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Finite-Element Method for Large-Scale Ab Initio Electronic-Structure Calculations

John Pask, Phil Sterne

The finite-element method is a powerful and general approach for solving partial differential and integral equations. The solutions of the Schrödinger and Poisson equations constitute the most time-consuming steps in density-functional based electronic-structure calculations. These calculations provide a means of determining materials properties completely from quantum-mechanical first principles (ab initio), with no adjustable parameters; and so provide a robust and system-independent means for understanding and predicting a wide range of properties in diverse materials systems. Many such electronic-structure methods provide accurate results for unit cells consisting of a small number of atoms (see Methods section for examples). Few methods, however, can provide accurate results for larger systems of several hundred atoms or more, a physically important regime where complex surfaces, interfaces, and materials defects can begin to be modeled. The goal of this research is to develop and apply a finite-element based method for large-scale ab initio electronic-structure calculations, extending the range of materials systems accessible by such rigorous, quantum mechanical means.

The finite-element method combines the significant advantages of both basis-oriented and real-space grid-based approaches. The finite-element method is an expansion method which uses a strictly local, piecewise polynomial basis (See Figure 1).

Steps in the construction of a finite-element basis.
Figure 1. Steps in the construction of a finite-element basis. 1) The domain (unit cell volume) is partitioned into subdomains called elements. 2) Polynomial basis functions are defined within each element. 3) Polynomials on neighboring elements are pieced together to form the strictly local piecewise-polynomial basis functions of the method. Because the basis functions are polynomials, the method is completely general and systematically improvable. Because they are strictly local, the method realizes the significant advantages of real-space grid approaches for the solution of large problems.

Because the method is basis-oriented, it is variational, allows for arbitrarily accurate integrations, and allows for increased efficiency by choosing basis functions based upon physical insight. Because the basis functions are polynomials, the method is completely general and systematically improvable. Because they are strictly local, the method achieves a number of additional advantages with respect to large-scale calculations: the method produces sparse, structured matrices which are well suited to solution by efficient iterative methods and require only O(N) storage, where N is the number of atoms; the basis functions can be concentrated where needed in real-space (where the solution varies most rapidly) in order to increase the efficiency of the representation; all computations are performed directly in real-space, eliminating the need for computationally expensive transforms such as Fourier transforms which can incur large communications costs on large-scale computational platforms; and the method is well suited to parallel implementation. The method therefore combines the advantages of basis-oriented approaches such as the planewave method with the significant advantages with respect to large-scale calculations of real-space grid approaches such as the finite-difference method.

We have developed and implemented a finite-element based approach for the solution of the equations of density functional theory. We have used this approach routinely to calculate positron distributions and lifetimes in support of LLNL's experimental research program on positron annihilation in materials. We have incorporated finite-element Schrödinger and Poisson solvers into a general self-consistent density-functional electronic-structure method applicable to materials of arbitrary composition and symmetry, and to metals and insulators alike. Recent efforts, in collaboration with N. Sukumar and W. Hu at U. C. Davis, focus on increasing the efficiency of the FE representation substantially by employing modern partition-of-unity FE (PUFE) techniques to build known atomic physics into the FE basis. Initial results show order-of-magnitude reductions in basis set size relative to established planewave based methods, and thus the potential for dramatic speedups in large-scale quantum molecular dynamics on massively parallel computers. In collaboration with Z. Bai at U. C. Davis, we have developed efficient parallel solvers and preconditioners for the large, sparse generalized eigenproblems produced by discretization of the density-functional equations in the FE and PUFE bases.

Most recently, in collaboration with L. Lin and C. Yang at the University of California, Berkeley and Lawrence Berkeley National Laboratory, we have undertaken the development and application of new discontinuous Galerkin based electronic structure methods to advance understanding of the chemistry and dynamics of Li-ion batteries.

Snapshot from a 2,014-atom quantum molecular dynamics simulation.
Snapshot from a 2,014-atom quantum molecular dynamics simulation of the anode-electrolyte interface in a lithium-ion battery, simulated using new Discontinuous Galerkin electronic structure method. View from the graphite anode. (Graphics: Liam Krauss)

Selected Publications

  1. "Adaptive local basis set for Kohn-Sham density functional theory in a discontinuous Galerkin framework II: Force, vibration, and molecular dynamics calculations," G. Zhang, L. Lin, W. Hu, C. Yang, and J.E. Pask, submitted, 2015. arXiv: 1510.06489
  2. "Lithium ion solvation and diffusion in bulk organic electrolytes from first principles and classical reactive molecular dynamics," M.T. Ong, O. Verners, E.W. Draeger, A.C.T. van Duin, V. Lordi, and J.E. Pask, J. Phys. Chem. B 119, 1535-1545 (2015).
  3. "A projected preconditioned conjugate gradient algorithm for computing many extreme eigenpairs of a Hermitian matrix," E. Vecharynski, C. Yang, and J.E. Pask, J. Comput. Phys. 290, 73-89 (2015).
  4. "Hybrid preconditioning for iterative diagonalization of ill-conditioned generalized eigenvalue problems in electronic structure calculations," Z. Bai, Y. Cai, J.E. Pask, and N. Sukumar, J. Comput. Phys. 255, 16–30 (2013). http://dx.doi.org/10.1016/j.jcp.2013.07.020 
  5. "dftatom: A robust and general Schrödinger and Dirac solver for atomic structure calculations," O. Certik, J.E. Pask, and J. Vackar, Comput. Phys. Commun. 184, 1777–1791 (2013). http://dx.doi.org/10.1016/j.cpc.2013.02.014 
  6. "Finite elements in electronic structure," J.E. Pask, in Encyclopedia of Applied and Computational Mathematics, Björn Engquist (Ed.), Springer, Heidelberg, 2015. (Review)
  7. "Efficient adaptive integration of functions with sharp gradients and cusps in n-dimensional parallelepipeds," S.E. Mousavi, J.E. Pask, and N. Sukumar, Int. J. Numer. Meth. Engng. 91, 343–357 (2012).
  8. "Linear scaling solution of the all-electron Coulomb problem in solids," J.E. Pask, N. Sukumar, and S.E. Mousavi, Int. J. Multiscale Comput. Engng. 10, 83–99 (2012). Preprint: arXiv:1004.1765v3
  9. "Origin of large moments in Mn(x)Si(1-x) at small x ," M. Shaughnessy, C.Y. Fong, R. Snow, K. Liu, J.E. Pask, and L.H. Yang, Appl. Phys. Lett. 95, 022515 (2009).
  10. "Classical and enriched finite element formulations for Bloch-periodic boundary conditions ," N. Sukumar and J.E. Pask, Int. J. Numer. Meth. Engng. 77, 1121 (2009)
  11. "Finite element methods in ab initio electronic structure calculations ," J.E. Pask and P.A. Sterne, Modelling Simul. Mater. Sci. Eng. 13, R71 (2005). (Review)
  12. "Real-space formulation for the electrostatic potential and total energy of solids ," J.E. Pask and P.A. Sterne, Phys. Rev. B, 71, 113101 (2005).
  13. "Calculation of positron observables using a finite element-based approach ," P.A. Sterne, J.E. Pask, and B.M. Klein, Appl. Surf. Sci. 149, 238 (1999).

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