Magneto-structural correlations in doubly hydroxo-bridged Cu(II)-dimers

The magneto-structural correlations of superexchange-coupled doubly hydroxo-bridged Cu(II)-dimers have been investigated. To this end, an analytical approach has been applied to [Cu2(OH)2F4]2- model complexes. This approach supplies an analytical scheme, based on orbital interactions, for calculating the transfer integral, HAB, which is shown to play the key role in the magnetic coupling constant J for understanding magneto-structural correlations. The single contributions to the transfer integral are calculated and described explicitly. Therefore, this approach supplies a detailed insight into the magnetic behavior and the interaction mechanisms of the hydroxo-bridged Cu-dimers. All analytical results are compared with experimental and numerical data.


Introduction
The doubly hydroxo-bridged Cu(II) dimers are one of the most extensively experimentally studied molecular magnets 1,2 . Since Cu(II) is a d 9 ion there is merely one unpaired electron (active electron) per metal center. Therefore, these complexes represent a relatively simple system for describing magnetostructural correlations in superexchange-coupled compounds. Beside the dependence of the magnetic coupling on the bridging angle, its correlations to the out-of-plane angle of hydrogen, the copper-oxygen bonding-distance and the bending angle of the molecules have been investigated both experimentally 3,4,5 and theoretically 1,6,7 . Additionally, empirical relations between the magnetic coupling constant J and geometrical parameters were derived from experimental data 3 without, however, giving any insight into the coupling mechanisms. Numerical calculations of J with density functional theory (DFT) could reproduce the experimental results and, based on calculations on model systems, they supplied important information about the relations between geometrical parameters and magnetic coupling 1,6 . An analysis and an interpretation of the magnetic behavior were usually based on the expression for the magnetic coupling constant derived by Hay et al. 7 4 understanding of the magnetic behavior is still missing. Therefore, the aim of this work is to investigate the magneto-structural correlations in the doubly hydroxo-bridged Cu(II)-dimers in more detail based on an orbital interaction scheme. To this end, an approach 9 is applied that enables the analytical calculation of the transfer integral H AB which was explicitly shown to be the crucial quantity for the magnetostructural correlations. H AB is represented as a function of the energies of metal and ligand orbitals and the overlap integrals. In the present work this analytical formalism is applied on [Cu 2 (OH) 2 F 4 ] 2modelcomplexes where the bulky terminal ligands of the real systems (e.g. ref. [2,3]) are substituted by fluorine enabling an easier analysis. Since the structural variations discussed here in principle only affect the interactions between Cu and OH, the magneto-structural correlations should be qualitatively independent of the choice of the terminal ligands, as long as the distorted square-planar coordination of copper is preserved. The magnetic behavior will be investigated for variations of the following structural parameters: bridging angle θ, out-of-plane angle of hydrogen τ, bending angle γ (bending about the axis parallel to the bridging (OH)-molecules), copper-oxygen bonding distance. All analytical calculations are supplemented by fully numerical calculations with the self-consistent charge (SCC)-Xα program 10,11 .

Computational details
For the planar [Cu 2 (OH) 2 F 4 ] 2model complex, the following structural parameters are used, if not stated otherwise: Cu-O = 1.95Å, Cu-F = 1.94Å, O-H = 0.96Å, F-Cu-F = 93°. The xz-plane is chosen as the molecular plane, where z is the internuclear axis of the two monomers (figure 1).

Fig. 1.
The transfer integral H AB may be defined as 7

( )
where ε ± are the orbital energies of the magnetic molecular orbitals of the dimer that contain the active electrons. These molecular orbitals (MOs) belong to a symmetrical (-) and an unsymmetrical (+) linearcombination of atomic orbitals (AOs). As basis set for the analytical calculations Cu(3d), F(2s,2p), O(2s,2p) and H(1s) AOs are chosen which are described by the product of a single Slater-type orbital (STO) and a real spherical harmonic. The orbital exponent can be taken as constant for different geometries 9 . The 4s and 4p orbitals of copper are assumed to be negligible.
For the analytical calculation of the magnetic MOs and their energies ε ± it is convenient to construct two sets of group-orbitals from the AOs of the basis set according to the symmetries of the two magnetic MOs.
The analytical calculation of H AB is performed in general in two steps 9,12 aiming to transform the full multi-center MO-Hamiltonian into an effective single-center problem. First, the Kohn-Sham equation is solved in linear combination of atomic orbital approximation with respect to Cu(3d) group orbitals that are Schmidt-orthogonalized to the ligand group orbitals. The latter are assumed to be orthogonal or have been orthogonalized among each other. In the second step, the Hamiltonian-matrix, calculated with the orthogonalized orbitals, is diagonalized analytically. This matrix is of block-diagonal form due to the use of the symmetry-adapted basis. This procedure was denoted as dimer approach 9 . In combination with a bridging-ligand-only method 9 , where matrix elements of the terminal ligands are removed from the blockdiagonalized Hamiltonian matrix, it leads to sufficiently low dimensional blocks that can be diagonalized analytically. Those eigenfunctions having predominantly Cu(3d) character correspond to the magnetic orbitals and their eigenvalues correspond to ε ± . These energies cannot be approximated via a perturbation calculation since some nondiagonal Hamiltonian matrix-elements are too large. The neglect of the terminal ligands in the diagonalization step is well justified in these complexes since the interactions between fluorine and copper are quite weak and cancel each other to a very high degree in the energydifference of eq.(2).
For an analytical solution the orbital energies of the AOs and the overlap integrals and the nondiagonal Hamiltonian matrix between the AOs are required. The first have turned out to be more or less constant with respect to geometrical changes of the dimers and are taken to be the orbital energies of the atoms in the molecule. The overlap integrals for STOs can be calculated analytically 13 . For the nondiagonal Hamiltonian matrix elements the following approximation is used 12 : ( ) ( ) correspond to the energies of the ferromagnetic and broken symmetry states, respectively, and S is the total spin. For the analytical calculations of J a parameterized form of eq.(1) is where C and f are constant parameters corresponding to the direct exchange and the Hubbard U, respectively. Inserting the analytical H AB into eq.(5) and choosing two reasonable values for C and f should supply a good description of J as a function of geometrical parameters. For comparison also numerical transfer integrals will be inserted.

