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Nearly compensated exchange in the dimer compound callaghanite Cu$_2$Mg$_2$(CO$_3$)(OH)$_6$$\cdot$2H$_2$O

A combined theoretical and experimental study of the natural Cu(2+)-mineral callaghanite is presented. Its crystal structure features well separated Cu$_2$(OH)$_6$ structural dimers with weakly bonded carbonate groups and water molecules in between. Susceptibility, field-dependent magnetization and specific heat measurements reveal a compound with a small spin gap of about 7 K. The observed magnetic properties are well described by a model of isolated antiferromagnetic spin dimers. Possible ferromagnetic interactions between the dimers amount to -1.5 K, at most. Different flavors of electronic structure calculations have been employed to locate the magnetic dimers in the crystal structure, i.e., to determine whether they coincide with the structural dimers or not. Calculations of the coupling between the structural dimers clearly show that magnetic and structural dimers are the same. For the intradimer coupling, however, the computational results confirmed a coupling strength close to zero but the sign of the coupling could not be determined unambiguously. Based on this finding, we then discuss how the reliability of the numerical methods depends on the characteristics of exchange pathways and on structural features of the compound in general. Eventually, we try to provide a minimum coupling strength that is needed for a reliable computational description.