i dont know this commit

source/theory.rst

@@ -23,15 +23,11 @@ which has been shown to yield accurate band structures for a wide range of mater
 
 with the random-phase approximation for the polarization function :math:`{P}`
 
-.. _eqP:
-
-:math:`{\displaystyle P(\omega)=-\frac{i}{\pi}\int_{-\infty}^\infty G(\omega-\omega')G(\omega')d\omega'\,.}`
+:math:`{\displaystyle P(\omega)=-\frac{i}{\pi}\int_{-\infty}^\infty G(\omega-\omega')G(\omega')d\omega'\,.}` [[#eq:P]]
 
 Usually, the Green function :math:`{G_0}` of a mean-field system is taken for :math:`{G}`, in which case it takes the form 
 
-.. _eqG0:
-
-:math:`{\displaystyle G_0(\mathbf{r},\mathbf{r}';\omega)=\sum_\mathbf{k}\sum_n \frac{\phi_{\mathbf{k}n}(\mathbf{r})\phi^*_{\mathbf{k}n}(\mathbf{r}')}{\omega-\epsilon_{\mathbf{k}n}+ i\eta\mathrm{sgn}(-\epsilon_{\mathbf{k}n}\mathrm{F}}\,,}`
+:math:`{\displaystyle G_0(\mathbf{r},\mathbf{r}';\omega)=\sum_\mathbf{k}\sum_n \frac{\phi_{\mathbf{k}n}(\mathbf{r})\phi^*_{\mathbf{k}n}(\mathbf{r}')}{\omega-\epsilon_{\mathbf{k}n}+ i\eta\mathrm{sgn}(-\epsilon_{\mathbf{k}n}\mathrm{F}}\,,}` [[#eq:G0]]
 
 where :math:`{\phi_{n\mathbf{k}}(\mathbf{r})}` and :math:`{\epsilon_{n\mathbf{k}}}` are the eigenfunctions and eigenvalues, respectively, of the mean-field system. Here, a natural choice is to employ the solution of the KS equations
 
@@ -39,23 +35,17 @@ where :math:`{\phi_{n\mathbf{k}}(\mathbf{r})}` and :math:`{\epsilon_{n\mathbf{k}
 
 often with the local-density approximation (LDA) for the exchange-correlation potential. In this respect, one says that LDA is the ''starting point'' for the ''GW'' calculation. Other starting points are generalized gradient approximation (GGA), LDA+''U'', hybrid functionals, Hartree-Fock, QSGW, et cetera. The explicit form of :math:`{G_0}` enables us to perform the frequency integration in [[#eq:P|:math:`{P}`]], which yields
 
-.. _eqPexp:
-
-:math:`{\displaystyle P\left(\mathbf{r},\mathbf{r}',\omega\right)=2\sum_{\mathbf{q},\mathbf{k}}^{\mathrm{BZ}}\sum_{n}^{\mathrm{occ}}\sum_{n'}^{\mathrm{unocc}}\phi_{\mathbf{q}n}\left(\mathbf{r}\right)\phi_{\mathbf{q}n}^{*}\left(\mathbf{r}'\right)\phi_{\mathbf{q}+\mathbf{k}n'}\left(\mathbf{r}'\right)\phi_{\mathbf{q}+\mathbf{k}n'}^{*}\left(\mathbf{r}\right)\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(\mathbf{r},\mathbf{r}';\omega')}{\omega-\omega'+i\eta\mathrm{sgn}(\omega')}d\omega'\,,}`
+:math:`{\displaystyle P\left(\mathbf{r},\mathbf{r}',\omega\right)=2\sum_{\mathbf{q},\mathbf{k}}^{\mathrm{BZ}}\sum_{n}^{\mathrm{occ}}\sum_{n'}^{\mathrm{unocc}}\phi_{\mathbf{q}n}\left(\mathbf{r}\right)\phi_{\mathbf{q}n}^{*}\left(\mathbf{r}'\right)\phi_{\mathbf{q}+\mathbf{k}n'}\left(\mathbf{r}'\right)\phi_{\mathbf{q}+\mathbf{k}n'}^{*}\left(\mathbf{r}\right)\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(\mathbf{r},\mathbf{r}';\omega')}{\omega-\omega'+i\eta\mathrm{sgn}(\omega')}d\omega'\,,}` [[#eq:Pexp]]
 
 where the last equality implicitly defines the spectral function :math:`{S(\omega)}`.
  
