HAL Id: hal-00003331
https://hal.archives-ouvertes.fr/hal-00003331
Submitted on 19 Nov 2004
HAL is a multi-disciplinary open access
archive for the deposit and dissemination of
sci-entific research documents, whether they are
pub-lished or not. The documents may come from
teaching and research institutions in France or
L’archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d’enseignement et de
recherche français ou étrangers, des laboratoires
Electron delocalization in mixed-valence Keggin
polyoxometalates. Ab initio calculation of the local
effective transfer integrals and its consequences on the
spin coupling.
Nicolas Suaud, Alejandro Gaita-Arino, Juan Modesto Clemente-Juan, Jose
Marin-Sanchez, Eugenio Coronado
To cite this version:
Nicolas Suaud, Alejandro Gaita-Arino, Juan Modesto Clemente-Juan, Jose Marin-Sanchez, Eugenio
Coronado. Electron delocalization in mixed-valence Keggin polyoxometalates. Ab initio calculation
of the local effective transfer integrals and its consequences on the spin coupling.. Journal of the
American Chemical Society, American Chemical Society, 2002, 124, pp.15134. �hal-00003331�
polyoxometalates. Ab initio al ulation of the lo al e e tive
transfer integrals and its onsequen es on the spin oupling.
Ni olas Suaud
, Alejandro Gaita-Ari~no, Juan Modesto Clemente-Juan, Jose San hez-Marn and Eugenio Coronado
. Instituto de Cien ia Mole ular, Universidad de Valen ia,
C/ Do tor Moliner 50, 46100 Burjassot, Spain. email: Ni olas.Suauduv.es, Eugenio.Coronadouv.es
We present a quantitative evaluation of the in uen e of the ele tron
trans-fer on the magneti properties of mixed-valen e polyoxometalates redu ed
by two ele trons. For that purpose, we extra t from valen e-spe tros opy ab
initio al ulations on embeddedfragments, the valueof the transfer integrals
between W nearest-neighbor atoms in a mixed-valen e PW 12
O 40
polyox-owolframate Keggin anion. In ontradi tion with what is usually assumed,
we show that the ele tron transfer between edge-sharing and orner-sharing
WO 6
o tahedra have very lose values. Considering fragments of various
ranges, we analyse the a ura y of al ulations on fragments based on only
two WO 5
pyramids whi h should allow a low ost general study of transfer
parametersin polyoxometalates. Finally, these parameters are introdu ed in
an extended Hubbard hamiltonian that models the whole anion. It permits
toprove thatele tron transfersindu ea largeenergy gapbetween the singlet
ground state and the lowest triplet states providing a lear explanation of
the diamagneti properties of the mixed-valen e Keggin ions redu ed by two
Polyoxometalates (POM) are a lass of inorgani ompounds with a remarkable degree of mole ularand ele troni tunabilities that impa tindis iplines asdiverse as atalysis [1℄, medi ine[2℄andmaterialss ien e [3℄. These ompoundsare mole ularmetal oxidesmainly based onV, Mo and W ions intheir highestoxidation states. Dueto the lusterstru ture, POM are spe ially useful as model systems for the studies of magneti and ele troni in-tera tions. Indeed, many of these stru tures allowthe in lusion of well-isolated lusters of paramagneti ions with various nu learities and de nite topologies and geometries [4℄. On the other hand, they permit ontrolled inje tion of ele trons, giving rise to mixed-valen e spe ies inwhi hdelo alizedele trons may oexistand intera twith lo alizedmagneti mo-ments. In this ontext they provide unique systems for the developmentof new theories in the mixed valen earea.
Some typi al stru tural types are the Lindquist [5℄ (M 6 O 2 19 ;M = Mo;W), Keggin [6℄ (XM 12 O m 40
;X = P;Si;Co;Ni;:::) and Dawson-Wells [7℄ (P 2 M 18 O n 62
). The Keggin stru -tureis onstitutedoffouredge-sharingtriadsofMO
6
o tahedraarrangedaroundtheXatom in su h a way that the resultingspe ies has a tetrahedral symmetry. The Keggin stru ture an also be viewed as formed by a XO
m 4
lathrate en apsulated into a neutral M 12
O 36 sphere based on orner sharing MO
5
square-pyramids whose api al oxygen atoms points outsidethe sphere [8,9℄(see Fig. 1a). They are thussometimenoted asXO
m 4 M 12 O 36 . Animportantpropertyofthepolyoxometalatesanionsisthattheiridentityisusually pre-served byreversibleredoxpro esses[10℄,forming"heteropolyblues"or"heteropolybrowns" redu tionprodu tsbyadditionofvariousele tronswhi haredelo alizedoverthesphere[11℄. Experimentallyithas been found thatwhen the redu ed spe ies ontain aneven numberof delo alized ele trons, their spins are always ompletely paired, even at roomtemperature. This result is general and has been found not only in the Keggin stru ture but also in the otherones. Itwasinitiallyattributedtoaverystrongantiferromagneti ouplingviaa mul-tiroute superex hange me hanism [12℄, but more re ently it has been theoreti ally shown that a ombination of ele tron repulsion and ele tron delo alization an also stabilize the singlet groundstate [13{15℄.
