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Physical review. B, Condensed matter

Canonical purification of the density matrix in electronic-structure theory

Adam H. R. Palser, David E. Manolopoulos

Published Nov 15, 1998• 181 citations
Publisher

Abstract

Two methods are suggested for performing ground-state electronic-structure calculations within the independent-electron approximation. Both methods involve the purification of a specified initial density matrix, which can be done either canonically (at a fixed electron count ${N}_{e})$ or grand canonically (at a fixed chemical potential \ensuremath{\mu}). Linear system-size scaling is achieved either way, as we illustrate in example tight-binding calculations on carbon nanotubes.

DOI
10.1103/physrevb.58.12704

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Comment on 'Transforms for idempotency purification of density matrices in linear-scaling electronic-structure calculations'

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•Jan 1, 2002•0 citations
Two methods are suggested for performing ground-state electronic-structure calculations within the independent-electron approximation. Both methods involve the purification of a specified initial density matrix, which can be done either canonically (at a fixed electron count Ne){N}_{e})Ne​) or grand canonically (at a fixed chemical potential \ensuremath{\mu}). Linear system-size scaling is achieved either way, as we illustrate in example tight-binding calculations on carbon nanotubes.
We introduce a method for the solution of the electronic-structure problem in the independent-electron approximation. The method is based upon a variational solution for the density matrix, which is truncated to zero beyond a real-space radius Rc{\mathit{R}}_{\mathit{c}}Rc​, and becomes exact as Rc{\mathit{R}}_{\mathit{c}}Rc​\ensuremath{\rightarrow}\ensuremath{\infty}. Most importantly, the computer time scales only linearly with system size. The method is tested in the context of tight-binding models in one and three dimensions.
A purification algorithm for expanding the single-particle density matrix in terms of the Hamiltonian operator is proposed. The scheme works with a predefined occupation and requires less than half the number of matrix-matrix multiplications compared to existing methods at low (10%) and high (>90%) occupancies. The expansion can be used with a fixed chemical potential, in which case it is an asymmetric generalization of and a substantial improvement over grand canonical McWeeny purification. It is shown that the computational complexity, measured as the number of matrix multiplications, essentially is independent of system size even for metallic materials with a vanishing band gap.
A renormalization-group method for electronic structure of large systems within a tight-binding framework is presented. A telescopic series of nested Hilbert spaces is constructed, having exponentially decreasing dimensions and electrons. The Hamiltonian matrices have exponentially converging energy ranges focusing to the Fermi level. The computational effort scales near linearly with system size even when the density matrix is highly nonlocal. This is illustrated by calculations on a model metal and a metallic finite carbon nanotube, for which standard linear scaling methods are inapplicable.
A newly developed energy renormalization-group method for electronic structure of large systems with small Fermi gaps within a tight-binding framework is presented in detail. A telescopic series of nested Hilbert spaces is constructed, having exponentially decreasing dimensions and electrons, for which the Hamiltonian matrices have exponentially converging energy ranges focusing to the Fermi level and in which the contribution to the density matrix is a sparse contribution. The computational effort scales near linearly with system size even when the density matrix is highly nonlocal. This is illustrated by calculations on a model metal, a small radius carbon-nanotube and a two-dimensional puckered sheet polysilane semiconductor.