Hybrid density functional study of organic magnetic crystals:bi-metallic CrIII cyanides and rhombohedral C60

J. A. CHANy, B. MONTANARI*z, W. L. CHANy and N. M. HARRISONy}

yDepartment of Chemistry, Imperial College London, South Kensington, London SW7 2AZ, UKzCCLRC, Rutherford Appleton Laboratory, Chilton, Didcot, Oxfordshire OX11 0QX, UK

}CCLRC, Daresbury Laboratory, Daresbury, Warrington, Cheshire WA4 4AD, UK

(Received 6 May 2005; in final form 12 May 2005)

Periodic hybrid-exchange density functional theory calculations have been used to investigatethe magnetic properties of two classes of organic magnets, namely the bi-metallic CrIII

cyanides and the polymerized rhombohedral C60 fullerenes (Rh-C60). For the systemsKMII[CrIII(CN)6] with MII ¼ V, Mn, Ni and CrIII[CrIII(CN)6], the magnetic ordering energies,Mulliken populations, and spin density plots are reported for the optimized geometries. Thequalitative nature of the magnetic coupling mechanism is consistent with that observed inprevious unrestricted Hartree–Fock calculations, but the coupling energies computed here aresignificantly higher. The increased coupling is found to be a result of both changes in thegeometry and the electronic structure resulting from the more reliable treatment of electronicexchange and correlation effects. The existence of long-range coupling between local spinmoments is investigated in three different defective Rh-C60 structures: (i) a previouslyproposed prototype structure, where an atom is removed from the C60 cage; (ii) a relatedstructure in which vacancies in nearby cages are brought closer together in pairs; and (iii) astructure where the intra-fullerene bond between the two inter-fullerene bonds is brokenspontaneously after applying isotropic pressure to one layer of the Rh-C60 structure. All ofthese structures are characterized by low flat spin polarized bands at the Fermi edge andlocalized spin moments around the defects, but no evidence of long-range magnetic couplingis found.

Keywords: Organic magnetic crystals; Hybrid density functional study; Bi-metallic CrIII

cyanides; Rhombohedral C60

1. Introduction

In recent years, a new class of magnetic materials, basedon organic compounds, has emerged. These compoundsare light-weight, often insulating, and can be trans-parent and soft, all properties that traditional magnets,composed mainly of transition metals, do not possess.Organic magnets thus provide a new class of materialswith huge potential for novel applications. The mainobstacle to achieving materials of practical use is the lowCurie temperature, TC, which is below 100K in mostorganic magnets. A fundamental understanding of theorigin of the magnetism in these materials and thephysical factors governing the strength of the magneticcoupling is essential to the engineering of materialssuitable for particular applications.

First principles calculations based on density func-tional theory (DFT) currently play an important role inthe characterization of many materials and, in partic-ular, in determining structure–property relationships.However, it has long been recognized that, in stronglyinteracting magnetic systems, the widely used localdensity (LDA) and gradient corrected (GGA) approx-imations fail to predict a qualitatively correct groundstate in many cases [1]. The introduction of non-localFock exchange produces a qualitatively correct descrip-tion of the ground state [2]. Mixing non-local andsemi-local exchange in hybrid-exchange functionals,as in the now very widely used B3LYP functional [3–5],yields a quantitatively good description of thermo-chemistry [6], optical band gaps [7], magnetic momentsand coupling constants [8–12] and metal insulatortransitions [12] in strongly correlated systems.

The CRYSTAL code provides a general purpose, all-electron implementation of both the Hartree–Fock (HF)*Corresponding author. Email: b.montanari@rl.ac.uk

Molecular Physics, Vol. 103, No. 18, 20 September 2005, 2573–2585

Molecular PhysicsISSN 0026–8976 print/ISSN 1362–3028 online # 2005 Taylor & Francis Group Ltd

http://www.tandf.co.uk/journalsDOI: 10.1080/00268970500178869

and DFT approximations in periodic boundary condi-tions and is thus ideally suited for the high precision,hybrid-exchange calculations required to study magneticinteractions in strongly correlated systems [13, 14].The current work concerns the use of hybrid exchange

DFT to examine the mechanisms for magnetic couplingin two classes of system which provide the mostpromising candidates for high-temperature organicmagnets. The bi-metallic CrIII cyanides are examinedas excellent examples of open shell d-electron systemswith local moments strongly coupled via organic ligandsand the defective rhombohedral C60 fullerene structuresas examples of the newly proposed metal-free carbon-based, room temperature ferromagnets.In the past few years bimetallic chromium-based

Prussian Blue analogues, with the general formulaCnAp[Cr(CN)6�q � xH2O, have proved to be very success-ful materials for building three-dimensional molecularmagnets with high and tunable Curie temperatures [15].This group of materials adopts highly symmetricstructures: most often a face-centered-cubic frameworkwith the metallic cations connecting CrIII(CN)6 units(figure 1). An example is Cr3[Cr

III(CN)6�2 � 10H2O,which is air-stable and has a TC of 240K [16].More recently, Curie temperatures above 300K havebeen achieved for chromium-based Prussian Blueanalogues containing vanadium [17–20]. Amongst these,the highest TC, as high as 376K, has been obtainedfor the compound KVII[CrIII(CN)6� � 2H2O [20].

