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A gene al p ocedu e o e alua e many-body spin ope a o ampli udes om pe iodic
calcula ions: applica ion o cup a es
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2007 New J. Phys. 9 369
(h p://iopscience.iop.o g/1367-2630/9/10/369)
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New Jou nal o Physics
A gene al p ocedu e o e alua e many-body
spin ope a o ampli udes om pe iodic
calcula ions: applica ion o cup a es
Ibé io de P R Mo ei a1,4, Ca men J Calzado2, Jean-Paul Mal ieu3
and F ancesc Illas1
1Depa amen de Química Física and Ins i u de Química Teò ica i
Compu acional (IQTCUB), Uni e si a de Ba celona and Pa c Cien í ic de
Ba celona, C/Ma í i F anquès 1, E-08028 Ba celona, Spain
2Depa amen o de Química Física, Uni e sidad de Se illa, C/P o . Ga cía
González s/n, E-41012 Se illa, Spain
3IRSAMC, Labo a oi e de Physique Quan ique, Uni e si é Paul Saba ie ,
118 Rou e de Na bonne, F-31062 Toulouse-Cedex, F ance
E-mail: i.mo ei [email protected], [email protected],[email p o ec ed] and
[email p o ec ed]
New Jou nal o Physics 9(2007) 369
Recei ed 18 May 2007
Published 12 Oc obe 2007
Online a h p://www.njp.o g/
doi:10.1088/1367-2630/9/10/369
Abs ac . A gene al p ocedu e is p esen ed which pe mi s he o m o an
ex ended spin Hamil onian o be es ablished o a gi en magne ic solid and
he magni ude o i s e ms o be e alua ed om spin pola ized, Ha ee–Fock
o densi y unc ional calcula ions ca ied ou o pe iodic models. The
compu a ional s a egy makes use o a gene al mapping be ween he ene gy
o pe inen b oken-symme y solu ions and he diagonal e ms o he spin
Hamil onian in a local ep esen a ion. F om his mapping i is possible o
de e mine no only he ampli ude o he well-known wo-body magne ic coupling
cons an s be ween nea -neighbo si es, bu also he ampli udes o ou -body
cyclic exchange e ms.Asc u iny o he on-si espin densi ies p o idesaddi ional
in o ma ion and con ol o he many b oken-symme y solu ions which can be
ound. The p ocedu e is applied o he La2CuO4, S 2CuO2F2, S 2CuO2Cl2and
Ca2CuO2Cl2squa e la ices and he S Cu2O3ladde compound. I is shown ha
a p ope desc ip ion o he magne ic s uc u e o hese compounds equi es ha
wo- and ou -body e ms a e explici ly included in he spin Hamil onian. The
implica ions o he in e p e a ion o ecen expe imen s a e discussed.
4Au ho o whom any co espondence should be add essed.
New Jou nal o Physics 9(2007) 369 PII: S1367-2630(07)50628-2
1367-2630/07/010369+25$30.00 © IOP Publishing L d and Deu sche Physikalische Gesellscha
2
Con en s
1. In oduc ion 2
2. Theo e ical backg ound and me hodology 6
3. Compu a ional de ails 13
4. Resul s 14
4.1. Spin Hamil onian pa ame e s o he 2D squa e la ices ............. 14
4.2. Spin Hamil onian pa ame e s o he S Cu2O3spin ladde compound ...... 17
4.3. C i ical analysis o he esul s ........................... 19
5. Summa y, conclusions and possible ex ensions 21
Acknowledgmen s 23
Re e ences 23
1. In oduc ion
The disco e y o he anomalous p ope ies o high-Tcsupe conduc ing cup a es (HTCSs) in
he la e eigh ies has igge ed a conside able in e es in he c ys al and elec onic s uc u e o
hese compounds om bo h expe imen al and heo e ical poin s o iew [1]–[5]. An eno mous
esea ch e o on ce amic ma e ials, mos ly based on coppe oxides, has been ca ied ou
con inuously o y o imp o e he p ope ies o known s uc u es and syn he ic pa hways and
has esul ed in he syn hesis o a wide a ie y o cup a es. The imp essi e ichness o low-
dimensional magne ic beha io o he di e en coppe compounds can be, o a la ge ex en ,
aced back o he s acking o he dis o ed CuX6oc ahed a (o CuX5py amidal o CuX4plana
uni s) in he la ice [4,6]. Mos o he cup a e based ma e ials a e o med by almos independen
CuO4uni s wi h dis an apical ligands o comple e he s ongly dis o ed CuO4X2o CuO4X
uni s and, hence, he s uc u e is domina ed by he link be ween CuO4uni s. Depending on
he na u e o he coun e ions and on he numbe o links be ween he di e en CuO4uni s,
di e en s uc u es can be o med anging om he ypical lamella wo-dimensional (2D)
s uc u e o he HTCSs o many lowe dimensional s uc u es by di e en combina ions o edge-
sha ing and co ne -sha ing CuO4plaque es (o CuO4uni s) ha can gi e ise o spin ladde s
(e.g. S n−1CunO2n−1se ies wi h n⩾2 [7]–[9]), zigzag spin chains (e.g. S CuO2[10]) and
quasi-1D sys ems (e.g. A2CuO2(A =Ca,S ) [11,12] o Li2CuO2[13]) o med by edge-sha ing
CuO4uni s.
The elec onic g ound s a e o his kind o ma e ials is usually desc ibed by he open
shell na u e o he Cu2+ ions a anged in he CuO4uni s in which he Cu(3d9)a omic
con igu a ion gi es ise o a dx2–y2 ype hole wi h he lobes poin ing owa ds he O ions.
The esul ing Cu–O–Cu pa hways ange om ∼90◦ o 180◦and hey a e esponsible o
he ich a ie y o low-dimensional magne ic s uc u es domina ed by mode a e e omagne ic
(FM) o s ong an i e omagne ic (AFM) in e ac ions. F om he heo e ical poin o iew, hese
sys ems a e s ongly co ela ed in na u e, making s anda d band heo y echniques based on
densi y unc ional heo y (DFT) unable o accu a ely desc ibe ei he hei alence o low ene gy
spec um [14]. Howe e , i has been shown ha hyb id exchange-co ela ion unc ionals can
p o ide eliable desc ip ions o s ongly co ela ed ansi ion me al magne ic sys ems ([15] and
e e ences he ein).
New Jou nal o Physics 9(2007) 369 (h p://www.njp.o g/)
3
Mos o he HTCS ma e ials ha e a lamella s uc u e in which s ong AFM in e ac ions
ake place along 180◦Cu–O–Cu bonds in edge sha ing Cu4O4plaque es leading o a 2D
ne wo k o e ec i e spin S=1/2 pa icles. These s ong magne ic in e ac ions obse ed in he
HTCS a e hough o be undamen al ing edien s o he high-Tcsupe conduc i i y mic oscopic
mechanism [5,16]. Since hese compounds may be ega ded as e ec i e S=1/2 spin la ices,
hei low ene gy spec um and collec i e p ope ies a e assumed o be go e ned by a simpli ied
Heisenbe g Hamil onian as in equa ion (1)
ˆ
H=X
hi,ji
Jij ˆ
Si·ˆ
Sj−1
4,(1)
which only accoun s o he magne ic coupling Jij be ween nea es -neighbo (NN) cen e s i
and j. This is in ag eemen wi h he widely accep ed gene al pic u e o HTC supe conduc i i y
in ol ing a ‘Heisenbe g sea’ whe e holes o elec ons a e in oduced by doping he pe ec
s uc u es. Howe e , i has been claimed ha o ully unde s and he magne ic exci a ions,
in a ed and neu on sca e ing spec a o 2D [17]–[23] and spin ladde cup a es [24,25] i
is necessa y o ex end he spin Hamil onian as in equa ion (2)
ˆ
H=X
i,j
Jij ˆ
Si·ˆ
Sj−1
4
+X
i,j,k,l
J ing ˆ
Si·ˆ
Sjˆ
Sk·ˆ
Sl+ˆ
Si·ˆ
Slˆ
Sj·ˆ
Sk−ˆ
Si·ˆ
Skˆ
Sj·ˆ
Sl−1
16
+· · · .(2)
In his exp ession he cons an s 1/4 and 1/16 ha e been in oduced o de ine he ze o o ene gy
as ha o he FM solu ion. In his spin model, he signs and ampli udes o he local in e si e
magne ic in e ac ions go e n he collec i e p ope ies o a spin la ice. They appea as he basic
ing edien s o he e ec i e spin Hamil onian which in ull gene ali y in ol es no only he
wo-body exchange Jij bu also o he in e ac ions, such as hose ep esen ed by he ou -body
cyclic e m J ing, o e en highe -o de e ms.
