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Extended Fe 4 butterfly complexes: theoretical analysis of magnetic properties and magnetostructural maps†
Silvia Gomez-Coca,
aThomas Cauchy
band Eliseo Ruiz*
aReceived 7th December 2009, Accepted 15th February 2010 First published as an Advance Article on the web 24th March 2010 DOI: 10.1039/b925699g
The inclusion of additional metal atoms in Fe4butterfly complexes drastically modifies their magnetic properties. Exchange interactions of a Fe4Y2complex have been calculated using theoretical methods based on density functional theory. The calculated values are in good agreement with experimental data showing that the change in the nature of bridging ligands induces a dramatic decrease of the
antiferromagnetic wing–body interaction while the body–body interaction between the two central iron atoms is ferromagnetic. Finally, we propose a new tool to facilitate the understanding of the magnetic properties in polynuclear iron complexes. Magnetostructural maps allow us to correlate the calculated exchange coupling constants with metal–metal distances for the dinuclear or polynuclear iron complexes that we have studied.
Introduction
Since the discovery by Gatteschi et al. of the single molecule magnet behaviour,1-3 many groups have tried to obtain new molecules with high total spin and large magnetic anisotropy in order to increase the energy barrier that fix the spin sign.4 Logically, the first attempts focused on increasing the number of paramagnetic metal centres. Following such an approach, new molecules with very high spins were obtained, however, they usually present small magnetic anisotropy. In order to prevent such a problem, some groups modified the synthesis of the complexes
aDepartament de Qu´ımica Inorg`anica and Institut de Recerca de Qu´ımica Te`orica i Computacional, Universitat de Barcelona, Diagonal, 647, E-08028, Spain. E-mail: [email protected]
bInstitut des Sciences et Technologies Mol´eculaires, CNRS UMR 6200 MOLTECH ANJOU, Universit´e d’Angers, 2 Bd Lavoisier, 49045, Angers, France
† Electronic supplementary information (ESI) available: Table S1 contain- ing the spin density values corresponding to the high spin state (S= 10) and the more stableS = 0 single-determinant. Such values were employed for the estimation of the value obtained with eqn (2). See DOI:
10.1039/b925699g
Fig. 1 Molecular structure of the [FeIII4Y2(m4-O)2(NO3)2(pivalate)6(Hedte)2] complex (H4edte=(N,N,N¢,N¢-tetrakis-(2-hydroxyethyl)ethylenedi-amine).
A typical Fe4complex was also plotted for comparison.14The bridging oxygen, iron, and yttrium atoms are represented by colour spheres red, green and orange, respectively.
in order to include different metal atoms. Thus, a new strategy is to introduce heavy metal cations, for instance lanthanides,5,6that usually can add large magnetic anisotropy in a system with 3d metals. One example of such an approach was recently provided by Akhtaret al.synthesizing new complexes Fe4X2(X=Y, Gd and Dy).7The FeIIIcations are distributed in such complexes in a similar way to the well-known Fe4butterfly complexes8-18but the FeIIIcations placed in the wings of the butterfly are now shifted out of plane of the central Fe2O2body by the presence of X metal atoms (see Fig. 1).
The study of the magnetic properties of such complexes is not easy in some cases due to the presence of some heavy metals, such as DyIIIcations, with large spin–orbit contributions.
However, the magnetic properties of complexes with YIII and GdIIIcations can be more easily rationalized. The experimental exchange coupling constants obtained for the Fe4Y2complex show values that are qualitatively different than those obtained for
“classical” Fe4 butterfly complexes.7,19Thus, despite YIII cations being diamagnetic, their inclusion can play a significant role in modifying the magnetic properties. Theoretical methods based on density functional theory have shown their ability to handle these
Table 1 Experimental structural parameters (A˚ and◦) and fittedJvalues (cm-1) for the Fe4Y2complex also indicating the limit values for the family of Fe4complexes
Feb–O Few–O a b g Jwb Jbb Ref.
Fe4Y2 2.008 2.153 97.3 76 0 -8.34 +2.2 7
Fe4family 1.93–197 1.81–1.87 92–97 12–48 0–33 -66 to-92 -2.4 to-22 8–18
kind of challenging problems by considering the complexity of the molecules and the tiny energy differences involved.20-22 The exchange coupling constants can be calculated with a quantitative accuracy allowing in some cases to corroborate the experimental values. Hence, the first goal of this manuscript is to study the magnetic properties of the Fe4Y2 complex and secondly, we want to present the use of magnetostructural maps for the FeIII complexes that we have studied in our group during the last few years. The goal of this kind of representation is not just to establish some magnetostructural correlations. The analysis of these figures also allows us to rationalize the magnetism of the existing molecules as well as to extract some conclusions about new systems. Recently, we published the magnetostructural maps corresponding to manganese complexes,23that are together with iron complexes studied more for their single molecule magnet behaviour.