Based on the use of a localized electron orbital model,Verdauger et al. [15] have provided a rationale forthe magnetic coupling mechanism in these materials.The HF approximation has been used in an initial theo-retical investigation of the magnetic coupling mechan-ism in the idealized cubic materials KMII[CrIII(CN)6](MII ¼ V,Mn,Ni) and CrIII[CrIII(CN)6] [21]. The com-puted magnetic ground state and relative orderingenergies were found to be consistent with those observedfor the parent materials, and the underlying mechanismwas found to be consistent with an ionic picture of thebonding and the superexchange model. In the currentwork the extension of these calculations to B3LYPtheory provides more quantitative energetics and cor-rects for the now well documented tendency of the HFapproximation to over-localize the d-electron states andthus significantly underestimate the magnetic couplingenergy [10, 22].

The second part of the paper concerns the recentobservation of high-temperature ferromagnetism inpure carbon materials. Strong ferromagnetism is asso-ciated with the strong coupling of electrons in relativelylocalized d- or f-orbitals, whilst intrinsic magnetism incarbon would have to rely on the s- and p-orbitalsalone. These materials therefore present both a majorchallenge to our current understanding of magnetismand provide a new class of potentially highly tunablematerials.

One such material has been synthesised via the highpressure high temperature treatment of the cubic phaseof C60 fullerenes. At specific temperatures and pressures,the fullerenes polymerise to form rhombobedral C60

(Rh-C60) shown in figure 4. This is a layered phasewhere each layer consists of C60 cages that have beencovalently bonded, through a 2+2 cycloaddition reac-tion, to six neighbouring cages. A ferromagnetic phase issynthesised within the conditions required for therhombobedral phase but only in a temperature regionclose to that at which Rh-C60 becomes unstable withrespect to hard graphitic phases [23, 24]. A remarkablyhigh Curie temperature of s500K was reported ininitial studies [23], with recent data indicating an evenhigher TC of s820K [25]. The magnetism appears to bean intrinsic property of carbon and not due to metallicimpurities such as iron [23, 26].

The experimental characterization of the minoritymagnetic phases in the predominantly non-magnetichost materials has proven difficult and currently thedetailed atomic structure is not known. Transmissionelectron microscopy reveals an apparently well-orderedcrystalline structure in which the C60 cages are largelyintact and still in a Rh-C60-like arrangement [24].It is possible that the ferromagnetism in these systemsstems from the creation of local defects during the

Figure 1. Basal plane of the prototype structure forbi-metallic CrIII cyanides.

2574 J. A. Chan et al.

high-pressure, high-temperature treatment. Such localdefects might give rise to unpaired electrons andtherefore local magnetic moments, and may lead to amagnetically ordered structure if the local magneticmoments interact via a long-range coupling mechanism.A few theoretical studies have so far addressed the

origins of magnetism specifically in Rh-C60. Boukhvalovet al. used local spin density calculations to computethe electronic structure of pristine polymerized Rh-C60

which indicate that it is not magnetic, in agreement withobservation [27]. Other authors have introduced variousdefects and noted the unsurprising result that, whenbonds are broken, localized spin density can form.Of course, the key issues are related to the likelihoodof the formation of such defects and the long-rangecoupling of the resultant spin density. For example, inthe work of Andriotis et al., a defect was created in a C60

fullerene cage simply by removing a carbon atom(see figure 3(a)) [28]. Tight-binding molecular dynamicsand cluster ab initio calculations were then used toanalyse the magnetic properties of the resulting defectiveC60 cages. An ionic model due to McConnell wasinvoked to suggest that inter-cage, through space,coupling could result in long-range coupling betweencages linked via the inter-fullerene bonds of thepolymerized structure [28, 29]. This hypothesis isre-examined in the current work.The work of Ribas et al. used B3LYP calculations to

simulate a C60 dimer, and CASSCF calculations tocompute the interactions within an isolated C60 cage

with hydrogen atoms replacing the inter-fullerene bondsof the Rh-C60 structure [30, 31]. In both cases theirmodel of single broken inter-fullerene bonds betweenthe cages led to a localized spin density, and in thelatter case ferromagnetic or antiferromagnetic statesdepending on the position of the broken inter-fullerenebonds. Various other Rh-C60 structures have beenconstructed with broken inter-fullerene bonds within asemi-empirical Hartree–Fock/AM1 formalism and theresultant band structure, which contains flat bands at theFermi edge, demonstrated a tendency towards magneticlocalization [32]. B3LYP calculations have also beenused to show that a distortion of the Rh-C60 structureabout the inter-fullerene bonds, while constraining allother bond lengths in the cage to 1.40 A, can also giverise to a spin polarized ground state [33].

In all of the above studies the possibility of localmoment formation around various types of defects hasbeen established and there are clearly a large number oflocal defect structures which will result in such localmoments. However, as noted above, an explanationof the observed strong ferromagnetism requires oneto establish both that the defects are credible and thatthere is strong ferromagnetic coupling between the localmoments. In a recent work, the present authorsaddressed both of these issues in a study that combinedreactive force field modelling of the defect formationprocess and the use of periodic B3LYP theory toexamine the inter-cage coupling [34]. In this study adefective polymerized Rh-C60 structure was identified,

(a) (b)

Figure 2. Spin density isosurfaces of the FI (a) and FO (b) states of KVII[CrIII(CN)6]. The light and dark isosurfaces are atþ0.010 and �0:010�B, respectively.

