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 development 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. And other hand, because these same tools are not good enough for many applications, and better methods in terms of accuracy or numerical cost are required.  These research is targeted on a broad range of aspects of electronic structure simulations: the development of new theoretical frameworks, new or improved methods to calculate properties within existing theories, or even new algorithms. In most cases, this theoretical work requires the development of test implementations to assess its properties. While the implementation of established methods is focused towards robustness, efficiency and generality. Given the experimentative nature of this work, 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 to simulate electrons.  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 to simulate electrons. When simulating electronsusing some level of approximation to quantum mechanics, like Hartree-Fock or density functional theory,  different fields needs to be represented numerically: numerically, for example  the ionic potential, the single-particle orbitals or states, and or  the electronic density. The most popular representations methods are based on the use of basis sets 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 a very efficient method as many integrals can be calculated from analytical formulae~\cite{szabo1996modern}. In condensed matter physics, on the other hand, the traditional basis is a set of plane waves, that correspond to  the eigenstates of the 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{Harrison_2004,Genovese_2011}. One alternative that does not use a basis set are real-space grids.  Using a mesh of points is one of the most intuitive and widely used method for numerically representing spacially resolved quantities, for this reason grids are an efficient and well-established method for solving partial differential equations in many areas of science and engineering.   Discretizing in a real-space grids does not benefit from this physical connection to the system being simulated. However the method has another advantages. In first place, grids are flexible enough to simulate different kinds of systems, both finite and periodic systems can be directly simulated (including systems with partial periodicity).