Density functional theory of spin-coupled models for …...structure of large ~e.g., 60–120 atoms! transition metal-containing complexes. The Kohn–Sham32,33 Hamiltonian takes into

JOURNAL OF CHEMICAL PHYSICS VOLUME 116, NUMBER 14 8 APRIL 2002

Density functional theory of spin-coupled models for diiron-oxo proteins:Effects of oxo and hydroxo bridging on geometry, electronicstructure, and magnetism

Jorge H. Rodrigueza) and James K. McCuskerb)

Department of Chemistry, University of California at Berkeley, Berkeley, California 94720

~Received 24 October 2001; accepted 17 January 2002!

We have performed a comprehensive study of the electronic structure and magnetic properties ofstructurally characterized models for diiron-oxo proteins. Results from Kohn–Sham densityfunctional theory show that two complexes, with formula Fe2(m-O!~m-O2CCH3)2(HBpz3)2 and@Fe2(m-OH!~m-O2CCH3)2(HBpz3)2#1, are strongly and weakly antiferromagnetically coupled,respectively, in agreement with experiment. The physical origin of the stronger and weakerexchange typically measured for oxo- and hydroxo-bridged diiron complexes, respectively, has beenelucidated. The main superexchange pathways giving rise to molecular antiferromagnetism in bothcomplexes have been identified. The dominant pathway in the oxo-bridged complex,Fe1(dxz):m-O(px):Fe2(dxz), was formed byp interactions whereas that of the hydroxo-bridged,Fe1(dz2):m-OH(pi):Fe2(dz2), was formed bys interactions. We also found a pathway mediated bythe bridging acetates, Fe1(dx22y2):bis~m-acetato!:Fe2(dx22y2), which induces weakantiferromagnetism in the oxo-bridged complex but is significantly more important in thehydroxo-bridged complex. The antiferromagnetic exchange constants that parameterize theHeisenberg HamiltonianH5JS1"S2 have been predicted for both, strongly and weakly, coupledcomplexes. Overall, the signs, trends, and magnitudes of the theoretical values~Jm-O

calc

51152.7 cm21, Jm-OHcalc 5123.3 cm21! were in excellent agreement with experiment. The

geometries of the complete molecular structures have been optimized inC2v symmetry and used tocalculate molecular properties such as atomic charges and spin densities. The electronicconfigurations (Fe:4s0.293d5.93,m-O:2s1.922p4.99;Fe:4s0.303d5.82,m-OH:2s1.822p5.25,H:1s0.51) ofthe respective binuclear cores revealed relatively high occupancies for the nominally ferric ions,thus reflecting a donating character of their immediate N3O3 coordination. In addition, thediiron-oxo protein hemerythrin has been discussed. Theoretical and structural considerationsindicated that the oxo-bridged diferric complex considered herein models extremely well theantiferromagnetic behavior of azidomet- and azidometmyo-hemerythrin. Finally, the magneticbehavior of closely related oxo-bridged diferric and hydroxo-bridged diferrous complexescontaining Me3TACN capping ligands has been explained in light of the results presented in thiswork. © 2002 American Institute of Physics.@DOI: 10.1063/1.1461363#

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I. INTRODUCTION

A number of homobinuclear1–6 and heterobinuclear5–7

first row transition metal complexes that display remarkaelectronic and magnetic properties have been synthesA general characteristic of these complexes is the preseof two paramagnetic spin centers that interact~m-O!bis~m-acetato! ligands and couple antiferromagnecally. Diiron complexes of this type are of particular biologcal relevance since their oxo-bis~acetato!-bridged cores~Scheme 1! constitute structural models for active sitesnon-heme diiron-oxo6,8,9 proteins. Magnetic studies~e.g.,susceptibility,1,2,5 Mossbauer10–12! of diiron centers in modecompounds and metalloproteins consistently show a strdependence on the nature of their bridging ligands. Indeupon protonation of their oxo bridges, the strength of an

a!Present address: Department of Physics, Purdue University, West Lette, IN 47907-1396. Electronic mail: [email protected]

b!Present address: Department of Chemistry, Michigan State University,Lansing, MI 48824-1322.

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ferromagnetic exchange, as quantified by the HeisenbHamiltonianHHB5JS1"S2 , generally drops by an order omagnitude:

Scheme 1.

In oxo-bridged binuclear complexes, such as those sied in this work~Fig. 1!, the magneticd orbitals ~i.e., thosehosting unpaired electrons! of the two metal sites can overlavia p orbitals of their ~nominally! diamagnetic bridgingligands. As a result of such indirect overlap, a number

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license or copyright, see http://ojps.aip.org/jcpo/jcpcr.jsp

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6254 J. Chem. Phys., Vol. 116, No. 14, 8 April 2002 J. H. Rodriguez and J. K. McCusker

FIG. 1. Fe31 – O–Fe31(HBpz3)2 andFe31 – OH–Fe31(HBpz3)2: structuresobtained from geometry optimizationat the U-BPW91/6-31G* level in C2vsymmetry. The views are perpendicular ~top! and parallel~bottom! to theFe1–mO–Fe2 or Fe1–mOH–Fe2planes.

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superexchange~i.e., bridge-mediated! pathways can beformed which favor an antiparallel alignment of the electrspins. Such antiferromagnetic pathways can be represeas idealized electronic configurations of the forFe1(d1i

↑ ):m-O(p↑↓):Fe2(d2i↓ ). Here, the atomic orbitals o

the iron sites can have equal~i.e., d1i5d2i! or mixed ~i.e.,d1iÞd2i) local symmetry.5,13,14 Depending on the oxidationand spin state of the cations involved,11 molecular symmetry,and geometry,15–19 there may be several pairs of interactinelectrons.

Although detailed papers5,13,14 have given insight abouthe mechanisms of spin coupling in oxo-bridged diiron coplexes, there remain fundamental questions to be answeIn particular: What are the physical origins and microscomechanisms giving rise to the dramatically weaker antifermagnetism of hydroxo-bridged complexes relative to thoxo-bridged counterparts? What are the compositions ofdominant exchange pathways before and after protonatiothe oxo bridge? What are the concurrent, but different,fects of geometric variations and proton addition that ocupon protonation of the oxo bridges? Furthermore, althoit is generally recognized that superexchange15,20–23via themain ~i.e., m-oxo or m-hydroxo! bridges is a dominanmechanism of molecular antiferromagnetism,13,15,24the pos-sible roles of the bridging acetates have not been studiedetail. To address these questions, it is necessary to percareful electronic structure calculations, preferably oncomplete molecular structures, at exactly the same levetheory on well characterized oxo- and hydroxo-bridgmodel compounds. It has been our aim to perform suchculations to elucidate several subtle, but fundamental, difences between the electronic and magnetic structures ofand hydroxo-bridged~bio!inorganic diiron complexes.

Despite significant computational efforts, many theorical determinations of exchange constants~J! have only af-forded the right order of magnitude for strongly coupl~e.g., oxo-bridged! diiron and dimanganese complexes.25–30

Greater accuracy has been obtained by Adamoet al.31 for


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bis~m-azido! dicopper complexes. However, theoretical esmates ofJ have generally been significantly less accurate aeven qualitatively incorrect~i.e., predicting ground state ferromagnetism instead of antiferromagnetism or vversa!25,28 for weakly coupled~e.g., hydroxo-bridged! com-plexes. The great difficulty encountered in the accurate pdiction of exchange constants can be traced to severaltors. These include:~i! the extremely small antiferromagnetenergy splittings to be calculated, about 1026– 1027 smallerthan the total molecular energies;~ii ! the inadequacy of someelectronic structure methods to account for inclusion of crelation or proper balance between exchange and correlain spin-polarized metal complexes; and~iii ! the oversimpli-fication of molecular models and geometries used in theculations.

In this work we apply gradient-corrected Kohn–Shadensity functional theory to elucidate the electronic structof two compounds synthesized by Armstrong and Lippard1,2

~Fig. 1!, hereafter referred to as Fe31 – O–Fe31(HBpz3)2

and Fe31 – OH–Fe31(HBpz3)2 . By studying the completemolecular structures and fully optimized geometries of thcomplexes we have obtained a detailed understanding oferal important and related topics. In particular:~i! Whichmetallic and bridging orbitals are involved in the antiferrmagnetism exhibited by these compounds;~ii ! The detailedphysical origin of strong and weak antiferromagnetism inoxo- and hydroxo-bridged complexes, respectively;~iii ! Thespecific and different roles of oxo, hydroxo, and acetbridges; and~iv! The accurate prediction of Heisenberg echange constants for both~i.e., strong and weak! antiferro-magnetic coupling regimes. In addition, by studying a thstructure, Fe31 – O* –Fe31(HBpz3)2 , which corresponds tothe deprotonated optimized geometryFe31 – OH–Fe31(HBpz3)2 , we have been able to differentate the related, but distinct, effects of bridge protonation aconcomitant Fe-mO distance variations. Finally, in light othe results presented in this work, we discuss the mecnisms of antiferromagnetic spin-coupling in~i! the diiron-


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6255J. Chem. Phys., Vol. 116, No. 14, 8 April 2002 Density functional theory for diiron-oxo proteins

oxo proteins azidomet and azidometmyo-hemerythrin~ii ! two closely related model compounds4 hereafter referredto as Fe31 – O–Fe31(Me3TACN)2 and Fe21 – OH–Fe21

(Me3TACN)2 (Me3TACN51,4,7-trimethyl-1,4,7-triazacyclononane!.

II. THEORETICAL BACKGROUND

Kohn–Sham density functional theory~KS-DFT!32–34 isa powerful method for studies of ground state electrostructure of large~e.g., 60–120 atoms! transition metal-containing complexes. The Kohn–Sham32,33 Hamiltoniantakes into account exchange and correlation effects35–37 thatare important for the description of many molecules andparticular12,38 of spin-polarized open-shell transition metclusters.

To allow for spin polarization one can apply a spin urestricted~U! formalism33,39 which allowsa andb electronsto occupy orbitals with different energy and spatial localiztion. The unrestricted Kohn–Sham~UKS! equations giverise to molecular orbitalsf i

UKS(r ) which have eigenvaluese i

anda or b spin index. Although Kohn–Sham orbitals havan auxiliary role,33,34 recent studies have pointed out theproperties. For example, Kohn, Becke, and Parr34 noticedthat all f i

KS(r ) and e i are of great semiquantitative valubecause they reflect correlation effects. BaerendsGritsenko40,41 also recommended the Kohn–Sham orbitand corresponding one-electron energies as tools in quative chemical considerations. They noticed that Kohn–Shorbitals retain their expected bonding or antibonding charter as well as their expected behavior under perturbat~e.g., geometrical distortions!.

To obtain orbitals that mimic those of a pure open shsinglet, it is convenient to impose self-consistent-field~SCF!convergence to a singlet wave function (C1

UKS) of brokenspatial and spin symmetry.42,43 Such wave functions havesymmetry lower than that of the geometric structure andpose fair localization of unpaireda electrons on one iron siteand unpairedb electrons on the other site. Kohn–Shambitals of broken symmetry are regarded30,43,44as approxima-tions to localizednatural magnetic orbitals22,45,46 and, assuch, their overlaps can provide a semiquantitative meaof magnetic interactions in weakly coupled systems.

In general, unrestricted Kohn–Sham wave functionstained from SCF calculations are eigenfunctions ofSz but arenot eigenfunctions of the total spin operatorS2. As a conse-quence, the singlet~i.e., antiferromagnetic! single determi-nant wave functions (C1

UKS) are, in general, spin contamnated and can have strong admixtures from higher sstates42 @i.e., ^S2&UKS.S(S11)#. However, the high spin~i.e., ferromagnetic! states of spin-coupled systems canwell represented21,22,47 by single Slater determinant(C11

UKS). Accordingly, spin unrestricted wave functionsferromagnetic states are generally found [email protected]., ^S2&UKS

'S(S11)# in conventionalab initio ~UHF and MP2!48,49

and also DFT38,47,50calculations.

