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Theoretical study of exchange coupling constants in an Fe19 complex Eliseo Ruiza,b,*, Antonio Rodrı´guez-Forteaa,b, Joan Canoa,c, Santiago Alvareza,b b

a Departament de Quı´mica Inorga`nica, Teo`rica Universitat de Barcelona, Diagonal 647, Barcelona 08028, Spain Centre Especial de Recerca en Quı´mica Teo`rica, Universitat de Barcelona, Diagonal 647, Barcelona 08028, Spain c Laboratoire de Chimie Inorganique, UMR 8613 Bat. 420-ICMO, Universite´ Paris-Sud, 91405 Orsay, France

Abstract The application of theoretical methods based on density functional theory using generalized-gradient approximation functionals provides reasonable estimates of the exchange coupling constants for polynuclear transition metal complexes. Calculations for the complete, nonmodeled Fe19 complex have been performed and a comparison with the experimental magnetic susceptibility values using Monte Carlo simulations is presented. q 2003 Elsevier Ltd. All rights reserved. Keywords: D. Magnetic properties

1. Introduction One of the most active areas of molecular magnetism is the synthesis and characterization of large polynuclear transition metal complexes [1,2]. Among these polynuclear complexes, the so-called Mn12 and Fe8 compounds have attracted the interest of many researchers, especially due to their single-molecule magnet (SMM) behavior and quantum magnetic tunneling effects [3,4]. The SMM character is associated to the total spin of the molecule and its magnetic anisotropy. Hence, new systems of high nuclearity are good synthetic targets. A clear example of this strategy is the Fe19 complex obtained by Powell et al. with the metheidi ligand (N-(1-hydroxymethylethyl)iminodiacetate [5,6]. From the magnetic susceptibility, values of S ¼ 33=2 and D ¼ 20:04 cm21 have been determined for this complex [7]. These compounds are structurally complex and usually present several exchange pathways between the paramagnetic centers (nine in the case of the Fe19 complex). This fact makes it almost impossible to fit the experimental magnetic susceptibility to a Heisenberg spin Hamiltonian using a unique set of exchange coupling constants. An additional problem is due to the size of the system: since the total number of states is 6.09 £ 1014, it is not possible to employ methods based on the diagonalization of the Hamiltonian matrix to study the magnetic behavior. * Corresponding author. Address: Departament de Quı´mica Inorga`nica, Centre Especial de Recerca en Quı´mica Teo`rica, Universitat de Barcelona, Diagonal 647, Barcelona 08028, Spain. 0022-3697/$ - see front matter q 2003 Elsevier Ltd. All rights reserved. doi:10.1016/j.jpcs.2003.11.033

From a theoretical point of view, studies of the magnetic behavior of polynuclear complexes have been limited usually to dinuclear complexes [8 –10]. Recently, we have extended the application of density functional theory (DFT) based methods to polynuclear complexes with several paramagnetic centers [11]. Usually, a very good estimate of the exchange coupling constants has been obtained using a hybrid functional (as B3LYP) and Gaussian functions. However, this approach is severely limited by the size of the system, and systems with up to 200 –300 atoms can be handled with a huge computational effort. An alternative approach that we have explored recently consists in using a computer code based on numerical basis sets instead of Gaussian functions [12]. This main difference, together with the inclusion of some approximations in the calculation of the Coulomb term and the use of generalized-gradient approximation functionals results in a considerable reduction of computer time that allows to treat very large systems [13,14]. The computer time needed with such an approach is around 30 – 50 times shorter than the equivalent calculation with the B3LYP functional and Gaussian functions. Although the results are not as accurate as with methods based on hybrid functionals, our experience indicates that the sign of the exchange coupling and the relative strength of the different coupling constants are correctly reproduced. The present paper has two goals. First, to present calculations of the exchange coupling constants for an Fe19 complex, and second to test the performance of such methods by comparing the magnetic susceptibility

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simulated from these calculated values using Monte Carlo methods [15,16] with the experimental one.

