Coulomb
\[\frac{1}{|\br-\br'|} =
\sum_\ell \sum_{m=-\ell}^\ell
\frac{4\pi}{2\ell+1}
\frac{r_<^\ell}{r_>^{\ell+1}}
Y_{\ell m}^*(\hat\br) Y_{\ell m}(\hat\br')\]
or
\[\frac{1}{r} = \int \frac{d\mathbf{G}}{(2\pi)^3}\frac{4\pi}{G^2}
e^{i\mathbf{G}\cdot\br}.\]
Gaussians
\[n(r) = (\alpha/\pi)^{3/2} e^{-\alpha r^2},\]
\[\int_0^\infty 4\pi r^2 dr n(r) = 1\]
Its Fourier transform is:
\[n(k) = \int d\br e^{i\mathbf{k}\cdot\br} n(r) =
\int_0^\infty 4\pi r^2 dr \frac{\sin(kr)}{kr} n(r) =
e^{-k^2/(4a)}.\]
With \(\nabla^2 v=-4\pi n\), we get the potential:
\[v(r) = \frac{\text{erf}(\sqrt\alpha r)}{r},\]
and the energy:
\[\frac12 \int_0^\infty 4\pi r^2 dr n(r) v(r) =
\sqrt{\frac{\alpha}{2\pi}}.\]
Note: \(\text{erf}(x) \simeq x\sqrt{4/\pi}\) for small \(x\).
Shape functions
GPAW uses Gaussians as shape functions for the PAW compensation charges:
\[g_{\ell m}(\br) =
\frac{\alpha^{\ell + 3 / 2} \ell ! 2^{2\ell + 2}}
{\sqrt{\pi} (2\ell + 1) !}
e^{-\alpha r^2}
Y_{\ell m}(\hat{\br}).\]
They are normalized as:
\[\int d \br g_{\ell m}(\br) Y_{\ell m}(\hat{\br}) r^\ell = 1.\]
Hydrogen
The 1s orbital:
\[\psi_{\text{1s}}(r) = 2Y_{00} e^{-r},\]
and the density is:
\[n(r) = |\psi_{\text{1s}}(r)|^2 = e^{-2r}/\pi.\]
Radial Schrödinger equation
With \(\psi_{n\ell m}(\br) = u(r) / r Y_{\ell m}(\hat\br)\), we have the
radial Schrödinger equation:
\[-\frac12 \frac{d^2u}{dr^2} + \frac{\ell(\ell + 1)}{2r^2} u + v u
= \epsilon u.\]
We want to solve this equation on a non-equidistant radial grid with
\(r_g=r(g)\) for \(g=0,1,...\). Inserting \(u(r) = a(g) r^{\ell+1}\), we
get:
\[\frac{d^2 a}{dg^2} (\frac{dg}{dr})^2 r^2 +
\frac{da}{dg}(r^2 \frac{d^2g}{dr^2} + 2 (\ell+1) r \frac{dg}{dr}) -
2 r^2 (v - \epsilon) a = 0.\]
Including Scalar-relativistic corrections
The scalar-relativistic equation is:
\[-\frac{1}{2 M} \frac{d^2u}{dr^2} + \frac{\ell(\ell + 1)}{2Mr^2} u -
\frac{1}{(2Mc)^2}\frac{dv}{dr}(\frac{du}{dr}-\frac{u}{r}) + v u
= \epsilon u.\]
where the relativistic mass is:
\[M = 1 - \frac{1}{2c^2} (v - \epsilon).\]
With \(u(r) = a(g) r^\alpha\), \(\kappa = (dv/dr)/(2Mc^2)\) and
\[\alpha = \sqrt{\ell^2 + \ell + 1 -(Z/c)^2},\]
we get:
\[\frac{d^2 a}{dg^2} (\frac{dg}{dr})^2 r^2 +
\frac{da}{dg}(r^2 \kappa \frac{dg}{dr} + r^2 \frac{d^2g}{dr^2} +
2 \alpha r \frac{dg}{dr}) +
[2 M r^2 (\epsilon - v) +
\alpha (\alpha - 1) - \ell (\ell + 1)
+ \kappa (\alpha - 1) r] a = 0.\]