Fig. 1. Diagram illustrating a classification of disordered magnets based onfrustration and the degree of disorder. For details see Refs. [7,27].

F. Cimpoesu et al. / Computational Materials Science 91 (2014) 320–328 321

enabled the direct measurement of the occupied electronic struc-ture [11,12].

Magnetic experimental studies of the V[TCNE]x�y(S) systems(with S being acetonitrile or tetrahydrofuran) revealed [13,14] sta-tic critical behaviour consistent with existing models for randomanisotropy magnets (RAM), previously applied only for site-dilutedand amorphous metallic alloys [15]. It was speculated that themagnetic disorder was due to the structural and substitutional dis-order resulting from the incorporation of the spinless solvent, aswell as the random coordination topologies of the vanadium ions[13,14]. Similar scaling studies performed for other members ofthe family M[TCNE]x�y(CH2Cl2), with M = Mn, and Fe, showed avery rich magnetic behaviour for both compounds, with double-transition random anisotropy characteristics for the Fe-based sys-tem [16] and re-entrant spin-glass behaviour for the Mn-basedmaterial [17]. More recently, photoinduced magnetization studiesemphasized [18] the light-induced increase of random magneticanisotropy due to increased structural disorder.

To date, little theoretical work has been done on the structureand electronic properties of the V[TCNE]x system. Recently, amodel structure was proposed [19] and on that basis the structural,electronic, and magnetic properties were investigated using den-sity functional theory (DFT) with periodic boundary conditions(PBC). However, to date, disorder and anisotropy have not beenstudied theoretically for V[TCNE]x. Despite piling experimental evi-dence indicating random anisotropy and not spin glass behaviour,no theoretical explanation has been provided so far.

We explore here possible sources of structural disorder andanalyze their impact on the magnetic properties of the system.By means of DFT calculations we determine the strength and signof the exchange coupling constants and are able to explain bothwhy irreversibilities are present and why the spin glass/randomexchange model is not applicable. Furthermore, based on multi-configurational methods, considering the additional spin–orbitinteraction, and on in house routines for the numerical differenti-ation of the energy, containing the Zeeman interaction due to anexternal magnetic field, we can determine the uniform as well asthe random single-ion-anisotropy of the V(II) ions. We show thatour results explain the experimental data, clarifying the interplayof ferrimagnetism and random magnetic anisotropy in V[TCNE]x

and indicating correlated sperimagnetic and not spin glassbehaviour.

2. Background on pure and disordered magnets

The exchange coupling constant J , for a system of two elec-trons is related to the difference between the energies of the twopossible spin configurations (parallel and antiparallel). The resultis an effective interaction, in the spin space, described by the Hei-senberg exchange Hamiltonian [6]:

bHH ¼ �2J Si � Sj ð1Þ

The ferromagnetic state, characterized by parallel alignment ofthe spins, occurs when the exchange constant, J , is positive,whereas the antiferromagnetic state, characterized by antiparallelspin alignment, arises when the exchange integral is negative.

The single-ion anisotropy (SIA) arises if the symmetry of theenvironment is lower than cubic. In second order of perturbationtheory, the additional spin–orbit (SO) interaction leads to a zero-field splitting matrix which has different elements, even in thediagonal form. In the diagonal basis one gets a spin Hamiltonianthat describes the single-ion-anisotropy [20,21]:

HSIA ¼ D S2z �

13

SðSþ 1Þ� �

þ EðS2x � S2

yÞ ð2Þ

In the case of axially symmetric environments E = 0 and the single-ion anisotropy is just [20] HSIA ¼ DS2

z .This contribution to the spin Hamiltonian expresses the fact

that in the absence of an applied magnetic field the spins tend toorient along preferential directions, induced by the anisotropicenvironment through the crystal field. We note that in most prac-tical cases for transition-metal ions some distortions from the idealsymmetry are present causing some anisotropy to arise [21]. Thesingle-ion anisotropy constant, D, is proportional to the square ofthe SO coupling constant (which has typical values of 102 cm�1)and inverse proportional to the crystal field splitting (�104 cm�1)being of the order of 1 cm�1 or even smaller for many transitionmetal ion systems [20,21].