The la ges wo-body couplings a e expec ed o occu be ween NN si es al hough nex -NN
(NNN) in e ac ions may be non-negligible o e en o he same o de o magni ude (c CuGeO3
sys em [26,27]). Rega ding he ou -body ope a o e ms, hey may be impo an in Cu4O4
plaque es since hei o igin lies in he cyclic ci cula ion o elec ons a ound he ing. Thei
impo ance and ha o analogous cyclic six-body e ec s ha e been poin ed ou in o he ypes
o hal - illed band sys ems such as he πsys em o conjuga ed o ganic molecules [28]. Simila
ou -body e ms a e c ucial o desc ibe he g ound s a e p ope ies o 3He [29].
The di ec de e mina ion o he ampli ude o he many-body e ms o an ex ended spin
Hamil onian such as he one in equa ion (2) om expe imen is, in gene al, impossible. Spin
ladde s wi h a a ie y o in e si e dis ances ep esen an especially di icul case. In gene al, he
expe imen al de e mina ion o he coupling cons an s is based on a se ies o hypo heses abou
he negligibili y o in e ac ions be ween ‘ emo e’ si es. F om hese hypo heses, a gi en spin
model is assumed and alida ed only om a nume ical i o he he modynamic o spec al
p ope ies. I is cus oma y o conside NN in e ac ions only al hough his may be an excessi e
simpli ica ion and e en ually can lead o con adic o y es ima es o he dominan couplings.
This is p ecisely he case o he cup a e spin ladde s o which con lic ing alues o he J ung/Jleg
a ios anging om 0.5 o 1.0 we e p oposed. In e ac ions in ol ing si es a a longe dis ance o
New Jou nal o Physics 9(2007) 369 (h p://www.njp.o g/)
4
in ol ing a ious si es had o be in oked o a ionalize he di e en expe imen al esul s a ising
om di e en echniques. One may hink o ins ance in in e -ladde in e ac ions, diagonal
e ms in he plaque es o ou -body cyclic e ec s. Indeed, se e al heo e ical s udies seem o
consis en ly indica e ha ou -body e ms (J ing) a e c ucial [21]–[24]. He e, i is impo an
o poin ou ha e y ecen ly Toade e al [30] p o ided s ong expe imen al e idence o
he impo ance o he J ing e m in La2CuO4wi h J ing ≈0.5J. This alue is compa able o
he pai ing ene gies and s ongly sugges s ha he esul ing ci cula ing cu en s could ha e an
impo an ole in he mechanism o supe conduc i i y. This is in con as wi h p e ious es ima es
o J ing o 2D and spin ladde cup a es, ob ained ei he om indi ec measu emen s o om
nume ical simula ions wi h an ex ended Heisenbe g model, which p opose subs an ially smalle
ampli udes wi h J ing ∼0.3J[19,21,22,24,31]. The o igin o hese disc epancies elies on he
choice made by di e en au ho s o he magni ude o he o he coupling cons an s in he spin
Hamil onian o equa ion (2). Hence, he J ing e m as e alua ed by Toade e al elies on NN
and NNN coupling cons an s o J=111.8meV and Jd= −11.4meV, espec i ely, ex ac ed
om one o he a ious i ings o he magnon spec um [22]. Howe e , i is impo an o
poin ou ha his alue o Jis smalle han ano he expe imen al es ima e o 135±6meV
ob ained wi h a NN Heisenbe g Hamil onian [32]. Mo eo e , one should no e ha he p esen
es ima e o a FM Jddi e s om p e ious heo e ical p edic ions [33] and om i ing o
expe imen al measu emen s on ma e ials wi h simila exchange pa hs [34]. Howe e , one should
also ecognize ha he heo e ical s udy by Anne e al [33], akes only in o accoun wo-
elec on p ocesses o e alua e his Jd e m while, as clea ly explained by Toade e al [30]
and e e ences he ein, he FM cha ac e o Jda ises p ecisely om h ee-elec on exchange
p ocesses a ound a plaque e, which a e o he same o de o magni ude as ou -elec on
exchange p ocesses a ound he same plaque e and which a e no aken in o accoun in [33,34].
The discussion abo e illus a es he di icul ies aced by expe imen alis s when a emp ing o
ex ac he magni ude o he impo an e ms and he need o independen accu a e and unbiased
heo e ical p edic ions. Fo he S Cu2O3ladde compound a simila si ua ion is encoun e ed; he
ecen Raman esponse expe imen s by Schmid e al sugges J ung =140meV, J ing/J ung =0.2
and Jleg/J ung =1.5 [35], in con as wi h p e ious wo k indica ing a mo e iso opic beha io
be ween ungs and legs [36].
F om he p eceding discussion i is clea ha an accu a e p edic ion o he a ious magne ic
in e ac ions en e ing in he spin Hamil onian as in equa ion (2) is no only highly desi able o
unde s and he g ound s a e p ope ies o his kind o sys ems bu u gen ly needed. One could,
o ins ance, s a om some app oxima e elec onic Hamil onian such as a single-band model
in ol ing only he magne ic cen e s o a wo-band model in ol ing also he elec ons o he
b idging ligands. Howe e , his is likely o in oduce e en mo e p oblems since i is di icul
o assess he accu acy o such a mul ipa ame ic app oach. An al e na i e and s aigh o wa d
way o an unbiased es ima e is he di ec e alua ion o he ampli ude o he ele an magne ic
in e ac ions on a ealis ic model sys em ea ing all he elec ons in a la ge enough basis se .
I p ac icable, his p agma ic app oach will p esen he ad an age o p o iding an independen ,
unbiased and consis en p edic ion o he ampli ude o ele an exchange pa ame e s ha can
sol e he di icul ies encoun e ed by he usual i ing echniques used by expe imen alis s.
Un o una ely, he di ec calcula ion o he impo an e ms is no a simple ask since i
also equi es he use o a model o he solid. In a i s app oach, one may neglec ansla ional
symme y and de ine a p ope ly embedded ini e clus e , a agmen o he pe iodic la ice wi h
wo, h ee o ou magne ic si es wi h hei coo dina ion ligands and pe o m he bes ab ini io
New Jou nal o Physics 9(2007) 369 (h p://www.njp.o g/)

5
(using he exac Hamil onian) explici ly co ela ed calcula ions. This s a egy, in pa icula when
using he so-called di e ence dedica ed con igu a ion in e ac ion (DDCI), has p o ided e y
consis en and eliable e alua ions o he wo-body in e ac ions in a la ge se ies o pe o ski es,
oxides and 2D cup a es [37]–[41]. Howe e , his app oach apidly aces compu a ional limi s
due o he need o e y la ge embedded clus e models. This is especially he case when
a emp ing o ex ac ou -body e ms in ladde compounds [42]–[45]. Al e na i ely, one may
exploi he pe iodic symme y o he c ys al bu in his case explici ly co ela ed calcula ions
a e no easible and one mus ely on spin-pola ized mean- ield ype app oaches. In such a case,
only he FM solu ion may be conside ed as a alid app oxima ion o an eigens a e o he p oblem
wi h p ope ly de ined spin quan um numbe s [46,48]. The solu ions wi h lowe alues o he
squa e o he o al spin ope a o S2(lowe magne iza ion) canno be p ope ly ea ed. Fixing
a p io i di e en localized pe iodic spin dis ibu ions one may ob ain a se o dis inc sel -
consis en solu ions o di e en ene gies. Howe e , hese solu ions a e no eigen unc ions o he
S2ope a o ; hey a e (spin) b oken-symme y solu ions. We will show ha assigning he ene gy
o he b oken-symme y solu ions o he expec a ion alue o he co esponding spin dis ibu ion
o a Heisenbe g Hamil onian enables one o ob ain es ima es o he magne ic coupling cons an s.
