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Title: Self-consistent predictor/corrector algorithms for stable and efficient integration of the time-dependent Kohn-Sham equation

ORCiD logo [1]; ORCiD logo [1]
  1. Department of Chemistry and Biochemistry, and Chemical Physics Program, The Ohio State University, Columbus, Ohio 43210, USA
Publication Date:
Sponsoring Org.:
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Journal Article: Publisher's Accepted Manuscript
Journal Name:
Journal of Chemical Physics
Additional Journal Information:
Journal Volume: 148; Journal Issue: 4; Related Information: CHORUS Timestamp: 2018-02-15 00:38:36; Journal ID: ISSN 0021-9606
American Institute of Physics
Country of Publication:
United States

Citation Formats

Zhu, Ying, and Herbert, John M. Self-consistent predictor/corrector algorithms for stable and efficient integration of the time-dependent Kohn-Sham equation. United States: N. p., 2018. Web. doi:10.1063/1.5004675.
Zhu, Ying, & Herbert, John M. Self-consistent predictor/corrector algorithms for stable and efficient integration of the time-dependent Kohn-Sham equation. United States. doi:10.1063/1.5004675.
Zhu, Ying, and Herbert, John M. 2018. "Self-consistent predictor/corrector algorithms for stable and efficient integration of the time-dependent Kohn-Sham equation". United States. doi:10.1063/1.5004675.
title = {Self-consistent predictor/corrector algorithms for stable and efficient integration of the time-dependent Kohn-Sham equation},
author = {Zhu, Ying and Herbert, John M.},
abstractNote = {},
doi = {10.1063/1.5004675},
journal = {Journal of Chemical Physics},
number = 4,
volume = 148,
place = {United States},
year = 2018,
month = 1

Journal Article:
Free Publicly Available Full Text
This content will become publicly available on January 31, 2019
Publisher's Accepted Manuscript

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  • The stationary internal density functional theory (DFT) formalism and Kohn-Sham scheme are generalized to the time-dependent case. It is proven that, in the time-dependent case, the internal properties of a self-bound system (such as an atomic nuclei or a helium droplet) are all defined by the internal one-body density and the initial state. A time-dependent internal Kohn-Sham scheme is set up as a practical way to compute the internal density. The main difference from the traditional DFT formalism and Kohn-Sham scheme is the inclusion of the center-of-mass correlations in the functional.
  • The research described in this product was performed in part in the Environmental Molecular Sciences Laboratory, a national scientific user facility sponsored by the Department of Energy's Office of Biological and Environmental Research and located at Pacific Northwest National Laboratory. In previous research (J. Chem. Phys. 111, 3800 (1999)) a Hessian-based integration algorithm was derived for performing direct dynamics simulations. In the work presented here, improvements to this algorithm are described. The algorithm has a predictor step based on a local second-order Taylor expansion of the potential in Cartesian coordinates, within a trust radius, and a fifth-order correction to thismore » predicted trajectory. The current algorithm determines the predicted trajectory in Cartesian coordinates, instead of the instantaneous normal mode coordinates used previously, to ensure angular momentum conservation. For the previous algorithm the corrected step was evaluated in rotated Cartesian coordinates. Since the local potential expanded in Cartesian coordinates is not invariant to rotation, the constants of motion are not necessarily conserved during the corrector step. An approximate correction to this shortcoming was made by projecting translation and rotation out of the rotated coordinates. For the current algorithm unrotated Cartesian coordinates are used for the corrected step to assure the constants of motion are conserved. An algorithm is proposed for updating the trust radius to enhance the accuracy and efficiency of the numerical integration. This modified Hessian-based integration algorithm, with its new components, has been implemented into the VENUS/NWChem software package and compared with the velocity-Verlet algorithm for the H₂CO→H₂+CO, O₃+C₃H₆, and F -+CH₃OOH chemical reactions.« less
  • We discuss techniques for accelerating the self consistent field (SCF) iteration for solving the Kohn-Sham equations. These techniques are all based on constructing approximations to the inverse of the Jacobian associated with a fixed point map satisfied by the total potential. They can be viewed as preconditioners for a fixed point iteration. We point out different requirements for constructing preconditioners for insulating and metallic systems respectively, and discuss how to construct preconditioners to keep the convergence rate of the fixed point iteration independent of the size of the atomistic system. We propose a new preconditioner that can treat insulating andmore » metallic system in a unified way. The new preconditioner, which we call an elliptic preconditioner, is constructed by solving an elliptic partial differential equation. The elliptic preconditioner is shown to be more effective in accelerating the convergence of a fixed point iteration than the existing approaches for large inhomogeneous systems at low temperature.« less
  • The density functional approach in the Kohn-Sham approximation is widely used to study properties of many-electron systems. Due to the nonlinearity of the Kohn-Sham equations, the general self-consistence searching method involves iterations with alternate solving of the Poisson and Schroedinger equations. One of problems of such an approach is that the charge distribution renewed by means of the solution of the Schroedinger equation does not conform to boundary conditions of the Poisson equation for the Coulomb potential. The resulting instability or even divergence of iterations manifests itself most appreciably in the case of infinitely extended systems. The known attempts tomore » deal with this problem are reduced in fact to abandoning the original iterative method and replacing it with some approximate calculation scheme, which is usually semi-empirical and does not permit to evaluate the extent of deviation from the exact solution. In this work, we realize the iterative scheme of solving the Kohn-Sham equations for extended systems with inhomogeneous electron gas, which is based on eliminating the long-range character of Coulomb interaction as the cause of tight coupling between charge distribution and boundary conditions. The suggested algorithm is employed to calculate energy the spectrum, self-consistent potential, and electrostatic capacitance of the semi-infinite degenerate electron gas bounded by an infinitely high barrier, as well as the work function and surface energy of simple metals in the model with homogeneous distribution of positive background. The difference between self-consistent Hartree solutions and those taking into account the exchange-correlation interaction is analyzed. The comparison with the results previously published in the literature is carried out. The case study of the metal-semiconductor tunnel contact shows this method as applied to an infinitely extended system where the steady-state current can flow.« less