Heisenberg exchange interaction

Ground state antiferromagnetism in binuclear complearises from interactions between pairs of unpaired electr


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each member of the pair being localized on a different meion. Although the phenomenon of antiparallel spin coupliresembles a magnetic interaction between two spins,physical origin is electrostatic in nature~Heisenberg51! and aconsequence of the antisymmetry requirement of the Pexclusion principle.13,51 Dirac52,53 showed that this interaction, although electrostatic in origin, could be written asscalar product of the spin operators of two electrons. Sformulation was readily extended and applied by VVleck54 to exchange interactions between atoms in mecules and solids. As such, the net spin coupling betwtwo metal ions with single-ion operatorsS1 and S2 can berepresented and quantified by the Heisenberg–Dirac–Vleck Hamiltonian

HHB5JS1"S2 . ~1!

In general, the net spin coupling within a binuclear coplex results from a complex admixture of antiferromagneand ferromagnetic interactions.15,20,21,55Often, the former aredominant and the spin ground state corresponds to the lo~i.e., Smin5S12S2! eigenvalue of the total spin operatorS5S11S2 . In this case, theJ constant of Eq.~1! is positiveand the binuclear complex is said to have an antiferromnetic ground state.

The eigenstates of the Heisenberg Hamiltonian@Eq. ~1!#have energies given by

E2S11HB 5 1

2J$S~S11!2S1~S111!2S2~S211!%. ~2!

It follows from Eq.~2! that the exchange interaction patially removes the degeneracy of the various spin manifogiving rise to the energy splittings shown in Scheme 1.calculate the exchange constant from density functionalculations we can use Eq.~2! in conjunction with spinprojections38,49,56 on the spin unrestricted wave functionIn a previous paper38 we presented expressions for the cculation of exchange constants in complexes where aS5 3

2

cation is bound to aS5 12 radical. Here, we followed

Yamaguchiet al.49,56and used a general expression for symetric or asymmetric binuclear complexes. As shown inAppendix57

J52E2Smax11

UKS 2E2Smin11UKS

Smax~Smax11!2^S2&2Smin11UKS , ~3!

where we approximated the energies of the calculated~UKS!and pure~PUKS! Kohn–Sham wave functions for the highspin state as being equal~i.e., E2Smax11

UKS 'E2Smax11PUKS ). Such

approximation is valid since high-spin~i.e., ferromagnetic!wave functions are essentially free from spin contaminat~vide supra!. An important feature of Eq.~3! is that it isbased on expectation values of energy and spin operatorscan be evaluated directly from converged SCF wave futions. We notice, however, that our application of Eq.~3! isfor Kohn–Sham wave functions from which the proper coputation of^S2& is not straightforward.58

III. MATERIALS AND METHODS

Several combinations of exchange and correlation futionals were used in the DFT calculations. The geome


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optimizations were performed with two different methodThe first is a combination of Becke59 gradient-corrected exchange and Perdew–Wang37,60 nonlocal correlation ~U-BPW91!. The second method uses the three parameterchange of Becke35,59,61,62 in conjunction withLee–Yang–Parr36 correlation ~U-B3LYP!. This latter func-tional constitutes a hybrid approach that combinHartree–Fock39,63–65 with Slater32,66,67 and gradient-corrected Becke35,59,61 exchange. The correlation iU-B3LYP is introduced by a combination of the local annonlocal functionals of Vosko–Wilk–Nusair62 andLee–Yang–Parr,36 respectively. Further single point calculations designed to study ground state electronic structuwere performed with U-BPW91, U-B3LYP, anU-MPW1PW91.68 The Heisenberg exchange couplings wecalculated with the latter functional and a 6-311G* basis.

All electron basis sets were used during the course ofcalculations, namely 6-31G*69,70 and 6-311G*.39,71,72 Foriron we used the basis optimized by Wachters73 and Hay74

using the scaling factors of Raghavachari and Trucks.75 Thex-ray crystallographic structures1,2 were used as startinpoints to perform full geometry optimizations foFe31 – O–Fe31(HBpz3)2 and Fe31 – OH–Fe31(HBpz3)2 .The optimizations imposed convergence toC2v symmetryand used 6-31G* which included 696 basis functions. Addtional single point calculations performed on the optimizgeometries used 6-311G* and included 911 basis functions

Separate calculations were carried out to determineergies and Kohn–Sham wave functions corresponding tolow-spin (C2S1151

UKS ) and high-spin (C2S11511UKS ) states of

Fe31 – O–Fe31(HBpz3)2 and Fe31 – OH–Fe31(HBpz3)2 .The former were similar in nature to the broken-symmewave functions described by Noodleman,47,42,43whereas thelatter were performed in full C2v symmetry. TheGaussian76,77 electronic structure packages were usedcarry out geometry optimizations and single point calcutions. The total energies were obtained using a tight78 con-vergence criterion. The wavefunctions were analyzed wMOLDEN79 to plot electronic density contours. Analysesatomic charge densities, atomic spin densities, and electrconfigurations were performed within the natural populatframework~NPA! developed by Weinholdet al.80–82 In thisapproach, an atom-centered basis set used to expand threstricted Kohn–Sham orbitals is transformed into a coplete orthonormal set of natural localized atomic orbit~NAO!.80,81Computations were carried out on a local doubprocessor SGI-Octane, the HP-Convex SPP-2000 of thetional Center for Supercomputer Applications~NCSA!, andthe IBM-SP2 cluster of Mahui High Performance ComputiCenter~MHPCC!.

IV. RESULTS

A. Geometry optimizations

An accurate knowledge of geometric parameters, in pticular those related to the~m-O!bis~m-acetato! diiron cores,is important for understanding the magnetic properties ofspin-coupled complexes. For instance, it is known tFe–mO bond lengths correlate with the strength of antifer


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magnetic coupling, the magnitude of that coupling decaynearly exponentially with increasing distances.83 Some of themost reliable DFT functionals~e.g., U-BPW91, U-B3LYP!are known to predict high quality, yet slightly different, gometries for the same molecular structures.84 Therefore, toestablish subtle, but fundamental, differences betweenmagnetic properties of Fe31 – O–Fe31(HBpz3)2 andFe31 – OH–Fe31(HBpz3)2 , it was essential to obtain highquality geometries from optimizations performed at exacthe same level of theory.

While performing optimizations of transition metacontaining molecules of some 20–30 atoms is considecomputationally practical,85 the optimization of larger com-plexes is less common due to the considerable computatiexpense. Nevertheless, we have performed full geometrytimizations for Fe31 – O–Fe31(HBpz3)2 ~69 atoms! andFe31 – OH–Fe31(HBpz3)2 ~70 atoms! using an all-electronbasis set. Only optimizations corresponding to the high-s(C11

UKS) states are reported here since such states can berepresented by a single determinant Kohn–Sham wave fution ~Sec. II!. The optimizations were carried out inC2vsymmetry and produced the structures shown in Fig. 1 wparameters given in Table I. The Cartesian coordinates ofoptimized geometries are given as Supplementary tables.bond lengths and angles optimized at the U-BPW91/6-31*and U-B3LYP/6-31G* levels were fairly close. However, thU-B3LYP parameters were, in general, in closer agreemwith the x-ray structures.1,2 In what follows we mainly referto geometries obtained from this latter method.

1. Fe3¿– O – Fe3¿„HBpz 3…2

Distances between individual metal sites and their mbridge ~i.e., m-O! are of special interest. The calculated~U-B3LYP! Fe–mO ~i.e., Fe1–O3 and Fe2–O3! bond lengthswere 1.830 Å, whereas the average x-ray value is 1.7862

While the predicted Fe–mO bonds were about 0.044 Ålonger than the x-ray values, the agreement between theical and experimental values for other bonds~Table I! wasmuch closer. In particular, for Fe1–O4 and Fe1–O5 the cculated values were only 0.01 Å longer whereas for Fe1–Fe1–N9, and Fe1–N10 the calculated values were ab0.02 Å longer. Thus, the trends observed in the x-ray strture were generally well reproduced by the optimizatioFor example, both optimized and x-ray structures displaFe1–N9 and Fe2–N15 bonds which were longer thanother Fe–N bonds.

In terms of magnetic properties, the Fe1–mO–Fe2 angleis the most interesting since it largely determines whd orbitals localized on one iron site can interact, via briding p orbitals, with d orbitals localized on the other ironsite.13,15–19 To some approximation, the nature of thpossible intermetallicd orbital interactions ~i.e., anti-ferromagnetic or ferromagnetic! are given by theGoodenough–Kanamori16–19 rules as extended by Andersoand others.13,15 One rule predicts that when two ions havlobes of magnetic orbitals pointing toward each other in sua way that the orbitals would have a reasonable large oveintegral, the exchange is antiferromagnetic.15 However, suchsimple rules are often used to make qualitative predicti



TABLE I. Selected bond lengths~Å! and angles~deg.! obtained from geometry optimizations at the U-BPW91/6-31G* and U-B3LYP/6-31G* levels inC2v

symmetry for theF11UKS state.

Fe31 – O–Fe31(HBpz3)2 Fe31 – OH–Fe31(HBpz3)2

U-BPW91 U-B3LYP X-raya U-BPW91 U-B3LYP X-raya

Fe1–O3 1.846 1.830 1.783 2.010 1.994 1.960Fe1–O4 2.064 2.051 2.041 2.016 2.004 1.994Fe1–O5 2.064 2.051 2.042 2.016 2.004 2.001Fe1–N8 2.175 2.172 2.153 2.139 2.132 2.108Fe1–N9 2.214 2.217 2.197 2.106 2.103 2.110Fe1–N10 2.175 2.172 2.154 2.139 2.132 2.088

Fe2–O3 1.846 1.830 1.788 2.010 1.994 1.953Fe2–O6 2.064 2.051 2.049 2.016 2.004 2.002Fe2–O7 2.064 2.051 2.042 2.016 2.004 2.000Fe2–N14 2.175 2.172 2.149 2.139 2.132 2.094Fe2–N15 2.214 2.217 2.177 2.106 2.103 2.105Fe2–N16 2.175 2.172 2.150 2.139 2.132 2.108

O3–H44 0.977 0.970 0.702Fe1–O3–Fe2 127.7 126.2 123.5 124.3 123.5 123.0

O3–Fe1–O4 93.4 94.4 96.6 90.6 91.0 91.1O3–Fe1–O5 93.4 94.4 97.1 90.6 91.0 92.8O4–Fe1–O5 92.8 92.3 91.9 90.7 91.2 91.1

O3–Fe2–O6 93.4 94.4 96.4 90.6 91.0 92.2O3–Fe2–O7 93.4 94.4 97.1 90.6 91.0 92.5O6–Fe2–O7 92.8 92.3 90.1 90.7 91.2 90.5

aParameters reported by Armstrong and Lippard~Refs. 1, 2!.

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for the limiting cases of linear~e.g., Fe1–O–Fe25180°! orperpendicular ~e.g., Fe1–O–Fe2590°! geometries. TheFe1–O3–Fe2 angle~'126°! was far from these limiting val-ues. This underscored the need to perform the detailed sof exchange interactions presented in this work~Secs. IV–V!.

The distorted octahedral environment around the invidual iron sites was illustrated by the somewhat differevalues of the various O–Fe–O angles. The calculated OFe1–O4 and O3–Fe1–O5 angles were 2.1° larger thancalculated O4–Fe1–O5 angle. This latter trend was mpronounced in the x-ray structure, which shows O3–Fe1–and O3–Fe1–O5 angles 4.7°–5.2° larger than O4–Fe1–

2. Fe3¿– OH – Fe3¿„HBpz 3…2

The optimized~U-B3LYP! Fe1–O3 and Fe2–O3 bonlengths were 1.994 Å, only 0.01 Å shorter than the oth~i.e., m-acetato! Fe–O bonds. This was in sharp contrastFe31 – O–Fe31(HBpz3)2 where the former bonds were 0.22Å shorter than the latter. Therefore, contraryFe31 – O–Fe31(HBpz3)2 , the Fe1–O3 and Fe2–O3 bondsFe31 – OH–Fe31(HBpz3)2 might not be expected to substatially dominate the electronic structure of its iron sites, tlonger bonds giving rise to a more ionic character and alsa weaker antiferromagnetic interaction. The predicFe1–O3 and Fe2–O3 distances were 0.034 Å and 0.04longer than the x-ray values, respectively. This discrepasuggests that more accurate distances for the pseudo tcenter-bond ~Fe1–mO–Fe2! may be predicted by usinglarger basis sets. However, the other bond lengths giveTable I were in closer agreement with the experimental str


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ture. For instance, for Fe1–O4 and Fe1–O5 the avercalculated-experimental difference was only 0.001whereas for Fe1–N8, Fe1–N9, and Fe1–N10 the calculavalues were within experimental error.