Hybrid density functional study of organic magnetic crystals 2575

which possessed an extra carbon bridge bond betweenthe C60 cages and hydrogen atoms covalently bonded tothe cages. For this structure, long-range couplingbetween the magnetic moments located on each C60

was found, leading to a ferromagnetic ground state. Thestrength of the coupling in this particular geometry wasfound to be insufficient to account for the high Curietemperatures observed, but the study established thepossibility of a ferromagnetic carbon material resultingfrom a realistic defect structure involving inter-cagechemical bridges.In the current work a series of defective structures

where the inter-cage bonding that exists in the pristinepolymers is preserved, and no other inter-cage bondis present, are studied. The structure proposed byAndriotis et al. [28], where the same carbon atom isremoved from each fullerene, is used as a prototypedefective structure; a related structure, where the vacan-cies of two adjacent C60 cages migrate closer to eachother, is also investigated. In addition, a new defectivestructure that is generated spontaneously by applyingisotropic pressure in the plane of the 2D polymerizedlayer is also studied. In all cases, local spin momentsaround the defects arose, but no evidence of long-rangeinteraction between these moments was found.The structure of the paper is as follows. The

methodology employed is described in section 2, thetransition metal cyanides are discussed in section 3 andthe defective fullerenes in section 4, and finally theconclusions are summarized in section 5.

2. Computational details

The first principles calculations presented here wereperformed using the hybrid exchange density functionalB3LYP as implemented in the CRYSTAL package [13].In CRYSTAL, the crystalline wavefunctions areexpanded as a linear combination of atom centredGaussian orbitals (LCAO) with s, p, or d symmetry.In the current study, all-electron calculations areperformed which make no assumptions about theshape of the ionic potential or electron charge density.

For the cyanides calculations, a high-quality, all-electron basis set is used, of double and triple valencequality (that is, two or three independent functions todescribe each valence orbital). This basis set was used in aprevious unrestricted Hartree–Fock (UHF) study ofthese compounds, and has been described in detailelsewhere [21]. To ensure that the total energy isconverged to better than 3meV per formula unit withrespect to reciprocal space sampling, a (8,8,8)Monkhorst–Pack grid was used. The structural optimi-zation was performed in the ferromagnetic configurationusing a modified BFGS algorithm [35].

In the fullerene calculations the starting geometrywas the experimentally determined crystalline structureof Rh-C60, for which the lattice parameters area ¼ 9:19 A, c ¼ 24:5 A in the hexagonal unit cell [36].As the current work focuses on the study of the covalentinter-cage bonds as a mechanism for long-rangecoupling, the system was restricted to a single layer of

(a) (b)

Figure 3. Spin density isosurfaces of the FI (a) and FO (b) states of KNiII[CrIII(CN)6]. The light and dark isosurfaces are atþ0.010 and �0:010�B, respectively.

2576 J. A. Chan et al.

the two-dimensional polymerized cages, that isdescribed in (1� 1) and (2� 1) cells without anyconstraints of symmetry.In order to generate realistic defect structures,

reactive force field simulations of the fullerenes undervarious high-pressure conditions were used previouslyto produce a number of different defect structures.The details of these simulations have been describedpreviously [34]. Defect structures generated in thisway were optimized at the UHF level in the ferro-magnetic configuration using an algorithm proposedby Schlegel [37]. Basis sets of double valence quality(6-21G� for C and 6-31G� for H) were used. A recip-rocal space sampling on a (4� 4) Monkhorst–Packgrid was adopted and was found to be sufficient toconverge the total energy to within 0.1meV per C60

cage in the parent material. The Gaussian overlapcriteria which control the truncation of the Coulomband exchange series in direct space have been docu-mented elsewhere [13]. In the current work they wereset to 10�7, 10�7, 10�7, 10�7, and 10�14.In order to compare the energies of various mag-

netically ordered states, it is necessary to converge stableself-consistent field solutions for different electronicspin configurations. This was achieved by using asuperposition of spin-polarized ionic charge and spindensities for particular atomic states to provide asuitable initial wavefunction. In some cases, more directcontrol of the spin occupancy patterns has been used tocontrol the initial conditions, such as, for instance, thetotal spin. We emphasize, however, that this onlyaffected the initial wavefunctions and that all solutionspresented are unconstrained.

