skip to main content
OSTI.GOV title logo U.S. Department of Energy
Office of Scientific and Technical Information

Title: Partition of unity finite element method for quantum mechanical materials calculations

Publication Date:
Sponsoring Org.:
OSTI Identifier:
Grant/Contract Number:
Resource Type:
Journal Article: Publisher's Accepted Manuscript
Journal Name:
Extreme Mechanics Letters
Additional Journal Information:
Journal Volume: 11; Journal Issue: C; Related Information: CHORUS Timestamp: 2017-10-08 21:51:54; Journal ID: ISSN 2352-4316
Country of Publication:

Citation Formats

Pask, J. E., and Sukumar, N. Partition of unity finite element method for quantum mechanical materials calculations. Netherlands: N. p., 2017. Web. doi:10.1016/j.eml.2016.11.003.
Pask, J. E., & Sukumar, N. Partition of unity finite element method for quantum mechanical materials calculations. Netherlands. doi:10.1016/j.eml.2016.11.003.
Pask, J. E., and Sukumar, N. Wed . "Partition of unity finite element method for quantum mechanical materials calculations". Netherlands. doi:10.1016/j.eml.2016.11.003.
title = {Partition of unity finite element method for quantum mechanical materials calculations},
author = {Pask, J. E. and Sukumar, N.},
abstractNote = {},
doi = {10.1016/j.eml.2016.11.003},
journal = {Extreme Mechanics Letters},
number = C,
volume = 11,
place = {Netherlands},
year = {Wed Feb 01 00:00:00 EST 2017},
month = {Wed Feb 01 00:00:00 EST 2017}

Journal Article:
Free Publicly Available Full Text
Publisher's Version of Record at 10.1016/j.eml.2016.11.003

Save / Share:
  • Abstract not provided.
  • Over the course of the past two decades, quantum mechanical calculations have emerged as a key component of modern materials research. However, the solution of the required quantum mechanical equations is a formidable task and this has severely limited the range of materials systems which can be investigated by such accurate, quantum mechanical means. The current state of the art for large-scale quantum simulations is the planewave (PW) method, as implemented in now ubiquitous VASP, ABINIT, and QBox codes, among many others. However, since the PW method uses a global Fourier basis, with strictly uniform resolution at all points inmore » space, and in which every basis function overlaps every other at every point, it suffers from substantial inefficiencies in calculations involving atoms with localized states, such as first-row and transition-metal atoms, and requires substantial nonlocal communications in parallel implementations, placing critical limits on scalability. In recent years, real-space methods such as finite-differences (FD) and finite-elements (FE) have been developed to address these deficiencies by reformulating the required quantum mechanical equations in a strictly local representation. However, while addressing both resolution and parallel-communications problems, such local real-space approaches have been plagued by one key disadvantage relative to planewaves: excessive degrees of freedom (grid points, basis functions) needed to achieve the required accuracies. And so, despite critical limitations, the PW method remains the standard today. In this work, we show for the first time that this key remaining disadvantage of real-space methods can in fact be overcome: by building known atomic physics into the solution process using modern partition-of-unity (PU) techniques in finite element analysis. Indeed, our results show order-of-magnitude reductions in basis size relative to state-of-the-art planewave based methods. The method developed here is completely general, applicable to any crystal symmetry and to both metals and insulators alike. We have developed and implemented a full self-consistent Kohn-Sham method, including both total energies and forces for molecular dynamics, and developed a full MPI parallel implementation for large-scale calculations. We have applied the method to the gamut of physical systems, from simple insulating systems with light atoms to complex d- and f-electron systems, requiring large numbers of atomic-orbital enrichments. In every case, the new PU FE method attained the required accuracies with substantially fewer degrees of freedom, typically by an order of magnitude or more, than the current state-of-the-art PW method. Finally, our initial MPI implementation has shown excellent parallel scaling of the most time-critical parts of the code up to 1728 processors, with clear indications of what will be required to achieve comparable scaling for the rest. Having shown that the key remaining disadvantage of real-space methods can in fact be overcome, the work has attracted significant attention: with sixteen invited talks, both domestic and international, so far; two papers published and another in preparation; and three new university and/or national laboratory collaborations, securing external funding to pursue a number of related research directions. Having demonstrated the proof of principle, work now centers on the necessary extensions and optimizations required to bring the prototype method and code delivered here to production applications.« less
  • A new algorithm for calculating the Hamiltonian matrix elements with all-electron explicitly correlated Gaussian functions for quantum-mechanical calculations of atoms with two p electrons or a single d electron have been derived and implemented. The Hamiltonian used in the approach was obtained by rigorously separating the center-of-mass motion and it explicitly depends on the finite mass of the nucleus. The approach was employed to perform test calculations on the isotopes of the carbon atom in their ground electronic states and to determine the finite-nuclear-mass corrections for these states.
  • An algorithm for quantum-mechanical nonrelativistic variational calculations of L = 0 and M = 0 states of atoms with an arbitrary number of s electrons and with three p electrons have been implemented and tested in the calculations of the ground {sup 4}S state of the nitrogen atom. The spatial part of the wave function is expanded in terms of all-electrons explicitly correlated Gaussian functions with the appropriate pre-exponential Cartesian angular factors for states with the L = 0 and M = 0 symmetry. The algorithm includes formulas for calculating the Hamiltonian and overlap matrix elements, as well as formulasmore » for calculating the analytic energy gradient determined with respect to the Gaussian exponential parameters. The gradient is used in the variational optimization of these parameters. The Hamiltonian used in the approach is obtained by rigorously separating the center-of-mass motion from the laboratory-frame all-particle Hamiltonian, and thus it explicitly depends on the finite mass of the nucleus. With that, the mass effect on the total ground-state energy is determined.« less