The calculated Fe1–O3–Fe2 angle~123.5°! was fairlyclose to its x-ray value~123°! and slightly more acute thanthe corresponding angle of Fe31 – O–Fe31(HBpz3)2

~126.2°!. We note that for Fe31 – OH–Fe31(HBpz3)2 allO–Fe–O angles had nearly identical values. This was in ctrast to the results obtained for Fe31 – O–Fe31(HBpz3)2

which showed O4–Fe1–O5 and O6–Fe2–O7 being sowhat more acute than the other O–Fe–O angles. This icated that, in comparison to Fe31 – O–Fe31(HBpz3)2 , theiron sites of Fe31 – OH–Fe31(HBpz3)2 had a less distortedoctahedral environment.

In general, there was close agreement betwthe theoretical and experimental parametersFe31 – OH–Fe31(HBpz3)2 . However, there was a significandifference between optimized~0.970 Å! and reported experi-mental~0.702 Å! O3–H44 distances. We have also reportdiscrepancies in the hydrogen positions of anotcomplex.38 Typically, hydrogen positions determined fromx-ray diffraction data correspond to a minimization of rsidual densities and do not necessarily correspond to prpositions.38 By contrast, the calculated distance for thproton corresponded, within the accuracy of the methto its true equilibrium position in a particular molecular eletronic state. Knowledge of the proton positionFe31 – OH–Fe31(HBpz3)2 was important for the accuratprediction of the exchange constant of this weakly coupcomplex.


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B. Energies and composition of delocalized molecularorbitals

The coordinate systems of Fig. 2 have been used toscribe all molecular and atomic orbitals. There were similaties but also significant differences between the electrostructures of Fe31 – O–Fe31(HBpz3)2 , Fe31 – O* –Fe31

(HBpz3)2 , and Fe31 – OH–Fe31(HBpz3)2 . We now de-scribe the energies and most salient features of some mrich frontier orbitals.

1. Fe3¿– O – Fe3¿„HBpz 3…2

Figure 3 and Supplementary Table I show the main coposition and energies of MOs corresponding to the high-sstate (C11

UKS) of Fe31 – O–Fe31(HBpz3)2 . The table alsoshows the labels corresponding to theC2v irreducible repre-sentations. The spin-unrestricted calculations alloweda andb orbitals to have, in general, different energies and spalocalizations. Exchange interactions between majority~a!electrons significantly stabilized their frontier MOs with rspect to the corresponding unoccupiedb orbitals. Within theenergy scale of Fig. 3 there were two groups of occupieaorbitals. The first included the five highest MOs~i.e., 173a-177a! which were mainly composed of Fe(d) with or with-out admixture fromm-O(p) orbitals. The second group comprised orbitals delocalized throughout the HBpz3 ligands~i.e., 166a-172a!. Some lower energy orbitals~not shown!had some metal character but also substantial contribufrom the bridging or capping ligands.

The metal-rich molecular orbitals corresponded to symetric ~S! and antisymmetric~A! combinations of twod or-bitals of equal local symmetry, each localized on a differiron center~Fig. 3!. The occupieda set included MOs witheither dominant or relatively minor contributions from thmetals. By contrast, the corresponding unoccupiedb set con-sisted entirely of MOs mainly centered on the iron sites.of these ~168b-177b! were grouped together significanthigher in energy than the two nearly degenerate, highestcupied, minority orbitals~166b,167b!. For these reasons, wfocused on the unoccupiedb set ~Table II!. The most rel-evant orbitals have been extensively described in the Supmentary material and are shown in Figs. 4–6.

FIG. 2. Local coordinate systems centered on Fe1, Fe2,m-O, andm-OH

used to describe the atomic and molecular orbitals. Thez axes are defined bythe Fe–mO or Fe–mOH bonds and theyz plane coincides with the Fe1–mO–Fe2 or Fe1–mOH–Fe2 planes. Oi and O' are the axes parallel and perpedicular to the Fe1–Fe2 axis, respectively, in the Fe1–mO–Fe2 orFe1–mOH–Fe2 planes.x andOx ~not shown! are perpendicular to the samplanes.


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2. Fe3¿– O* – Fe3¿„HBpz 3…2

Figure 7 and Supplementary Table II show the macompositions and energies of MOs corresponding tohigh-spin state (C11

UKS) of Fe31 – O* –Fe31(HBpz3)2 . Theenergy scheme of Fig. 7 resembled that of Fig. 3. The retive order of the frontier orbitals for both oxo-bridged strutures was nearly the same. The exceptions were the Mof Fe(dyz) and Fe(dxz) composition, which switched ordebetween 169b and 170b, and the MOs of m-O(px)and m-O(p') composition that were higher in energfor Fe31 – O* –Fe31(HBpz3)2 . A comparison of the energysplittings between symmetric and antisymmetric cobinations of Fe(di) orbitals,D i

b5(eSi2eAi), revealed somemeaningful differences. In general, the splittings welarger for Fe31 – O–Fe31(HBpz3)2 than Fe31 – O* –Fe31

(HBpz3)2 . The greatest changes between these two sttures were seen forDyz

b ~0.388 eV!, Dxzb ~0.342 eV! and, to a

lesser extent, forDz2b ~0.111 eV!. By contrast,Dxy

b andDx22y2

b were only slightly smaller for the first structure.Concurrent with variations inD i

b splittings, somemeaningful changes in orbital composition were anoted ~Supplementary Tables I and II!. The metal-richorbitals ~168b-177b! of Fe31 – O* –Fe31(HBpz3)2 had, ingeneral, greater metallic character than thoseFe31 – O* –Fe31(HBpz3)2 . Again, the main differences be

FIG. 3. Energies and main composition of frontier molecular orbitals cresponding to the high-spin~C11

UKS , U-BPW91/6-311G* ! wave function ofFe31 – O–Fe31(HBpz3)2 . Orbital energies andC2v labels are given inSupplementary Table I. S and A indicate symmetric and antisymmetric cbinations of metald orbitals.



TABLE II. Main composition of magnetic orbitals corresponding toF2S11511UKS ~ferromagnetic! U-BPW91/6-311G* wave function of

Fe31 – O–Fe31(HBpz3)2 .

MO Composition Typea % Metal % Bridging unit

168b Fe1(dx22y2):bis~m-acetato!:Fe2(dx22y2) S 57.6 6.2169b Fe1(dyz),Fe2(dyz) A 63.8170b Fe1(dxz),Fe2(dxz) A 60.8171b Fe1(dx22y2):bis~m-acetato):Fe2(dx22y2) A 61.0 5.2172b Fe1(dyz):m-O:Fe2(dyz) S 51.2 9.9173b Fe1(dxz):m-O:Fe2(dxz) S 45.0 20.5174b Fe1(dxy),Fe2(dxy) A 53.2175b Fe1(dxy),Fe2(dxy) S 51.0176b Fe1(dz2):m-O:Fe2(dz2) S 48.4 12.9177b Fe1(dz2):m-O:Fe2(dz2) A 47.0 11.6

aS and A represent symmetric and antisymmetric combinations of locald orbitals, respectively.

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tween the two structures were noticed for MOs ofdxz anddyz symmetry which were more metallic in the first case.contrast, the orbitals ofdxy character~174b and 175b! hadnearly equal composition in both structures and also neequal Dxy

b splittings. Supplementary Table II also revealthat although MOs 166b and 167b were mainly localized onthe bridging oxygen, there was delocalization toward the icenters.

3. Fe3¿– OH – Fe3¿„HBpz 3…2

Figure 8 and Supplementary Table III show the orbienergies of the hydroxo-bridged complex. The relative orof the one electron orbitals were somewhat different frthose of the two oxo-bridged structures~Figs. 3 and 7!. Forinstance, theb orbitals of t2g parentage were more closepacked in the energy scale. Similarly, the orbitals ofeg par-entage were more closely spaced.

A comparison with the energy scheme of Fig. 7 allow


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us to distinguish the effects of bridge protonation from thoof related geometric variations. Upon protonation, the sptings between symmetric and antisymmetric combinationsFe(d) orbitals were, in general, very close to thoseFe31 – O* –Fe31(HBpz3)2 . However, there was an appreciable difference in the value ofDxz

b : this splitting was 0.470eV smaller for Fe31 – OH–Fe31(HBpz3)2 due to the stabili-zation of MO 173b. At the same time, the Fe(dxz) contribu-tion to MO 173b ~&55%! increased substantially relative tthat of the two oxo-bridged geometries. Inversely, the conbution of O(px) to MO 173b ~&8.8%! was significantlylower for Fe31 – OH–Fe31(HBpz3)2 .

C. Broken symmetry calculations onFe3¿– O–Fe3¿

„HBpz3…2 and Fe3¿– OH–Fe3¿„HBpz3…2

The singlet wave functions of broken space and ssymmetry (C1

UKS) were spin contaminated and correspondto an admixture of states with various spin multiplicities47

Accordingly, the U-BPW91/6-311G* expectation values o

s
FIG. 4. Fe31 – O–Fe31(HBpz3)2 : Delocalized MOs 169b and 172b of the high-spin~C11UKS , U-BPW91/6-311G* ! wave function. The isovalue contour plot
correspond to antisymmetric~A, left! and symmetric~S, right! combinations of Fe(dyz) orbitals. The plots are slices taken in the Fe1–mO–Fe2 (yz) plane.


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the total spin operator86,87 were ^S2&154.35 and ^S2&1

54.84 for Fe31 – O–Fe31(HBpz3)2 and Fe31 – OH–Fe31(HBpz3)2 , respectively.88

The single determinants of broken symmetry were chacterized by metal-rich MOs mainly localized on a singiron site. However, important features of some orbitals wtheir additional contributions from bridging ligands and dlocalization tails toward the other iron site. Figures 9–show somed-type a orbitals mainly localized on the lef~Fe1! site of the oxo- and hydroxo-bridged complexes. Tcorresponding set ofb orbitals ~not shown! were mainly lo-calized on the right~Fe2! site and, otherwise, had an equivlent description.

1. Localized molecular orbitals ofFe3¿– O – Fe3¿

„HBpz 3…2

Three occupied MOs involved bonding interactioof Fe atomic orbitals (dz2,dyz ,dxz) with the threem-O(pi ,p' ,px) orbitals. Three additional occupied, highenergy MOs involved the corresponding antibonding intertions. The other two iron orbitals did not interact strongwith the oxo bridge. Instead, MO 145a had mainlydx22y2

composition and, although nominally nonbonding, was decalized toward the acetato bridges. MOs 139a and 171a haddxy character and included bonding and antibonding inter

FIG. 5. Fe31 – O–Fe31(HBpz3)2 : Delocalized MOs 170b and 173b of thehigh-spin~C11

UKS , U-BPW91/6-311G* ! wave function. The isovalue contouplots correspond to antisymmetric~A, left! and symmetric~S, right! combi-nations of Fe(dxz) orbitals. The plots are slices taken 0.5 Å above the FemO–Fe2 (yz) plane.

FIG. 6. Fe31 – O–Fe31(HBpz3)2 : Delocalized MOs 176b and 177b of thehigh-spin~C11

UKS , U-BPW91/6-311G* ! wave function. The isovalue contouplots correspond to symmetric~S, left! and antisymmetric~A, right! combi-nations of Fe(dz2) orbitals. The plots are slices taken in the Fe1–mO–Fe2(yz) plane.


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c-

tions, respectively, with the nitrogen andm-acetato ligands inthe local axial~i.e., xy! planes. The occupieda orbitals in-volved in molecular magnetism are shown in Figs. 9–12 ahave been extensively described in the Supplementary mrial.

