Accu a e ab ini io de e mina ion o magne ic in e ac ions and hopping in eg als in La
2−x S x CuO 4 sys ems
Ca men J. Calzado, Ja ie F. Sanz, and Jean Paul Mal ieu
Ci a ion: The Jou nal o Chemical Physics 112, 5158 (2000); doi: 10.1063/1.481093
View online: h p://dx.doi.o g/10.1063/1.481093
View Table o Con en s: h p://sci a ion.aip.o g/con en /aip/jou nal/jcp/112/11? e =pd co
Published by he AIP Publishing
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Accu a e
ab ini io
de e mina ion o magne ic in e ac ions
and hopping in eg als in La2À
x
S
x
CuO4sys ems
Ca men J. Calzadoa)
Depa amen o de Quimica Fisica, Uni e sidad de Se illa, E-41012 Se illa, Spain and Labo a oi e de
Physique Quan ique, IRSAMC, Uni esi e
´Paul Saba ie , F-31062 Toulouse Cedex, F ance
Ja ie F. Sanz
Depa amen o de Quimica Fisica, Uni e sidad de Se illa, E-41012 Se illa, Spain
Jean Paul Mal ieu
Labo a oi e de Physique Quan ique, IRSMAC, Uni e si e
´Paul Saba ie , F-31062 Toulouse Cedex, F ance
共Recei ed 13 July 1999; accep ed 14 Decembe 1999兲
The na u e o magne ic in e ac ions and elec on ans e p ocesses in La2⫺xS xCuO4sys ems a e
s udied, by means o an ab ini io embedded clus e model app oach, using a di e ence dedica ed
con igu a ion in e ac ion 共DDCI兲p ocedu e. Fo he undoped sys em, he c ucial ole played by he
ligand o me al cha ge ans e 共LMCT兲con igu a ions in he magne ic p ocess makes necessa y he
use o an enla ged DDCI space, which explici ly akes accoun o he elaxa ion o hese LMCT
con igu a ions. This p ocedu e allows us o app oach he expe imen al magne ic coupling cons an
alue. In hole-doped sys ems, he alue ob ained o he elec on ans e in eg al, , is o 0.54–0.57
eV. The ex a hole, cha ac e ized om he na u e and occupa ion o di e en ial na u al o bi als,
has a s ong pcha ac e 共app oxima ely 50%兲and is essen ially localized in CuO2planes. These
esul s a e in ag eemen wi h he expe imen al e idence abou hese kinds o compounds. Nei he
he alue o no he na u e o he ex a hole a e se iously a ec ed by he op imiza ion o he
o bi als used in he CI expansion. This sugges s ha a –Je ec i e Hamil onian is an adequa e
model o s udy he elec onic p ope ies o hese sys ems. © 2000 Ame ican Ins i u e o Physics.
关S0021-9606共00兲30210-0兴
I. INTRODUCTION
Al hough supe conduc i i y phenomena was de ec ed in
1911 by Kame lingh Onnes,1 he disco e y o he new high-
empe a u e supe conduc ing ma e ials by Bedno z and
Mu
¨lle 2in 1986 ga e a new impe us o he ield o supe con-
duc i i y, s imula ing nume ous expe imen al and heo e ical
wo ks.
The p esence o coppe oxide planes wi h a uni cell
CuO2is a common ea u e o he high c i ical empe a u e
共Tc兲supe conduc o s and i is belie ed ha supe conduc i -
i y is closely ela ed wi h p ocesses which ake place wi hin
hese planes. These ma e ials a e also cha ac e ized by he
p esence o mixed alency, i.e., he supe conduc i i y is in-
duced by doping wi h holes o elec ons, in such a way ha
he o mal coppe oxida ion s a e is no wo.3
Unde s anding he beha io o hese s ongly co ela ed
ma e ials equi es he use o model Hamil onians, which ex-
plici ly ea he la ge Coulomb in e ac ions. Mean- ield ap-
p oxima ions, as local densi y app oxima ion 共LDA兲calcula-
ions, ex emely success ul a desc ibing many c ys alline
solids, a e no ele an o such ma e ials. Fo ins ance, se -
e al LDA calcula ions ha e ailed o e eal he an i e omag-
ne ic na u e o La2CuO4.4
Ande son5was he i s o p opose a single d-band e ec-
i e Hamil onian as an adequa e model o analyze he elec-
onic s uc u e o cup a es. Howe e , he con o e sy was
opened when di e en expe imen s showed he s ong oxy-
gen cha ac e o he ex a hole in doped cup a es and he
absence o e idences o mixed- alen coppe a oms.6These
esul s ha e sugges ed he necessi y o in oduce he 2poxy-
gen band in he model Hamil onian. These a e he g ounds o
he h ee-band model o Eme y,7which pe mi s one o suc-
cess ully in e p e expe imen al da a, such as x- ay pho o-
emission spec a 共XPS兲and elec on ene gy loss spec a
共EELS兲spec a, especially in he high-ene gy limi .8A dis-
ad an age is ha h ee-band Hamil onians equi e mo e pa-
ame e s han a e accessible om expe imen s.
