  Here we have performed detailed first principles calculations for the
electronic structure and magnetic properties of Ba$_6$Cr$_2$S$_{10}$ to study
the origin of the anti-ferromagnetic exchange interaction between spins on Cr
ions for the spin-chain compound Ba$_6$Cr$_2$S$_{10}$ synthesised recently.
Most importantly, we have found the co-existence of a charge density wave phase
along one line and an anti-ferromagnetic spin chain along another. The
dimerization of sulfur atoms loosely bonded with Ba atoms drives the system
into an insulating state owing to the formation of charge density wave.
Meanwhile, the small size of the effective Hubbard $U$ parameter ($\sim 0.5$
eV) due to electrostatic screening mainly accounts for the anti-ferromagnetic
ground state. This co-existence equips us with a platform to tune the charge
and spin degrees of freedom independently. Moreover, there exists a
next-nearest-neighbouring anti-ferromagnetic interaction along the chain, which
could bring forward spin frustration and hence quantum spin liquid.