2. Localized molecular orbitals ofFe3¿– OH – Fe3¿

„HBpz 3…2

There were similarities but also significant differencbetween interactions of Fe(d) and bridging orbitals in theoxo- and hydroxo-bridged complexes. In both cases, merich MOs involved bonding and antibonding interactionsFe(dz2) and Fe(dxz) with bridging O3(p) orbitals. However,in sharp contrast to Fe31 – O–Fe31(HBpz3)2 , orbitals ofFe1(dyz) symmetry did not interact strongly with O3(p).The other two orbitals of each iron were essentially isolafrom the hydroxo bridge and their MOs had similar compsitions to those corresponding to Fe31 – O–Fe31(HBpz3)2 :MO 145a had substantialdx22y2 character and was delocaized toward the acetato bridges, whereas MOs 127a and172a haddxy character and revealed bonding and antiboing interactions, respectively, with nitrogen andm-acetatooxygens in the axial (xy) planes. The magnetic orbitals havbeen described in the Supplementary material.


UKS , U-BPW91/6-311G* ! wave function ofFe31 – O* –Fe31(HBpz3)2 . Orbital energies andC2v labels are given inSupplementary Table II. S and A indicate symmetric and antisymmecombinations of metald orbitals.


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D. Atomic charges and spin densities

The distributions of molecular charge are shown in TaIII. We found fairly close agreement between results deriv


UKS , U-BPW91/6-311G* ! wave function ofFe31 – OH–Fe31(HBpz3)2 . Orbital energies andC2v labels are given inSupplementary Table III. S and A indicate symmetric and antisymmecombinations of metald orbitals.


ed

from U-BPW91, U-B3LYP, and U-MPW1PW91 wave functions. The latter two~i.e., hybrid! methods produced aslightly greater charge polarization; however, all functionyielded similar trends.

The molecular (C2v) symmetry imposed charge equivalence between the two iron cations of each complex. Forhigh-spin (C11

UKS) states the positive charge was mainly asigned to the Fe ions. Carbons fromm-acetato bridges werealso found to be positively charged. By contrast, all sixoms coordinated to the cations exhibited negative charOxygens corresponding tom-oxo and m-hydroxo groups

-

c

FIG. 9. Localized orbitals of the broken-symmetry~C1UKS ,

U-BPW91/6-311G* ! wave function. The isovalue contour plots are in thFe1–mO–Fe1 or Fe1–mOH–Fe1 (yz) planes. Top: MOs 125a and 172aof Fe31 – O–Fe31(HBpz3)2 . Bottom: MOs 124a and 171a ofFe31 – OH–Fe31(HBpz3)2 .

TABLE III. NPA atomic charge and spin densities calculated from broken symmetry (F1UKS) and high-spin (F11

UKS) wave functions at the U-BPW91/6-311G*level.

Atom

Fe31 – O–Fe31(HBpz3)2 Fe31 – OH–Fe31(HBpz3)2

F1UKS F11

UKS F1UKS F11

UKS

Charge Spin Charge Spin Charge Spin Charge Spin

Fe1 11.769 13.834 11.838 14.103 11.868 14.080 11.886 14.150O4 20.726 10.061 20.733 10.109 20.724 10.092 20.726 10.135O5 20.726 10.061 20.733 10.109 20.724 10.092 20.726 10.135N8 20.399 10.067 20.402 10.074 20.427 10.090 20.427 10.096N9 20.396 10.039 20.401 10.052 20.423 10.098 20.426 10.103N10 20.399 10.067 20.402 10.074 20.427 10.090 20.427 10.096

Fe2 11.769 23.834 11.838 14.103 11.868 24.080 11.886 14.150O6 20.726 20.061 20.733 10.109 20.724 20.092 20.726 10.135O7 20.726 20.061 20.733 10.109 20.724 20.092 20.726 10.135N14 20.399 20.067 20.402 10.074 20.427 20.090 20.427 10.096N15 20.396 20.039 20.401 10.052 20.423 20.098 20.426 10.103N16 20.399 20.067 20.402 10.074 20.427 20.090 20.427 10.096

O3 20.916 0.000 21.021 10.797 21.072 0.000 21.094 10.276H44 10.490 0.000 10.486 10.005C38 10.808 0.000 10.810 10.001 10.823 0.000 10.824 10.003C40 10.808 0.000 10.810 10.001 10.823 0.000 10.824 10.003


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were the most negative whereas those correspondinm-acetatos had lesser negative charges. The nitrogens oHBpz3 ligands were the least negative. Similar trends wfound for both complexes, Fe31 – OH–Fe31(HBpz3)2 beingslightly more polarized in the Fe1–O3–Fe2 plane. This wconsistent with the longer Fe–O3 bond lengths and, thfore, more ionic nature of the hydroxo-bridged complex. Rmarkably, consistent with molecular geometries, the NP82

charge analysis made a slight distinction between N9,15


U-BPW91/6-311G* ! wave function. The isovalue contour plots arethe Fe1–mO–Fe1 or Fe1–mOH–Fe1 (yz) planes. Top: MOs 143a and 164aof Fe31 – O–Fe31(HBpz3)2 . Bottom: MOs 107a and 144a ofFe31 – OH–Fe31(HBpz3)2 .


U-BPW91/6-311G* ! wave function. The isovalue contour plots are 0.5above the Fe1–mO–Fe1 or Fe1–mOH–Fe1 (yz) planes. Top: MOs 144aand 168a of Fe31 – O–Fe31(HBpz3)2 . Bottom: MOs 135a and 152a ofFe31 – OH–Fe31(HBpz3)2 .


tothee

se--

ndthe other nitrogens bound to iron. Charge distributions cresponding to low-spin broken-symmetry (C1

UKS) stateswere qualitatively similar to those of high-spin states but,the case of Fe31 – O–Fe31(HBpz3)2 , somewhat less polarized. This was only slightly noticeable for the low-spin staof Fe31 – OH–Fe31(HBpz3)2 which had a distribution ofcharge fairly close to that of the high-spin state.

The polarization of molecular charge was also reflecin the occupancy of thes, p, andd shells of each binucleacore. Electronic configurations were calculated within tnatural population~NPA! framework.82 The results ob-tained for Fe31 – O–Fe31(HBpz3)2 (Fe1,2:4s0.293d5.93,mO:2s1.922p4.99) and those of Fe31 – OH–Fe31(HBpz3)2

(Fe1,2:4s0.303d5.82,mOH:2s1.822p5.25,H:1s0.51) showedsubtle, but meaningful, differences. However, the configutions of both complexes revealed relatively high occupancfor their nominally ferric ions. This reflected the donatincharacter of the N3O3 coordination.

Table III also shows the distribution of molecular spiAll atoms in the table displayed positive~a! spin densityfor the high-spin (C11

UKS) states. Most of this density walocalized on each of the two equivalent iron ions:14.103and 14.150 units for Fe31 – O–Fe31(HBpz3)2 andFe31 – OH–Fe31(HBpz3)2 , respectively. The bridging O3had significant density localized in the first comple~10.797! but a lesser amount in the second~10.276!. Thespin associated with the other atoms bound to iron was r

FIG. 12. Fe31 – O–Fe31(HBpz3)2 : Localized orbitals of the broken symmetry wave function~C1

UKS , U-BPW91/6-311G* !. The isovalue contourplots are:~a! Slice taken in the O6–Fe1–O7 plane showing the main ctribution from Fe1(dx22y2) to MO 145a and delocalization toward thebis~m-acetato! O7(p) and O8(p) orbitals. ~b! Slice taken in the O6–Fe2–O7 plane showing the delocalization of MO 145a hosted by bis~m-acetato! O7(p) and O8(p) orbitals. ~c! Slice taken in the O4–Fe2–O5plane showing the main contribution from Fe2(dx22y2) to MO 145b anddelocalization toward the bis~m-acetato! O4(p) and O5(p) orbitals. ~d!Slice taken in the O6–Fe2–O7 plane showing the main contribution frthe Fe2(dx22y2) to MO 145b which isp-bonding with a small contributionfrom bis~m-acetato! O7(p) and O8(p) orbitals. The corresponding MOs~145a,145b! of Fe31 – OH–Fe31(HBpz3)2 are qualitatively similar.



Downloaded 27

TABLE IV. Absolute values of overlaps calculated from broken symmetry molecular orbitals ofC1UKS

U-BPW91/6-311G* wave function.

Main character of MOsa:b Fe31 – O–Fe31(HBpz3)2 Fe31 – OH–Fe31(HBpz3)2

Fe1(dz2):Fe2(dz2) ^172au172b&50.2114 ^171au171b&50.2754Fe1(dxy):Fe2(dxy) ^171au171b&50.0685 ^172au172b&50.0948Fe1(dxz):Fe2(dxz) ^144au144b&50.4761 ^135au135b&50.2247Fe1(dyz):Fe2(dyz) ^164au164b&50.1819 ^144au144b&50.1115Fe1(dx22y2):Fe2(dx22y2) ^145au145b&50.1991 ^145au145b&50.2051

Fe1(dyz):Fe2(dz2) ^164au172b&50.2349 ^144au171b&50.1461Fe1(dz2):Fe2(dyz) ^172au164b&50.2349 ^171au144b&50.1461

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tively minor, on the order of10.05 to 10.14 units. Thusalthough the irons in the hydroxo-bridged complex weslightly more polarized, the main difference between the tcomplexes lied in the spin density associated with thbridging oxygens.

The low-spin wave functions (C1UKS) converged to sin-

glet states of broken spin symmetry. In contrast to the palel polarizations of the high-spin states, the singlet solutioshoweda density localized on Fe1 andb density localizedon Fe2. The bridging O3 was strictly diamagnetic, as dtated by symmetry, whereas the other ligands bound toand Fe2 had slighta andb character, respectively. Most othe density in the low-spin states was localized on the mions with 63.834 units for Fe31 – O–Fe31(HBpz3)2 and64.080 units for Fe31 – OH–Fe31(HBpz3)2 .

V. DISCUSSION

A. Microscopic mechanisms of spin coupling

To identify some main atomic orbital interactions ivolved in antiferromagnetic superexchange we first conered delocalized MOs of the fully symmetric (C2v) high-spin wave functions (C11

UKS). In particular, we focused on thdifferences in energy,D i5(eSi2eAi), of MOs correspondingto symmetric ~S! and antisymmetric~A! combinations ofFe(di) orbitals. According to a model presented by HaHoffman et al.,21 such energy differences are particularsensitive to geometric distortions and substituent effectsbimetallic molecules.21 These authors also showed21 that thesquares of these parameters, (eSi2eAi)

2 are directly propor-tional to the pairwise contributions to the antiferromagnecomponent (JAF) of the Heisenberg exchange constant~Sec.II !.21,22 Here, the subscriptsi represent the local~i.e., mono-meric! symmetry of the atomic orbitals.

By applying the (eSi2eAi)2 model to high-spin fully

symmetric wave functions we identified the main interations between Fe1(di) and Fe2(di) orbitals of equal localsymmetry~i.e., same-symmetry pathways!. The analysis oflow-spin broken-symmetry wave functions constitutes andependent and complementary model that can provide ational information. In particular, such wave functions creveal intermetallic interactions between atomic orbitalsdifferent local symmetry~i.e., mixed-symmetry pathways!.Metal-rich broken-symmetry orbitals of different spin inde~i.e., a or b! are not, in general, orthogonal to each othand are mainly localized on Fe1 (f1i

a ) or Fe2 (f2 jb ). Thus

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broken-symmetry orbitals constitute an approximation tonatural magnetic orbitals~NMO! defined by Girerd andKahn22,45,46and, to this extent, can be analyzed in termstheir overlapsS1i ,2j5^f1i

a uf2 jb &. Within the NMO frame-

work, the overlapsS1i ,2j are intimately related to the strengtof antiferromagnetic spin coupling.46

1. Fe3¿– O – Fe3¿„HBpz 3…2

The D ib splittings calculated for the oxo-bridged com

plex ~Fig. 3! and the Hay–Hoffman model21 indicated thepresence of four main antiferromagnetic same-symmesuperexchange pathways. The main three involved intertallic interactions via the oxo bridge and contributedthe molecular antiferromagnetism in the following oder: Fe1(dxz):m-O(px):Fe2(dxz), Fe1(dyz):m-O(p'):Fe2(dyz), and Fe1(dz2):m-O(pi):Fe2(dz2). According to the

D ib values, the first pathway was clearly dominant. The la

two, however, were more comparable and made fairly silar contributions to the antiferromagnetic ordering. In adtion, a fourth pathway had a lesser but noticeable contrition and involved the other bridging ligands: Fe1(dx22y2):bis~m-acetato!:Fe2(dx22y2).

