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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.
For the simulations of electronic systems, such as molecules or solids, When simulating electrons using some
approximated level of approximation to quantum
mechanical mechanics, like Hartree-Fock or density functional theory
(DFT)~\cite{Hohenberg_1964,Kohn_1965}, different fields needs to be represented numerically: the
ionic potential, the single-particle orbitals and the electronic density. The most popular
representations methods are based on the use of basis sets that 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 an analytical formula~\cite{szabo1996modern}. In condensed matter physics, on the other hand, the traditional basis is a set of plane waves, the eigenvectors of the Hamiltonian 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.
Discretizing in a real-space grids does not benefit from this physical connection to the system being simulated. However they are flexible enough to simulate different kinds of systems.