Density Functional Theory (DFT)

The Schrodinger equation is fundamental for describing the quantum mechanical electronic structure of matter, since it does not require any empirical input. However, the solution of the Schrodinger equation is exceedingly expensive, and this limits the size of systems that can be directly evaluated to tens of electrons. Numerous approaches have been proposed to reduce the computational cost of the solution of the Schrodinger equation. These approaches include the widely used DFT of Hohenberg and Kohn.

In their seminal work, Hohenberg and Kohn proved the existence of a one-to-one correspondence between the ground state electron density and the ground state wavefunction of a many-particle system. By this correspondence, the electron density replaces the many-body electronic wavefunction as the fundamental unknown field, thereby greatly reducing the dimensionality and computational complexity of the problem. The most common present-day implementation of DFT is through the Kohn-Sham method, in which the intractable many-body problem of interacting electrons is reduced to a tractable problem of non-interacting electrons moving in an effective potential.

Our research on DFT has focussed on the following areas

Real-space formulations of DFT

Traditionally, plane waves has been the basis of choice for solving DFT. However, the  need for periodic boundary conditions and uniform resolution in space, limits their effectiveness in studying localized systems including defects. Further, plane waves have global support which restricts their use in the development of multiscale methods. To overcome the aforementioned limitations, we are interested in developing real-space formulations for DFT for both the all-electron problem and the pseudopotential approximations.

Publications

  • Suryanarayana, P., Gavini, V., Blesgen, T., Bhattacharya, K., Ortiz, M., 2010. Non-periodic finite-element formulation of Kohn-Sham density functional theory. Journal of the Mechanics and Physics of Solids. 58(2), 256-280. [LINK]
  • Suryanarayana, P., Bhattacharya, K., Ortiz, M., 2011. A mesh-free convex approximation scheme for Kohn-Sham density functional theory. Journal of Computational Physics 230 (13), 5226 - 5238. [LINK]

Linear-scaling methods for DFT

Traditional implementations of DFT solve for the orbitals through diagonalization, a procedure which scales cubically with respect to the number of atoms. This places serious limitations on the size of the system which can be studied. To overcome this, we are interested in developing efficient linear-scaling methods for DFT.

Publications

 

  • Suryanarayana, P., 2013. On spectral quadrature for linear-scaling Density Functional Theory. Chemical Physics Letters  584, 182-187 [LINK]
  • Suryanarayana, P., 2013. Optimized Purification for Density Matrix Calculation. Chemical Physics Letters 555, 291-295. [LINK]
  • Suryanarayana, P., Bhattacharya, K., Ortiz, M., 2013. Coarse-graining Kohn-Sham Density Functional Theory. Journal of the Mechanics and Physics of Solids 61(1), 38-60. [LINK]

Coarse-graining DFT

Defects, though present in relatively minute concentrations, play a significant role in determining macroscopic properties. Even vacancies, the simplest and most common type of defect, are fundamental to phenomena like creep, spall and radiation ageing. This necessitates an accurate characterization of defects, which represents a unique challenge since both the electronic structure of the defect core as well as the long range elastic field need to be resolved simultaneously. Further, this has to be achieved at physically relevant defect concentrations, which is typically in parts per million. Coupled with the fact that accurate electronic structure calculations are limited to a few hundred atoms, this represents a truly challenging multiscale problem. To overcome this, We are interested in developing methods which allow tremendous reduction in the computational effort required to study defects using DFT.

Publications

  • Suryanarayana, P., Bhattacharya, K., Ortiz, M., 2013. Coarse-graining Kohn-Sham Density Functional Theory. Journal of the Mechanics and Physics of Solids 61(1), 38-60. [LINK]