The new positions in coordinate space are given as the solutions of the Langevin equation using Euler's method, namely, we go from the Langevin equation

$$\frac{\partial x(t)}{\partial t} = DF(x(t)) +\eta,$$

with $\eta$ a random variable, yielding a new position

$$y = x+DF(x)\Delta t +\xi\sqrt{\Delta t},$$

where $\xi$ is gaussian random variable and $\Delta t$ is a chosen time step. The quantity $D$ is, in atomic units, equal to $1/2$ and comes from the factor $1/2$ in the kinetic energy operator. Note that $\Delta t$ is to be viewed as a parameter. Values of $\Delta t \in [0.001,0.01]$ yield in general rather stable values of the ground state energy.