The local-density approximation (LDA), together with the half-occupation (transition state) is notoriously successful in the calculation of atomic ionization potentials. When it comes to extended systems, such as a semiconductor infinite system, it has been very difficult to find a way to halfionize because the hole tends to be infinitely extended (a Bloch wave). The answer to this problem lies in the LDA formalism itself. One proves that the half-occupation is equivalent to introducing the hole self-energy (electrostatic and exchange-correlation) into the Schroedinger equation. The argument then becomes simple: the eigenvalue minus the self-energy has to be minimized because the atom has a minimal energy. Then one simply proves that the hole is localized, not infinitely extended, because it must have maximal self-energy. Then one also arrives at an equation similar to the SIC equation, but corrected for the removal of just 1/2 electron. Applied to the calculation of band gaps and effective masses, we use the self-energy calculated in atoms and attain a precision similar to that of GW, but with the great advantage that it requires no more computational effort than standard LDA. PACS numbers: 71.15.-m 31.15.-p 71.20.Mq

The very old and successful density-functional technique of half-occupation is revisited [J. C. Slater, Adv. Quant. Chem. 6, 1 (1972)]. We use it together with the modern exchange-correlation approximations to calculate atomic ionization energies and band gaps in semiconductors [L. G. Ferreira et al., Phys. Rev. B 78, 125116 (2008)]. Here we enlarge the results of the previous paper, add to its understandability, and show when the technique might fail. Even in this latter circumstance, the calculated band gaps are far better than those of simple LDA or GGA. As before, the difference between the Kohn-Sham ground state one-particle eigenvalues and the half-occupation eigenvalues is simply interpreted as the self-energy (not self-interaction) of the particle excitation. In both cases, that of atomic ionization energies and semiconductor band gaps, the technique is proven to be very worthy, because not only the results can be very precise but the calculations are fast and very simple.

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We study six different two-dimensional (2D) allotropes of carbon, silicon, germanium, and tin by means of the ab initio density functional theory for the ground state and approximate methods to calculate their electronic structures, including quasiparticle effects. Four of the investigated allotropes are based on dumbbell geometries, one on a kagome lattice, and one on the graphenelike hexagonal structure for comparison. Concerning carbon, our calculations of the cohesive energies clearly show that the hexagonal structure (graphene) is most stable. However, in the case of Si and Ge, the dumbbell structures, particularly the large honeycomb dumbbell (LHD) geometries, are energetically favored compared to the sp 2 /sp 3 -bonded hexagonal lattice (i.e., silicene and germanene). The main reason for this is the opening of a band gap in the honeycomb dumbbell arrangements. The LHD sheet crystals represent indirect semiconductors with a K → gap of about 0.5 eV. In the Sn case we predict the MoS 2 -like symmetry to be more stable, in contrast to the stanene and LHD geometries predicted in literature. Our results for freestanding group-IV layers shine new light on recent experimental studies of group-IV overlayers on various substrates.