diff --git a/BSEdyn.tex b/BSEdyn.tex index b0faeea..81c7f9b 100644 --- a/BSEdyn.tex +++ b/BSEdyn.tex @@ -531,7 +531,7 @@ From a more practical point of view, Eq.~\eqref{eq:BSE-final} can be recast as a \begin{pmatrix} \bA{}(\Om{s}{}) & \bB{}(\Om{s}{}) \\ - -\bB{}(\Om{s}{}) & -\bA{}(\Om{s}{}) + -\bB{}(-\Om{s}{}) & -\bA{}(-\Om{s}{}) \\ \end{pmatrix} \cdot @@ -565,8 +565,8 @@ Now, let us decompose, using basic perturbation theory, the non-linear eigenprob \begin{multline} \label{eq:LR-PT} \begin{pmatrix} - \bA{}(\Om{s}{}) & \bB{}(\Om{s}{}) \\ - -\bB{}(\Om{s}{}) & -\bA{}(\Om{s}{}) \\ + \bA{}(\Om{s}{}) & \bB{}(\Om{s}{}) \\ + -\bB{}(-\Om{s}{}) & -\bA{}(-\Om{s}{}) \\ \end{pmatrix} \\ = @@ -578,8 +578,8 @@ Now, let us decompose, using basic perturbation theory, the non-linear eigenprob \end{pmatrix} + \begin{pmatrix} - \bA{(1)}(\Om{s}{}) & \bB{(1)}(\Om{s}{}) \\ - -\bB{(1)}(\Om{s}{}) & -\bA{(1)}(\Om{s}{}) \\ + \bA{(1)}(\Om{s}{}) & \bB{(1)}(\Om{s}{}) \\ + -\bB{(1)}(-\Om{s}{}) & -\bA{(1)}(-\Om{s}{}) \\ \end{pmatrix}, \end{multline} with @@ -662,7 +662,7 @@ Thanks to first-order perturbation theory, the first-order correction to the $s$ \cdot \begin{pmatrix} \bA{(1)}(\Om{s}{(0)}) & \bB{(1)}(\Om{s}{(0)}) \\ - -\bB{(1)}(\Om{s}{(0)}) & -\bA{(1)}(\Om{s}{(0)}) \\ + -\bB{(1)}(-\Om{s}{(0)}) & -\bA{(1)}(-\Om{s}{(0)}) \\ \end{pmatrix} \cdot \begin{pmatrix} @@ -742,18 +742,18 @@ All the static and dynamic BSE calculations have been performed with the softwar & \tabc{$\Om{s}{\stat}$} & \tabc{$\Delta\Om{s}{\dyn}$(dTDA)} & \tabc{$\Delta\Om{s}{\dyn}$} & \tabc{$\Om{s}{\stat}$} & \tabc{$\Delta\Om{s}{\dyn}$(dTDA)} & \tabc{$\Delta\Om{s}{\dyn}$} \\ \hline - $^1\Pi_g(n \ra \pis)$ & Val. & 9.90 & -0.32 & -0.31 & 9.92 & -0.40 & -0.42 & 10.01 & -0.42 & -0.42 \\ - $^1\Sigma_u^-(\pi \ra \pis)$ & Val. & 9.70 & -0.33 & -0.34 & 9.61 & -0.42 & -0.40 & 9.69 & -0.44 & -0.44 \\ - $^1\Delta_u(\pi \ra \pis)$ & Val. & 10.37 & -0.31 & -0.31 & 10.27 & -0.39 & -0.40 & 10.34 & -0.41 & -0.40 \\ - $^1\Sigma_g^+$(R) & Ryd. & 15.67 & -0.17 & -0.12 & 15.04 & -0.21 & -0.10 & 14.72 & -0.21 & -0.16 \\ - $^1\Pi_u$(R) & Ryd. & 15.00 & -0.21 & -0.21 & 14.75 & -0.27 & -0.26 & 14.72 & -0.29 & -0.26 \\ - $^1\Sigma_u^+$(R) & Ryd. & 22.88\fnm[1] & -0.15 & -0.21 & 19.03 & -0.08 & -0.06 & 16.78 & -0.06 & -0.07 \\ - $^1\Pi_u$(R) & Ryd. & 23.62\fnm[1] & -0.11 & -0.10 & 19.15 & -0.11 & -0.13 & 16.93 & -0.09 & -0.09 \\ + $^1\Pi_g(n \ra \pis)$ & Val. & 9.90 & -0.32 & -0.33 & 9.92 & -0.40 & -0.42 & 10.01 & -0.42 & - \\ + $^1\Sigma_u^-(\pi \ra \pis)$ & Val. & 9.70 & -0.33 & -0.31 & 9.61 & -0.42 & -0.36 & 9.69 & -0.44 & - \\ + $^1\Delta_u(\pi \ra \pis)$ & Val. & 10.37 & -0.31 & -0.32 & 10.27 & -0.39 & -0.42 & 10.34 & -0.41 & - \\ + $^1\Sigma_g^+$(R) & Ryd. & 15.67 & -0.17 & -0.59 & 15.04 & -0.21 & -0.46 & 14.72 & -0.21 & - \\ + $^1\Pi_u$(R) & Ryd. & 15.00 & -0.21 & -0.26 & 14.75 & -0.27 & -0.30 & 14.72 & -0.29 & - \\ + $^1\Sigma_u^+$(R) & Ryd. & 22.88\fnm[1] & -0.15 & -0.20 & 19.03 & -0.08 & -0.07 & 16.78 & -0.06 & - \\ + $^1\Pi_u$(R) & Ryd. & 23.62\fnm[1] & -0.11 & -0.10 & 19.15 & -0.11 & -0.10 & 16.93 & -0.09 & - \\ \\ - $^3\Sigma_u^+(\pi \ra \pis)$ & Val. & 7.39 & -0.48 & -0.63 & 7.46 & -0.59 & -0.56 & 7.59 & -0.62 & -0.60 \\ - $^3\Pi_g(n \ra \pis)$ & Val. & 8.07 & -0.42 & -0.44 & 8.14 & -0.52 & -0.50 & 8.24 & -0.54 & -0.51 \\ - $^3\Delta_u(\pi \ra \pis)$ & Val. & 8.56 & -0.41 & -0.46 & 8.52 & -0.52 & -0.50 & 8.62 & -0.55 & -0.51 \\ - $^3\Sigma_u^-(\pi \ra \pis)$ & Val. & 9.70 & -0.33 & -0.34 & 9.61 & -0.42 & -0.40 & 9.69 & -0.44 & -0.44 \\ + $^3\Sigma_u^+(\pi \ra \pis)$ & Val. & 7.39 & -0.48 & -0.69 & 7.46 & -0.59 & -0.41 & 7.59 & -0.62 & - \\ + $^3\Pi_g(n \ra \pis)$ & Val. & 8.07 & -0.42 & -0.43 & 8.14 & -0.52 & -0.48 & 8.24 & -0.54 & - \\ + $^3\Delta_u(\pi \ra \pis)$ & Val. & 8.56 & -0.41 & -0.40 & 8.52 & -0.52 & -0.40 & 8.62 & -0.55 & - \\ + $^3\Sigma_u^-(\pi \ra \pis)$ & Val. & 9.70 & -0.33 & -0.31 & 9.61 & -0.42 & -0.36 & 9.69 & -0.44 & - \\ \hline \\ & & \mc{3}{c}{aug-cc-pVDZ ($\Eg^{\GW} = 19.49$ eV)} @@ -764,18 +764,18 @@ All the static and dynamic BSE calculations have been performed with the softwar & \tabc{$\Om{s}{\stat}$} & \tabc{$\Delta\Om{s}{\dyn}$(dTDA)} & \tabc{$\Delta\Om{s}{\dyn}$} & \tabc{$\Om{s}{\stat}$} & \tabc{$\Delta\Om{s}{\dyn}$(dTDA)} & \tabc{$\Delta\Om{s}{\dyn}$} \\ \hline - $^1\Pi_g(n \ra \pis)$ & Val. & 10.18 & -0.41 & -0.43 & 10.42 & -0.42 & -0.40 & 10.52 & -0.43 & -0.40 \\ - $^1\Sigma_u^-(\pi \ra \pis)$ & Val. & 9.95 & -0.44 & -0.44 & 10.11 & -0.45 & -0.45 & 10.20 & -0.45 & -0.45 \\ - $^1\Delta_u(\pi \ra \pis)$ & Val. & 10.57 & -0.41 & -0.40 & 10.75 & -0.42 & -0.41 & 10.85 & -0.42 & -0.42 \\ - $^1\Sigma_g^+$ & Ryd. & 13.72 & -0.04 & -0.04 & 13.60 & -0.03 & -0.03 & 13.55 & -0.02 & -0.02 \\ - $^1\Pi_u$ & Ryd. & 14.07 & -0.05 & -0.05 & 13.98 & -0.04 & -0.04 & 13.96 & -0.03 & -0.04 \\ - $^1\Sigma_u^+$ & Ryd. & 13.80 & -0.08 & -0.08 & 13.98 & -0.07 & -0.08 & 14.08 & -0.06 & -0.06 \\ - $^1\Pi_u$ & Ryd. & 14.22 & -0.04 & -0.03 & 14.24 & -0.03 & -0.03 & 14.26 & -0.03 & -0.02 \\ + $^1\Pi_g(n \ra \pis)$ & Val. & 10.18 & -0.41 & -0.43 & 10.42 & -0.42 & -0.41 & 10.52 & -0.43 & - \\ + $^1\Sigma_u^-(\pi \ra \pis)$ & Val. & 9.95 & -0.44 & -0.41 & 10.11 & -0.45 & -0.42 & 10.20 & -0.45 & - \\ + $^1\Delta_u(\pi \ra \pis)$ & Val. & 10.57 & -0.41 & -0.41 & 10.75 & -0.42 & -0.45 & 10.85 & -0.42 & - \\ + $^1\Sigma_g^+$ & Ryd. & 13.72 & -0.04 & -0.05 & 13.60 & -0.03 & -0.03 & 13.55 & -0.02 & - \\ + $^1\Pi_u$ & Ryd. & 14.07 & -0.05 & -0.05 & 13.98 & -0.04 & -0.04 & 13.96 & -0.03 & - \\ + $^1\Sigma_u^+$ & Ryd. & 13.80 & -0.08 & -0.10 & 13.98 & -0.07 & -0.10 & 14.08 & -0.06 & - \\ + $^1\Pi_u$ & Ryd. & 14.22 & -0.04 & -0.03 & 14.24 & -0.03 & -0.03 & 14.26 & -0.03 & - \\ \\ - $^3\Sigma_u^+(\pi \ra \pis)$ & Val. & 7.75 & -0.63 & -1.32 & 8.02 & -0.64 & -0.60 & 8.12 & -0.64 & -0.66 \\ - $^3\Pi_g(n \ra \pis)$ & Val. & 8.42 & -0.54 & -0.53 & 8.66 & -0.56 & -0.79 & 8.75 & -0.56 & -0.50 \\ - $^3\Delta_u(\pi \ra \pis)$ & Val. & 8.86 & -0.54 & -0.55 & 9.04 & -0.56 & -0.59 & 9.14 & -0.56 & -0.60 \\ - $^3\Sigma_u^-(\pi \ra \pis)$ & Val. & 9.95 & -0.44 & -0.45 & 10.11 & -0.45 & -0.45 & 10.20 & -0.45 & -0.45 \\ + $^3\Sigma_u^+(\pi \ra \pis)$ & Val. & 7.75 & -0.63 & -2.42 & 8.02 & -0.64 & -0.45 & 8.12 & -0.64 & - \\ + $^3\Pi_g(n \ra \pis)$ & Val. & 8.42 & -0.54 & -0.50 & 8.66 & -0.56 & -0.79 & 8.75 & -0.56 & - \\ + $^3\Delta_u(\pi \ra \pis)$ & Val. & 8.86 & -0.54 & -0.47 & 9.04 & -0.56 & -0.52 & 9.14 & -0.56 & - \\ + $^3\Sigma_u^-(\pi \ra \pis)$ & Val. & 9.95 & -0.44 & -0.41 & 10.11 & -0.45 & -0.42 & 10.20 & -0.45 & - \\ \end{tabular} \end{ruledtabular} \fnt[1]{Excitation energy larger than the fundamental gap.