2. Computational details The spin Hamiltonian for a general polynuclear complex is indicated in Eq. (1), X 1 ^2 2 ^ ^ ^ ^ H ¼ 2 Jij Si Sj þ D Sz 2 S þ EðS^ 2x 2 S^ 2y Þ ð1Þ 3 i.j where S^ i and S^ j are the spin operators of the paramagnetic centers i and j and S^ and S^ z are the total spin operator of the molecule and its axial component, respectively. The Jij values are the coupling constants for the different exchange pathways between all the paramagnetic centers of the molecule, while D and E are the axial and rhombic components of the anisotropy, respectively. For the calculation of zero field splitting D and E parameters it is indispensable to include spin –orbit coupling effects in the electronic structure calculations. In this work, we will focus only on the calculation of exchange coupling values. It is remarkable the work of Pederson and Khanna who have recently presented an approach based on a perturbative method to include the spin –orbit coupling that allows the calculation of the zero field splitting for different polynuclear complexes [17]. A more detailed description of the procedure to obtain the exchange coupling constants can be found in Ref. [11]. Basically, we need to calculate the energy of n þ 1 spin distributions if we have a system with n different exchange coupling constants. These values will allow us to build up a system of n equations where the J values are the unknowns. The computer code employed for the all calculations is the program SIESTA (Spanish Initiative for Electronic Simulations with Thousands of Atoms) [18]. This code has been recently developed and designed for efficient calculations in large and low symmetry systems [19]. We have employed the generalized-gradient functional proposed by Perdew et al. [20]. Only valence electrons are included in the calculations, with the core being replaced by normconserving scalar relativistic pseudo-potentials factorized in the Kleinman –Bylander form [21]. The pseudo-potentials are generated according to the procedure of Trouiller and Martins [22] from the ground state atomic configurations for H, C, N and O atoms and [Ar]3d74s1 for Fe atoms. The core radii for the s, p and d components for Fe are all 2.00 a.u. and we have included partial-core corrections for a better description of the core region [23]. The cutoff radii were 1.14 for oxygen, hydrogen and nitrogen atoms and 1.25 for carbon atoms. We have employed a numerical basis set of triple-z quality with polarization functions for the iron atoms and a double-z one with polarization functions for the main group elements. There are two parameters that control

the accuracy of the numerical calculation [19]. The numerical wavefunction vanishes at the chosen confinement radius rc ; whose value is different for each atomic orbital. The energy radii of different orbitals is determined by a single parameter, the energy shift, which is the energy increase of the atomic eigenstate due to the confinement. The integrals of the self-consistent terms are obtained with the help of a regular real space grid in which the electron density is projected. The grid spacing is determined by the maximum kinetic energy of the plane waves that can be represented in that grid. Previously, we have studied the influence of these two parameters in the calculated J value [12]. Thus, the values of 50 meV for the energy shift and 225 Ry for mesh cutoff provide a good compromise between accuracy and computer time to estimate exchange coupling constants. The simulated magnetic susceptibilities have been calculated using Monte Carlo simulations with the Metropolis algorithm [15,16]. A sampling of states was generated to calculate the average magnetization M using Eq. (2), that preferentially includes the configurations that bring important contributions at temperature T. Xn Mi e2Ei =kT i¼1 X kMl ¼ ð2Þ n e2Ei =kT i¼1 The magnetic susceptibility, xm ; can be obtained from the fluctuations in the magnetization by employing Eq. (3), where kMl and kM 2 l are the mean values of M and M 2 :

x ¼ ðkM 2 l 2 kMl2 Þ=kT

ð3Þ

The number of steps in the Monte Carlo simulation for each temperature is 107 =T (T in K). Thus, we include more steps in the sampling at low temperature because it is more difficult to describe correctly the magnetic behavior at such temperatures. A 10% of the steps are employed for the thermalization of the system.

3. Results and discussion The structure of [Fe19(metheidi)10(m3-OH)6(m-OH)8 (m3-O)6(H2O)12]NO3 that contains 323 atoms, represented in Fig. 1, can be described as an hexagon with one internal and two external Fe(III) cations, surrounded by ten Fe(III) cations. A scheme indicating the topology corresponding to the nine different exchange coupling constants is shown in Scheme 1 (see Appendix A for the description of the 10 different spin configurations employed for the calculation of the J values). The calculated J values are indicated in Table 1, all of them correspond to antiferromagnetic interactions. From these results, we can classify the nine interactions in three different sets. The first one would be constituted by the exchange coupling constants corresponding to doubly bridged centers (J1 ; J2 ; J3 and J8 ) that present weak

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Table 1 Description of the bridging ligands, metal–metal distances, calculated exchange coupling constants J (cm21) for the Fe19 complex using the PBE functional and the fitted values that reproduce the experimental data

J1 J2 J3 J4 J5 J6 J7 J8 J9

Fig. 1. Crystal structure of [Fe19(metheidi)10(m3-OH)6(m-OH)8(m3O)6(H2O)12]NO3 complex. The black spheres correspond to the iron atoms while the oxygen, carbon, nitrogen and hydrogen atoms are represented by different shades of gray, from dark to bright, respectively.

anti-ferromagnetic coupling and short metal –metal distances. The second set includes interactions through one bridging ligand (J4 ; J5 ; J6 and J7 ) that present stronger antiferromagnetic couplings and larger metal – metal distances. Finally, the case of J9 corresponds to an intermediate situation with one bridging ligand and a weak coupling. In the first publication devoted to this compound (see Ref. [5]), the authors already noticed that the antiferromagnetic coupling through one m-oxo ligand (J5 and J6 ) is stronger than with alkoxo or hydroxo bridges (J4 ; J7 and J9 ), as confirmed by our results. The comparison of such results with those proposed for other polynuclear Fe(III) complexes shows the same trends. For the Fe8 complex, the strongest anti-ferromagnetic coupling corresponds to the central interaction through an oxo bridging ligand and the weakest interactions are those with a double bridging ligand, while

Scheme 1.

