1 Introduction
Actually, many research groups working on Molecular Magnetism have regularly employed lanthanides due to the their large magnetic moment and magnetic anisotropy [1,2]. Recently, some mixed d/f compounds have shown single-molecule magnet properties [3], such as Cu2Tb2 [4], Mn6Dy6 [5], Mn11Dy4 [6], Mn2Dy2 [7], Fe2Dy2 [8], Fe2Ho2 [8] and Mn11Gd2 complexes [9]. Thus, since the seminal paper in 1985 of Bencini et al. [10] devoted to a CuIIGdIII heterodinuclear complex, many groups have dedicated much attention to this kind of systems; it is worth noting among others, those of Costes et al. [11,12] and Kahn et al. [13,14]. The GdIII complexes are probably the most studied complexes in this field due to the presence of the largest possible magnetic moment S = 7/2 and the lack of an important orbital contribution that facilitates the analysis of the magnetic properties. Also, the d/f GdIII complexes show another appealing feature, practically all of them present ferromagnetic coupling [15].
During the last years we have successfully employed theoretical methods based on density functional theory to calculate exchange coupling constants in dinuclear transition metal complexes [16–18] as well as in large polynuclear complexes [19–21] achieving a surprising accuracy when hybrid exchange correlation functionals, e.g. B3LYP [22], have been used [23]. Despite the large number of d/f complexes reported [15], few theoretical studies have been carried out especially for the difficulty to handle the spin–orbit effects in relatively large complexes. Yan et al. have studied the exchange coupling and the spin distributions in three MGdIII complexes (M = CuII, FeIII and CrIII) using density functional methods [24]. A detailed analysis of the ferromagnetic GdIIICuII coupling was performed by Paulovic et al. using CASSCF and CASPT2 methods combined with spin–orbit terms [25]. However, considering the large number of atoms present in some of the polynuclear complexes mentioned above, only theoretical methods based on density functional theory can handle this kind of system. Thus, our first goal is to check the accuracy of density functional methods to describe this type of exchange interactions in CuIIGdIII complexes, analyzing specifically the role of the relativistic effects even if the f7 configuration of the GdIII cation has no orbital degeneracy associated with the ground state. A second step will be an analysis of the exchange interactions in the CuIIGdIII complexes and the study of the magnetostructural correlations in this kind of systems.
The obvious choice to treat accurately the relativistic effects is the four-component Dirac equation [26]. Therefore, much effort has been spent to approximate the major relativistic contributions by two-component Hamiltonians, which are conceptually simple, variationally stable, sufficiently accurate and computationally more efficient than the four-component equation. Basically in the two-component equations, e.g. Breit–Pauli Hamiltonian, there are five terms, the relativistic kinetic and potential terms, two contributions usually known as scalar relativistic corrections (also called spin-free relativistic corrections), the mass–velocity and the Darwin corrections and finally, the spin–orbit term. In order to include scalar relativistic effects in this kind of calculation [27]; we have considered a set of possibilities: (i) the use of quasi-relativistic pseudopotentials; (ii) the Zeroth-Order Regular Approximation (ZORA), a two-component approximation to the Dirac equation [28]; and (iii) the Douglas–Kroll–Hess Hamiltonian (DKH) [29,30]. These approaches are limited to introducing only the scalar relativistic effects while we expect that the spin–orbit effects are of minor importance for the correct description of the f7 configuration of the GdIII cations.
2 Results and discussion
2.1 Validation of the methodology
In order to verify the methodology used to describe the exchange interactions in the CuIIGdIII complexes, we have selected one of them [31] (refcode ZUVTIT) that was also employed by Paulovic et al. in multiconfigurational calculations [25] (see Fig. 1). The calculated J values using the hybrid B3LYP functional are collected in Table 1 (see Section 4).
Molecular structures of [LCuGd(NO3)3]·Me2CO (L = 1,2-bis((3-methoxysalicylidene)amino)-2-methylpropanato) (refcode ZUVTIT). The CuII and GdIII cations, oxygen, nitrogen, and carbon atoms are represented as blue, pink, red, light blue, and brown spheres, respectively.
