Electronic Structure Calculations with Localized Orbitals: The Siesta Method.

Practical quantum mechanical simulations of materials, which take into account explicitly the electronic degrees of freedom, are presently limited to about 1000 atoms. In contrast, the largest classical simulations, using empirical interatomic potentials, involve over 109 atoms. Much of this 106-factor difference is due to the existence of well-developed order-N algorithms for the classical problem, in which the computer time and memory scale linearly with the number of atoms N of the simulated system. Furthermore, such algorithms are well suited for execution in parallel computers, using rather small interprocessor communications. In contrast, nearly all quantum mechanical simulations involve a computational effort which scales as O(N3), that is, as the cube of the number of atoms simulated. Such an intrinsically more expensive dependence is due to the delocalized character of the electron wavefunctions. Since the electrons are fermions, every one of the¡«N occupied wavefunctions must be kept orthogonal to every other one, thus requiring ¡«N2 constraints, each involving an integral over the whole system, whose size is also proportional to N. Despite such intrinsic difficulties, the last decade has seen an intense advance in algorithms that allow quantum mechanical simulations with an O(N) computational effort. Such algorithms are based on avoiding the spatially extended electron eigenfunctions and using instead magnitudes, such as the one-electron density matrix, that are spatially localized, thus allowing for a spatial decomposition of the electronic problem. This strategy exploits what has been called by Walter Kohn the nearsightedness of the electron-gas [1]. Its implementation requires, or is greatly facilitated, by the use of a spatially localized basis set, such as a linear combination of atomic orbitals (LCAO). This paper gives a brief overview of such methods and describes in some detail one of them, the Spanish Initiative for Electronic Simulations with Thousands of Atoms (SIESTA). [ABSTRACT FROM AUTHOR]