Given a generic trial-function , not orthogonal to the ground state it is possible to obtain a new one closer to the ground-state by applying the operator to this wave-function for a sufficient large . The idea of the Stochastic Reconfiguration is to change the parameters of the original trial-function in order to be as close as possible to the projected one.

For this purpose we define:

where is the projected one and is the new trail-function obtained changing variational parameters. We can write the equation eq. 2.2 as:

(2.3) |

where

Now we want to choose the new parameters in such a way that is as close as possible to . Thus we require that a set of mixed average correlation function, corresponding to the two wave-functions 2.2, 2.1, are equal. Here we impose precisely that:

(2.5) |

for . This is equivalent to the equation system:

(2.6) | |||

for | (2.7) |

Because the equation for is related to the normalization of the trial-function and this parameter doesn't effect any physical observable of the system, we can substitute from the first equation in the others:

where

The solution of this equation system defines a direction in the parameters space. If we vary parameters along this direction for a sufficient small step we will decrease the energy.

The matrix is calculated at each iteration through a standard variational Monte Carlo sampling; the single iteration constitutes a small simulation that will be referred in the following as ``bin''. After each bin the wave function parameters are iteratively updated according to

(2.10) |

SR is similar to a standard steepest descent (SD) calculation, where the expectation value of the energy is optimized by iteratively changing the parameters according to the corresponding derivatives of the energy (generalized forces):

namely:

(2.12) |

where is a suitable small time step, which can be taken fixed or determined at each iteration by minimizing the energy expectation value.

Indeed the variation of the total energy at each step is easily shown to be negative for small enough because, in this limit

Thus the method certainly converges at the minimum when all the forces vanish. In the SR we have

Using the analogy with the steepest descent, it is possible to show that convergence to the energy minimum is reached when the value of is sufficiently small and is kept constant for each iteration. Indeed the energy variation for a small change of the parameters is:

and it is easily verified that the above term is always negative because the reduced matrix , as well as , is positive definite, being an overlap matrix with all positive eigenvalues.

For a stable iterative method, such as the SR or the SD one, a basic ingredient is that at each iteration the new parameters are close to the previous according to a prescribed distance. The fundamental difference between the SR minimization and the standard steepest descent is just related to the definition of this distance. For the SD it is the usual one, that is defined by the Cartesian metric , instead the SR works correctly in the physical Hilbert space metric of the wave function , yielding namely the square distance between the two wave functions corresponding to the two different sets of variational parameters and . Therefore, from the knowledge of the generalized forces , the most convenient change of the variational parameters minimizes the functional , where is the linear change in the energy and is a Lagrange multiplier that allows a stable minimization with small change of the wave function . Then the final iteration (2.13) is easily obtained.

The advantage of SR compared with SD is obvious because sometimes a small change of the variational parameters corresponds to a large change of the wave function, and the SR takes into account this effect through the Eq. 2.13. In particular the method is useful when a non orthogonal basis set is used, as we have in this thesis. Moreover by using the reduced matrix it is also possible to remove from the calculation those parameters that imply some redundancy in the variational space, as it is shown in the following sections of this chapter.