Microscopic magnetic modeling for the spin-$\frac12$ kagome compound $[$NH$_4]_2[$C$_7$H$_{14}$N$][$V$_7$O$_6$F$_{18}]$

In the recently synthesised compound $[$NH$_4]_2[$C$_7$H$_{14}$N$][$V$_7$O$_6$F$_{18}]$, magnetic $S$=$\frac12$ V$^{4+}$ atoms form an ideal kagome lattice $[$1$]$. Very recent $\mu$SR studies indicate the emergence of a gapless spin liquid state $[$2$]$. Using density functional theory calculations, we address the microscopic magnetic model of this interesting compound. We show that its peculiar symmetry gives rise to two inequivalent nearest-neighbor couplings. The behavior of the resulting spin model is studied using exact diagonalization and compared to the experiments. \\ $[$1$]$ F.H.Aidoudi~\textsl{et~al.}, Nature~Chem.~\textbf{3}, 810 (2011). \\ $[$2$]$ L.Clark~\textsl{et~al.}, Phys.~Rev.~Lett.~\textbf{110}, 207208 (2013).