Calculated and experimental J exchange coupling constants (in cm−1) for the LCuGd(NO3)3·Me2CO complex (see Fig. 1, refcode ZUVTIT) including the scalar relativistic effects using the ZORA (ORCA code) and DKH (Gaussian code) approaches
| J | |
| B3LYP. SDD pseudo Gd (Gaussian) | −4.41 |
| B3LYP. all electron (Gaussian) | −0.66 |
| B3LYP. all electron, DKH (Gaussian) | +2.4 |
| B3LYP. all electron (ORCA) | −0.96 |
| B3LYP. all electron, ZORA (ORCA) | +2.4 |
| exp. | +7.0 |
From the obtained results, we can extract the following conclusions: (i) The inclusion of the scalar relativistic effects plays a crucial role to reproduce correctly the sign of the interaction; (ii) The results using the ZORA and DKH approximations are very similar as it is usually found [27,32]; (iii) The use of quasi-relativistic pseudopotentials seems to be not an alternative to the ZORA and DKH approximations because the sign of the interaction is not well reproduced. We have verified the influence of the acetone molecule in the model and despite the long Cu⋯O bond distance of 2.607 Å due to Jahn–Teller effect in the 4 + 1 coordination sphere, the B3LYP-DKH value using the Gaussian basis set is +1.7 cm−1. This value is relatively close to that obtained with the whole model, however, it is worth noting the influence of the acetone molecule, taking into account that the “magnetic orbital” [33] of the CuII cation is perpendicular to the Cu⋯O interaction and the long Cu⋯O distance.
2.2 Magnetostructural correlations
Our aim is to establish some magnetostructural correlations, thus, we have collected in Table 2 the main structural parameters and the exchange coupling constants for all reported dinuclear CuIIGdIII complexes showing a central CuIIGdIIIO2 framework with only two bridging ligands. The analysis of such results indicates the existence of a fair correlation between the J value and the Cu–O–Gd–O torsion angle [11,15,34] that measures the planarity of the central framework (see Fig. 2) rather than with the Gd⋯Cu distance [35]. Thus, for the “planar” structures the coupling becomes more ferromagnetic as was previously noticed by different authors [11,15,34]. However, there are two clear exceptions to such correlations that are the complexes [36,37] with NEBLUB and OBEMAK refcodes (see Fig. 3 and Table 2), in both cases, despite a structure relatively planar, the coupling is rather weak (left lower part in Fig. 2).
Structural data of the CuIIGdIII complexes, Cu⋯Gd distances, Cu–O–Gd bond angles and Cu–O–Gd–O torsion angle (in Å and degrees, respectively) and experimental J exchange coupling constants (in cm−1)
| Refcodes | Cu⋯Gd | Cu–O–Gd | Cu–O–Gd–O | Jexp | Ref. | |
| GdCu(OTf)(bdmap)2(H2O)·THF | NEBLUB | 3.310 | 104.7, 105.5 | 0.49 | −0.08 | [36] |
| Gd(hfa)3Cu(salen) | RINQAG | 3.230 | 95.8, 94.3 | 32.99 | 0.4 | [38] |
