Structural and magnetic properties of the $R{\text{Cr}}_{2}{\text{Si}}_{2}\text{C}$ compounds ($R=\text{Y}$, La--Sm, Gd--Er) with the carbon-filled ${\text{CeMg}}_{2}{\text{Si}}_{2}$-type structure (space group $P4/mmm$) have been studied by means of powder and single-crystal x-ray diffractions, magnetic measurements, and neutron powder diffraction. The compounds with $R=\text{Pr}$, Nd, Gd--Dy order ferromagnetically at low temperature $({T}_{C}\ensuremath{\le}35\text{ }\text{K})$, whereas those with $R=\text{Y}$, La, Ce, and Sm do not exhibit any magnetic ordering down to 2 K. On the contrary, the ``parent'' ${\text{ThCr}}_{2}{\text{Si}}_{2}$-type (space group $I4/mmm$) $R{\text{Cr}}_{2}{\text{Si}}_{2}$ compounds ($R=\text{Y}$, Sm, Tb--Lu) exhibit strong antiferromagnetic properties $({T}_{N}>600\text{ }\text{K})$ linked to a large magnetic moment $(\ensuremath{\sim}1.9\text{ }{\ensuremath{\mu}}_{B})$ on the Cr sublattice. In order to better understand the role played by carbon on structural and magnetic properties in these systems, ab initio electronic structure calculations of $R{\text{Cr}}_{2}{\text{Si}}_{2}$ and $R{\text{Cr}}_{2}{\text{Si}}_{2}\text{C}$ compounds with $R=\text{Y}$ and La have been performed, using the Korringa-Kohn-Rostoker (KKR) method, in the ${\text{ThCr}}_{2}{\text{Si}}_{2}$- and ${\text{CeMg}}_{2}{\text{Si}}_{2}$-types as well as their corresponding C-``filled'' types, with C atoms located in Cr (001) planes. This study has allowed elucidating the particular role of carbon in the breakdown of the local magnetic moment on the Cr sublattice. KKR calculations clearly evidence that strong hybridization between $d$ states on Cr atoms and $p$ states on C atoms leads to a marked decrease in the density of states in the vicinity of ${E}_{F}$ (well below the Stoner limit). Since similar electronic structure modifications are observed whatever the crystal structure, we tentatively conclude that the Cr-C interactions are more predominant in the disappearance of magnetism in $R{\text{Cr}}_{2}{\text{Si}}_{2}\text{C}$ than the effect of the rearrangement of Si and $R$ atomic planes when passing from the ${\text{ThCr}}_{2}{\text{Si}}_{2}$ type to the ${\text{CeMg}}_{2}{\text{Si}}_{2}$-type structure. Moreover, KKR calculations with the coherent-potential approximation undertaken for the ``model'' $R{\text{Cr}}_{2}{\text{Si}}_{2}{\text{C}}_{x}$ system $(0\ensuremath{\le}x\ensuremath{\le}1)$ well illustrate the evolution of the electronic structure and magnetic properties upon increasing carbon concentration.
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