Article pubs.acs.org/JPCA
Understanding Thermodynamic and Spectroscopic Properties of Tetragonal Mn12 Single-Molecule Magnets from Combined Density Functional Theory/Spin-Hamiltonian Calculations Shadan Ghassemi Tabrizi,* Alexei V. Arbuznikov, and Martin Kaupp* †
Institut für Chemie, Theoretische Chemie, Technische Universität Berlin, Sekr. C7, Strasse des 17. Juni 135, 10623 Berlin, Germany S Supporting Information *
ABSTRACT: We apply broken-symmetry density functional theory to determine isotropic exchange-coupling constants and local zero-field splitting (ZFS) tensors for the tetragonal Mn12tBuAc single-molecule magnet. The obtained parametrization of the many-spin Hamiltonian (MSH), taking into account all 12 spin centers, is assessed by comparing theoretical predictions for thermodynamic and spectroscopic properties with available experimental data. The magnetic susceptibility (calculated by the finite-temperature Lanczos method) is well approximated, and the intermultiplet excitation spectrum from inelastic neutron scattering (INS) experiments is correctly reproduced. In these respects, the present parametrization of the 12-spin model represents a significant improvement over previous theoretical estimates of exchange-coupling constants in Mn12, and additionally offers a refined interpretation of INS spectra. Treating anisotropic interactions at the third order of perturbation theory, the MSH is mapped onto the giant-spin Hamiltonian describing the S = 10 ground multiplet. Although the agreement with high-field EPR experiments is not perfect, the results clearly point in the right direction and for the first time rationalize the angular dependence of the transverse-field spectra from a fully microscopic viewpoint. Importantly, transverse anisotropy of the effective S = 10 manifold is explicitly shown to arise largely from the ZFSinduced mixing of exchange multiplets. This effect is given a thorough analysis in the approximate D2d spin-permutational symmetry group of the exchange Hamiltonian. been raised at the very start of the field of molecular magnetism.2 Various attempts have been made to determine exchangecoupling constants for Mn12, either by fitting experimental data18−20 or by performing electronic-structure calculations, where the LDA+U model,21 or broken-symmetry DFT22 have been deployed. From these earlier investigations, which generally yielded very different values for the coupling constants, the careful work by Chaboussant et al.20 still stands out in that the high-temperature magnetic susceptibility and (albeit to a lesser degree) the intermultiplet INS spectra are well described (the respective exchange Hamiltonian was in fact fitted to these data), whereas neither of the previous parameter sets could properly account for the experimental results.20,23 However, the exchange couplings of ref 20 lack corroboration from an electronicstructure perspective, and as we shall show in the present work, they require substantial revision. Furthermore, a detailed interpretation and rationalization of the INS spectra,20,24 which we provide here for the first time, needs to take into account magnetic anisotropy at the level of the individual Mn ions, and it requires an explicit consideration of intensities.
1. INTRODUCTION Since the discovery of its single-molecule magnet (SMM) nature,1−3 the family of closely related tetragonal Mn12 clusters of general formula [Mn12O12(RCO2)16(ROH)4] has developed into the most-investigated anisotropic molecular spin system.4−6 Among the exciting quantum effects associated with Mn12, we mention only the quantum tunneling of the magnetization (QTM) phenomenon, 7−9 Berry phase interference 10 (BPI),11−13 and magnetic avalanches.14−16 An immense multitude of refined experimental techniques has been applied to probe the spectroscopic and thermodynamic properties of the different (rather closely related) members of the Mn12 family. Ultimately, a detailed understanding of this complicated spin system on a microscopic level can be regarded as a major goal of such investigations. Theory is capable of addressing magnetic observables of multinuclear spin clusters in a two-step procedure. First, a suitable parametrization of the many-spin Hamiltonian (MSH), which treats the system in terms of interacting local spin centers, must be found. Subsequently, a variety of methods can be applied to calculate the properties of interest, the choice of a particular method depending on the specific system at hand.17 In most SMMs, isotropic exchange of Heisenberg type represents the dominant interaction in a many-spin model, and the question as to the strength of exchange coupling in Mn12 has © 2016 American Chemical Society
Received: July 10, 2016 Published: August 2, 2016 6864
DOI: 10.1021/acs.jpca.6b06896 J. Phys. Chem. A 2016, 120, 6864−6879
Article
The Journal of Physical Chemistry A
employing the BP86 functional45,46 and the lacv3p** basis-set/ Mn-pseudopotential combination.47 The molecular structure is shown in Figure 1. Accurate single-point calculations of a number
