Time step

The approximation (3.18) of the Green's function has first order accuracy in the timestep. High order approximations can be used. One of the possibilities to gain second order accuracy is to use the formula

$$e^{-\hat H t} = e^{-\hat H_3 t/2} e^{-\hat H_2 t/2} e^{-\hat H_1 t} e^{-\hat H_2 t/2} e^{-\hat H_3 t/2} + O(t^3)$$ (3.92)

The result for the energy in the DMC algorithm depends on the value of the timestep used. The exact ground-state energy is obtained by extrapolating the results to the zero timestep. Approximation (3.92) for the evaluation operator leads to a quadratic dependence of the energy on the timestep. The result of such a calculation is presented in Fig. 3.3. In this respect the use of a quadratic algorithm, such as (), is preferable because for small timestep the results are less sensitive to the choice of the timestep and with a judicious choice one does not need to extrapolate.

Figure 3.3: Hard spheres at $na^3 = 10^{-4}$. Dependence of the energy on the time step.

On one side the timestep has to be small so that the approximation in the Green's function is good, on the other side the larger is the timestep the faster the phase space is explored and less number of iterations are needed for the same statistical accuracy.