When combined with the Heisenberg exchange, the SIA canfavour, in its two limiting cases, either Ising (D ? �1) or XY(D ?1) anisotropy of the spin systems. While a Heisenberg spinis isotropic (one can picture the classical image of a vector thatcan point anywhere on a sphere) and has a spin-dimensionalityof n = 3, an XY spin can point anywhere on a circle (n = 2), and anIsing spin can point along a single given direction either up ordown (n = 1) [7,22].

Disorder can be quenched or annealed [7,23–25]. In the case ofrandom magnets, the disorder is quenched if the spatial distribu-tion of exchange is frozen in, due to fixed magnetic moments. Dis-order is considered annealed if the magnetic ions can diffuse in thesolid and change the distribution of exchange. The two types ofdisorder are distinct and lead to very different types of magneticbehaviour.

The main feature of the dynamic behaviour of systems with dis-order is the slower rate of relaxation processes. In ordered macro-scopic systems the thermodynamic equilibrium is regained veryquickly after the perturbation is removed such that for normalexperimental times, even at phase transitions when the relaxationtimes are the largest, one can ignore nonequilibrium effects [25]. Inrandom systems the relaxation is hindered by disorder and theevolution back to equilibrium is slower [25].

A magnetic system is frustrated if it cannot minimize the energyof each spin pair in the network, simultaneously [26]. The strengthof the frustration is characterized by the number of configurationsthat minimize the total energy equally well. Frustration is gener-ated by the competition between interactions of different kinds(e.g., competing ferro- and antiferromagnetic interactions or com-peting nearest-neighbor and next-nearest-neighbor interactions)or by the topology of the lattice (e.g., triangular, kagomé, pyroch-lore lattices with antiferromagnetic nearest-neighbor interactions,etc.) [27]. A simple diagram classifying disordered magnets basedon frustration and degree of disorder is shown in Fig. 1.

The dilute magnets [25,28] are systems with random exchangebut with no frustration. From a theoretical point of view, the diluteferromagnet, can be modeled by a random exchange coupling, suchthat the interaction varies only in strength, but not in sign. Theequilibrium behaviour of dilute ferromagnets is qualitatively thesame as that of pure systems [25], the subtle effects of disorder

322 F. Cimpoesu et al. / Computational Materials Science 91 (2014) 320–328

being more relevant for Ising systems, especially in three dimen-sions. The dynamic properties of unfrustrated disordered systemsare similar to the pure ones, however, the domain wall pinningat the disordered sites causes slower relaxation [29].

Spin glasses [23,24,30] are magnetic systems characterized by arandom, yet cooperative freezing of spins below a well definedtemperature. Theoretically, a spin glass is a frustrated randomexchange system [31,32]. Unlike the dilute disordered systems dis-cussed above, where the exchange was random in value but of thesame sign, frustrated random exchange systems have both ferro-and antiferromagnetic interactions. Such systems are describedby the Hamiltonian:

bHSG ¼ �2Xi<j

J ijSi � Sj; ð3Þ

where the exchange constants have both positive and negative val-ues. For instance, one simple implementation of this model consid-ers for J ij only two values, �J 0, whereas the typical continuousdistribution is a Gaussian centred in origin (J ¼ 0) [24].

The static response of a spin glass is dependent on the magnetichistory, with different field-cooled and zero-field-cooled magneti-zation curves [23,24]. The dynamics of the ac susceptibility close tothe critical temperature are very slow. Due to the strong random-ness present in the system the domain wall has a fractal shape andis pinned longer and more often, causing strong irreversibilities inthe magnetic behaviour [33]. One indicator of such irreversibilitiesis the rough energy landscape [34].