Such app oaches ha e been a he ex ensi ely used in he ield o molecula magne ism and may
also be employed on embedded clus e s o ex ac he wo-body e ms [48,49]. The mean- ield
calcula ions may use he exac Hamil onian, and he co esponding in o ma ion comes ei he
om un es ic ed b oken-symme y Ha ee–Fock (UHF) solu ions o om he simila ones
ob ained by means o DFT app oaches. In he i s case, he esul ing e alua ions o he AFM
couplings a e se e ely unde es ima ed [15], [50]–[52]. In he second one, he in oduc ion o
elec on co ela ion e ec s imp o es he esul bu he nume ical alues a e highly sensi i e o
he chosen exchange-co ela ion po en ials. No ice ha he local densi y app oxima ion (LDA)
and he di e en gene alized g adien app oxima ions (GGA) in DFT equen ly lead o me allic
solu ions, closed shell in na u e [15], which miss he main physical ea u es o he low ene gy
s a es and canno p o ide any in o ma ion abou he magne ic couplings. Ac ually, he bes
esul s a e ob ained wi h hyb id unc ionals ini ially p oposed by Becke [53,54], o which
a pe cen age o Fock exchange is mixed wi h a gi en DFT exchange-co ela ion unc ional.
Howe e , he esul s a e again s ongly dependen on he amoun o Fock exchange included
in he exchange po en ial [55,56]. P e ious calcula ions ha e shown ha he bes pe cen age,
a leas o NiO [15] and cup a es [57,58], is a ound 35% Fock. He e, i is wo h poin ing
ou ha hyb id DFT calcula ions need o include he non-local Fock con ibu ion which can be
accu a ely compu ed when he o al densi y is exp essed in e ms o Gaussian ype o bi als
(GTO) basis se s. Recen ly, howe e , i has been shown ha a easonable accu acy can be
eached employing plane wa es as well [59].
In he p esen wo k, we epo a gene aliza ion o he pe iodic symme y-b oken (UHF o
DFT) app oach which pe mi s one o ob ain no only he a ious NN and NNN mos impo an
wo-body e ms, as ini ially done o MF2(M =Cu, Ni, Co, Fe, Mn) compounds [52,60], bu
o p edic highe -o de e ms such as he ou -body cyclic e ms appea ing in equa ion (2). To
his end b oken-symme y hyb id DF compu a ions a e ca ied ou o he La2CuO4, S 2CuO2F2,
S 2CuO2Cl2and Ca2CuO2Cl22D squa e la ices and he S Cu2O3 wo-leg spin ladde . A gene al
heo e ical amewo k is p esen ed ha pe mi s one o ex ac accu a e alues o he many-body
e ms o ex ended spin Hamil onians om hese pe iodic i s -p inciple calcula ions. In addi ion,
i will be shown ha hese pe iodic calcula ions con i m he impo ance o he FM in e ladde
exchange and o he ou -body in a-plaque e ope a o .
New Jou nal o Physics 9(2007) 369 (h p://www.njp.o g/)
6
This pape is o ganized as ollows: in sec ion 2, we de elop he ele an me hodological
aspec s o ex ac he ampli ude o he pa ame e s o a gene al spin Hamil onian om pe iodic
all-elec on calcula ions o he elec onic s uc u e o he sys ems. Fi s , he logics leading o a
mapping be ween spin eigen unc ions and symme y-b oken solu ions is de eloped in a gene al
scheme s a ing om he Hubba d model o a plaque e wi h ew ema ks on he calcula ed
solu ions. Sec ion 3desc ibes he compu a ional de ails o he calcula ions. The esul s ob ained
o he plana cup a es and he ladde sys em a e epo ed in sec ion 4and, inally, in sec ion 5,
we p esen he conclusions o his in es iga ion and u he ex ensions o he heo y.
2. Theo e ical backg ound and me hodology
Le us assume ha one may calcula e a mean ield single de e minan desc ip ion o he FM
s a e in a gi en uni cell. This wa e unc ion is usually exp essed in e ms o symme y-adap ed
(delocalized) Bloch unc ions. An app op ia e localizing ans o ma ion o he singly occupied
unc ions will de ine a om-cen e ed Wannie magne ic o bi als [61,62]. The FM solu ion o
he sys em may be w i en as
ΦSz,max =|co e·abc · · · n|,(3)
whe e co e s ands o he closed-shell pa o he wa e unc ion and a, b, c, ...,n ep esen he
magne ic o bi als localized on he di e en magne ic cen e s. In his pic u e he e is one elec on
pe si e and one can e e o he co esponding de e minan as a alence bond (VB) neu al o m
(see also he de ini ion in [63]). In he ollowing, he expec a ion ene gy hΦSz,max|ˆ
H|ΦSz,maxi =
E0will be aken as ene gy o igin (E0=0). The nelec ons in no bi als de ine a hal - illed band
p oblem. Each o he possible dis ibu ions o he magne ic elec ons in he magne ic o bi als
de ines a de e minan and he se o hese de e minan s spans an o hogonal VB basis se . The
ma ix elemen s o he exac Hamil onian in his basis de ine a alence con igu a ion in e ac ion
ma ix. In o de o unde s and he physical e ec s a ising om he in e ac ion be ween he αand
β(o spin up and spin down) elec ons, we shall conside i s a ini e se o neighbo si es, he
pe iodici y being in oduced in a subsequen s ep. One mus i s ecall ha he lowes ene gy
de e minan s a e he neu al ones (in he VB sense), o which any magne ic o bi al is singly
occupied. The ionic de e minan s se , p esen ing posi i ely and nega i ely cha ged si es, a e o
highe ene gy. Le us conside a gi en VB neu al de e minan as
ΦI=co e·a· · · h¯
i jk¯
lm · · · n,(4)
buil wi h he local magne ic o bi als o he FM solu ion and whe e he ba s indica e, as usual,
be a spin elec ons (hence, iis synonymous o iαand ¯
io iβ).
The expec a ion ene gy (wi hin he exac non ela i is ic Hamil onian) o such a
de e minan is gi en by equa ion (5)
hΦI|ˆ
H|ΦIi − E0=X
i∈α(I)X
j∈β(I)
Kij,(5)
whe e α(I)and β(I)a e he se o spin o bi als o αand βspin in ΦIand iαand jβ he
co esponding spin o bi als. This expec a ion ene gy is highe han ha o ΦSz,max since i misses
he di ec (posi i e) exchange in eg als be ween he αand βspin o bi als. I is easonable o
assume ha he nonze o Kij in eg als conce n NN and NNN si e pai s only. These in eg als
New Jou nal o Physics 9(2007) 369 (h p://www.njp.o g/)
7
ha e been es ima ed in cup a es [42]–[45], [57,64,65] and play an impo an ole in de ining
he inal alue o he magne ic coupling cons an . Howe e , o he pe u ba i e de elopmen
gi en below we conside ha he di e ence among he di e en Kij in eg als is small enough o
be neglec ed in on o he exci a ion ene gies o he VB ionic de e minan and a common ze o-
o de ene gy E0will be kep o he VB neu al de e minan s in he pe u ba i e de elopmen s.
The s a egy will now consis o es ablishing a ela ion be ween he diagonal elemen s
o he Heisenbe g Hamil onian and he ene gies o symme y-b oken UHF solu ions. This is
accomplished by means o sui able pe u ba ion expansions.
