From cef3f2065649a215fca324b3238b4e11e5bcc24b Mon Sep 17 00:00:00 2001 From: Pierre-Francois Loos Date: Tue, 27 Oct 2020 14:01:50 +0100 Subject: [PATCH] saving work --- sfBSE.rty | 5 +++-- sfBSE.tex | 66 ++++++++++++++++++++++++++++++++----------------------- 2 files changed, 42 insertions(+), 29 deletions(-) diff --git a/sfBSE.rty b/sfBSE.rty index a8a97d2..35374b7 100644 --- a/sfBSE.rty +++ b/sfBSE.rty @@ -26,7 +26,8 @@ \newcommand{\Hxc}{\text{Hxc}} \newcommand{\xc}{\text{xc}} \newcommand{\Ha}{\text{H}} -\newcommand{\co}{\text{x}} +\newcommand{\co}{\text{c}} +\newcommand{\x}{\text{x}} % \newcommand{\Norb}{N_\text{orb}} @@ -92,7 +93,7 @@ \newcommand{\tW}[2]{\widetilde{W}_{#1}^{#2}} \newcommand{\Wc}[1]{W^\text{c}_{#1}} \newcommand{\vc}[1]{v_{#1}} -\newcommand{\Sig}[1]{\Sigma_{#1}} +\newcommand{\Sig}[2]{\Sigma_{#1}^{#2}} \newcommand{\SigC}[1]{\Sigma^\text{c}_{#1}} \newcommand{\SigX}[1]{\Sigma^\text{x}_{#1}} \newcommand{\SigXC}[1]{\Sigma^\text{xc}_{#1}} diff --git a/sfBSE.tex b/sfBSE.tex index 4584900..b4eb8a6 100644 --- a/sfBSE.tex +++ b/sfBSE.tex @@ -40,20 +40,19 @@ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \alert{Here comes the introduction.} Unless otherwise stated, atomic units are used, and we assume real quantities throughout this manuscript. -In the following, we consider systems with collinear spins and a spin-independent hamiltonian without contributions such as spin-orbit interaction. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{Unrestricted $GW$ formalism} \label{sec:UGW} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -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. +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, assuming no linear dependencies in the one-electron basis set, 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 $\dw$) 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. +Let us denote as $\MO{p_\sig}(\br)$ the $p$th (spin)orbital of spin $\sig$ (where $\sig =$ $\up$ or $\dw$) and $\e{p_\sig}{}$ its one-electron energy. 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 $\sig$. In a spin-flip excitation, the hole and particle states, $\MO{i_\sig}$ and $\MO{a_\bsig}$, have opposite spins, $\sig$ and $\bsig$. +In the following, we assume real quantities throughout this manuscript, $i$ and $j$ are occupied orbitals, $a$ and $b$ are unoccupied orbitals, $p$, $q$, $r$, and $s$ indicate arbitrary orbitals, and $m$ labels single excitations. +Moreover, we consider systems with collinear spins and a spin-independent hamiltonian without contributions such as spin-orbit interaction. %================================ \subsection{The dynamical screening} @@ -61,9 +60,10 @@ In a spin-flip excitation, the hole and particle states, $\MO{i_\sig}$ and $\MO{ The pillar of Green's function many-body perturbation theory is the (time-ordered) one-body Green's function, which has poles at the charged excitations (i.e., ionization potentials and electron affinities) of the system. \cite{ReiningBook} The spin-$\sig$ component of the one-body Green's function reads \cite{ReiningBook,Bruneval_2016a} \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} +\label{eq:G} + 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} where $\eta$ is a positive infinitesimal. Based on the spin-up and spin-down components of $G$, one can easily compute the non-interacting polarizability (which is a sum over spins) @@ -72,17 +72,17 @@ Based on the spin-up and spin-down components of $G$, one can easily compute the \end{equation} and subsequently 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 + \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} where $\delta(\br_1 - \br_2)$ is the Dirac function. 