The extended-Hubbard modelHamiltonianused for these al ulations handles with the e e tive parameters orrespondingto the main mi ros opi intera tions:
6
the t 0
ele tron transfer (hopping) parameter between edge-sharing o tahedra ,
the U on-site ele tron repulsion between ele trons belongingto the magneti orbital of asame metal enter,
the ve V 1
:::V 5
inter-site ele tron repulsion orresponding to the ve inequivalent pairs of metal enters.
Usingthis modelit maybepossibletoqualitativelyexplainthe strongantiferromagneti oupling between the pair of ele trons in the redu ed Keggin anion. However, the spa e spanned by this number of parameters is by far too large to permit an univoque solution of the problem. In order to redu e the size of this spa e and thus to draw a pi ture of the oupling between the two delo alized ele trons, independent information on the values of theseparametersisessential. Themainaimofthepresentarti leistoevaluatetheoreti ally someof theseparameters. Torea hthis goalwewill al ulatethetransferparameters tand t
0
by using very a urate ab initio methods. These values will be then introdu ed in the extended Hubbard model in order to obtain information on the lowest lying spin levels of the system and on the e e tive ouplingbetween the pair of ele trons that are delo alized over the Kegginstru ture.
II. EMBEDDED FRAGMENTS
Thetransfer (hopping) e e tiveintegralsare essentiallylo alparameters [16℄. Therefore fragment spe tros opy al ulations an be used to a urately evaluate their values. The fragments are embedded in a bath adapted to reprodu e the main e e ts of the rest of the rystal,namely,the short-rangePauliex lusionand thelong-rangeMadelungpotential. Thisbath onsistsinalargenumberofpun tual hargesand total-ionpseudopotentials[17℄ (TIP's). A quasi-spheri al embedding of pun tual harges is obtained by repla ing all the atoms surrounding the fragment (those loser than 20
A from the enter of the onsidered Keggin anion) by pun tual harges. The total-ion pseudopotentials TIP's are put in the position of all the atoms of the rst and se ond shells en losing the fragment. In refer-en es [18, 19℄ a omplete des ription of embedded fragment spe tros opy al ulations is reported aswell asa dis ussion of the a ura yof the embeddingpro edure.
represented by the PW 4
O 20
tetramer (Figure 1b). This fragment is based onfour adja ent WO
6
o tahedra and on the PO 4
en apsulated lathrate. W1 and W2 belong to the same triad (and thus belong to edge-sharingWO
6
o tahedra); W3 and W4 are part of the same triad (and thus belong to edge-sharing WO
6
o tahedra) adja ent to the triad of W1 and W2. W1andW4,ononeside, andW2andW3,ontheotherside, belongto orner-sharing o tahedra. Su h a fragment thus permits the ele tron delo alization along orner-sharing o tahedra (t parameter), along edge-sharing o tahedra (t
0
parameter) and between se ond neighborso tahedra (t
d
parameter). Figure1 representstheW
4 O
16
fragment,extra tedfromthepre edentPW 4 O 20 fragment by substituting the PO 4
mole ule by the appropriated pun tual harges and TIP's. It is thus omposed of four orner-sharing WO
5
pyramids. A omparison between the values of the transfer parametersobtained onthe PW
4 O 20 and W 4 O 16
fragments permitstoevaluate the role that the PO
4
plays on the ele tron delo alization. This is so be ause pun tual hargesandTIP's donotsupportany basisset,andthus annot heldanybridgingpathway between W enters. Therefore, assuming that pun tual harges and TIP's mimi properly theele trostati andPauliex lusione e ts,theroleofthePO
4
ontheele trondelo alization is the di eren e between the values of the transfer parameters al ulated on the PW
4 O 20 and on the W 4 O 16 fragments.
The smallest fragments we have onsidered are formed by the dimeri units W 2 O 9 and W 2 O 10
represented in gure 1d. In these three ases the W, O and P ions that belong to the 4W-based fragments but not to the 2W-based fragments are modelled by pun tual harges and TIP's added to the total embedding. The two W
2 O
9
fragments, based on orner-sharing pyramids, are extra ted from edge-sharing o tahedra (rightmost fragment) and orner-sharing o tahedra (fragment onthe left). The two pyramids of the W
2 O
10 frag-mentdonotshareany atom. Thesefragmentsa ordthe independent evaluationoft
0
, tand t
d
parameters, respe tively. Thus, a omparison with the previous results should permit to he k the a ura yof these al ulations.
