saving work

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)}

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}