The main antiferromagnetic orbital interactionsFe31 – O–Fe31(HBpz3)2 were also inferred from overlapcalculated from the broken-symmetry wave function~TableIV !. The broken-symmetry results were consistent with thinferred from the full-symmetry wave function since thformer also included the four pathways of equal local symetry. Significantly, the broken-symmetry results indicatthat the relative importance of the four same-symmetry paways followed, in general, the same trends indicated aboNamely, that Fe1(dxz):m-O(px):Fe2(dxz) was the domi-nant pathway, whereas Fe1(dyz):m-O(p'):Fe2(dyz) andFe1(dz2):m-O(pi):Fe2(dz2) contributed to a lesser extenand were fairly comparable. However, the calculated ovlaps suggested that the antiferromagnetic contributionFe1(dx22y2):bis~m-acetato!:Fe2(dx22y2) was comparable tothat of the latter two interactions. The broken-symmetry cculations also provided insight about ground state intertions between Fe(d) orbitals of different local symmetryIn particular, the contours of Figs. 10 and 9 showed M164a and 172a of Fe31 – O–Fe31(HBpz3)2 as mixed-symmetry pathways with approximate compositiFe1(dyz):m-O(p'):Fe2(dz2) and Fe1(dz2):m-O(pi):Fe2(dyz), respectively. The overlaps corresponding to the


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mixed-symmetry pathways were equal and their magnitsuggested that their efficiencies were slightly greater tthree out of four same-symmetry interactions.

In summary, orbital interactions inferred from both fusymmetry and broken-symmetry models were fairly constent with each other. Based on the combined results of hspin and low-spin calculations we have identified six mantiferromagnetic superexchange~i.e., bridge-mediated!pathways. Four of these corresponded to interactionstween Fe(d) orbitals of equal local symmetry and two tinteractions between Fe(d) orbitals of different local symmetry. Furthermore, five out of six pathways originated frointermetallic interactions via the oxo bridge and one viabridging acetates. For Fe31 – O–Fe31(HBpz3)2 , both mod-els indicated that the main agent of antiferromagnetismthe same-symmetry pathway Fe1(dxz):m-O(px):Fe2(dxz).

Overlaps derived from broken-symmetry calculatioalso provided insight about interactions betweenm-O(p)and Fe(d) electrons of opposite spin. FoFe31 – O–Fe31(HBpz3)2 the larger overlaps correspondedinteractions with the following approximate symmetriem-O(p'

a ) – Fe2(dz2b ) ~^143au172b&!, m-O(px

a) – Fe2(dxzb )

~^168au144b&!, and m-O(pia) – Fe2(dz2

b ) ~^125au172b&!. Allof these were visualized as bonding interactions betweenbridging oxygens and minor contributions from Fe2 in MO143a, 168a, and 125a, respectively~Figs. 9–11!. The mag-nitude of the overlaps strongly suggested that the firstinteractions contribute to the antiferromagnetic orderingthe complex to a larger extent than the third. Three equlent interactions also existed betweenm-O(pb) andFe1(da). Thus the broken-symmetry calculations also pvided a detailed approximation to antiferromagnetismduced bym-O(pa,b)→Fe(db,a) delocalization~i.e., ligandspin polarization!. An analysis of the relative importancesligand spin polarization and superexchange is beyondscope of the present work. We simply note that superchange is considered by the pioneering authors20,24,89–92to bethe dominant mechanism of spin coupling.

2. Fe3¿– O* – Fe3¿„HBpz 3…2

Conversion from oxo- to hydroxo-bridging is associatwith variations of two intimately related, but different, strutural variables:~i! geometric parameters~e.g., Fe–mO dis-tances and Fe1–mO–Fe2 angles! and~ii ! the presence of theproton. One main objective of our study was to probedistinct effects of these two variables. Toward this end,performed calculations on the oxo-bridged structuFe31 – O* –Fe31(HBpz3)2 which corresponded to the optmized geometry of Fe31 – OH–Fe31(HBpz3)2 with the pro-ton on the bridge being absent. By focusing on the absovalues ofD i

b for the three structures studied in this work wobtained a semiquantitative measure of the isolated effecthe two structural variables. The effects of the first varia~i.e., geometry! on the delocalized orbitals were clearly oserved by comparing Figs. 3 and 7. As inferred below~Sec.V C!, these effects were essentially related to the differFe–mO bond lengths of Fe31 – O–Fe31(HBpz3)2 andFe31 – O* –Fe31(HBpz3)2 . Based on the differentD i

b split-


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tings of these two structures we identified significant vartions in the efficiency of three exchange pathways. The meffects were seen for Fe1(dyz):m-O(p'):Fe2(dyz) andFe1(dxz):m-O(px):Fe2(dxz) whoseD i

b splittings decreasedwith increasing Fe–mO distances. A similar but less pronounced effect was found for Fe1(dz2):m-O(pi):Fe2(dz2).Therefore, these three pathways, in that order, becaless efficient vias of molecular antiferromagnetism asgeometric parameters of Fe31 – O* –Fe31(HBpz3)2 wereimposed. By contrast, there was only a slight incremenDx22y2

b indicating that the intermetallic interaction corresponding to Fe1(dx22y2):bis~m-acetato!:Fe2(dx22y2) wasnot significantly affected by the same geometric vartions. This latter pathway, however, was of greater relatimportance for the antiferromagnetic orderingFe31 – O* –Fe31(HBpz3)2 sinceDx22y2

b was comparable tosplittings corresponding to other~i.e., oxo-bridged! interme-tallic interactions.

3. Fe3¿– OH – Fe3¿„HBpz 3…2

The D ib splittings calculated for the hydroxo-bridge

complex~Fig. 8! indicated the presence of four main samsymmetry superexchange pathways. Three of thesevolved intermetallic interactions via the hydroxo bridgand one through them-acetato groups. In terms of thantiferromagnetic ordering, the relative importancethese same-symmetry pathways was as folloFe1(dz2):m-OH(pi):Fe2(dz2), Fe1(dx22y2):bis~m-acetato!:Fe2(dx22y2), Fe1(dxz):m-OH(px):Fe2(dxz), andFe1(dyz):m-OH(p'):Fe2(dyz). The first interaction wasdominant, whereas the latter three had more comparabletributions. Quite remarkably, the main trends derived frothe broken-symmetry model were the same. In fact,Fe31 – OH–Fe31(HBpz3)2 , the overlaps~Table IV! betweenbroken-symmetry orbitals also indicated that, out of fosame-symmetry interactions, Fe1(dz2):m-OH(pi):Fe2(dz2)was the strongest, whereas Fe1(dyz):m-OH(p'):Fe2(dyz)was the weakest.

To distinguish the effects of the presence of a proon the oxo bridge from those of concomitant geometvariations that occur upon protonation~i.e., second struc-tural variable!, we compared results obtained foFe31 – OH–Fe31(HBpz3)2 and Fe31 – O* –Fe31(HBpz3)2 .The isolated effects of this second variable were clearlysessed by comparing theD i

b splittings of Figs. 7 and 8. Sucha comparison showed that a main effect was to significastabilize MO 173b, thus reducingDxz

b . This effect was re-lated to a noticeable localization of MO 173b on the metalsand, correspondingly, a smaller contribution from the briding m-OH ~Supplementary Tables II and III!. Therefore,within the framework of the (eSi2eAi)

2 model, the solepresence of the proton~i.e., not including geometric variations! was accompanied by a significantly less efficieFe1(dxz):m-OH(px):Fe2(dxz) pathway and, consequently,weaker molecular antiferromagnetism. This appears to bedominant effect as we noted only a minor variation in theD i

b

splittings of the other same-symmetry pathways. Furthersight was given by the composition of broken-symme


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MO 135a of Fe31 – OH–Fe31(HBpz3)2 ~Supplementarymaterial! which, in contrast to MO 144a ofFe31 – O–Fe31(HBpz3)2 , also showed some delocalizatiotoward the HBpz3 groups and, consequently, a smaller ovlap of Fe1 with the other iron site.

The combined effects of geometry and protonation wvisualized by comparison of results obtained fFe31 – O–Fe31(HBpz3)2 and Fe31 – OH–Fe31(HBpz3)2 .Both models, that is (eSi2eAi)

2 ~Figs. 3 and 8! and broken-symmetry~Table IV!, revealed some significant differencebetween the antiferromagnetic interactions of the two coplexes. Although both molecules displayed similar samsymmetry interactions, the absolute and/or relative contritions to their antiferromagnetic couplings were markedifferent. The main difference lied in the same-symmeinteractions involving Fe(dxz) orbitals. At a qualitative level,this was seen in the broken-symmetry MOs of Fig. 11 whprovided a pictorial and intuitive representation of thep in-teractions between Fe(dxz) andm-O(px). The figure showshow MO 135a of Fe31 – OH–Fe31(HBpz3)2 was morelocalized on the metals than MO 144a ofFe31 – O–Fe31(HBpz3)2 . In addition to qualitative in-sight about bridge-mediated intermetallic interactions,two models also provided a semiquantitative measuretheir importance. For example, the strongest intaction in Fe31 – O–Fe31(HBpz3)2 , Fe1(dxz):m-O(px):Fe2(dxz), was associated with aDxz

b value of 1.104 eVand a broken-symmetry overlap of 0.476. By contrathe strongest interaction in Fe31 – OH–Fe31(HBpz3)2 ,Fe1(dz2):m-OH(pi):Fe2(dz2), was associated with aDz2

b

value of 0.469 eV and a broken-symmetry overlap of 0.2Important differences were also noted in the mixe

symmetry pathways of Fe31 – O–Fe31(HBpz3)2 andFe31 – OH–Fe31(HBpz3)2 . We did not find, by simple in-spection of the Fe31 – OH–Fe31(HBpz3)2 orbitals, an occu-pied mixed-symmetry pathway with compositioFe1(dyz):m-OH(p'):Fe2(dz2) and which was mainly centered on Fe1. Instead, Fe31 – OH–Fe31(HBpz3)2 displayedthe highly stabilized MO 107a ~Fig. 10!, an orbital that wasmainly localized on the bridging hydroxide. Such stabiliztion and bridge localization were clearly due to a strobonding interaction betweenm-OH(p') and H(s) orbitals.At the same time, protonation resulted in a fair localizatiof MO 144a ~Fig. 10! which had main Fe1(dyz) characterand only a very minor contribution from the bridgingm-OH.Therefore, the composition of MOs 107a and 144a stronglysuggested that two vias of antiferromagnetism~i.e., themixed-symmetry pathways! which were very important, al-though not dominant, in Fe31 – O–Fe31(HBpz3)2 , weresignificantly less efficient in Fe31 – OH–Fe31(HBpz3)2 .This view derived from simple inspection of the molecuorbitals was consistent with the more quantitative insiprovided by overlaps calculated between MOsmainly Fe1,2(dyz) and Fe2,1(dz2) composition. Theseoverlaps showed significantly lower values fFe31 – OH–Fe31(HBpz3)2 ~Table IV!. It should be empha-sized that the previous comparisons were made betwbroken-symmetry wavefunctions calculated at exactlysame level of theory for both complexes. Such comparis


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The one-electron energies~Figs. 3 and 8! stronglysuggested a different ordering for the atomic orbitof t2g parentage in Fe31 – O–Fe31(HBpz3)2 andFe31 – OH–Fe31(HBpz3)2 . The main antiferromagnetic superexchange pathways, which were consistent with (eSi

2eAi)2 splittings and f1i

a uf2 jb & overlaps, are summarized i

Scheme 2~not drawn to scale!. Here,mO andmAcet. repre-sent the specific intervening bridges and the dashed boidentify the dominant interaction in each complex. A modetailed estimate of the relative importance of the variopathways can be inferred from Figs. 3 and 8 and Table IVdiscussed above.

Scheme 2.