Asthe stru ture of the PW 12
O 40
luster only hangesslightly when the harge ompen-sating ationsofthesaltare hanged, allthe al ulationsreportedinthisworkare basedon theX-ray rystallographi al oordinates[8℄ofthe(H
5 O + 2 ) 3 (PW 12 O 3 40
)salt. Inthis ase,the T
d
of all the W ions indu e the equivalen e of the intera tions W1 $ W2 and W3 $ W4 (alongedge-sharingo tahedra)aswellasthe intera tionsW1$W4andW2$W3(along orner-sharingo tahedra)andoftheintera tionsW1$W3andW2$W4betweense ond neighboro tahedra.
III. MODEL HAMILTONIANSAND EXTRACTIONOF PARAMETERS
In the redu ed Keggin ion, the unpaired ele trons are essentially delo alized over the d
xy
-like orbital of ea h of the W ions (pointing in between the equatorial O ions of the o tahedron). Hen e, a model hamiltonian that takes into a ount the transfer (hopping) parameter between d
xy
-likeorbitalsof adja ent W ions and the Coulombrepulsionbetween the two extra ele trons is well suited to reprodu e the ele tron delo alization and spin oupling in this system [14℄. We fo us in this arti le on the al ulation of the transfer parameters. The extra tion of the e e tive ele trostati repulsions from similar ab initio al ulations willbethe aimof a forth oming paper. For anevaluation of these parameters based onpure ele trostati onsideration,see referen e[14℄.
A. 4-W-ions based fragments
Letus onsiderone ele trondelo alizedoverthe4-W-ionsbasedfragments(fragments1b and 1 ). The model hamiltonianindu es the following doubletstates:
1 = d 1 +d 2 +d 3 +d 4 2 (1) 2 = d 1 d 2 d 3 +d 4 2 (2) 3 = d 1 +d 2 d 3 d 4 2 (3) 4 = d 1 d 2 +d 3 d 4 2 (4) whered 1 ,d 2 ,d 3 andd 4
standfortheSlaterdeterminants onstru tedwhentheextraele tron is situatedonW1,W2,W3and W4, respe tively.
1 4 2 3 t 0 =hd 1 jHjd 2 i=hd 3 jHjd 4 i (6) t d =hd 1 jHjd 3 i=hd 2 jHjd 4 i (7)
The transition energies are:
E 1!2 =E 2 E 1 = 2t 0 2t d (8) E 1!3 =E 3 E 1 = 2t 2t d (9) E 1!4 =E 4 E 1 = 2t 2t 0 (10) where E 1 , E 2 , E 3 and E 4
are respe tively the energy of the states 1 , 2 , 3 and 4 ; E 1!2 , E 1!3 and E 1!4
being respe tively the transition energy between the states 1 and 2 , 1 and 3 , 1 and 4 .
The transfer parameters are extra ted from the omparison of the transition energies al ulated by ab-initio methodswith equations 8,9 and 10.
B. 2W-ions based fragments
Ea h one of the three 2W-ions based fragments (Fig. 1d) supports only one kind of ele trontransfer. Thus, thefragmentontheleftisused forextra tingt,whilethe rightmost one allows the evaluationof t
0
; nally,from the fragmentat the bottom we an extra t the value of t
d .
The transition energy E DD
between symmetri and antisymmetri doublet states predi tedbythemodelhamiltonianisthusrelatedtothetransferparameterbytherelation:
E DD =E D+ E D =2t i
where t stands for t, t or t depending on the fragment we are dealing with. E D+
and E
D
are the energy of the doublet states D
+
and D
, respe tively. The symmetry of these two fun tions are those of d
1 +d 2 and d 1 d 2
. Thus, forthe W 2 O 9 fragments D + is the antisymmetri doublet and
D
is the symmetri one, whereas in the W 2 O 10 fragment D +
isthe symmetri doublet and D
is the antisymmetri one.
Hen e, the energies of the symmetri and antisymmetri doublet states obtained by ab initio al ulationsdire tly givethe values of the transfer parameters.
IV. COMPUTATIONAL METHODOLOGY
Onea hofthepreviouslydes ribedfragmentsweperformedbothCASSCFandCASPT2 al ulations using the MOLCAS suite of programs [20℄.
