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CERMICS, Ecole Nationale des Ponts et Chaussées 

Chemistry, University of Southern California 

Lawrence Berkeley National Laboratory 

Physics, Tulane University 
Electronic structure calculations are the very core of quantum chemistry and play an increasingly important role in nanotechnologies, molecular biology and materials science.
This workshop will focus on two topics:
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 thousandatom 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(N^{3}), 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 linearscaling methods that reduce the order of complexity for DFT algorithms, largescale 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:
The workshop will discuss the mathematical and algorithmic aspects of the above in the context of coupledcluster (including equationofmotion) and multireference methods.
Methodological developments in the Density Functional Theory:
The density functional theory (DFT) of Hohenberg, Kohn and Sham is a way to find the groundstate density n(r) and energy E of a manyelectron system (atom, molecule, condensed material) by solving a constrained minimization problem whose first order optimality conditions (the KohnSham equations) can be written as a nonlinear eigenvalue problem. It resembles the HartreeFock theory, but is formally exact because it includes the effects of electron correlation as well as exchange in the density functional for the exchangecorrelation energy E_{x}c[n] and in its functional derivative, the exchangecorrelation potential v_{x}c([n],r). Timedependent properties and excited states are also accessible through a timedependent version of DFT. Density functional theory is much more computationally efficient than correlatedwavefunction theory, especially for large systems, but has the disadvantage that in practice E_{x}c[n] and v_{x}c([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 E_{x}c[n] and v_{x}c([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 manyelectron selfinteractionfree, (ii) recover full exact exchange under uniform density scaling to the highdensity 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, v_{x}c([n],r) can be constructed by the optimized effective potential method. For timedependent DFT, a selfinteractionfree v_{x}c with memory is needed. These and related problems may be explored in this workshop, with emphasis on their mathematical aspects.
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