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

For equal-spin $s_1$ tetramer single molecule magnets with ionic point groups $g=T_d, D_{4h}, D_{2d}, C_{4h}, C_{4v}$, and $S_4$, we write the group-invariant single-ion, Dzaloshinskii-Moriya, and symmetric anisotropic near-neighbor and next-nearest-neighbor exchange Hamiltonians using the respective local axial and azimuthal vector groups. The anisotropic exchange renormalized near-neighbor and next-nearest-neighbor isotropic exchange interactions are $\tilde{J}_g$ and $\tilde{J}_g'$. Using our exact, compact forms for the single-ion matrix elements, we evaluate the eigenstates of the full Hamiltonian to first order in the various interactions. There are two types of ferromagnetic (FM) and antiferromagnetic (AFM) tetramers. In Type I, $\tilde{J}_g'-\tilde{J}_g>0$, the tetramers act as two dimers with the maximal pair quantum numbers $s_{13}=s_{24}=2s_1$ at low temperature. Type II tetramers with $\tilde{J}_g'-\tilde{J}_g<0$ are frustrated, with minimal values of the pair quantum numbers $s_{13}$ and $s_{24}$ at low temperature. For both Type I and Type II AFM tetramers, we evaluate the first-order level-crossing inductions analytically for arbitary $s_1$, and illustrate the results for $s_1=1/2, 1, 3/2$. Accurate Hartree expressions for the thermodynamics, electron paramagnetic resonance (EPR) absorption and inelastic neutron scattering cross-section are given. A procedure to extract the effective microscopic parameters for Types I and II tetramers using EPR is given.