The random anisotropy magnets are systems with uniformexchange, while some kind of anisotropy is randomly distributed,determining the spins to orient along directions that vary ran-domly from site to site. The main model studied is the random uni-axial anisotropy model [35–37]:

bHRAM ¼ �2JXi<j

Si � Sj þ Dr

Xi

ðni � SiÞ2 þ Dc

Xi

ðN � SiÞ2

ð4Þ

The first term is the Heisenberg exchange of strength J , the sec-ond is a random uniaxial anisotropy term of strength Dr and localorientation ni, the third is a coherent (uniform, non-random) uni-axial anisotropy term of strength Dc and fixed orientation, N.

In the case of such systems there is a competition between twoenergies: the exchange and the anisotropy. The random anisotropy,no matter how small, destroys the long range ferromagnetic orderfor systems with magnetic lattice dimensionality of three or less[38]. The random anisotropy model leads, depending on thestrength of the anisotropy and of the external magnetic field, tovarious magnetic regimes, like speromagnet, correlated speromag-net, ferromagnet with wandering axis, etc. [39,40].

We note that, at some point, similar terminology (in particular‘spin glasses’) was used for different types of systems. To avoid anyconfusion the notion of a ‘spin glass’ is used in connection to therandom frustrated exchange model whereas ‘speromagnet’ and‘correlated speromagnet’ are used regarding the states of randomanisotropy systems [22,24,41].

When the random anisotropy is large compared to exchangeand the coherent anisotropy is negligible, each spin tends to orientalong the anisotropy axis at its site, the collective behaviour beinga secondary phenomenon. The resulting speromagnetic state[22,39,40] is described by only short range ferromagnetic orderand the spin orientation may resemble that of a spin glass. Whenthe random anisotropy is weak (and any uniform anisotropy and/or applied field are negligible) the system behaves cooperatively,however it has no net magnetization [37]. The local ferromagneticmagnetization changes direction significantly at distances on theorder of the finite ferromagnetic correlation length, leading to acorrelated speromagnetic state [40].

In the presence of a moderate applied field a magnetic momentis produced, leading to a field induced new regime, called ferro-magnet with wandering axis [39] or asperomagnet [40]. In thelimit of high applied magnetic fields the noncollinear structurealigns even further toward the applied magnetic field, the randomanisotropy being able only to slightly tilt the spins from the direc-tion of the applied field. A coherent uniaxial anisotropy can havethe effect of an applied field, leading to either one of the statesmentioned above, depending on its strength [39].

We note that in all of the previous discussion of terminology weassumed a ferromagnetic exchange interaction between spins. Ifthe exchange coupling constant is negative, in the case of two-sub-network magnetic structures, the terminology changes accord-ingly. Continuing the parallel between ferro- and ferrimagnetism,in the case of random systems with strong anisotropy we contrastspero- and sperimagnetism [22,40], as illustrated in Fig. 2.

The term sperimagnet refers to a random ferrimagnet withstrong anisotropy, a system with two subnetworks of spins, suchthat the moments of one or both subnetworks are frozen in ran-dom orientations. Continuing the analogy, when the randomanisotropy is weak compared to the exchange interaction, the sys-tem has no net magnetization but behaves cooperatively over lar-ger distances, leading to a ‘correlated sperimagnetic’ state.

3. Computational details

The DFT [42] band structure calculations with periodic bound-ary conditions (PBC) were performed with the Gaussian 03 package[43], using the BLYP [44,45] functional and the 6-31G� basis set[46]. The complete active space self consistent field (CASSCF)[46] calculations and the subsequent spin–orbit (SO) interactiontreatments, were performed with the GAMESS package [47] usingthe 6-31G� basis set.

