From f45416efc594e6ca336f419e23aca2ba8d475cf6 Mon Sep 17 00:00:00 2001 From: anoop chandran Date: Mon, 17 Dec 2018 09:31:06 +0100 Subject: [PATCH] first attempt to fixing overflowing equation --- source/tutorials.rst | 3 ++- 1 file changed, 2 insertions(+), 1 deletion(-) diff --git a/source/tutorials.rst b/source/tutorials.rst index a4974ce..91f84a3 100644 --- a/source/tutorials.rst +++ b/source/tutorials.rst @@ -389,7 +389,8 @@ Polarization function ===================== The polarization function gives the linear change in the electronic density of a non-interacting system with respect to changes in the effective potential. It is, thus, a fundamental quantity in the calculation of screening properties of a many-electron systems. For example, the dielectric function, instrumental in the calculation of spectroscopic quantities (e.g. EELS) and the screened interaction needed in GW, is related to the polarization matrix through :math:{\varepsilon(\mathbf{k},\omega)=1-P(\mathbf{k},\omega)v(\mathbf{k})}, here given in matrix notation. The corresponding explicit formula for matrix elements of P in the mixed product basis is -:math:{P_{\mu\nu}(\mathbf{k},\omega)=2\sum_{\mathbf{q}}^{\mathrm{BZ}}\sum_{n}^{\mathrm{occ}}\sum_{n'}^{\mathrm{unocc}}\langle M_{\mathbf{k}\mu} \phi_{\mathbf{q}n} | \phi_{\mathbf{k+q}n'} \rangle\langle \phi_{\mathbf{k+q}n'} | \phi_{\mathbf{q}n} M_{\mathbf{k}\nu} \rangle \cdot\left(\frac{1}{\omega+\epsilon_{\mathbf{q}n}-\epsilon_{\mathbf{q}+\mathbf{k}n'}+i\eta}-\frac{1}{\omega-\epsilon_{\mathbf{q}n}+\epsilon_{\mathbf{q}+\mathbf{k}n'}-i\eta}\right) =\int_{-\infty}^\infty \frac{S_{\mu\nu}(\mathbf{k},\omega')}{\omega-\omega'+i\eta\mathrm{sgn}(\omega')}d\omega'\,.} [[#Eq:P]] +:math:{P_{\mu\nu}(\mathbf{k},\omega)=2\sum_{\mathbf{q}}^{\mathrm{BZ}}\sum_{n}^{\mathrm{occ}}\sum_{n'}^{\mathrm{unocc}}\langle M_{\mathbf{k}\mu} \phi_{\mathbf{q}n} | \phi_{\mathbf{k+q}n'} \rangle\langle \phi_{\mathbf{k+q}n'} | \phi_{\mathbf{q}n} M_{\mathbf{k}\nu} \rangle \cdot\left(\frac{1}{\omega+\epsilon_{\mathbf{q}n}-\epsilon_{\mathbf{q}+\mathbf{k}n'}+i\eta}-\frac{1}{\omega-\epsilon_{\mathbf{q}n}+\epsilon_{\mathbf{q}+\mathbf{k}n'}-i\eta}\right)} +:math:{\phantom{P_{\mu\nu}(\mathbf{k},\omega)}=\int_{-\infty}^\infty \frac{S_{\mu\nu}(\mathbf{k},\omega')}{\omega-\omega'+i\eta\mathrm{sgn}(\omega')}d\omega'\,.} [[#Eq:P]] We have implicitly defined the spectral function S in the last equation, an explicit expression for which is basically given by the formula in the middle with the :math:{1/(\omega...)} replaced by :math:{\delta(\omega...)}. (Technically, the :math:{M_{\mathbf{k}\mu}(\mathbf{r})} form the eigenbasis of the Coulomb matrix, so they are linear combinations of the mixed product basis functions.) -- GitLab