B. Atomic spin densities

The distinct spatial localization of spin density on eaof their paramagnetic fragments~e.g., metal cations, organiradicals! is an important characteristic of spin couplecomplexes.12,38 In particular, in ferromagnetic states thindividual fragments have parallel and, at the satime, significant net spin density.38 Thus, forFe31 – O–Fe31(HBpz3)2 and Fe31 – OH–Fe31(HBpz3)2 ,the positive signs and magnitudes of the spin densities inC11

UKS states~Table III! were consistent with ferromagnetispin coupling between two high-spin ferric ions. The nomnal high-spin configuration of oxo-bridged diferric cores~inunits of unpaired spins! is Fe1(S155) – O(S50) – Fe2(S2

55). This useful, but simple, description idealizes a systhaving five unpaired electrons localized on each metaland no unpaired electrons on the diamagnetic bridgligand. By contrast, results derived from density functioncalculations allowed us to describe the high-spin states wsignificantly greater detail, namely Fe1(S1'14.10↔14.32) – O(S'10.80↔10.66) – Fe2(S2'14.10↔14.32)for Fe31 – O–Fe31(HBpz3)2 and Fe1(S1'14.15↔14.39 ) – OH (S'10.28↔ 1 0.20) – Fe2 (S2' 1 4.15↔14.39) for Fe31 – OH–Fe31(HBpz3)2 . Here, the numbersto the left ~U-BPW91! and right ~U-MPW1PW91! of thearrows represent a range determined from two highly repsentative exchange-correlation density functionals.

As discussed in Sec. II, the spin distributionthe singlet states (C1

UKS) mimicked the open-shell singlestructure of the antiferromagnetically coupled binuclecomplexes. The neta and b spin localized on the otherwise equivalent iron ions was consistent with the experimtally established antiferromagnetism of the complexes.1,2 The magnitude of the local densities~Table III!


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Downloaded 27

TABLE V. Spin expectation valuesS2&,a total energy differences (E11UKS-E1

UKS), and Heisenberg exchangconstants~J!b calculated from high-spin (F11

UKS) and broken symmetry (F1UKS) wave functions.

^S2&1 ^S2&11

E11UKS-E1

UKS

~cm21!JUKSc

~cm21!JExpd

~cm21!

Fe31 – O–Fe31(HBpz3)2 4.86 30.02 1919.9 152.7 242Fe31 – OH–Fe31(HBpz3)2 4.99 30.01 290.9 23.3 34

aExpectation values before annihilation of the first spin contaminant~Refs. 86, 87!.bValues reported for theHHB5JS1"S2 representation of the Heisenberg operator~Ref. 92!.cCalculated from Eq.~3! and parameters given in this table.dExperimental values reported by Armstrong and Lippard~Refs. 2, 3!.

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allowed us to describe the antiferromagnetic coreFe31 – O–Fe31(HBpz3)2 as Fe1(S1'63.83↔64.28)–O(S50) – Fe2(S2'73.83↔74.28! and that ofFe31 – OH–Fe31 (HBpz3)2 as Fe1(S1'64.08↔64.39)– OH(S50) – Fe2(S2'74.08↔74.39). The 6 and 7signs of Fe1 and Fe2, respectively, represented the twosible solutions of broken symmetry. There was a meaningincrement in the absolute values of metallic spin densupon protonation~'0.25 units for each ion!. This reflectedthe more ionic character of Fe31 – OH–Fe31(HBpz3)2 rela-tive to Fe31 – O–Fe31(HBpz3)2 and was consistent with thlonger Fe–O3 bond lengths~Table I! and slightly greatermetallic charge of the former~Table III!.

In this regard, two points should be mentioned:~i! Thetotal calculated molecular spin was110 units forC11

UKS and0 units forC1

UKS. These were necessary and exact resultsour calculations since they imposed convergence to hspin and low-spin states, respectively. The spatial distrition of spin, however, was not knowna priori and corre-sponded to the net molecular spin density,rs(r )5ra(r )2rb(r ), of each complex. A useful and highly informatividealization of rs(r ) is given by atomic densities derivefrom partitioning analyses such as NPA80,81 ~Table III!. ~ii !Spin-unrestricted density functional theory is capableyielding nearly fully spin-polarized descriptions for spicoupled complexes with fairly localized unpaireelectrons.12,38For such complexes, the calculated densitieseach individual paramagnetic fragment are very close to tnominal spin state. Therefore, we view as significant thethat densities calculated for individual iron ionsFe31 – O–Fe31(HBpz3)2 and Fe31 – OH–Fe31(HBpz3)2

were somewhat lower than their nominal (S55) values. Thisreflected the fair degree of delocalization of some valeFe(d) andm-O(p) electrons, particularly in the former complex.

C. Heisenberg exchange constants and magneto-structural correlations

The exchange couplings calculated from Eq.~3! aregiven in Table V. The predicted values had positive sigindicating that both complexes had antiferromagnetic grostates.92 This was in agreement with the experimetally established antiferromagnetism of the complexes1–3

The predicted values for Fe31 – O–Fe31(HBpz3)2 andFe31 – OH–Fe31(HBpz3)2 were92 1152.7 cm21 and 123.3cm21, respectively (Jm-OH /Jm-O50.15). These values can bcompared with those determined from susceptibi

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experiments,2,3 namely1242 cm21 and 134 cm21, respec-tively (Jm-OH /Jm-O50.14). Therefore, the theoretical valuereproduced extremely well the trends observed in thecomplexes. Similar trends have been observed experimtally in a variety of binuclear complexes11 and diiron-oxoproteins6 where the exchange constant drops by an ordemagnitude upon protonation of them-oxo bridge.

The predictedJ values for the oxo- and hydroxo-bridgecomplexes differed by 37% and 27%, respectively, from thexperimental values. These differences between theoreand experimental values, the former being lower, wmainly attributed to the slightly longer calculated Fe–mOdistances~Table I!. Accordingly, the theoretical-experimentadiscrepancy was lower for Fe31 – OH–Fe31(HBpz3)2 forwhich closer agreement was obtained between calculatedx-ray Fe–mOH distances. In spite of these differences, theretical and experimentalJ values were remarkably close. Thfact that fairly close theoretical-experimental agreement wobtained with the same methodology for two different copling regimes~i.e., strong and weak antiferromagnetism! wasparticularly striking. This may be further appreciated if oconsiders that for Fe31 – OH–Fe31(HBpz3)2 the absolutevalue of the total molecular~SCF-DFT! energy was 98.03107 cm21 whereasJ was only 23.3 cm21. Consequently,the parameters of Table V involved the prediction of energthat, in comparison to the total molecular energies, were vsmall.

The prediction of Heisenberg exchange constants fornuclear metal complexes has traditionally been a challengtheoretical problem. While fairly accurate parameters habeen obtained from experiment~e.g., magnetic susceptibility!, theoretical values have mostly been an approximatoften giving only the right order of magnitude. This has beparticularly true for weakly coupledm-hydroxo-bridgedcomplexes, such as Fe31 – OH–Fe31(HBpz3)2 , for whicheven prediction of the correct sign ofJ was not generallyachieved. Thus the values presented in this work~Table V!represented significant progress toward the accurate pretion of signs, trends, and magnitudes of Heisenberg exchaconstants. To our knowledge, the combined accuracy ofpredictions for ~m-O!bis~m-acetato! and ~m-OH!bis~m-acetato! diiron complexes is unprecedented. This suggethat the exchange-correlation and spin projection schedescribed in Sec. II constitute a robust methodology wwide applicability.

The optimized U-BPW91 geometries~Table I! displayeda Fe–O3 bond 0.164 Å longer and an Fe1–O3–Fe2 an


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3.4° more acute for the protonated complex. Further calctions on simple three (Fe31 – O–Fe31) and four(Fe31 – OH–Fe31) atom models, with varying geometric parameters, revealed that a Fe–mO increment of 0.164 Å had alarge effect on the strength of spin coupling~i.e., decreasingJ by 85.4%!, whereas the corresponding angular variation23.4° had only a minor effect~i.e., increasingJ by 1.6%!.The calculatedJ values for the actual complexes and simpfied three and four atom models followed similar trendThus in analogy to the distance and angular dependencFe31 – O–Fe31, we could attribute the effects of geometon the antiferromagnetism of Fe31 – O–Fe31(HBpz3)2 andFe31 – OH–Fe31(HBpz3)2 to be almost entirely related tvariations in Fe–mO distances83 and not to small Fe1–mO–Fe2 angular variations.

D. The magnetism of diiron-oxo proteins: Azidomet-hemerythrin

While part of our previous discussion should applythe physiologically important6,9,93,94 oxy-Hr and deoxy-Hr,one should notice that these forms have someferences in comparison to Fe31 – O–Fe31(HBpz3)2 andFe31 – OH–Fe31(HBpz3)2 , respectively. In oxy-Hr, one oits iron sites is bound to a terminal hydroperoxide whicbeing a fairly strong-field ligand, should perturb the eletronic structure of that iron.12,25 Deoxy-Hr is in the diferrousstate and one of its iron sites is only five coordinate. Fthese reasons, we shall focus on the azido forms of herythrin which, in comparison to Fe31 – O–Fe31(HBpz3)2 ,have similar~i.e., N3O3! coordination around the immediatenvironment of their iron sites.

Azidomet-Hr95 and azidometmyo-Hr contain an azidanion bonded to the iron site which is otherwise occupiedhydroperoxide in oxy-Hr. Recent theoretical studies12 haveshown that, in comparison to stronger-field ligands suchNO or O2, the azido anions behave as weak-field liganthat do not significantly perturb the electronic structure ofiron sites. This is consistent with observed Mo¨ssbauer isomeshifts and quadrupole splittings which are the same fortwo iron sites of azidomet-Hr.96 Thus a fairly similar elec-tronic structure is expected for the diferric cores of azidomHr, azidometmyo-Hr, and Fe31 – O–Fe31(HBpz3)2 , all oftheir iron sites having N3O3 coordination. This fact, in conjunction with the similarity in geometric and susceptibiliparameters among these three structures,1 implied that thesame essential antiferromagnetic mechanisms outlined a~Scheme 2-a! for Fe31 – O–Fe31(HBpz3)2 are also at workin the two biological systems. Therefore, our results allowus to conclude that six main superexchange pathwayspresent in azidomet-Hr and azidometmyo-Hr. The five minteractions should occur via the oxo bridge and one viam-carboxylato groups. In particular, the strongest and weest interactions should be Fe1(dxz):m-O(px):Fe2(dxz) andFe1(dx22y2):bis~m-carboxylato!:Fe2(dx22y2), respectively.

E. The magnetism of Fe 3¿– O–Fe3¿„Me3TACN…2

and Fe2¿– OH–Fe2¿„Me3TACN…2

Oxo-bis~acetato!-bridged diferric and diferrous complexes containing facially coordinating Me3TACN ligands


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Both complexes display a distorted bioctahedral structwith all iron sites having N3O3 coordination. Thus the bi-nuclear cores of the compounds containing Me3TACN cap-ping ligands are structural analogs of the HBpz3-ligatedcomplexes studied in this work. We note that identical eperimental Fe–mO average distances~1.785 Å! and nearlyidentical exchange constants~J51238 cm21 and 1242cm21! have been reported for Fe31 – O–Fe31(Me3TACN)2 ,4

and Fe31 – O–Fe31(HBpz3)2 ,3 respectively. It fol-lowed from theoretical, geometric, and experimental conserations, similar to those made above for azidomet-that the same six antiferromagnetic interactionsFe31 – O–Fe31(HBpz3)2 were also present inFe31 – O–Fe31(Me3TACN)2 . These interactions are displayed in Scheme 3-a in which Fe1(dxz):m-O(px):Fe2(dxz)and Fe1(dx22y2):bis~m-acetato!:Fe2(dx22y2) are the domi-nant and weakest, respectively.

Scheme 3.

