Lanthanide-based single-ion magnetic molecules can have large magnetic hyperfine interactions as well as large magnetic anisotropy. Recent experimental studies reported tunability of these properties by changes of chemical environments or by application of external stimuli for device applications. In order to provide insight onto the origin and mechanism of such tunability, here we investigate the magnetic hyperfine and nuclear quadrupole interactions for $^{159}$Tb nucleus in TbPc$_2$ (Pc=phthalocyanine) single-molecule magnets using multireference ab-initio methods including spin-orbit interaction. Since the electronic ground and first-excited (quasi)doublets are well separated in energy, the microscopic Hamiltonian can be mapped onto an effective Hamiltonian with an electronic pseudo-spin $S=1/2$. From the ab-initio-calculated parameters, we find that the magnetic hyperfine coupling is dominated by the interaction of the Tb nuclear spin with electronic orbital angular momentum. The asymmetric $4f$-like electronic charge distribution leads to a strong nuclear quadrupole interaction with significant non-axial terms for the molecule with low symmetry. The ab-initio calculated electronic-nuclear spectrum including the magnetic hyperfine and quadrupole interactions is in excellent agreement with experiment. We further find that the non-axial quadrupole interactions significantly influence the avoided level crossings in magnetization dynamics and that the molecular distortions affect mostly the Fermi contact terms as well as the non-axial quadrupole interactions.
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