 The Dyson equation can be rewritten in the form of an effective equation of motion of a single particle, the quasiparticle equation
 
-.. _eqPexq:
-
-:math:`{\displaystyle\left\{-\frac{\nabla}{2}+v^\mathrm{ext}(\mathbf{r})+v^\mathrm{H}(\mathbf{r})\right\}\psi_{\mathbf{k}n}(\mathbf{r})+\int \Sigma^\mathrm{xc}(\mathbf{r},\mathbf{r}';E_{\mathbf{k}n})\psi_{\mathbf{k}n}(\mathbf{r}')d^3 r=E_{\mathbf{k}n}\psi_{\mathbf{k}n}(\mathbf{r})}`
+:math:`{\displaystyle\left\{-\frac{\nabla}{2}+v^\mathrm{ext}(\mathbf{r})+v^\mathrm{H}(\mathbf{r})\right\}\psi_{\mathbf{k}n}(\mathbf{r})+\int \Sigma^\mathrm{xc}(\mathbf{r},\mathbf{r}';E_{\mathbf{k}n})\psi_{\mathbf{k}n}(\mathbf{r}')d^3 r=E_{\mathbf{k}n}\psi_{\mathbf{k}n}(\mathbf{r})}` [[#eq:qpeq]]
 
 with the (non-orthonormal) quasiparticle wavefunctions and (complex-valued) quasiparticle energies. It can be shown that the interacting Green function takes the same form as [[#eq:G0|:math:`{G_0}`]] (Lehmann representation) if the eigenfunctions and eigenvalues are replaced by the latter. The fact that this equation of motion is formally similar to the KS equation motivates to use perturbation theory of first order and write the quasiparticle energies as
 
-.. _eqqppert:
-
-:math:`{\displaystyle E_{n\mathbf{k}}=\epsilon_{n\mathbf{k}}+\langle\phi_{n\mathbf{k}}|\Sigma^\mathrm{xc}(E_{n\mathbf{k}})-v^\mathrm{xc}|\phi_{n\mathbf{k}}\rangle\approx\epsilon_{n\mathbf{k}}+Z_{n\mathbf{k}}\langle\phi_{n\mathbf{k}}|\Sigma^\mathrm{xc}(\epsilon_{n\mathbf{k}})-v^\mathrm{xc}|\phi_{n\mathbf{k}}\rangle}`
+:math:`{\displaystyle E_{n\mathbf{k}}=\epsilon_{n\mathbf{k}}+\langle\phi_{n\mathbf{k}}|\Sigma^\mathrm{xc}(E_{n\mathbf{k}})-v^\mathrm{xc}|\phi_{n\mathbf{k}}\rangle\approx\epsilon_{n\mathbf{k}}+Z_{n\mathbf{k}}\langle\phi_{n\mathbf{k}}|\Sigma^\mathrm{xc}(\epsilon_{n\mathbf{k}})-v^\mathrm{xc}|\phi_{n\mathbf{k}}\rangle}` [[#eq:qppert]]
 
 with the renormalization constant :math:`{Z_{n\mathbf{k}}=[1-\partial\Sigma^{\mathrm{xc}}/\partial\omega(\epsilon_{n\mathbf{k}})]^{-1}}`. Both expressions can be taken to evaluate the quasiparticle energy, the first one is a non-linear equation in :math:`{E_{n\mathbf{k}}}` and requires an iterative solution. We thus have three ways to calculate the quasiparticle energies: (a) by solving the full [[#eq:qpeq|quasiparticle equation]], (b) with the [[#eq:qppert|non-linear equation]], and (c) using the [[#eq:qppert|linearized solution]]. The mathematical complexity, the computational cost, and the accuracy decreases in this order. For example, solution (a) requires the full self-energy matrix (i.e., including the off-diagonal elements) to be evaluated. Once the full matrix :math:`{\Sigma^\mathrm{xc}_{\mathbf{k},nn'}(\omega)}` is known, one can proceed to perform a self-consistent calculation within the QSGW approach. In this approach, the self-energy matrix is ''hermitianized'' and made frequency independent. This self-energy can then be used to replace :math:`{v^\mathrm{xc}}` in the KS equation, defining a new mean-field system. A self-consistent solution of this mean-field system (using a DFT code) is then employed as a new starting point and so on until self-consistency in the quasiparticle energies is achieved.