The –Jmodel o Zhang and Rice9 ies o keep he
ad an ages o bo h one- and h ee-band Hubba d models. I
consis s o a single-band e ec i e Hamil onian, which im-
plici ly accoun s o Cu 3dandO2phyb idiza ion, wi h
simila esul s as a h ee-band model in he low-ene gy po -
ion o he spec um.10,11 Indeed, Hybe sen e al.12 ha e
shown ha he –Jwa e unc ions ha e an o e lap la ge
han 90% wi h he eigen unc ions o a h ee-band Hamil-
onian. The –JHamil onian only con ains wo pa ame e s:
H⫽⫺ 兺
具
ij
典
␴
关ci
␴
⫹共1⫺ni⫺
␴
兲共1⫺nj⫺
␴
兲cj
␴
⫹h.c.兴
⫹J兺
具
ij
典
共SiSj⫺1
4ninj兲,共1兲
whe e e e s o he hopping in eg al be ween adjacen iand
jcen e s and Jis he e ec i e magne ic coupling be ween
a兲Au ho o whom co espondence should be add essed. Elec onic mail:
[email p o ec ed]
JOURNAL OF CHEMICAL PHYSICS VOLUME 112, NUMBER 11 15 MARCH 2000
51580021-9606/2000/112(11)/5158/10/$17.00 © 2000 Ame ican Ins i u e o Physics
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hem. The p esen a icle, which de elops he con en o a
ecen ly epo ed le e ,13 shows ha hese pa ame e s can be
accu a ely ob ained om high le el ab ini io quan um
chemical calcula ions. The app oach goes h ough he con-
side a ion o a ini e clus e , in ol ing a leas wo Cu a oms
and he neighbo oxygen a oms, p ope ly embedded in he
ield o he o he la ice ions. Ab ini io CI echniques may be
applied o such a ini e clus e . Fi s he undoped sys ems a e
conside ed, in which each coppe a om bea s an unpai ed
elec on, and we de e mine he co esponding magne ic o -
bi als by he SCF calcula ion o he uppe mul iple . In o de
o de e mine he e ec i e magne ic coupling cons an be-
ween he wo unpai ed elec ons, he ene gy di e ence be-
ween he lowes single and iple s a es mus be accu a ely
calcula ed. The de e mina ion o he exci a ion ene gies may
ollow he logic o he so-called di e ence-dedica ed con-
igu a ion in e ac ion 共DDCI兲,14 which gene a es he lis o
de e minan s con ibu ing o he second-o de o pe u ba ion
o he exci a ion ene gy. Fo a sys em wi h wo-elec on in
wo o bi als 共aand b兲in he magne ic limi 共weak delocal-
iza ion兲, he model space would consis o he wo neu al
alence-bond de e minan s,
兩
ab
¯
兩
and
兩
ba
¯
兩
, and he co e-
sponding lis 共DDCI-2 lis 兲in ol es de e minan s p esen ing
up o wo inac i e o bi als only 共i.e., one hole and one pa -
icle, o wo holes and wo pa icles兲. P e ious applica ions
o ha s a egy o La2CuO4sys em had only eco e ed a
alue o he magne ic coupling o ⫺95 meV,15 while he
expe imen al alue is close o ⫺135 meV.16,17 The p esen
wo k eexamines he physics o his p oblem and shows he
necessi y o enla ge he lis o de e minan s in o de o allow
he dynamical pola iza ion o he ligand→me al cha ge
ans e con igu a ions, LMCT, o ake place. The enla ged
CI spaces p oduce accu a e alues o he coupling cons an
共⫺140 meV兲. The sugges ion has led o a ecen calcula ion
o he magne ic coupling in pe o ski es o Ni and Cu,18
which happens o be ex emely success ul.
Then he hole-doped analog 共i.e., he La2⫺xS xCuO sys-
em兲is conside ed. In o de o de e mine he hopping in e-
g al , which delocalizes he hole共s兲on he la ice, we ha e
conside ed he same clus e s as be o e (Cu2O7and Cu2O11)
bu wi h one elec on less, and de e mined he ene gy di e -
ence be ween he lowes double s a es o gand usymme y,
he co esponding ene gy di e ence being conside ed as
wice he hopping in eg al in he –Jmodel. The calcula ion
o he ene gy di e ence ollows he same logic as abo e, i.e.,
is based on a di e ence dedica ed CI. The model space is he
minimal one 共one elec on in wo ac i e molecula o bi als兲,
and he CI lis again includes all de e minan s con ibu ing o
he ene gy di e ence a he second o de o pe u ba ion.19
The calcula ion has been pe o med i s using he molecula
o bi als o he iple s a e o he undoped clus e , in ol ing
holes which a e, a he ze o-o de le el, essen ially localized
on he Cu a oms. The CI expansion shows an impo an de-
localiza ion o he holes on he oxygen a oms 共abou hal Cu
and hal O兲, bu al eady p o ides a easonable alue o he
hopping in eg al 共0.54–0.57 eV兲. Then he ac i e space, i.e.,
he gand uo bi als a e op imized by de e mina ion o ap-
p oxima ed na u al o bi als. This elaxa ion ac ually esul s
in a s ong mixing o he 3dCu o bi als wi h he 2po bi als
o he su ounding o bi als, which become sligh ly dominan
in he hole. Bu he esul ing alue o , calcula ed om hese
op imized o bi als, emains he same. This con i ms he a-
lidi y o he Zhang–Rice analysis and he possibili y o map-
ping he local in e ac ions in an e ec i e Hamil onian o –J
cha ac e , spanned by he 3dx2⫺y2Cu o bi als wi h app o-
p ia e delocaliza ion ails, as discussed in he inal sec ion.
II. CLUSTER MODELS AND ENVIRONMENT
REPRESENTATION: COMPUTATIONAL DETAILS
La2⫺xS xCuO4has a pe o ski e-like s uc u e and c ys-
allizes in a body-cen e ed e agonal s uc u e. I consis s o
CuO2laye s sepa a ed by wo LaO planes, which o m he
cha ge ese oi . Cu a oms a e su ounded by co ne -sha ing
oxygen oc ahed a.20 In he undoped ma e ial, neighbo ing
oc ahed a along a 共110兲di ec ion a e sligh ly il ed in oppo-
si e di ec ions wi h espec o he c-axis. Howe e , his dis-
o ion will no be aken in o accoun o model he c ys al,
and he undis o ed La1.85S 0.15CuO4s uc u e will be
conside ed,21 expec ing ha o ho hombic dis o ions do no
ha e signi ican e ec s on he and J alues. Fo he de e -
mina ion o hese alues and he p ope ies o doped and
undoped sys ems, bime allic clus e s we e used o model he
sys em, including he nea es -neighbo oxygens, in such a
way ha hey cons i u e wo- and h ee-dimensional clus e s
(Cu2O7and Cu2O11, espec i ely, Fig. 1兲. The minimum
clus e ep esen a ion, Cu2O, is an inadequa e model in bo h
cases, and he co esponding esul s ob ained o Jand we e
no included in his wo k. The clus e s a e su ounded by an
a ay o poin cha ges, in o de o p o ide an adequa e ep-
esen a ion o he c ys al Madelung po en ial. The cha ge
alues a e ixed acco ding o he me hod o E jen.22 In he
FIG. 1. Two- and h ee-dimensional clus e s used o model he
La2⫺xS xCuO4sys em.
