  In the isostructural oxides Ca3CoMO6 (M = Co, Rh, Ir), the CoMO6 chains made
up of face-sharing CoO6 trigonal prisms and MO6 octahedra are separated by Ca
atoms. We analyzed the magnetic and electronic properties of these oxides on
the basis of density functional theory calculations including on-site repulsion
and spin-orbit coupling, and examined the essential one-electron pictures
hidden behind results of these calculations. Our analysis reveals an intimate
interplay between Jahn-Teller instability, uniaxial magnetism, spin
arrangement, metal-metal interaction, and spin-orbit coupling in governing the
magnetic and electronic properties of these oxides. These oxides undergo a
Jahn-Teller distortion but their distortions are weak, so that their
trigonal-prism Con+ (n = 2, 3) ions still give rise to strong easy-axis
anisotropy along the chain direction. As for the d-state split pattern of these
ions, the electronic and magnetic properties of Ca3CoMO6 (M = Co, Rh, Ir) are
consistent with d0 < (d2, d-2) < (d1, d-1), but not with (d2, d-2) < d0 < (d1,
d-1). The trigonal-prism Co3+ ion in Ca3Co2O6 has the L = 2 configuration
(d0)1(d2, d-2)3(d1, d-1)2 because of the metal-metal interaction between
adjacent Co3+ ions in each Co2O6 chain, which is mediated by their z2 orbitals,
and the spin-orbit coupling of the trigonal-prism Co3+ ion. The spins in each
CoMO6 chain of Ca3CoMO6 prefer the ferromagnetic arrangement for M = Co and Rh,
but the antiferromagnetic arrangement for M = Ir. The octahedral M4+ ion of
Ca3CoMO6 has the (1a)1(1e)4 configuration for M = Rh but the (1a)2(1e)3
configuration for M = Ir, which arises from the difference in the spin-orbit
coupling of the M4+ ions and the Co...M metal-metal interactions.
