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Physical review. B, Condensed matter

Density-matrix electronic-structure method with linear system-size scaling

X.-P. Li, Ricardo Wagner Nunes, David H. Vanderbilt

Published Apr 15, 1993• 705 citations
Publisher

Abstract

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 ${\mathit{R}}_{\mathit{c}}$, and becomes exact as ${\mathit{R}}_{\mathit{c}}$\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.

DOI
10.1103/physrevb.47.10891

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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.
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.
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.
A time-dependent density-functional tight-binding method in real time domain is developed to calculate the absorption spectra of very large systems. The time-dependent first-order response of the density matrix due to an external perturbation is solved using Chebyshev method with high efficiency and accuracy. Linear scaling of CPU time and memory usage with the system size is achieved by exploring the sparsity of the involving matrices as well as by introduction of a cutoff for the first-order density matrix. The compressed sparse row scheme is used to store the matrices, and SPARSEKIT is employed for sparse matrix multiplication. The absorption spectra of three-dimensional water clusters (H2O)216{({\mathrm{H}}_{2}\mathrm{O})}_{216}(H2​O)216​, (H2O)432{({\mathrm{H}}_{2}\mathrm{O})}_{432}(H2​O)432​, (H2O)648{({\mathrm{H}}_{2}\mathrm{O})}_{648}(H2​O)648​, and (H2O)864{({\mathrm{H}}_{2}\mathrm{O})}_{864}(H2​O)864​ are calculated using the present approach. The error due to the cutoff of density matrix is negligible. It is shown from these calculations that the presented method is very efficient and capable of calculating the absorption spectra for very large three-dimensional systems.
Density matrix electronic structure theory is used in many quantum chemistry methods to "alleviate" the computational cost that arises from directly using wave functions. Although density matrix based methods are computationally more efficient than wave function based methods, significant computational effort is involved. Because the Schrödinger equation needs to be solved as an eigenvalue problem, the time-to-solution scales cubically with the system size in mean-field type approaches such as Hartree-Fock and density functional theory and is solved as many times in order to reach charge or field self-consistency. We hereby propose and study a method to compute the density matrix by using a quadratic unconstrained binary optimization (QUBO) solver. This method could be useful to solve the problem with quantum computers and, more specifically, quantum annealers. Our proposed approach is based on a direct construction of the density matrix using a QUBO eigensolver. We explore the main parameters of the algorithm focusing on precision and efficiency. We show that, while direct construction of the density matrix using a QUBO formulation is possible, the efficiency and precision have room for improvement. Moreover, calculations performed with quantum annealing on D-Wave's new Advantage quantum computer are compared with results obtained with classical simulated annealing, further highlighting some problems of the proposed method. We also suggest alternative methods that could lead to a more efficient QUBO-based density matrix construction.
Large strongly correlated systems provide a challenge to modern electronic structure methods, because standard density functionals usually fail and traditional quantum chemical approaches are too demanding. The density-matrix renormalization group method, an extremely powerful tool for solving such systems, has recently been extended to handle long-range interactions on real-space grids, but is most efficient in one dimension where it can provide essentially arbitrary accuracy. Such 1d systems therefore provide a theoretical laboratory for studying strong correlation and developing density functional approximations to handle strong correlation, if they mimic three-dimensional reality sufficiently closely. We demonstrate that this is the case, and provide reference data for exact and standard approximate methods, for future use in this area.