5159
J. Chem. Phys., Vol. 112, No. 11, 15 Ma ch 2000 Magne ic in e ac ion in La2⫺
x
S
x
CuO4
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clus e -poin cha ge in e ace, some o al ion po en ials
共TIPs兲a e included o ep esen he Cu⫹2and La⫹3ca ions
nea es o he clus e a oms. They p e en an a i icial pola -
iza ion o he clus e bounda y oxygen elec ons owa ds he
posi i e poin cha ges. P e ious heo e ical s udies o
Ma in23 and Wang e al.24 ha e shown he impo ance o
including bo h poin cha ges and e ec i e po en ials o co -
ec ly desc ibe he occupa ion o he 3do bi als, he ene gy
o he Cu 4sand 4po bi als, and e en he Cu–O bond
leng hs in geome y op imiza ion calcula ions.
TIPs a e ep esen ed by la ge co e Hay–Wad e ec i e
co e po en ials25 共ECPs兲wi h o mal cha ges 共⫹2 and ⫹3 o
Cu and La, espec i ely兲. Fo me allic a oms, he ECPs p o-
posed by S e ens, Basch, and K aus26 a e used o ep esen
he en inne mos elec ons. Cu alence elec ons a e de-
sc ibed using a (8s8p6d)/关4s4p3d兴basis se . Fo he cen-
al oxygen a om o he clus e , an all-elec on basis se is
employed. P elimina y calcula ions es ing he in luence o
basis unc ion quali y ha e shown he necessi y o include
pola iza ion unc ions, a leas in he cen al oxygen a om, in
o de o ob ain he co ec ela i e s abiliza ion o single and
iple s a es in he undoped sys ems. The basis se inally
used o his a om is (10s5d1d) con ac ed o 关3s2p1d兴.27
Fo he es o he oxygen a oms, which do no con ibu e
di ec ly o he spin and elec on ans e mechanisms, a
(6s6p) con ac ed o 关2s2p兴basis se is used o he a-
lence elec ons, while he co e 1sones a e desc ibed by an
ECP.28 We ha e checked ha he pola izabili y o he oxy-
gen dianions O⫺2in he la ice, calcula ed wi h his 2s2p
basis is 1.3 Å3, in good ag eemen wi h he expe imen al da a
共0.5–3.2 Å3兲共Re . 29兲 o oxygen ions in me allic oxides.
As a s a ing molecula o bi al se , he es ic ed open
shell Ha ee–Fock 共ROHF兲wa e unc ion o he iple s a e
o he undoped sys em is used. In he D2hsymme y poin
g oup, his s a e is o B2usymme y and can be w i en as
3
␾
u⫽
兩
co egu
兩
. The wo ac i e o bi als gand ua e essen-
ially he symme ic (a1g) and an isymme ic (b2u) combi-
na ion o he dx2⫺y2o bi als o each o he wo Cu clus e
a oms 共 espec i ely, 93% and 90% din cha ac e 兲. As is well
known, and as p e iously no iced by Ma in23 in he case o
La2CuO4, Ha ee–Fock app oxima ion p oduces an o e es-
ima ion o he on-si e Coulomb epulsion (Ud) and an ex-
cessi ely ionic ep esen a ion. We will show below ha con-
igu a ion in e ac ion can co ec his de iciency and eco e
he p ope me al–ligand co alency.
III. UNDOPED SYSTEMS: MAGNETIC COUPLING
CONSTANT DETERMINATION
The La2CuO4is an an i e omagne ic insula o wi h an
exchange cons an expe imen ally de e mined by means o
neu on sca e ing 关J⫽⫺134⫾5 meV兴共Re . 16兲and Raman
measu emen s 关J⫽⫺128⫾6 meV兴.17 In he undoped sys-
em, each Cu⫹2ca ion con ains an unpai ed elec on essen-
ially localized in a dx2⫺y2o bi al, acco ding o he e ag-
onal ligand ield. Each Cu⫹2ion can be conside ed as a
magne ic pa icle wi h spin S⫽1/2. The Heisenbe g–Di ac–
Van Vleck Hamil onian30 is he mos used model o deal
wi h he in e ac ion be ween magne ic pa icles. Fo wo pa -
icles ha ing o al spin S, his Hamil onian has he ollowing
exp ession:
H
ˆ⫽⫺JS
ˆ1S
ˆ2⫽⫺J关1
2共S
ˆ1
⫹S
ˆ2
⫺⫹S
ˆ1
⫺S
ˆ2
⫹兲⫹S
ˆz1S
ˆz2兴,共2兲
whe e Jis he magne ic coupling cons an , Siis he o al spin
ope a o o cen e i,Si
⫹and Si
⫺a e he spin-up and spin-
down ope a o s o pa icle i, and Szi is he ope a o o he z
componen o spin o pa icle i. The eigen alues o he
Heisenbe g Hamil onian a e exp essed as a unc ion o J,in
such a way ha he di e ence be ween he lowes single and
iple eigen alues gi es he alue o J:
ES⫺ET⫽J.共3兲
Fo an i e omagne ic sys ems, he single is he g ound s a e
and Jis nega i e. The p oblem lies in he accu a e de e mi-
na ion o he ene gy alues o hese wo s a es. Fo a wo-
cen e p oblem, ep esen ed by he localized dx2⫺y2mag-
ne ic o bi al, aand b, hese wo s a es can be exp essed a a
ze o-o de le el as
⌿T
0⫽1
&共
兩
ab
¯
兩
⫺
兩
ba
¯
兩
兲,⌿S
0⫽1
&共
兩
ab
¯
兩
⫹
兩
ba
¯
兩
兲.共4兲
Using delocalized magne ic o bi als g⫽(a⫹b)/&and u
⫽(a⫺b)/& hese s a es a e
⌿T
0⫽1
&共
兩
gu
¯
兩
⫺
兩
ug
¯
兩
兲,⌿S
0⫽1
&共
兩
gg
¯
兩
⫺
兩
uu
¯
兩
兲.共5兲
As has been p e iously es ablished,14,31 he
␾
1⫽
兩
ab
¯
兩
and
␾
2⫽
兩
ba
¯
兩
de e minan s cons i u e a model space, o e which
i is possible o build an e ec i e 2⫻2 Hamil onian. A he