As the CASSCF + SO calculations cannot provide all the relevantanisotropy constants, we designed and implemented additionalroutines that allow a complete description of the single ion anisot-ropy, both in terms of strength and in terms of sign. This methodwas recently reported [48–50] for a different class of molecularmagnets, and is based on the numerical differentiation of theenergy eigenvalues containing an additional Zeeman term [20],with respect to the magnetic field along various space orientations.The numerical derivative supplies a polar diagram of the magneti-zation function for the lowest states, representation that helpsidentifying the spin of the lowest multiplet and the strength andsign of the single-ion-anisotropy [50].

4. Results and discussion

This section consists of three parts. The first explores varioussources of structural disorder, the second deals with DFT calcula-tions providing the exchange constants and addressing the applica-bility of the random exchange model, whereas the third reveals therole of magnetic anisotropy by multiconfigurational calculations.

4.1. Structural disorder in V[TCNE]x

The structure consistent with the available experimental data[10] must ensure the correct V[TCNE]2 stoichiometry and aquasi-octahedral coordination sphere, with almost homogeneousbond lengths of about 2 Å [19]. The simplest 2D structure fulfillingthe V[TCNE]2 stoichiometry and the trans V-TCNE-V bridging is aquasi-rectangular pattern with the V(II) ion placed at the tetra-coordinated nodes of the lattice and the TCNEs on the edges ofthe tetragonal network. Then, the 3D lattice is completed by thecorresponding stacking of such 2D planes along the c crystallo-

Fig. 2. Schematic representation of spin configurations in some disordered magnets: (a) dilute ferromagnet, (b) speromagnet, (c) dilute ferrimagnet, and (d) sperimagnet. Fordetails see Ref. [22].

Fig. 3. Schematic representation of the ideal model structure of V[TCNE]2. (A) Themolecular V[TCNE]2 unit. (B) The 2D expansion of a lattice accomplishing the 1:2metal to ligand stoichiometry. (C) The 3D stacking along the c direction of the 2Dplains. The vanadium, carbon, and nitrogen atoms are represented as yellow, grey,and blue, respectively. (For interpretation of the references to colour in this figurelegend, the reader is referred to the web version of this article.)

Fig. 4. Perspective view of V[TCNE]2 structures along the c axis: the DFT optimizedstructure (A) and the structure altered by a rotation of the bi-connected TCNE unitsfrom the upper sheet around the b axis by 90� (B) and 180� (C).

F. Cimpoesu et al. / Computational Materials Science 91 (2014) 320–328 323

graphic axis, by sliding along one of the two axes (say a), so thatthe upper and lower TCNEs can complete the axial coordinationof V(II) sites from adjacent planes (see Fig. 3). In this way, theligands along the a and b crystallographic axes become non-equiv-alent, the first ones being tetra-connected, along both the a and cdirections, with the other ligands remaining bi-connected bridges,along the b directions only, with pending non-coordinated -CNgroups on the c direction.

The model structure idealized initially as a monoclinic cell, bytaking orthogonal and identical a and b translation vectors, wassubmitted to the DFT optimization with PBCs, using the BLYP func-tional and the 6-31G� basis set. The result for the doublet ground

324 F. Cimpoesu et al. / Computational Materials Science 91 (2014) 320–328

state of the molecular unit (i.e. the ferrimagnetic overall ordering)was a triclinic structure with the unit cell parameters as follows:a = 10.191 Å, b = 9.963 Å, c = 7.213 Å, a = 81.57�, b = 135.22�, andc = 85.81� (see Fig. 4(A)). The cell parameters are slightly differentfrom those previously reported [19], however, the differences aremostly due to a distinct choice of the molecular orientation andcell definition.

We focus in the following on the sources of disorder in CVDfilms, noting that the presence of the intercalated solvent in thesamples synthesized chemically may cause additional defects.One source of disorder originates in the equivalence of the a andb sliding directions when building the 3D network with successivelayers. Since in normal syntheses (chemical routes or vapour depo-sition) there is no reason to expect a chiral asymmetry, we expect aracemic global result, with regions having the tetra-connectedTCNE along the a or the b axes occurring equally probable. It islikely that the local chirality is changed from one region to thenext, with the cost of lattice defects, such as TCNE vacancies (seeFig. 5). The influence of TCNE vacancies on the magnetic anisotropyof the system will be studied in a following section.