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} -which is spin independent as the bare Coulomb interaction $\abs{\br_1 - \br_2}^{-1}$ does not depend on spin coordinates. +which is naturally spin independent as the bare Coulomb interaction $\abs{\br_1 - \br_2}^{-1}$ does not depend on spin coordinates. -Within the $GW$ formalism, the is computed at the RPA level by considering only the manifold of the spin-conserved neutral excitation. -In the orbital basis, the spectral representation of $W$ reads +Within the $GW$ formalism, the dynamical screening is computed at the random-phase approximation (RPA) level by considering only the manifold of the spin-conserved neutral excitations. +In the orbital basis, the spectral representation of $W$ is \begin{multline} \label{eq:W} W_{p_\sig q_\sig,r_\sigp s_\sigp}(\omega) = \ERI{p_\sig q_\sig}{r_\sigp s_\sigp} @@ -92,14 +92,14 @@ In the orbital basis, the spectral representation of $W$ reads \end{multline} where the bare two-electron integrals are \cite{Gill_1994} \begin{equation} -\label{eq:sERI} \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} and the screened two-electron integrals (or spectral weights) are explicitly given by \begin{equation} +\label{eq:sERI} \ERI{p_\sig q_\sig}{m} = \sum_{ia\sigp} \ERI{p_\sig q_\sig}{r_\sigp s_\sigp} (\bX{m}{\spc,\RPA}+\bY{m}{\spc,\RPA})_{i_\sigp a_\sigp} \end{equation} -In Eqs.~\eqref{eq:W} and \eqref{eq:sERI}, the RPA spin-conserved neutral excitations $\Om{m}{\spc,\RPA}$ and their corresponding eigenvectors $(\bX{m}{\spc,\RPA}+\bY{m}{\spc,\RPA})$ are obtained by solving a linear response system of the form +In Eqs.~\eqref{eq:W} and \eqref{eq:sERI}, the RPA spin-conserved neutral excitations $\Om{m}{\spc,\RPA}$ and their corresponding eigenvectors, $\bX{m}{\spc,\RPA}$ and $\bY{m}{\spc,\RPA}$, are obtained by solving a linear response system of the form \begin{equation} \label{eq:LR-RPA} \begin{pmatrix} @@ -120,6 +120,7 @@ In Eqs.~\eqref{eq:W} and \eqref{eq:sERI}, the RPA spin-conserved neutral excitat \end{equation} where the expressions of the matrix elements of $\bA{}{}$ and $\bB{}{}$ are specific of the method and of the spin manifold. The spin structure of these matrices, though, is general +\begin{subequations} \begin{align} \label{eq:LR-RPA-AB} \bA{}{\spc} & = \begin{pmatrix} @@ -143,6 +144,7 @@ The spin structure of these matrices, though, is general \bB{}{\dwup,\updw} & \bO \\ \end{pmatrix} \end{align} +\end{subequations} In the absence of instabilities, the linear eigenvalue problem \eqref{eq:LR-RPA} has particle-hole symmetry which means that the eigenvalues are obtained by pairs $\pm \Om{m}{}$. In such a case, $(\bA{}{}-\bB{}{})^{1/2}$ is positive definite, and Eq.~\eqref{eq:LR-RPA} can be recast as a Hermitian problem of half the dimension \begin{equation} @@ -153,9 +155,11 @@ where the excitation amplitudes are \begin{equation} \bX{}{} + \bY{}{} = \bOm{-1/2} \cdot (\bA{}{} - \bB{}{})^{1/2} \cdot \bZ{}{} \end{equation} -Within the Tamm-Dancoff approximation (TDA), the coupling terms between the resonant and anti-resonant parts, $\bA{}{}$ and $-\bA{}{}$, are neglected, which consist in setting $\bB{}{} = \bO$. -In such a case, Eq.