The Heisenbe g Hamil onian is ob ained om he exac one by conside ing he space o all
VB de e minan s ΦIas a model space [66] in he ame o he e ec i e Hamil onian heo y o
Bloch o des Cloizeaux [67,68]. Le us call ˆ
P he p ojec o on his subspace o dimension N
ˆ
P=X
I=1,N
|ΦIi hΦI|,(6)
he e ec i e Hamil onian ˆ
He is de ined by he Neigen equa ions
ˆ
He |ˆ
P9mi = Em|ˆ
P9mi,m=1,N(7)
whe e he ec o s |9mia e he Neigen ec o s o ˆ
H
ˆ
H|9mi=Em|9mi,m=1,N,(8)
ha ing he la ges p ojec ions on he model space, and Em he co esponding eigen alues. Since
all VB neu al de e minan s ha e he same spa ial pa , he e ec i e Hamil onian may be w i en
in e ms o spin ope a o s and, in p inciple, i will in ol e many-spin ope a o s.
Due o he la ge ene gy gap be ween he neu al and VB ionic de e minan s he e ec i e
Hamil onian may be app oached h ough he quasi-degene a e pe u ba ion heo y (QDPT). Le
us conside he diagonal ma ix elemen associa ed wi h a de e minan ΦI. To he second-o de
DΦI
ˆ
He 
ΦIE=DΦI
ˆ
H
ΦIE+X
α∈ionic DΦI
ˆ
HΦαEDΦα
ˆ
HΦIE
E0−E0
α
=E0+X
iα, jβ Kij −2 2
ij
U!,(9)
whe e we ha e conside ed ha he di ec exchange in eg als Kij a e small in on o he ene gy
di e ences be ween he neu al and he VB ionic con igu a ions in he e alua ion o he second-
o de co ec ions. The de e minan ΦIin e ac s wi h all he de e minan s ob ained by cha ge
ans e s be ween αand βsi es leading o ionic de e minan s o he o m
Φi+j−I=a+
jaiΦI;DΦi+j−I
ˆ
H
ΦIE= ij,(10)
and
Φi−j+I=a+
¯
ia¯
jΦI;DΦi−j+I
ˆ
H
ΦIE= ij.(11)
The ampli ude o he hopping in eg al ij apidly dec eases wi h he dis ance be ween si es iand
j. Compa ing hΦI|ˆ
He |ΦIiwi h equa ion (2) one ob ains
−Jij =2 Kij −2 2
ij
U!,(12)
New Jou nal o Physics 9(2007) 369 (h p://www.njp.o g/)
8
which is a a he well-known ela ion. Le us suppose now ha one ies o op imize he ene gy
o a symme y-b oken de e minan Φ0
Iob ained om ΦIby elaxing he spin o bi als. To he
i s -o de his op imiza ion is equi alen o a CI be ween ΦIand he singly exci ed de e minan s
leading o
9I=ΦI−X
iα, jβ
c(1)
ij Φi+j−I+Φi−j+I=ΦI−X
iα, jβ
ij
UΦi+j−I+Φi−j+I,(13)
he spino bi al o a ions
i0=i−X
jβ
ij
Uj,(14)
j0=j−X
iα
ij
Ui,(15)
gi ing delocaliza ion ails o he αspin o bi al on si e ion o he o bi als o he neighbo si es
occupied by βspin elec ons (o equi alen ly, c(1)
ij = ij/U). F om 9Ione can de ine a UHF
single de e minan
Φ0
I=co e·a0· · · h0¯
i0j0k0¯
l0m0· · · n0,(16)
which only di e s om 9Iby second-o de e ec s
Φ0
I≈9I+O(2)(17)
and i s ene gy only di e s om he second-o de co ec ed ene gy associa ed wi h 9Iby hi d-
o de e ms
DΦ0
I
ˆ
H
Φ0
IE≈DΦI
ˆ
H
ΦIE−X
iα, jβ
2 2
ij
U+O(3). (18)
Hence,
EUHF
I0=hΦI|ˆ
He |ΦIi+O(3)≈E0−1
2P
iα, jβ
Jij.(19)
Equa ion (19) ela es he calcula ed ene gy o he I’ b oken symme y solu ion o he exac
Hamil onian, ela i e o he FM s a e, o he co esponding ene gy exp ession o he Heisenbe g
Hamil onian in equa ion (1) and i p o ides a i s use ul ela ionship o ex ac he ele an wo-
body magne ic coupling cons an s om he di e en b oken-symme y solu ions. Since his
ela ion is also alid when ΦI ep esen s a pe iodic spin dis ibu ion, equa ion (19) p o ides a
p ac ical way o ex ac hese e ec i e pa ame e s om pe iodic calcula ions. This is p ecisely
he p ocedu e ollowed in p e ious wo ks [52,60]. Howe e , i is possible o use he pe u ba ion
de elopmen s abo e o gene alize he p ocedu e in such a way ha ex ac ion o highe -o de
e ms in he spin Hamil onian as in equa ion (2) becomes s aigh o wa d. Be o e con inuing he
de elopmen a ca ea is necessa y since he Wannie unc ions as in equa ion (3) o (4) e e o
he whole c ys al. In p ac ice, one has o limi he numbe o spin dis ibu ions o a ini e numbe
which is gi en by all possible spin pe mu a ions in a la ge enough supe cell. Con e gence o
he esul s wi h espec o he model chosen can be indeed e i ied by ex ending he supe cell.
Howe e , since he magne ic coupling cons an s a e local pa ame e s [37], he esul s a e
no expec ed o a y signi ican ly upon enla ging he supe cell used o ex ac he ele an
pa ame e s. The e o e, s a ing om nnon equi alen spin dis ibu ions ΦI,ΦJ, . . . , ΦM, one
may, in p inciple, de e mine mquan i ies Jij and, o cou se, disc imina e he leading e ms om
New Jou nal o Physics 9(2007) 369 (h p://www.njp.o g/)
15
Table 1. Ene gy exp essions pe Cu a om ( ela i e o FM phase) o he magne ic
solu ions in igu e 1used o ex ac J=Jij(NN),Jd=Jij(NNN)and J ing in
he 2D La2CuO4, S 2CuO2F2, S 2CuO2Cl2and Ca2CuO2Cl2laye ed cup a es.
The h ee en ies o each magne ic phase show he co esponding calcula ed
alues (in meV) o he Fock-35, (B3LYP), and [UHF] po en ials.
Phase Ene gy exp ession La2CuO4S 2CuO2F2S 2CuO2Cl2Ca2CuO2Cl2
140.1 153.7 130.8 146.7
AFM J(182.8) (214.6) (182.0) (196.6)
[31.0] [33.1] [26.2] [32.4]
78.9 84.7 70.1 79.1
AFM1 J/2+ Jd/2+ J ing/8 (102.5) (125.1) (103.1) (109.4)
[15.4] [16.9] [13.2] [16.3]
78.8 82.2 68.2 78.3
AFM2 J/2+ Jd(100.3) (118.1) (99.4) (105.5)
[15.1] [17.0] [13.3] [16.0]
0.0 0.0 0.0 0.0
FM 0 (0.0) (0.0) (0.0) (0.0)
[0.0] [0.0] [0.0] [0.0]
coupling ampli udes (J) i is wo h no ing ha UHF la gely unde es ima es hem, as ound
in many p e ious wo ks [57,76,80]. Densi y unc ional calcula ed alues gene ally ep esen
an imp o emen o e UHF esul s al hough hey exhibi he ypical s ong dependence on he
exchange-co ela ion unc ional al eady poin ed ou by Ma in and Illas [55,56]. Hence, B3LYP
o e es ima es he J alues by a ac o close o 1.5 whe eas he alues p edic ed by he Fock-35
po en ial a e close o expe imen as also expec ed om p e ious wo k on simila sys ems [15,
76,80]. The e o e, one can ake he Fock-35 alues as a eliable p edic ion. In all cases, he
NNN diagonal in e ac ion is p edic ed o be small and o AFM cha ac e , in con as o wha
is sugges ed by some i s o expe imen al da a [81]. Howe e , he mos impo an esul o he
p esen wo k conce ns he cyclic exchange J ing ampli ude. The gene al p ocedu e desc ibed in
de ail in sec ion 2has pe mi ed he i s di ec es ima e o he ampli ude o he J ing e ms om
ab ini io pe iodic calcula ions [82].
