IMA Mathematical and Algorithmic Challenges in Electronic Structure Theory
http://www.ima.umn.edu/2008-2009/W9.29-10.3.08/
University of Minnessota, Minneapolis, MN
September 29-October 3, 2008

Organizers:
Eric Cances CERMICS, Ecole Nationale des Ponts et Chausses
Anna I. Krylov  Chemistry, University of Southern California
Juan C. Meza  Lawrence Berkeley National Laboratory
John P. Perdew  Physics, Tulane University

Description:

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 Schrdinger 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 Schrdinger 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.

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.

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