~\eqref{eq:LR-RPA} reduces to $\bA{}{} \cdot \bX{m}{} = \Om{m}{} \bX{m}{}$. - +Within the Tamm-Dancoff approximation (TDA), the coupling terms between the resonant and anti-resonant parts, $\bA{}{}$ and $-\bA{}{}$, are neglected, which consists in setting $\bB{}{} = \bO$. +In such a case, Eq.~\eqref{eq:LR-RPA} reduces to straightforward Hermitian problem of the form: +\begin{equation} + \bA{}{} \cdot \bX{m}{} = \Om{m}{} \bX{m}{} +\end{equation} At the RPA level, the matrix elements of $\bA{}{}$ and $\bB{}{}$ are \begin{subequations} \begin{align} @@ -194,26 +198,35 @@ for the spin-flip excitations. Within the acclaimed $GW$ approximation, \cite{Hedin_1965,Golze_2019} the exchange-correlation (xc) part of the self-energy \begin{equation} \begin{split} - \Sig{}^{\text{xc},\sig}(\br_1,\br_2;\omega) - & = \Sig{}^{\text{x},\sig}(\br_1,\br_2) + \Sig{}^{\text{c},\sig}(\br_1,\br_2;\omega) + \Sig{}{\xc,\sig}(\br_1,\br_2;\omega) + & = \Sig{}{\x,\sig}(\br_1,\br_2) + \Sig{}{\co,\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{split} \end{equation} is, like the one-body Green's function, spin-diagonal, and its spectral representation reads \begin{gather} - \SigX{p_\sig q_\sig} - = - \frac{1}{2} \sum_{i\sigp} \ERI{p_\sig i_\sigp}{i_\sigp q_\sig} + \Sig{p_\sig q_\sig}{\x} + = - \sum_{i} \ERI{p_\sig i_\sig}{i_\sig q_\sig} \\ \begin{split} - \SigC{p_\sig q_\sig}(\omega) + \Sig{p_\sig q_\sig}{\co}(\omega) & = \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_{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{gather} -which has been split in its exchange (x) and correlation (c) contributions. -The Dyson equation linking the Green's function and the self-energy holds separately for each spin component, and the quasiparticle energies $\eGW{p_\sig}$ are obtained by solving the frequency-dependent quasiparticle equation +which the self-energy has been split in its exchange (x) and correlation (c) contributions. +The Dyson equation linking the Green's function and the self-energy holds separately for each spin component +\begin{multline} + \qty[ G^{\sig}(\br_1,\br_2;\omega) ]^{-1} + = \qty[ G_{\KS}^{\sig}(\br_1,\br_2;\omega) ]^{-1} + \\ + + \Sig{}{\xc,\sig}(\br_1,\br_2;\omega) - v^{\xc}(\br_1) \delta(\br_1 - \br_2) +\end{multline} +where $G_{\KS}^{\sig}$ is the Kohn-Sham Green's function built with Kohn-Sham orbitals and one-electron energies according to Eq.~\eqref{eq:G} and $v^{\xc}(\br)$ is the Kohn-Sham local exchange-correlation potential. + +The quasiparticle energies $\eGW{p_\sig}$ are obtained by solving the frequency-dependent quasiparticle equation \begin{equation} \omega = \e{p_\sig}{} - V_{p_\sigma}^{\xc} + \SigX{p\sigma} + \SigC{p\sigma}(\omega) \end{equation} @@ -221,8 +234,7 @@ with \begin{equation} V_{p_\sigma}^{\xc} = \int \MO{p_\sig}(\br) v^{\xc}(\br) \MO{p_\sig}(\br) d\br \end{equation} -where $v^{\xc}(\br)$ the Kohn-Sham exchange-correlation. -\alert{Adding the Dyson equation? Introduce linearization of the quasiparticle equation and different degree of self-consistency.} +\alert{Introduce linearization of the quasiparticle equation and different degree of self-consistency.} %================================ \subsection{The Bethe-Salpeter equation formalism}