Electronic structure and magnetism in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>P</mml:mi><mml:mn>4</mml:mn><mml:mo>/</mml:mo><mml:mi>n</mml:mi><mml:mi>m</mml:mi><mml:mi>m</mml:mi></mml:mrow><mml:mo> </mml:mo><mml:msub><mml:mi>KCoO</mml:mi><mml:mn>2</mml:mn></mml:msub></mml:math>

${\mathrm{KCoO}}_{2}$ has been found in 1975 to exist in a unique structure with $P4/nmm$ space group featuring Co atoms in a square pyramidal coordination. The Co atoms in the plane are linked by O in a square arrangement reminiscent of the cuprates. However, its electronic structure has not been studied until now. Unlike Co atoms in ${\mathrm{LiCoO}}_{2}$ and ${\mathrm{NaCoO}}_{2}$, which are in octahedral coordination and are nonmagnetic band structure insulators, the unusual coordination of ${d}^{6}\phantom{\rule{4pt}{0ex}}{\mathrm{Co}}^{3+}$ in ${\mathrm{KCoO}}_{2}$ is here shown to lead to a magnetic stabilization of an insulating structure with high magnetic moments of $4{\ensuremath{\mu}}_{B}$ per Co. The electronic band structure is calculated using the quasiparticle self-consistent ($\mathrm{QS})GW$ method and the basic formation of magnetic moments is explained in terms of the orbital decomposition of the bands. The optical dielectric function is calculated using the Bethe-Salpeter equation including only transitions between equal spin bands. The magnetic moments are shown to prefer an antiferromagnetic ordering along the [110] direction. Exchange interactions are calculated from the transverse spin susceptibility and a rigid spin approximation. The N\'eel temperature is estimated using the mean-field and Tyablikov methods and found to be between $\ensuremath{\sim}100$ and $\ensuremath{\sim}250\phantom{\rule{0.16em}{0ex}}\mathrm{K}$. The band structure in the AFM ordering can be related to the FM ordering by band folding effects. The optical spectra are similar in both structures and show evidence of excitonic features below the quasiparticle gap of $\ensuremath{\sim}4\phantom{\rule{0.16em}{0ex}}\mathrm{eV}$.

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