The statistical error of green's function Monte Carlo

The statistical error in the ground state energy as calculated by Green's Function Monte Carlo (GFMC) is analyzed and a simple approximate formula is derived which relates the error to the number of steps of the random walk, the variational energy of the trial function, and the time step of the random walk. Using this formula it is argued that as the thermodynamic limit is approached with N identical molecules, the computer time needed to reach a given error per molecule increases as N ^h where 0.5 <b < 1.5 and as the nuclear charge Z of a system is increased the computer time necessary to reach a given error grows as Z ^5.5. Thus GFMC simulations will be most useful for calculating the properties of low Z elements. The implications for choosing the optimal trial function from a series of trial functions is also discussed.