formulas.rst 3.58 KB
Newer Older
 jensj committed Jul 14, 2010 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 ======== 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::  jensj committed Sep 23, 2011 21  \frac{1}{r} = \int \frac{d\mathbf{G}}{(2\pi)^3}\frac{4\pi}{G^2}  jensj committed Jul 14, 2010 22 23 24  e^{i\mathbf{G}\cdot\br}.  jensj committed Oct 03, 2011 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 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).  jensj committed Oct 03, 2011 42 43 44 45 46 .. note:: .. math:: e^{i \mathbf{G} \cdot \br} =  jensj committed Oct 28, 2014 47  4 \pi \sum_{\ell m} i^\ell j_\ell(Gr) Y_{\ell m}(\hat{\br})  jensj committed Oct 03, 2011 48 49  Y_{lm}(\hat{\mathbf{G}}).  jensj committed Oct 03, 2011 50 51 52 53 54 55 56 57 58 59 60 61 62 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:  jensj committed Oct 03, 2011 63 .. autofunction:: gpaw.atom.radialgd.fsbt  jensj committed Oct 03, 2011 64 65 66 67 68 69  .. _spherical Bessel function: http://en.wikipedia.org/wiki/Bessel_function #Spherical_Bessel_functions:_jn.2C_yn  jensj committed Jul 14, 2010 70 71 72 73 74 75 76 Gaussians ========= .. math:: n(r) = (\alpha/\pi)^{3/2} e^{-\alpha r^2}, .. math:: \int_0^\infty 4\pi r^2 dr n(r) = 1  jensj committed Oct 03, 2011 77 Its Fourier transform is:  jensj committed Jul 14, 2010 78 79 80 81 82 83 84  .. 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)}.  jensj committed Sep 23, 2011 85 With \nabla^2 v=-4\pi n, we get the potential:  jensj committed Jul 14, 2010 86   jensj committed Sep 23, 2011 87 .. math:: v(r) = \frac{\text{erf}(\sqrt\alpha r)}{r},  jensj committed Jul 14, 2010 88 89 90 91 92 93 94 95  and the energy: .. math:: \frac12 \int_0^\infty 4\pi r^2 dr n(r) v(r) = \sqrt{\frac{\alpha}{2\pi}}.  jensj committed Jun 08, 2012 96 97 Note: \text{erf}(x) \simeq x\sqrt{4/\pi} for small x.  jensj committed Jul 14, 2010 98   jensj committed Oct 03, 2011 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 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.  jensj committed Jul 14, 2010 119 120 121 122 123 Hydrogen ======== The 1s orbital:  jensj committed Jan 26, 2011 124 .. math:: \psi_{\text{1s}}(r) = 2Y_{00} e^{-r},  jensj committed Jul 14, 2010 125 126 127 128 129  and the density is: .. math:: n(r) = |\psi_{\text{1s}}(r)|^2 = e^{-2r}/\pi.  jensj committed Sep 19, 2011 130 131 132 133 134 135 136 137 138 139 140 141 142  Radial Schrödinger equation =========================== With \psi_{n\ell m}(\br) = u(r) / r Y_{\ell m}(\hat\br), we have the radial Schrödinger 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  jensj committed Dec 19, 2014 143 r_g=r(g) for g=0,1,.... Inserting u(r) = a(g) r^{\ell+1}, we  jensj committed Sep 19, 2011 144 145 146 147 get: .. math::  jensj committed Oct 28, 2014 148  \frac{d^2 a}{dg^2} (\frac{dg}{dr})^2 r^2 +  jensj committed Dec 19, 2014 149 150  \frac{da}{dg}(r^2 \frac{d^2g}{dr^2} + 2 (\ell+1) r \frac{dg}{dr}) - 2 r^2 (v - \epsilon) a = 0.  jensj committed Sep 19, 2011 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169  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).  jensj committed Dec 19, 2014 170 171 172 173 174 175 176 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:  jensj committed Sep 19, 2011 177 178 179  .. math::  jensj committed Oct 28, 2014 180 181  \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} +  jensj committed Dec 19, 2014 182 183 184 185  2 \alpha r \frac{dg}{dr}) + [2 M r^2 (\epsilon - v) + \alpha (\alpha - 1) - \ell (\ell + 1) + \kappa (\alpha - 1) r] a = 0.