Xavier Andrade edited Introduction.tex  over 9 years ago

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Several alternatives to atomic-orbital and plane-wave basis sets exist~\cite{White_1989,Tsuchida_1995,Harrison_2004,16008435,Genovese_2011}. One particular approach that does not depend on a basis set uses a grid to directly represent fields in real-space. The method was pioneered by Becke~\cite{Becke_1989}, who used a combination of radial grids centered around each atom. In 1994 Chelikowsky, Troullier and Saad~\cite{Chelikowsky_1994} presented a practical approach for the solution of the Kohn-Sham (KS) equations using uniform grids combined with pseudo-potentials. What made the approach competitive was the use of high-order finite differences to control the error of the Laplacian without requiring very dense meshes. From that moment, several real-space implementations have been presented~\cite{Seitsonen_1995,Hoshi_1995,Gygi_1995,Briggs_1996,Fattebert_1996,Beck_1997,Ono_1999,Beck_2000,Nardelli_2001,Marques_2003,Pask_2005,Kronik_2006,Schmid_2006,Krotscheck_2007,Bernholc_2008,Shimojo_2008,Goto_2009,Enkovaara_2010,Iwata_2010,Sasaki_2011,Ono_2011}.  Discretizing in real-space grids does not benefit from a direct physical connection to the system being simulated. However, the method has other advantages. In first place, a real-space discretization is, in most cases, straight-forward to perform starting from the continuum description of the electronic problem. Operations like integration are directly translated into sums over the grid and differential operators can be discretized using finite differences. In fact, most electronic structure codes must rely on an auxiliary real-space discretization used, for example, for the calculation of the exchange and correlation (xc)  term of DFT. Grids are flexible enough to directly simulate different kinds of systems: finite, and fully or partially periodic. It is also possible to perform simulations with reduced (or increased) dimensionality. Additionally the discretization error can be systematically and continuously controlled by adjusting the spacing between mesh points, and the physical extension of the grid.