3. Cyanides

For each of the systems KMII[CrIII(CN)6] (MII ¼ V,Mn, Ni) and CrIII[CrIII(CN)6] a full geometrical relaxa-tion was performed. The resultant equilibrium latticeconstants and inter-atomic distances are reported intable 1. The lattice constants calculated within theB3LYP approximation are consistently s3% smaller

than UHF values previously reported [21]. However, theUHF structures are not just expanded versions of theB3LYP structures. The analysis of the bond lengthsshows that whilst the Cr–C and M–N distances areconsistently shorter when calculated with B3LYP, theC–N distances are longer. This indicates a significantdifference in the B3LYP and HF descriptions of theelectronic structure and bonding in these materials.As will be illustrated below, the Cr–C and M–N bondsshow a predominantly ionic nature, whereas theC–N bond is mainly covalent. X-ray diffraction datafor the compounds Cr3½CrðCNÞ6�2 � 10H2O [16] andCsNi[Cr(CN)6� � 2H2O [38] yield lattice parameters

equal to 10.35 and 10.57 A, respectively. Although a

direct comparison with the model compounds examinedin this work is complicated by the residual water inthese materials, it is notable that the agreement betweenthe measured and computed lattice parameters of theCr and Ni compounds is excellent in the B3LYPapproximation and a significant improvement over theprevious UHF treatment. This is an indication that, asexpected from previous studies, the B3LYP approxima-tion provides an improved description of the groundstate electronic structure and bonding in these materials.

The magnetic ordering energies, computed asthe difference between the ferromagnetic (FO) andferrimagnetic (FI) states, are given in table 2. Themagnetic ground state is correctly reproduced for eachmaterial—the Ni compound being weakly ferro-magnetic, the V and Cr compounds strongly ferrimag-netic and the Mn compound weakly ferrimagnetic.

Table 1. Equilibrium structural parameters (A) for the FO state of CrIII½CrIIIðCNÞ6� andKMII½CrIIIðCNÞ6�, where M¼V, Mn, and Ni. The values in parentheses are the corresponding

UHF values from [21].

M a C–N Cr–C M–N

Cr 10.448 (10.719) 1.167 (1.146) 2.038 (2.154) 2.019 (2.060)

V 10.723 (11.012) 1.171 (1.148) 2.069 (2.152) 2.122 (2.216)

Mn 10.815 (11.089) 1.171 (1.155) 2.060 (2.184) 2.176 (2.206)

Ni 10.517 (10.850) 1.170 (1.149) 2.041 (2.138) 2.048 (2.139)

Table 2. Magnetic ordering energies (FO–FI) (eV)compared with the UHF values reported in [21]. A positive

energy favours the FI state.

M d configuration B3LYP UHF

Cr t32g 0.198 0.118

V t32g 0.378 0.120

Mn t32ge2g 0.059 0.0215

Ni t32ge2g �0.073 �0.048

Hybrid density functional study of organic magnetic crystals 2577

As has been discussed previously, within single electronUHF or DFT approaches, this behaviour can beunderstood as a direct consequence of the ionic modeland the operation of the superexchange mechanism viathe delocalized p system of the cyanide ligand [21]. Ingeneral, the B3LYP coupling energies are all larger inmagnitude than those obtained with UHF. The increasein the magnetic ordering energies, however, is not thesame for all metals. The smallest increases, which arestill above 50%, occur in the Mn and Cr systems. In thecase of the V and Ni systems, the B3LYP values aremore than double the UHF values.The difference between the UHF and the B3LYP

predictions is to be ascribed not only to the difference inthe equilibrium structure, where the shorter bondlengths in the B3LYP formalism naturally lead tohigher magnetic coupling, but also to the differenttreatment of the electron–electron interactions. Asdocumented in section 1, hybrid functionals such asB3LYP are generally much more reliable in predictingmagnetic coupling energies than either UHF or GGAformalisms. Here it is clear from the bond lengthsreported in table 1 and bond populations in table 3that the B3LYP approximation leads to a weakeningof the C–N bond and strengthening of the Cr–C bondin all of the materials considered. The bond populationsalso reveal a significant increase in covalency of themetal–N bond in the Ni and V compounds, while theCr and Mn compounds are relatively insensitive. Itappears to be this partial delocalization of the d-electrondensity in the Ni and V compounds that results in thesignificant increase in the magnetic coupling over thatpredicted in the UHF approximation.The strong coupling in the V and Cr materials can

be compared with the critical temperatures of 376Kand 240K observed in the amorphous compoundsKVII[CrIII(CN)6� � 2H2O [20] and Cr3½CrðCNÞ6�2�

10H2O [16], respectively. The relatively weaker couplingin the Ni and Mn materials is consistent with the criticaltemperatures of 53K and 60K, respectively, observedin the AII

3 [CrIII(CN)6�2 � nH2O materials [15] (where

AII is either Ni or Mn). Also, the ratio between the

B3LYP magnetic ordering energies of the M ¼ Vand M ¼ Cr d3 systems reflects more closely thanUHF the ratio between the observed critical tempera-tures for closely related systems. Such comparison ismade meaningful by the fact that the formal state ofthe metal ions is the same in these two systems. TheB3LYP value for this ratio is 1.9, which compareswell with the experimental value of 1.6, whilst the UHFvalue is 1.0.

