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arXiv:cond-mat/0311469 · Physical review. B, Condensed matter

Implicit purification for temperature-dependent density matrices

Anders M. N. Niklasson

Published Dec 30, 2003• 39 citations
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Abstract

An implicit purification scheme is proposed for calculation of the temperature-dependent, grand canonical single-particle density matrix, given as a Fermi-Dirac operator expansion in terms of the Hamiltonian. The computational complexity is shown to scale with the logarithm of the polynomial order of the expansion, or equivalently, with the logarithm of the inverse temperature. The system of linear equations that arise in each implicit purification iteration is solved efficiently by a conjugate gradient solver. The scheme is particularly useful in connection with linear scaling electronic structure theory based on sparse matrix algebra. The efficiency of the implicit temperature expansion technique is analyzed and compared to some explicit purification methods for the zero temperature density matrix.

DOI
10.1103/physrevb.68.233104

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An implicit purification scheme is proposed for calculation of the temperature-dependent, grand canonical single-particle density matrix, given as a Fermi-Dirac operator expansion in terms of the Hamiltonian. The computational complexity is shown to scale with the logarithm of the polynomial order of the expansion, or equivalently, with the logarithm of the inverse temperature. The system of linear equations that arise in each implicit purification iteration is solved efficiently by a conjugate gradient solver. The scheme is particularly useful in connection with linear scaling electronic structure theory based on sparse matrix algebra. The efficiency of the implicit temperature expansion technique is analyzed and compared to some explicit purification methods for the zero temperature density matrix.
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.
We present a method to compute the Fermi function of the Hamiltonian for a system of independent fermions based on an exact decomposition of the grand-canonical potential. This scheme does not rely on the localization of the orbitals and is insensitive to ill-conditioned Hamiltonians. It lends itself naturally to linear scaling as soon as the sparsity of the system's density matrix is exploited. By using a combination of polynomial expansion and Newton-like iterative techniques, an arbitrarily large number of terms can be employed in the expansion, overcoming some of the difficulties encountered in previous papers. Moreover, this hybrid approach allows us to obtain a very favorable scaling of the computational cost with increasing inverse temperature, which makes the method competitive with other Fermi operator expansion techniques. After performing an in-depth theoretical analysis of computational cost and accuracy, we test our approach on the density functional theory Hamiltonian for the metallic phase of the LiAl alloy.
Abstract. In a recent paper we have suggested that the finite temperature density matrix can be computed efficiently by a combination of polynomial expansion and iterative inversion techniques. We present here significant improvements over this scheme. The original complex-valued formalism is turned into a purely real one. In addition, we use Chebyshev polynomials expansion and fast summation techniques. This drastically reduces the scaling of the algorithm with the width of the Hamiltonian spectrum, which is now of the order of the cubic root of such parameter. This makes our method very competitive for applications to ab-initio simulations, when high energy resolution is required.
In a recent paper we have suggested that the finite temperature density matrix can be computed efficiently by a combination of polynomial expansion and iterative inversion techniques. We present here significant improvements over this scheme. The original complex‐valued formalism is turned into a purely real one. In addition, we use Chebyshev polynomials expansion and fast summation techniques. This drastically reduces the scaling of the algorithm with the width of the Hamiltonian spectrum, which is now of the order of the cubic root of such parameter. This makes our method very competitive for applications to ab‐initio simulations, when high energy resolution is required.
Linear-scaling electronic structure methods based on the calculation of moments of the underlying electronic Hamiltonian offer a computationally efficient and numerically robust scheme to drive large-scale atomistic simulations, in which the quantum-mechanical nature of the electrons is explicitly taken into account. We compare the kernel polynomial method to the Fermi operator expansion method and establish a formal connection between the two approaches. We show that the convolution of the kernel polynomial method may be understood as an effective electron temperature. The results of a number of possible kernels are formally examined, and then applied to a representative tight-binding model.
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.