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Physical Review B

Linear scaling time-dependent density-functional tight-binding method for absorption spectra of large systems

Fan Wang, Chi Yung Yam, GuanHua Chen, XiuJun Wang, Kangnian Fan, Thomas A. Niehaus, Thomas Frauenheim

Published Jul 24, 2007• 22 citations
Publisher

Abstract

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 ${({\mathrm{H}}_{2}\mathrm{O})}_{216}$, ${({\mathrm{H}}_{2}\mathrm{O})}_{432}$, ${({\mathrm{H}}_{2}\mathrm{O})}_{648}$, and ${({\mathrm{H}}_{2}\mathrm{O})}_{864}$ 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.

DOI
10.1103/physrevb.76.045114

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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.
A linear-scaling first-principles quantum mechanical method is developed to evaluate the optical responses of large molecular systems. Instead of a many-body wave function, the equation of motion is solved for the reduced single-electron density matrix in the time domain. The locality of the reduced single-electron density matrix is utilized to ensure that computational time scales linearly with system size. The two-electron Coulomb integrals are evaluated with the fast multipole method, and the calculation of exchange-correlation quadratures utilizes the locality of an exchange-correlation functional and the integral prescreening technique. As an illustration, the resulting time-dependent density-functional theory is used to calculate the absorption spectra of polyacetylene oligomers and linear alkanes. The linear-scaling of computational time versus the system size is clearly demonstrated.
Basing on our hierarchical equations of motion for time-dependent quantum transport [X. Zheng, G. H. Chen, Y. Mo, S. K. Koo, H. Tian, C. Y. Yam, and Y. J. Yan, J. Chem. Phys. 133, 114101 (2010)], we develop an efficient and accurate numerical algorithm to solve the Liouville-von-Neumann equation. We solve the real-time evolution of the reduced single-electron density matrix at the tight-binding level. Calculations are carried out to simulate the transient current through a linear chain of atoms, with each represented by a single orbital. The self-energy matrix is expanded in terms of multiple Lorentzian functions, and the Fermi distribution function is evaluated via the Padè spectrum decomposition. This Lorentzian-Padè decomposition scheme is employed to simulate the transient current. With sufficient Lorentzian functions used to fit the self-energy matrices, we show that the lead spectral function and the dynamics response can be treated accurately. Compared to the conventional master equation approaches, our method is much more efficient as the computational time scales cubically with the system size and linearly with the simulation time. As a result, the simulations of the transient currents through systems containing up to one hundred of atoms have been carried out. As density functional theory is also an effective one-particle theory, the Lorentzian-Padè decomposition scheme developed here can be generalized for first-principles simulation of realistic systems.
The poor scaling of many existing quantum mechanical methods with respect to the system size hinders their applications to large systems. In this tutorial review, we focus on latest research on linear-scaling or O(N) quantum mechanical methods for excited states. Based on the locality of quantum mechanical systems, O(N) quantum mechanical methods for excited states are comprised of two categories, the time-domain and frequency-domain methods. The former solves the dynamics of the electronic systems in real time while the latter involves direct evaluation of electronic response in the frequency-domain. The localized density matrix (LDM) method is the first and most mature linear-scaling quantum mechanical method for excited states. It has been implemented in time- and frequency-domains. The O(N) time-domain methods also include the approach that solves the time-dependent Kohn-Sham (TDKS) equation using the non-orthogonal localized molecular orbitals (NOLMOs). Besides the frequency-domain LDM method, other O(N) frequency-domain methods have been proposed and implemented at the first-principles level. Except one-dimensional or quasi-one-dimensional systems, the O(N) frequency-domain methods are often not applicable to resonant responses because of the convergence problem. For linear response, the most efficient O(N) first-principles method is found to be the LDM method with Chebyshev expansion for time integration. For off-resonant response (including nonlinear properties) at a specific frequency, the frequency-domain methods with iterative solvers are quite efficient and thus practical. For nonlinear response, both on-resonance and off-resonance, the time-domain methods can be used, however, as the time-domain first-principles methods are quite expensive, time-domain O(N) semi-empirical methods are often the practical choice. Compared to the O(N) frequency-domain methods, the O(N) time-domain methods for excited states are much more mature and numerically stable, and have been applied widely to investigate the dynamics of complex molecular systems.
A linear‐scaling time‐dependent density functional theory is developed. The equation of motion is solved for the reduced single‐electron density matrix in the real time domain. Chebyshev expansion is used for integration in time domain. Filter diagonalization is implemented to determine the excited state energies. The locality of the reduced single‐electron density matrix is utilized to ensure computational time scales linearly with system size. We summarize these methods in this brief review.
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 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.