Fe21 – OH–Fe21(Me3TACN)2 has an average x-raFe–mOH distance of 2.065 Å and an experimentexchange constant J5126 cm21.4 By comparison,Fe31 – OH–Fe31(HBpz3)2 has an average Fe–mOH distanceof 1.956 Å and an exchange constant ofJ5134 cm21.2

Thus the somewhat longer Fe–mOH distance of the former isqualitatively consistent with its somewhat lower exchanconstant.83 Nevertheless, considering that one complexhigh-spin diferrous whereas the other is high-spin diferricis at first surprising that their exchange constants aremarkedly different. High-spin Fe21 has four unpaired electrons whereas high-spin Fe31 has five. Thus, one may initially expect that the former will form binuclear cores withlesser number of exchange pathways and, consequently,stantially weaker antiferromagnetism than the latter. Hoever, this is not observed experimentally.

The magnetic behavior of Fe21 – OH–Fe21

(Me3TACN)2 and Fe31 – OH–Fe31(HBpz3)2 is consistentwith and can be explained by the mechanism proposedRodriguezet al.11 These authors pointed out that large dferences in coupling strength between oxo- and hydrobridged diiron centers are mainly related to the type of briding ligand and not to the iron oxidation state. Mospecifically, they stated that a change in oxidation st~i.e., from Fe31 to Fe21! should not drastically alter themagnitude ofJ as long as the main bridging ligand and echange pathways remain the same.11 With the added insightprovided by the present calculations, their argument canapplied here as follows. The main pathwayFe31 – OH–Fe31(HBpz3)2 is formed by Fe(dz2) orbitals


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which, as dictated by the local ligand fields, have substtially higher energy than the ground-state Fe(d) orbitals(10Dq'104 cm21).97,98 Upon reduction, at low temperatures, the extra~i.e., sixth! electron of each high-spin Fe21

will populate the ligand field ground state and pair wanother electron. This will in turn give rise to a nonmanetic orbital Fe(d↑↓) and, consequently, to the disappearanof the corresponding antiferromagnetic pathway~s!. Assum-ing that the ordering of thed orbitals is the samein Fe31 – OH–Fe31(HBpz3)2 and Fe21 – OH–Fe21

(Me3TACN)2 , Fe(dyz) will become the nonmagnetic orbital upon reduction to the diferrous state. Sinfor Fe31 – OH–Fe31(HBpz3)2 the same-symmetry anmixed-symmetry pathways formed by Fe(dyz) orbitals werenot dominant contributors, their disappearanceFe21 – OH–Fe21(Me3TACN)2 should not drastically altethe strength of antiferromagnetic coupling. Thus tmain antiferromagnetic interactions of Fe21 – OH–Fe21

(Me3TACN)2 can be represented by Scheme 3-b in whFe1(dz2):mOH(pi):Fe2(dz2) is dominant.

VI. CONCLUSION

We have performed a comprehensive study ofelectronic structures of Fe31 – O–Fe31(HBpz3)2 , Fe31 –O*–Fe31(HBpz3)2 , and Fe31 – OH–Fe31(HBpz3)2 . The firstgradient-corrected density functional calculations oncompletemolecular structures of these complexes areported here. The optimized geometries, energies, frontierbital compositions, atomic charges, and spin densitiesemerged from these calculations provided a detailed destion of the electronic configurations and magnetic propties of the compounds. We have gained novel insight abthe specific and different roles of oxo, hydroxo, and acetbridges on the molecular magnetism of these diiron coplexes. The understanding obtained from this work ahelped to explain much of the electronic structure amagnetism of biological diiron centers, in particulathose present in the diiron-oxo proteins azidomet- aazidometmyo-hemerythrin. In addition, our results alsoplained the different antiferromagnetic behavior of tstrongly and weakly coupled Fe31 – O–Fe31(Me3TACN)2

and Fe21 – OH–Fe21(Me3TACN)2 complexes, respectivelySigns and magnitudes of Heisenberg exchange constanoxo- and hydroxo-bis~acetato!-bridged diiron complexeshave been predicted with an unprecedented degree accufor both strong- and weak-coupling regimes. In what flows, we summarize some of our findings and conclusio

~i! The strong antiferromagnetism oFe31 – O–Fe31(HBpz3)2 arises from the presence of smain superexchange pathways. Four of these corresponinteractions between atomic orbitals of equal local symmewhereas two arise from interactions between orbitals offerent local symmetry. Furthermore, five out of these sixtermetallic interactions are mediated by the oxo bridwhereas one is mediated by the acetato groups. The dnant pathway was Fe1(dxz):m-O(px):Fe2(dxz) whereasFe1(dx22y2):bis~m-acetato!:Fe2(dx22y2) was the weakest.

~ii ! The weak antiferromagnetism oFe31 – OH–Fe31(HBpz3)2 originates in six main superex


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change pathways that are qualitatively similar to thopresent in Fe31 – O–Fe31(HBpz3)2 . However, the ab-solute contributions and/or relative importances of tpathways in these two complexes are significantly differeIn Fe31 – OH–Fe31(HBpz3)2 , the contributions to anti-ferromagnetic ordering from all six pathways are mocomparable. Nevertheless, Fe1(dz2):m-OH(pi):Fe2(dz2)can be considered the dominant pathway whereas thof same-symmetry and mixed-symmetry involvinFe(dyz) orbitals are weakest. The relative impotance of Fe1(dx22y2):bis~m-acetato!:Fe2(dx22y2) inFe31– OH–Fe31(HBpz3)2 is approximately the same as thof Fe1(dxz):m-OH(px):Fe2(dxz).

~iii ! Superexchange interactions mediated by mixsymmetry pathways contribute significantly to the antifermagnetism of Fe31 – O–Fe31(HBpz3)2 but are appreciablyless important for Fe31 – OH–Fe31(HBpz3)2 . To the extentthat the in-planedyz orbitals become nonmagnetic~i.e., dou-bly occupied! in some bioctahedral diferrous complexesuch as Fe21 – OH–Fe21(Me3TACN)2 , the mixed-symmetry pathways vanish.

~iv! Significantly, magneto-structural correlations drived from the two semiquantitative methods used in twork @i.e., based on (eSi2eAi)

2 energy differences and^f1i

a uf2 jb & overlaps# are in very close agreement. The latt

method also provided an estimate of antiferromagnetic ctributions from mixed-symmetry pathways.

~v! In most cases, the bonding and antibonding MOsbroken symmetry follow traditional trends of MO theory, thformer being of main ligand character and the latter of mecharacter. One notable exception are thep-bonding interac-tions between Fe(dxz) and m-O(px) ~Fig. 11! which aremainly localized on the metals.

~vi! A variety of geometric parameters and physicquantities, such as Fe–mOH bond lengths, atomic chargeand spin densities, reflect a more ionic nature of the Fe31

ions in Fe31 – OH–Fe31(HBpz3)2 relative to those inFe31 – O–Fe31(HBpz3)2 .

~vii ! Optimized Fe–mO and Fe–mOH distances as welas the overall electronic structures~e.g., atomic charges! in-dicate thatm-O behaves as a strong-field ligand wherem-OH is a relatively weak-field ligand. The one-electron eergies~Figs. 3 and 8! strongly suggest a different orderinfor the ligand-field ~d! orbitals of t2g parentage inFe31 – O–Fe31(HBpz3)2 and Fe31 – OH–Fe31(HBpz3)2

~Scheme 2!.~viii ! Considerations based on electronic and geome

structures, as well as magnetic susceptibility data, indicthat Fe31 – O–Fe31(HBpz3)2 models extremely well the mi-croscopic interactions leading to the strong antiferromnetism of azidomet-Hr and azidometmyo-Hr.

~ix! Schemes 2 and 3 constitute a general guidelineunderstanding, qualitatively and semiquantitatively, the aferromagnetism in a variety of~fairly! symmetric bioctahe-dral diferric and diferrous centers. Scheme 2a, in particumodels not only the interactions of~m-O!bis~m-acetato! di-ferric proteins but, by removing the corresponding pathwthose of~m-O! biological units that may not have bridgincarboxylatos. A judicious application of the results presen


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s


in this work should provide insight into the electronic struture and magnetism of a variety of~bio!inorganic diironcomplexes.

~x! The accurate determination of Heisenberg excha~J! in molecular antiferromagnets has traditionally beenrealm of experimental techniques. We have performed calations that allow the prediction of antiferromagnestrengths with an accuracy that approaches that of theexperimental methods~e.g., SQUID susceptometry!. Theseresults suggest that the density functional and spin projecmethods described in this work constitute an alternamethod for the accurate determination of exchange cstants. The results for weakly and strongly coupled diircomplexes also suggest that our methods are robust andhave wide applicability for studies of~bio!molecular magne-tism.

Supplementary material is available for four tables cotaining the optimized Cartesian coordinatesFe31 – O–Fe31(HBpz3)2 and Fe31 – OH–Fe31(HBpz3)2 .Extensive descriptions of molecular orbitals involved in mlecular magnetism.99

ACKNOWLEDGMENTS

The authors thank Dr. Michael Lee, Dr. Lou NoodlemaDr. James P. Kirby, and Professor Kenneth Sauer for helcomments. Assistance from Brandon Weldon and JereMonat is also gratefully acknowledged. The authors wishthank the National Institutes of Health, National InstituteGeneral Medical Sciences~Grant No. GM58551-02!, and theAlfred P. Sloan Foundation~J.K.M.! for financial support.The computational work was made possible by Grant NCHE970037N from the National Center for SupercompuApplications~NCSA! and allocations from Mahui High Performance Computing Center~MHPCC!.

APPENDIX: CALCULATION OF HEISENBERGEXCHANGE CONSTANTS

To calculate Heisenberg exchange constants we firssumed that the spin-contaminatedC2Smin11

UKS wave function

was a linear combination of the pure~projected! and normal-ized wave functions of all accessible spin states:

F2Smin11UKS 5 (

2Smin11

2Smax11

C2S11F2S11PUKS. ~A1!

Second, we made use of the normalization condition tfollows from Eq.~A1!:

(2Smin11

2Smax11

C2S112 51. ~A2!

Third, the energy and spin expectation values of the ctaminated low-spin~e.g., singlet! states were expanded iterms of the corresponding quantities of the projected sta

E2Smin11UKS 5^F2Smin11

UKS uHUKSuF2Smin11UKS &

~A3!

5 (2Smin11

2Smax11

C2S112 E2S11

PUKS,


eeu-

st

nen-nay

-f

-

,uly

of

.r

s-

at

-

s:

^S2&2Smin11UKS 5^F2Smin11

UKS uS2uF2Smin11UKS &

~A4!

5 (2Smin11

2Smax11

C2S112 S~S11!.

The energies (E2S11PUKS) of all C2S11

PUKS states are shiftedfrom those of their corresponding Heisenberg states (E2S11

ex )by some constantd. From Eqs.~1!, ~A3!, and~A4! we wrote

E2Smin11UKS 5 (

2Smin11

2Smax11

C2S112

3$ 12J$S~S11!2S1~S111!2S2~S211!%1d%

~A5!

5 12 J^S2&2Smin11

UKS 2 (2Smin11

2Smax11

C2S112

3$ 12 J$S1~S111!1S2~S211!%2d% ,

E2Smax11PUKS 5 1

2 J$Smax~Smax11!2S1~S111!

2S2~S211!%1d%. ~A6!

Finally, from Eqs. ~A5! and ~A6! we determinedE2Smax11

PUKS 2E2Smin11UKS . Then, assumingE2Smax11

UKS 'E2Smax11PUKS

~Sec. II! and using Eq.~A2!, we arrived at Eq.~3!:

J52E2Smax11

UKS 2E2Smin11UKS

Smax~Smax11!2^S2&2Smin11UKS .

1W. H. Armstrong and S. J. Lippard, J. Am. Chem. Soc.105, 4837~1983!.2W. H. Armstrong and S. J. Lippard, J. Am. Chem. Soc.106, 4632~1984!.3W. H. Armstrong, A. Spool, G. C. Papaefthymiou, R. B. Frankel, andJ. Lippard, J. Am. Chem. Soc.106, 3653~1984!.

4J. A. R. Hartman, R. L. Rardin, P. Chaudhuri, K. Pohl, K. Wieghardt,Nuber, J. Weiss, G. C. Papaefthymiou, R. B. Frankel, and S. J. LipparAm. Chem. Soc.109, 7387~1987!.

