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
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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.
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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
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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.
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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
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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.
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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
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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.
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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
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la ions in Exchange Coupled Sys ems, edi ed by R. D. Wille , D. Ga es-
chi, and O. Khan, NATO Ad anced S udies Se ies, C 共Reidel, Do d ech ,
1985兲, Vol. 140, p. 87.
34J. Mi alles, O. Cas ell, R. Caballol, and J. P. Mal ieu, Chem. Phys. 172,33
共1993兲.
35C. J. Calzado, J. Cab e o, J. P. Mal ieu, and R. Caballol, J. Chem. Phys.
116, 2728 共2002兲.
36P. Du and, J. Mol. S uc .: THEOCHEM 120, 443 共1985兲.
37P. Du and and J. P. Mal ieu, in Ab ini io Me hods in Quan um Chemis y I,
edi ed by K. P. Lawley 共Wiley, New Yo k, 1987兲, p. 320.
38C. Bloch, Nucl. Phys. 6, 329 共1958兲.
39See, o ins ance, W. H. P ess, B. P. Flanne y, S. A. Teukolsky, and W. T.
Ve e ling, Nume ical Recipes 共Camb idge Uni e si y P ess, Camb idge,
1986兲.
40J. P. Mal ieu, P. Du and, and J. P. Daudey, J. Phys. A 18, 809 共1985兲.
41P. E. Sulewski, P.A. Fleu y, K. B. Lyons, S. W. Cheong, and Z. Fisk, Phys.
Re . B 41,225共1990兲; R. P. Singh, P. A. Fleu y, K. B. Lyons, and P. C.
Sulewski, Phys. Re . Le . 62, 2736 共1989兲.
42G. Aeppli, S. M. Hayden, H. A. Mook, Z. Fisk, S. W. Cheong, D. Ry z, J.
P. Remeika, G. P. Espinosa, and A. S. Coope , Phys. Re . Le . 62, 2052
共1989兲; Y. Endoh, K. Yamada, R. J. Bi geneau e al., Phys. Re . B 37,
7443 共1988兲; S. M. Hayden, G. Aeppli, R. Osbo n, A. D. Taylon, T. G.
Pe ing, S. W. Cheong, and Z. Fisk, Phys. Re . Le . 67, 3622 共1991兲.
43R. D. Wille , in Magne o-S uc u al Co ela ion in Exchange Coupled
Sys ems, edi ed by R. D. Wille , D. Ga eschi, and O. Khan, NATO Ad-
anced S udies Se ies C 共Reidel, Do d ech , 1985兲, Vol. 140. p. 389; G.
Maass, B. Ge s ein, and R. D. Wille , J. Chem. Phys. 46, 401 共1967兲;G.
O’Bannon and R. D. Wille , Ino g. Chim. Ac a 53, 6131 共1983兲.
44P. Chaudhu i, K. Ode , K. Wiegha d , B. Nube , and J. Weiss, Ino g.
Chem. 25, 2818 共1986兲.
45B. N. Figgis and R. L. Ma in, J. Chem. Soc. 3837 共1956兲.
46H. U. Gu
¨del, A. S eble , and A. Fu e , Ino g. Chem. 18, 1021 共1979兲.
47K. Ande sson, M. R. A. Blombe g, M. P. Fu
¨lsche e al.,MOLCAS e sion
4, Lund Uni e si y, Sweden 共1997兲.
48CASDI p og am, N. Ben Amo and D. Maynau, Chem. Phys. Le . 286,211
共1998兲.
49NATURAL p og am, V. M. Ga cı
´a and O. Cas ell 共1997兲.
50M. J. F isch, G. W. T ucks, H. B. Schlege e al.,GAUSSIAN 98, Re ision
A.3, Gaussian, Inc., Pi sbu gh, PA, 1998.
51A. D. Becke, J. Chem. Phys. 98,5648共1993兲.
52P. J. Hay and W. R. Wad , J. Chem. Phys. 82,299共1985兲.
53A. D. McLean and G. S. Chandle , J. Chem. Phys. 72, 5639 共1980兲;R.
K ishnan, J. S. Binkley, R. Seege , and J. A. Pople, ibid. 72,650共1980兲.
54R. Di ch ield, W. J. Heh e, and J. A. Pople, J. Chem. Phys. 54,724共1971兲;
W. J. Heh e, R. Di ch ield, and J. A. Pople, ibid. 56,2257共1972兲.
55W. S e ens, H. Basch, and M. K auss, J. Chem. Phys. 81, 6026 共1984兲;W.
S e ens, M. K auss, H. Basch, and P. G. Jasieu, Can. J. Chem. 70,612
共1992兲.
56L. Noodleman and J. G. No man, J . J. Chem. Phys. 70,4903共1979兲;L.
Noodleman, ibid. 74, 5737 共1981兲; L. Noodleman and E. R. Da idson,
Chem. Phys. 109, 131 共1986兲; L. Noodleman, C. Y. Peng, D. A. Case, and
J. M. Mouesca, Coo d. Chem. Re . 144,199共1995兲.
57J. des Cloizeaux, Nucl. Phys. 20,321共1960兲.
58J. H. Van Vleck, Phys. Re . 33, 467 共1929兲.
59B. H. B andow, In . J. Quan um Chem. 15, 207 共1979兲.
60This ema k only conce ns he ou -dimensional model space, bu i does
no in alida e he expansion om he neu al VB de e minan s 共 wo-
dimensional s ic ly degene a e model space兲which has been success ully
employed in he pas 关Re . 32; P. De Lo h, P. Ka a iloglou, J. P. Daudey,
and O. Kahn, J. Am. Chem. Soc. 110, 5676 共1988兲; P. De Lo h, J. P.
Daudey, H. As heime , L. Walz, and W. Haase, J. Chem. Phys. 82,5048
共1985兲兴.
61The usual QDPT expansion when con e ged leads o he Bloch e ec i e
Hamil onian.
62J. Cab e o, C. J. Calzado, D. Maynau, R. Caballol, and J. P. Mal ieu,
submi ed.
63R. Caballol, O. Cas ell, F. Illas, I. de P. R. Mo ei a, and J. P. Mal ieu, J.
Phys. Chem. A 101, 7860 共1997兲.
64C. Adamo, V. Ba one, A. Bencini, F. To i, and I. Cio ini, Ino g. Chem. 38,
1996 共1999兲.
65J. F. Sanz, C. J. Calzado, and A. Ma
´ quez, In . J. Quan um Chem. 76,458
共2000兲.
66C. J. Calzado and J. P. Mal ieu, Chem. Phys. Le . 317,404共2000兲.
67R. L. Ma in and F. Illas, Phys. Re . Le . 79, 1539 共1997兲; F. Illas and R.
L. Ma in, J. Chem. Phys. 108, 2519 共1998兲.
4000 J. Chem. Phys., Vol. 116, No. 10, 8 Ma ch 2002 Calzado
e al.
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