A Comparison of Linear Scaling Tight Binding Methods
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"Physics
ABSTRACT Four linear scaling tight binding methods (the density matrix method, bond order potentials, the global density of states method, and the Fermi operator expansion) are described and compared to show relative computational efficiency for a given accuracy. The density matrix method proves to be most efficient for systems with narrow features in their energy gaps, while recursion based moments methods prove to be most efficient for metallic systems. INTRODUCTION During computer simulations, the need for accuracy under diverse conditions implies that a quantum mechanical description of interatomic interactions is required. However, the need to handle many atoms implies that computationally efficient implementations are necessary. These two considerations are in competition with one another. A good compromise solution is to use tight binding (TB) which is a simple, but often accurate, quantum mechanical model. While this model can be implemented efficiently on a computer, there is still a bottleneck, namely the cubic scaling with respect to number of atoms of the computational effort required for matrix diagonalisation. This effectively limits the system size to around 100 atoms for molecular dynamics simulations. Recently, however, a number of schemes have been proposed for which the computational effort scales linearly with number of atoms (so called O(N) methods) [1-11]. In this paper, the density matrix method [1, 2] (DMM), the bond order potential method [6, 7] (BOP), the global density of states method [10] (GDOS), and the Fermi operator expansion method [8] (FOE) are compared for two example systems. The aim of the investigation is to discover which method is computationally most efficient for a given level of accuracy for a given system.
METHODS The linear scaling methods currently available for TB can be broken down into two broad areas: variational methods and moments based methods. In this section a brief overview of the methods is given according to the two categories. Density Matrix Method This is a variational method. This method was proposed simultaneously by Li et al. [1], and by Daw [2], though from different arguments. The method revolves around the density matrix. The number of electrons in the system, the band energy for the ground state and the corresponding contribution to the forces from the band energy can all be written in terms of the density matrix p: Ne
=
2Tr[p]
(1)
U
=
2Tr[pH]
(2)
Fi
=
-2Tr
pka
I,
(3)
where Tr indicates taking the trace of a matrix, Hý is the Hamiltonian, and p is the chemical potential. The ground state energy can be found by minimising U with respect to p subject to two constraints: 417
Mat. Res. Soc. Symp. Proc. Vol. 491 ©1998 Materials Research Society
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idempotency of the density matrix (p = p, which is equivalent to p having eigenvalues of 0 and 1) and constant number of particles (Ne = constant). In the density matrix method, the elements of the density matrix are used as variational degrees of freedom with respect to which the ene
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