The main objective of the present proposal is to develop efficient algorithms for electronic structure analysis that are applicable for both metals and insulators. This will be done by developing multipole representation of the Fermi operator, which is a fundamental object in electronic structure analysis and more generally quantum theories of matter. In addition, the PI also proposes to develop efficient algorithms for representing and computing the Green's functions that arise in this context. A second component of the project is the numerical analysis of the algorithms in electronic structure analysis. Topics to be studied include the accuracy of linear scaling algorithms, convergence and convergence rates of self-consistent iterations. This will be done by developing and using simple but canonical model problems that capture the essential aspects of the problem but allow explicit analytical calculations.
Electronic structure analysis is at the foundation of chemistry and material science, as well as some aspects of biology. Our ability to understand chemical reactions and fundamental aspect of materials relies heavily on efficient numerical algorithms for solving models from quantum chemistry or density functional theory. Existing algorithms are much more effective for insulators than for metals. This is particularly true for the recently developed linear scaling algorithms which relies heavily on the exponential decay property of the wave functions or density matrices, a property that holds for insulators but not for metals. The present project is aimed at bringing powerful mathematical tools to bear on the problem of electronic structure analysis. The proposed work will explore the mathematical features of the electronic structure problem in a way that has never been done before. By doing so, new insights and new algorithms will result that greatly advance our ability to analyze the electronic structure of matter.
Density functional theory is at the present time the most popular tool for analyzing the electronic structure of materials and molecules. It is at the very foundation of material science, chemistry, biology and physics. It is of great importance to the techological innovations in energy, the environment and new forms of materials. Issues in solar cells, batteries, semi-conductors, etc, all rely on our better command of density functional theory. One major difficulty associated with the application of density functional theory has always been its computational cost, which scales cubically as the number of electrons grow in the systems. For this reason, application of the density functional theory has been limited, to a large extend, to either homogeneous systems in materials (i.e. a unit cell of homogenous crystal) or molecules. We have developed an algorithm, called Pepsi (pole expansion plus selected inversion), which for the first time reduces the cost of density functional theory algorithms from cubic scaling to quadratic scaling. For low dimensional systems such as sheets or tubes, the algorithm is even more efficient. Pepsi has been implemented in popular softwares for electronic structure analysis such as Siesta and CPMD. This will allow us to handle inhomogeneous systems, and in some cases, full macromolecules. More importantly, first principle-based design of materials and molecules may become a possibility. For these to become a reality, we still need to couple these algorithms with more coarse-grained models in order to model realistic environmental conditions.