Fo La2CuO4, he ou come o he p esen pe iodic app oach using he Fock-35
pa ame e iza ion is in ag eemen wi h he o he only a ailable heo e ical es ima es o
his compound a ising om clus e calcula ions pe o med by ei he explici ly co ela ed
wa e unc ions [42,43] o DFT based calcula ions [45,76] and expe imen al es ima ions
o J[22,32]. In addi ion, he p esen Fock-35 es ima e o J ing ∼35.8meV is in excellen
ag eemen wi h he indi ec e alua ions o Coldea e al (J ing ∼38±8meV) [22] and he
simula ions o Mizuno e al (J ing ∼40meV) [31]. No ice, howe e , ha he J ing/J=0.25 a io
p edic ed he e is smalle han ha epo ed by Toade e al [30], which is o 0.5, bu in excellen
ag eemen wi h a mo e gene ally accep ed a io J ing/J=0.3 [21,22,31] o his compound.
Finally, we would like o poin ou ha he expe imen al de e mina ion o his e m is s ill unde
discussion and e en he exis ence and impo ance o ou -body e ms in La2CuO4ha e aised
some con o e sy [83]. The p esen wo k p o ides unbiased i s p inciples esul s ha ully
New Jou nal o Physics 9(2007) 369 (h p://www.njp.o g/)

16
Table 2. Nume ical alues (in meV) o he wo- and ou -body exchange
ampli udes in La2CuO4, S 2CuO2F2, S 2CuO2Cl2and Ca2CuO2Cl22D cup a es
ob ained using he Fock-35, (B3LYP), and [UHF] po en ials. A ailable
expe imen al da a a e discussed in sec ions 1and 4.1.
Sys em J JdJ ing J ing/J Tc(max)(K)
140.1 8.8 35.8 0.25
La2CuO4(182.8) (8.9) (53.1) (0.29) 42a
[30.9] [−2.8] [0.7] [0.02]
153.7 5.4 41.1 0.27
S 2CuO2F2(214.6) (10.7) (99.5) (0.46) 46b
[33.1] [0.4] [1.4] [0.04]
130.0 2.8 26.4 0.20
S 2CuO2Cl2(182.0) (8.4) (62.8) (0.34) –
[26.2] [0.1] [0.2] [0.01]
146.7 4.9 26.3 0.18
Ca2CuO2Cl2(196.6) (7.2) (60.0) (0.31) 28c
[32.4] [−0.2] [1.4] [0.04]
aF om [87], b[71], c[88].
suppo he a gumen s o Toade e al in hei eply o Raymond e al [84]. I is also wo h
poin ing ou ha he Fock-35 calcula ions p edic s an AFM Jd∼8.8meV again consis en wi h
he alues p o ided by embedded clus e calcula ions [42,44,77]. Hence, i will be o g ea
in e es o epea he i in Toade e al expe imen s [30] by using he p esen es ima e o bo h
Jand o Jdo , in a bo om up app oach, use he p esen alues o he h ee e ec i e pa ame e
o check consis ency wi h espec o expe imen .
Fo Ca2CuO2Cl2, S 2CuO2F2and S 2CuO2Cl2, he e a e no p e ious heo e ical o
expe imen al alues o he ampli ude o he ing exchange e m. P esen calcula ions p edic
alues o Ca2CuO2Cl2and S 2CuO2Cl2 ha a e sligh ly smalle han he co esponding ones in
La2CuO4. This is consis en wi h la ge NN and NNN dis ances in Ca2CuO2Cl2and S 2CuO2Cl2
compa ed o La2CuO4(3.87 and 3.97 e sus 3.81Å, o NN and ∼5.47 and ∼5.62 e sus
5.38Å o NNN dis ances). Fo S 2CuO2Cl2 he ag eemen be ween he es ima e o he NN
in e ac ion (J∼130meV) and he expe imen al alue (J=125±6meV [85,86]) is excellen ,
he Jdcoupling is also p edic ed o be AFM and he es ima ed J ing/J a io ∼0.20 is somewha
smalle han ha o La2CuO4bu s ill in a ealis ic ange. Fo S 2CuO2F2 he esul s a e close
o La2CuO4ones despi e he di e ences in c ys al s uc u e (3.86 e sus 3.81Å, o NN and
∼5.45 and ∼5.62 e sus 5.38Å o NNN dis ances).
Be o e closing his sec ion i is in e es ing o no e ha he magni ude o Jalone does
no disc imina e S 2CuO2F2and Ca2CuO2Cl2 om La2CuO4(s oichiome ic S 2CuO2Cl2
co esponds o a ex emely s able s uc u e and a emp s o syn hesize doped phases by ca ion
subs i u ion o in e s i ial anion excess ha e been unsuccess ul). Taking he eliable Fock-35
alues, o La2CuO4Tc=42K [87] and J=140.1meV and, simila ly, S 2CuO2F2Tc=
46K [71] bu J=153.7meV whe eas o Ca2CuO2Cl2Tc=28K [88] bu J=146.7meV.
A la ge J alue does no co espond o a la ge Tc. Howe e , J ing poin s owa ds a di e en
New Jou nal o Physics 9(2007) 369 (h p://www.njp.o g/)
17
beha io since J ing =35.8meV o La2CuO4and J ing =41.1meV o S 2CuO2F2whe eas
J ing =26.3meV o Ca2CuO2Cl2. This is in line wi h he esul s o Toade e al sugges ing
ha he J ing plays an impo an ole in de ining he p ope ies o he supe conduc ing phase.
We would like o poin ou ha his is also in line wi h p e ious esul s indica ing ha a linea
ela ionship exis s be ween J/ and Tc[41,76]. This is because, a co ec alue o J/ also
indica es a co ec alue o /Uand hence o J ing/J(∼( /U)2, c equa ion (22) and [45]).
4.2. Spin Hamil onian pa ame e s o he S Cu2O3spin ladde compound
Fo he S Cu2O3spin ladde , en di e en magne ic solu ions ha e been ob ained om he spin
dis ibu ions shown in igu e 2. Due o an excessi e spin us a ion be ween he ladde s, he FM
solu ion becomes exceedingly high in ene gy and indeed physically meaningless. The sho -
ange in a-ladde epulsion o ces he unpai ed elec on o be excessi ely delocalized on he
oxygen a oms. The e o e, he AFM gs solu ion is used as ene gy o igin. Fo he emaining nine
b oken-symme y solu ions wi h di e en magne ic o de s, we de i ed a se o nine equa ions
which a e shown in able 3. These equa ions ha e been de i ed assuming ha only i e magne ic
in e ac ions may ha e non-negligible ampli udes: hese a e NN in e ac ions along he legs Jl,
o he ungs J , he NNN Jdin e ac ions in he plaque es, he in e ac ion be ween legs o
di e en ladde s Ji, and he ou -body ope a o J ing ampli ude. A a iance om he abo e-
discussed cup a es, he esul ing equa ions a e linea ly dependen , he sys em o equa ions is
o e de e mined and a leas -squa e p ocedu e has been used o ind he op imum se o alues
o he i e magne ic in e ac ions conside ed. The op imum alues a e epo ed in able 4. The
consis ency o he se o equa ions is almos comple e since he s anda d de ia ion be ween he
DF calcula ed ene gies and hose ob ained om he equa ions in able 3and he so-ob ained se
o magne ic in e ac ions is o 1.5meV o he B3LYP esul s and e en smalle o he Fock-35
se . This consis ency indica es ha all non-negligible in e ac ions ha e been included in he spin
model Hamil onian.
Fo all he me hods he dominan in e ac ions a e he couplings ac oss he ungs (J ) and
along he legs (Jl) which all me hods p edic o be nea ly equal in magni ude—i.e. (J /Jl)≈1—
as expec ed om he simila i y o he Cu–O–Cu exchange pa hs in ag eemen wi h p e ious wo
magne ic si es clus e model calcula ions [36] al hough a mode a e clus e size dependence
on he magne ic in e ac ion along he leg has been obse ed [77]. Ne e heless, enla ging
he clus e model by in oducing a hi d coppe a om belonging o he e y close neighbo
ladde esul s again in J /Jl≈1 [71]. In any case, his dependence e idences he di icul y
o embedding ini e clus e s o ce ain ypes o ma e ials and suppo s he p esen pe iodic
app oach ha does no depend on he clus e design used o ep esen he eal ma e ial.