Bridging ligands

˚) M· · ·M distance (A

Jcalc (cm21)

Jfit (cm21)

(m-OH)2 (m-OH)2 m-OH, m-O m-OC m-O m-O m-OC m-OC, m-O m-OH

3.180, 3.189 3.206, 3.201 3.097 3.596, 3.592 3.415, 3.437 3.412, 3.428 3.513 2.986, 2.976 3.629, 3.622

25.9 215.4 219.6 235.2 250.2 276.4 240.4 222.1 216.2

27.9 219.0 228.9 239.6 274.2 299.0 265.5 230.8 226.2

the external interactions with one alkoxo-type ligand present intermediate strength [24,25]. The analysis of the calculated DFT energies for the different spin configurations (see Appendix A) shows that the second case corresponds to the most stable spin distribution. In order to verify this point, we have calculated the energies for all the possible spin configurations (524,288 cases) confirming that this case is the most stable one while a spin distribution with S ¼ 25=2 is the following one. The most stable spin configuration can be easily described as the inversion of the spin of the atoms of the central hexagon giving a S value of 35/2 (see Scheme 2), not too far from the 33/2 assigned experimentally. The presence in the structure of several m3-oxo and m3-hydroxo bridging ligands produces frustration of some couplings and, consequently, this fact could be responsible of a change of the ms value of some Fe(III) centers to form a ground state with S ¼ 33=2: The comparison between experimental and theoretical data for this kind of systems can be performed by simulating (or calculating) the magnetic susceptibility from the calculated J values. For the Fe19 complex, the direct calculation is not possible actually due to the huge number of states, being a good alternative the classical Monte Carlo simulations.

Scheme 2.

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antiferromagnetic. The relative strength of the calculated J values is in good agreement with those in other Fe(III) complexes described in the literature. Such values are underestimated, as seen by comparison of the simulated magnetic susceptibility with the experimental one. This result was expected because it is well known that the generalized-gradient functional provides less accuracy than the hybrid functionals for the estimation of J values in complex systems. However, such methodology allows handling huge systems and provides the correct description of the sign and strength of the exchange interactions, and offer a good starting point to obtain a best set of J values through a fitting using Monte Carlo simulations. Acknowledgements

Fig. 2. Magnetic susceptibility curves of the Fe19 complex. The black circles correspond to experimental data while the white circles are also the experimental data but eliminating the u contribution (see Refs. [5–7]). The dashed and solid lines have been generated directly from the DFT J values and from a fitting using the calculated values as starting point (see both sets of values in Table 1).

The curve corresponding to the simulated magnetic susceptibility is shown in Fig. 2, where two sets of experimental data are represented. In one case we have eliminated the u contribution, probably due to the intermolecular interaction that causes the rapid decay at low temperatures, to facilitate the straight comparison with the theoretical results. The agreement between the experimental data and theoretical results is relatively poor as expected. The use of PBE functional reproduces usually correctly the sign and strength of the exchange coupling constants but the numerical value is not as accurate as in the case of the hybrid functionals, as discussed in Section 1. Thus, we have performed a fitting using the Monte Carlo simulations with the calculated J values as starting point and keeping the relative strength of the interactions. The best-fit parameters are relatively close to those obtained from the calculation being the fitted coupling constants larger than the DFT ones (see Table 1). This conclusion seems to apply in general to other complexes with several coupling constants (such as Fe8) that have complex expressions for the energy of the different distributions while for wheel-shape complexes with few different exchange pathways, the opposite behavior is found [26].

Financial support came from Ministerio de Ciencia y Tecnologı´a through project number DGI BQU2002-04033C02-01 and from Comissio´ Interdepartamental de Cie`ncia i Tecnologia (CIRIT) through grant 2001SGR-0044. One of us (J.C.) wants to thank the European Community for covering the expenses of his stay in Barcelona through the IHP Programme and also the Molnanomag network. Appendix A For the calculation of the nine exchange coupling constants, 10 different spin configurations have been employed. We indicate here only the atoms (see Scheme 3) with spin down for each configuration: 1. 2. 3. 4. 5. 6. 7. 8. 9. 10.

all atoms spin up 2,3,4,11,12,13 3,11,13 6,8,9,14,16,18 8,9,10,16,17,18 1,5,7,15,19 2,3,8,10,12,14,15,18,19 5,6,7,14,15,19 5,7,15,19 3,11,12

4. Conclusions The analysis of the calculated exchange coupling constants for an Fe19 complex using the PBE functional together with a numerical basis set indicates that the nine exchange coupling interactions present in the molecule are

Scheme 3.

E. Ruiz et al. / Journal of Physics and Chemistry of Solids 65 (2004) 799–803

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