Approximation and Learning Density Matrices

In this project, we investigate the manipulation, approximation, and learning of density matrices arising in electronic structure calculation in computational chemistry, material science, and solid-state physics. Electronic structure calculation aims at finding the state of the electrons of a given molecular system specified by the nuclear coordinates. The density matrix can be seen as the solution to the governing non-linear eigenvalue problem in the context of Density-Functional Theory (DFT). Therefore, there is an intrinsic map between the nuclear coordinates and the corresponding density matrix. A key challenge is thus to learn this map and/or to interpolate between given density matrices along given reaction coordinates. One challenge is that the set of admissible densities is a non-linear, but differential, manifold that is isomorphic to the Grassmann manifold. We will use tools from computational differential geometry to propose approximate density matrices that lie exactly on the manifold.