diff --git a/CCvsMBPT.tex b/CCvsMBPT.tex index 4e5cf61..243dc9f 100644 --- a/CCvsMBPT.tex +++ b/CCvsMBPT.tex @@ -230,7 +230,7 @@ Substituting Eq.~\eqref{eq:RPA_1} into Eq.~\eqref{eq:RPA_2} yields the following \begin{equation} \bB{}{} + \bA{}{} \cdot \bT{}{} + \bT{}{} \cdot \bA{}{} + \bT{}{} \cdot \bB{}{} \cdot \bT{}{} = \bO \end{equation} -that matches the well-known rCCD equations +that matches the well-known rCCD amplitude equations \begin{multline} \label{eq:rCCD} \dbERI{ij}{ab} @@ -277,25 +277,28 @@ and matches the rCCD correlation energy because $\Tr(\bOm{}{} - \bA{}{}) = \Tr(\bR{}{} - \bA{}{}) = \Tr(\bB{}{} \cdot \bT{}{})$, as evidenced by Eq.~\eqref{eq:RPA_1}. Note that the same expression as Eq.~\eqref{eq:EcRPAx} can be derived from the adiabatic connection fluctuation dissipation theorem.\cite{Angyan_2011} -This simple and elegant proof was subsequently extended to excitation energies by Berkerbach, \cite{Berkelbach_2018} who showed that similitudes between EE-EOM-rCCD and RPAx exist when the EOM space is restricted to 1h1p configurations and only the two-body terms are dressed by rCCD correlation. +This simple and elegant proof was subsequently extended to excitation energies by Berkerbach, \cite{Berkelbach_2018} who showed that similitudes between EE-EOM-rCCD and RPAx exist when the EOM space is restricted to the 1h1p configurations and only the two-body terms are dressed by rCCD correlation. -To be more specific, restricting ourselves to CCD, \ie, $\hT = \hT_2$, the elements of the 1h1p block of the EOM Hamiltonian read +To be more specific, restricting ourselves to CCD, \ie, $\hT = \hT_2$, the elements of the 1h1p block of the EOM Hamiltonian read \cite{Stanton_1993} \begin{equation} \mel*{ \Psi_{i}^{a} }{ \bHN }{ \Psi_{j}^{b} } = \cF_{ab} \delta_{ij} - \cF_{ij} \delta_{ab} + \cW_{jabi} \end{equation} where $\bHN = e^{-\hT} \hH e^{\hT} - E_\text{CCD}$ is the normal-ordered similarity-transformed Hamiltonian, $\Psi_{i}^{a}$ are singly-excited determinants, the one-body terms are \begin{subequations} \begin{align} +\label{eq:cFab} \cF_{ab} & = \e{a}{} \delta_{ab} - \frac{1}{2} \sum_{klc} \ERI{kl}{bc} t_{kl}^{ac} \\ +\label{eq:cFij} \cF_{ij} & = \e{i}{} \delta_{ij} + \frac{1}{2} \sum_{kcd} \ERI{ik}{cd} t_{jk}^{cd} \end{align} \end{subequations} and the two-body term is \begin{equation} +\label{eq:cWibaj} \cW_{ibaj} = \ERI{ib}{aj} + \sum_{kc} \ERI{ik}{ac} t_{kj}^{cb} \end{equation} -Neglecting the effect of $\hT_2$ on the one-body terms and relying on the rCCD ampltiudes in the two-body terms yields +Neglecting the effect of $\hT_2$ on the one-body terms [see Eqs.~\eqref{eq:cFab} and \eqref{eq:cFij}] and relying on the rCCD amplitudes in the two-body terms, Eq.~\eqref{eq:cWibaj}, yields \begin{equation} \begin{split} \mel*{ \Psi_{i}^{a} }{ \tHN }{ \Psi_{j}^{b} } @@ -305,10 +308,10 @@ Neglecting the effect of $\hT_2$ on the one-body terms and relying on the rCCD a \end{split} \end{equation} which exactly matches Eq.~\eqref{eq:RPA_1}. -Although the excitation energies of this approximate EE-EOM-rCCD scheme are equal to the RPAx ones, it has been shown that the transition amplitudes (or residues) are distincts and only agrees at the lowest order in the Coulomb interaction. \cite{Emrich_1981,Berkelbach_2018} +Although the excitation energies of this approximate EE-EOM-rCCD scheme are equal to the RPAx ones, it has been shown that the transition amplitudes (or residues) are distinct and only agrees at the lowest order in the Coulomb interaction. \cite{Emrich_1981,Berkelbach_2018} This does not hold only for RPAx as it is actually quite general and can be applied to any ph problem with the structure of Eq.~\eqref{eq:RPA}, such as TD-DFT, BSE, and many others. -This can be be also extended to the pp and hh sectors as shown by Peng \textit{et al.} \cite{Peng_2013} and Scuseria \textit{et al.} \cite{Scuseria_2013} (see also Ref.~\onlinecite{Berkelbach_2018} for an extension to excitation energies). +This has been also extended to the pp and hh sectors by Peng \textit{et al.