Construction of Giant‐Spin Hamiltonians from Many‐Spin Hamiltonians by Third‐Order Perturbation Theory and Application to an Fe₃Cr Single‐Molecule Magnet

Abstract

A general giant-spin Hamiltonian (GSH) describing an effective spin multiplet of an exchange-coupled metal cluster with dominant Heisenberg interactions was derived from a many-spin Hamiltonian (MSH) by treating anisotropic interactions at the third order of perturbation theory. Going beyond the existing second-order perturbation treatment allows irreducible tensor operators of rank six (or corresponding Stevens operator equivalents) in the GSH to be obtained. Such terms were found to be of crucial importance for the fitting of high-field EPR spectra of a number of single-molecule magnets (SMMs). Also, recent magnetization measurements on trigonal and tetragonal SMMs have found the inclusion of such high-rank axial and transverse terms to be necessary to account for experimental data in terms of giant-spin models. While mixing of spin multiplets by local zero-field splitting interactions was identified as the major origin of these contributions to the GSH, a direct and efficient microscopic explanation had been lacking. The third-order approach developed in this work is used to illustrate the mapping of an MSH onto a GSH for an S=6 trigonal Fe3 Cr complex that was recently investigated by high-field EPR spectroscopy. Comparisons between MSH and GSH consider the simulation of EPR data with both Hamiltonians, as well as locations of diabolical points (conical intersections) in magnetic-field space. The results question the ability of present high-field EPR techniques to determine high-rank zero-field splitting terms uniquely, and lead to a revision of the experimental GSH parameters of the Fe3 Cr SMM. Indeed, a bidirectional mapping between MSH and GSH effectively constrains the number of free parameters in the GSH. This notion may in the future facilitate spectral fitting for highly symmetric SMMs.

Exact Mapping from Many‐Spin Hamiltonians to Giant‐Spin Hamiltonians

“…Befored iscussing the results of exact spin projection we briefly digress to shed some light on the symmetryp roperties of anisotropicm any-spin models, which have rarely been addressed explicitly, [48,88] but weree mployed at least in aq ualitative way (see, for example, Refs. [43], [45], [89]).…”

“…[48], [52], [73]): for the Fe 3 Cr SMM [93] (S = 6),S; M E transforms like the product S; M ji  y A 2 ,w here y A 2 is an arbitrary spatialf unctiont ransforming according to irrep A 2 in the D 3 symmetry group of the MSH. [43] Unless the magnetic field is applieda long as ymmetry axis of the molecule,o rp erpendicular to am irror plane, the Zeeman term completelyb reaks SPS (C i symmetry,i fp resent, Figure 1. Classical magnetic anisotropy energy surfaces, caused by localZFS interactions (in arbitrary units)f or an MSH with D 2d SPS symmetry( rhombic ZFS tensors, e = d/6, d < 0).…”

“…The absence of the latter type of mixing is impliedb yi ncluding only linear field-dependent terms when fitting experimental spectra;a second-rank spin term that dependsq uadratically on field strength B,s ymbolically S 2 B 2 , [20] does not occur through third order when local g-tensorsa re isotropic. [43] If necessary, terms of this kind coulds till be determined by performing exact spin projection in zero-field, at dB,a nd at 2dB (whereg enerally differento rientationso ft he small magnetic field dB would need to be sampled to determine the tensor quantities that parameterize the S 2 B 2 interaction term).…”

“…The numerical search for DPs between energy levels n jiand n þ 1 ji was conducted by followingt he Berry curvature of level n ji . [43,103] As of yet, BPI effects have not been observed for Ni 4 ,b ut the possibility of detection of certain field-dependent tunnelq uenchings (e.g., D AE2 )b yl ow-field EPR experiments has been indicated recently. [56] In full analogy to tetragonal Mn 12 , [45] the GSH used to fit experimental data [cf.…”

“…[36]- [42]). General analytical equations to take S-mixing into account for spin projection have been derived in the second [20] and third order [43] of perturbation theory with respectt oa nisotropic zero-field MSH terms, based on the van Vleck transformation; [44] third-order spin projection was recently applied to the famous Mn 12 SMM. [45] However,w hile computationally attractive, perturbation theory up to order n yields zero-field GSH terms only up to rank k = 2n, [46] and it is difficult to judge in advance if ag iven order can properly account for observations.…”