Systems of adatoms on semiconductor surfaces display competing ground states and exotic spectral properties typical of two-dimensional correlated electron materials which are dominated by a complex interplay of spin and charge degrees of freedom. We report a fully ab initio derivation of low energy Hamiltonians for the adatom systems Si(111):X, with X=Sn, Si, C, Pb, that we solve within self-consistent combined GW and dynamical mean field theory ("GW+DMFT"). Calculated photoemission spectra are in agreement with available experimental data. We rationalize experimentally observed tendencies from Mott physics towards charge-ordering along the series as resulting from substantial long-range interactions.PACS numbers: 71.15. Mb, 73.20.At, 71.10.Fd, 71.30.h Understanding the electronic properties of materials with strong electronic Coulomb correlations remains one of the biggest challenges of modern condensed matter physics. The interplay of delocalization and interactions is not only at the origin of exotic ground states, but also determines the excitation spectra of correlated materials. The "standard model" of correlated fermions, the Hubbard model, in principle captures these phenomena. Yet, relating the model to the material on a microscopic footing remains a challenge. Even more importantly, the approximation of purely local Coulomb interactions can become severe in realistic materials, where long-range interactions and charge fluctuation physics cannot be neglected.Systems of adatoms on semiconducting surfaces, such as Si (111):X with X=Sn, C, Si, Pb, have been suggested [1] to be good candidates for observing low-dimensional correlated physics. Commonly considered to be realizations of the one-band Hubbard model and toy systems for investigating many-body physics on the triangular lattice, such surfaces have been explored experimentally [2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18] and theoretically [19][20][21][22][23][24][25][26][27][28][29][30][31].These so-called α-phases show a remarkable variety of interesting physics including commensurate charge density wave (CDW) states [5,6,9] and isostructural metal to insulator transitions (MIT) [14]. However, while specific systems and/or phenomena have been investigated also theoretically, a comprehensive understanding including materials trends is still lacking. A central goal of our work is to present a unified picture that relates, within a single framework, different materials (adatom systems), placing them in a common phase diagram.We derive low-energy effective Hamiltonians ab initio from a combined density functional and constrained random phase approximation (cRPA) scheme [32] in the implementation of [33] (see also the extension to surface systems in [34]). While the first surprise are the relatively large values of the onsite interactions which we find to be of the order of the bandwidth (≈ 1 eV), most importantly we show that non-local interactions are large (nearestneighbor interaction of ≈ 0.5 eV) and, hence, an essential part of t...