| 3.199 | 93.6, 93.8 | 34.56 | ||||
| Cu(salabza)Gd(hfac)3 | JOGQAX | 3.248 | 93.1, 98.5 | 34.28 | 0.8 | [39] |
| Cu(salen)Gd(pta)3 | OFELAM | 3.288 | 95.2, 99.8 | 23.72 | 1.21 | [40] |
| Cu(acacen)Gd(hfa)3 | OFELIU | 3.288 | 98.1, 96.6 | 28.46 | 1.25 | [40] |
| 3.313 | 99.8, 96.7 | 25.59 | ||||
| LCuGd(NO3)3·Me2CO | OBEMAK | 3.444 | 106.8, 105.1 | 4.05 | 1.3 | [37] |
| Gd(hfa)3Cu(salen)(Meim) | RINQIO | 3.252 | 95.0, 98.3 | 27.72 | 1.42 | [38] |
| Cu(acacen)Gd(pta)3 | OFELEQ | 3.274 | 98.0, 94.6 | 29.81 | 1.47 | [40] |
| Cu(ehphi)Gd(hfac)3 | MIDHIQ | 3.252 | 99.7, 94.5 | 30.78 | 1.91 | [41] |
| CuGd(hmp)2(NO3)3(H2O)2 | XAYTIB | 3.346 | 102.8, 102.0 | 15.32 | 3.36 | [15] |
| [CuGd(ems)(NO3)·3H2O]Cu(ems) | XOZZUH | 3.306 | 98.9, 100.1 | 17.77 | 3.76 | [34] |
| LCuGd(NO3)3·CH3OH | WIXYIL | 3.224 | 97.0, 97.1 | 23.31 | 4.33 | [42] |
| LCu(C3H6O)Gd(NO3)3 | NEVHIF | 3.523 | 106.5, 109.0 | 13.52 | 4.8 | [43] |
| LCuGd(NO3)3 | AWUQUE | 3.443 | 103.7, 103.4 | 14.12 | 4.98 | [44] |
| LCu(H2O)Gd(NCS)3·Me2CO | BERPAQ | 3.454 | 106.6, 106.3 | 15.44 | 5.5 | [45] |
| LCuGd(NO3)3·Me2CO | QEMYAI | 3.425 | 109.0, 109.8 | 9.60 | 5.6 | [46] |
| LCu(MeOH)Gd(NO3)3 | NEVHEB | 3.484 | 106.5, 109.0 | 10.00 | 6.8 | [43] |
| LCuGd(NO3)3·Me2CO | ZUVTIT | 3.428 | 107.4, 105.9 | 10.15 | 7.0 | [31] |
| LCuGd(NO3)3·Me2CO | NAMDIP | 3.475 | 106.9, 107.3 | 10.05 | 7.3 | [47] |
| LCuGd(NO3)3·Me2CO | EZAPAW | 3.473 | 107.3, 107.6 | 9.29 | 7.4 | [48] |
| 3.477 | 107.8, 107.8 | 11.06 | ||||
| LCuGd(NO3)3 | GANFIL | 3.498 | 107.4, 108.9 | 7.45 | 7.6 | [49] |
| 3.499 | 106.7, 108.7 | 10.99 | ||||
| LCuGd(NO3)3 | LOYTOI | 3.400 | 103.7, 106.6 | 3.36 | 8.08 | [50] |
| LCuGd(NO3)3·Me2CO | FAKLOT | 3.454 | 106.7, 107.3 | 8.96 | 8.63 | [12] |
| [LCuCl2Gd(H2O)4]Cl·2H2O | KEQRED | 3.512 | 108.8, 108.7 | 1.38 | 10.1 | [11] |
| LCuGd(NO3)3 | LAMBOR | 3.449 | 105.3, 107.8 | 5.50 | 10.8 | [51] |
| LCuGd(NO3)3 | LAMBUX | 3.401 | 105.8, 104.2 | 1.61 | 12.6 | [51] |
Representation of the dependence of the J values for CuIIGdIII dinuclear complexes with a central CuGdO2 framework with the Cu–O–Gd–O torsion angle for the complexes indicated in Table 2.
Molecular structures of the four studied CuII–GdIII complexes: [LCuGd(NO3)3]·Me2CO (L = N,N′-bis((5,6-dimethoxyphenolato-2-yl)methylene)ethylenediamine) (refcode OBEMAK), [LCuGd(CF3SO3)(H2O))3]·THF (L = 1,3-bis (dimethylamino)-2-propanol (refcode NEBLUB), [LCuGd(hfac)3] (L = N,N′-bis(salicylidene)-2-aminobenzylamine; hfac = hexafluoroacetylacetanato) (refcode JOGQAX) and [LCuGd(H2O)3Cl2]Cl·H2O (L = 1,2-bis((3-methoxysalicylidene)amino)-2,2′-dimethylpropanato) (refcode KEQRED). The CuII and GdIII cations, oxygen, nitrogen, sulphur, fluorine, chlorine and carbon atoms are represented as blue, pink, red, light blue, yellow, orange, green and brown spheres, respectively.
