Large Scale Quantum Simulations Using Tight-Binding Hamiltonians and Linear Scaling Methods
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ABSTRACT We describe linear scaling methods for electronic structure calculations and quantum molecular dynamics simulations, which are based on an orbital formulation of the electronic problem. In particular, we discuss some open problems which need to be addressed to improve the performances of these methods, and briefly review some applications to carbon and silicon systems, within a Tight-Binding framework. INTRODUCTION Quantum simulations are aimed at modeling materials at the microscopic level, by solving numerically the equations governing the atomic motion. In order to obtain an accurate microscopic description of most materials properties, the interaction between atoms must be described using the laws of quantum mechanics. Interatomic forces can be computed by solving the Schr6dinger equation for electrons, thus determining the electronic ground state at given positions of the nuclei.
In many cases of interest the nuclei can
be considered as classical objects. Once atomic trajectories are determined, using, e.g., molecular dynamics, a variety of materials properties can be calculated. The computer time required by a quantum simulation, and ultimately its feasibility, are mainly determined by the time necessary to solve the Schrhdinger equation for electrons. Standard approaches[l] to the solution of this equation require a workload proportional to the cube of the number of atoms involved in the simulation: Doubling the size of the system amounts to multiplying by eight the computing time. This unfavorable scaling poses severe limitations to the kind of problems which can be tackled with quantum simulations. Recently new methods for solving the Schr6dinger equation have been developed, which imply a workload growing linearly with the system-size. These approaches, called linear scaling methods[2], allow one to simulate systems much larger than previously accessible, widening the range of materials science issues that can be addressed. At present, linear scaling methods using Tight-Binding Hamiltonians allow one to perform simulations involving up to thousands of atoms on small workstations, and up to ten thousand atoms for tens of picoseconds when using supercomputers. This has made it possible to study problems such as large organic molecules in water[3], thin film growth[4, 5], extended defects[6] and dislocations[7] in semiconductors. Although the implementation of first-principles linear scaling methods is less advanced than that of semi-empirical methods, promising results[8] have already appeared in the literature. In this paper we first summarize the key features of quantum simulations based on 425 Mat. Res. Soc. Symp. Proc. Vol. 491 ©1998 Materials Research Society
linear scaling approaches (section I), and then discuss technical problems involved in linear scaling electronic structure calculations, such as convergence in iterative minimizations (section II). Then we address specific issues involved in molecular dynamics simulations (section III), and give a brief review of applications within a
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