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The Role of Anisotropic Exchange in Single Molecule Magnets: A CASSCF/NEVPT2 Study of the Fe_{4} SMM Building Block [Fe_{2}(OCH_{3})_{2}(dbm)_{4}] Dimer

^{1}

^{2}

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## Abstract

**:**

## 1. Introduction

## 2. Results and Discussion

#### 2.1. Method Assessment

#### 2.1.1. X-ray Structure

**1-Ph**${}_{\mathbf{xray}}$ from now on) has been used for the first benchmark calculation at the post-HF level. Both CASSCF and NEVPT2 approaches (see Theory and Computational Methods and Computational Details sections) have been applied and results are listed in Table 1. As expected [6,10,23,25], the inclusion of correlation only in the five 3d orbitals of the two Fe(III) leads to a very poor agreement with the experimental value. Indeed, a ferromagnetic ${J}_{12}$ value of −9.12 cm${}^{-1}$ was found (see Equation 5) versus an experimental antiferromagnetic value of 15.4 cm${}^{-1}$. The introduction of dynamical correlation via NEVPT2 recovered some of the important contributions given by the bridging ligand orbitals to the super-exchange mechanism and both the experimental sign and an acceptable magnitude of 5.32 cm${}^{-1}$ for ${J}_{12}$ have been then reached. Possible strategies to further improve this result would require the introduction of a dedicated CI procedure [33] or equivalent [34] on top of the CASSCF instead of NEVPT2. Another possible source of the discrepancy can reside in the limited flexibility of the wavefunction resulting by a CAS(5,5), i.e., the so called 3d double-shell effect [35] or the missing of the explicit inclusion of bridging ligands orbitals in the CAS [35]. Even in these cases, the dimension of the CAS should be at least doubled. However, the computational demand for such an approach would require a severe reduction in the size of the system and will not be addressed here. Assessed that the inclusion of the dynamical correlation via perturbative approach is mandatory for a better description of the electronic states energy ladder we used the same protocol for the calculation of the anisotropic Spin Hamiltonian terms. In our treatment, only the SOC contribution is included while it is worth to remember that experimental single ion values contain both spin-spin and SOC contributions. Nevertheless, on the basis of the results reported in ref. [17,36], the lacking of the spin-spin contribution could not account for the big difference between the computed and the experimental single ion anisotropy values and the simulation protocol employed leads to a slight underestimation of the single ions contribution. However, the computed anisotropic exchange parameters result in fairly good agreement with the experimental ones both in sign and magnitude [19]. In Table 1 we reported the computed values for anisotropic exchange anisotropy tensors. In conclusion, the computational set up can be considered to be reliable enough to be used for the magneto-structural analysis.

#### 2.1.2. Simplified X-ray Structure

**1-Me**${}_{\mathbf{xray}}$ model). Such a choice was supposed to slightly affect the ligand field felt by the two iron ions. Indeed, the modelization consists only in the substitution of an sp${}^{2}$ carbon (phenyl) with an sp${}^{3}$ one (methyl) as functional group in alpha position to the CO groups of the dbm ligands, which is the actual moiety that binds the metals. In fact, the computed ${J}_{12}$ was practically unaffected by this substitution (${J}_{12}$ = 5.52 cm${}^{-1}$). On the other hand, ${D}_{12}$ was found, unexpectedly, to change both in sign and magnitude. The change of the outer coordination shell showed to be enough to turn the easy-axis kind of the exchange anisotropy to an easy-plane one with ${D}_{12}$ = 0.411 cm${}^{-1}$ and E/D = 0.135 (see Table 1). In order to confirm the reliability of this result, we checked its convergence with respect to the number of excited spin multiplets introduced in the SOC diagonalization adding up to 99 quintuplets states. Although the ${D}_{12}$ value gets reduced to 0.344 cm${}^{-1}$, its easy-plane anisotropy was maintained. Therefore, the variation from easy-axis to easy-plane anisotropy can be ascribed to the variation of the π contribution induced by the substitution of the CO groups. Such a result is very interesting since it shed some light on how subtle the effects on the anisotropic exchange terms can be and how important the chemical tailoring in designing magnetic materials should be.

