Future developments

• Reduction of the number of parameters
  In the present implementation the JAGP wave-function is parametrized with an exceedingly large number of parameters. In fact both in the three-body and in the pair determinant the number of coefficients $\lambda$ increases as $N(N-1)/2$ where $N$ is the number of orbitals in the basis set. So even for a system with only $54$ hydrogen atoms we have to optimize about $3000$ parameters. Although it is still possible (using the strategies showed in Chapter 6, to optimize this wave-function following the ion dynamic with a reasonable accuracy the computational cost) the amount of memory needed to work with so large matrices make impossible to extend this approach to system larger than 54 protons.
  At this stage we are testing different strategies to overcome this problem: the most promising is to use a parametric form for the $\lambda$ coefficients that, to a first approximation, can be chosen to be dependent only from the atomic distances.

• Improving the wave-function optimization.
  Because QMC calculation are very computer demanding, one has to accelerate the ionic dynamics as much as possible. But this is not possible up to a certain threshold otherwise the optimization procedure is not able anymore to follow the Bohr-Oppenheimer ionic dynamics. In order to overcome this difficulty it is possible to change the optimization procedure to converge not to the minimum of the current ionic configuration, but to be as close as possible to the one of the following ionic configuration. This approach can be partially realized with the information we have from the Hessian matrix, and should allow the use of much larger time steps in the GLE.

• Size effects and TABC.
  The computational cost of the Quantum Monte Carlo integration does not allow to study very large system. In order to make the QMC competitive one has to reduce as much as possible the size effects. The size effects derive from the kinetic and the potential energy. We are planing to apply the Twisted Average Boundary Conditions (76) to our system using a Twist sampling in such a way to integrate dynamically the boundary conditions during the simulation. Moreover in these years different strategies were proposed to reduce the finite size effect due to the long-range potential, and we are planing to use some of these strategies (108).