Preconditioning of self-consistent-field cycles in density-functional theory: The extrapolar method

The number of self-consistent-field iterations needed for the density-functional theory treatment of metallic systems grows with the size of the unit cell, not only for basic algorithms like simple mixing, but also for more advanced schemes, in which results from several past steps are mixed. Preconditioning techniques have the potential to suppress this growth, although the available methods have strong limitations: either they deliver little improvement in case of mixed systems with metallic and nonmetallic regions, or the computation of the preconditioner scales badly with the size of the system, with a large prefactor. We propose an approximate preconditioner, with tremendously reduced prefactor, that makes the number of self-consistent cycle nearly independent of the size of the system, and bears little overhead up to the one hundred atom range. The susceptibility matrix, a key ingredient in our scheme, is approximated thanks to the closure relation. Instead of using the exact formulation of the dielectric matrix, we rely on the random-phase approximation, that allows to further decrease the prefactor thanks to a very low wave vector cutoff, even for systems with both vacuum and a metallic region. We test this algorithm for systems of increasing size and demonstrate its practical usefulness.