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We present the \emph{ab initio} calculation of the electronic structure andmagnetic properties of Bi$_2$Fe$_4$O$_9$. This compound crystallizes in theorthorhombic crystal structure with the Fe$^{3+}$ ions forming the Cairopentagonal lattice implying strong geometric frustration. The neutrondiffraction measurements reveal nearly orthogonal magnetic configuration, whichat first sight is rather unexpected since it does not minimize the total energyof the pair of magnetic ions coupled by the Heisenberg exchange interaction.Here we calculate the electronic structure and exchange integrals of Bi2Fe4O9within the LSDA+U method. We obtain three different in-plane (J3=36 K, J4=73 K,J5=23 K) and two interplane (J1=10 K, J2=12 K) exchange parameters. The derivedset of exchange integrals shows that the realistic description of Bi2Fe4O9needs a more complicated model than the ideal Cairo pentagonal lattice withonly two exchange parameters in the plane. However, if one takes into accountonly two largest exchange integrals, then according to the ratio x\equivJ3/J4=0.49<\sqrt{2} (a critical parameter for the ideal Cairo pentagonallattice, see. Ref.~1) the ground state should be the orthogonal magneticconfiguration in agreement with experiment. The microscopic origin of differentexchange interactions is also discussed.

Ab initio investigation of the exchange interactions in Bi$_2$Fe$_4$O$_9$: The Cairo pentagonal lattice compound