} diff --git a/Notes/BSEdyn-notes.tex b/Notes/BSEdyn-notes.tex new file mode 100644 index 0000000..33f713d --- /dev/null +++ b/Notes/BSEdyn-notes.tex @@ -0,0 +1,467 @@ +\documentclass{article} +\usepackage{graphicx,dcolumn,bm,xcolor,microtype,multirow,amscd,amsmath,amssymb,amsfonts,physics,longtable,wrapfig} +\usepackage[version=4]{mhchem} + +%\usepackage[utf8]{inputenc} +%\usepackage[T1]{fontenc} +%\usepackage{txfonts} + +\usepackage[ + colorlinks=true, + citecolor=blue, + breaklinks=true + ]{hyperref} +\urlstyle{same} + +\newcommand{\ie}{\textit{i.e.}} +\newcommand{\eg}{\textit{e.g.}} +\newcommand{\alert}[1]{\textcolor{red}{#1}} +\definecolor{darkgreen}{HTML}{009900} +\usepackage[normalem]{ulem} +\newcommand{\titou}[1]{\textcolor{red}{#1}} +\newcommand{\trashPFL}[1]{\textcolor{red}{\sout{#1}}} +\newcommand{\PFL}[1]{\titou{(\underline{\bf PFL}: #1)}} + +\newcommand{\mc}{\multicolumn} +\newcommand{\fnm}{\footnotemark} +\newcommand{\fnt}{\footnotetext} +\newcommand{\tabc}[1]{\multicolumn{1}{c}{#1}} +\newcommand{\SI}{\textcolor{blue}{supplementary material}} +\newcommand{\QP}{\textsc{quantum package}} +\newcommand{\T}[1]{#1^{\intercal}} + +% coordinates +\newcommand{\br}{\mathbf{r}} +\newcommand{\dbr}{d\br} + +% methods +\newcommand{\evGW}{ev$GW$} +\newcommand{\qsGW}{qs$GW$} +\newcommand{\GOWO}{$G_0W_0$} +\newcommand{\Hxc}{\text{Hxc}} +\newcommand{\xc}{\text{xc}} +\newcommand{\Ha}{\text{H}} +\newcommand{\co}{\text{x}} + +% +\newcommand{\Norb}{N_\text{orb}} +\newcommand{\Nocc}{O} +\newcommand{\Nvir}{V} +\newcommand{\IS}{\lambda} + +% operators +\newcommand{\hH}{\Hat{H}} + +% methods +\newcommand{\KS}{\text{KS}} +\newcommand{\HF}{\text{HF}} +\newcommand{\RPA}{\text{RPA}} +\newcommand{\BSE}{\text{BSE}} +\newcommand{\TDABSE}{\text{BSE(TDA)}} +\newcommand{\dBSE}{\text{dBSE}} +\newcommand{\TDAdBSE}{\text{dBSE(TDA)}} +\newcommand{\GW}{GW} +\newcommand{\stat}{\text{stat}} +\newcommand{\dyn}{\text{dyn}} +\newcommand{\TDA}{\text{TDA}} + +% energies +\newcommand{\Enuc}{E^\text{nuc}} +\newcommand{\Ec}{E_\text{c}} +\newcommand{\EHF}{E^\text{HF}} +\newcommand{\EBSE}{E^\text{BSE}} +\newcommand{\EcRPA}{E_\text{c}^\text{RPA}} +\newcommand{\EcBSE}{E_\text{c}^\text{BSE}} + +% orbital energies +\newcommand{\e}[1]{\eps_{#1}} +\newcommand{\eHF}[1]{\eps^\text{HF}_{#1}} +\newcommand{\eKS}[1]{\eps^\text{KS}_{#1}} +\newcommand{\eQP}[1]{\eps^\text{QP}_{#1}} +\newcommand{\eGW}[1]{\eps^{GW}_{#1}} +\newcommand{\Om}[2]{\Omega_{#1}^{#2}} + +% Matrix elements +\newcommand{\Sig}[1]{\Sigma_{#1}} +\newcommand{\MO}[1]{\phi_{#1}} +\newcommand{\ERI}[2]{(#1|#2)} +\newcommand{\sERI}[2]{[#1|#2]} + +% excitation energies +\newcommand{\OmRPA}[1]{\Omega_{#1}^{\text{RPA}}} +\newcommand{\OmRPAx}[1]{\Omega_{#1}^{\text{RPAx}}} +\newcommand{\OmBSE}[1]{\Omega_{#1}^{\text{BSE}}} + +\newcommand{\spinup}{\downarrow} +\newcommand{\spindw}{\uparrow} +\newcommand{\singlet}{\uparrow\downarrow} +\newcommand{\triplet}{\uparrow\uparrow} + +% Matrices +\newcommand{\bO}{\mathbf{0}} +\newcommand{\bH}{\mathbf{H}} +\newcommand{\bV}{\mathbf{V}} +\newcommand{\bI}{\mathbf{1}} +\newcommand{\bb}{\mathbf{b}} +\newcommand{\bA}{\mathbf{A}} +\newcommand{\bB}{\mathbf{B}} +\newcommand{\bx}{\mathbf{x}} + +% units +\newcommand{\IneV}[1]{#1 eV} +\newcommand{\InAU}[1]{#1 a.u.