Diffusion Monte Carlo importance sampling

Diffusion Monte Carlo with Importance Sampling... 

There are five important sources of error in these first diffusion Monte Carlo calculations (1) statistical or sampling error associated with the limited number of independent sample energies used in determining the energy from an average of variable potential energies, (2) the use of a finite time step At rather than an infinitesimal time step as required for the exact simulation of a differential equation, (3) numerical error associated with trimcation and/or round-off... 

The first applications in diffusion Monte Carlo were made for the nodeless ground state of the molecular ion The effect was a substantial improvement in accuracy from -1.3414 0.0043 hartree (-841.74 kcal/mol) for the total energy in an earlier calculation to -1.3439 0.0002 hartree in a similar calculation using importance sampling. The statistical error is reduced by a factor of about 20, and any systematic error is presumed to be similarly reduced. 

To obtain the importance sampling version of diffusion quantum Monte Carlo, we first multiply the basic equation, Eq. [9] by a trial wavefunction /q and define a new function f - x A /o, which is the product of the true wavefunction and the trial wavefunction. After several pages of rearrangement, one may obtain the basic equation for DQMC with importance sampling, ... 

In general, Monte Carlo methods refer to any procedures which involve sampling from random numbers. These methods are used in simulations of natural phenomena, simulation of experimental apparatus, and numerical analysis. An important feature is the simple structure of the computational algorithm. The method was developed by von Neuman, Ulam, and Metroplois during World War II to study the difiiision of neutrons in fissionable materials (ie, atomic bomb design)- Let us consider atom diffusion and demonstrate the principle of the Monte Carlo method. A two-dimensional square grid (Fig. 3.20A) represents interstitial sites in a sofid. 

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