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We present a theory of electronic properties of gated triangular graphenequantum dots with zigzag edges as a function of size and carrier density. Wefocus on electronic correlations, spin and geometrical effects using acombination of atomistic tight-binding, Hartree-Fock and configurationinteraction methods (TB+HF+CI) including long range Coulomb interactions. Thesingle particle energy spectrum of triangular dots with zigzag edges exhibits adegenerate shell at the Fermi level with a degeneracy N_{edge} proportional tothe edge size. We determine the effect of the electron-electron interactions onthe ground state, the total spin and the excitation spectrum as a function of ashell filling and the degeneracy of the shell using TB+HF+CI for N_{edge} < 12and approximate CI method for N_{edge}\geq 12. For a half-filled neutral shellwe find spin polarized ground state for structures up to N=500 atoms inagreement with previous {\it ab initio} and mean-field calculations, and inagreement with Lieb's theorem for a Hubbard model on a bipartite lattice.Adding a single electron leads to the complete spin depolarization forN_{edge}\leq 9. For larger structures, the spin depolarization is shown tooccur at different filling factors. Away from half-fillings excesselectrons(holes) are shown to form Wigner-like spin polarized triangularmolecules corresponding to large gaps in the excitation spectrum. The validityof conclusions is assessed by a comparison of results obtained from differentlevels of approximations. While for the charge neutral system all methods givequalitatively similar results, away from the charge neutrality an inclusion ofall Coulomb scattering terms is necessary to produce results presented here.

Electronic properties of gated triangular graphene quantum dots: Magnetism, correlations, and geometrical effects