} +\newcommand{\InAA}[1]{#1 \AA} +\newcommand{\kcal}{kcal/mol} + +\DeclareMathOperator*{\argmax}{argmax} +\DeclareMathOperator*{\argmin}{argmin} + +% orbitals, gaps, etc +\newcommand{\updw}{\uparrow\downarrow} +\newcommand{\upup}{\uparrow\uparrow} +\newcommand{\eps}{\varepsilon} +\newcommand{\IP}{I} +\newcommand{\EA}{A} +\newcommand{\HOMO}{\text{HOMO}} +\newcommand{\LUMO}{\text{LUMO}} +\newcommand{\Eg}{E_\text{g}} +\newcommand{\EgFun}{\Eg^\text{fund}} +\newcommand{\EgOpt}{\Eg^\text{opt}} +\newcommand{\EB}{E_B} + +\newcommand{\LCPQ}{Laboratoire de Chimie et Physique Quantiques (UMR 5626), Universit\'e de Toulouse, CNRS, UPS, France} + +\title{Notes on the Dynamical Bethe-Salpeter Equation} + +\author{Pierre-Fran\c{c}ois Loos} + +\begin{document} + +\maketitle + + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\section{The concept of dynamical quantities} +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +As a chemist, it is maybe difficult to understand the concept of dynamical properties, the motivation behind their introduction, and their actual usefulness. +Here, we will try to give a pedagogical example showing the importance of dynamical quantities and their main purposes \cite{ReiningBook}. +To do so, let us consider the usual chemical scenario where one wants to get the neutral excitations of a given system. +In most cases, this can be done by solving a set of linear equations of the form +\begin{equation} + \label{eq:lin_sys} + \bA \bx = \omega \bx +\end{equation} +where $\omega$ is one of the neutral excitation energies of the system associated with the transition vector $\bx$. +If we assume that the operator $\bA$ has a matrix representation of size $K \times K$, this \textit{linear} set of equations yields $K$ excitation energies. +However, in practice, $K$ might be very large, and it might therefore be practically useful to recast this system as two smaller coupled systems, such that +\begin{equation} + \label{eq:lin_sys_split} + \begin{pmatrix} + \bA_1 & \T{\bb} \\ + \bb & \bA_2 \\ + \end{pmatrix} + \begin{pmatrix} + \bx_1 \\ + \bx_2 \\ + \end{pmatrix} + = \omega + \begin{pmatrix} + \bx_1 \\ + \bx_2 \\ + \end{pmatrix} +\end{equation} +where the blocks $\bA_1$ and $\bA_2$, of sizes $K_1 \times K_1$ and $K_2 \times K_2$ (with $K_1 + K_2 = K$), can be associated with, for example, the single and double excitations of the system. +Note that this \textit{exact} decomposition does not alter, in any case, the values of the excitation energies, not their eigenvectors. + +Solving separately each row of the system \eqref{eq:lin_sys_split} yields +\begin{subequations} +\begin{gather} + \label{eq:row1} + \bA_1 \bx_1 + \T{\bb} \bx_2 = \omega \bx_1 + \\ + \label{eq:row2} + \bx_2 = (\omega \bI - \bA_2)^{-1} \bb \bx_1 +\end{gather} +\end{subequations} +Substituting Eq.~\eqref{eq:row2} into Eq.~\eqref{eq:row1} yields the following effective \textit{non-linear}, frequency-dependent operator +\begin{equation} + \label{eq:non_lin_sys} + \Tilde{\bA}_1(\omega) \bx_1 = \omega \bx_1 +\end{equation} +with +\begin{equation} + \Tilde{\bA}_1(\omega) = \bA_1 + \T{\bb} (\omega \bI - \bA_2)^{-1} \bb +\end{equation} +which has, by construction, exactly the same solutions than the linear system \eqref{eq:lin_sys} but a smaller dimension. +For example, an operator $\Tilde{\bA}_1(\omega)$ built in the basis of single excitations can potentially provide excitation energies for double excitations thanks to its frequency-dependent nature, the information from the double excitations being ``folded'' into $\Tilde{\bA}_1(\omega)$ via Eq.