Another source of disorder comes from the orientation of theTCNE molecules. Evidence for orientational disorder of TCNE moi-eties has been reported [51] in the case of manganese-porphyrin-TCNE compounds [52,53]. In that case, the two orientations ofTCNE are related by a 180� rotation about the diagonal axis joiningthe trans nitrogen atoms bound to the metal ion. The two formshave occupancies of 83.3% and 16.7%, respectively [51].

In the case of V[TCNE]x, one can similarly imagine different con-figurations of the bi-connecting l2-TCNE ligands around the V���Vaxis, keeping the tetra-connected units, l4-TCNE, rigid. Schematicsof such different binding configurations are shown in Fig. 4(B) and(C). The optimized structure shown in Fig. 4(A) is altered by a rota-tion around the b axis of all the bi-connected TCNE units from theupper sheet, by 90� (B) and 180� (C). In the next section we willstudy the influence of the various TCNE conformations on the signand strength of the magnetic exchange interaction.

Fig. 5. (A) Schematic drawing of the simplest lattice conceivable for V[TCNE]x, resultingblue building blocks represent the tetra-connected TCNEs, whereas the yellow ones thmutually differing by the sliding direction achieving the tetra-connected TCNEs. (B) Schemone region to the next, a racemic global mixture of the two enantiomeric local struracemisation, with relative small areas of local ordering, is proposed as the reason for theto colour in this figure legend, the reader is referred to the web version of this article.)

Schematic views of the V[TCNE]2 structures showing some gen-eric configurations resulting from the rotation of the bi-connectedTCNE units around the b axis and the corresponding space groupare displayed in Fig. 6. The propensity for rotational disorder ofthe bi-connected TCNE molecule allows a study of the magneticexchange constant as a function of the rotation angle.

At the end of this section we note that in the case of the quasi-one-dimensional manganese-porphyrin-TCNE compounds alreadymentioned, [53] the disorder caused by the various orientationsof the TCNE ligand [51] as well as by the presence of various sol-vents intercalated in the structure lead to slow relaxation pro-cesses and cluster-glass behaviour [54,55].

4.2. Energy landscape and applicability of the spin glass model

To explore the influence of the rotation of the bi-connectedTCNE molecule on the energy landscape as well as on the strengthand sign of the exchange coupling constant we performed calcula-tions at DFT/BLYP/6-31G� level with PBCs. The energy of the low-spin state, ELS, with ferrimagnetic spin alignment, and the magneticexchange coupling constant, J, as a function of the dihedral angle,w, which defines the rotation of the plane of the bi-connected TCNEaround the line connecting the N-atoms bound to V(II), are dis-played in Fig. 7.

The energy landscape is relatively rough, with local minima atabout 90� and 180�. The energy differences at these angles withrespect to the optimized geometry (w = 0) are of about 2693 Kand 4916 K, respectively. Reaching such geometries, however,requires passing over large barriers of more than 5420 K. In con-trast, the dependence of the energy on the angle w is relativelyweak, near 0� and 360�. For instance, rotations by 15�, 30� or 45�require overcoming energy barriers of about 176 K, 369 K and751 K, respectively. We note that the large energy at w = 244� isdue to an unphysical situation, with two non-coordinated N atomsfrom neighbouring cells getting too close, at 1.3408 Å, despite the

from vertical stacking and horizontal sliding of the planes represented in Fig. 3. Thee bis-connected TCNEs. The two mirror images represent enantiomeric networksatic drawing illustrating the static disorder. By switching the sliding direction from

ctures is obtained. Gray indicates a background reference layer. The disorderedamorphousness of the samples obtained so far. (For interpretation of the references

Fig. 6. Schematic view of the V[TCNE]2 model structures showing some generic configurations resulted from the rotation of the bi-connected TCNE units around the b axisand the corresponding space group.

