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arXiv:cond-mat/9703224 · Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics

Calculating the linear response functions of noninteracting electrons with a time-dependent Schrödinger equation

Toshiaki Iitaka, Shintaro Nomura, Hideki Hirayama, Xinwei Zhao, Yoshinobu Aoyagi, Takuo Sugano

Published Jul 1, 1997• 72 citations
arXivHTMLPDF

Abstract

An $O(N)$ algorithm is proposed for calculating linear response functions of noninteracting electrons. This algorithm is simple and suitable to parallel and vector computation. Since it avoids ${O(N}^{3})$ computational effort of matrix diagonalization, it requires only $O(N)$ computational efforts, where $N$ is the dimension of the state vector. The use of this $O(N)$ algorithm is very effective since, otherwise, we have to calculate a large number of eigenstates, i.e., the occupied one-electron states up to the Fermi energy and the unoccupied states with higher energy. The advantage of this method compared to the Chebyshev polynomial method recently developed by Wang and Zunger [L. W. Wang, Phys. Rev. B 49, 10 154 (1994); L. W. Wang and A. Zunger, Phys. Rev. Lett. 73, 1039 (1994)] is that our method can calculate linear response functions without any storage of huge state vectors on external storage.

DOI
10.1103/physreve.56.1222

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An O(N)O(N)O(N) algorithm is proposed for calculating linear response functions of noninteracting electrons. This algorithm is simple and suitable to parallel and vector computation. Since it avoids O(N3){O(N}^{3})O(N3) computational effort of matrix diagonalization, it requires only O(N)O(N)O(N) computational efforts, where NNN is the dimension of the state vector. The use of this O(N)O(N)O(N) algorithm is very effective since, otherwise, we have to calculate a large number of eigenstates, i.e., the occupied one-electron states up to the Fermi energy and the unoccupied states with higher energy. The advantage of this method compared to the Chebyshev polynomial method recently developed by Wang and Zunger [L. W. Wang, Phys. Rev. B 49, 10 154 (1994); L. W. Wang and A. Zunger, Phys. Rev. Lett. 73, 1039 (1994)] is that our method can calculate linear response functions without any storage of huge state vectors on external storage.
An O(N) algorithm is proposed for calculating linear response functions of non-interacting electrons. This program is simple and suitable to parallel- and vector- computation. Since it avoids O(N 3) computational effort of matrix diagonalization, it requires only O(N) computational efforts where N is the dimension of the statevector. The use of this O(N) algorithm is very effective since otherwise we have to calculate large number of eigenstates, i.e., the occupied one-electron states up to the Fermi energy and the unoccupied states with higher energy. The advantage of this method compared to the Chebyshev polynomial method recently developed by Wang (L.W. Wang, Phys. Rev. B 49, 10154 (1994);L.W. Wang, Phys. Rev. Lett. 73, 1039 (1994) ) is that our method can calculate linear response functions without any storage of huge statevectors on external storage while their method should store many statevectors. 1
We introduce an efficient and numerically stable method for calculating linear response functions chi(q,omega) of quantum systems at finite temperatures. The method is a combination of numerical solution of the time-dependent Schrödinger equation, random vector representation of trace, and Chebyshev polynomial expansion of Boltzmann operator. This method should be very useful for a wide range of strongly correlated quantum systems at finite temperatures. We present an application to the ESR spectrum of s=1 / 2 antiferromagnet Cu benzoate.
A numerical method is developed for calculating the real time Green's functions of very large sparse Hamiltonian matrices, which exploits the numerical solution of the inhomogeneous time-dependent Schroedinger equation. The method has a clear-cut structure reflecting the most naive definition of the Green's functions, and is very suitable to parallel and vector supercomputers. The effectiveness of the method is illustrated by applying it to simple lattice models. An application of this method to condensed matter physics will be found in H. Tanaka, Phys. PRB 57, 2168 (1998).
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.
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.
Recent significant improvements of the numerical solutions of the time-dependent Schrödinger equation beg the question as to whether these recent methods are comparable in efficacy (in terms of accuracy and computational time) to the current "method of choice," i.e., the Chebyshev expansion of the time-evolution operator and the fast-Fourier-transform method of determining the kinetic energy. In this paper we review the methods in question and, by studying the time development of a coherent wave packet in an oscillator well, we are able to assess the effectiveness of the various methods. It turns out that the new generalizations come close (to within an order of magnitude) to being able to generate solutions as precisely and efficiently as the Chebyshev-fast-Fourier-transform method. The strict unitarity of the generalized methods may be an advantage. We also show that the fast-Fourier-transform approach to calculating the kinetic energy can be replaced by straightforward numerical differentiation to obtain the same precision.