In order to study theoretically the exchange coupling in this family of complexes, we have selected three CuIIGdIII complexes [11,31,39] (JOGQAX, ZUVTIT and KEQRED, see Fig. 3) to cover a wide range of Cu–O–Gd–O torsion angle values, together with the two exceptions to the magnetoestructural correlation (NEBLUB and OBEMAK) [36,37] indicated previously (see results in Table 3). From the calculated J values, we can extract the following remarks: (i) The calculated J values for the two complexes (NEBLUB and OBEMAK), where the experimental J values do not follow the magnetoestructural correlation with the Cu–O–Gd–O torsion angle, are larger than the experimental ones, and thus, they are closer to the predictions of the correlation; (ii) For the other three complexes, the calculated exchange coupling constants reproduce correctly the sign and the relative strength of the exchange interaction, only, the value obtained for the ZUVTIT complex is slightly smaller than the experimental one or that expected for its Cu–O–Gd–O torsion angle value.
Calculated and experimental J exchange coupling constants (in cm−1) for the four studied complexes (see Fig. 1) including the scalar relativistic effects (DKH approach in Gaussian code) using the B3LYP functional with an all electron basis set
| Cu–O–Gd–O | Jcalc | Jexp | |
| NEBLUB | 0.49 | +8.9 | −0.08 |
| JOGQAX | 34.28 | +1.5 | +0.8 |
| OBEMAK | 4.05 | +4.6 | +1.3 |
| ZUVTIT | 10.15 | +2.4 | +7.0 |
| KEQRED | 1.38 | +8.5 | +10.1 |
To check the existence of the magnetostructural correlation, we have performed calculations using the KEQRED complex [11] as reference modifying the Cu–O–Gd–O torsion angle just by doing a vertical displacement in the same direction of the two bridging oxygen atoms. The results are plotted in Fig. 4, showing a nice agreement with the experimental data confirming that planar CuGdO2 frameworks are more ferromagnetic. As final conclusion, we can mention that the applied methodology gives very good results for the exchange interactions for the CuIIGdIII complexes and corroborates the existence of a correlation between the magnetic properties and the Cu–O–Gd–O torsion angle.
Representation of the variation of the calculated J values (black circles) with the Cu–O–Gd–O torsion angle for a model structure based on KEQRED complex. The experimental data (empty circles) for the complexes of Table 2 are provided for comparison.
2.3 Analysis of the exchange interactions
The ferromagnetic nature of the CuIIGdIII in this kind of complex has been attributed by Kahn et al. to influence the excited state produced for the electron jump from the 3d CuII electrons to the 5d GdIII shell. Originally, Gatteschi et al. proposed the participation of the 6s orbital of the GdIII cation through a spin polarization model [52,53], however, recent CASPT2 calculations [25] seem to confirm the mechanism proposed by Kahn et al. [13]. Despite the role of the stabilization of the ferromagnetic ground state due to the interaction with excited states, we want to study the influence of the antiferromagnetic contribution analyzing the overlap between the 3d CuII and 4f GdIII magnetic orbitals. Hence, we have calculated such overlap values for the previously studied KEQRED complex [11] (see Fig. 3) and the same complex with a distorted Cu–O–Gd–O torsion angle of 40° corresponding to the weakest ferromagnetic case (see Fig. 4). As noticed previously by us and other authors [54,55], the empty “magnetic orbitals” of the low spin wavefunction are the best choice in order to avoid problems in the determination of the “magnetic orbitals” due to a strong mixing with the ligand orbitals that appear in the occupied orbitals. The analysis of the calculated overlap values using such orbitals shows that for the original KEQRED complex, with an almost planar CuGdO2 framework and relatively strong calculated ferromagnetic coupling (+8.5 cm−1), there are only two non-orthogonal 3d–4f interactions (see left column in Fig. 5, taking a threshold overlap value of 0.01), and consequently five “orthogonal” ferromagnetic contributions, while for the distorted complex with a J = +2.5 cm−1, the opposite situation is found with five non-orthogonal 3d–4f interactions (see right column in Fig. 5). Thus, the sum of the absolute value of the seven overlap integrals for the 3d–4f interactions for the original KEQRED complex is 0.086, considerably lower than the value of 0.115 for the distorted complex, despite the longer Cu–O bond distance in the distorted model. This increase of the overlap matches with the weakening of the ferromagnetic coupling for larger Cu–O–Gd–O torsion angles due to a larger antiferromagnetic contribution as proposed in the Kahn–Briat model [56,57].
