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@ -257,7 +257,9 @@
*** Numerical orbitals *** Numerical orbitals
Trexio supports numerical atom centered orbitals. The implementation is Trexio supports numerical atom centered orbitals. The implementation is
based on the approach of FHI-aims [Reference to paper]. These orbitals are based on the approach of FHI-aims [Blum, V. et al; Ab initio molecular
simulations with numeric atom-centered orbitals; Computer Physics
Communications 2009]. These orbitals are
defined by the atom they are centered on, their angular momentum and a defined by the atom they are centered on, their angular momentum and a
radial function $R_s$, which is of the form radial function $R_s$, which is of the form
\[ \[
@ -266,9 +268,20 @@
Where $u_i(\mathbf{r})$ is numerically tabulated on a dense logarithmic Where $u_i(\mathbf{r})$ is numerically tabulated on a dense logarithmic
grid. It is constructed to vanish for any $\mathbf{r}$ grid. It is constructed to vanish for any $\mathbf{r}$
outside of the grid. The reference points are stored in ~numgrid_r~ outside of the grid. The reference points are stored in ~numgrid_r~
and ~numgrid_phi~. Additionaly, a separate spline for the gradient and Laplacian and ~numgrid_phi~. Additionaly, a separate spline for the first and second
can be stored in ~numgrid_grad~ and ~numgrid_lap~. The index of the first data point for derivative of $u(\mathbf{r})$ can be stored in ~numgrid_grad~ and ~numgrid_lap~.
each shell is stored in ~numgrid_start~, the number of data points per spline Storing them in this form allows to calculate the actual gradients and
Laplacian easily as follows:
\[
\grad_{x_i} \phi = \frac{x_i}{r^2}\left u\prime(r) - \frac{u(r)}{r}\right
\]
\[
\Delta \phi = \frac{1}{r^3}\left x^2 u''(r) + \left3x^2-r^2\right \left \frac{u(r)}{r^2} - \frac{u'(r)}{r}\right \right
\]
The index of the first data point for each shell is stored in
~numgrid_start~, the number of data points per spline
is stored in ~numgrid_size~ for convenience. is stored in ~numgrid_size~ for convenience.
What kind of spline is used can be provided in the ~interpolator_kind~ field. What kind of spline is used can be provided in the ~interpolator_kind~ field.