#### 2.2. Magneto-Structural Correlations

**1-Me**${}_{\mathbf{xray}}$ geometry in vacuum (

**1-Me**${}_{\mathbf{opt}}$) followed by the calculation of its hessian. Both calculations have been done with DFT (see Computational Details). The root mean square displacement (RMSD) between

**1-Me**${}_{\mathbf{opt}}$ and

**1-Me**${}_{\mathbf{xray}}$ is only 0.143 Å. The optimised main geometrical parameters are reported in Table 2 with their correspondent X-ray geometrical values.

**1-Me**${}_{\mathbf{opt}}$ model in place of the

**1-Ph**${}_{\mathbf{xray}}$ model and the magneto-structural correlations will be referred to this geometry. For the study of the magneto-structural correlations, we produced six distorted structures corresponding at ±1q, ±2q and ±3q displacements, expressed in terms of normal mode unit, with respect to the equilibrium geometry. The ensemble of angles spanned by these structures is rather large (94.9${}^{\circ}$÷ 106.4${}^{\circ}$) and fully includes all the experimental scenarios encountered so far in both propeller-shaped SMMs and this oxo-bridged Fe2 dimer family [18,37].

**1-Ph**${}_{\mathbf{xray}}$ model.

## 3. Theory and Computational Methods

## 4. Computational Details

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

- Heisenberg, W. Zur Theorie des Ferromagnetismus. Z. Phys.
**1928**, 49, 619–636. [Google Scholar] [CrossRef] - Dirac, P. Quantum Mechanics of Many-Electrons Systems. Proc. R. Soc.
**1929**, 123, 714–733. [Google Scholar] [CrossRef] - Van Vleck, J. The Dirac Vector Model in Complex Spectra. Phys. Rev.
**1934**, 45, 405–419. [Google Scholar] [CrossRef] - Pederson, M.; Khanna, S. Magnetic anisotropy barrier for spin tunneling in Mn
_{12}O_{12}molecules. Phys. Rev. B**1999**, 60, 9566–9572. [Google Scholar] [CrossRef] - Vahtras, O.; Loboda, O.; Minaev, B.; Ågren, H.; Ruud, K. Ab initio calculations of zero-field splitting parameters. Chem. Phys.
**2002**, 279, 133–142. [Google Scholar] [CrossRef] - Calzado, C.J.; Cabrero, J.; Malrieu, J.P.; Caballol, R. Analysis of the magnetic coupling in binuclear complexes. I. Physics of the coupling. J. Chem. Phys.
**2002**, 116, 2728–2747. [Google Scholar] [CrossRef] - Ganyushin, D.; Neese, F. First-principles calculations of zero-field splitting parameters. J. Chem. Phys.
**2006**, 125, 24103. [Google Scholar] [CrossRef] [PubMed] - Neese, F. Importance of direct spin-spin coupling and spin-flip excitations for the zero-field splittings of transition metal complexes: A case study. J. Am. Chem. Soc.
**2006**, 128, 10213–10222. [Google Scholar] [CrossRef] [PubMed] - Maurice, R.; Bastardis, R.; Graaf, C.; Suaud, N.; Mallah, T.; Guihéry, N. Universal theoretical approach to extract anisotropic spin Hamiltonians. J. Chem. Theory Comput.
**2009**, 5, 2977–2984. [Google Scholar] [CrossRef] [PubMed] - Bencini, A.; Totti, F.; Daul, C.C.A.; Doclo, K.; Barone, V. Density functional calculations of magnetic exchange interactions in polynuclear transition metal complexes. Inorg. Chem.
**1997**, 1669, 5022–5030. [Google Scholar] [CrossRef] - Bencini, A.; Totti, F. A few comments on the application of density functional theory to the calculation of the magnetic structure of oligo-nuclear transition metal clusters. J. Chem. Theory Comput.
