# Grassmann manifold, gauge, and quantum chemistry

The Grassmann manifold Gr(m,n) is the set of n-dimensional subspaces in R^m (assuming m>n), and is used in many science and engineering applications. A point in Gr(m,n) can be represented by an orthogonal matrix of size m by n, multiplied by another arbitrary orthogonal matrix of size n by n. In quantum chemistry and in particular the widely used density functional theory (DFT), this arbitrary orthogonal matrix is referred to as the gauge. Physical quantities such as energies and electron densities should be independent of the gauge choice. In this talk, I am going to discuss the interplay between gauge-dependent and gauge-independent quantities in quantum chemistry along three recent directions: time-dependent density functional theory, electron localization, and self-consistent field iteration. In each case, the focus on the gauge-independent representation of the Grassmann manifold brings interesting, and sometimes surprising numerical benefits.

*Lin Lin is currently an assistant professor at the Department of Mathematics at UC Berkeley, and a faculty scientist at LBNL. His current research focuses on the development and analysis of efficient numerical methods for electronic structure calculations. He has recently received the Sloan fellowship, the NSF CAREER award, the DOE Early Career award, and the SIAM Computational Science and Engineering (CSE) early career award.*