The accu acy o he p esen calcula ions can be judged om he ampli ude p edic ed
o J h ough he mos ealis ic Fock-35 po en ial (∼155meV) which is consis en wi h he
ecen Raman esponse expe imen s [35]. The es ima ed Jd alue is AFM as in he 2D cup a es
and a non-negligible in e ladde FM exchange (Ji≈ −0.22Jl) is ound and, consequen ly, one
should no conside his sys em as o med by non-in e ac ing ladde s. Rega ding J ing, bo h
i s ampli ude and he J ing/J a io a e la ge han o 2D cup a es ( able 2), bu s ill close o
0.3 in ag eemen wi h a ious se s o expe imen al in o ma ions [21]–[24]. The ag eemen is
pa icula ly good wi h he alues epo ed by Schmid e al [35] al hough hese au ho s do no
conside Ji. This neglec is p obably he eason o he s ong aniso opy in hei i ing which
esul s in a Jl/J =1.5 a io, a alue which is di icul o accep om he Cu–O–Cu dis ances.
New Jou nal o Physics 9(2007) 369 (h p://www.njp.o g/)
18
Figu e 2. Schema ic ep esen a ion o he magne ic solu ions calcula ed o
ex ac Jl,J ,Jd,Jiand J ing pa ame e s in he ladde cup a e S Cu2O3.
Con inuous (dashed) lines co espond o FM (AFM) alignmen s o Jland J
in he ladde s. The g ay zone de ines he cell used o he symme y-b oken
solu ion.
Such la ge aniso opy be ween Jland J is no suppo ed om he p esen pe iodic calcula ions.
We also ound ha epea ing he leas squa e i ing o he ene gies in able 2bu neglec ing he
e ms in ol ing J ing esul s in a la ge aniso opy in he hus J and Jlcalcula ed alues. These
esul s s ongly sugges e ising p e ious i ings using he p esen es ima es as s a ing poin s.
New Jou nal o Physics 9(2007) 369 (h p://www.njp.o g/)
19
Table 3. Ene gy exp essions pe Cu a om ( ela i e o AFM gs phase) o he
magne ic solu ions calcula ed o ex ac Jl,J ,Jd,Jiand J ing in he S Cu2O3
ladde compound ( o he de ini ion o hese in e ac ions see sec ion 2). These
co espond o he magne ic solu ions gi en schema ically in igu e 2. The las
column shows he co esponding calcula ed ene gy alues (in meV) o he
Fock-35, (B3LYP), and [UHF] po en ials.
Phase Ene gy exp ession Values in S Cu2O3
AFM9 Jl/6+ J /6−3Jd/2−0.05J ing 44.98 (57.15) [10.79]
AFM8 Jl/4−Jd/4+ Ji/8 31.92 (48.85) [6.86]
AFM7 Jl/2−Jd/2+ Ji/4 67.38 (96.23) [14.23]
AFM6 Jl/2−Jd/2−Ji/4 83.78 (114.37) [18.53]
AFM5 Jl/4+ J /8−Jd/4−Ji/8−0.075J ing 57.68 (75.75) [13.35]
AFM4 J /4−Jd/2 38.87 (50.22) [8.14]
AFM3 Jl/4+ J /8−Jd/4−0.075J ing 53.43 (70.73) [12.24]
AFM2 J /8−Jd/4 16.20 (20.41) [4.07]
AFM1 Jl/8+ J /16−Jd/8−0.0375J ing 26.68 (35.39) [6.15]
AFM gs 0 0.0 (0.0) [0.0]
Table 4. Nume ical alues (in meV) o he wo- and ou -body exchange
ampli udes in S Cu2O3ladde compound ob ained using he Fock-35, (B3LYP),
and [UHF] po en ials. A ailable expe imen al da a a e discussed in sec ions 1
and 4.
Sys em JlJ JdJiJ ing J ing/J
153.1 155.6 2.7 −34.2 48.8 0.31
S Cu2O3(216.3) (204.1) (5.4) (−36.0)(97.0) (0.47)
[32.7] [32.7] [0.1] [−8.8] [0.04] [ < 0.01]
4.3. C i ical analysis o he esul s
Tables 2and 4summa ize he esul s o he e ec i e spin Hamil onian pa ame e s o he
2D cup a es and he ladde compound, espec i ely, ob ained by means o he UHF me hod,
B3LYP and Fock-35 hyb id po en ials. Since he Fock-35 alues a e in ag eemen wi h a ailable
expe imen al da a i is clea ha B3LYP alues a e o e es ima ions. This esul me i s a u he
deepe analysis. As well known, he NN magne ic coupling Jscales as 2/U(c equa ion (12)).
P e ious wo ks ha e shown ha alues a e no e y sensi i e o he exchange po en ial
[57,77]. This sugges s ha he exceedingly la ge J alue p edic ed by he B3LYP me hod
e ec i ely implies a oo small U alue and hence an insu icien on-si e wo-elec on epulsion,
as discussed elsewhe e [57,80]. This inco ec desc ip ion o elec on–elec on co ela ions in
he B3LYP unc ional has a much mo e d ama ic consequence. In ac , i has been shown by
Mal ieu and Maynau [28] ha J ing scales as 4/U3(c equa ion (22)) and, hence, a oo small U
alue will esul in an e en la ge o e es ima ion o J ing as clea ly shown in ables 2and 4. This
in e p e a ion is suppo ed by he esul s ob ained om he UHF me hod. In his case he alues
New Jou nal o Physics 9(2007) 369 (h p://www.njp.o g/)
20
La2CuO4Ca2CuO2Cl2
S 2CuO2F2
S 2CuO2Cl2
Figu e 3. Rep esen a ion o a omic spin densi ies on cen e i(ρi(I)), de ined
acco ding o Mulliken popula ion analysis, e sus he numbe o opposed
spins on NN si es (nNNβ(I)) o he di e en Imagne ic ‘phases’ o he 2D
cup a es using UHF, Fock-35 and B3LYP me hods. A linea i , as sugges ed
by equa ion (21), is also shown.
o Ja e la gely unde es ima ed due o he lack o elec onic co ela ion e ec s, a well-known
e ec [39,55,56,90,91]. The e o e, one may conclude ha UHF la gely o e es ima es he
on si e wo-elec on epulsion and, consequen ly, J ing ob ained a his le el o heo y mus be
e y small. This is indeed he case, he UHF es ima e o J ing is anishingly small. These esul s
a e mani es a ions o he sensi i i y o J ing/Jwi h espec o he | /U| a io; his is J ing/J
scaling as | /U|2. I J(calc)/J(exp)=λ, hen J ing(calc)/J ing(exp)≈λ3. To conclude, he DFT
based app oaches e ec i ely include elec onic co ela ion e ec s ha lead o a dec ease o
he e ec i e Upa ame e which was se e ely o e es ima ed by he mean- ield Ha ee–Fock
me hod. Howe e , he B3LYP po en ial appea s o lead o unphysical oo low U alues wi h
impo an consequences in he esul ing physical desc ip ion.
