From 7fa481882be5644cf0dd05f267d59c55ec8872f1 Mon Sep 17 00:00:00 2001 From: Anthony Scemama Date: Wed, 13 Mar 2024 14:25:02 +0100 Subject: [PATCH] Fixed formulas for NAOs --- trex.org | 16 ++++++++-------- 1 file changed, 8 insertions(+), 8 deletions(-) diff --git a/trex.org b/trex.org index 93dce8c..a6f55ee 100644 --- a/trex.org +++ b/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 \[