Ferromagnetic Interactions in Non-Kekulé Polymers. II

Abstract

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...

Magnetic Ordering in Porous Graphenes

“…This coincidence is due to the 1,3-linkage of m -phenylene moieties. As is well-known, m -phenylene skeletons serve as a ferromagnetic coupler through the nonbonding character of the frontier level. , In two-dimensional systems such as 9 – 12 , the 1,3,5-phenylene unit serves as a nonbonding coupler of the fragmental amplitude patterns, similar to the case in Mataga polymer . Though the hexagon-pored graphenes are not non-Kekulé molecules, the conservation of amplitude patterns in each edge also comes from the nonbonding interactions between each butadiene or divinylpolyphenylene unit, as seen from the nodal points.…”

“…Dimer models in the hexagon-pored graphene are convenient for description of the local spin preference of oligomers cut along the peculiar axis. However, for prediction of global spin preference, Wannier analysis is a more powerful tool to grasp the whole ferromagnetic interactions of the polymeric systems . It has been shown that one-dimensional porous graphene ribbons always have flat bands at the frontier levels, regardless of the number of porous ladders .…”

“…Though the neutral state of this porous graphene is a closed-shell system, we can expect flat-band ferromagnetism in the cationic or anionic states, if it is properly oxidized or reduced. This is reminiscent of flat-band ferromagnetism of non-Kekulé-type polymers. − In general, porous graphenes with large pore sizes are probably not planar by themselves. Nevertheless, as Bieri et al showed, some porous graphenes can be grown on metal surfaces by aryl–aryl coupling reactions, conserving the skeletal planarity .…”

Band structures of defective graphenes

“…The mechanism for the ferromagnetic interactions resembles those of organic ferromagnets such as non-Kekulé polymers in which the direct exchange interactions between the flat-band electrons mainly contribute to the stability of magnetic states. This situation has been well established by the Wannier functions localized at each unit cell [12,13]. That is, in organic polyradicals, itinerant characters of the non-bonding electrons cause direct exchange interactions between the adjacent cells.…”

Band structures of porous graphenes

“…The corresponding eigenvalues are a ± 0.618b. Although this is a closed-shell system, the interesting degeneracy in the frontier bands is akin to non-Kekulé extended systems [17,18], in which non-bonding crystal orbitals are degenerate. We note that the dispersion between X and M is completely zero for all the bands.…”