Results and discussion
Exchange coupling constants in Fe4Y2complex
From the experimental point of view one of the problems in analyzing the exchange coupling constants of polynuclear transition metal complexes with several exchange constants is the existence of many sets of J values that would perfectly fit the measured magnetic susceptibility. Due to such limitations, in some papers the authors prefer to use a reduced set ofJvalues (Table 1) instead of the real set of exchange constants; in this case, for the Fe4Y2complex there are four different values (see Fig. 2 and eqn (1)). The expression of the Heisenberg–Dirac–Van Vleck Hamiltonian considering only the exchange terms for the Fe4Y2
complex is the following:
Hˆ =-JbbSˆ1Sˆ2-Jwb1[Sˆ1Sˆ4+Sˆ2Sˆ3]-Jwb2[Sˆ1Sˆ3+Sˆ2Sˆ4]
-JwwSˆ3Sˆ4 (1)
whereSˆiis the local spin operator of each paramagnetic centre.
Magnetic and structural data for the Fe4Y2complex are col- lected in Table 1 together with the same information for the family of Fe4complexes.8-18It is worth noting that a structural difference between such complexes is the presence of an axial alkoxo bridging ligand between the central and external iron atom while in the Fe4
complexes, there are carboxylato axial ligands. This is a crucial
Fig. 2 Structural parameters and topology of the exchange interactions for the Fe4Y2complex taking into account the presence of two non-equiv- alent wing–body interactions.
change because now, the wing–body interactions have a double oxo-type bridging ligand as well as the body–body exchange coupling. Hence, as mentioned above, the magnetic behaviour of the Fe4Y2 complex is completely different, thus, the Jwb
interactions that are relatively strong in the Fe4complexes are also antiferromagnetic but much weaker due to the existence of the double bridging ligand. This kind of double oxo-bridge exchange pathway usually provides weak antiferromagnetic interactions (see next section, Fig. 5). The Jbb interaction, that for the Fe4
complexes is almost impossible to determine accurately due to the presence of a much stronger antiferromagneticJwbinteraction,19is weakly ferromagnetic for the Fe4Y2complex. The analysis of the structural parameters (see Table 1) indicate two main differences between the Fe4Y2 and Fe4 complexes; longer Fe–O distances and a considerable increase of theb angle due to the presence of the two yttrium atoms distorting the Fe4butterfly modifying the coordination number of the oxo bridging ligands.
The calculatedJ values for the Fe4Y2 complex are indicated in Table 2 (see Computational details section). It is important to
Table 2 Average structural parameters (in A˚ and◦) and calculated and experimental exchange coupling constants7(in cm-1, see Fig. 2) corresponding to the Fe4Y2complex. The range of equivalentJvalues for the family of Fe4complexes is provided for comparison8-18
Bridging ligand d(Fe◊ ◊ ◊Fe) Fe–O–Fe d(Fe–O) Jcalc Jexp JcalcFe4 JexpFe4
Jbb 2m4-O 3.029 97.9 2.008 +3.5 +2.2 +8.3 to-15.2 -2.4 to-22
Jwb1 m4-O,m2-OR 3.185 103.3 2.032 -9.6 -8.34 -67 to-85 -66 to-92
Jwb2 m4-O,m2-OR 3.127 100.3 2.038 -3.8
Jww m-Fe2O2 5.539 — — -1.0 — -5.6 to-7.2 —
remark that despite very smallJvalues, the employed methodol- ogy provides an excellent agreement with the experimental data.
