Three Body Jastrow term

In order to describe well the correlation between electrons the simple two-body Jastrow factor is not sufficient. Indeed it takes into account only the electron-electron separation and not the individual electronic position $\vec{r}_i$ and $\vec{r}_j$ . It is expected that close to atoms the correlation effects deviate significantly from the translational invariant Jastrow. For this reason we introduce a factor, often called three body (electron-electron-nucleus) Jastrow, that explicitly depends on both electronic positions $\vec{r}_i$ and $\vec{r}_j$ . The three body Jastrow is chosen to satisfy the following requirements:

• The cusp conditions set up by the two-body Jastrow term and by the one-body term are preserved.
• Similarly to the two-body we do not include any spin dependency in the three-body Jastrow. In the way the wave-function remains a spin singlet.
• Whenever the atomic distances are large it factorizes into a product of independent contributions located near each atom, an important requirement to satisfy the size consistency of the variational wave function.

Analogously to the pairing trial function in Eq. 1.19 we define a three body factor as:

$$J_3(\vec{r}_1,...,\vec{r}_N) = \exp \left( \sum_{i<j} \Phi_J(\vec{r}_i,\vec{r}_j) \right)$$

$$\Phi_J(\vec{r}_i,\vec{r}_j) = \sum_{l,m,a,b} g_{l,m}^{a,b}\psi_{a,l} (\vec{r}_i)\psi_{b,m} (\vec{r}_j),$$ (1.28)

where indices $l$ and $m$ indicate different orbitals located around the atoms $a$ and $b$ respectively. Each Jastrow orbital $\psi_{a,l}(\vec{r})$ is centred on the corresponding atomic position $\vec{R}_a$ . We have used Gaussian and exponential orbitals multiplied by appropriate polynomials of the electronic coordinates, related to different spherical harmonics with given angular momentum, as in the usual Slater basis.

The chosen form for the 3-body Jastrow (1.28) has very appealing features: it easily allows to include the symmetries of the system by imposing them on the matrix $g_{l,m}^{a,b}$ exactly as it is possible for the pairing part (e.g. by replacing $\lambda_{m,n}^{a,b}$ with $g_{m,n}^{a,b}$ in Eq. 1.20). It is size consistent, namely the atomic limit can be smoothly recovered with the same trial function when the matrix terms $g_{l,m}^{a,b}$ for $a \ne b$ approach zero in this limit (see Ref. (15)). Notice that a small non zero value of $g_{l,m}^{a,b}$ for $a \ne b$ acting on p-wave orbitals can correctly describe a weak interaction between electrons such as the Van der Waals forces.