Highly efficient method for Kohn-Sham density functional calculations of $500–10000$ atom systems

A method for the solution of the self-consistent Kohn-Sham equations using Gaussian-type orbitals is presented. Accurate relative energies and forces are demonstrated to be achievable at a fraction of the computational expense for large systems. With this approach calculations involving around $1000$ atoms can easily be performed with a serial desktop computer and $\sim 10000$ atom systems are within reach of relatively modest parallel computational resources. The method is applicable to arbitrary systems including metals. The approach generates a minimal basis on the fly while retaining the accuracy of the large underpinning basis set. Convergence of energies and forces are given for clusters as well as cubic cells of silicon and aluminum, for which the formation energies of defects are calculated in systems up to 8000 and 4000 atoms, respectively. For these systems the method exhibits linear scaling with the number of atoms in the presently important size range of $\sim 500–3000$ atoms.

Figure 1

Schematic of a primitive matrix with $N=36$. The shaded rows and columns are the rows and columns in the set $F_{\alpha }$ and $n_{\alpha }=5$. Here, for example, $F_{\alpha }=\left\{6,11,20,23,30\right\}$. The black elements at the intersection of the rows and columns prescribed by $F_{\alpha }$ are the elements that form the $n_{\alpha }\times n_{\alpha }$ matrices $\overline{\mathbf{H}}$ and $\overline{\mathbf{S}}$.

Figure 2

(Color online) Upper: timings of the time dominant contributions to the SCF iteration for the ideal vacancy in silicon in a range of cubic unit cells. The numbers above the line give the number of cores used. Lower: a comparison of dominant (eigenproblem) memory consumption. The solid line is for the conventional algorithm and the dashed line relates to the subspace method presented in this work.