**2009**, 5, 144–154. [Google Scholar] [CrossRef] [PubMed] - Rocha, A.R.; García-Suárez, V.M.; Bailey, S.W.; Lambert, C.J.; Ferrer, J.; Sanvito, S. Towards molecular spintronics. Nat. Mater.
**2005**, 4, 335–339. [Google Scholar] [CrossRef] [PubMed] - Wernsdorfer, W. A long-lasting phase. Nat. Mater.
**2007**, 6, 174–176. [Google Scholar] [CrossRef] [PubMed] - Mannini, M.; Pineider, F.; Sainctavit, P.; Danieli, C.; Otero, E.; Sciancalepore, C.; Talarico, A.M.; Arrio, M.A.; Cornia, A.; Gatteschi, D.; et al. Magnetic memory of a single molecule quantum magnet wired to a gold surface. Nat. Mater.
**2009**, 8, 194–197. [Google Scholar] [CrossRef] [PubMed] - Urdampilleta, M.; Nguyen, N.V.; Cleuziou, J.P.; Klyatskaya, S.; Ruben, M.; Wernsdorfer, W. Molecular quantum spintronics: Supramolecular spin valves based on single molecule magnets and carbon nanotubes. Int. J. Mol. Sci.
**2011**, 12, 6656–6667. [Google Scholar] [CrossRef] [PubMed] - Westrup, K.C.M.; Boulon, M.; Totaro, P.; Nunes, G.G.; Back, D.F.; Barison, A.; Jackson, M.; Paulsen, C.; Gatteschi, D.; Sorace, L.; et al. Adding remnant magnetization and anisotropic exchange to propeller like single molecule magnets through chemical design. Chem. Eur. J.
**2014**, 20, 13681–13691. [Google Scholar] [CrossRef] [PubMed] - Lunghi, A.; Totti, F. DFT magnetic characterization of a Fe
_{4}SMMs series: From isotropic exchange interactions to multi-spin zero field splitting. J. Mater. Chem. C**2014**, 2, 8333–8343. [Google Scholar] [CrossRef] - Le Gall, F.; Fabrizi de Biani, F.; Caneschi, A.; Cinelli, P.; Cornia, A.; Fabretti, A.C.; Gatteschi, D. Synthesis, crystal structures and magnetic characterization of four β-diketonate-alkoxide iron (III) dimers. Dependence of the magnetic properties on geometrical and electronic parameters. Inorg. Chim. Acta
**1997**, 262, 123–132. [Google Scholar] [CrossRef] - Ter Heerdt, P.; Stefan, M.; Goovaerts, E.; Caneschi, A.; Cornia, A. Single ion and molecular contributions to the zero-field splitting in an iron(III)-oxo dimer studied by single crystal W-band EPR. J. Magn. Reson.
**2006**, 179, 29–37. [Google Scholar] [CrossRef] [PubMed] - Bencini, A.; Gatteschi, D. EPR of Exchange Coupled Systems; Springer Science + Business Media: Berlin, Germany, 2012. [Google Scholar]
- Angeli, C.; Cimiraglia, R.; Malrieu, J.P. N-electron valence state perturbation theory: A fast implementation of the strongly contracted variant. Chem. Phys. Lett.
**2001**, 350, 297–305. [Google Scholar] [CrossRef] - Atanasov, M.; Aravena, D.; Suturina, E.; Bill, E.; Maganas, D.; Neese, F. First principles approach to the electronic structure, magnetic anisotropy and spin relaxation in mononuclear 3d-transition metal single molecule magnets. Coord. Chem. Rev.
**2015**, 289–290, 177–214. [Google Scholar] [CrossRef] - Maurice, R.; Guihéry, N.; Bastardis, R.; Graaf, C.D. Rigorous extraction of the anisotropic multispin Hamiltonian in bimetallic complexes from the exact electronic Hamiltonian. J. Chem. Theory Comput.
**2009**, 6, 55–65. [Google Scholar] [CrossRef] [PubMed] - Maurice, R.; de Graaf, C.; Guihéry, N. Magnetic anisotropy in binuclear complexes in the weak-exchange limit: From the multispin to the giant-spin Hamiltonian. Phys. Rev. B
**2010**, 81, 214427. [Google Scholar] [CrossRef] - Maurice, R.; Sivalingam, K.; Ganyushin, D.; Guihéry, N.; de Graaf, C.; Neese, F. Theoretical determination of the zero-field splitting in copper acetate monohydrate. Inorg. Chem.
