Xavier Andrade edited Introduction.tex  over 9 years ago

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The development of theoretical methods for the simulation of electronic system is an active area of study. This interest has been fueled, on one hand, by the success of theoretical tools like density functional theory (DFT)~\cite{Hohenberg_1964,Kohn_1965}, that can predict many properties with good accuracy at a relatively modest computational cost. On the other hand, these same tools are not good enough for many applications~\cite{Cohen_2008}, and more accurate and more efficient methods are required.  Current research is targeted on in the area covers  a broad range of aspects of electronic structure simulations: the development of new novel  theoretical frameworks, new or improved methods to calculate properties within existing theories, or evennew,  more efficient or scalable, and scalable  algorithms. In most cases, this theoretical work requires the development of test implementations to assess the properties and predictive power of the new methods. Given the experimentative nature of the development of methods for the simulations of electrons, the translation to code of new theory needs to be easy to implement and to modify. This is a factor that is not usually considered when analyzing and comparing the broad range of methods and codes used by chemists, phyisicst and material scientist.The most popular representations rely on basis sets, that usually have a certain physical connection to the system being simulated.  The most popular representations for electronic structure rely on basis sets, that usually have a certain physical connection to the system being simulated.  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_2004,16008435,Genovese_2011}. One particular approach that does not depend on a basis set uses a grids grid  to directly represent  fields in real-space 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 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 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. 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 term of DFT.  Grids are flexible enough to directly simulate different kinds of systems, finite systems systems: finite,  and fully fully,  or partially 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. The simple discretization and flexibility of the real space grids makes them an ideal framework to implement, develop and test new ideas. Of course, an additional ingredient is required. Modern electronic structure codes are quite complex, this means that researchers seldomly can write code from the scratch, instead they need to resort to existing codes to implement their developments.