One body term

Another important term of our trial-function it is the one-body term. In fact as pointed out in Ref. (36), it is easier to optimize a one-body term explicitly rather than including more orbitals in the determinantal basis set.
Moreover, even if it is possible to satisfy nuclear cusp conditions (see Appendix C) with the pairing determinant, this has to be done iteratively during the optimization process adding constraints to the variational parameters or approximately disregarding the term of the eq.1.22. In order to solve efficiently this problem we included nuclear cusp conditions explicitly in the one-body term, in the same way of ref. (37):

$$J_1(\vec{r}_1,...,\vec{r}_N) = \exp{\left [ \sum_{i,a}^{N} \left ( \xi_a(\vec{r}_{i}) + \Xi_a(\vec{r}_{ia}) \right ) \right ]},$$ (1.23)

where $\xi_a(\vec{r}_{i})$ orbital is used to satisfy the nuclear cusp conditions on nucleus $a$ :

$$\xi_a (r)= { \frac{-Z_a r}{ (1 + b r )} }$$ (1.24)

and the $\Xi(\vec{r}_{ia}) = \sum_l \lambda_l \psi_{a,l}(\vec{r}_{ia})$ is a linear combination of atomic orbitals centered on the nucleus $a$ , and that do not effect the nuclear cusp condition. We have used Gaussian and exponential orbitals such to have a smooth behaviour close to the corresponding nuclei, namely as:

$$\psi_{a,i} (\vec{r}) -\psi_{a,i} ( \vec{R}_a ) \simeq \vert\vec{r}-\vec{R}_a\vert^2,$$ (1.25)

or with larger power, in order to preserve the nuclear cusp conditions (1.24).
The basis set $\psi_{a,i} (\vec{r})$ is the same used in the so-called three-body term that we are going to describe in the following. The same kind of behavior has been imposed for the orbitals appearing in the determinant. In this way the nuclear cusp conditions are very easily satisfied for a general system containing many atoms, in a simple and efficient way.