mirror of
https://github.com/TREX-CoE/trexio.git
synced 2024-08-24 14:11:44 +02:00
Fixed formulas for NAOs
This commit is contained in:
parent
0c001fd5a3
commit
7fa481882b
16
trex.org
16
trex.org
@ -272,19 +272,19 @@
|
||||
outside of the grid. The reference points are stored in ~nao_grid_r~
|
||||
and ~nao_grid_phi~. Additionaly, a separate spline for the first and second
|
||||
derivative of $u(\mathbf{r})$ can be stored in ~nao_grid_grad~ and ~nao_grid_lap~.
|
||||
Storing them in this form allows to calculate the actual gradients and
|
||||
Laplacian easily as follows:
|
||||
Storing them in this form allows to calculate the actual first and
|
||||
second derivatives easily as follows:
|
||||
|
||||
\[
|
||||
\nabla_{x_i} \phi = \frac{x_i}{r^2}\left( u^\prime\left(r\right) - \frac{u\left(r\right)}{r}\right)
|
||||
\frac{\partial \phi}{\partial x} = \frac{x}{r^2}\left( u^\prime\left(r\right) - \frac{u\left(r\right)}{r}\right)
|
||||
\]
|
||||
\[
|
||||
\Delta \phi = \frac{1}{r^3}\left(x^2 u^{\prime\prime}(r) + \left( 3x^2-r^2\right) \left( \frac{u(r)}{r^2} - \frac{u'(r)}{r}\right) \right)
|
||||
\frac{\partial^2 \phi}{\partial x^2} = \frac{1}{r^3}\left(x^2 u^{\prime\prime}(r) + \left( 3x^2-r^2\right) \left( \frac{u(r)}{r^2} - \frac{u'(r)}{r}\right) \right)
|
||||
\]
|
||||
|
||||
The index of the first data point for each shell is stored in
|
||||
~nao_grid_start~, the number of data points per spline
|
||||
is stored in ~nao_grid_size~ for convenience.
|
||||
The index of the first data point for each shell is stored in
|
||||
~nao_grid_start~, the number of data points per spline is stored
|
||||
in ~nao_grid_size~ for convenience.
|
||||
|
||||
What kind of spline is used can be provided in the ~interpolator_kind~ field.
|
||||
For example, FHI-aims uses a cubic spline, so the ~interpolator_kind~ is
|
||||
@ -299,7 +299,7 @@
|
||||
will return the value of $u(\mathbf{r})$ at the $i$th reference point.
|
||||
A radius is converted to this scale by (note the zero-indexing)
|
||||
\[
|
||||
i_{log} = \frac{1}{c} \cdot \log{\frac{r}{r_0}}
|
||||
i_{\log} = \frac{1}{c} \cdot \log \left( \frac{r}{r_0} \right)
|
||||
\]
|
||||
where
|
||||
\[
|
||||
|
Loading…
Reference in New Issue
Block a user