Xavier Andrade edited Introduction.tex  over 9 years ago

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In chemistry the method of choice is to use atomic orbitals as a basis to describe the orbitals of a molecule. When these atomic orbitals are expanded in Gaussian functions, it leads to an efficient method as many integrals can be calculated from analytical formulae~\cite{szabo1996modern}. In condensed matter physics, the traditional basis is a set of plane waves, that correspond to the eigenstates of a homogeneous electron gas. These physics-inspired basis sets have, however, some limitations. For example, it is not trivial to simulate crystalline systems using atomic orbitals~\cite{Dovesi_2014}, and, on the other hand, in plane wave approaches finite systems must be approximated as periodic system using a super cell approach.   Several alternatives to atomic-orbital and plane-wave basis sets exist~\cite{White_1989,Tsuchida_1995,Harrison_2004,Pask_2005,16008435,Genovese_2011}. exist~\cite{White_1989,Tsuchida_1995,Harrison_2004,16008435,Genovese_2011}.  One particular approach that does not depend on a basis set uses a grids to directly 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 equations using uniform grids combined with pseudo-potentials. What made the approach competitive was the use of high-order finite differences, that keep control the error of the Laplacian without requiring very dense meshes. From that moment, several real-space implementation 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 a real-space grids does not benefit from a direct physical connection to the system being simulated. However, the method has another 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 overs the grid and differential operators can be discretized using finite-differences. Grids are flexible enough to simulate different kinds of systems, both finite and periodic systems can be directly simulated, including systems with partial periodicity.