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

Abstract

Thermodynamic and spectroscopic data of exchange-coupled molecular spin clusters (e.g. single-molecule magnets) are routinely interpreted in terms of two different models: the many-spin Hamiltonian (MSH) explicitly considers couplings between individual spin centers, while the giant-spin Hamiltonian (GSH) treats the system as a single collective spin. When isotropic exchange coupling is weak, the physical compatibility between both spin Hamiltonian models becomes a serious concern, due to mixing of spin multiplets by local zero-field splitting (ZFS) interactions ('S-mixing'). Until now, this effect, which makes the mapping MSH→GSH ('spin projection') non-trivial, had only been treated perturbationally (up to third order), with obvious limitations. Here, based on exact diagonalization of the MSH, canonical effective Hamiltonian theory is applied to construct a GSH that exactly matches the energies of the relevant (2S+1) states comprising an effective spin multiplet. For comparison, a recently developed strategy for the unique derivation of effective ('pseudospin') Hamiltonians, now routinely employed in ab initio calculations of mononuclear systems, is adapted to the problem of spin projection. Expansion of the zero-field Hamiltonian and the magnetic moment in terms of irreducible tensor operators (or Stevens operators) yields terms of all ranks k (up to k=2S) in the effective spin. Calculations employing published MSH parameters illustrate exact spin projection for the well-investigated [Ni(hmp)(dmb)Cl] ('Ni ') single-molecule magnet, which displays weak isotropic exchange (dmb=3,3-dimethyl-1-butanol, hmp is the anion of 2-hydroxymethylpyridine). The performance of the resulting GSH in finite field is assessed in terms of EPR resonances and diabolical points. The large tunnel splitting in the M=± 4 ground doublet of the S=4 multiplet, responsible for fast tunneling in Ni , is attributed to a Stevens operator with eightfold rotational symmetry, marking the first quantification of a k=8 term in a spin cluster. The unique and exact mapping MSH→GSH should be of general importance for weakly-coupled systems; it represents a mandatory ultimate step for comparing theoretical predictions (e.g. from quantum-chemical calculations) to ZFS, hyperfine or g-tensors from spectral fittings.

Analyzing Anisotropic Exchange in a Pentanuclear Os2Ni3 Complex

“…Effective Hamiltonian theory is the basis for connecting microscopic (ab initio) and phenomenological (spin) Hamiltonians. [ 33 , 34 , 35 , 36 ] Since CASOCI is not a perturbative method but diagonalises the CI matrix, the model space is formed by selecting, when the CASOCI calculation has completed, some states well separated from the others. The effective (giant) spin is then implicitly determined by the number of model space functions which amounts to .…”

“…For example, the Zeeman part of the spin Hamiltonian can only have matrix elements between model space spin states where the values of M differ by 1 at most, while it is possible that the ab initio model space functions have a non‐zero matrix elements between corresponding model space functions where M differs by more than one. An exact match can always be found if one includes higher‐order spin operators in the spin Hamiltonian, [35] but this leads to a spin Hamiltonian with a very large set of parameters. Instead, our approach is to perform a least‐squares procedure to define the lowest‐order spin Hamiltonian (Eq.…”

Propeller‐Shaped Fe4 and Fe3M Molecular Nanomagnets: A Journey from Crystals to Addressable Single Molecules

“…Allowing for the presence of an applied magnetic field B , the complete SH for a Fe 4 complex can be written as Equation : where both i and j run from 1 to 4 and an isotropic g factor has been assumed for simplicity in the Zeeman term [this assumption is totally realistic for high‐spin iron(III) ions, whose g factors are usually quasi‐isotropic and very close to 2.00]. Equation (2) is called a “ many‐spin ” or “ multi‐spin ” (MS) Hamiltonian as it explicitly considers individual spin centers . As long as superexchange couplings dominate over magnetic anisotropies, as is the case in Fe 4 complexes, the spin levels pertaining to the ground total‐spin state are accurately captured by a so‐called “ giant‐spin ” Hamiltonian, as in Equation : …”

“…A number of experimental, , and theoretical, data on Fe 4 complexes and their heterometallic analogues confirm such a scenario. Information on the arrangement of local anisotropies can be extracted by analyzing departures from Equation (3) caused by residual mixing between total‐spin multiplets . These deviations occur in the form of high‐order anisotropy components [see second term in Equation (10)] and were clearly detected in an angle‐resolved EPR study on Cr 3+ ‐centered complex 3 Cr,Me .…”

“…level anticrossings), thereby triggering spin transitions between opposite sides of the barrier. They can consist in: (a) rhombic anisotropy components described by the ZFS parameter E , which is expected to be nonzero whenever threefold symmetry is not strictly obeyed; (b) higher‐order anisotropy components described by extended Stevens operators Ô k q and the associated B k q parameters, which may appear even in threefold‐symmetric molecules due to residual mixing between total‐spin multiplets;, (c) transverse Zeeman terms produced by misaligned crystalline domains or dislocations, intermolecular dipolar fields in the crystal, or intramolecular hyperfine interactions with atomic nuclei. These terms are included in Hamiltonian Ĥ ′′ of Equation , in the same order as they have been listed: …”