Towards Understanding Prolate 4$f$ Monomers: Numerical Predictions and Experimental Validation of Electronic Properties and Slow Relaxation in a Muffin-shaped Er$^\mathrm{III}$ Complex

We report the synthesis, crystal structure and magnetic properties of the muffin-shaped complex [Er(PPTMP)$_2$(H$_2$O)][OTf]$_3$ (PPTMP = (4-(6-(1,10-phenanthrolin-2-yl)pyridin-2-yl)-1H-1,2,3-triazol-1-yl)methyl pivalate) (\one). Complex \one\ is shown to exhibit field-induced slow relaxation of the magnetisation at $B = 0.1$~T via two distinct relaxation paths. Using tunable high-frequency/high-field electron paramagnetic resonance spectroscopy we experimentally determine the effective $g$-factors and zero field splittings of the two energetically lowest Kramers doublets (KD). Our data reveal that the distorted muffin-shaped ligand field favours an $m \simeq \pm 9/2$ magnetic ground state, while the main contribution to the first excited KD at $\Delta_{1 \rightarrow 2} = 780(5)$~GHz is suggested to be $m \simeq \pm 5/2$. The ground state $g$-tensor has generally axial form but hosts significant transversal components, which we conclude to be the source of SMM-silent behaviour in zero field. Our findings are backed up by \textit{ab-initio} spin-orbit configuration interaction calculations showing excellent agreement with the experimental data and in particular highlight that the counterions should be included in the numerical modelling of the crystalline structure.