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.