Possible Violation of the Spin-Hamiltonian Concept in Interpreting the Zero-Field Splitting

Abstract

The majority of experimental data in electron spin resonance and molecular magnetism are interpreted in terms of the spin-Hamiltonian (SH) formalism. However, this is an approximate theory that requires a proper testing. In the older variant, the multielectron terms are used as a basis in which the D-tensor components are evaluated by employing the second-order perturbation theory (PT) for nondegenerate states; here, the spin–orbit interaction expressed via the spin–orbit splitting parameter λ serves for the perturbation. The model space is restricted only to the fictitious spin functions |S, M⟩. In the case of the orbital (quasi) degeneracy of the ground term, the PT tends to diverge and the subtracted D, E, and g parameters are false. In the second variant working in the “complete active space” (CAS), the spin–orbit coupling operator is involved by the variation method resulting in the spin–orbit multiplets (energies and eigenvectors) The multiplets can be evaluated either by applying ab initio CASSCF + NEVPT2 + SOC calculations or by using semiempirical generalized crystal-field theory (with the one-electron SOC operator depending upon ξ). The resulting states can be projected onto the subspace of the spin-only kets in the way that the eigenvalues stay invariant. Such an effective Hamiltonian matrix can be reconstructed using six independent components of the symmetric D-tensor from which the D and E values are obtained by solving linear equations. The eigenvectors of the spin–orbit multiplets in the CAS allow determining the dominating composition of the spin projectioncumulative weights of |±M⟩. These are conceptually different from those generated by the SH alone. It is shown that in some cases, the SH theory works satisfactorily for a series of transition-metal complexes; however, sometimes it fails. The ab initio calculations on the SH parameters are compared with the approximate generalized crystal-field theory conducted at the experimental geometry of the chromophore. In total, 12 metal complexes have been analyzed. One of the criteria that assesses the validity of SH is the projection norm N for spin multiplets (this has not to be far from 1). Another criterion is the gap in the spectrum of the spin–orbit multiplets that separates the hypothetical (fictitious) spin-only manifold from the rest of the states.

Limitations on the D-Parameter in Ni(II) Complexes

“…There is a very large negative contribution from the first excitation energy Δ 1 = 3363 cm –1 , D * = −109 cm –1 , that brings a question about the effect of the quasi-degeneracy on the symmetry descent T d → D 2 d → C 2 v . The analysis of the norm of the projected states N 1 = 0.96, N 2 = 0.97, and N 3 = 0.98 confirms that the spin-Hamiltonian approach is approved …”

“…These are useful in generating magnetization and magnetic susceptibility. Details on the implementation of this method along with evaluation of the zero-field splitting parameters D and E can be found elsewhere …”

Large Magnetic Anisotropy in Mono- and Binuclear cobalt(II) Complexes: The Role of the Distortion of the Coordination Sphere in Validity of the Spin-Hamiltonian Formalism

“…Such a parameter accounts for the separation of the model space defined by the SH from the rest of the energy spectrum. The value of ∑ = 313 ≫ 50 for 1 indicates that the SH formalism works in the present case. , The spin–orbit multiplets can be analyzed in more detail to get their compositions from the pure spin functions w 1 |±1/2⟩ + w 2 |±3/2⟩ by using the weights calculated from the corresponding transformation matrix.…”

Easy-axis magnetic anisotropy in tetragonally elongated cobalt(ii) complexes beyond the spin-Hamiltonian formalism

“…It is to be noted that, the ground multi-electronic term (at the zero energy) 4 Δ 0 = 0 is split owing to the spin-orbit coupling to δ 1,2 and δ 3,4 multiplets; the first excited term 4 Δ 1 is split into δ 5,6 and δ 7,8 so that X = δ 5,6 − δ 3,4 indicates whether the projected 4 × 4 space referring to the fictitious spin manifolds is well separated from the rest of the energy spectrum. 33 The greater the score, the better the quality of the spin Hamiltonian formalism. In the present case S(3a) = 91 approves the utilization of the spin Hamiltonian formalism.…”