Two body Jastrow term

As it is well known, the Jastrow term plays a crucial role in treating many body correlation effects. One of the most important correlation contribution arises from the electron-electron interaction. Therefore it is important to use at least a two-body Jastrow factor in the trial wave function. Moreover this term reduces the probability for two electrons to be close, and so decreases the average value of the repulsive interaction, providing a clear energy gain. The two-body Jastrow function reads:

$$J_2(\vec{r}_1,...,\vec{r}_N) = \exp{\left (\sum_{i<j}^{N}u(r_{ij}) \right )},$$ (1.26)

where $u(r_{ij})$ depends only on the relative distance $r_{ij}= \vert\vec{r}_i -\vec{r}_j\vert$ between two electrons and allows to fulfill the cusp conditions for opposite spin electrons as long as $u(r_{ij})\to \frac{r_{ij}}{2}$ for small electron-electron distance. The pair correlation function $u$ can be parametrized successfully by few variational parameters.

We have adopted a functional form $u$ proposed by Fahy (38), that we found particularly convenient:

$$u(r)= { \frac{r}{2 (1 + b r )} },$$ (1.27)

where the variational parameter $b$ has been optimized for each system. In this functional form the cusp condition for anti-parallel spin electrons is satisfied, whereas the one for parallel spins is neglected in order to avoid the spin contamination (for more details about spin contamination see Ref. (39) ). This allows to remove the singularities of the local energy due to the collision of two opposite spin electrons, yielding a smaller variance and a more efficient VMC calculation.