Title: Density Functional Theory and Block Tridiagonal Matrix Inversion

Speaker: Dan Erik Petersen, M.Sc.Engineering, Ph.D. Student at the Department of Computer Science, Copenhagen University, Denmark

Date/Time: Wednesday, March 26, 2008, 9:00 - 10:00am

Location: CSRI Building, Room 279 (NM)

Brief Abstract: Simulation software for the analysis of electronic structure at the atomic level is a problem of fundamental interest in today's world of ever increasing use of nano-devices designed at the atomic level. At these length scales, the usual methods of classical physics to predict device characteristics fails in the face of quantum mechanical effects, and new methods rooted in quantum mechanics are the norm.

Since the seminal work of Kohn, Sham, Mermin, and others, the method of Density Functional Theory
(DFT) has become one of the most studied and employed methods of theoreticians and experimentalists
alike in the modeling of properties of materials where quantum mechanical effects matter. The leaps and
bounds of computing power and availability has enabled the useful modeling of larger and more complex
systems and as computing resources improve, simulation will likely become the norm for assessing and
designing most, if not all, nanoscopic devices, materials and molecules in the future.

One of the most important and costly calculation steps that arises in DFT is that of the generating the
Green's function matrix, used both in the process of determining ground state properties of materials or
devices, as well as the transmission, conductance, and other non-equilibrium properties of these modeled
systems. The Green's function matrix is essentially the inverse of a large matrix with block tridiagonal
structure. However, as only the same block tridiagonal structure of the original matrix is desired from the
inverse, algorithms can be designed to take advantage of both the sparsity and structure of the input matrix as well as the desired output matrix.

This talk will give a short introduction to DFT, the origin and properties of the block tridiagonal matrix for
which we need to calculate the Green's function matrix, some algorithms - in serial and parallel - for this
calculation, as well as a much improved method for the calculation of transmission of nanodevices over the usual method found in DFT literature.