======== Formulas ======== Coulomb ======= .. math:: \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 .. math:: \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: .. math:: f(G)Y_{\ell m}(\hat G) = \int d\br e^{i\mathbf{G}\cdot\br} f(r)Y_{\ell m}(\br), where .. math:: f(G) = 4\pi i^\ell \int_0^\infty r^2 dr j_\ell(Gr) f(r). .. note:: .. math:: 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: .. math:: 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: .. autofunction:: gpaw.atom.radialgd.fsbt .. _spherical Bessel function: http://en.wikipedia.org/wiki/Bessel_function #Spherical_Bessel_functions:_jn.2C_yn Gaussians ========= .. math:: n(r) = (\alpha/\pi)^{3/2} e^{-\alpha r^2}, .. math:: \int_0^\infty 4\pi r^2 dr n(r) = 1 Its Fourier transform is: .. math:: 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: .. math:: v(r) = \frac{\text{erf}(\sqrt\alpha r)}{r}, and the energy: .. math:: \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: .. math:: 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: .. math:: \int d \br g_{\ell m}(\br) Y_{\ell m}(\hat{\br}) r^\ell = 1. Hydrogen ======== The 1s orbital: .. math:: \psi_{\text{1s}}(r) = 2Y_{00} e^{-r}, and the density is: .. math:: n(r) = |\psi_{\text{1s}}(r)|^2 = e^{-2r}/\pi. Radial SchrÃ¶dinger equation =========================== With \psi_{n\ell m}(\br) = u(r) / r Y_{\ell m}(\hat\br), we have the radial SchrÃ¶dinger equation: .. math:: -\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: .. math:: \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: .. math:: -\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: .. math:: M = 1 - \frac{1}{2c^2} (v - \epsilon). With u(r) = a(g) r^\alpha, \kappa = (dv/dr)/(2Mc^2) and .. math:: \alpha = \sqrt{\ell^2 + \ell + 1 -(Z/c)^2}, we get: .. math:: \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.