Representation of the “non-orthogonal” magnetic orbitals for the KEQRED complex (left column) and for the distorted model of the same complex with a Cu–O–Gd–O torsion angle of 40°. The Sab value corresponds to the overlap integral with the magnetic orbital centered on the CuII cation represented above on each column.
2.4 Spin density
The spin density corresponding to the ferromagnetic S = 4 ground state of the KEQRED complex [11] is plotted in Fig. 6. The two metals show a very different behavior; thus, the unpaired electron of the CuII cation is located in an antibonding orbital that is considerably mixed with the ligand orbitals resulting in a spin delocalization over the neighboring atoms [58,59]. However, the spin density of the GdIII cation remains almost integrally in the lanthanide due to the internal character of the 4f orbitals caused by their poor screening of the nuclear charge. The calculated spin density in the GdIII cations ranges from 6.94 to 6.97 e− for the five calculated complexes. The more surprising fact is the predominance of the spin polarization mechanism on the terminal atoms directly coordinated to the GdIII cation. Despite the presence of a spherical spin density in such cations, the mixture of the 4f orbitals with those of the ligands (see also the highly localized Gd orbitals in Fig. 5) is so weak that the spin delocalization is smaller than the spin polarization. This behavior is completely different than that usually found with transition metal cations if the symmetry of the ligand orbitals allows the mixing with those bearing the unpaired electrons of the metal, the spin delocalization mechanism is the predominant one [58,59].
Representation of the spin density corresponding to the ferromagnetic S = 4 ground state of the KEQRED CuIIGdIII complex. The isodensity surface represented corresponds to a value of 0.0015 e−/bohr3 (white and blue regions indicate positive and negative spin populations, respectively).
3 Conclusions
From the methodological point of view for the magnetic properties of the CuIIGdIII complexes, the inclusion of the relativistic effects using ZORA or DKH approaches is crucial to obtain a good agreement with the experimental exchange coupling constants. The calculations combining such approaches with B3LYP functional and all electron basis set provide an accurate description of the ferromagnetic coupling present in the studied complexes. For the CuIIGdIII complexes with a central CuGdO2 framework, the exchange coupling constant seems to be directly related with the Cu–O–Gd–O torsion angle. Thus, for planar CuGdO2 frameworks the ferromagnetic coupling becomes stronger. The analysis of the overlap integrals for the 3d–4f interactions confirms such magnetostructural correlation due to an increase of the overlap for the non-planar structures, and consequently a larger antiferromagnetic contribution. Finally, it is worth noting that the spin density of the CuIIGdIII complexes shows surprisingly that the spin polarization effects are predominant over the spin delocalization despite that mixing of the 4f orbitals of the GdIII cations is allowed by symmetry with the orbital ligands.
4 Computational details
Since a detailed description of the computational strategy adopted in this work can be found elsewhere [18,23,60,61] we will only briefly sketch its most relevant aspects here. Using a phenomenological Heisenberg Hamiltonian to describe the exchange coupling in a dinuclear compound, where J is the coupling constant, and S1 and S2 the local spins on centers 1 and 2, respectively. It has been found that, when using DFT-based wavefunctions, a reasonable estimate of the exchange coupling constants can be obtained from the energy difference between the low spin wavefunction, ELS (traditionally called broken-symmetry solution for symmetric complexes) and the state with highest spin, EHS through the following equation:
| (1) |
| (2) |
Acknowledgments
The research reported here was supported by the Dirección General de Investigación del Ministerio de Educación y Ciencia and Comissió Interdepartamental de Ciència i Tecnologia (CIRIT) through grants CTQ2005-08123-C02-02/BQU and 2005SGR-00036, respectively. The computing resources were generously made available in the Centre de Supercomputació de Catalunya (CESCA) with a grant provided by Fundació Catalana per a la Recerca (FCR) and the Universitat de Barcelona.