Sc u iny o he a omic spin densi ies, de ined acco ding o Mulliken popula ion analysis,
b ings consis en addi ional in o ma ion. In o de o check he alidi y o equa ion (21), we
epo in igu e 3 he dependence o he spin densi ies on he me al cen e s as a unc ion o
he numbe o opposed spins on NN si es. No ice ha his in o ma ion is ex ac ed om he
New Jou nal o Physics 9(2007) 369 (h p://www.njp.o g/)

21
a ious magne ic ‘phases’ o each compound. As sugges ed by equa ion (20), a good linea
dependence o he local spin densi y on he numbe o spin al e na ions is obse ed o each o
he Hamil onians. This esul e idences he di e en s eng hs o he elec onic delocaliza ion
in hese h ee Hamil onians and he slope o he co ela ion can be used o es ima e he /U
a io. Fo La2CuO4, he /Ues ima ed a ios a e 0.044, 0.093 and 0.110 o UHF, Fock-35 and
B3LYP, espec i ely. These esul s nicely compa e wi h p e ious es ima es o his a io om
ini e clus e calcula ions: 0.080 and 0.120 o Fock-35 and B3LYP [57]. Also, hese esul s
con i m he gene al end discussed abo e, namely an excessi e localiza ion in UHF, esul ing
in a oo la ge unsc eened e ec i e U, and a sligh ly oo s ong delocaliza ion in B3LYP wi h
a concomi an unde es ima ion o he e ec i e U. In he ladde compound, he analysis o he
co ela ion wi h espec o he numbe o spin al e na ions is no uni ocal since he bonds a e
no iden ical and, he e o e, is no ca ied ou .
As men ioned in sec ion 2, i is obse ed ha some highly us a ed and high ene gy sel -
consis en solu ions p o ide local spin densi ies which de ia e signi ican ly om he beha iou
expec ed om equa ion (21). This is due o he physically meaningless s ong me al- o-ligand
delocaliza ion o he magne ic elec ons. Hence, he spin densi y analysis p o ides a c i e ion
o elimina e hese spu ious solu ions. This p oblem has been essen ially obse ed in he ladde
whe e he J alues a e la ge as a esul o he sho e Cu–Cu dis ances and, also, due o he FM
in e ladde in e ac ion. As a consequence, he o al spin us a ion becomes exceedingly cos ly.
Acco dingly, we s ongly ecommend using a su icien ly la ge se o low ene gy solu ions in
he i ing p ocedu e.
5. Summa y, conclusions and possible ex ensions
The ex ac ion o NN spin couplings in magne ic la ices om symme y-b oken mean- ield
calcula ions has been equen ly pe o med on molecula sys ems and solids (see [48] and
e e ences he ein). The p esen wo k uses he bene i o he la ge mul iplici y o symme y-
b oken pe iodic solu ions and p oposes a new, gene al and unbiased scheme o p edic
he ampli ude o he pa ame e s de ining a gene al spin Hamil onian om DFT pe iodic
calcula ions. An impo an p ope y o he p esen p ocedu e is ha one does no need o
make any assump ion on he ela i e ampli ude o hese e ms. Ins ead, i elies on a mapping
app oach be ween he ene gy o pe inen magne ic solu ions and he diagonal e ms o he spin
Hamil onian in a local ep esen a ion. In addi ion, he o e de e mined se o equa ions p o ides
a es o he comple eness o he in e ac ions conside ed in he spin model. I has also been
shown ha he spin densi ies o he symme y-b oken solu ions may be a ionalized and hey
u nish addi ional in o ma ion on he /U a io h ough equa ion (20). Hence, he analysis o he
spin densi y dis ibu ions o hese a ious solu ions p o ides use ul and consis en in o ma ion
on he /U(o J/ ) a io and, hence, on he elec onic delocaliza ion. As a co olla y one can
poin ou ha he obse ed co ela ion be ween Tca op imal doping and he J/ a io [41,
76] can be o mula ed as well as a co ela ion wi h /U a io ha b ings in a mo e physically
in ui i e desc ip ion. Finally, his wo k also poin s ou he ex eme sensi i i y o he ou -body
ope a o s ampli udes o he choice o he densi y unc ional and especially o he a io o Fock
exchange. Fo he ime being he use o a 35% a io has led o consis en esul s o cup a es and
o he ansi ion me al compounds. The ques ion o he ans e abili y o his pa ame ic quan i y
o o he magne ic ions emains open.
New Jou nal o Physics 9(2007) 369 (h p://www.njp.o g/)
22
A e y impo an ou come o he p esen wo k conce ns he p edic ion o he ou -body
cyclic ope a o s in he gene al spin Hamil onian in equa ion (2). The applica ion o he gene al
p ocedu e discussed a leng h in sec ion 2 o a se ies o compounds p o ides an independen
con i ma ion o he impo ance o ou -body e ms in hese ma e ials. In pa icula , o La2CuO4
he p esen esul s p o ide u he suppo o he conclusions by Toade e al [30] based
on he analysis o neu on di ac ion expe imen s. Mo eo e , we supply eliable alues o
spin Hamil onian pa ame e s o o he key compounds such as Ca2CuO2Cl2, S 2CuO2F2and
S 2CuO2Cl2. These pe mi us o conclude ha he impo ance o ou -body e ms is likely o be
simila o mos o he HTCS ela ed cup a es and e en mo e impo an in ladde compounds,
in ag eemen wi h he ecen expe imen s o Schmid e al [35]. Likewise, i is also sugges ed
ha he i o neu on sca e ing da a should be e ised by conside ing al e na i e alues o bo h
Jand Jdmagne ic coupling e ms. In addi ion, he p esen s udy p o ides u he e idence ha
he ou -body e m and he FM in e -ladde exchange in oducing spin us a ion be ween legs
in he S Cu2O3ladde should no be neglec ed.
To conclude his s udy, i is wo h men ioning some possible gene aliza ions o he
p ocedu e p esen ed in sec ion 2. One may, o ins ance, wonde whe he he six-body
spin ope a o s would ha e signi ican ampli udes in such la ices. The cyclic ci cula ion o
elec ons in six-membe ings— ela ed o he so-called chemical a oma ici y—is esponsible
o he appea ance o such ope a o s a he six h-o de o pe u ba ion heo y. I has been
demons a ed [28] ha while he ou -body ope a o esul s in a coupling
= 40
4
/U
3
H
ˆ
he six-body ma ix elemen esul ing in a o a ion o six spins is gi en by
= 504
6
/U .
5
H
ˆ
The la ge p e ac o is due o he huge numbe o p ocesses leading o a ull pe mu a ion o
spins in a six-membe ed ing. Now, no ice ha a se o wo used plaque es de ines such a six-
membe ed ing. I would be wo h checking whe he , despi e he smallness o he /U a io,
such many-body ope a o s a e negligible in cup a e la ices. F om he pe u ba i e a gumen s
abo e, and in iew o he calcula ed /U a io, one expec s he a io Jsix- ing/J ou - ing ∼0.1.
Consequen ly, his ques ion has no been add essed in he p esen s udy.
Ye ano he gene aliza ion would conce n la ices wi h S>1/2 magne ic si es such as
Ni(d8)ions, NiO being he pa adigm o hese ma e ials. The same s a egy is applicable
o e alua e no only he NN in e ac ions bu ou -body exchange ampli udes as well. This
will be conside ed in u he wo k which will also show ha one may, in p inciple,
use symme y-b oken solu ions o e alua e he possible impo ance o biquad a ic spin
exchange e ms.
New Jou nal o Physics 9(2007) 369 (h p://www.njp.o g/)
23
Acknowledgmen s
Financial suppo om he F ench CNRS (UMR5626), Spanish Minis e io de Educacion y
Ciencia (Ramon y Cajal esea ch con ac (I de PRM) and p ojec s CTQ2005-08459-CO2-01,
UNBA05-33-001), he Gene ali a de Ca alunya (p ojec s 2005SGR00697, 2005 PEIR 0051/69
and Dis inció pe a la P omoció de la Rece ca Uni e si à ia g an ed o FI), Eu opean
Communi y (p ojec COST-D26/0008/02) and F ench–Ca alan in e na ional coope a ion p ojec
(PICS 1458) is ully acknowledged.
