Efficiency of algorithms for Kohn-Sham density functional theory

Abstract

Recent developments in numerical methods for computing Kohn-Sham electronic structure have substantially increased the sizes of systems which can be studied. In addition to fast Car-Parinello and conjugate gradient algorithms, there have been several innovative proposals aimed at achieving order N scaling in computer time, where N is the number of atoms in the cell. These developments demand a better understanding of the convergence properties of different numerical approaches to Kohn-Sham density functional calculations. In particular the non-linearities of the Kohn-Sham equations lead to non-trivial dependence of convergence times on the system size. The number of iterations required to converge the self-consistent Kohn-Sham equations is shown to grow like N 2 3 for insulators and like N for metals. If each iteration can be carried out in order N time, the total computational effort to solve a given system will thus scale like N 5 3 or N 2 for insulators or metals respectively. A similar analysis of conjugate gradient minimization methods shows that, in the worst case, of order O(N 1 3 ) energy and gradient evaluations are needed to converge the energy. Only in insulating systems which are “Wannier representable” (WR) is the number of iterations of O(1), allowing true order N calculations. A different approach, based upon conjugate gradient minimization with respect to variations in the single particle potential rather than the wave functions, is also found to scale like N 1 3 for insulators and scales as N for metals. Slower scaling with N will also be found in systems in which one dimension of the unit cell is much longer than the others, or for metals where the Fermi energy lies close to a van Hove singularity. These results are quite independent of calculational details, such as choice of basis set, but simply follow from the scaling with N of the relevant Jacobian or Hessian matrices.