Ferromagnetic interactions in two-dimensional non-Kekulé polymers were theoretically analyzed under periodic boundary conditions. We constructed two-dimensional Wannier functions in the non-bonding crystal orbitals (NBCOs), and PNBCO (product of NBCO) were their products. Ferromagnetic interactions were attributed to anti-parallel-spin instabilities in PNBCO, similar to one-dimensional non-Kekulé polymers. The instabilities consisted of on-site terms and through-space terms. The former resulted from the squared amplitude of the same atomic site in PNBCO, and the latter resulted from antibonding-through-space interactions between second-nearest-neighbor carbon atomic sites in PNBCO.There has been increasing interest in organic ferromagnets. Mataga theoretically proposed possible organic ferromagnets based on molecular orbital methods. 2 The theory is based on topological degeneracy 3 of non-bonding molecular orbitals (NBMOs) characteristic of non-Kekulé systems. In particular, m-phenylene (1) in Figure 1 has been regarded as a robust ferromagnetic coupler in organic ferromagnets.2,3 The carbon atomic sites can be divided into two groups, that is, so-called starred atoms and unstarred atoms. 4 They can be chosen so as to be not adjacent each other. Then, the spin-quantum number of non-Kekulé systems predicted by Hund's rule becomes ðN c À 2TÞ=2, where N c and T are the number of carbon atoms and classical double bonds, respectively. 4 Ovchinnikov also proposed possible organic ferromagnets based on non-Kekulé polymers in view of valence bond theory. 5 The spin-quantum number of non-Kekulé systems predicted by Ovchinnikov is jn A À n Aþ j=2, where n A and n Aþ are identical to the number of starred and unstarred atoms, respectively. 5 This theory has been related to the spin-polarization concept in non-Kekulé systems.6 Tyutyulkov and co-workers also have designed many non-Kekulé polymers based on band theory. 7-9 A typical one-dimensional non-Kekulé polymer, poly(m-phenylene) (2), is shown in Figure 1. High-spin states of polymers and oligomers based on m-phenylene skeletons have been theoretically well established. 7,8,[10][11][12] Thus, non-Kekulé chain polymers, including poly(m-phenylene), have been promising candidates for organic ferromagnets. Recent, heteroatom-containing systems have also been investigated due to their stabilities. [13][14][15][16] Non-Kekulé chain polymers are extended systems of nonKekulé molecules. High-spin non-Kekulé molecules have ''nondisjoint'' NBMOs 17,18 which span common atoms. The ferromagnetic spin alignment in non-Kekulé systems essentially results from an exchange integral between nondisjoint NBMOs.19-21 Through-space interactions in nondisjoint systems also have been pointed out as another ferromagnetic interaction. 22 Similarly, possible organic ferromagnets based on non-Kekulé polymers have infinite NBMOs to form nonbonding crystal orbitals (NBCOs) which are nondisjoint. Recently, ferromagnetic interactions in non-Kekulé systems have been interpreted in view of anti-paral...