Reducing the number of parameters

After each ionic move it is important to optimize the trial function to be as close as possible to the variational ground state. The point is that there are some parts of the wave function that vary a bit between different ionic configurations, as for example the two-body Jastrow factor or the core orbitals for large ions. So it is not important to move all parameters at each step, but it is fundamental to recognize which ones have to be optimized because they are far from the minimum. Moreover since we are performing an optimization in the presence of noise it is possible to know the exact minimum only within a given statistical error due to the finite sampling. So we fix a tolerance $\xi$ and move only such parameters whose generalized forces satisfy:

$$\frac{\left\vert F_i \right\vert}{\Delta F_i} \ge \xi$$ (5.29)

we have chosen $\xi \simeq 4$ that amounts to change on average the $20\%$ of the variational parameters. This procedure has allowed us to follow the ion dynamic with a stable and fast optimization on the variational parameters. Moreover to stabilize the optimization procedure other two cautions were used:
1. It is very difficult to parametrize an highly correlated wave-function with many parameters. In fact it happens often that in this case the stochastic matrix becomes singular because there are too many parameters redundant to describe the wave-function and some of them have to be eliminated for a stable optimization scheme, see section 2.1.2.
2. It can happen that the Hessian Matrix is not positive definite, this can be due to different reasons: a statistical fluctuation due the finite sampling or because the trial-function is far from the minimum, in this case the correction, proposed in Ref. (50) and described at the end of section 2.3, is used. In our simulations we have used $0.2$ as a threshold value for the $r$ parameters that rules the stability of the SRH optimization (see section 2.3). This has achieved a fast and stable convergence of the wave-function.