second o de o he quasidegene a e pe u ba ion heo y
共QDPT兲, he o -diagonal elemen He o he ma ix ep e-
sen a ion o his Hamil onian is equal o14,31
具
␾
1
兩
H
ˆe 共2兲
兩
␾
2
典
⫽
具
␾
1
兩
H
ˆ
兩
␾
2
典
⫹兺
␾
␣
苸S
具
␾
1
兩
H
ˆ
兩
␾
␣
典具
␾
␣
兩
H
ˆ
兩
␾
2
典
E1
0⫺E
␣
0
⫽J/2, 共6兲
whe e he de e minan s
兩
␾
␣
典
in e ac simul aneously wi h
bo h o he wo model space de e minan s, and hey a e he
only ones which con ibu e o he ene gy di e ences ES
⫺ETa his le el. The con igu a ion in e ac ion ma ix buil
o e his space is called DDCI-2.14 DDCI s ands o di e -
ence dedica ed con igu a ion in e ac ion and he ‘‘2’’ indi-
ca es ha his space includes exci a ions up o wo deg ees o
eedom, i.e., consis s o all he de e minan s buil om he
model space as exci a ions in ol ing wo inac i e holes o
wo inac i e pa icles, o one inac i e hole and one inac i e
pa icle. As p e iously desc ibed,14,32 he DDCI space is in-
a ian unde he uni a y ans o ma ion o he ac i e o bi -
als, and i is possible o gene a e he di e en ial space in he
o iginal symme y o he clus e , ha is, using as ac i e o -
bi als he delocalized ones and Eq. 共5兲as ze o-o de desc ip-
ion o single and iple s a es. Bo h Cu2O7and Cu2O11
belong o he D2hsymme y poin g oup, and single and
iple a e o A1gand B2usymme ies, espec i ely.
The diagonaliza ion o he DDCI-2 ma ices gi es he co ec
5160 J. Chem. Phys., Vol. 112, No. 11, 15 Ma ch 2000 Calzado, Sanz, and Mal ieu
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sign o he Jcons an and oughly 80% o i s alue, as has
been p e iously epo ed by Casano as e al.15 o La2CuO4
sys em and also compu ed in his wo k 共Table I兲.Asis
shown in his able, he alues ob ained o he 2D and 3D
clus e s a e p ac ically iden ical, in ag eemen wi h he ex-
pe imen ally es ablished wo-dimensional na u e o magne-
ism in his sys em.16,17 Since he di e ence be ween expe i-
men al and heo e ical alues canno be associa ed o
collec i e e ec s, as Illas e al.33 ha e ecen ly demons a ed,
his unde es ima ion has o be ela ed o an inadequa e ea -
men o elec onic co ela ion e ec s.
Ac ually, he alue o he an i e omagne ic in e ac ion
is qui e la ge, much la ge han in many binuclea sys ems
o which DDCI-2 appea ed o be o su icien accu acy.31
We hough ha he disc epancy in he p esen case migh
come om he la ge coe icien 共⬇0.1, see below兲o he
ionic alence–bond s uc u e in he single s a e, which
should be conside ed as belonging o he model space, now
enla ged o in ol e beside
␾
1and
␾
2,
␾
3⫽
兩
aa
¯
兩
and
␾
4
⫽
兩
bb
¯
兩
.
␾
1and
␾
2only di e by one o bi al om
␾
3and
␾
4
and in such a case and acco ding o Sla e ’s ules, he di e -
ence dedica ed CI space has o be enla ged up o a DDCI-3
lis , in ol ing wo inac i e holes and one inac i e pa icle
(2h–1p) and one inac i e hole and wo inac i e pa icles
(1h–2p) con igu a ions. This lis inc eases he size o he CI
ma ix by a ac o o 50–60, bu as shown in Table I, gi es a
heo e ical es ima e which p ac ically ma ches he expe i-
men al alue. In Table II he analyses o he single and
iple wa e unc ions a e p esen ed, bo h o DDCI-2 and
DDCI-3 spaces. Toge he wi h he comple e ac i e space
共CAS兲de e minan s, o he con ibu ions a e epo ed, co e-
sponding o ligand o me al cha ge ans e con igu a ions,
LMCT, whe e single exci a ions om sp ligand-like o bi als
o he gand uac i e o bi als ake place 共schema ically ep-
esen ed by Lg→dand Lu→d, whe e Lgand Lu ep esen ,
espec i ely, symme ic and an isymme ic combina ions o s
and pligand basis unc ions, wi h mino con ibu ions o he
d-me al unc ions, scheme I兲.
SCHEME I.
Two main e ec s can be dis inguished when DDCI-3 space
is used:
共i兲 he inc ease o he coe
兩
gg
¯
兩
/coe
兩
uu
¯
兩
a io in he sin-
gle s a e,
共ii兲 he appea ance o con ibu ions ex e nal o he CAS,
bo h o single and iple s a es.
The o me co esponds o an inc ease o he weigh o ionic
alence–bond de e minan s,
␾
3and
␾
4in he single de-
sc ip ion. The CAS p ojec o PCAS is de ined as
PCAS⫽兺
i苸CAS
兩
␾
i
典具
␾
i
兩
.共7兲
Ope a ing o e
兩
⌿S
典
and
兩
⌿T
典
, we ob ain
PCAS
兩
⌿s
典
⫽
␣
兩
gg
¯
兩
⫺
␤
兩
uu
¯
兩
,
共8兲
PCAS
兩
⌿T
典
⫽
␥
关
兩
gu
¯
兩
⫺
兩
ug
¯
兩
兴.
Assuming ha gand ua e composed only by he symme ic
and an isymme ic linea combina ion o he localized dx2
⫺y2o bi als, Eq. 共8兲 esul as
PCAS
兩
⌿s
典
⫽
␣
⫹
␤
&
冉
兩
ab
¯
兩
⫹
兩
ba
¯
兩
&
冊
⫹
␣
⫺
␤
&
冉
兩
aa
¯
兩
⫹
兩
bb
¯
兩
&
冊
,
共9兲
TABLE I. Exchange coupling cons an alues J, meV, ob ained wi h di e en DDCI spaces.