A ording to the physi s of mixed-valen e ompounds that permits to di erentiate the orbitalsdependingof their ontributiontoele trontransfer,the CASSCFpro eduredivides the mole ularorbitals intothree subspa es:
the ina tive orbitals that remain doubly o upied;
the a tive orbitals whose o upation is allowed to hange;
the virtual orbitals that remain uno upied.
TheCompleteA tiveSpa e(CAS)isthende ned, foragivennumberofa tiveele trons (those o upying the a tive orbitals),as the set of all the Slater determinants that an be builta ordingtothepreviouso upationrules. TheCASSCF onsistsintheself- onsistent optimizationofallthe orbitalsandofthe oeÆ ientsofthewavefun tiondevelopped onthe CAS.Polarizationand orrelationoftheele tronsofthea tiveorbitalsarethusvariationally taken intoa ount in the mean eldof the ele trons of the ina tive orbitals.
Fromthiszeroth-orderCASPT2goesfurtherandtakesintoa ountthedynami al polar-izationand orrelatione e tsinase ond-orderperturbativetreatment,thatisasasumation of intera tions that do not intera t between them. Even if CASPT2 is more pre ise than CASSCF,wepresentbothresults. Indeed,the di eren esbetween thetwomethodsa ount for the in uen e of the dynami al e e ts onthe transfer parameters providingat the same time a he king of the suitability of the CASSCF to bea goodzeroth-order forCASPT2.
methodbutoftenrequiresmu hlarger al ulationsthanCASPT2. IntheDDCImethodthe dynami alpolarizationand orrelatione e ts are taken into a ountvariationallystillonly those e e ts that ontribute(at the se ond-order of perturbation) tothe energy di eren es between the states of the CAS are taken into a ount. This method requires the diago-nalization of the matrix representative of the DDCI spa e generated by all the following ex itationson allthe determinantsof the CAS:
1-hole(one ele tron is ex ited fromanina tive orbitalto ana tive orbital),
1-parti ule (a tive!virtual), 1-hole-1-parti ule(ina t.!virt.), 2-holes (ina t.+ina t.!a tive+a tive), 2-parti ules (a t.+a t.!virt.+virt.), 2-holes-1-parti ule(ina t.+ina t.!a t.+virt.), 1-hole-2-parti ules(ina t.+a t.!virt.+virt.).
Besidestheex itationswithintheDDCIspa e,otherdoubleex itationsarealsopossible, the2-holes-2-parti ulesex itationswhi hbythewayarethemostnumerous(foras hemati representation of the determinants of the DDCI spa e and of the 2-holes-2-parti ules ex i-tations ofthe CAS, see Figure2). If we only onsider energy di eren es, these out of spa e ex itations an be ignored when a ommon set of Mole ular Orbitals (MO) are used for all the al ulations [21℄. Indeed in this ase their e e t is only to shift the energy of all the states of the CAS by the same energy (at the se ond-order of perturbation). All the al ulations were thus performed onthe set of MO optimized atthe CASSCF level for the lowest doubletstate.
Due to the size of the matri es to diagonalize, the DDCI al ulations have only been performedonthe 2W-basedfragmentsinorder toobtain resultsasa urate aspossibleand to ompare these resultswith thoseobtained with theCASPT2 method. TheDDCIresults were obtained with the CASDI suite of programs [23℄.
xy
based on one ele tron and 4 orbitals in the ase of 4W-based fragment al ulations (the orresponding states are those of se tionIIIA); it is redu ed to one ele tron and 2 orbitals in the ase of 2W-basedfragment al ulations.
In all the al ulations, the inner- ore ele trons ([1s 2 2s 2 2p 6 3s 2 3p 6 4s 2 3d 10 4p 6 5s 2 4d 10 4f 14 ℄ fortheW atomsand[1s
2
℄fortheOandP atoms)arerepresentedbye e tive orepotentials (ECP). The outer- ore and valen e ele trons are represented using a 13s10p9d5f primitive basisset ontra tedto3s3p4d2f fortheW,a5s6p1dprimitivebasisset ontra tedto2s4p1d forthe Oatomsthatbelongtothe oordination sphereoftheW treatedinthefragments,a 5s6p1dprimitive basis set ontra tedto 1s2p1dfor the other O atoms (the two O the PO
4 mole ulenot linked tothe W of the fragments) anda 7s6p1dprimitivebasis set ontra ted to 1s1p1d for the P atom. Exa t expressions of the basis sets and ECP an be found in referen e [24℄.
V. RESULTS AND DISCUSSION
A. Results on 4-W-ions fragments
The results obtained on the 4-W-ions fragments (fragments b and in gure 1) are summarized intable I.
