In this chapter we present a real-space approach to density functional calculations. Real-space calculations [28, 134, 207, 291, 4, 202, 118, 39, 116, 76, 123, 325, 122, 117, 326, 361, 124, 223, 257, 244, 125, 138, 245] are being rapidly developed as alternatives to plane wave calculations. In this chapter we will use a real-space grid with a finite difference representation for the kinetic energy operator. The advantage of real-space grid calculations is their simplicity and versatility (e.g., there are no matrix elements to be calculated and the boundary conditions are more easy imposed). As with plane wave basis sets, the accuracy can be improved easily and systematically. In fact, there exists a rigorous cutoff for the plane waves, which can be represented in a given grid without aliasing, that provides a convenient connection between the two schemes. Pseudopotentials, developed in the plane wave context, can be applied equally well in grid-based methods, resulting in an accurate and efficient evaluation of the electron–ion potential.
Unlike in the case of plane waves, the evaluation of the kinetic energy using finite differences is approximate, but it can be significantly improved by using high-order representations of the Laplacian operator. However, an important difference between finite difference schemes and basis set approaches is the lack of a Rayleigh–Ritz variational principle in the finite difference case.