Clus e Space n°de (T,S) O bi al se J共meV兲
Cu2O7
DDCI-2 7680,7687 HF ⫺96
IDDCI ⫺103
DDCI-2⫹Single 兵LMCT其12 454,12 461 HF ⫺135
DDCI-3 411 084,411 091 HF ⫺138
IDDCI ⫺142
Cu2O11
DDCI-2 11 772,11 779 HF ⫺99
IDDCI ⫺111
DDCI-2⫹Single 兵LMCT其22 586,22 593 HF ⫺131
DDCI-3 804 828,804 835 HF ⫺139
IDDCI ⫺145
Expe imen al alues: ⫺128⫾6meV
a,⫺134⫾5meV
b
aRe e ence 17.
bRe e ence 16.
TABLE II. Single and iple wa e unc ions o DDCI-2 and DDCI-3
spaces in he canonical MO basis se . ROHF MO as ze o-o de desc ip ion.
Only he con ibu ions equal o la ge han 0.05 a e epo ed.
Clus e Con .
DDCI-2 DDCI-3
TSTS
Cu2O7gg
¯
¯0.7703 ¯0.7380
uu
¯
¯⫺0.6275 ¯⫺0.5717
gu
¯
⫺0.7037 ¯⫺0.6629 ¯
ug
¯
0.7037 ¯0.6629 ¯
Lu→d¯¯0.1152 0.1393
Lg→d¯¯0.0933 0.0721
Cu2O11 gg
¯
¯0.7723 ¯0.7379
uu
¯
¯⫺0.6246 ¯⫺0.5688
gu
¯
⫺0.7037 ¯⫺0.6616 ¯
ug
¯
0.7037 ¯0.6616 ¯
Lu→d¯¯0.1002 0.1217
Lg→d¯¯0.0862 0.0660
5161
J. Chem. Phys., Vol. 112, No. 11, 15 Ma ch 2000 Magne ic in e ac ion in La2⫺
x
S
x
CuO4
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PCAS
兩
⌿T
典
⫽&
␥
冉
兩
ab
¯
兩
⫺
兩
ba
¯
兩
&
冊
.
The alues o he coe icien s (
␣
⫹
␤
)/&and (
␣
⫺
␤
)/&,
which conce n, espec i ely, he neu al and ionic alence–
bond componen s, and &
␥
a e p esen ed in Table III. As is
easy o check, he neu al/ionic a io in he single s a e di-
minishes when he DDCI-3 space is used. The ques ion now
is why he e ec o he ionic alence–bond single is en-
hanced when he DDCI-3 space is used.
The second e ec is he appea ance o con ibu ions as-
socia ed o cha ge ans e p ocesses om o bi als cen e ed
in sp combina ions o oxygen basis unc ions o o bi als
3dCu in cha ac e . Ac ually, Van Oos en e al.34 had ob-
ained qui e accu a e esul s o exchange coupling in
La2CuO4sys em 共⫺120 meV兲 om a low-dimensional non-
o hogonal con igu a ion in e ac ion 共NOCI兲calcula ion. The
me hod includes, in addi ion o he de e minan s
␾
1,
␾
2,
␾
3, and
␾
4, each o hese wi h a speci ically pola ized co e,
he ligand→me al cha ge ans e de e minan s, in ol ing
he o bi als o he b idging ligand:
␾
5⫽...Cu⫹共↑↓兲⫺L⫹共↑兲⫺Cu⫹2共↓兲..,
␾
6⫽...Cu⫹共↑↓兲⫺L⫹共↓兲⫺Cu⫹2共↑兲..,
whe e L⫹is an O⫺ion (L⫽O⫽). These de e minan s appea
in he DDCI-2 lis as one-hole s a es. Thei ole is c ucial o
acili a e he Ande son mechanism. This mechanism is a
second-o de e ec in h ough-space in e ac ions 共scheme II,
M⫽Cu⫹2),
SCHEME II.
wi h he exchange coupling Jp opo ional o ⫺( 2/U),
whe e Uis he exci a ion ene gy om
␾
1 o
␾
3and is he
hopping in eg al be ween he wo magne ic o bi als, bu i
becomes a ou h-o de e ec when he ligand is on he
M–M axis, acco ding o he ollowing p ocess 共scheme III,
M⫽Cu⫹2),
SCHEME III.
which in ol es ligand o me al hopping in eg als ( LM) and
gi es a con ibu ion o he exchange p opo ional o
⫺( LM)4/(U⬘)2U, whe e U⬘and Ua e he exci a ion ene -
gies om
␾
1 o
␾
5and
␾
1 o
␾
3, espec i ely. In his
scheme, he coe icien s o he cha ge- ans e s a es and o
he ionic o ms,
␾
3, a e p opo ional o LM /U⬘and
( LM)2/UU⬘, espec i ely.
The p ocess will be unde es ima ed i one does no in-
oduce he speci ic dynamic epola iza ion occu ing on he
LMCT con igu a ions. These epola iza ion e ec s a e
b ough by he single exci a ions on he op o he de e mi-
nan s
␾
5,
␾
6..., i.e., by 2h–1pde e minan s which do no
belong o he DDCI-2 lis bu o he DDCI-3 one. This ex-
plains why he LMCT de e minan s ca y signi ican weigh s
in he wa e unc ion desc ip ions only when he DDCI lis is
enla ged. The in e ac ion o hese de e minan s wi h hei
single exci a ions, single 兵LMCT其, p oduces an impo an
lowe ing o he e ec i e U⬘ene gies o he LMCT s a es
and la ge con ibu ions o he single and iple wa e unc-
ions. I is wo h while o no ice ha LMCT de e minan s a e
also single exci a ions wi h espec o he ionic
␾
3and
␾
4
con igu a ions, in such a way ha he elaxa ion o he
LMCT s a es p oduces a s abiliza ion o he ionic alence
bond con igu a ion and an inc ease on i s weigh on he sin-
gle desc ip ion. Reducing he e ec i e ene gy U⬘, he coe -
icien s o he LMCT o ms and o he ionic de e min-
an s,
␾
3, inc ease 共 espec i ely, Lg→dand Lu→din Table
II, and (
␣
–
␤
)/&in Table III兲. The change o (
␣
⫺
␤
)/&
⫽( LM)2/UU⬘by 20% can be ela ed o he change by 40%
o he an i e omagne ic coupling, J⬃( LM)4/U(U⬘)2
⫽(
␣
⫺
␤
)2/2U.