As discussed in detail previously [21], the interpreta-tion of the magnetic interactions in these systemsdepends on whether an ionic or covalent view of themetal–ligand interaction is adopted [39]. Mulliken bondpopulations can be used here as an indicator of the typeof bonding present. Bond populations equal or greaterthan 0:1jej in magnitude are indicative of bonding oranti-bonding interactions, i.e. covalency [40]. Analysisof the Mulliken bond populations, reported in table 3,shows that the C–N bonds are strongly covalent.Conversely, Cr–C and M–N bonds are ionic in nature.The reported values show that the bond populationsdepend only weakly on the magnetic order. The contri-bution of the covalent bonding to the magnetic energyis therefore small, indicating that the ionic pictureprovides a more appropriate basis for an explanation ofthe magnetic interactions in these systems. In the ionicpicture, the superexchange model is appropriate [41].The present calculations confirm that, for the Cr and Vcompounds, where the electronic configuration of theM site is t32g, the magnetic interaction is strong andantiferromagnetic in nature. Analysis of the spin densityplots displays the operation of the long-range super-exchange interaction in a DFT calculation. In theferrimagnetic configuration the energy of the system islowered significantly by the polarization of the cyanideligands p system (figure 4(a)). In the ferromagnetic statethe polarization is suppressed (figure 4(b)). This is themean field description of the superexchange interactionand yields an effective repulsion between the localizedt32g orbitals on the metal centres via the ligand whichdestabilizes the FO state relative to the FI state.The dependence of this mechanism on the symmetry

Table 3. The Mulliken bond populations (|e|) for the metal–N, Cr–C, and C–N nearest neighbour interactions in the FO and FIstates. The corresponding UHF values from [21] are shown in parentheses.

M

M–N Cr–C C–N

FO FI FO FI FO FI

Cr 0.052 (0.033) 0.054 (0.034) 0.052 (0.034) 0.054 (0.035) 0.633 (0.697) 0.629 (0.689)

V 0.047 (0.011) 0.054 (0.012) 0.050 (0.034) 0.054 (0.035) 0.632 (0.714) 0.623 (0.706)

Mn 0.003 (�0.014) 0.005 (�0.013) 0.051 (0.034) 0.052 (0.034) 0.652 (0.714) 0.651 (0.712)

Ni 0.019 (0.003) 0.019 (0.002) 0.052 (0.035) 0.052 (0.034) 0.634 (0.709) 0.636 (0.712)

2578 J. A. Chan et al.

of the localized d-orbitals is clear from an examinationof the spin density in the FI and FO states of the Nicompound (figure 5). For the Ni d8 state the unpairedelectrons are of eg symmetry and interact far less withthe p system of the ligand. The polarization of the ligandin the FO state is due almost entirely to the repulsionwith the t32g orbitals of the Cr ion and thus very little spindensity localizes on the C atom. The absence of a strongsuperexchange interaction leaves a net FO magneticcoupling.The higher degree of electronic delocalization and

dependence of the electronic polarization of the CN�

group on the spin state are also reflected in thecomparison between the FO and FI Mulliken bondpopulations. The population of the M–N and Cr–Cbonds is larger in the FI states for M ¼ Cr, V, and Mn.For all these systems the population of the C–N bonddecreases slightly in going from the FO to the FIconfiguration. The operation of the superexchangemechanism is reflected in small changes in the bondpopulations. In the transition from the FO to the FIstate for the Cr, V, and Mn compounds, there is a smallcharge transfer from the C�N p system to the M–Crand Cr–C � bonds. The influence of the spin stateon the bond populations is almost negligible in the Nicompound.The Mulliken spin populations for each atom are

reported in table 4. In all cases, the C and N atoms havea much smaller spin population than the transitionmetal atoms, as expected. The magnitude of spinpopulations on C and N is particularly small in theFO states for M ¼ Cr and V, and much larger in theFI case. Conversely, in these systems the spin popula-tions on the M and Cr sites are larger in the FOstates than in the FI states, and this again indicatesthe higher degree of delocalization in the FI state.These differences are most pronounced in the Vcompound.

4. Polymerized fullerenes

The electronic band structure of the 3D pristine,polymerized Rh-C60 at the experimental geometry [36]was calculated within the B3LYP approximation, andhas an indirect fundamental band gap of 1.48 eV.Previous DFT calculations in the local densityapproximation (LDA) also described the system as anon-spin-polarized semiconductor with an indirectfundamental gap, but the value of the gap was muchsmaller, at 0.35 eV [42]. A reliable measurement of thegap is complicated by the highly defective nature of thematerial, but it is highly likely that the B3LYP valueis more reliable as the LDA generally significantly

Figure 5. Central vacancy structure proposed in [28] withthe undercoordinated atoms labelled C1, C2 and C3 (a). Spindensity maps of the vacancy structure in its ground state withparallel (FO) inter-cage spin configuration (b), and antiparallel(AF) inter-cage spin configuration (c). The light and darkisosurfaces are at þ0.015 and �0:015�B, respectively.

Figure 4. The pristine rhombohedral phase of polymericC60.

Hybrid density functional study of organic magnetic crystals 2579

underestimates band gaps while the B3LYP approxima-tion has recently been shown to reproduce the observedband gaps in a wide variety of materials [7]. For thesingle layer calculations the lattice constants of one layerare relaxed within the reactive force-field formalism,yielding an initial structure with a ¼ 9:23 A. This is anincrease of just 0.4% from the observed lattice constantof the bulk crystal.