Fig. 7. Energy of the low-spin (ferrimagnetic) ground state as a function of therotation angle, w, of the bi-connected TCNE, obtained by DFT band calculations ofV[TCNE]2 structures.

F. Cimpoesu et al. / Computational Materials Science 91 (2014) 320–328 325

large negative Mulliken charge of �0.80 a.u. on each. Suchconfiguration is highly unlikely to occur experimentally.

Based on these results we argue that small angle rotations caneasily be thermally activated at room temperature, causing somedynamic disorder. However, if during the chemical vapor deposi-tion process the bi-connected TCNE molecule has a deviation of90� or 180–210� with respect to the optimal geometry, the cyano-carbon ligand may be trapped into such position due to the largebarrier to rotation, causing quenched disorder to occur.

One such possible configuration, at 180� is shown in Fig. 4,along with the optimized geometry. The TCNE units from the uppersheet are all rotated by the same angle, whereas those of the lowersheet are kept fixed. However, in principle, random orientations ofdifferent fragments can be conceived, resulting in a even higherdegree of structural disorder.

The exchange coupling constant displayed in Fig. 8 was calcu-lated following the broken symmetry DFT approach with PBCs[56], based on a simple Ising Hamiltonian, also used for otherTCNE-based systems [57]. The energy of the Ising spin system,assuming, regardless of the lattice direction, the same exchangecoupling between V(II) and [TCNE], for the high-spin and broken-symmetry states, is [58]:

Fig. 8. Exchange coupling constant, J , as a function of w, determined for V[TCNE]2

model structures by the BS-DFT approach through band structure calculations.

EðHS=BSÞ ¼ ðþ=�Þ2 � 3 � ð�2J ÞSz;V � Sz;TCNE:

A factor of 2 comes from the two VII-[TCNE] interactions perunit cell, a factor of 3 comes from the summation over the threecrystallographic directions and the other factor of 2 comes fromthe expression of the Heisenberg interaction. Consequently, themagnetic exchange interaction strength can be determined basedon the expression:

J ¼ EBS � EHS

24SV STCNEð5Þ

where the energy difference in the numerator is between the bro-ken symmetry (BS) and the high-spin (HS) states, 1/24 factorincludes a 1/6 factor to account for the double counting when con-sidering the PBCs.

Fig. 9. (A) The 4FJ state is split by the octahedral ligand field into A2, T1 and T2

quartet states and distortions lead to further splittings. Spin–orbit interactions insecond order partially remove the degeneracy of the 4A2 leading to two Kramers’doublets. (B) Zeeman interactions cause a linear response to the applied field.

326 F. Cimpoesu et al. / Computational Materials Science 91 (2014) 320–328

The exchange constant is negative for all angles, indicating fer-rimagnetic spin alignment of the spin SV = 3/2 on V(II) with respectto the spin STCNE = 1/2 on each of the two TCNEs. The absolutestrengths of over 700 K, consistent with previous calculations[19], are significantly larger than the values expected given thecritical temperature of 370 K. Such overestimations were foundfor other TCNE-based molecular magnets [57,59].

Important to note is that, regardless of the TCNE orientation, theexchange is always negative. In the absence of competing ferro-and antiferromagnetic exchange interactions, the applicability ofthe spin glass/random exchange model [24] is highly unlikely, inagreement with the experimental evidence [13,14,18].

