# Density-Functional Theory and its Generalizations: Legendre Transform, Constrained Search, Open Problems

Wednesday, August 1, 2007 - 10:15am - 11:15am

EE/CS 3-180

Paul Ayers (McMaster University)

The quantum many-electron problem is easy in principle (solve the

There is a “catch.” Reducing the number of dimensions leads to other problems associated with approximating the energy functional and/or associated with restricting the domain of the variational procedure. Two powerful techniques for resolving these difficulties are the Legendre transform and constrained-search formulations of density functional theory. This talk will discuss these formulations, and show how they can be extended to define generalized density-functional theories. I'll conclude with some of my favorite open problems in density-functional theory.

*N*-electron Schrödinger equation) and hard in practice (because the cost of numerical methods typically grows exponentially with the number of variables). However, there are simplifying features. First, the dimensionality can be reduced because electronic Hamiltonians contain only 1-body and 2-body terms. (This leads to reduced density-matrix methods.) Second, the dimensionality can be reduced because electrons are identical particles: if you know everything about one electron, then you know everything about all of the electrons. (This leads to electron-propagator theory and density-functional theory.)There is a “catch.” Reducing the number of dimensions leads to other problems associated with approximating the energy functional and/or associated with restricting the domain of the variational procedure. Two powerful techniques for resolving these difficulties are the Legendre transform and constrained-search formulations of density functional theory. This talk will discuss these formulations, and show how they can be extended to define generalized density-functional theories. I'll conclude with some of my favorite open problems in density-functional theory.