Planar complex with varying bridging angle θ
In AOs of oxygen come into play. However, with respect to the simplifications of the analytical description this error is acceptable.

Fig. 2.
The great advantage of the analytical calculation compared with, e.g. DFT-calculations, is the separability of the single contributions to H AB . This was already demonstrated for Cu-F complexes 9 . The expressions for the single contributions to H AB applied there have been somewhat adapted for the dimer approach of complexes with stronger orbital interactions as this is the case for the hydroxo-bridged dimers. In order the keep the expressions handy and simple, some small terms entering into to calculation of ε ± described above are neglected so that the sum of all contributions may deviate slightly from the total analytical value of H AB . The contribution from the direct d-d interaction is where ± denotes contributions to the symmetrical (-) and unsymmetrical (+) magnetic MOs, respectively.
H kd , S kd are nondiagonal elements between Cu(3d) and bridging ligand orbitals of the Hamiltonian and overlap matrix and H kk is the energy of a bridging ligand orbital. Small contributions from diagonalization (e.g. from s-and H-eigenfunctions) may be included as perturbations whereas large ones are taken from direct diagonalization ( ) ( ) Table 1.
It has to be kept in mind that the bridging ligand orbitals contributing to the symmetrical magnetic MO are   The curves obtained from inserting the analytical and numerical H AB into eq.(5) are shown in figure 5 together with the results from the broken symmetry calculations.

Bent complex
Experimental and theoretical data reveal a decreasing strength of the antiferromagnetic coupling with increasing bending angle γ 1,4,5 , finally leading to a transition to ferromagnetic coupling. For the investigation of the bent dimers again the model-complex with a bridging angle of 95° is used. The monomers of this complex will then be rotated into opposite directions by an angle γ about the axis connecting the OH-bridges ( figure 6). The dihedral angle of the dimer is defined as 180°-2γ. The calculation procedure is the same as in the previous case. However, the monomers are not in the xz-plane, so that, the Cu(3d)-orbital is not anymore of pure d xz -character but is a linear combination of d xz and d yz .
As discussed in ref. [9] for the singly bridged [ ]   The transfer integrals strongly decrease with increasing bending angle. The single contributions to the analytical H AB are listed in table 3. Table 3.
Due to the bending the Cu-Cu distance is reduced with corresponding increase of the d-d interaction. The  Indeed, the linear combinations become successively different for increasing γ basically due to the different types of bridging orbitals contributing to these MOs. Nevertheless, the analytical calculation supply a correct description of the transition from antiferromagnetic to ferromagnetic coupling and predict at least qualitatively the return to an antiferromagnetic behavior for large bending angles.
In the work of Ruiz et al. 1 the first transition appears at about 20° for a bridging angle of 95°. figure 8 shows this transition at about 13°. The difference may again be assigned to the different terminal ligands.
Charlot et al. 4,5 explain this transition with a stabilization of the antisymmetric MO with increasing γ. The symmetric MO should be unaffected by the structural variations and since it is lower in energy the energy difference to the antisymmetric MO and thus H AB decreases leading to ferromagnetic coupling. Our results yield a similar stabilization of the antisymmetric MO due to the decreasing z-d interaction (table 3).
However, due to the increasing strength of the y-d interaction the symmetric orbital does not remain constant in energy but is destabilized. A constant energy is obtained only if neglecting the contribution from the y-eigenfunction.
If complexes with bridging angles of about 90° are bent one would start with a ferromagnetic complex (see figure 3) and would observe a transition to antiferromagnetic coupling. For even smaller bridging angles only an increasing strength of the antiferromagnetic coupling would be obtained.

Variation of the Cu-O bonding-distance
Numerical calculations and experimental results 1,3 reveal a transition from antiferromagnetic to ferromagnetic coupling when reducing the Cu-O distance which is attributed to an increase of the direct ferromagnetic exchange 1 . If this were indeed the case, the parameterized formula for J, eq.(5), cannot be applied.
For the analytical calculations the planar complex [Cu 2 (OH) 2 F 4 ] 2is chosen with a bridging angle fixed at 95°. The Cu-O bonding distance is varied between 1.70 and 2.20 Å. The calculation procedure for this planar complex is analogous to that for varying bridging angles with the only difference that terminal and bridging p z -orbitals are orthogonalized to each other. This has turned out to be important for small bonding distances. Fully numerical and analytical transfer integrals are compared in figure 9.

Conclusion
In the present paper it has been shown that the simple [Cu 2 (OH) 2 Figure 1.