As itis generallyobserved inmixed-valen e ompounds [16, 18, 19℄, we an observethat the dynami al e e ts (the di eren e between CASPT2 and CASSCF results)remain small and donot hange signi antly the values of the transfer parameters.
In ontrast with what may be expe ted, t and t 0
have very similar values. This is a surprising resultasthe W O W angle between the two W ions and thebridgingoxygen (the oxygen that doesnot belong to the PO
4
mole ule)has very di erent values. For edge-sharing o tahedra this angle is 127and for orner-sharing o tahedra it is 153, whereas theW Odistan e havevery losevalues,1.964
Aand 1.957
Arespe tively. Thus, thisangle seems not to be the only relevant parameter involved in the ele tron transfer. It would be interesting toget abetterunderstanding of allthe relevantgeometri alparametersthat a t on the ele tron transfer integrals and an explain why t ' t
0
. Cal ulations in this ontext are presented inthe subse tion VC.
The value oft isabout 4timessmallerwith respe t tothoseof tand t, andof thesame sign.
Finally, the very small di eren es between the results extra ted on the PW 4 O 20 and W 4 O 16
fragments,less than 15meV, permit toextra t two important on lusions:
thebridginge e tsduetotheOanionsofthePO 4
mole uleareverysmall,asit ould have been expe ted due to the large distan e ('2:5
A) between these anions and the W atoms. Note that the ele tron transfer pathways from one W atom to the other through the PO
4
mole uleare allowed inthe PW 4
O 20
fragmentbut not inthe W 4
O 16 fragment.
the e e ts of the atoms of the PO 4
mole ule on the transfer integrals are a urately reprodu ed by the hoosen harges and TIP's. Whereasin the W
4 O
16
fragmentthese atomsare modelledby pun tual harges and TIP's, they are treatedwith basis set in the PW
4 O
20
fragment.
These results learly show the aptness of al ulations on four orner-sharing pyramids embedded fragment but, even for this redu ed fragment, the al ulations are too large to treat systems where the symmetry between W ions is redu ed. Another problem arises fromthe meaningofthe t
d
parameter. Indeed,aswe extra ted threeparametersfromthree energy di eren es, we ould not he k the aptnessof this parameter tomodelthe system.
Weshowinthe next se tionhow al ulationsondimerfragmentspermittoliftthesetwo problems.
B. Dimer al ulations
Theresultsofthe al ulationsperformedonthe three2W-basedfragmentsare presented in tableII.
We an see that the results on t and t 0
ompare very well with the results extra ted from the W
4 O
16
fragment with di eren es smaller than 3% at the CASSCF level as well as at the CASPT2 level. On the one hand, this on rms that, as usual in mixed-valen e ompounds, the ele tron transfer in the Keggin anion is essentially a lo al parameter and an be a urately al ulated on fragments based on the intera ting metal ions and on the
ofthe WO 5
pyramidsontheatomsofthe fragmentare orre tlyreprodu edbythe hoosen pun tual harges and TIP's. As the same pun tual harges and TIP's were used in the whole embedding, we thus an assume that the whole embedding orre tly reprodu e the main e e ts of the rystal onthe fragments [19℄ ( he kings of the aptness of anembedding based on a limited number of pun tual harges to reprodu e the Madelung eld will be presented in a forth omingpaper [25℄ in the ase of another polyoxometalate). Hen e, the dimer fragment an be onsidered aquitegoodmodelto extra tthe transfer parameters in a very eÆ ient and simple way.
Con erning t d
, we an see that the values extra ted from dimer al ulations are signi - antlydi erentfromthoseextra tedfrom4W-basedfragments, 102meV versus 125meV at the CASPT2 level. Indeed, whereas in the two other fragments all the losest neighbors of the bridging O anions are in the fragment, the environment of the bridging O anions in this ase is not so a urately treated sin e part of the losest neighbors of these anions are modelled by harges and TIP's. Nevertheless, the dimer al ulations prove that the t
d parameter has a non-negligible value and justi ate our hoi e to take this parameter into a ount in the model hamiltonian. The t
d
value obtained from 4W-based fragment is not anartifa t due tothe extra tionof three parameters fromthree transition energies and we an trust its value.
Thus, a urate values ofele tron tranfers an be obtained from al ulationson orre tly embedded smallfragments based onlyon two WO
5
pyramids.
