Density Functional Theory (DFT) calculations with computational effort which increases linearly with the number of atoms (linear-scaling DFT) have been successfully developed for insulators, taking advantage of the exponential decay of the one-particle density matrix. For metallic systems, the density matrix is also expected to decay exponentially at finite electronic temperature and linear-scaling DFT methods should be possible by taking advantage of this decay. Here we present a method for DFT calculations at finite electronic temperature for metallic systems which is effectively linear-scaling (O(N)). Our method generates the elements of the one-particle density matrix and also finds the required chemical potential and electronic entropy using polynomial expansions. A fixed expansion length is always employed to generate the density matrix, without any loss in accuracy by the application of a high electronic temperature followed by successive steps of temperature reduction until the desired (low) temperature density matrix is obtained. We have implemented this method in the ONETEP linear-scaling (for insulators) DFT code which employs local orbitals that are optimised in situ. By making use of the sparse matrix machinery of ONETEP, our method exploits the sparsity of Hamiltonian and density matrices to perform calculations on metallic systems with computational cost that increases asymptotically linearly with the number of atoms. We demonstrate the linear-scaling computational cost of our method with calculation times on palladium nanoparticles with up to ∼13 000 atoms.

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