A comparison of linear scaling tight-binding methods

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. Various example systems are explored: an insulator (carbon in the diamond structure), a semiconductor (silicon), a transition metal (titanium) and a molecule (benzene). 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.

Properties of Complex Inorganic Solids

“…The BOP method is a tight-binding scheme in which the computational effort scales linearly with the number of particles considered (Horsfield and Bratkovsky 1996;Horsfield, et al 1996a, b;Bowler, et al 1997;Nguyen-Manh, et al 1998). The energy of a system of atoms is divided into three parts and can be written as U = U p^r +U bond +U env (1) Upair represents the electrostatic interaction between the atoms and the overlap repulsion between the valence d and p orbitals.…”

“…However, they are adjusted self-consistently to maintain local charge neutrality with respect to each atom via the so-called promotion energy. This condition reflects the perfect screening properties of metallic materials and it is achieved efficiently within the BOP scheme as described in (Horsfield, et al 1996a, b;Bowler, et al 1997).…”

“…The semi-empirical method in which the required covalent character of bonding is included explicitly is the tight-binding method. In recent years this method has been reformulated in terms of Bond-order Potentials (BOP) Pettifor and Aoki 1991;Horsfield, et al 1996a, b;Bowler, et al 1997) using the orthogonal basis and two-centre bond (hopping) integrals. In this case a very important problem is the transferability of the bond integrals.…”

Nonorthogonal density-matrix perturbation theory

“…During the last decade a new computational paradigm has evolved in electronic structure theory, where no critical part of a calculation is allowed to increase in complexity more than linearly with system size [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34]. Linear scaling electronic structure theory extends tightbinding, Hartree-Fock, and Kohn-Sham schemes to the study of large complex systems beyond the reach of conventional methods.…”

Linear-scaling quantum Monte Carlo technique with non-orthogonal localized orbitals

“…Very recently, a procedure has been suggested [7] for achieving at least partial linear scaling for QMC, based on the idea of "maximally localized Wannier functions" [8]. The purpose of this report is to propose and test a simpler alternative method, which appears to have important advantages.The O(N ) techniques that have been developed for tight binding (TB) [9], DFT [10, 11] and Hartree-Fock [12] calculations all depend ultimately on the fact that the density matrix ρ(r, r ′ ) associated with the single-electron orbitals decays to zero as |r − r ′ | → ∞, and the manner of this decay has been extensively studied ([13] and references therein). Briefly, the decay is algebraic for metals and exponential for insulators, with the decay rate increasing with band gap, so that there is more to be gained from O(N ) techniques for wide-gap insulators.…”