Mixed Approach to Incorporate Self-Consistency Into Order-N Lcao Methods
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ABSTRACT We present a method for selfconsistent Density Functional Theory calculations in which the effort required is proportional to the size of the system, thus allowing the aplication to problems with a very large size. The method is based on the LCAO approximation, and uses a mixed approach to obtain the Hamiltonian integrals between atomic orbitals with Order-N effort. We show the performance and the convergence properties of the method in several silicon and carbon systems, and in a DNA periodic chain. INTRODUCTION In the last few years, the realization of the possibility of devising algorithms with Order-N scaling (which scale linearly with the system size) for the electronic structure problem has led to a flurry of activity [1]. These Order-N methods would allow the calculation of total energies and forces for systems with thousands of atoms, opening the possibility of performing ab initio molecular dynamics (MD) simulations in systems with unprecedented size. Several approaches [1] have been thoroughly tested in the context of parametrized tight-binding models or in simplified non-selfconsistent versions of the Local Density Approximation (LDA). The application to full selfconsistent Density Functional Theory (DFT) is far less well established, due to the fact that the cost of computing the selfconsistent Hamiltonian is so large that it seriously overrides the gains obtained by the use of an Order-N algorithm. In this work, we present a selfconsistent LDA formulation with linear scaling and small computational demands, so that systems with many hundreds of atoms can be treated in modest workstations. The main ingredients of the method are: (i) Use of the linear combination of atomic orbitals (LCAO) approximation, as a basis of expansion of the electronic states. (ii) Mixed approach for the calculation of the matrix elements of the selfconsistent LDA Hamiltonian between atomic orbitals. Some of the terms are computed by numerical integration in a real-space grid, whereas others are stored in tables and interpolated. (iii) Computation of total energies and forces from the LDA Hamiltonian in Order-N operations by using truncated Wannier-like localizad wave functions (LWF) as electronic states, and a band-energy functional which is minimized with no orthogonality constraints [1]. CALCULATION APPROACH The valence electrons are treated selfconsistently in the LDA to DFT. The core electrons are replaced by the non-local, norm-conserving pseudopotentials Vp, in the Bachelet et al. form [2]. We use the parametrization of Perdew and Zunger [3] for the exchange-correlation (XC) potential. In this work we use minimal basis sets of one s and three p orbitals per atom. These are extremely size-efficient, reducing the number of variables dramatically, compared to plane-wave or real-space grid approaches, so that larger systems can be studied. The errors implied by the choice of a basis can be reduced at the expense of increasing its size, with the corresponding increase of computational effort. However, the error magnit
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