From the analysis of the results, we can extract the following conclusions: (i) there is a substantial reduction of the strength of the antiferromagnetic Jwb coupling in the Fe4Y2 complex in comparison to the Fe4 complexes. This is basically due to the modifications caused by double oxo-type bridging ligands and also, some structural changes by the presence of yttrium cations. (ii) The difference between the twoJwbexchange constants for the asymmetry of the Fe4Y2 complex seems to be related to the larger average bridging angle Fe–O–Fe values for the Jwb1 exchange pathway (103.3◦) than for the Jwb2 interaction (100.3◦). There is a nice correlation between the three calculated J values with a double oxo-type bridging ligand and the Fe–
O–Fe angle (see Table 2) – the larger angle value the stronger the antiferromagnetic contribution. (iii) The ferromagnetic Jbb
value seems to corroborate one magnetostructural trend proposed in a previous study,19 that the presence of twom4-oxo bridging ligands and the relatively long Feb–O bond distances can induce ferromagnetic coupling for such interaction. (iv) The calculated Jwwcoupling for the Fe4Y2complex is of similar nature but weaker than that obtained in the Fe4complexes. A nice agreement with the experiment is found by comparing the thermal variation of the magnetic susceptibility obtained with the calculatedJvalues using the MAGPACK code,24 taking into account the extreme sensitivity of the shape of the curve to small changes in theJ values (see Fig. 3).
Fig. 3 Thermal variation of thecmTproduct for the Fe4Y2complex. The calculated values obtained using the DFTJvalues with the MAGPACK code24 (and the experimentalgvalue of 2.01) are indicated using black circles while the experimental data is indicated with white circles.
In the previous paper devoted to the Fe4 complexes,19 we have shown that the Kahn–Briat model25,26 that relates the antiferromagnetic contribution to the exchange interaction with the square of the overlap between the “magnetic orbitals” of the two involved paramagnetic centres (A,B) can be easily estimated through the relationship:
DAB HSA
LSA
HSB
LSB
=⎛ − + − =
⎝
⎜⎜⎜⎜⎜⎜
⎜⎜
⎞
⎠
⎟⎟⎟⎟⎟⎟⎟⎟⎟
2 2 2 2 2
(r ) (r ) (r ) (r ) 44
1 n 2
i n ai bi
=∑ | (2)
where rA,BHS,LS are the different spin populations of the para- magnetic centers A or B involved in the exchange interaction in the highest (HS) configuration while the lowest spin (LS) configurations correspond to those obtained with the spin in- version of the Fe3 and Fe4 centres (see ESI for the detailed spin population values†), n is the number of unpaired elec- trons at paramagnetic centres A and B, and ai andbi are the magnetic orbitals analogous to those proposed in the Kahn–
Briat model. Thus, the application of this equation to quantify the importance of the antiferromagnetic contributions for Fe4
complexes provided DAB values between 1.5 and 3, using the spin population obtained using a Mulliken approach, while the equivalent value for the Fe4Y2complex is only 0.3 in agreement with the substantial decrease of the antiferromagnetic coupling (see Table 2).
The spin density distributions corresponding to theS=0 more stable single determinant is shown in Fig. 4. The spin distribution is almost spherical at the paramagnetic centers due to the d5 electronic configuration of the FeIIIcations, and the delocalization mechanism is predominant at the ligand atoms coordinated to the metals.27,28The spin population on the iron atoms is around 4.2 e- and the missing spin density, relative to five unpaired electrons, appears delocalized over the ligands mainly in the oxygen atoms of the Fe4O2framework and in the terminal oxygen atoms coordinated to the FeIII cations (see ESI for the detailed spin population values†). It is worth noting that in the central oxygen atoms there are two lobes with spin densities of different signs that appear due to the presence of two neighboring FeIII cations with opposite spin density. This spin density is probably an artifact due to the single-determinant wavefunction considered in this case. Finally, there is no spin density in the yttrium atoms indicating the non-participation of the yttrium atomic orbitals in the “magnetic orbitals”.
Fig. 4 Representation of the spin density distribution map for the [FeIII4Y2(m4-O)2(NO3)2(pivalate)6(Hedte)2] complex corresponding to the more stableS=0 single-determinant solution. The isodensity surface represented corresponds to a value of 0.005 e-/bohr3(positive and negative values are represented as white and blue surfaces, respectively).
Magnetostructural maps
Previously, we have employed a new kind of representation to rationalize the calculatedJvalues for manganese complexes.23The goal of this kind of representation is not just to establish some magnetostructural correlations.19 The analysis of these figures should also allow us to extract some new conclusions because we are including a larger group of complexes than in the typical analysis of magnetostructural correlations that is usually limited to one family of complexes. The dependence of the calculated J values for iron complexes with at least one or two oxo-type bridging ligands with the Fe◊ ◊ ◊Fe distance for each interaction is represented in Fig. 5.19,22,29-36We have selected the Fe◊ ◊ ◊Fe distance instead of the angles involving the bridging ligands because we need a universal structural parameter and for the angles we would need different definitions depending on the number of atoms in the exchange pathway. Thus, it would be impossible to make a comparison between complexes with a different number of atoms in the exchange pathway.