4.1. A central vacancy structure

Here the defect structure proposed by Andriotis et al.is re-examined. The structure is generated by taking theexperimental geometry [36] and removing one carbonatom from each fullerene cage, as shown in figure 3(a).The lattice constants were optimized using the reactiveforce-field approximation, resulting in a ¼ 9:22 A, thenthe internal coordinates were optimized in the ferro-magnetic configuration in the UHF approximation.The spin density map of the ground state is reported

in figure 3(b) where spin density is shown to localize atthe dangling bonds of the undercoordinated carbonatoms, as expected. The total magnetic moment per C60

cage is 2:0�B.The spin alternation model developed for planar

p-conjugated systems [43–45] has recently been shownto be valid also for fullerene systems [34]. Within themodel the spin polarization simply alternates betweenbonded carbon atoms due to local exchange repulsion.The ground state configuration for the vacancy isentirely consistent with this model; there is spin-updensity on atoms C1 and C2, spin down density on C3,and alternating spin polarization on neighbouring atoms(figure 3(b)). A number of other metastable spinconfigurations can be generated but are found to bebetween s0.06 and 0.44 eV per cage higher in energy.The Mulliken charge and spin bond populations

between the undercoordinated atoms are reported intable 5 together with the inter-atomic distances. Theundercoordinated atoms are separated by over 2.7 A,almost twice as large as a typical C–C distance on a C60

fullerene and, as one might expect, the direct chargeand spin overlap populations are small.

In order to test whether the magnetic moments oneach C60 cage interact with those localized on the nearbyC60, a (2� 1) supercell was considered, which containstwo C60 units. Table 6 summarizes the findings for thissystem. Two spin configurations where examined. In theFO state the spin configuration is identical to the oneobtained for the (1� 1) cell and described above. In theantiferromagnetic (AF) state, all spin orientations onone C60 cage in the supercell are reversed, as shown infigure 3(c). There is no significant difference between theenergies of the inter-cage FO and AF configurations andtherefore there is no significant coupling between thespin moments of neighbouring cages. This indicatesthat, if this vacancy defect structure could be realizedin practice, it would result in a paramagnetic groundstate and not an ordered magnetic ground state.No significant change in the Mulliken charge or spinoverlap populations of the inter-fullerene bonds between

Table 4. The Mulliken spin populations (|e|) for the metal M, Cr, C, and N sites in the FO and FI states. The corresponding UHFvalues from [21] are shown in parentheses.

M

FO FI

M Cr C N M Cr C N

Cr 2.91 (3.04) 2.95 (3.14) �0.02 (�0.06) 0.05 (0.03) 2.90 (3.07) �2.94 (�3.18) �0.16 (0.28) 0.16 (�0.27)

V 2.75 (2.97) 2.95 (3.14) 0.01 (�0.09) 0.04 (0.07) �2.66 (2.98) 2.88 (�3.17) �0.18 (0.29) 0.14 (�0.26)

Mn 4.76 (4.91) 2.95 (3.16) �0.06 (�0.17) 0.11 (0.15) �4.75 (4.93) 2.92 (�3.17) 0.12 (0.22) �0.10 (�0.18)

Ni 1.76 (1.93) 2.92 (3.17) �0.11 (�0.24) 0.16 (0.22) �1.75 (1.94) 2.90 (�3.15) �0.08 (0.16) 0.05 (�0.13)

Table 5. Inter-atomic distance, and Mulliken charge andspin populations between the undercoordinated atoms ofthe defective Rh-C60, central vacancy structure in the

ground state configuration.

Interaction Distance (A) Charge (|e|) Spin (|e|)

C1–C2 2.924 0.010 þ0.004

C2–C3 2.723 0.023 0.000

C3–C1 2.893 0.025 �0.002

Table 6. Total magnetic moment per C60 cage (S), Mullikenspin population of the undercoordinated atoms and totalenergy differences for the defective Rh-C60, central vacancy

structure.

Configuration S ð�BÞ Atoms Spin (|e|) �Etot ðeV=C60Þ

FO 2.0 C1 1.22 0.000

C2 1.26

C3 �0.83

AF 2.0 C1 �1.20 0.000

C2 �1.24

C3 0.84

2580 J. A. Chan et al.

FO and AF spin configurations is seen, in agreementwith the lack of coupling between the cages.It is notable that the two-dimensional band structure

of the ground state has low lying flat bands at theFermi edge separated by a spin-flip indirect band gapof 1.66 eV. However, in this case the flat bands aresimply a result of the localized spin orbitals with nosignificant inter-cage interaction and are not associatedwith a magnetic instability and thus are not indicative ofa magnetically ordered ground state.