F om a p ac ical poin o iew, i is impo an o no ice
ha hese singly exci ed 兵LMCT其de e minan s a e no nu-
me ous, since one o he inac i e holes is he Lgo Luo bi al
o he ligands. The DDCI-2⫹single兵LMCT其spaces a e 40–
50- old smalle han DDCI-3 spaces bu , as is shown in
Table I, he expe imen al alue o Jis also eco e ed.
Wi h ega d o he na u e o he molecula o bi als Lg
and Lu, Fig. 2 ep esen s he ROHF elec onic densi y maps
o gand uac i e o bi als and Lgand Luo bi als o he
Cu2O7clus e . As can be seen in his igu e, Lgand Luco -
espond o he bonding combina ions o he Cu dx2⫺y2and
sp oxygen o bi als 共scheme I兲, whe eas gand ua e he an i-
bonding combina ions, wi h a s ong dcha ac e . The elec-
onic dis ibu ion o he wo o bi al pai s 共gand Lg,uand
Lu) a e qui e simila , making possible an op imal ligand–
me al cha ge ans e in bo h hese cases.
A. Dependence o he
J
alue on he molecula
o bi al choice
In o de o o e come he possible dependence o ou
esul s on he s a ing molecula o bi al se , we ha e ob ained
he S–Ta e age na u al o bi als by means o an i e a i e
p ocedu e, p oposed by Ga cia e al.35 and implemen ed in
he DDCI scheme. An a e age densi y ma ix o he single
and iple s a es is buil : R
¯
⫽(RS⫹RT)/2. A e he diago-
naliza ion o his ma ix, a na u al o bi al se is ob ained.
Wi h his new se , he DDCI p ocedu e is epea ed 共i e a i e
dDDCI, IDDCI兲un il bo h he o bi als and he S–Tene gy
TABLE III. Valence–bond p ojec ions o single and iple wa e unc ions.
ROHF MO as ze o-o de desc ip ion.
Clus e Coe . DDCI-2 DDCI-3 DDCI-2⫹single 兵LMCT其
Cu2O7&
␥
0.995 0.938 0.986
(
␣
⫹
␤
)/&0.988 0.926 0.975
(
␣
⫺
␤
)/&0.101 0.117 0.119
Cu2O11 &
␥
0.995 0.935 0.986
(
␣
⫹
␤
)/&0.987 0.924 0.975
(
␣
⫺
␤
)/&0.104 0.119 0.120
5162 J. Chem. Phys., Vol. 112, No. 11, 15 Ma ch 2000 Calzado, Sanz, and Mal ieu
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i e ence a e sel -consis en . Beside hei in e es o he de-
e mina ion o op imal molecula o bi als in he ene gy di -
e ence calcula ion, he densi y ma ices and he na u al o -
bi als a e c ucial ins umen s in he analysis o he wa e
unc ion and hey will be used o ha pu pose all h ough
his wo k. Ac ually, he na u al o bi als and occupa ion num-
be s make possible some concen a ion o in o ma ion e-
ga ding wa e unc ions de eloped on housands o
N-elec on con igu a ions.
The alues o he exchange coupling ob ained wi h he
con e ged a e age na u al o bi al se is p esen ed in Table I,
o DDCI-2 and DDCI-3 spaces. The J alues ob ained wi h
he canonical and na u al o bi al se s a e simila and only a
sligh inc ease o he absolu e alue is obse ed when he
na u al o bi al se is used. This e ec is la ge in he DDCI-2
space han in he DDCI-3. The IDDCI p ocedu e only p o-
duces mino modi ica ions on he magne ic o bi als 共gand u
ha e 92% and 89% o dcha ac e 兲. Figu es 3共a兲and 3共b兲
co espond, espec i ely, o he iple ROHF and na u al ID-
DCI localized ac i e o bi al o Cu2O7, ob ained om a
␲
/4
o a ion o he symme y-adap ed gand uo bi als. The mino
change in he slope o hese o bi als can be seen as a majo
pa icipa ion o he poxygen o bi als in he na u al o bi als.
IV. DOPED SYSTEMS: NATURE OF THE HOLES
In La2CuO4, he subs i u ion o La⫹3ion by S ⫹2p o-
duces a de ec o elec ons in he CuO2planes, i.e., in o-
duces holes in he sys em. When he S concen a ion 共x兲
inc eases, di e en phase ansi ions ake place. Fo x
⬍0.05 he sys em is an an i e omagne ic insula o , while o
0.05⬍x⬍0.3 i becomes a supe conduc o . The maximum
alue o he c i ical empe a u e (Tc⫽40 K) is obse ed a
he op imal doping x⫽0.15. Me allic beha io is obse ed
o S doping la ge han 0.3.36
The na u e and dis ibu ion o ca ie s de e mine he su-
pe conduc i i y. This has mo i ed nume ous s udies, expe i-
men al as well as heo e ical. Di e en expe imen al e i-
dence has shown ha ca ie s in La2⫺xS xCuO4ha e a s ong
2poxygen cha ac e . Bo h XANES 共x- ay abso p ion nea
edge s uc u e兲6and x- ay pho oemission spec a6ha e indi-
ca ed ha he p edominan Cu a oms con igu a ion in doped
coppe oxides is 3d,9wha e e x. The e is no e idence o
mixed alency in hese sys ems. Also he 1soxygen EELS
spec a sugges ha holes a e essen ially placed in he 2p
oxygen band.6Se e al heo e ical wo ks, as he in e media e
neglec o di e en ial o e lap 共INDO兲 esul s o Wang
e al.24 o he ab ini io SCF/CI calcula ions epo ed by
Ma in23 show ha he addi ional hole has a ound 50% p
cha ac e . In acuum he ioniza ion po en ial o Cu⫹2ions is
much la ge han ha o he O⫺2anions. In in ini e ionic
la ices, howe e , his pic u e is no longe alid since he
elec os a ic po en ial pushes he Cu dband abo e he O p
band, in a way ha he highes pa ially occupied o bi als
共 he magne ic ac i e o bi als in undoped sys ems兲a e essen-
ially Cu dx2⫺y2in cha ac e . This delocalized monoelec-
onic pic u e would play in a o o a pa icipa ion o he d
shell in he ioniza ion. Ou wo k will examine he ela i e
O/Cu composi ion o he holes.