Fig. 5 Magnetostructural maps of the exchange interactions in FeIII complexes including oxo-type (OX) bridging ligands. Empty and filled symbols indicate if the calculations were performed using numerical GGA calculations (Siesta code)37,38or B3LYP results using Gaussian basis sets (Gaussian and NWChem codes).39-41Blue and red colours were employed to indicate the presence of complexes with at least one or two oxo-type bridging ligands.
From Fig. 5, we can extract the following conclusions: (i) the FeIII◊ ◊ ◊FeIII interactions with the presence of oxo-type double bridging ligands (blue symbols) have relatively shorter Fe◊ ◊ ◊Fe distances than those with one oxo-type bridging ligand inter- action (red symbols) showing usually weak antiferromagnetic interactions and in a few cases weak ferromagnetic coupling.
(ii) Oxo-type single bridging ligand interactions show antiferro- magnetic interactions which in some cases are relatively strong.
The dispersion of oxo-type single bridging ligand interactions is much larger than that found for the oxo-type double ones that appear more concentrated in a region of the figure. Clearly, the presence in FeIII cations of one unpaired electron on each d orbital makes in general difficult the presence of ferromag-
netic couplings because they can easily interact with all kinds of ligands (s or p) to achieve an efficient exchange pathway to overlap the “magnetic orbitals” of the second FeIII cation.
(iii) It is worth noting a significant difference in comparison with the manganese complexes.23For the manganese complexes, there is an asymptotic behaviour that cancels the interaction for large M◊ ◊ ◊M distances. Here, the increase of the Fe◊ ◊ ◊Fe distance does not correlate with a weakening of the exchange interaction.
In the range of Fe◊ ◊ ◊Fe distances considered in Fig. 5, basically, the opposite behaviour is found. As a conclusion, it is very clear from Fig. 5 the dramatic change in theJwb interactions studied in the previous section, those interactions in the Fe4complexes belong to the one oxo-type bridging ligand (red symbols) with relatively strong antiferromagnetic couplings while the Fe4Y2
complex with two oxo-type bridging ligands (blue symbols) has weak antiferromagneticJwbinteractions.
A more detailed analysis of the oxo-type single bridging ligand interactions is provided in Fig. 6. The presence of more bridging ligands clearly reduces the Fe◊ ◊ ◊Fe distance, and for such complexes the interactions are relatively strong antiferromagnetic.
The carboxylato bridging ligands do not have a very important influence, thus, the complexes with two bridging carboxylato ligands (violet symbols in Fig. 6) have shorter Fe◊ ◊ ◊Fe distances but similar magnetic coupling than those with only one carboxy- lato bridging ligand (green symbols). It is also worth noting that only the hydroxo bridging ligand seems to be able to give ferromagnetic couplings while a wide dispersion of the strength of the interactions is found with the oxo bridging ligands (red symbols). However, the interactions corresponding to double or triple bridging ligands show similarJvalues.
Fig. 6 Magnetostructural maps of the exchange interactions in FeIII complexes including only those with one oxo-type (OX) bridging ligand.
Empty and filled symbols indicate if the calculations were performed using numerical GGA calculations (Siesta code) or B3LYP results using Gaussian basis sets (Gaussian and NWChem codes). The different colours have been employed to classify the complexes by families taking into account the non oxo-type bridging ligands.
Concluding remarks
We have employed theoretical methods based on density func- tional theory to study the magnetic properties of a Fe4Y2complex.
The presence of the diamagnetic YIII cation induces structural changes, and the magnetism of this complex is completely different to the “typical” Fe4butterfly complexes. Basically, there is one important change in the bridging ligands corresponding to the wing–body interactions; for the Fe4Y2complex the presence of a second oxo-type bridging ligand while one is present in the Fe4
complexes. Thus, the wing–body interaction in the Fe4 complex is strongly antiferromagnetic (from-70 to-90 cm-1) while in the Fe4Y2complex it is one order of magnitude lower. The calculated values confirm the decrease of theJvalue obtained from the mea- sures of the magnetic susceptibility, and also corroborate the weak ferromagnetic character of the body–body interaction. The wing–
wing interaction that was neglected in the analysis of the ex- perimental data is considerably smaller in the Fe4Y2 complex than in Fe4 butterflies. The dramatic change of the wing–body interaction is clearly reflected in a very small value of the sum of the square of overlap between the magnetic orbitals, that according to the Kahn–Briat model is proportional to the antiferromagnetic contribution.
