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IMA Annual Program Year Workshop
Mathematical and Algorithmic Challenges in Electronic Structure Theory
September 29 - October 3, 2008

Eric CancesCERMICS
Anna KrylovUniversity of Southern California
Juan MezaLawrence Berkeley Laboratory
John PerdewTulane University

Electronic structure calculations are the very core of quantum chemistry and play an increasingly important role in nano-technologies, molecular biology and materials science.

This workshop will focus on two topics:

  • the mathematical challenges in developing accurate, efficient, and robust algorithms for electronic structure calculations of large systems;
  • the latest methodological developments and the remaining open problems in Density Functional Theory.

Algorithms for electronic structure calculations:

Density functional theory (DFT) is the most widely used ab initio method in material simulations. DFT can be used to calculate the electronic structure, the charge density, the total energy and the atomic forces of a material system, and with the advance of new algorithms and supercomputers, DFT can now be used to study thousand-atom systems. But there are many problems that either require much larger systems (more than 100,000 atoms), or many total energy calculation steps (molecular dynamics or atomic relaxations). Some possible applications include the study of nanostructures and the design of novel materials.

Unfortunately, conventional DFT algorithms scale as O(N3), where N is the size of the system (e.g., the number of atoms) putting many problems beyond the reach of even planned petascale computers. Therefore understanding the electronic structures of larger systems will require new mathematical advancements and algorithms. Some areas that will be addressed in this workshop include linear-scaling methods that reduce the order of complexity for DFT algorithms, large-scale nonlinear eigenvalue problems, and optimization techniques for solving the Schrödinger equation. In addition, we will discuss the implementation and parallelization of these methods for large supercomputer systems.

Contrarily to DFT, wavefunction theory provides us with a series of increasingly refined systematic approximations to the exact solution of the electronic Schrödinger equation. Wave function based electronic structure methods, which are implemented in a variety of packaged programs, can now be routinely employed to predict structures, spectra, properties and reactivity of molecules, sometimes with accuracy rivaling that of the experiment. However, due to the steep computational scaling, mathematical and algorithmic complexity, the following challenges remain:

  • properties calculation for correlated wave functions;
  • extending efficient and predictive methods and algorithms for open-shell and electronically excited species;
  • reducing the computational cost and scaling.

The workshop will discuss the mathematical and algorithmic aspects of the above in the context of coupled-cluster (including equation-of-motion) and multi-reference methods.

Methodological developments in the Density Functional Theory:

The density functional theory (DFT) of Hohenberg, Kohn and Sham is a way to find the ground-state density n(r) and energy E of a many-electron system (atom, molecule, condensed material) by solving a constrained minimization problem whose first order optimality conditions (the Kohn-Sham equations) can be written as a nonlinear eigenvalue problem. It resembles the Hartree-Fock theory, but is formally exact because it includes the effects of electron correlation as well as exchange in the density functional for the exchange-correlation energy Exc[n] and in its functional derivative, the exchange-correlation potential vxc([n],r). Time-dependent properties and excited states are also accessible through a time-dependent version of DFT. Density functional theory is much more computationally efficient than correlated-wavefunction theory, especially for large systems, but has the disadvantage that in practice Exc[n] and vxc([n],r) must be approximated (usually through a nonsytematic "educated guess"), leading in many cases to moderate but useful accuracy. Used almost exclusively in condensed matter physics since the 1970's, DFT became popular in quantum chemistry in the 1990's due to the development of more accurate approximations.

Besides the algorithmic challenges discussed above, the principal challenges facing DFT are (a) better understanding of the exact theory itself and derivation of further exact properties of Exc[n] and vxc([n],r), and (b) improved approximations that satisfy known exact constraints and sometimes are also fitted to known data. For example, it has been argued that the approximations should (i) be one- and many-electron self-interaction-free, (ii) recover full exact exchange under uniform density scaling to the high-density limit, and (iii) include nonlocal correlation effects, including static correlation and the van der Waals interaction between nonoverlapping densities. For implicit density functionals that are explicit orbital functionals, vxc([n],r) can be constructed by the optimized effective potential method. For time-dependent DFT, a self-interaction-free vxc with memory is needed. These and related problems may be explored in this workshop, with emphasis on their mathematical aspects.


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