A very recent LDA+U density functional study by Mazurenko et al.25 reported isotropic exchange couplings, site ZFS tensors for Mn centers, and notably, antisymmetric (Dzyaloshinsky− Moriya) exchange tensors between pairs of centers. This parametrization was subsequently evaluated by Hanebaum and Schnack26 in terms of its compatibility with thermodynamic data (the magnetic susceptibility is clearly the most diagnostic quantity in this respect). Unfortunately, however, in ref 26 the coupling constants from ref 25 were accidentally misrepresented by choosing them too small by a factor of 2. Yet, as we shall show below, even the correctly represented MSH still remains highly deficient. In summary, thus far there appears to be no MSH from firstprinciples calculations that can account for magnetic-susceptibility data and INS spectra, precluding an in-depth rationalization of the extraordinary quantum properties displayed by the paradigmatic Mn12 SMMs. The local origin of magnetic anisotropy had been addressed before,27−30 but we are aware of only one previous explicit investigation of how site ZFS would map onto the S = 10 ground multiplet.31 However, the giant-spin Hamiltonian (GSH) describing this multiplet was constructed in the strong-exchange limit only31 (employing overall unrealistic sets of exchange-coupling constants), which does not allow for a qualitative explanation of HF-EPR data,30,32 or of BPI effects in magnetic quantum tunneling.13 We note in passing that a couple of pioneering and instructive studies33−35 of many-spin effects in Mn12 started from the “Florentine”36 coupling scheme, a simplified eight-spin model2 that was later shown to be inadequate.18,37 Here, we shall fill the gap in the understanding of magnetism in the Mn12 family of spin clusters by providing a set of exchangecoupling constants, calculated by broken-symmetry DFT, that provides a good description of the magnetic susceptibility and, in conjunction with the calculated site ZFS tensors, also correctly accounts for the INS spectra.20,24 In fact, our combined DFT/ spin-Hamiltonian approach allows for a refined interpretation of the experimental data.20 The question of the microscopic origin of transverse anisotropy of the S = 10 ground multiplet of Mn12 has not yet been touched upon in an explicit fashion (i.e., from a realistic 12spin model). By mapping our MSH onto the S = 10 GSH at the third order of perturbation theory,38 we can largely rationalize the transverse magnetic anisotropy (compatible with molecular symmetry) in terms of the ZFS-induced admixture of excited spin multiplets (S-mixing38−41), thus providing a direct link to HF-EPR experiments30 and magnetic relaxation studies.13
Figure 1. Molecular structure employed in DFT calculations (hydrogen atoms omitted for clarity; tBuAc ligands in Mn12tBuAc30 were replaced by Ac ligands, as explained in the text). Color code: MnIV (s = 3/2), blue spheres (sites 1−4); MnIII (s = 2), cyan spheres (sites 5−12); O, red sticks; C, gray sticks. The O atoms along the (elongated) Jahn−Teller axes of the eight MnIII ions are shown in pink.
of broken-symmetry states,48,49 and of the high-spin state, were performed with Jaguar at the B3LYP/lacv3p** level of theory. The B3LYP hybrid functional45,50 represents a very reasonable choice for the present type of system (and a widely employed one, see, e.g., refs 51−55). Following ref 25, we assume a model with seven inequivalent coupling constants (the coupling graph is displayed in Figure 2). Choosing a nonminimal number of spin configurations (Ising states), mapping their DFT energies onto the Heisenberg Hamiltonian (eq 1) sets up an overdetermined system of linear equations that we solved for the exchange-coupling constants {J} and the mean energy E0 in a least-squares sense.56,57 Notably, the quality of this mapping is considerably better compared to that in an earlier work22 that included only four different exchange couplings in the model. Specifically, in comparison with ref 22, the mean absolute error in the solution to the overdetermined system is reduced by one-third, confirming the importance of the added flexibility that a larger number of inequivalent couplings provides. Further details on this point, and a list of the chosen spin configurations, together with their DFT and Ising energies, are given in section 1 of Supporting Information. For the determination of site zero-field splitting tensors, we performed accurate single-point calculations with flexible IGLO-II58 basis sets on the ligand atoms and a 9s7p4d allelectron basis set on Mn,59 using the Turbomole program.60 SCF orbitals were exported to our in-house program MAG61 for the calculation of ZFS tensors by the second-order treatment of spin−orbit coupling devised by Pederson and Khanna (PK),62 with corrected prefactors by van Wüllen.28,63 Employing the atomic-mean-field approximation,64,65 we obtain each individual site ZFS tensor by including effective atomic spin−orbit operators only for the specific Mn ion under consideration. Due to the S4 symmetry, only ZFS tensors for inequivalent sites Mn1, Mn5, and Mn9 (Figure 1) need to be calculated by DFT. The ZFS tensors of the other sites are determined by molecular symmetry; i.e., they are obtained by rotations about the S4 axis
2. COMPUTATIONAL DETAILS AND THEORETICAL BACKGROUND DFT Calculations. The molecular structure of Mn12tBuAc = [Mn12O12(tBu−CH2CO2)16(CH3OH)4]·CH3OH, determined by X-ray diffraction, was taken from ref 30. Whereas solventrelated disorder causes rhombic distortions in the “original”42 Mn12Ac species29 (where Mn12Ac = [Mn12O12(CH3CO2)16(H2O)4]·2CH3CO2H·4H2O), complicating an analysis of spectroscopic experiments probing the zerofield split S = 10 ground multiplet (for a review, see ref 9), there is no sizable disorder in Mn12tBuAc, and it has thus truly axial symmetry (see the reviews by Bagai and Christou5 and by Hill,43 and references cited therein). For the computations, the large t BuAc ligands were truncated to acetates, and the positions of hydrogen atoms were optimized with the Jaguar 7.8 software,44 6865
DOI: 10.1021/acs.jpca.6b06896 J. Phys. Chem. A 2016, 120, 6864−6879
Article
The Journal of Physical Chemistry A
estimated by ∼30% on average.69 The small magnitude of the MnIV ZFS tensors (which contribute merely ca. 1% to the effective D) justifies their scaling by the same factor of ∼2, which is expected (and confirmed, see below) to hold for MnIII. Many-Spin Hamiltonian and Construction of the GSH (“Spin Projection”). The MSH used to describe the Mn12 spin cluster is represented by the sum of an isotropic part Ĥ (0) and an anisotropic contribution Ĥ (1). The isotropic Heisenberg exchange Hamiltonian is parametrized by a set of coupling constants {Jij}: (0) Ĥ =
∑ Jij sî ·sĵ (1)
i