5R. Hotzelmann, K. Wieghardt, U. Flo¨rke, H.-J. Haupt, D. Weatherburn, JBonvoisin, G. Blondin, and J. Girerd, J. Am. Chem. Soc.114, 1683~1992!.

6L. Que, Jr. and A. E. True, Prog. Inorg. Chem.38, 97 ~1990!.7P. Chaudhuri, M. Winter, H. J. Ku¨ppers, K. Wieghardt, B. Nuber, and JWeiss, Inorg. Chem. Commun.26, 3302~1987!.

8D. M. Kurtz, JBIC2, 159 ~1997!.9D. M. Kurtz, Chem. Rev.90, 585 ~1990!.

10J. H. Rodriguez, H. N. Ok, Y. M. Xia, P. G. Debrunner, B. E. Hinrichs,Meyer, and N. Packard, J. Phys. Chem.100, 6849~1996!.

11J. H. Rodriguez, Y. M. Xia, P. G. Debrunner, P. Chaudhuri, andWieghardt, J. Am. Chem. Soc.118, 7542~1996!.

12J. H. Rodriguez, Y. M. Xia, and P. G. Debrunner, J. Am. Chem. Soc.121,7846 ~1999!.

13A. P. Ginsberg, Inorg. Chim. Acta5, 45 ~1971!.14C. A. Brown, G. J. Remar, R. L. Musselman, and E. I. Solomon, Ino

Chem.34, 688 ~1995!.15P. W. Anderson, inMagnetism, edited by G. T. Rado and H. Suhl~Aca-

demic, New York, 1963!, Vol. 1.16J. B. Goodenough, Phys. Rev.100, 564 ~1955!.17J. B. Goodenough, J. Phys. Chem. Solids6, 287 ~1958!.18J. B. Goodenough,Magnetism and the Chemical Bond~Interscience,

New York, 1963!.19J. Kanamori, J. Phys. Chem. Solids10, 87 ~1959!.20P. W. Anderson, Phys. Rev.115, 2 ~1959!.21P. J. Hay, J. C. Thibeault, and R. J. Hoffmann, J. Am. Chem. Soc.97,

4884 ~1975!.22O. Kahn,Molecular Magnetism~VCH, New York, 1993!.23A. Bencini and D. Gatteschi,EPR of Exchange Coupled System

~Springer Verlag, Berlin, 1990!.


O.

L.

.

.

l-

o

m

em

et

hy

-

sio

elf

.

en-is-

. A

n-

ret-nc-

otthis

n-

ningularage


24J. Owen and E. A. Harris, inElectron Paramagnetic Resonance, editedby S. Geschwind~Plenum, New York, 1972!.

25T. C. Brunold and E. I. Solomon, J. Am. Chem. Soc.121, 8277~1999!.26K. Fink, R. Fink, and V. Staemmler, Inorg. Chem.33, 6219~1994!.27C. A. Brown, M. A. Pavlosky, T. E. Westre, Y. Zhang, B. Hedman, K.

Hodgson, and E. I. Solomon, J. Am. Chem. Soc.117, 715 ~1995!.28X. G. Zhao, W. H. Richardson, J. L. Chen, J. Li, L. Noodleman, H.

Tsai, and D. N. Hendrickson, Inorg. Chem.36, 1198~1997!.29J. E. McGrady and R. Stranger, J. Am. Chem. Soc.119, 8512~1997!.30V. Barone, A. Bencini, I. Ciofini, C. A. Daul, and F. Totti, J. Am. Chem

Soc.120, 8357~1998!.31C. Adamo, V. Barone, A. Bencini, F. Totti, and I. Ciofini, Inorg. Chem

38, 1996~1999!.32W. Kohn and L. J. Sham, Phys. Rev.140, A1133 ~1965!.33R. G. Parr and W. Yang,Density-Functional Theory of Atoms and Mo

ecules~Clarendon, Oxford, 1989!.34W. Kohn, A. D. Becke, and R. G. Parr, J. Phys. Chem.100, 12974~1996!.35A. D. Becke, J. Chem. Phys.98, 5648~1993!.36C. Lee, W. Yang, and R. G. Parr, Phys. Rev. B37, 785 ~1988!.37J. P. Perdew and Y. Wang, Phys. Rev. B45, 13244~1992!.38J. H. Rodriguez, D. E. Wheeler, and J. K. McCusker, J. Am. Chem. S

120, 12051~1998!.39W. J. Hehre, L. Radom, P. v.R. Schleyer, and J. A. Pople,Ab Initio

Molecular Orbital Theory~Wiley, New York, 1986!.40E. J. Baerends and O. V. Gritsenko, J. Phys. Chem. A101, 5383~1997!.41O. V. Gritsenko and E. J. Baerends, Theor. Chem. Acc.96, 44 ~1997!.42L. Noodleman, D. A. Case, and E. J. Baerends, inDensity Functional

Methods in Chemistry, edited by J. K. Labanowski and J. W. Andzel~Springer-Verlag, New York, 1991!.

43L. Noodleman, C. Y. Peng, D. A. Case, and J. M. Mouesca, Coord. ChRev.144, 199 ~1995!.

44C. Adamo and V. Barone, Chem. Phys. Lett.274, 242 ~1997!.45J. J. Girerd, M. F. Charlot, and O. Kahn, Mol. Phys.34, 1063~1977!.46J. J. Girerd, Y. Journaux, and O. Kahn, Chem. Phys. Lett.82, 534~1981!.47L. Noodleman, J. Chem. Phys.74, 5737~1981!.48K. Yamaguchi, H. Fukui, and T. Fueno, Chem. Lett.149, 625 ~1986!.49K. Yamaguchi, F. Jensen, A. Dorigo, and K. N. Houk, Chem. Phys. L

149, 537 ~1988!.50S. Yamanaka, T. Kawakami, H. Nagao, and K. Yamaguchi, Chem. P

Lett. 231, 25 ~1994!.51W. Heisenberg, Z. Phys.49, 619 ~1928!.52P. A. M. Dirac, Proc. R. Soc. London, Ser. A123, 714 ~1929!.53P. M. Levy, Phys. Rev.177, 509 ~1969!.54J. H. Van Vleck, Phys. Rev.45, 405 ~1934!.55P. W. Anderson, Phys. Rev.79, 350 ~1950!.56K. Yamaguchi, Y. Takahara, and T. Fueno, inApplied Quantum Chemis

try, edited by Jr. V. H. Smith, H. F. Schaefer III, and K. Morokuma~D.Reidel, Dordrecht, 1986!.

57Several relatively similar expressions forJ have been reported by variouauthors~Refs. 47, 48, 100!. Equation~3! was reported by Yamaguchet al. ~Ref. 50!. In the Appendix we present a more detailed derivationthis expression.

58J. Wang, A. D. Becke, and V. H. Smith, Jr., J. Chem. Phys.102, 3477~1995!.

59A. D. Becke, Phys. Rev. A38, 3098~1988!.60J. P. Perdew, Phys. Rev. B33, 8822~1986!.61A. D. Becke, J. Chem. Phys.98, 1372~1993!.62S. H. Vosko, L. Wilk, and M. Nusair, Can. J. Phys.58, 1200~1980!.63D. R. Hartree, Proc. Cambridge Philos. Soc.24, 89 ~1927!.64V. Fock, Physik61, 126 ~1930!.65D. R. Hartree,The Calculation of Atomic Structure~Wiley, New York,

1957!.66P. Hohenberg and W. Kohn, Phys. Rev.136, B864 ~1964!.67J. C. Slater,Quantum Theory of Molecules and Solids. Vol. 4: The S

Consistent Field for Molecules and Solids~McGraw-Hill, New York,1974!.


c.

.

t.

s.

f

-

68C. Adamo and V. Barone, J. Chem. Phys.108, 664 ~1998!.69R. Ditchfield, W. J. Hehre, and J. A. Pople, J. Chem. Phys.54, 724

~1971!.70W. J. Hehre, R. Ditchfield, and J. A. Pople, J. Chem. Phys.56, 2257

~1972!.71R. Krishnan, J. S. Binkley, R. Seeger, and J. A. Pople, J. Chem. Phys72,

650 ~1980!.72R. Krishnan, M. J. Frisch, and J. A. Pople, J. Chem. Phys.72, 4244

~1980!.73A. J. H. Wachters, J. Chem. Phys.52, 1033~1970!.74P. J. Hay, J. Chem. Phys.66, 4377~1977!.75K. Raghavachari and G. W. Trucks, J. Chem. Phys.91, 1062~1989!.76M. J. Frisch, G. W. Trucks, H. B. Schlegelet al., GAUSSIAN 94, Revision

E.2, Gaussian, Inc., Pittsburgh, PA, 1995.77M. J. Frisch, G. W. Trucks, H. B. Schlegelet al., GAUSSIAN 98, Revision

A.4, Gaussian, Inc., Pittsburgh, PA, 1998.78M. J. Frisch, A Frisch, and J. B. Foresman,GAUSSIAN 94User’s Manual

~Gaussian Inc., Pittsburgh, 1995!.79MOLDEN, Molecular Density Package, G. Schaftenaar~1991!, CAOS/

CAMM Center Nijmegen, The Netherlands.80A. E. Reed and F. Weinhold, J. Chem. Phys.78, 4066~1983!.81A. E. Reed, R. B. Weinstock, and F. Weinhold, J. Chem. Phys.83, 735

~1985!.82NBO 4.0, E. D. Glendening, J. K. Badenhoop, A. E. Reed, J. E. Carp

ter, and F. Weinhold, Theoretical Chemistry Institute, University of Wconsin, Madison~1996!.

83S. M. Gorun and S. J. Lippard, Inorg. Chem.30, 1625~1991!.84D. E. Wheeler, J. H. Rodriguez, and J. K. McCusker, J. Phys. Chem

103, 4101~1999!.85P. E. M. Siegbahn, inNew Methods in Computational Quantum Mecha

ics, edited by I. Prigogine and S. A. Rice~Wiley, New York, 1996!.86H. B. Schlegel, J. Chem. Phys.84, 4530~1986!.87H. B. Schlegel, J. Phys. Chem.92, 3075~1988!.88Popleet al. ~Ref. 101! have argued that the expectation values^S2& aris-

ing from Kohn–Sham wave functions should be larger than the theoical values. However, this is allowed since the Kohn–Sham wave fution is not the wave function of the real~interacting! system.

89J. Kondo, Prog. Theor. Phys., Japan18, 541 ~1957!.90N. L. Huang and R. Orbach, Phys. Rev.154, 487 ~1967!.91D. E. Rimmer, J. Phys. C1, 329 ~1969!.92The Hex5JS1"S2 representation of the Heisenberg Hamiltonian is n

universal. All the results given in this work are expressed in terms ofrepresentation. Other notations include:Hex52JS1"S2 , and Hex

522JS1"S2 . To convert to these latter two notations, simply multiplyJ

by 21 or 212, respectively.

93A. L. Feig and S. J. Lippard, Chem. Rev.94, 759 ~1994!.94S. J. Lippard and J. M. Berg,Principles of Bioinorganic Chemistry~Uni-

versity Science Books, Mill Valley, CA, 1994!.95M. A. Holmes and R. E. Stenkamp, J. Mol. Biol.220, 723 ~1991!.96J. L. York and A. J. Bearden, Biochemistry9, 4549~1970!.97C. J. Ballhausen,Introduction to Ligand Field Theory~McGraw-Hill,

New York, 1962!.98A. Abragam and B. Bleaney,Electron Paramagnetic Resonance of Tra

sition Ions~Dover, New York, 1986!.99See EPAPS Document No. E-JCPSA6-116-707214 for tables contai

optimized cartesian coordinates and extensive descriptions of molecorbitals. This document may be retrieved via the EPAPS homep~http://www.aip.org/pubservs/epaps.html! or from ftp.aip.org in thedirectory /epaps/. See the EPAPS homepage for more information.

100A. A. Ovchinnikov and J. K. Labanowski, Phys. Rev. A53, 3946~1996!.101J. A. Pople, P. M. W. Gill, and N. Handy, Int. J. Quantum Chem.56, 303

~1995!.


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Density functional theory of spin-coupled models for …...structure of large ~e.g., 60–120 atoms! transition metal-containing complexes. The Kohn–Sham32,33 Hamiltonian takes into