From 413b3f37d14dee2c6b30e7db8277319c936ff072 Mon Sep 17 00:00:00 2001 From: Pierre-Francois Loos Date: Fri, 23 Oct 2020 15:13:15 +0200 Subject: [PATCH] saving work --- sfBSE.rty | 2 ++ sfBSE.tex | 62 ++++++++++++++++++++++++++++++++++++++++++++++++------- 2 files changed, 57 insertions(+), 7 deletions(-) diff --git a/sfBSE.rty b/sfBSE.rty index 6895954..3830d4d 100644 --- a/sfBSE.rty +++ b/sfBSE.rty @@ -95,6 +95,8 @@ \newcommand{\Sig}[1]{\Sigma_{#1}} \newcommand{\SigGW}[1]{\Sigma^{GW}_{#1}} \newcommand{\SigC}[1]{\Sigma^\text{c}_{#1}} +\newcommand{\SigX}[1]{\Sigma^\text{x}_{#1}} +\newcommand{\SigXC}[1]{\Sigma^\text{xc}_{#1}} \newcommand{\Z}[1]{Z_{#1}} \newcommand{\MO}[1]{\phi_{#1}} \newcommand{\ERI}[2]{(#1|#2)} diff --git a/sfBSE.tex b/sfBSE.tex index 7a3957d..e9e8f2c 100644 --- a/sfBSE.tex +++ b/sfBSE.tex @@ -48,18 +48,36 @@ Unless otherwise stated, atomic units are used, and we assume real quantities th Let us consider an electronic system consisting of $n = n_\up + n_\dw$ electrons (where $n_\up$ and $n_\dw$ are the number of spin-up and spin-down electrons respectively) and $N$ one-electron basis functions. The number of spin-up and spin-down occupied orbitals are $O_\up = n_\up$ and $O_\dw = n_\dw$, respectively, and there is $V_\up = N - O_\up$ and $V_\dw = N - O_\dw$ spin-up and spin-down virtual (\ie, unoccupied) orbitals. The number of spin-conserved single excitations is then $S^\spc = S_{\up\up}^\spc + S_{\dw\dw}^\spc = O_\up V_\up + O_\dw V_\dw$, while the number of spin-flip excitations is $S^\spf = S_{\up\dw}^\spf + S_{\dw\up}^\spf = O_\up V_\dw + O_\dw V_\up$. -Let us denote as $\MO{p_\sig}$ the $p$th orbital of spin $\sig$ (where $\sig =$ $\up$ or $\dw$). +Let us denote as $\MO{p_\sig}$ the $p$th orbital of spin $\sig$ (where $\sig =$ $\up$ or $\dw$) and $\e{p_\sig}{}$ its one-electron energy. +In the present context these orbitals can originate from a HF or KS calculation. In the following, $i$ and $j$ are occupied orbitals, $a$ and $b$ are unoccupied orbitals, $p$, $q$, $r$, and $s$ indicate arbitrary orbitals, and $m$ labels single excitations. It is important to understand that, in a spin-conserved excitation the hole orbital $\MO{i_\sig}$ and particle orbital $\MO{a_\sig}$ have the same spin $\sig$. -A bra and ket composed by these two orbitals will be denoted as $\rbra{ia\sig}$ and $\rket{ia\sig}$. -In a spin-flip excitation, the hole has a spin $\sig$ and the particle has the opposite spin $\bsig$. +In a spin-flip excitation, the hole and particle states, $\MO{i_\sig}$ and $\MO{a_\bsig}$, have opposite spins, $\sig$ and $\bsig$. %================================ \subsection{The dynamical screening} %================================ +The one-body Green's function is defined as +\begin{equation} + G^{\sig}(\br_1,\br_2;\omega) + = \sum_i \frac{\MO{i_\sig}(\br_1) \MO{i_\sig}(\br_2)}{\omega - \e{i_\sig}{} - i\eta} + + \sum_a \frac{\MO{a_\sig}(\br_1) \MO{a_\sig}(\br_2)}{\omega - \e{a_\sig}{} + i\eta} +\end{equation} +Based on this Green's function, one can easily compute the non-interacting polarizability +\begin{equation} + \chi_0(\br_1,\br_2;\omega) = - \frac{i}{2\pi} \sum_\sig \int G^{\sig}(\br_1,\br_2;\omega+\omega') G^{\sig}(\br_1,\br_2;\omega') d\omega' +\end{equation} +and subseauently the