Once the level of theory (DFT-LDA, Hartree–Fock, or other) has been chosen, the differences between electronic structure methods are essentially due to the choice of basis set. Pseudopotentials may or may not be part of the package. The main difference is the replacement of the bare Coulomb potential of the nucleus by a softer potential and some technical issues regarding the angular dependence of the pseudopotential, but the abundance of electronic structure methods in the market is mostly due to the quest for the ultimate basis set.

The central and computationally most intensive aspect of an electronic structure calculation is the self-consistent solution of the one-electron eigenvalue equation. This involves the calculation of the Kohn–Sham or the Fock matrix elements and the corresponding energy. In addition, geometry optimization and molecular dynamics simulations require the calculation of forces on the nuclear degrees of freedom. In solid-state applications, the optimization of lattice parameters and constant-pressure molecular dynamics simulations require also the calculation of the stress tensor.

In this chapter we shall describe how the Hamiltonian and the total energy are calculated in practice, in the most widely used methods. We start in Section 9.1 by introducing the KKR method as an approach derived from multiple scattering theory, where basis sets expansions are bypassed by using Green's function techniques. Section 9.2 is devoted to describing the most relevant aspects of allelectron schemes based on augmentation spheres. These are considered amongst the most accurate approaches because they do not approximate the behavior of core electrons through pseudopotentials, and the basis sets are flexible and adjust themselves according to the eigenvalue energies.