This page will describe our work involving developing new methods and algorithms for solving the Schrodinger equation. An aim is to be able to model large systems at an accurate quantum mechanical level.
In the mid-1990s, our group started working on new approaches for solving the Kohn-Sham equations of Density Functional Theory (DFT). Previously, most electronic structure calculations were done by representing the wave functions with a basis set expansion, often of gaussian form (other forms include Slater-type orbitals and plane waves). We, along with a few other groups, developed methods in which the wave function is represented on a grid in real-space. In addition, taking ideas from applied mathematics, we developed multigrid methods that accelerate the solution process. The work has been summarized in a Rev. Mod. Phys. article in 2000 and a chapter in Rev. Comput. Chem. v. 26 (below).
Our work has turned in an alternative direction based on a collaboration with Joel Dedrick, a chip design expert and venture capital developer in Silicon Valley. He contacted TLB several years ago after reading the literature on electronic structure methods. He thought the multigrid approach held promise for large-scale applications directed at drug design (in the long term). We had a meeting at Los Alamos with several experts there, and decided that Dedrick's hardware design did not mesh so well with the multigrid approach. We would like an algorithm that is directed at massively parallel calculations on millions of processors.
This led to some new ideas that link Quantum Monte Carlo methods and the Feynman-Kac approach for solving PDEs. TLB and Dedrick are completing a book chapter for "Solving the Schrodinger Equation: Has Everything Been Tried?" edited by Paul Popelier, Imperial College Press, London. That chapter summarizes some of our work so far on this problem. The goal is to use a density matrix representation and obtain the 1-DM (and perhaps 2-DM) using random walks and the Feynman-Kac theory. The work is in progress.
A long-term goal is to develop computational methods that are relatively local in space. One 'radical' approach has been proposed recently based on Bohm's formulation of quantum mechanics, namely "quantum mechanics without wavefunctions". See the paper by Schiff and Poirier, J. Chem. Phys. 136, 031102 (2012). They apply their approach to relatively simple one-particle problems. Is there a way to extend this theory to many-particle systems including fermions (electrons)? Other approaches might include cellular automata. Can we invent a 'game' that in the end effectively solves the Schrodinger equation without having to solve for the huge-dimensional wavefunction? An interesting book from 1993 delves into some of these issues: "The quantum theory of motion" by Peter Holland, Cambridge, Cambridge.
Some publications concerning electronic structure methods:
T. L. Beck, Real Space Mesh Techniques in Density Functional Theory, Rev. Mod. Phys., 72, 1041-1080 (2000).
T. L. Beck, Real-Space and Multigrid Methods in Chemistry, Reviews in Computational Chemistry Volume 26 (Wiley, New York, 2009).
T. L. Beck and J. H. Dedrick, "Solving the Schrodinger Equation on Real-Space Grids and with Random Walks," in Solving the Schrodinger Equation: Has Everything Been Tried, ed. P. Popelier, Imperial College Press, London, 2011.