Formulas

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

\[\frac{1}{r} = \int \frac{d\mathbf{G}}{(2\pi)^3}\frac{4\pi}{G^2} e^{i\mathbf{G}\cdot\br}.\]

Fourier transforms

The Fourier transform of a radial function multiplied by a spherical harmonic is:

\[f(G)Y_{\ell m}(\hat G) = \int d\br e^{i\mathbf{G}\cdot\br} f(r)Y_{\ell m}(\br),\]

where

\[f(G) = 4\pi i^\ell \int_0^\infty r^2 dr j_\ell(Gr) f(r).\]

Note

\[e^{i \mathbf{G} \cdot \br} = 4 \pi \sum_{\ell m} i^\ell j_\ell(Gr) Y_{\ell m}(\hat{\br}) Y_{lm}(\hat{\mathbf{G}}).\]

The spherical Bessel function is defined as:

\[j_\ell(x) = \text{Re}\{ \frac{e^{ix}}{x} \sum_{n=0}^\ell \frac{(-i)^{\ell+1-n}}{n!(2x)^n} \frac{(\ell+n)!}{(\ell-n)!} \}.\]

This is implemented in this function:

gpaw.atom.radialgd.fsbt(l, f_g, r_g, G_k)

Fast spherical Bessel transform.

Returns:

 oo
/ 2
|r dr j (Gr) f(r),
/      l
 0

using l+1 fft’s.

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:

\[\frac{d^2 a}{dg^2} (\frac{dg}{dr})^2 r^2 + \frac{da}{dg}(r^2 \kappa \frac{dg}{dr} + r^2 \frac{d^2g}{dr^2} + 2 \alpha r \frac{dg}{dr}) + [2 M r^2 (\epsilon - v) + \alpha (\alpha - 1) - \ell (\ell + 1) + \kappa (\alpha - 1) r] a = 0.\]