As emphasized in the previous chapter, localized functions provide an intuitive description of electronic structure and bonding. This chapter is devoted to quantitative methods in which the wavefunction is expanded as a linear combination of localized atomic(-like) orbitals, such as gaussians, Slater-type orbitals, and numerical radial atomic-like orbitals. Such calculations can be very efficient; they can also be very accurate, as shown by the highly developed codes used in chemistry; and they provide the basis for creation of new methods, such as “order-N” (Ch. 23) and Green's function approaches. There is a cost, however: full self-consistent DFT calculations require specification of the basis, and the price paid for efficiency is loss of generality (in contrast to the “one basis fits all” philosophy of plane wave methods). Since details depend upon the basis, we can only describe general principles with limited examples.

It is instructive to note that there are important connections to localized muffin-tin orbitals (MTOs) (Ch. 16), the linear LMTO method (Ch. 17). This has led to an “ab initio tight-binding” method (Sec. 17.6) in which a minimal basis of orthogonal localized orbitals is derived from the Kohn–Sham hamiltonian.