4.3. Uniform and random magnetic anisotropy in V[TCNE]x

The magnetic anisotropy is caused by the asymmetry of theimmediate coordination environment, through mechanisms inter-playing the ligand field (LF) splitting and the spin orbit (SO) cou-pling. The ground state of the V(II) d3 ion in an octahedral field is4A2g and the excited states are lying higher in energy by amountsmuch larger than the SO coupling [21,60]. The LF splitting

Fig. 10. Polar diagrams of the state-specific magnetization functions, for (a) {V[MCNCASSCF + SO calculations. The M(h,u) functions are represented in Cartesian frames, in B

parameter 10Dq can be extracted [21] as the gap between the4A2g ground state and the first excited multiplet, 4T2g. In lowersymmetry, a conventional measure of the 10Dq parameter can beobtained as the gap between the ground state and the average ofthe first three exited states, taken as a barycentre of the parentT2g level (see Fig. 9). In a distorted octahedral environment the spindegeneracy of the 4A2g state is partly removed into two Kramers’doublets.

In the case of V[TCNE]x, the magnetic anisotropy may be uni-form, the same at each V(II) site, as well as random, because of lat-tice defects [13,16]. To account for both the uniform and therandom single-ion anisotropy, we consider two simplified models,a hexa- and a penta-coordinated structure, respectively, shown inthe background, in Fig. 10. The two structures are obtained by cut-ting from the extended lattice {V[MCNE]6}2+ and {V[MCNE]5}2+

structures (MCNE = monocyanoethylene). The pentacoordinatedfragments were created by removing a MCNE ligand placed alongthe c axis, on the assumption that the simplest and most relevantdefect from the perspective of random anisotropy is the vacancy ofone TCNE ligand. Such vacancies occur in the trend of the system toachieve a non-chiral bulk status on the expense of local shiftsresulting from the mutual sliding of planar substructure fragments,as already discussed above.

The complete active space self consistent field (CASSCF) calcula-tions and the subsequent SO treatments, were performed using the6-31G� basis set for both fragments studied. The CASSCF(3,5) calcu-lations considered 3 electrons in 5 orbitals, corresponding to the d3

configuration of V(II). With state averaging on the seven levelsoriginating from the LF split of the 4F multiplet, we obtained forthe 10Dq parameter values of about 11,900 cm�1 and18,875 cm�1 for the {V[MCNE]5}2+ and {V[MCNE]6}2+ molecularmodels, respectively [61]. The {V[MCNE]6}2+ model yielded a valuerelatively close to that provided by experiments, of about20,000 cm�1 (2.3 eV) [12].

Adding the SO perturbation to the previous CASSCF calculations,we obtained a full spectrum of 28 states, due to the degeneracy of

E]5}2+ and (b) {V[MCNE]6}2+, and the corresponding ZFS schemes, obtained viaohr magneton units. The molecular skeleton is immersed with arbitrary scaling.

F. Cimpoesu et al. / Computational Materials Science 91 (2014) 320–328 327

the 4FJ multiplet (3/2 6 J 6 9/2). The lowest four levels of the CASS-CF + SO calculations, with J = 3/2, are of interest for the zero-field-splitting (ZFS) calculation of the spin-quartet ground state, as theylead to the two Kramers’ doublets corresponding to the Sz = ± 1/2and Sz = ± 3/2 components.

However, the output of the ab initio calculation does not offerdirect information about the Sz values of the lowest two doublets.To completely describe the magnetic anisotropy, we numericallycalculated the state-specific magnetization functions, as the deriv-ative of the energy of each CASSCF + SO state with respect to theapplied field along various directions, defined by the polar anglesh and u [49]. The dependence on the applied field, H, is achievedby additionally considering the Zeeman Hamiltonian: HZ = �glB-

H�J, where g is the Landé factor and lB the Bohr magneton.The numerical derivative of the energy with respect to the mag-

netic field is represented as a polar diagram (see Fig. 10), yieldingthe magnetization functions of the lowest doublet states of{V[MCNE]5}2+ and {V[MCNE]6}2+ model structures. The polar repre-sentations help identifying the Sz components of the ZFS multiplet,as the maximal radial extension has approximately the gSz value(in Bohr magnetons). In Fig. 10 the lower state of {V[MCNE]5}2+,has the M(h,u) representation in the shape of a torus of radiusgSz � 1, corresponding to the Sz = ± 1/2 components, as the Landéfactor is g � 2. The higher doublet shows a two-lobe shape andhas maximal extension of the lobes amounting to gSz � 3, in linewith the Sz = ± 3/2 components. In the case of {V[MCNE]6}2+ theZFS pattern is reversed.

