deletions | additions       diff --git a/Introduction.tex b/Introduction.tex  index 26f7713..333e7a5 100644  --- a/Introduction.tex  +++ b/Introduction.tex 
...
When simulating electrons different fields needs to be represented numerically, for example the ionic potential, the single-particle orbitals or states, or the electronic density. The most popular representations methods are based on the use of 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 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 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}. exist~\cite{Harrison_2004,Pask_2005,16008435,Genovese_2011}.  One alternative that does not use a basis set are is to represent fields on  real-space grids. mesh or grid.  Using a meshof 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).