Re e ences
[1] Ma is D C 2006 The Theo y o Magne ism Made Simple (Singapo e: Wo ld Scien i ic)
[2] Kahn O 1993 Molecula Magne ism (New Yo k: Wiley)
[3] Yosida K 1998 Theo y o Magne ism, Sp inge Se ies in Solid-S a e Sciences ol 122 (Be lin: Sp inge )
[4] Ca a R J 2000 J. Am. Ce am. Soc. 83 5
[5] Dago o E 1994 Re . Mod. Phys. 66 763
[6] Rice T M 1998 Physica B 241
[7] Hi oi Z, Azuma M, Takano M and Bando Y 1991 J. Solid S a e Chem. 95 230
[8] Takahashi H, Mo i N, Nakanishi T, Naga a T, Ueha a M, Akimi su J and Kinoshi a K 1997 Physica B
112 237
[9] Azuma M, Hi oi Z, Takano M, Ishida K and Ki aoka Y 1994 Phys. Re . Le . 73 3463
[10] Teske C L and Mülle -Buschbaum H 1970 Z. Ano g. Allg. Chem. 379 234
[11] Hjo h M and Hyld o J 1990 Ac a Chem. Scand. 44 516
[12] Ami T, C aw o d M K, Ha low R L, Wang Z R, Johns on D C, Huang Q and E win R W 1995 Phys. Re . B
51 5994
[13] Sapiña F, Rod íguez-Ca ajal J, Sanchís M J, Ibañez R, Bel án A and Bel án D 1990 Solid S a e Commun.
74 779
[14] Picke W E 1989 Re . Mod. Phys. 61 433
Picke W E 1989 Re . Mod. Phys. 61 749 (co ec ion)
[15] Mo ei a I de P R, Illas F and Ma in R L 2002 Phys. Re . B65 155102
[16] Nagaosa N 1997 Science 275 1078
[17] Takahashi M 1977 J. Phys. C: Solid S a e Phys. 10 1289
[18] Roge M and Del ieu J M 1989 Phys. Re . B39 2299
[19] Schmid H and Ku amo o Y 1990 Physica C167 263
[20] Honda Y, Ku amo o Y and Wa anabe T 1993 Phys. Re . B47 11329
[21] Lo enzana J, E oles J and So ella S 1999 Phys. Re . Le . 83 5122
[22] Coldea R, Hayden S M, Aeppli G, Pe ing T G, F os C D, Mason T E, Cheong S-W and Fisk Z 2001 Phys.
Re . Le . 86 5377
[23] Sugai S, Sa o M, Kobayashi T, Akimi su J, I o T, Takagi H, Uchida S, Hosoya S, Kaji ani T and Fukuda T
1990 Phys. Re . B42 R1045
[24] B ehme S, Mikeska H-J, Mülle M, Nagaosa N and Uchida S 1999 Phys. Re . B60 329
[25] Ma suda M, Ka suma a K, Eccles on R S, B ehme S and Mikeska H-J 2000 Phys. Re . B62 8903
[26] Cas illa G, Chak a a y S and Eme y V J 1995 Phys. Re . Le . 75 1823
[27] Fab icius K, Klümpe A, Löw U, Büchne B, Lo enz T, Dhalenne G and Re cole schi A 1998 Phys. Re . B
57 1102
[28] Mal ieu J P and Maynau D 1982 J. Am. Chem. Soc. 104 3021
[29] Roge M, Del ieu J M and He he ing on J H 1983 Re . Mod. Phys. 55 1
[30] Toade A M, Go J P, Roge M, Shannon N, S ewa J R and Ende le M 2005 Phys. Re . Le . 94 197202
New Jou nal o Physics 9(2007) 369 (h p://www.njp.o g/)
24
[31] Mizuno Y, Tohyama T and Maekawa S 1998 Phys. Re . B58 R14713
Mizuno Y, Tohyama T and Maekawa S 1999 J. Low Temp. Phys. 117 389
[32] K eime B e al 1992 Phys. Re . B46 14034
[33] Anne J F, Ma in R M, McMahan A K and Sa pa hy S 1989 Phys. Re . B40 2620
[34] Kim Y J e al 1999 Phys. Re . Le . 83 852
[35] Schmid K P, Gössling A, Kuhlmann U, Thomsen C, Lö e A, G oss C and Assmus W 2005 Phys. Re . B
72 094419
[36] de G aa C, Mo ei a I de P R, Illas F and Ma in R L 1999 Phys. Re . B60 3457
[37] Mo ei a I de P R, Illas F, Calzado C J, Sanz J F, Mal ieu J P, Ben Amo N and Maynau D 1999 Phys. Re . B
59 R6593
[38] Calzado C J, Sanz J F, Mal ieu J P and Illas F 1999 Chem. Phys. Le . 307 102
[39] Calzado C J, Sanz J F and Mal ieu J P 2000 J. Chem. Phys. 112 5158
[40] Muñoz D, Illas F and Mo ei a I de P R 2000 Phys. Re . Le . 84 1579
[41] Mo ei a I de P R, Muñoz D, Illas F, de G aa C and Ga cia-Bach M A 2001 Chem. Phys. Le . 345 183
[42] Calzado C J and Mal ieu J P 2001 Phys. Re . B63 214520
[43] Calzado C J and Mal ieu J P 2001 Eu . Phys. J. B21 375
[44] Calzado C J, de G aa C, Bo das E, Caballol R and Mal ieu J P 2003 Phys. Re . B67 132409
[45] Calzado C J and Mal ieu J P 2004 Phys. Re . B69 094435
[46] Illas F, Mo ei a I de P R, Bo ill J M and Fila o M 2004 Phys. Re . B70 132414
[47] Illas F, Mo ei a I de P R, Bo ill J M and Fila o M 2006 Theo . Chem. Acc. 115 587
[48] Mo ei a I de P R and Illas F 2006 Phys. Chem. Chem. Phys. 81645
[49] Mo ei a I de P R and Illas F 1997 Phys. Re . B55 4129
[50] Rica J M, Do esi R, Roe i C and Saunde s V R 1995 Phys. Re . B52 2381
Rica J M, Do esi R, Roe i C and Saunde s V R 1997 Phys. Re . B55 15942 (e a um)
[51] Do esi R, Rica J M, Saunde s V R and O lando R 1995 J. Phys.: Condens. Ma e 77997
[52] Reinha d P, Mo ei a I de P R, de G aa C, Illas F and Do esi R 2000 Chem. Phys. Le . 319 625
[53] Becke A D 1993 J. Chem. Phys. 98 1372
[54] Becke A D 1988 Phys. Re . A38 3098
[55] Ma in R L and Illas F 1997 Phys. Re . Le . 79 1539
[56] Illas F and Ma in R L 1998 J. Chem. Phys. 108 2519
[57] Calzado C J, Cab e o J, Mal ieu J P and Caballol R 2002 J. Chem. Phys. 116 3985
[58] Mo ei a I de P R and Do esi R 2004 In . J. Quan um Chem. 99 811
[59] F anchini C, Baye V, Podloucky R, Paie J and K esse G 2005 Phys. Re . B72 045132
[60] Mo ei a I de P R, Do esi R, Roe i C, Saunde s V R and O lando R 2000 Phys. Re . B62 7816
[61] Wannie G H 1937 Phys. Re . 52 191
[62] Ha ison W A 1989 Elec onic S uc u e and he P ope ies o Solids (New Yo k: Do e ) p 87
[63] Mo ei a I de P R, Suaud N, Guihé y N, Mal ieu J P, Caballol R, Bo ill J M and Illas F 2002 Phys. Re . B66
134430
[64] S echel E B and Jennison D R 1988 Phys. Re . B38 4632
[65] Hybe sen M S, S echel E B, Schlu e M and Jennison D R 1990 Phys. Re . B41 11068
[66] Du and P, Mal ieu J P and Lawley K P 1987 Ab Ini io Me hods in Quan um Chemis y ol 1 (New Yo k:
Wiley) chap e 4
[67] Bloch F 1958 Nucl. Phys. 6329
[68] des Cloizeaux J 1960 Nucl. Phys. 20 321
[69] Mille L L, Wang X L, Wang S X, S assis C, Johns on D C, Fabe J J and Loong C K 1990 Phys. Re . B41
1921
[70] A gy iou D N, Jo gensen J D, Hi e man R L, Hi oi Z, Kobayashi N and Takano M 1995 Phys. Re . B51
8434
[71] Al-Mamou i M, Edwa ds P P, G ea es C and Slaski M 1994 Na u e 369 382
New Jou nal o Physics 9(2007) 369 (h p://www.njp.o g/)