Finally, we propose the use of magnetostructural maps in which we have included all the calculated exchange coupling constants for di- or polynuclear iron complexes studied by our group during the last few years. These maps are a representation of the correlation between calculatedJvalues and the Fe◊ ◊ ◊Fe distance.
The magnetostructural maps clearly indicate the difficulty to have ferromagnetic interactions with the FeIIIcomplexes showing such behaviour in only some complexes with double oxo-type bridging ligands. There is also a fundamental difference between the magnetostructural maps of iron and manganese complexes.
The exchange interaction in the manganese complexes show an asymptotic dependence with the Mn◊ ◊ ◊Mn distance, thus, short distances give strong antiferromagnetic interactions and with the increase of the distance, the interaction can become weakly antiferromagnetic or even ferromagnetic. However, in the case of the iron complexes there is not a clear trend, usually there are stronger interactions with longer Fe◊ ◊ ◊Fe distance. Finally, it is important to remark that the magnetostructural map represented in Fig. 5 clearly explains the change of behaviour found in the wing–body exchange couplings between the Fe4Y2 and Fe4
complexes, due to the presence of a second oxo-type bridging ligand for such interactions in the Fe4Y2complex, as discussed in the first section.
Computational details
Since a detailed description of the computational strategy adopted in this work can be found elsewhere we will only sketch briefly its most relevant aspects here.42-45 A phenomenological Heisenberg Hamiltonian is used, excluding the terms related with the magnetic anisotropy, to describe the exchange coupling in the polynuclear complex:
ˆ ˆ ˆ
H= −a bJabSaSb
∑< (3)
whereSˆa and Sˆb are the spin operators of the different para- magnetic centers. TheJab parameters are the pairwise coupling
constants between the paramagnetic centers of the molecule.
Basically, we need to calculate the energy ofn+1spin distributions for a system with n different exchange coupling constants.42,45 These energy values allow us to build up a system ofnequations in which theJvalues are the unknowns. In the present study, six calculations were performed in order to obtain the four exchange coupling constantsJwb1,Jwb2,JbbandJww of the Fe4Y2complex (see Fig. 2). They correspond to the high spinSz=10 solution, threeSz=0 configurations with negative spin at two FeIIIcations (see Fig. 2){Fe3, Fe4}, and{Fe2, Fe3}and two with the inversion of only one spin{Fe2}and{Fe4}.
Gaussian0339 and NWChem40,41 calculations were performed with the hybrid B3LYP functional46 using a guess function generated with the Jaguar 7.0 code,47employing a procedure that allow to individually determine the local charges and multiplicities of the atoms including the ligand field effects.48 A triple-z all electron Gaussian basis set has been used for all the atoms49 while an all electron basis set was employed for yttrium atoms with the following contraction pattern [633321/53211/531].50The obvious choice to treat accurately the relativistic effects is the four-component Dirac equation.51Therefore, much effort has been spent to approximate the major relativistic contributions by two- component Hamiltonians, which are conceptually simple, vari- ationally stable, sufficiently accurate and computationally more efficient than the four-component equation. We have employed the Douglas–Kroll–Hess Hamiltonian (DKH).52,53This approach in the employed implementation is limited to introduce only the scalar relativistic effects while we expect that the spin–orbit effects are of minor importance for the correct description of the d0configuration of the YIIIcations. The influence of the DKH corrections in the calculatedJvalues is rather small. TheJvalues without the DKH correction are the following:Jbb=+4.7 cm-1, Jwb1 = -9.3 cm-1,Jwb2 = -3.2 cm-1andJww = -1.0 cm-1. Thus, all the values are more “antiferromagnetic” with the inclusion of the DKH method providing a slightly better agreement with the experimental data and the largest difference is 1.2 cm-1for theJbb
value.
Acknowledgements
We want to thank Annie K. Powell and Chris Anson for providing us with the crystal structures prior to its publication.
The research reported here was supported by the Ministerio de Ciencia e Innovaci´on and Generalitat de Catalunya through grants CTQ2008-06670-C02-01 and 2009SGR-1459, respectively.
The authors thankfully acknowledge the computer resources, technical expertise and assistance provided by the Barcelona Supercomputing Center (Centro Nacional de Supercomputaci´on).
S. G. thanksMinisterio de Educaci´on y Cienciafor a predoctoral fellowship AP2008-01005.
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