~\eqref{eq:row2} \cite{ReiningBook}. + +How have we been able to reduce the dimension of the problem while keeping the same number of solutions? +To do so, we have transformed a linear operator $\bA$ into a non-linear operator $\Tilde{\bA}_1(\omega)$ by making it frequency dependent. +In other words, we have sacrificed the linearity of the system in order to obtain a new, non-linear systems of equations of smaller dimension. +This procedure converting degrees of freedom into frequency or energy dependence is very general and can be applied in various contexts. +Thanks to its non-linearity, Eq.~\eqref{eq:non_lin_sys} can produce more solutions than its actual dimension. +However, because there is no free lunch, this non-linear system is obviously harder to solve than its corresponding linear analogue given by Eq.~\eqref{eq:lin_sys}. +Nonetheless, approximations can be now applied to Eq.~\eqref{eq:non_lin_sys} in order to solve it efficiently. + +One of these approximations is the so-called \textit{static} approximation, which corresponds to fix the frequency to a particular value. +For example, as commonly done within the Bethe-Salpeter formalism, $\Tilde{\bA}_1(\omega) = \Tilde{\bA}_1 \equiv \Tilde{\bA}_1(\omega = 0)$. +In such a way, the operator $\Tilde{\bA}_1$ is made linear again by removing its frequency-dependent nature. +This approximation comes with a heavy price as the number of solutions provided by the system of equations \eqref{eq:non_lin_sys} has now been reduced from $K$ to $K_1$. +Coming back to our example, in the static approximation, the operator $\Tilde{\bA}_1$ built in the basis of single excitations cannot provide double excitations anymore, and the only $K_1$ excitation energies are associated with single excitations. + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\section{A two-level model} +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +Let us consider a two-level quantum system made of two orbitals \cite{Romaniello_2009b}. +We will label these two orbitals as valence ($v$) and conduction ($c$) orbitals with respective one-electron energies $\e{v}$ and $\e{c}$. +In a more quantum chemical language, these correspond to the HOMO and LUMO orbitals (respectively). +The ground state has a one-electron configuration $v\bar{v}$, while the doubly-excited state has a configuration $c\bar{c}$. +There is then only one single excitation which corresponds to the transition $v \to c$. +As usual, this can produce a singlet singly-excited state of configuration $(v\bar{c} + c\bar{v})/\sqrt{2}$, and a triplet singly-excited state of configuration $(v\bar{c} - c\bar{v})/\sqrt{2}$ \cite{SzaboBook}. + +Within many-body perturbation theory (MBPT), one can easily compute the quasiparticle energies associated with the valence and conduction orbitals. +Assuming that the dynamically-screened Coulomb potential has been calculated at the random-phase approximation (RPA) level and within the Tamm-Dancoff approximation (TDA), the expression of the $\GW$ quasiparticle energy is +\begin{equation} + \e{p}^{\GW} = \e{p} + Z_{p} \Sig{p}(\e{p}) +\end{equation} +where $p = v$ or $c$, +\begin{subequations} +\begin{align} + \label{eq:Sigv} + \Sig{v}(\omega) & = \frac{2 \ERI{vv}{vc}^2}{\omega - \e{v} + \Omega} + \frac{2 \ERI{vc}{cv}^2}{\omega - \e{c} + \Omega} + \\ + \label{eq:Sigc} + \Sig{c}(\omega) & = \frac{2 \ERI{vc}{cv}^2}{\omega - \e{v} + \Omega} + \frac{2 \ERI{vc}{cc}^2}{\omega - \e{c} + \Omega} +\end{align} +\end{subequations} +are the correlation parts of the self-energy associated with the valence of conduction orbitals, +\begin{equation} + Z_{p} = \qty( 1 - \left. \pdv{\Sig{p}(\omega)}{\omega} \right|_{\omega = \e{p}} )^{-1} +\end{equation} +is the renormalization factor, and +\begin{equation} + \ERI{pq}{rs} = \iint p(\br) q(\br) \frac{1}{\abs{\br - \br'}} r(\br') s(\br') d\br d\br' +\end{equation} +are the usual (bare) two-electron integrals. +In Eqs.~\eqref{eq:Sigv} and \eqref{eq:Sigc}, $\Omega = \Delta\e{} + 2 \ERI{vc}{vc}$ is the sole (singlet) RPA excitation energy of the system, with $\Delta\e{} = \eGW{c} - \eGW{v}$. + +One can now build the dynamical Bethe-Salpeter equation (dBSE) Hamiltonian, which reads +\begin{equation} \label{eq:HBSE} + \bH^{\dBSE}(\omega) = + \begin{pmatrix} + R(\omega) & C(\omega) + \\ + -C(-\omega) & -R(-\omega) + \end{pmatrix} +\end{equation} +with +\begin{subequations} +\begin{align} + R(\omega) & = \Delta\e{} + 2 \sigma \ERI{vc}{cv} - W_R(\omega) + \\ + C(\omega) & = 2 \sigma \ERI{vc}{cv} - W_C(\omega) +\end{align} +\end{subequations} +($\sigma = 1$ for singlets and $\sigma = 0$ for triplets) and +\begin{subequations} +\begin{align} + W_R(\omega) & = \ERI{vv}{cc} + \frac{4 \ERI{vv}{vc} \ERI{vc}{cc}}{\omega - \Omega - \Delta\e{}} + \\ + W_C(\omega) & = \ERI{vc}{cv} + \frac{4 \ERI{vc}{cv}^2}{\omega - \Omega} +\end{align} +\end{subequations} +are the elements of the dynamically-screened Coulomb potential for the resonant and coupling blocks of the dBSE Hamiltonian. +It can be easily shown that solving the equation +\begin{equation} + \det[\bH^{\dBSE}(\omega) - \omega \bI] = 0 +\end{equation} +yields 6 solutions (per spin manifold): 3 pairs of frequencies opposite in sign, which corresponds to the 3 resonant states and the 3 anti-resonant states. +As mentioned in Ref.~\cite{Romaniello_2009b}, spurious solutions appears due to the approximate nature of the dBSE kernel. +Indeed, diagonalizing the exact Hamiltonian would produce two singlet solutions corresponding to the singly- and doubly-excited states, while there is only one triplet state (see discussion earlier in the section). +Therefore, there is one spurious solution for the singlet manifold and two spurious solution for the triplet manifold. + +Within the static approximation, the BSE Hamiltonian is +\begin{equation} + \bH^{\BSE} = + \begin{pmatrix} + R^{\stat} & C^{\stat} + \\ + -C^{\stat} & -R^{\stat} + \end{pmatrix} +\end{equation} +with +\begin{align} + R^{\stat} & = R(\omega = \Delta\e{}) = \Delta\e{} + 2 \sigma \ERI{vc}{vc} - W_R(\omega = \Delta\e{}) + \\ + C^{\stat} & = C(\omega = 0) = 2 \sigma \ERI{vc}{vc} - W_C(\omega = 0) +\end{align} +In the static approximation, only one pair of solutions (per spin manifold) is obtained by diagonalizing $\bH^{\BSE}$. +There are, like in the dynamical case, opposite in sign. +Therefore, the static BSE Hamiltonian does not produce spurious excitations but misses the (singlet) double excitation. + +Enforcing the TDA, which corresponds to neglecting the coupling term between the resonant and anti-resonant part of the BSE Hamiltonian, \ie, $C(\omega) = 0$, allows to remove some of these spurious excitations. +In this case, the excitation energies are obtained by solving the simple equation $R(\omega) - \omega = 0$, which yields two solutions for each spin manifold. +There is thus only one spurious excitation in the triplet manifold, the two solutions of the singlet manifold corresponding to the single and double excitations. + +Another way to access dynamical effects while staying in the static framework is to use perturbation theory. +To do so, one must decompose the dBSE Hamiltonian into a (zeroth-order) static part and a dynamical perturbation, such that +\begin{equation} + \bH^{\dBSE}(\omega) = \underbrace{\bH^{\BSE}}_{\bH^{(0)}} + \underbrace{\qty[ \bH^{\dBSE}(\omega) - \bH^{\BSE} ]}_{\bH^{(1)}} +\end{equation} +Thanks to (renormalized) first-order perturbation theory, one gets +\begin{equation} + \omega_{1,\sigma}^{\BSE1} = \omega_{1,\sigma}^{\BSE} + Z_{1} \T{\bV} \cdot \qty[ \bH^{\dBSE}(\omega = \omega_{1,\sigma}^{\BSE}) - \bH^{\BSE} ] \cdot \bV +\end{equation} +where +\begin{equation} + \bV = + \begin{pmatrix} + X \\ Y + \end{pmatrix} +\end{equation} +are the eigenvectors of $\bH^{\BSE}$, and +\begin{equation} + Z_{1} = \qty{ 1 - \T{\bV} \cdot \left. \pdv{\bH^{\dBSE}(\omega)}{\omega} \right|_{\omega = \omega_{1,\sigma}^{\BSE}} \cdot \bV }^{-1} +\end{equation} +This corresponds to a dynamical correction to the static excitations, and the TDA can be applied to the dynamical correction, a scheme we label as dTDA in the following. + +We now take a numerical example by considering the singlet ground state of the \ce{He} atom in the 6-31G basis set. +This system contains two orbitals and the numerical values of the various quantities defined above are +\begin{align} + \e{v} & = -0.914\,127 + & + \e{c} & = + 1.399\,859 + \\ + \ERI{vv}{cc} & = 0.858\,133 + & + \ERI{vc}{cv} & = 0.227\,670 + \\ + \ERI{vv}{vc} & = 0.255\,554 + & + \ERI{vc}{cc} & = 0.316\,490 +\end{align} +which yields +\begin{align} + \Omega & = 2.769\,327 + & + \eGW{v} & = -0.863\,700 + & + \eGW{c} & = +1.373\,640 +\end{align} + +%%% FIGURE 1 %%% +\begin{figure} + \includegraphics[width=\linewidth]{dBSE} + \caption{ + $\det[\bH^{\dBSE}(\omega) - \omega \bI]$ as a function of $\omega$ for both the singlet (red) and triplet (blue) manifolds. + \label{fig:dBSE} + } +\end{figure} +%%% %%% %%% %%% + +Figure \ref{fig:dBSE} shows the three resonant solutions (for the singlet and triplet spin manifold) of the dynamical BSE Hamiltonian $\bH(\omega)$ defined in Eq.~\eqref{eq:HBSE}, the curve being invariant with respect to the transformation $\omega \to - \omega$ (electron-hole symmetry). +Numerically, we find +\begin{align} + \omega_{1,\updw}^{\dBSE} & = 1.90527 + & + \omega_{2,\updw}^{\dBSE} & = 2.78377 + & + \omega_{3,\updw}^{\dBSE} & = 4.90134 +\end{align} +for the singlet states, and +\begin{align} + \omega_{1,\upup}^{\dBSE} & = 1.46636 + & + \omega_{2,\upup}^{\dBSE} & = 2.76178 + & + \omega_{3,\upup}^{\dBSE} & = 4.91545 +\end{align} +for the triplet states. +it is interesting to mention that, around $\omega = \omega_1^{\sigma}$ ($\sigma =$ $\updw$ or $\upup$), the slope of the curves depicted in Fig.