Eric Cances
École des Ponts ParisTech
and INRIA Paris
France
Electronic structure calculation is one of the major application fields of scientific
computing. It is used daily in any chemistry or materials science department, and it accounts
for a high percentage of machine occupancy in supercomputing centers. Current challenges
include the study of complex molecular systems and processes (e.g. photosynthesis,
high-temperature superconductivity...), and the building of large, reliable databases for the
design of materials and drugs.
The most commonly used models are the Kohn-Sham Density Functional Theory (DFT), and the (post) Hartree-Fock models. The Hartree-Fock and Kohn-Sham models have similar mathematical structures. They consist in minimizing an energy functional on the Sobolev space H1(R3)N under L2-orthonormality constraints. The associated Euler-Lagrange equations are systems on nonlinear elliptic PDEs. After discretization in a Galerkin basis, one obtains smooth optimization problems on matrix manifolds, or on convex hulls of matrix manifolds.
Solving these problems is easy for small simple molecular systems, but very difficult for large or complex systems. Two classes of numerical methods compete in the field: constrained direct minimization of the energy functional, and self-consistent field (SCF) iterations to solve the Euler-Lagrange equations. In this talk, I will present a comparative study of these two approaches, as well as new efficient algorithms for systems with spin symmetries.