The movies to the left illustrate the basic features of the diffusion Monte Carlo (DMC) algorithm for a very simple example, the Neon dimer. The DMC method allows, in its simplest implementation, determination of the stationary ground state energy and structural expectation values of many-body quantum systems.

A little bit of background on Neon clusters:

At low enough temperatures noble Neon atoms can bond, forming molecules of two atoms (dimers), three atoms (trimers), or more. For illustrative purposes, we consider the Neon dimer. A black line in the two movies on the left shows the Born-Oppenheimer potential curve V(R) as a function of the interparticle distance R. The functional form we have chosen is a Lennard-Jones potential, attaining a minimum of Vmin = -35.53 Kelvin at Re = 5.875 Bohr radii. The energy of two separated atoms is zero, that is, V(R) = 0 for R->infinity. In the ground state, which is of interest here, the Neon dimer "oscillates" about the potential minimum, so it is around the distance Re that we expect to find the molecule with the largest probability. The black line labeled P(R) shows the density (or equivalently, the probability) of the system, obtained by solving the radial time-independent Schrödinger equation numerically. The movies show how the DMC algorithm determines this de nsity by "throwing dice". The first movie illustrates the equilibration process, while the second movie illustrates the accumulation of the density profile.

About the DMC simulation:

The DMC algorithm solves the time-independent Schrödinger equation by propagating so-called "walkers" in imaginary time. Changing to the time-dependent Schrödinger equation, written in terms of imaginary time, can be viewed as a numerical trick; only the stationary ground state energy and ground state structural properties can be determined by the DMC algorithm - not time-dependent observables. The first movie on the left shows how the DMC algorithm, starting with 1000 initial configurations with interparticle distance R around 7 Bohr radii, equilibrates towards the true ground state density P(R). The second movie on the left shows how the DMC algorithm builds up the ground state density using a set of "equilibrated walkers", that is, using a walker distribution that has been propagated for a sufficiently large imaginary time. The walker distribution reflects the ground state density, implying that expectation values can be accumulated by propagating the equilibrat ed walker distribution.

Typically, we use a walker ensemble consisting of about 1000 walkers. For illustrative purposes, however, the second movie on the left shows the histogram accumulation for five walkers only. At a given time step (movie frame), each walker is represented by a filled red dot illustrating the interparticle distance R of the walker (note that the simulation itself uses Cartesian coordinates). Horizontal lines show the energy of each of the five walkers. At the beginning of the movie, the histogram looks "rugged", not yet resembling the smooth density, calculated by alternative numerical means (see above). After each time step, the histogram is recalculated: The new walker positions are "binned" into a histogram, and added to the previous histogram with the appropriate weights; finally, the new histogram is normalized appropriately. Thus, as the movie, or equivalently the propagation in imaginary time progresses, the histogram becomes smoother, eventually closely tracing the smoo th density. (Note that this movie shows the histogram only for every 200 or so time steps, ensuring uncorrelated sampling.) Since the DMC sampling is stochastic in nature, the calculated density has an errorbar. This errorbar can be reduced by propagating for longer imaginary times. The histogram can also be "smoothed" by chosing a smaller bin size. The energy of the system can be calculated by averaging the energies of all walkers.

Outlook:

Since the wavefunction of the Neon dimer depends only on the interparticle distance R, its determination through the DMC algorithm can be visualized easily. Although not the most efficient for calculations on this particular system, the DMC method becomes incomparably powerful for systems with large numbers of interacting particles.