New natural spin-1/2 kagom\'{e} systems --- kapellasite Cu$_3$Zn(OH)$_6$Cl$_2$ and haydeeite Cu$_3$Mg(OH)$_6$Cl$_2$

New natural spin-1/2 systems with kagom\'{e} layers --- kapellasite Cu$_3$Zn(OH)$_6$Cl$_2$ and haydeeite Cu$_3$Mg(OH)$_6$Cl$_2$ --- are studied by full potential density functional calculations using the fplo6.00-24 code. The band structure, obtained by a paramagnetic calculation, was used to solve a tight-binding model. The transfer integrals were mapped subsequently to a Hubbard model and to a Heisenberg model, giving an estimate for the antiferromagnetic (AF) exchange. The total exchange, containing AF and ferromagnetic (FM) parts, was derived from LSDA + \textsl{U} supercell calculations. As the main result, we find that in both compounds only two exchange integrals are relevant: the nearest neighbour exchange \textsl{J}$_1$ and the interaction \textsl{J}$_{\mathrm{d}}$ along the diagonals of the Cu$^{2+}$ hexagons. Surprisingly, the size of these integrals depends strongly on the O---H bond length which was therefor optimized with respect to the total energy, resulting in about 1 \AA for both compounds. Using the optimized O---H bond length, we find \textsl{J}$_{1}>\sl{J}_{\mathrm{d}}$ in kapellasite and \textsl{J}$_{1}\sim\textsl{J}_{\mathrm{d}}$ in haydeeite. According to our results, kapellasite can be described as a modified kagom\'{e} lattice, while interpenetrating chains should be considered for haydeeite. Our results should encourage new experimental studies of these interesting materials.