FIG. 2. Isodensi y cu es o magne ic gand uo bi als, and he mos pa -
icipa ing ligand o bi al Lgand Lu o Cu2O7clus e , plo ed on he xy-plane.
5163
J. Chem. Phys., Vol. 112, No. 11, 15 Ma ch 2000 Magne ic in e ac ion in La2⫺
x
S
x
CuO4
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As men ioned abo e, i is necessa y o include p ope ly
he elec onic co ela ion in o de o adequa ely desc ibe he
g ound s a e o he doped sys em. This will be he objec o
he p esen sec ion.
A. The wo s a e model
Le us conside one coppe ion su ounded by ou oxy-
gen a oms. Acco ding o he o iginal ideas o Zhang and
Rice,9a hole a he oxygen can be in a symme ic o an i-
symme ic s a e wi h espec o he cen al hole a he Cu ion.
Holes in oxygen and Cu a oms will be coupled, esul ing in a
spin single o iple s a e. These au ho s demons a ed ha
he spin single s a e has he lowe ene gy, and assumed ha
i is possible o wo k in his single subspace wi hou chang-
ing he physics o he p oblem. In his con ex , he mo emen
o he hole is equi alen o he displacemen o a Cu 1/2 spin
and o a wo-cen e clus e , he p ocess can be summa ized
in Scheme IV,
SCHEME IV.
which ep esen s wo degene a ed si ua ions, whe e an un-
pai ed elec on 共o a hole兲oscilla es be ween wo equi alen
posi ions. They can be exp essed by wo localized s a es:
␾
a
0⫹⫽
兩
co e..a⬘
兩
,
␾
b
0⫹⫽
兩
co e..b⬘
兩
.共10兲
Thei symme y-adap ed linea combina ions a e
⌿g
0⫹⫽1
&共
␾
a
0⫹⫹
␾
b
0⫹兲⫽
兩
co e..g⬘
兩
;
共11兲
⌿u
0⫹⫽1
&共
␾
a
0⫹⫺
␾
b
0⫹兲⫽
兩
co e..u⬘
兩
,
whe e he o bi als a⬘and b⬘共 espec i ely, g⬘and u⬘) should
esemble, o some ex en , hose o he magne ic p oblem.
This pai o de e minan s de ines a new model space. As in
he magne ic p oblem, a he second o de o he quasidegen-
e a e pe u ba ion heo y, he o -diagonal elemen o he ma-
ix ep esen a ion o he e ec i e Hamil onian buil o e
his space is ela ed o he elec onic coupling cons an , :19
⫽Hab
e ⫽Hab
共0兲⫹兺
␣
具
␾
a
0⫹
兩
H
ˆ
兩
␾
␣
典具
␾
␣
兩
H
ˆ
兩
␾
b
0⫹
典
Ea
0⫺E
␣
0.共12兲
Since
␾
a
0⫹and
␾
b
0⫹di e by only one spin-o bi al, i is easy
o show ha he se o
兩
␾
␣
典
de e minan s include all he
exci a ions up o h ee deg ees o eedom 共DDCI-3 lis :
single and double exci a ions excep hose which in ol e wo
inac i e holes and wo inac i e pa icles兲.
As in he magne ic p oblem, he a ia ional ea men o he
DDCI lis in oduces highe o de co ec ions and a oids he
inaccu acies due o he smallness o he denomina o s in he
second-o de exp essions. Using symme y-adap ed con igu-
a ions, is gi en by
⫽Eg⫺Eu
2,共13兲
whe e Egand Eua e he eigen alues o he ⌿g
⫹and ⌿u
⫹
s a es, which in he D2hpoin g oup a e o A1gand B2u
symme y, espec i ely. Once he DDCI lis is buil , he di-
agonaliza ion o he di e ence dedica ed CI ma ix gi es us
he a ia ional es ima e o he hopping in eg al.
In Table IV he alues ob ained o 2D and 3D clus e s
a e collec ed. The size o he clus e does no a ec he alue
o he ans e in eg al. This esul sugges s he wo-
FIG. 3. Isodensi y cu es o ac i e localized o bi als in Cu2O7clus e , plo -
ed on he xy-plane: 共a兲ROHF o bi al, 共b兲na u al IDDCI magne ic o bi al,
共c兲na u al IDDCI doped-sys em o bi al, and 共d兲localized di e en ial na u-
al o bi al.