4.2. A migrated vacancy structure

It is possible that a (meta)stable structure exists wherethe carbon vacancies on neighbouring fullerenes areclose enough to each other so that the local spinmoments can directly couple. Such a structure might becreated as a minority phase under extreme temperatureand pressure conditions, which facilitate defect migra-tion. In order to examine this possibility a new structurehas been created based on the central vacancy proposalin which the defect sites on neighbouring C60 cages arepositioned closer to each other (see figure 6(a)). Thisstructure is described in a (2� 1) supercell of the parentstructure. The lattice parameters are constrained to bethe same as those in the central vacancy defect structurein order to facilitate a meaningful comparison oftheir total energies. The internal coordinates are fullyoptimized at the UHF level. Figure 6(b) shows thespin density map of the spin configuration with thelowest energy amongst all the configurations found.The intra-cage geometric and electronic arrangement ofthe three undercoordinated C atoms around eachvacancy, reported in detail in table 7, is similar to thatof the central vacancy structure. The majority of thespin polarisation is described by the spin alternationrule, though with exceptions on undercoordinatedatoms C4 and C5. If the spin density on these atomswere described by the spin alternation rule, the spindensity on atom C5 would be parallel to that on C6, andantiparallel to that on C4, contrary to what is observed(see figure 6). This suggests that the dominant spininteractions around these defects are through spacerather than through bond and, therefore, favour anantiparallel spin alignment between the atoms C5 andC6 and between C4 and C6. The polarization of the spindensity also decays somewhat more slowly and overallthe local moments cancel so that the net magneticmoment per C60 cage for the ground state is zero.Clearly, this geometry does not result in a magneticallyordered ground state; nevertheless, it is interesting toexamine the inter-cage interactions of the localized spindensity.

Table 8 shows the results for this state, called AF1,and another state constructed starting from the AF1configuration and flipping the spins of the unpairedelectrons on one C60 cage in the supercell. This

Table 7. Inter-atomic distance, and Mulliken chargeand spin populations between the undercoordinated carbonatoms of the defective Rh-C60, migrated vacancy structure

in the AF1 configuration.

Interaction Distance (A) Charge (|e|) Spin (|e|)

C4–C5 2.766 0.008 �0.003

C5–C6 2.877 0.025 þ0.002

C6–C4 2.945 0.022 0.000

Figure 6. Migrated vacancy structure with the undercoordi-nated atoms labelled C4, C5 and C6 (a). Spin density maps ofthe migrated vacancy structure with antiparallel (AF1) inter-cage spin configuration (b), and parallel (AF2) inter-cage spinconfiguration (c). The light and dark isosurfaces are at þ0.015and �0:015�B, respectively.

Hybrid density functional study of organic magnetic crystals 2581

configuration, shown in figure 6(c), and is referred to asAF2, also possesses zero magnetic moment per cage.The energy difference of 16meV per cage between theAF1 and AF2 states shows an increased inter-cage spincoupling compared to the central vacancy defect, due tothe closer proximity of the defects. However, the moststable configuration found has antiferromagneticcoupling between the major spin centres within thecages, as is evident from figure 6(b). It is notable thatthis antiferromagnetic coupling is consistent with thespin alternation model. Consequently, even if the defectson each cage were such that they would carry a netmagnetic moment, still the inter-cage coupling betweenthem would be antiferromagnetic.In addition, the energy of this migrated vacancy

configuration is 0.291 eV per cage higher than that ofthe ground state of the central vacancy structure,indicating that there is a strong repulsion betweenvacancies. These facts indicate that this structure doesnot compete as a candidate for the ferromagnetic phase.

4.3. Spontaneous formation of defectsfrom isotropic pressure

In order to generate more realistic defective structures,reactive force field simulations are performed in whichisotropic pressure is applied by gradually reducing thea and b lattice parameters in the (1� 1) cell, whilstrelaxing the structure at each step. Defective structuresappear spontaneously for a 5–6% reduction in the latticeconstants. A number of different structures can begenerated in this way as the total strain and its rate ofapplication are varied. A common structure, generatedunder these conditions, has a fractured intra-fullerenebond. The bond that breaks is between the two inter-fullerene bonds, as shown in figure 7(a). This is incontrast to the usual assumption that the inter-fullerenebond would be the most likely to break [30–32].However, the single crystal X-ray diffraction studiesperformed by Chen et al. [46] identified this intra-fullerene bond as the longest in the structure, thus

suggesting that it may be the weakest bond in thepolymerized structure. Table 9 reports the distance andthe Mulliken overlap populations between the twoundercoordinated atoms, which show a distance increaseof 62%, from 1.594 A to 2.583 A, as the bond breaks.

The resulting defective structure exhibits localmoments on the undercoordinated atoms and is, again,characterized by flat, spin polarized bands at the Fermiedge. Various spin configurations were analysed in a(2� 1) cell in order to investigate the intra-cageand inter-cage magnetic coupling. The FO configura-tion is one where the spin moments on all theundercoordinated atoms labelled as C1, C2, C3, andC4 in figure 7(a) are coupled parallel to each other. Thespin density map in figure 7(b) refers to the AF1

Table 8. Total magnetic moment per C60 cage (S), Mullikenspin population of the undercoordinated atoms and total

energy differences of the defective Rh-C60, migrated vacancystructure. The energies are relative to the central vacancy defect.

Configuration S ð�BÞ Atoms Spin (|e|) �Etot ðeV=C60Þ

AF1 0.0 C4 �0.89 þ0.291

C5 �0.67

C6 þ1.18

AF2 0.0 C4 þ0.90 þ0.307

C5 þ0.68

C6 �1.16

Figure 7. Defective structure generated by isotropic pressurewith the undercoordinated atoms labelled C1, C2, C3 andC4 (a), and the spin density maps of AF1 (b), and AF2(c) configurations. The light and dark isosurfaces are at þ0.010and �0:010�B, respectively.