~\ref{fig:dBSE} is small, while the two other solutions, $\omega_2^{\sigma}$ and $\omega_3^{\sigma}$, stem from poles and consequently the slope is very large around these frequency values. + +Diagonalizing the static BSE Hamiltonian yields the following singlet and triplet excitation energies: +\begin{align} + \omega_{1,\updw}^{\BSE} & = 1.92778 + & + \omega_{1,\upup}^{\BSE} & = 1.48821 +\end{align} +which shows that the physical single excitation stemming from the dynamical BSE Hamiltonian is the lowest one for each spin manifold, \ie, $\omega_1^{\updw}$ and $\omega_1^{\upup}$. + +%%% FIGURE 2 %%% +\begin{figure} + \includegraphics[width=\linewidth]{dBSE-TDA} + \caption{ + $\det[\bH^{\TDAdBSE}(\omega) - \omega \bI]$ as a function of $\omega$ for both the singlet (red) and triplet (blue) manifolds within the TDA. + \label{fig:dBSE-TDA} + } +\end{figure} +%%% %%% %%% %%% + +Figure \ref{fig:dBSE-TDA} shows the same curves as Fig.~\ref{fig:dBSE} but in the TDA. +As one can see, the spurious solution $\omega_2^{\sigma}$ has disappeared, and two pairs of solutions remain for each spin manifold. +Numerically, we have +\begin{align} + \omega_{1,\updw}^{\TDAdBSE} & = 1.94005 + & + \omega_{3,\updw}^{\TDAdBSE} & = 4.90117 +\end{align} +for the singlet states, and +\begin{align} + \omega_{1,\upup}^{\TDAdBSE} & = 1.47070 + & + \omega_{3,\upup}^{\TDAdBSE} & = 4.91517 +\end{align} +while the static values are +\begin{align} + \omega_{1,\updw}^{\TDABSE} & = 1.95137 + & + \omega_{1,\upup}^{\TDABSE} & = 1.49603 +\end{align} + +It is now instructive to provide the exact results, \ie, the excitation energies obtained by diagonalizing the exact Hamiltonian in the same basis set. +A quick configuration interaction with singles and doubles (CISD) calculation provide the following excitation energies: +\begin{align} + \omega_{1}^{\updw} & = 1.92145 + & + \omega_{1}^{\upup} & = 1.47085 + & + \omega_{3}^{\updw} & = 3.47880 +\end{align} +This evidences that BSE reproduces qualitatively well the singlet and triplet single excitations, but quite badly the double excitation which is off by more than 1 hartree. +All these numerical results are gathered in Table \ref{tab:Ex}. + +The perturbatively-corrected values are also reported, which shows that this scheme is very efficient at reproducing the dynamical value. +Note that, although the BSE1(dTDA) value is further from the dBSE value than BSE1, it is quite close to the exact excitation energy. + +%%% TABLE I %%% +\begin{table} + \caption{Singlet and triplet excitation energies at various levels of theory. + \label{tab:Ex} + } + \begin{center} + \small + \begin{tabular}{|c|ccccccc|c|} + \hline + Singlets & BSE & BSE1 & BSE1(dTDA) & dBSE & BSE(TDA) & BSE1(TDA) & dBSE(TDA) & Exact \\ + \hline + $\omega_1$ & 1.92778 & 1.90022 & 1.91554 & 1.90527 & 1.95137 & 1.94004 & 1.94005 & 1.92145 \\ + $\omega_2$ & & & & 2.78377 & & & & \\ + $\omega_3$ & & & & 4.90134 & & & 4.90117 & 3.47880 \\ + \hline + Triplets & BSE & BSE1 & BSE1(dTDA) & dBSE & BSE(TDA) & BSE1(TDA) & dBSE(TDA) & Exact \\ + \hline + $\omega_1$ & 1.48821 & 1.46860 & 1.46260 & 1.46636 & 1.49603 & 1.47070 & 1.47070 & 1.47085 \\ + $\omega_2$ & & & & 2.76178 & & & & \\ + $\omega_3$ & & & & 4.91545 & & & 4.91517 & \\ + \hline + \end{tabular} + \end{center} +\end{table} +%%% %%% %%% %%% + +% BIBLIOGRAPHY +\bibliographystyle{unsrt} +\bibliography{../BSEdyn} + +\end{document} diff --git a/Notes/dBSE-TDA.pdf b/Notes/dBSE-TDA.pdf new file mode 100644 index 0000000..dd2d38c Binary files /dev/null and b/Notes/dBSE-TDA.pdf differ diff --git a/Notes/dBSE.pdf b/Notes/dBSE.pdf new file mode 100644 index 0000000..5832965 Binary files /dev/null and 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