5164 J. Chem. Phys., Vol. 112, No. 11, 15 Ma ch 2000 Calzado, Sanz, and Mal ieu
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dimensional cha ac e o he ca ie s in he doped coppe
oxides. The alue o oscilla es be ween 0.53–0.55 eV in
good ag eemen wi h he usually accep ed alue o 0.5 eV
共Re . 37兲共 o ins ance, 0.57–0.59 eV by Wang e al.,24 0.65
eV by Ma in,23 and 0.4 eV by Hybe sen e al.兲.12
In he basis o he canonical ROHF molecula o bi als,
he p ojec ions in he CAS model space o he ⌿g
⫹and ⌿u
⫹
DDCI-3 wa e unc ions a e 0.77 and 0.76, espec i ely. This
means ha only he 50% o he wa e unc ions can be ep-
esen ed by he ex a hole localized in one o he Cu dx2
⫺y2o bi als. The es o he con ibu ions comes om di -
e en O 2p→Cu3dcha ge- ans e exci a ions. An appeal-
ing poin is ha he mos pa icipa ing O 2pcen e ed o bi als
co espond o he same Lgand Luo bi als which play an
impo an ole in he magne ic p oblem. The e a e h ee
kinds o de e minan s pa icipa ing in he g ound s a e wa e
unc ion desc ip ion:
共i兲 he model space de e minan s, M–L–M⫹,M
⫹–L–M,
whe e he e is an ex a hole in an ac i e localized
o bi al:
␾
a
0⫹⫽Cu⫹2共↑兲⫺L共↑↓兲⫺Cu⫹3共.兲,
␾
b
0⫹⫽Cu⫹3共.兲⫺L共↑↓兲⫺Cu⫹2共↑兲;
共ii兲neu al cha ge- ans e exci a ions, M–L⫹–M, wi h
an ex a hole in a ligand-cen e ed o bi al:
␾
1
⫹⫽Cu⫹2共↑兲⫺L⫹共↓兲⫺Cu⫹2共↑兲,
␾
2
⫹⫽Cu⫹2共↑兲⫺L⫹共↑兲⫺Cu⫹2共↓兲;
共iii兲ionic cha ge- ans e exci a ions, M⫹–L⫹–M⫺,
M⫺–L⫹–M⫹, whe e he ex a hole is in an oxygen
o bi al, and a me al–me al dA→dB ans e akes
place simul aneously:
␾
3
⫹⫽Cu⫹3共.兲⫺L⫹共↑兲⫺Cu⫹1共↑↓兲,
␾
4
⫹⫽Cu⫹1共↑↓兲⫺L⫹共↑兲⫺Cu⫹3共.兲.
The i s class 共 he model space兲in oduces he ex a hole in
a3do bi al. Bu his ep esen s only 50% o he doped
g ound s a e desc ip ion. Cha ge ans e exci a ions pu a
hole in o bi als wi h s ong cha ac e in 2poxygen, inc eas-
ing he elec onic densi y in 3dx2⫺y2, and he e o e dimin-
ishing he 3dna u e o he hole 共sc eening he wo holes in
3do bi als兲. This e ec has ecei ed he name o elec onic
pola on,23 in he sense ha an ex a hole in a coppe si e
p oduces no a nuclea dis o ion bu an elec onic dis o ion
o he neighbo oxygen, and a s ong p–d ehyb idiza ion
occu s 共scheme V兲. In ac , he pcha ac e o ac i e o bi als
has inc eased a e he hole doping. Fo ins ance, in he
doped Cu2O7clus e he ac i e gand uo bi als ha e a 83%
and 76% o dcha ac e , espec i ely, o be compa ed wi h
he s a e-a e aged IDDCI na u al o bi als ob ained o he
undoped sys em 共86% and 87%兲. Fo he Cu2O11 clus e , a
simila e ec is obse ed in Scheme V.
SCHEME V.
B. Dependence o he
alue on he molecula o bi al
choice
I should be no ed he e ha all he calcula ions so a
ha e been ca ied ou using as ac i e o bi als he magne ic
o bi als de e mined o J. This is in ag eemen wi h he ideas
behind he –Jmodel, whe e a unique se o ac i e o bi als
is used. Howe e , he e has been a lo o con o e sy abou
he na u e o he hole. Ac ually, he e exis s unanimi y abou
i s p edominan oxygen cha ac e . This means ha hole-
adap ed o bi als, a⬘and b⬘, should be di e en han he
o iginal aand bo bi als. Mo eo e , he possible dependence
o he DDCI esul s on he ini ial o bi als has been p e i-
ously men ioned. To o e come hese di icul ies, some
DDCI-3 calcula ions ha e been pe o med i e a ing he na u-
al o bi als o he doped p oblem, using he i e a i e di e -
ence dedica ed CI 共IDDCI兲me hod. This p ocedu e does no
change he alue o , as can be seen in Table IV. The mos
impo an e ec o hese hole adap ed o bi als is he inc ease
o he weigh o he CAS de e minan s in he ⌿g
⫹and ⌿u
⫹
desc ip ion 共 he coe icien s o he p ojec ions become 0.91–
0.90兲, i.e., he concen a ion o he wa e unc ions in an im-
p o ed model space. The i e a ed gand uo bi als inco po a e
by hemsel es he p–dhyb idiza ion, ha ing a 71% and
66% dcha ac e , espec i ely, o he doped g ound s a e o
Cu2O7, as can be seen compa ing he ac i e magne ic and
cha ge- ans e o bi als, aand a⬘关Figs. 3共b兲and 3共c兲兴. This
explains he la ge con ibu ions o he CAS de e minan s in
he wa e unc ion desc ip ion, inc easing he alidi y o he
wo s a e model.
C. Na u e o he hole
The de e mina ion o he hole na u e in doped sys ems is
no an easy ask. Two di e en elec onic a angemen s, neu-
al 共undoped兲and ionic 共doped兲ones, based on di e en
molecula o bi al basis se s, need o be compa ed. The ‘‘hole
language,’’ usual in his ield, is no s aigh o wa d since i
implici ly e e s o an ideal closed shell desc ip ion wi h d10
occupancy o Cu a oms. Mo eo e , he cha ge ans e p o-
cesses, pa ially esponsible o hese co alen e ec s, a e
unlike in he neu al and ionic sys ems.
A new s a egy o o e come hese di icul ies has been
de eloped. I consis s o building a di e en ial densi y ma-
ix, in he common a omic basis se , which explici ly de-
sc ibes he hole doping p ocess. I is a gene al p ocedu e,
ha could be use ul o s udy o he phenomena, whe e s a es
ela ed by hole doping a e in ol ed. The a omic-based den-
si y ma ix o he neu al and ionic sys ems a e buil sepa-
TABLE IV. Absolu e alues o he hopping in eg al, ob ained wi h DDCI-3
spaces and di e en MO se s.
Clus e T iple ROHF 共eV兲IDDCI 共eV兲n°de (g,u)
Cu2O70.553 0.575 255520, 255432
Cu2O11 0.532 0.544 526767, 526639
5165
J. Chem. Phys., Vol. 112, No. 11, 15 Ma ch 2000 Magne ic in e ac ion in La2⫺
x
S
x
CuO4
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