C. In uen e of the W O W angle on the transfer parameter
We have shown that those O ions shared by the o taedraand the PO 4
mole uledo not supportany pathwayforthe ele tron transfer. Thus, the delo alizationonlyo ursthrough the orbitalsof the other O ions sharedby twoo taedra. We an wonder if the W O W angleformedbythelateroxygenandthetungstenionsistheonlyrelevantparameterforthe ele tron transfer, ifit an explain why the valuesof t and t
0
onlydi erby about 40-50meV, and if it ould be possible to predi t the value of these parameters (or at least the ratio between them) onlyfromthese rystallogra data.
valuesof tandt. Thuswewilluse thesefragmentstogetmoreinformationsonthe e e tof the W O W angleonthe valuesof tand t
0
. Forthat purpose, weperformed al ulations onmodel fragmentsobtained by rotatingea hofthe WO
5
pyramids aroundthe bridgingO atomsin the plane ontainingthese atomsand the two W atoms. 20modelfragments were formed (ten for the tranfer between edge-sharing o taedra and ten for the tranfer between orner-sharing o taedra) orresponding tovariationsof the W O W angles of-5, -4, -3, -2, -1, +1, +2, +3, +4and +5around the real angles given by the x-ray stru ture (152.4for orner-sharing WO
6
o taedra and 126.8for edge-sharing o taedra). Wedid not perform al ulationonmore distorted fragmentsas they shouldbe toofar from the real stru ture togiverelevant values. The results are represented in gure 3.
We observe that when the W O W angle in reases the transfer parameters in rease in absolute value. This is due to the in rease of the overlap between the magneti orbital of the W ions and the bridging orbital of the O ion. Su h a variation is of the same orderof magnitudefor ornerand edge-sharingfragmentswithatangentat152.4of about
4:6meV/and of about 3:9meV/at 126.8(atthe CASPT2 level).
Ifthe W O W angle was the only geometri al parameter a ting on the transfer pa-rameter, model fragments basedeither on orner-sharing o taedraoredge-sharing o taedra should give the same value for the transfer parameter when the W O W angle is the same. Extrapolations of the urves presented in gure 3 learly show that this is not the ase.
Thus, other stru tural parameters than the W O W angle have strong e e ts on the ele tron transfer and should be taken intoa ounteither toexplain why the values of t and t
0
are lose ortoroughly predi tthe values of the ele tron transfer parametersinother polyoxometalate lusters.
D. Comparison between CASPT2 and DDCI results.
We have now established the validity of 2W-based fragments omparing, at the same level of al ulations, the values for the transfer parameters extra ted from fragments of various nu learities. It is thus possible on these small fragments to go further than the perturbative treatment of the dynami al e e ts (CASPT2 method) and perform a variational al ulation of the transfer parameters with the DDCI methods. Sin e the
2W-basedfragments al ulations givea goodevaluation ofthe order of magnitudeof the t parameter but with errors of about 20%, it is useless to intend to perform more a urate al ulations on su h a fragment (and it is at the moment te hni ally impossible to obtain DDCI results on 4W-based fragments). We thus only performed DDCI al ulations to evaluate the t and t
0
parameterswhi h are the onlyrelevant ones in the theoreti al models proposed to des ribe the magneti behavior of two-ele tron-redu ed Keggin ions [14℄. The results are the following:
t= 467meV t
0
= 507meV
Thesearethemosta urateresultspresented inthiswork. Di eren eswiththeCASPT2 al ulations (on the same fragments)are very small, of the order of 20meV, less than 5%.
Thus, the transferele tronbetweenW of orner-sharing WO 6
o tahedraandbetweenW of edge-sharing o tahedra have similar intensity, whatever the level of al ulation and the fragmentused for these al ulations.
E. Predi tion of the Singlet-Triplet energy gap.
Theoreti almodels[14℄were previouslydevelopped topredi t,infun tionof thetransfer and ele trostati repulsion parameters, the magneti behaviour of a two-ele tron-redu ed Keggin anion. Figure 4 represents the energy levels of the supported singlet and triplet lowest states.
Thevaluesofthetransferintegralsobtainedfromabinitio al ulationsgivearatiot 0
=jtj' 1:086(verti aldashedlineinFigure4). Usingtheseresultsthemodelpredi tsasinglet
1 A
1 ground state and that the rst ex ited state is a
3 T
2
triplet state. The energy gap between thesestatesisthusabout0:6t,thatis280meV. This verylargevaluepermitstounderstand the diamagneti properties of two-ele tron-redu ed Keggin anion ompounds [12℄.
VI. CONCLUSION
In this arti le we have quantitatively shown for the rst time how the ele tron hopping o urringinmixed-valen eKegginpolyoxometalates anpromoteastrongantiferromagneti ouplingbetween the pair of delo alized ele trons.
extended Hubbardtypemodelthatallowstoevaluatethe e e t ofthe ele tron transferand ele tron repulsion on the properties of the whole mixed-valen e luster, with ab initio al- ulationsthatprovidea urate values ofthe transfer parametersfrom al ulationsonsmall fragments of the Keggin luster. These fragments permit large on guration intera tion DDCI al ulations,and thus omparison with multireferen e se ond-order pertubative the-ory CASPT2 results, providing at the same time a general and eÆ ient method to extra t these parameters inall the polyoxometalatestru tures.