**2011**, 50, 6229–6236. [Google Scholar] [CrossRef] [PubMed] - Maurice, R.; Graaf, C.D.; Guihéry, N. Theoretical determination of Spin Hamiltonians with isotropic and anisotropic magnetic interactions in transition metal and lanthanide complexes. Phys. Chem. Chem. Phys.
**2013**, 15, 18784–18804. [Google Scholar] [CrossRef] [PubMed] - Singh, S.K.; Rajaraman, G. Probing the origin of magnetic anisotropy in a dinuclear Mn(III)Cu(II) single molecule magnet: The role of exchange anisotropy. Chem. Eur. J.
**2014**, 20, 5214–5218. [Google Scholar] [CrossRef] [PubMed] - Tereniak, S.J.; Carlson, R.K.; Clouston, L.J.; Young, V.G.; Bill, E.; Maurice, R.; Chen, Y.S.; Kim, H.J.; Gagliardi, L.; Lu, C.C. Role of the metal in the bonding and properties of bimetallic complexes involving manganese, iron, and cobalt. J. Am. Chem. Soc.
**2014**, 136, 1842–1855. [Google Scholar] [CrossRef] [PubMed] - Neese, F. The ORCA program system. Wiley Interdiscip. Rev. Comput. Mol. Sci.
**2012**, 2, 73–78. [Google Scholar] [CrossRef] - Weigend, F.; Ahlrichs, R.; Gmbh, F.K. Balanced basis sets of split valence, triple zeta valence and quadruple zeta valence quality for H to Rn: Design and assessment of accuracy. Phys. Chem. Chem. Phys.
**2005**, 7, 3297–3305. [Google Scholar] [CrossRef] [PubMed] - Weigend, F. Accurate coulomb-fitting basis sets for H to Rn. Phys. Chem. Chem. Phys.
**2006**, 22, 1057–1065. [Google Scholar] [CrossRef] [PubMed] - Perdew, J.P.J.; Burke, K.; Wang, Y. Generalized gradient approximation for the exchange-correlation hole of a many-electron system. Phys. Rev. B
**1996**, 54, 533–539. [Google Scholar] [CrossRef] - Garcia, V.M.; Castell, O.; Caballol, R.; Malrieu, J.P. An iterative difference-dedicated configuration interaction. Proposal and test studies. Chem. Phys. Lett.
**1995**, 238, 222–229. [Google Scholar] [CrossRef] - Neese, F. A spectroscopy oriented configuration interaction procedure. J. Chem. Phys.
**2003**, 119, 9428–9443. [Google Scholar] [CrossRef] - Malrieu, J.P.; Caballol, R.; Calzado, C.J.; de Graaf, C.; Guihéry, N. Magnetic interactions in molecules and highly correlated materials: Physical content, analytical derivation, and rigorous extraction of magnetic hamiltonians. Chem. Rev.
**2014**, 114, 429–492. [Google Scholar] [CrossRef] [PubMed] - Retegan, M.; Collomb, M.N.; Neese, F.; Duboc, C. A combined high-field EPR and quantum chemical study on a weakly ferromagnetically coupled dinuclear Mn(III) complex. A complete analysis of the EPR spectrum beyond the strong coupling limit. Phys. Chem. Chem. Phys.
**2013**, 15, 223–234. [Google Scholar] [CrossRef] [PubMed] - Gregoli, L.; Danieli, C.; Barra, A.L.; Neugebauer, P.; Pellegrino, G.; Poneti, G.; Sessoli, R.; Cornia, A. Magnetostructural correlations in tetrairon(III) single molecule magnets. Chem. Eur. J.
**2009**, 15, 6456–6467. [Google Scholar] [CrossRef] [PubMed] - Lunghi, A.; Iannuzzi, M.; Sessoli, R.; Totti, F. Single molecule magnets grafted on gold: Evolution of magnetic properties from Ab initio molecular dynamics. J. Mater. Chem. C
**2015**, 3, 7294–7304. [Google Scholar] [CrossRef] - Fernandez Garcia, G.; Lunghi, A.; Totti, F.; Sessoli, R. Toward mesoscale properties of self-assembled monolayers of SMM on Au(111): An integrated Ad Hoc FF and DFT study. J. Phys. Chem. C
**2016**, 120, 14774–14781. [Google Scholar] [CrossRef]

