Numerical integration

Numerical Gauss-Hermite integration is used to calculate the radial integrals occurring in the expressions for matrix elements [Eqs. (113) and (114)] and total energies [Eq. (120)]. This kind of integration is for integrals of the form $\int_{-\infty}^{\infty}e^{-x^{2}}f\left(x\right)dx$, and in order to obtain this form our integrals are transformed by using $x'=\left(\sqrt{2}b\right)x$. The integrals can then be written as:

$$\begin{eqnarray*} \int_{0}^{\infty}e^{-2\left(bx\right)^{2}}f\left(bx\right)x^{2... ...{i=1}^{N/2} w_{i} f\left( x''_i \right) \left( x''_i \right)^{2}.\end{eqnarray*}$$

Where $f_{sym}\left(x\right)=f\left(x\right)\theta\left(x\right)+f\left(-x\right)\theta\left(-x\right)$ was used in an intermediate step. To reduce the number of grid points by half to $N_{\mbox{\scriptsize {grid}}}=N/2$ it was also used that for Gauss-Hermite integration, the weight functions $w_i$ and grid points $x''_i =\frac{x'_i}{\sqrt{2}} =b r_i$ are symmetric about the origin. The integrals for matrix elements and total energies of most terms become exact when $N_{\mbox{\scriptsize {grid}}}=N_{0}+2$, where $N_{0}$ denotes the maximum HO shell included in the basis. But in general more points are needed when the integrand cannot be expressed as a product of four basis states, e.g., in the case for the Coulomb interaction and also for the density-dependent terms.