Corrected penalty-functional method for linear-scaling calculations within density-functional theory

We present a new method for the calculation of ground-state total energies within density-functional theory, based upon the single-particle density-matrix formulation, which requires a computational eeort which scales only linearly with system-size. The diicult idempotency constraint is imposed approximately using a penalty-functional constructed to allow eecient minimization. The resulting error in the total energy due to the violation of idempotency is removed by an analytic correction. The results for a system comprising 216 atoms of crystalline silicon are compared with those from a standard plane-wave code. Linear scaling to 512 atoms is also demonstrated on a workstation. Within density-functional theory (DFT), the complexity of the problem of calculating the ground-state energy of the inhomogeneous electron gas in an external potential 1] scales linearly with system-size N (i.e. the number of electrons). However, `traditional' methods based upon the Kohn-Sham (KS) formulation of DFT 2] require a computational eeort which scales asymptotically as N 3 , either because of the cost of diagonalizing the Hamiltonian, or as a result of the orthogonality requirement for the extended KS single-particle wave-functions. This O(N 3) scaling restricts the size of systems which can currently be treated. Much interest has therefore been shown in using the single-particle density-matrix (DM) to calculate the total energy 3]. Since the DM is short-ranged in real-space 4] and free from orthogonality constraints, it provides the basis of a linear-scaling method for KS-DFT 5{7] (see 8] for a review of some of these methods). However, most linear-scaling schemes have so far been applied only in the context of tight-binding or restricted basis-set calculations. In contrast, the method presented here has been applied to fully self-consistent DFT calculations with an arbitrarily complete basis-set, a task so far attempted by only one other group 6].