The quantum physics of many interacting electrons lies at the foundation of chemistry and condensed matter physics. A direct treatment of the many-electron problem is impossible due to its shear complexity: dealing with N interacting electrons requires solving partial differential equations in 3N dimensions. Equilibrium and non-equilibrium Density Functional Theories (DFT) are rigorous and formally exact theories which map the interacting N-electron problem into a non-interacting N-electron problem. The non-interacting electrons move in an effective potential that has a universal functional dependence on the total electron density. As a result, the problem is reduced to a problem in dimension 3, amenable for computation. In this proposal the PIs propose to study a number of dynamical problems in many-body quantum mechanics within an interdisciplinary environment of mathematicians and physicists. In particular, the PIs propose to develop further the mathematical foundations of density-functional theory, for equilibrium as well as the time-dependent case. The mathematical structure of the theory and its solutions will be further investigated and the insight from this analysis will be used to develop efficient numerical simulations. Particular emphasis will be given to the treatment of the spin-orbit interaction, within the full relativistic formulations and in non-relativistic formulations that include relativistic corrections. The PIs also plan to establish the foundations of the Dissipative Time-Dependent Density Functional Theory, and to apply the theory to the problem of charge and spin transport in materials.
The present technological progress is in great part based on design and discovery of new materials. Nowadays, the design of advanced materials involves laboratory work and computer simulations. Enhancing the accuracy and efficiency of computer simulations will reduce the costs, broaden the array of interesting and potentially useful materials, and speed up the process of testing and characterization. This is the target of the proposed research. The plan is to combine rigorous mathematical analysis, the insights from physics, chemistry and computer simulations in order to push the boundaries of theoretical simulations of advanced materials such as nano-structured materials, topological insulators and molecular electronic devices. The proposed research could have significant technological impact in applications such as nano-science and other areas of interest such as solar cell devices and energy conversion and storage. The PIs propose to integrate research and education by involving undergraduate and graduate students, and post-doctoral associates, in an interdisciplinary environment. Special attention will be paid to the recruitment of women and students from other underrepresented groups through the utilization of a diverse number of programs at the participating institutions.
Electronic structure analysis is one of the most fundational problems in computational science, particularly in material science, biology and chemistry. However, this task is notoriously hard due to the high cost of the algorithms. Existing algorithms for general systems scales at least like N^3 for a system with N atoms. Through this project, we have developed and implemented the PEXSI (pole expansion and selected inversion) algorithm for analyzing the electronic structure of materials based on density functional theory (DFT). The computational complexity of the PEXSI algorithm scales like N^2 for general systems including metallic systems (for lower dimensional systems such as graphene, the complexity is even lower). This is the first time that a less than cubic scaling algorithms has been developed. The algorithm has been implemented on parallel computers as well as popular software packages such as Siesta. The results are very promising: It appears that one can now analyze general systems with more than 10,000 atoms with relatively ease. This is at least an increase by a factor of 10, compared with the capabilities of previous algorithms and packages. In addition to PEXSI, we have also developed adaptive and optimal base functions for DFT. This allows us to substantially lower the degrees of freedom needed to solve DFT problems. Another new development is the extension to Dirac-Kohn-Sham problem that takes into account the relativistic effects. This is important for heavy atoms.