**Figure 1.**Fe${}_{2}$(OCH${}_{3}$)${}_{2}$(dbm)${}_{4}$ X-ray structure. Fe, O, C, and H atoms are blue, red, green, and white colored, respectively.

**Figure 2.**Contribution to normal modes of the two FeOFe bending internal displacements as function of frequencies. The solid black vertical line at ∼ 250 cm${}^{-1}$ points to the contribution of the symmetric double FeOFe bending chosen as normal mode reference q.

**Figure 3.**Scheme of the cartesian displacements corresponding to the normal mode q. Fe, O and C atoms are blue, red and green colored, respectively. H atoms are here not shown for clarity reasons.

**Figure 4.**Magneto-structural correlation plot for ${J}_{12}$ (cm${}^{-1}$) as function of the FeOFe angle (Deg.).

**Figure 5.**Magneto-structural correlation plot for ${D}_{12}$ (cm${}^{-1}$) as function of the FeOFe angle (Deg.).

J${}_{12}$ | D${}_{\mathit{SI}}$ | E${}_{\mathit{SI}}$/D${}_{\mathit{SI}}$ | D${}_{\mathit{EX}}$ | E${}_{\mathit{EX}}$/D${}_{\mathit{EX}}$ | |
---|---|---|---|---|---|

1-Ph${}_{\mathbf{xray}}$ | 5.32 | 0.329 | 0.113 | −0.108 | 0.321 |

1-Me${}_{\mathbf{xray}}$ | 5.52 | 0.090 | 0.124 | 0.411 | 0.135 |

1-Me${}_{\mathbf{opt}}$ | 2.92 | 0.197 | 0.105 | 0.116 | 0.131 |

Exp. ${}^{a}$ | 15.4 | 0.749 | 0.097 | −0.159 | 0.176 |

Fe–Fe (Å) | Fe–O (Å) | FeOFe (Deg.) | |
---|---|---|---|

1-Ph${}_{\mathbf{xray}}$ | 3.15 | 1.99 | 100.69 |

1-Me${}_{\mathbf{opt}}$ | 3.12 | 2.03 | 102.69 |

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**MDPI and ACS Style**

Lunghi, A.; Totti, F. The Role of Anisotropic Exchange in Single Molecule Magnets: A CASSCF/NEVPT2 Study of the Fe_{4} SMM Building Block [Fe_{2}(OCH_{3})_{2}(dbm)_{4}] Dimer. *Inorganics* **2016**, *4*, 28.
https://doi.org/10.3390/inorganics4040028

**AMA Style**

Lunghi A, Totti F. The Role of Anisotropic Exchange in Single Molecule Magnets: A CASSCF/NEVPT2 Study of the Fe_{4} SMM Building Block [Fe_{2}(OCH_{3})_{2}(dbm)_{4}] Dimer. *Inorganics*. 2016; 4(4):28.
https://doi.org/10.3390/inorganics4040028

**Chicago/Turabian Style**

Lunghi, Alessandro, and Federico Totti. 2016. "The Role of Anisotropic Exchange in Single Molecule Magnets: A CASSCF/NEVPT2 Study of the Fe_{4} SMM Building Block [Fe_{2}(OCH_{3})_{2}(dbm)_{4}] Dimer" *Inorganics* 4, no. 4: 28.
https://doi.org/10.3390/inorganics4040028