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Potential Enrichment

Double-well potential and enrichment functions for pseudoatomic wavefunctions

(Research-work supported by DOE, NSF-DMS and UC Lab Fees Research Program, 2007–2011)

First principles (ab initio) quantum mechanical simulations based on Kohn-Sham density functional theory (DFT) are a vital component of modern materials research. The parameter free, quantum mechanical nature of the theory facilitates both fundamental understanding and robust predictions across the gamut of materials systems, from metallic actinides to insulating organics. However, the solution of the equations of DFT (coupled Schrodinger and Poisson equations) is a formidable task and this has severely limited the range of materials systems that can be investigated by such rigorous, quantum mechanical means. Current state-of-the-art approaches for DFT calculations extend to more complex problems by adding more grid points (finite-difference methods) or basis functions (planewave and finite-element methods) without regard to the nature of the complexity, leading to substantial inefficiencies in the treatment of highly inhomogeneous systems such as those involving first-row, transition-metal or actinide atoms. This work, initiated with John Pask at LLNL in 2006, attempts to overcome this basic limitation of current approaches by employing partition-of-unity enrichment techniques in finite-element analysis to build the known atomic physics into the solution process, thus substantially reducing the degrees of freedom required and increasing the size of problems that can be addressed. We are collaborating with Zhaojun Bai (project) on the development of efficient and scalable finite element solvers for generalized eigenproblems in quantum mechanics.

Reports and Publications

  • C. Albrecht, C. Klaar, J. E. Pask, M. A. Schweitzer, N. Sukumar and A. Ziegenhagel (2018), "Orbital-enriched Flat-top Partition of Unity Method for the Schrödinger Eigenproblem," Computer Methods in Applied Mechanic and Engineering, Vol. 342, pp. 224–239. [PDF] Available at arXiv:1801.09596
  • Y. Cai, Z. Bai, J. E. Pask and N. Sukumar (2018), "Convergence Analysis of a Locally Accelerated Preconditioned Steepest Descent Method for Hermitian-Definite Generalized Eigenvalue Problems," Journal of Computational Mathematics, Vol. 36, Number 5, pp. 739–760. [PDF]
  • J. E. Pask and N. Sukumar (2017), "Partition of Unity Finite Element Method for Quantum Mechanical Materials Calculations," Extreme Mechanics Letters, Vol. 11, pp. 8–17. [PDF] Available at arXiv:1611.00731
  • Y. Cai, Z. Bai, J. E. Pask and N. Sukumar (2013), "Hybrid Preconditioning for Iterative Diagonalization of Ill-Conditioned Generalized Eigenvalue Problems in Electronic Structure Calculations," Journal of Computational Physics, Vol. 255, pp. 16–30. [PDF] Available at arXiv:1308.2445
  • S. E. Mousavi, J. E. Pask and N. Sukumar (2012), "Efficient Adaptive Integration of Functions with Sharp Gradients and Cusps in n-Dimensional Parallelepipeds," International Journal for Numerical Methods in Engineering, Vol. 91, Number 4, pp. 343–357. [PDF] Available at arXiv:1202.5341
  • J. E. Pask, N. Sukumar and S. E. Mousavi (2012), "Linear Scaling Solution of the All-Electron Coulomb Problem in Solids," International Journal for Multiscale Computational Engineering, Vol. 10, Number 1, pp. 83–99. Available at arXiv:1004.1765
  • J. E. Pask, N. Sukumar, M. Guney and W. Hu (March 2011), "Partition-Of-Unity Finite-Element Method for Large Scale Quantum Molecular Dynamics on Massively Parallel Computational Platforms," Department of Energy LDRD Grant 08-ERD-052, Report No. LLNL-TR-470692. [PDF]
  • N. Sukumar and J. E. Pask (2009), "Classical and Enriched Finite Element Formulations for Bloch-Periodic Boundary Conditions," International Journal for Numerical Methods in Engineering, Vol. 77, Number 8, pp. 1121–1138. [Errata: a = 5.7 bohr for the harmonic oscillator problem in Sec. 4.2.1; a = 4 bohr and σ = 1.5 for the periodic Gaussian problem in Sec. 4.2.2] [PDF]
  • N. Sukumar (February 2009), "Finite Element Methods in Quantum Mechanics" in iMechanica Web Blog: Journal Club Theme of February 2009. [HTML]


Review Articles

  • T. Torsti et al. (2006), "Three Real-Space Discretization Techniques in Electronic Structure Calculations,"Physica Status Solidi. B, Basic Research, Vol. 243, Number 5, pp. 1016–1053. [PDF]
  • J. E. Pask and P. A. Sterne (2005), "Finite Element Methods in Ab Initio Electronic Structure Calculations," Modelling and Simulation in Materials Science and Engineering, Vol. 13, pp. R71–R96. [PDF]
  • T. L. Beck (2000), "Real-Space Mesh Techniques in Density-Functional Theory," Reviews of Modern Physics, Vol. 72, Number 4, pp. 1041–1080. [PDF]
  • T. A. Arias (1999), "Multiresolution Analysis of Electronic Structure: Semicardinal and Wavelet Bases," Reviews of Modern Physics, Vol. 71, Number 1, pp. 267–311. [PDF]

Electronic-Structure Methods

Disclaimer: Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author and do not necessarily reflect the views of the National Science Foundation

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