dielectric function +\begin{equation} + \epsilon(\br_1,\br-2;\omega) = \delta(\br_1 - \br_2) - \int \frac{\chi_0(\br_1,\br_3;\omega) }{\abs{\br_2 - \br_3}} d\br_3 +\end{equation} +Based on this latter ingredient, one can access the dynamically-screened Coulomb potential +\begin{equation} + W(\br_1,\br_2;\omega) = \int \frac{\epsilon^{-1}(\br_1,\br_3;\omega) }{\abs{\br_2 - \br_3}} d\br_3 +\end{equation} Within the $GW$ formalism, the dynamical screening $W(\omega)$ is computed at the RPA level using the spin-conserved neutral excitations. -The matrix elements of $W(\omega)$ read +In the orbital basis, the spectral representation of $W(\omega)$ read \begin{multline} W_{p_\sig q_\sig,r_\sigp s_\sigp}(\omega) = \ERI{p_\sig q_\sig}{r_\sigp s_\sigp} + \sum_m \ERI{p_\sig q_\sig}{m}\ERI{r_\sigp s_\sigp}{m} @@ -68,7 +86,7 @@ The matrix elements of $W(\omega)$ read \end{multline} where the two-electron integrals are \begin{equation} - \ERI{p_\sig q_\tau}{r_\sigp s_\taup} = \iint \MO{p_\sig}(\br) \MO{q_\tau}(\br) \frac{1}{\abs{\br - \br'}} \MO{r_\sigp}(\br') \MO{s_\taup}(\br') d\br d\br' + \ERI{p_\sig q_\tau}{r_\sigp s_\taup} = \int \frac{\MO{p_\sig}(\br_1) \MO{q_\tau}(\br_1) \MO{r_\sigp}(\br_2) \MO{s_\taup}(\br_2)}{\abs{\br_1 - \br_2}} d\br_1 d\br_2 \end{equation} \begin{equation} @@ -154,12 +172,23 @@ for the spin-flip excitations. \subsection{The $GW$ self-energy} %================================ +\begin{equation} + \Sig{}^{\sig}(\br_1,\br_2;\omega) + = \frac{i}{2\pi} \int G^{\sig}(\br_1,\br_2;\omega+\omega') W(\br_1,\br_2;\omega') e^{i \eta \omega'} d\omega' +\end{equation} + +\begin{equation} + \SigX{p_\sig q_\sig}(\omega) + = - \frac{1}{2} \sum_{i\sigp} \ERI{p_\sig i_\sigp}{i_\sigp q_\sig} +\end{equation} + + \begin{equation} \begin{split} \SigC{p_\sig q_\sig}(\omega) - & = \sum_i \sum_m \frac{\ERI{p_\sig i_\sig}{m} \ERI{q_\sig i_\sig}{m}}{\omega - \e{i_\sig} + \Om{m}{\spc,\RPA} - i \eta} + & = \sum_{im} \frac{\ERI{p_\sig i_\sig}{m} \ERI{q_\sig i_\sig}{m}}{\omega - \e{i_\sig} + \Om{m}{\spc,\RPA} - i \eta} \\ - & + \sum_a \sum_m \frac{\ERI{p_\sig a_\sig}{m} \ERI{q_\sig a_\sig}{m}}{\omega - \e{a_\sig} - \Om{m}{\spc,\RPA} + i \eta} + & + \sum_{am} \frac{\ERI{p_\sig a_\sig}{m} \ERI{q_\sig a_\sig}{m}}{\omega - \e{a_\sig} - \Om{m}{\spc,\RPA} + i \eta} \end{split} \end{equation} @@ -171,6 +200,25 @@ The quasiparticle energies $\eGW{p}$ are obtained by solving the frequency-depen %================================ \subsection{The Bethe-Salpeter equation formalism} %================================ + +\begin{multline} + L^{\sig\sigp}(\br_1,\br_2;\br_1',\br_2';\omega) + = L_{0}^{\sig\sigp}(\br_1,\br_2;\br_1',\br_2';\omega) + \\ + + \int L_{0}^{\sig\sigp}(\br_1,\br_4;\br_1',\br_3;\omega) + \Xi^{\sig\sigp}(\br_3,\br_5;\br_4,\br_6;\omega) + \\ + \times L^{\sig\sigp}(\br_6,\br_2;\br_5,\br_2';\omega) + d\br_3 d\br_4 d\br_5 d\br_6 +\end{multline} + +\begin{multline} + i \Xi^{\sig\sigp}(\br_3,\br_5;\br_4,\br_6;\omega) + = \frac{\delta(\br_3 - \br_4) \delta(\br_5 - \br_6) }{\abs{\br_3-\br_6}} + \\ + - \delta_{\sig\sigp} W(\br_3,\br_4;\omega) \delta(\br_3 - \br_6) \delta(\br_4 - \br_6) +\end{multline} + Defining $W^{\stat}_{p_\sig q_\sig,r_\sigp s_\sigp} = W_{p_\sig q_\sig,r_\sigp s_\sigp}(\omega = 0)$, we have \begin{subequations} \begin{align}