\section{Conclusions}

In this article, we have shown several recent developments in the realm of electronic-structure theory that have been based on the Octopus real-space code and made possible in part by the flexibility and simplicity of working with real-space grids. Most of them go beyond a mere implementation of existing theory and represent new ideas in their respective areas. We expect that many of these approaches will become part of the standard tools of physicists, chemists and material scientists, and in the future will be integrated into other electronic-structure codes.

These advances also illustrate the variety of applications of real-space electronic structure, many of which going beyond the traditional calculation schemes used in electronic structure, and might provide a way forward to tackle current and future challenges in the field. 

What we have presented also shows some of the current challenges in real-space electronic structure. One example is the use of pseudo-potentials or other forms of projectors to represent the electron-ion interaction. Non-local potentials introduce additional complications on both the formulation, as shown by the case of magnetic response, and the implementation. Pseudo-potentials also include an additional, and in some cases, not well-controlled approximation. It would be interesting to study the possibility of developing an efficient method to perform full-potential calculations without additional computational cost, for example by using adaptive or radial grids.

Another challenge for real-space approaches is the cost of the calculation of two-body Coulomb integrals that appear in electron-hole linear response, RDMFT or hybrid xc functionals. In real-space these integrals are calculated in linear or quasi-linear time by considering them as a Poisson problem. However, the actual numerical cost can be quite large when compared with other operations. A fast approach to compute these integrals, perhaps by using an auxiliary basis, would certainly make the real-space approach more competitive for some applications.

The scalability of real-space grid methods makes them a good candidate for electronic-structure simulations in the future exaflop supercomputing systems expected for the end of the decade. In this aspect, the challenge is to develop high-performance implementations that can run efficiently on these machines.

%In general the number of grid points is large (in the range of \(10^4\) to \(10^6\)) in comparison with the number of expansion coefficients in localized basis set method. This is usually not an issue, since for real-space DFT/TDDFT the amount of work per coefficient is small and scales linearly with the number of grid points. However, some other methods are formulated in terms of objects that depend on two, or more, coordinates. For these systems, real-space methods become impractical even for moderately sized systems. In this case, alternative formulations are required that avoid these expensive many-body objects. This idea has been applied, for example, for the calculation of response functions~\cite{Nguyen_2012}.


