Random walks to Solve Schrödinger's Equation

Quantum Monte Carlo methods yield a "statistical" solution to the Schördinger equation. The method employed here is know as "fixed node" Greens Function Monte Carlo (GFMC). Subject to the conditions of fixed (chosen) nodal surfaces for the wave function, the GFMC incorporates ALL many-body correlations in the solution.

The animation depicts the evolution of the positions of electrons in a Lithium hydride (LiH) molecule during a quantum mechanical simulation. The spheres represent positions of an electron. This simulation begins with an arbitrary collection or "ensemble" of 1000 different sets of electrons distributed in the unlikely arrangement of a uniform sphere midway between the Li and H atoms.

The simulation seeks a ground state wave function, that is a description of the distribution of electric charge in the molecule, which is represented by an ensemble of possible electron positions distributed in the vicinity of the Li and H nuclei. The animation shows a random walk whose path takes it through all possible electron positions, and the dynamics are controlled by the Schrödinger equation. The time axis shows the evolution as the simulation proceeds toward the final LiH wave function. Time, in this case is best viewed as the number of simulation time steps rather than true time.

The colors indicate a quality measure of an importance function (or guiding function) that represents the best analytic approximation to the true wave function. The quality measure shown is the quantum mechanical energy.

From the animation of the pseudo-dynamics calculation one can see how fast an ensemble evolves spatially towards the true ground state electron distribution. This convergence is quite fast, only a few dozen iterations which corresponds to the time that the random walk would require to move an electron approximately once around the entire molecule.

Most importantly, the quality of the analytic approximation to the true wave function, described by the energy of the system, as the electron ensemble is moved, is easily seen. When a set of 4 electrons gets to a region of configuration space, the color changes dramatically, which indicates that the particular arrangement of electrons is poorly described by the analytic approximation.

The simulation corrects the errors in the approximate analytic wave function, however, the efficiency of the simulation is greatly improved when a highly accurate analytic function is known in advance. Using this visual approach, trouble spots in the analytic approximation are more easily determined, then by sectioning the simulation and viewing different energies (colors) individually, one can see that when electrons are bunched together the quality of the approximation is poor. Thus, high order multiparticle correlations in this region is necessary for improvement. Separate theoretical investigations tell us how to go about making such improvements.