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We present a method for total energy minimizations and molecular dynamicssimulations based either on tight-binding or on Kohn-Sham hamiltonians. Themethod leads to an algorithm whose computational cost scales linearly with thesystem size. The key features of our approach are  (i) an orbital formulation with single particle wavefunctions constrained tobe localized in given regions of space, and (ii) an energy functional whichdoes not require either explicit orthogonalization of the electronic orbitals,or inversion of an overlap matrix.  The foundations and accuracy of the approach and the performances of thealgorithm are discussed, and illustrated with several numerical examplesincluding Kohn-Sham hamiltonians. In particular we present calculations withtight-binding hamiltonians for diamond, graphite, a carbon linear chain andliquid carbon at low pressure. Even for a complex case such as liquid carbon --a disordered metallic system with differently coordinated atoms -- theagreement between standard diagonalization schemes and our approach is verygood. Our results establish the accuracy and reliability of the method for awide class of systems and show that tight binding molecular dynamicssimulations with a few thousand atoms are feasible on small workstations

Electronic-structure calculations and molecular-dynamics simulations with linear system-size scaling