The gap between the two doublet states is 2|D|, providing theZFS (or SIA) parameter, D. The D parameter is positive when the±1/2 doublet lays lower, indicating XY anisotropy, and negativewhen the ±3/2 doublet is the ground state, corresponding to Isinganisotropy [7,21]. For the {V[MCNE]5}2+ unit the SIA is 0.388 cm�1

(about 0.56 K), whereas for {V[MCNE]6}2+ it is �0.019 cm�1 (about�0.03 K).

The small value obtained for the SIA of the {V[MCNE]6}2+ modelmimicking the bulk of the material is consistent with the smalluniform Ising anisotropy proposed based on experimental data[13,18]. The {V[MCNE]5}2+ unit obtained by removal of one ligandalong the c axis displays XY anisotropy. We note that the removalof the ligand along the other two axes, a or b, leads to similar angu-lar distributions of the magnetization and values of the ZFS gaps(�0.62 K) [61].

The values of the exchange coupling (j J j � 700K), comparedwith the SIA parameters in the case of a perfect crystal,(|Dc| � 0.03 K), and in the presence of disorder (Dr � 0.56 K), areconsistent with the criteria that validate the RAM model [35,39].Following the standard RAM terminology [40], we argue thatV[TCNE]x can be considered a molecule-based correlated sperimag-net, as the spins interacting antiferromagnetically are located in atleast two sublattices and at least one sublattice has randomlyfrozen orientations.

5. Conclusions

We identified possible sources of structural disorder and ana-lyzed their impact on the magnetic properties of the V[TCNE]x sys-tem. We emphasized two types of defects, the various orientationsof the bi-connected TCNE molecule and the TCNE vacancies, anddescribed their effect on the magnetic behaviour of the system.

We reported various DFT band calculations providing theenergy of the low-spin state and the exchange coupling ofV[TCNE]x for various configurations of the bi-connected TCNE mol-ecule. We showed that the energy landscape is relatively rough,allowing for the TCNE molecule to be trapped in nonoptimal con-figurations, leading to static structural disorder. Moreover, our bro-ken symmetry DFT calculations provided only antiferromagnetic

exchange constants making highly unlikely the spin glass-likebehaviour.

CASSCF calculations yielded LF parameters in relative agree-ment with experiments. Subsequent SO treatments determinedthe SIA (both the coherent/uniform as well as the random magneticanisotropies) of the V(II) ions. The relative values of the calculatedparameters j J j � j Dr j > Dc j are consistent with the criteria thatvalidate the RAM model. Consequently, we have been able toexplain on theoretical grounds older experimental results as wellas more recent ones suggesting correlated sperimagnetism overspin glass behaviour for V[TCNE]x.

We believe that our approach, correlating the sources of disor-der with their effects on the magnetic properties of the molecularmagnet, goes beyond the material of interest here, as it allows thecalculation of exchange and the discussion of the relevance of thespin-glass/random exchange model as well as an analysis of the SIAand a discussion of the applicability of the random anisotropymodel. Such theoretical structure–property correlations are a pow-erful tool allowing for the explanation of the experimental data forother molecular magnetic materials.

Acknowledgments

The authors acknowledge the financial support from CNCS-UEFISCDI through the research grants PN2-ID-PCCE-239/2010and PN2-ID-PCE-14/2013.

Appendix A. Supplementary material

Supplementary data associated with this article can be found, inthe online version, at http://dx.doi.org/10.1016/j.commatsci.2014.04.062.

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