2 of size 10 ˚ A × 16 ˚ A with edges possessing sp 3 hybridization are found to be characterized by degenerate bands and by a band gap of 7.51 eV (the highest and lowest molecular orbitals are HOMO=-6.09 eV and LUMO=1.42 eV, re- spectively), as shown in Fig. (a). For comparison we have examined flakes of size 18 ˚ A × 16 ˚ A and found a decrease in gap to 7.15 eV (HOMO=-5.87 eV and LUMO=1.27 eV). The edges in the sp 2 hybridization, for which the edge carbon atoms are terminated by a single hydro- gen, possess the localized states. In this case the or- bital degeneracy is lifted and the gap decreases to 3.04 eV (HOMO= − 4 . 64 eV and LUMO= − 1 . 58 eV). In the available experiment [16, 17], only a transformation of graphene from the conductor to an insulator due to its hydrogenation was reported, but the size of the gap and the type of edge hybridization were not indicated. Since we found that the gap is sensitive to edge hybridization and can be increased from 3.04 eV to a maximum of 7.51 eV by its transformation from sp 2 to sp 3 type, this is- sue should be the top priority for further experimental investigations. A single H-vacancy defect in the graphane lattice gen- erates an unsaturated dangling bond on the carbon atom – π unpaired electron (perpendicular p$_{z}$ orbital). More- over, bonding of the carbon atom carrying the defect with its neighbors is changed from the sp 3 hybridization to sp $^{2}$, thus providing a modification of the bond length from 1.55 ˚ A to 1.52 ˚ A. The perpendicular p$_{z}$ orbital and the C-C bonds possessing sp 2 hybridization participate in the formation of the localized state characterized by an unpaired spin (see the spin density in Fig. ). There- fore, this localized state is spin-polarized and generates a defect level in the band gap (see the bands in Fig. (b)). For the α -spin state, the defect level ( π ) appears close to the valence band (HOMO= − 4 . 55 eV) thereby sup- pressing the size of the gap to Δ$_{α}$=6.06 eV, while for the β -spin state the π ∗ level occurs closer to the conduction band (LUMO= − 0 . 84 eV) and the gap is Δ$_{β}$=5.29 eV. spin-down tails. Therefore, according to Lieb’s theorem [5] for the localized states occupying the same sublattice a ferromagnetic ordering of their spins would be energet- ically favored, but for the states on different sublattices, one expects the antiferromagnetic ordering [6–9]. For vacancies equally distributed between both sides of the graphane plane ( AB -distribution), the energet- ically favorable spin ordering is the antiparallel align- ment of spins between one side (A sublattice) and the other (B sublattice). Thus, for even number of vacan- cies in the AB -distribution, all spins are paired but the band degeneracy can be slightly lifted. In Fig. (c) we present the bands for two H-vacancy defects in the AB - distribution separated by a distance d = 3 a$_{C}$$_{−}$$_{C}$ , where a$_{C}$$_{−}$$_{C}$ is the length of the C-C bond. Therefore for the α - and β -spin states, the obtained band gap of the size Δ$_{α,β}$ = 5 . 41 eV is defined by the energy gap between the π (HOMO= − 4 . 56 eV) and the π ∗ (LUMO= − 0 . 85 eV) defect levels. For odd number of defects, one localized state would have unpaired spin which generates an extra level π for the α - and π ∗ for the β -spin states. How- ever, when several H-vacancy defects are located on the same side of the graphane plane the parallel alignment of their spins would be preferred because they belong to the same sublattice ( AA -distribution). In Fig. (d) we show the energetics of the bands for graphane with two H- vacancy defects separated by a distance of d = 4 a$_{C}$$_{−}$$_{C}$ in its triplet state. Each spin state induces a defect level in the valence band of the α -spin state ( π states) and in the conduction band of the β -spin state ( π ∗ states). There- fore, the size of the bandgap for the α -state is defined by the conduction band and the defect level π in the valence band, while for the β -spin state by the valence band and the defect level π ∗ in the conduction band (Δ$_{α}$=6.08 eV, Δ$_{β}$=5.24 eV) that is similar to the case of the single H- vacancy (see Fig. (b)). However, the state characterized by antiparallel alignment of two spins (the singlet state), possesses the π and π ∗ defect levels for both the α - and β -spin states and the size of the bandgap is defined by the energy gap between these defect levels, π and π ∗ , i.e., Δ$_{α}$ ≃ Δ$_{β}$ (for example see Fig. (c)). The destructive and constructive contributions of the spin tails of the localized states decrease with increasing distance between the defects [22]. Therefore, we have calculated the difference in the total energy between the triplet and singlet states ( E$_{(}$ 2 $_{2}$; 1 $_{$_{2}$)}$) depending on the dis- tance between the two H-vacancy defects. If E$_{(}$ 2 $_{2}$; 1 $_{2}$) en- ergy is negative the ferromagnetic ordering of spins is energetically preferred, but an antiferromagnetic order- ing otherwise. We have calculated the two components: the energy E 0 ( 2 $_{2}$; 1 $_{2}$) is considered before relaxation of the lattice induced by the presence of defects and the E$_{(}$ 2 $_{2}$; 1 $_{2}$) component after relaxation. These energies for the AA - and AB -distributions and splitting of the π levels in the valence band ( ε$_{1}$ − ε$_{2}$ ) are presented in Table I. For DIFFERENT DISTRIBUTION OF THE H-VACANCY DEFECTS For several H-vacancy defects, ordering of spins of the localized states and the size of the bandgap are defined by the distance between the defects and the distribution of those defects between the sides of the graphane plane. The side dependence is related to the sublattice symme- try. For the chair conformation of graphane, the carbon atoms belonging to different sublattices are terminated by the hydrogen atoms from different sides of the plane. It was already known that in graphene [22], when the localized states occupy the same sublattice and if their spins have antiparallel alignment, then the contribution of the π states to the total energy diminishes as a result of the destructive interference between the spin-up and