2582 J. A. Chan et al.

configuration, where the spin moments on C1 and C2are antiparallel to each other as are those between C3and C4. The AF2 configuration, shown in figure 7(c), isobtained from AF1 by flipping the spins on C3 and C4.The results for each of these possibilities are quantifiedin table 10. By comparing the total energies of the FOand AF1 case, it is clear that the spin densities betweenatoms C1 and C2 and between C3 and C4 couple witheach other antiferromagnetically. This configuration isin agreement with the spin alternation model and themagnitude of the coupling is s0.7 eV per broken bond.Although the bond overlap populations are very small(table 9) there is significant increase in the overlapcharge in going from the FO to the AF1 case. The spindensity map in figure 7(b) shows that the spin density islocalized in p-orbitals that are directed toward eachother. The coupling results in a defective structure withno net moment per cage.In order to investigate the inter-cage coupling, the

total energies of the AF1 and AF2 configurations arecompared and found to differ by only 0.5meV, indi-cating that no significant inter-cage interactions arepresent. Therefore, this structure can also be discardedas a possible explanation of the ferromagnetic characterof this material.

5. Conclusions

Periodic hybrid exchange density functional simulationshave been used to investigate the magnetic properties oftwo classes of organic magnets, namely bi-metallic CrIII

cyanides and polymerized rhombohedral C60 fullerenes(Rh-C60).

In the bi-metallic CrIII cyanides the hybrid exchangefunctional results in cell volumes that are consistentlysmaller than those computed previously in the UHFapproximation and in better agreement with availableX-ray diffraction measurements. The hybrid exchangeapproach also predicts significantly larger magneticcoupling constants than UHF, partially due to theshorter bond lengths but also due to significant dif-ferences in the description of the electronic structure.For the V and Cr systems the ratio of the computedcoupling constants can be compared to the observedratio between the transition temperatures and indicatesthat the hybrid exchange formalism is in better agree-ment with experiment that the UHF approximation.

These results, in addition to the rapidly growingliterature on the use of the hybrid exchange in magnetictransition metal oxides, indicate that the methodprovides a reliable description of the magnetic groundstate in a wide range of magnetically ordered systems,including those containing organic ligands.

A class of defective Rh-C60 fullerenes, where the inter-fullerene bonding is unchanged compared to the pristinepolymers, has also been studied in order to examinethe proposal that long-range magnetic order is presentin the absence of additional inter-fullerene bonds. Thedefect structures examined here are all insulatingstates with spin-polarized, flat bands at the Fermi level.These states are not found to be generally indicativeof a magnetic instability with respect to a ferromagneticground state, but simply a result of the presence ofunpaired electrons in localized states with no significantinter-cage interaction. The most likely mechanism formagnetic coupling appears therefore to be direct throughspace exchange or superexchange mediated by thedelocalized p system of the cage, both of which aredescribed reliably by the formalism adopted in thecurrent work.

A fully periodic version of a recently proposedvacancy structure was used as a prototype defect. Theground state has a stable spin configuration with amagnetic moment of 2:0�B per cage, but the couplingwith the spin moments on nearby cages is negligible.The possibility of vacancy migration under extremeconditions resulting in structures with direct throughspace inter-cage coupling is also investigated. However,in this structure the magnetic moment on the C60 cagesis zero and the inter-cage coupling that does exist

Table 9. Mulliken charge and spin overlap populations (|e|)and distances (A) between the undercoordinated atoms C1 andC2 in FO and AF1 configurations of the defective Rh-C60

structure generated by isotropic pressure.

Configuration Distance Charge Spin

FO 2.583 0.006 0.000

AF1 2.583 0.045 0.000

Table 10. Spin configuration, total magnetic moment perC60 cage, S, Mulliken spin population of the undercoordinatedatoms and total energy differences of the defective Rh-C60

structure generated by isotropic pressure.

Configuration S ð�BÞ Atoms Spin (|e|) �Etot ðeV=C60Þ

FO 2.0 C1 0.74 0.000

C2 0.66

C3 0.74

C4 0.66

AF1 0.0 C1 �0.28 �0.735

C2 0.25

C3 �0.28

C4 0.25

AF2 0.0 C1 �0.29 �0.736

C2 0.25

C3 0.29

C4 �0.25

Hybrid density functional study of organic magnetic crystals 2583

between the local moments is antiferromagnetic innature. A more realistic defect structure generated byapplying isotropic pressure results in the fracture of theintra-cage bond between the two inter-cage bonds of thepolymerized structure. The resulting unpaired electronslocalized on each cage couple antiferromagnetically,leading to a zero magnetic moment per cage and, again,the inter-cage coupling between the unpaired electrons isnegligible.In summary, none of the structures examined here is

consistent with the observed ferromagnetic ground statemeasured in this material. The only structure studiedthus far which produces an inter-cage ferromagneticcoupling involves additional inter-cage bonds [34].Another important conclusion of this work is that the

simple alternating spin rule based on the spin polari-zation effect, originally formulated for p-conjugated,planar organic structures, generally retains its validityfor these systems where a significant curvature of thestructure is present. The understanding gained isvaluable for anticipating which defect structures mightgenerate ferromagnetic ground states.

Acknowledgements

The authors would like to thank the EPSRC forprovision of computer time under the MaterialsChemistry Consortium project, GR/S13422/01, and alsoProfessor J. Gale for useful discussions.

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