Asaperspe tive, we wantto fo us on the omputationalresour es needed forthis work. Whereas the al ulations on 4W-based fragment were onlypossible thanks tothe power of the enter of al ulation of the University of Valen ia,those performedon dimerfragments (at CASSCF, CASPT2 and DDCI levels) requires less then 10Gb of hard disk, 300Mb of memory and about 30 hours on a 866Mhz-PIII "home" omputer. Our work thus intents tobethe groundingof a low ost general study of the transfer parameters inmixed-valen e polyoxometalates ompounds,and in other kindsof high nu learity mixed-valen e system.
A knowledgments
Thisresear hwas nan iallysupportedbyEuropeanCommunity(NetworkMolnanomag, nHPRN-CT-1999-00012), by the spanish Misterio de Cien ia y Te hnologia (MAT2001-3507) and by the Generalitat Valen iana (GV01-312). We thank Marie-Bernadette Lepetit for many fruitfuldis ussions.
[1℄ I. V. Kozhevnikov, Chem. Rev. 98, 171 (1998); N. Mizuno and M. Misono, Chem. Rev. 98, 199 (1998); M. Sadakane and E.Ste khan,Chem. Rev.98, 219 (1998).
[2℄ J. T. Rhule,C. L.Hill,D. A.Juddand R.F. S hinazi,Chem. Rev. 98, 327(1998).
[3℄ A. Muller,F. Peters, M. T. Pope andD.Gattes hi, Chem. Rev.98, 239 (1998); E.Coronado andC.J.Gomez-Gar a,Chem.Rev.98,273(1998); W.G.KlempererandC.G.Wall,Chem. Rev. 98, 297 (1998); M. T. Pope and A.Muller,"Polyoxometalates: From Platoni Solids to Anti-RetroviralA tivity", KluwerA ademi Publishers: Dordre ht, TheNetherlands,1994.
Clemente-Juanand E.Coronado,Coord. Chem. Rev.XX, 361 (1999); [5℄ I. Lindquist,Arkiv. Kemi.5,247 (1952).
[6℄ J. F.Keggin, Nature131, 908 (1933). [7℄ B. Dawson, A ta Cryst. 6,113 (1953).
[8℄ G. M. Brown, M.-R. Noe-Spirlet, W. R. Busing and H. A. Levy, A ta Cryst. B 33, 1038 (1977).
[9℄ C. Ro hi ioli-Delt he , R. Thouvenot and R. Fran k, Spe tro him. A ta 32A, 587 (1976); R. A erete, N. Casa~n-Pastor, J. Bas-Serra and L. C. W. Baker, J. Am. Chem. So . 111, 6049 (1989); L. P. Kazansky and B. R. M Garvey, Coord. Chem. Rev. 188, 157 (1999); J. M. Maestre, X. Lopez, C. Bo, J.-M. Poblet and N. Casa~n-Pastor, J. Am. Chem. So . 123, 3749 (2001);X.Lopez, J.M. Maestre, C. Boand J.-M Poblet, J.Am. Chem. So .123, 9571 (2001).
[10℄ N. Casa~n-Pastor, P.Gomez-Romero, G.B. Jamesonand L.C.W. Baker,J. Am.Chem. So . 113, 5658 (1991).
[11℄ M. T. Pope,Isopoly and Heteropoly Metalates,Springer-Verlag,Berlin, 1983.
[12℄ M. Kozik, C. F. Hammer, and L. C. W. Baker, J. Am. Chem. So . 108, 2748 (1986); M. Kozik,andL.C.W.Baker, J.Am. Chem.So .109,3159(1987); M.Kozik,N.Casa~n-Pastor, C. F. Hammer, and L.C. W. Baker, J.Am. Chem. So .110,1697 (1988); M. Kozik, and L. C. W. Baker, J. Am. Chem. So .112,7604 (1990). N.Casa~n-Pastor,and L.C. W.Baker, J. Am. Chem. So .114,10384 (1992);
[13℄ S.A. Borsh h and B.Bigot, Chem. Phys. Letters 212, 398 (1993).
[14℄ J. J.Borras-Almenar, J.M. Clemente-Juan,E.Coronado andB. S.Tsukerblat, Chem.Phys. 195, 1(1995).
