. Phys., Vol. 116, No. 7, 15 Feb ua y 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 09:45:28 dedica ed CI echnique. Howe e he esul essen ially con- sis ed in a numbe , and any physical analysis o he ac o s leading o he inal alue o Jwas lacking. This p e en ed any con on a ion wi h he quali a i e models, popula among he specialis s o he domain, and equen ly used as a ool in he design o new magne ic a chi ec u es. The p esen wo k is an a emp o o e come his di icul y and o show ha i is possible om accu a e calcula ions o analyze he physical e ec s con ibu ing o he obse able. While pe u - ba i e app oaches p o ide a na u al way o such analysis 共and we ac ually ha e ollowed his way o analy ic de i a- ion and quali a i e conside a ions兲, hey a e no quan i a- i ely eliable, since many pa ial se ies a e oo slowly con- e gen 共 o ins ance he dynamical pola iza ion o ionic VB s uc u es兲. We ha e combined localiza ion o he magne ic o bi als in o a om cen e ed magne ic o bi als and app op ia e pa i ions o he CI space, in o de o each he desi ed in o - ma ion. The mos ele an conclusions a e he ollowing: 共1兲The physics canno be kep es ic ed o he alence space, wi h he simple balance be ween he di ec 共K兲 and kine ic (⫺4 2/U) exchanges, wha e e he de ini- ion o he alence space 共mean ield a ia ional o na u- al magne ic o bi als兲, he ac ion o he exac Hamil- onian in his es ic ed space leads o alues o Jwhich a e one o de o magni ude smalle han expe imen 共and some imes e en o inco ec sign兲. 共2兲The spin pola iza ion is non-negligible, al hough is no he main e ec beyond he alence space. I s na u e 共 e o o an i e o兲is sys em-dependen . 共3兲The main e ec beyond he CAS and belonging o he DDCI2 lis , 共buil om second o de a gumen s兲comes om 1h⫹1pou e space de e minan s and is a ou h 共and highe 兲o de co ec ion. I consis s in he dynami- cal epola iza ion o he ionic VB s uc u es. 共4兲The p ocesses in ol ing wo inac i e holes o wo inac- i e pa icles ha e a much smalle e ec . 共5兲Beyond he DDCI2 space, which is no su icien o each a quan i a i e ag eemen wi h expe imen , one mus in ol e 2h⫹1pand 1h⫹2pexci a ions. Thei e - ec is la ge. I is no a dynamical pola iza ion o he ligand o me al cha ge ans e s a es, bu i p oceeds h ough a dynamical coupling o he ligand-me al ansi- ions dipole wi h ansi ion dipoles o he su ounding elec ons, and an inc ease o he e ec i e hopping in e- g al o dispe si e o igin. A his s age, i may be in e es ing o see whe he and how hese di e en physical e ec s a e included in al e na- i e ab ini io echniques. The NOCI me hod ce ainly does no include he spin pola iza ion e ec s since i wo ks wi h es ic ed open-shell SCF desc ip ions o each VB o m. Since i only in oduces a limi ed numbe o l→acha ge ans e s a es 共wi h hei speci ic h→p elaxa ions兲, selec ed on a he in ui i e conside a ions, i ce ainly lacks pa o he 2h⫹1pand 1h⫹2pe ec s, which ha e opposi e ends. The impo ance o he lacking con ibu ions may depend on he na u e o he b idging-ligands. The CASPT2 app oach, when s a ing om he alence CI space 共minimal CAS兲, in oduces he e ec s o all DDCI pe u be s and in p inciple does no miss any o he abo e conside ed e ec s. Howe e , i is a con ac ed scheme and does no e ise he composi ion o he alence pa o he wa e unc ion, namely he ionic/neu al a io in he single s a e. As shown in pape II,58 he dynamical co ela ion e - ec s d ama ically inc ease his a io 共mul iplied by a ac o be ween 2 and 5兲 h ough highe o de e ec s, which a e inco po a ed in a a ia ional ea men . Using con ac ed schemes esul s in an unde es ima ion o he pe u ba ion, especially when pola izable ligands a e p esen in he mo- lecula s uc u e. In hese cases, he ecipe consis s in enla g- ing he CAS o include in his way pa o he highe o de e ec s. A de ailed discussion compa ing CASPT2 and DDCI me hods will be gi en elsewhe e. This pape has pe o med he abo e analysis wi h wo a he di e en se s o o bi als, he Ha ee–Fock ones, and quasina u al MOs. The magne ic o bi als a e signi ican ly mo e delocalized on he ligands in he second se , as phe- nomenologically obse ed,57 and a ionalized he e. This e - ec leads o la ge ze o-o de alues o he di ec exchange, K, and he hopping in eg al . The DDCI2 alues become mo e accu a e when na u al o bi als a e used, bu he 2h ⫹1pand 1h⫹2pexci a ions s ill ha e a non-negligible e - ec on he inal alue o J. The p esen analysis o he physical e ec s ha con ib- u e o he magne ic coupling may seem qui e complex. A TABLE VII. Con ibu ions o he coupling cons an , J共in cm⫺1兲, using na u al o bi als. 关Cu2Cl6兴2⫺关Cu2(N3)2(NH3)6兴2⫹Cu2(CH3COO)4(H2O)2Cu2O7 Di ec exchange 162 720 66 334 Kine ic exchange ⫺84 ⫺973 ⫺99 ⫺786 1h⫹1p⫺87 ⫺492 ⫺81 ⫺451 2h,2p⫺12 ⫺70 ⫺6⫺49 2h⫹1p,1h⫹2p6⫺310 ⫺118 ⫺185 To al ⫺15 ⫺1125 ⫺238 ⫺1137 Jexp 0,⫺40a⬍⫺800b⫺286c,⫺294⫾4d⫺1032⫾48e ⫺1081⫾40 aRe e ence 62. dRe e ence 38. bRe e ence 63. eRe e ence 40. cRe e ence 64. Re e ence 41. 2745J. Chem. Phys., Vol. 116, No. 7, 15 Feb ua y 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 09:45:28 his s age i seems o in alida e he models ha essen ially s ay wi hin he alence-only space. Howe e he nex pape 58 shows ha he applica ion o he e ec i e Hamil onian heo y, makes i possible o e u n o a simpli ied pic u e, i.e., o he alence model space. The use o e ec i e ene gies and in e ac ions modi ied in o de o inco po a e he complex and massi e e ec s o he ou e space exci a ions allows us o e u n close o quali a i e pic u es. ACKNOWLEDGMENTS These wo wo ks ha e been s imula ed by he lec u es gi en 共in pa icula by M. Ve dague 兲in he ESF Tu o ial on Theo e ical Aspec s o Molecula Magne ism, Vienna, No embe 2000. The au ho s a e indeb ed o C. de G aa o many commen s and sugges ions. They hank he Spanish- F ench Scien i ic Coope a ion 共In eg a ed Ac ion HF2000- 0030兲. J.C. and R.C. wan o hank he DGICYT o he Min- is e io de Educacio ´n y Cul u a o Spain 共P ojec No. PB98- 1216-CO2-02兲and he CIRIT o he Gene ali a de Ca alunya 共G an No. SGR99-182兲 o hei inancial sup- po . C.J.C. acknowledges he inancial suppo h ough he TMR ac i i y ‘‘Ma ie Cu ie esea ch aining g an s’’ G an No. HPMF-CT-1999-00285 o he Eu opean Commission. The Labo a oi e de Physique Quan ique is Uni e ´Mix e de Reche che 共UMR 5626兲du CNRS. 1共a兲M. Ve dague , A. 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