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Effect of the Wyckoff position of K atom on the crystal structure and electronic properties of KFe$_2$Se$_2$ compound

By the first-principle electronic structure calculations, we study the effect of Wyckoff position of K atom on the crystal and electronic structures of KFe$_2$Se$_2$ compound.When the K atoms take up the Wyckoff position $2a$, $2b$ or $4c$ (the related structure of KFe$_2$Se$_2$ is called as Struc-2a, Struc-2b or Struc-4c), the calculated lattice constant $c$ is in the range of 13.5$\sim$14.5 Å, 15.5$\sim$16.7 Å or 18.6$\sim$19.1 Å.Three concentric cylinder-like Fermi surfaces emerge around $Γ$-Z in Brillouin Zone for the Struc-2b in nonmagnetic state, different from the Struc-2a and Struc-4c.The Fe-Se-Fe angle is 107.8$^\circ$, 108.8$^\circ$ or 110.7$^\circ$ in the collinear anti-ferromagnetic state and the superexchange interaction $J_2$ between two next neighbor Fe moments is 13.08 meV/S$^2$, 20.75 meV/S$^2$ or 11.86 meV/S$^2$ in Struct-2a, Struc-2b or Struc-4c structure respectively.The Struc-2b and Struc-4c have a good correspondence to the newly discovered superconducting phases with T$_c$=40 K and T$_c$=30 K in KFe$_2$Se$_2$.Our findings provide a reasonable approach to understand the existence of multiple superconducting phases in alkali metal intercalated FeSe superconductor.