[15℄ J. J.Borras-Almenar, J.M. Clemente-Juan,E.Coronado andB. S.Tsukerblat, Chem.Phys. 195,17(1995);J.J.Borras-Almenar,J.M.Clemente-Juan,E.CoronadoandB.S.Tsukerblat, Chem. Phys. 195,29 (1995).
[16℄ I. de P. R. Moreira, F.Illas, C. J. Calzado, J. F. Sanz, J. P. Malrieu, N. Ben Amor and D. Maynau,Phys. Rev. B59, 6593 (1999).
[17℄ P.Durandet al,Theor. Chim. A ta 38, 283 (1975).
JimenezCalzado, J.Fernandez SanzandJ. P.Malrieu,J.Chem. Phys. 112,5158 (2000). [19℄ N.SuaudandM.-B.Lepetit,Phys.Rev.B62, 402(2000); N.SuaudandM.-B.Lepetit,Phys.
Rev. Lett. 88, 056404(2002).
[20℄ MOLCAS suite of programs. K. Anderson, M. P. Fuls her, G. Karlstrom, R. Lindh, P. A. Malqvist,J.Olsen,B.Roos,A.J.Sadlej,M.R.A.Blomberg,P.E.M. Siegbahn,V.Kello,J. Noga, M. Urban,P.O. Widmark,MOLCAS Version 4,Dptof Theor. Chem.,Chem. Center, Univ. of LUND,P.O.B. 124,S-221 00 Lund,Sweden, Lund1994.
[21℄ J. P.Malrieu,J.Chem. Phys. 47,4555 (1967).
[22℄ R. Broer and W.J.A. Maaskant, Chem. Phys. 102, 103 (1986); J. Miralles, O. Castell, R. Caballol and J. P. Malrieu, Chem. Phys. 172, 33 (1993); J. Cabrero, N. Ben Amor, C. de Graaf, F. Illasand R.Caballol,J.Phys. Chem. A 104,9983 (2000).
[23℄ CASDI suite of programs. D. Maynau, N. Ben Amor and J. V. Pitar h-Ruiz, University of Toulouse,Fran e(1999);N.BenAmorandD.Maynau,Chem. Phys.Letters286,211(1998); J. V.Pitar h-Ruiz, J.San hez-Marnand D. Maynau, J.Comp. Chem, inpress.
[24℄ Z.Barandiaran and L.Seijo,Can. J. Chem.70, 409 (1992).
(b)
(c)
(d)
W1
W2
W4
W3
P
O
O
O
O
W1
W2
W4
W3
W4
W3
W1
W4
W2
W4
(a)
Figure 1: (a) S hemati stru tureof a Keggin anion. (b) ThePW 4 O 20 fragment. ( ) The W 4 O 16 fragment. (d) The three 2W-based fragments. The oxygen atoms o upy the orners of the o tahedraorpyramids. The atomsmodelledbypun tual harges andTIP's arenotrepresented.
Virtual
Orbitals
Orbitals
Orbitals
Active
Inactive
CAS
1-hole
1-part.
1-h + 1-p
2-h
2-p
2-h + 1-p
1-h + 2-p
2-h + 2-p
Figure 2: S hemati representation of the single and double ex itations on the CAS. The spa e spannedbyfCAS, 1-h,1-p, 1-h+1-p,2-h, 2-p, 2-h+1-p,1-h+2-pg orrespond to theDDCI spa e. The numberof orbitalsand ele tronsinthe gure is arbitrary.
126.8 130
140
150152.4
W-O-W angle in degrees
-550
-500
-450
Electron transfer value in meV
CASSCF
CASPT2
CASSCF
CASPT2
Figure 3: Variations of the value of the ele tron transfer parameter with the W O W angle aroundtherealstru ture. The urvesontheleftrepresentthevariationobtainedfromedge-sharing fragments, the urvesontherightrepresent thevariationobtainedfrom orner-sharingfragments.
0
1
0
0.5
-0.5
0
-1.086
t’ / | t | ( for t < 0 )
-6
-5
-4
-3
-2
-1
0
E/|t|
1
A
1
3
T
2
1
T
2
1
E ,
3
3
T
1
Figure 4: Theoreti alenergy levelsof thedoublyredu edKeggin ion. Thedotted lines standsfor tripletstates, thesolidlinesforsingletstates(seeref.[14 ℄forthedenominationofthestates). The verti aldashed linerepresentsthevalueof the al ulatedt
0
4 20 4 16 t CASSCF -506 -510 CASPT2 -428 -443 t' CASSCF -551 -551 CASPT2 -470 -479 t d CASSCF -87 -89 CASPT2 -123 -125
Table I: ResultsinmeV of the al ulationson the4W-based fragments.
CASSCF CASPT2 t -510 -445 t' -560 -490 t d -80 -102
