Single-ion and exchange anisotropy in high-symmetry tetramer single molecule magnets

We study the effects of single-ion and both symmetric and antisymmetric exchange anisotropy in equal-spin $s_1$ tetramer single molecule magnets exhibiting the molecular group symmetries $g=C_{4h}$, $D_{4h}$, $C_{4v}$, $S_4$, $D_{2d}$, and $T_d$. The near-neighbor and next-nearest-neighbor isotropic exchange interactions are $J$ and $J'$, respectively. From the vector basis used to diagonalize the general quadratic spin-spin interaction Hamiltonian ${\cal H}$ for each site and site pairs, we impose the symmetries characteristic of each $g$ upon ${\cal H}$. Using our exact, compact forms for the four-spin single-ion matrix elements, we calculate the eigenstate energies to first order in the anisotropy interactions. Type I tetramers with $J'-J>0$ act as two dimers with maximal pair quantum numbers $s_{13}=s_{24}=2s_1$ at low temperature $T$. Type II tetramers with $J'-J<0$ are frustrated, with minimal low-$T$ pair quantum numbers. For both Type-I and Type-II antiferromagnetic tetramers, we calculate the first-order level-crossing inductions analytically. Accurate Hartree expressions for the thermodynamics, electron paramagnetic resonance (EPR) and inelastic neutron scattering cross-section are given. An EPR procedure to extract the effective microscopic parameters is provided.