} \cite{Peng_2013} and Scuseria \textit{et al.} \cite{Scuseria_2013} (see also Ref.~\onlinecite{Berkelbach_2018} for the extension to excitation energies for the pp and hh channels). At the BSE level, and within the static approximation, one must solve a very similar linear eigenvalue problem \begin{equation} @@ -402,7 +405,7 @@ where the elements of the correlation part of the dynamical self-energy are \end{equation} and the renormalization factor is \begin{equation} - Z_p = \qty[ 1 - \eval{\pdv{\SigC{p}(\omega)}{\omega}}_{\omega = \e{p}{\HF}} ]^{-1} + Z_p = \qty[ 1 - \eval{\pdv{\SigC{pp}(\omega)}{\omega}}_{\omega = \e{p}{\HF}} ]^{-1} \end{equation} @@ -410,15 +413,13 @@ Therefore, following a similar procedure, one can show that the BSE correlation \begin{equation} \Ec^\text{BSE} = \frac{1}{4} \Tr(\bOm{}{\BSE} - \bA{}{\BSE}) \end{equation} -can be obtained via a set of rCCD equations \eqref{eq:rCCD}, where one substitutes all the two-electron integrals $\dbERI{pq}{rs}$ by $\dbERI{pq}{rs} - W_{ps,qr}^\text{stat}$. -Similarly to the diagonalization of the eigensystem \eqref{eq:BSE}, these rCCD-based equations can be solved in $\order*{N^6}$ cost. -\titou{Comments on cost and extension to gradients.} +can be obtained via a set of rCCD equations, where one substitutes in Eq.~\eqref{eq:rCCD} all the two-electron integrals $\dbERI{pq}{rs}$ by $\dbERI{pq}{rs} - W_{ps,qr}^\text{stat}$. +Similarly to the diagonalization of the eigensystem \eqref{eq:BSE}, these rCCD-based equations can be solved in $\order*{N^6}$ cost, and provides a clear path for the computation of ground- and excited-state properties (such as nuclear gradients) within the BSE framework. - -Following Berkelbach's analysis, one can extend the connection to excitation energies. -However, there is a significant difference as the BSE equations involves $GW$ quasiparticle energies [see Eq.~\eqref{eq:A_BSE}], where some of the correlation has been already dressed, while the RPAx equations only involves (undressed) HF orbital energies +Following Berkelbach's analysis, \cite{Berkelbach_2018} one can extend the connection to excitation energies. +However, there is a significant difference as the BSE involves $GW$ quasiparticle energies [see Eq.~\eqref{eq:A_BSE}], where some of the correlation has been already dressed, while the RPAx equations only involves (undressed) one-electron orbital energies It is therefore interesting to investigate if it possible also the recast the $GW$ equations as a set of CC-like equations. -Connections between EOM-CC and the $GW$ approximation have been already studied by Lange and Berkelbach, \cite{Lange_2018} but we believe that the present work proposes a different perspective on this particular subject. +Connections between approximate IP- and EA-EOM-CC schemes and the $GW$ approximation have been already studied in details by Lange and Berkelbach, \cite{Lange_2018} but we believe that the present work proposes a different perspective on this particular subject, as we derive genuine CC equations and we do not decouple the 2h1p and 2p1h sectors. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %\section{$GW$ approximation} @@ -459,6 +460,8 @@ and the corresponding coupling blocks read & V^\text{2p1h}_{p,kcd} & = \ERI{pk}{dc} \end{align} +Going beyond the Tamm-Dancoff approximation is possible, but more cumbersome. \cite{Bintrim_2021} + Introducing $\bT{}{\text{2h1p}} = \bY{}{\text{2h1p}} \cdot \bX{}{-1}$ and $\bT{}{\text{2p1h}} = \bY{}{\text{2p1h}} \cdot \bX{}{-1}$, we have \begin{equation} \begin{pmatrix} @@ -521,7 +524,7 @@ Substituting Eq.~\eqref{eq:R} into Eqs.~\eqref{eq:T1R} and \eqref{eq:T2R} yields \end{split} \end{align} \end{subequations} -In the CC language, in order to determine the 2h1p and 2p1h amplitudes, $t_{ij}^{a}$ and $t_{i}^{ab}$, it means that one must solve the following amplitude equations +In the CC language, in order to determine the 2h1p and 2p1h amplitudes, $t_{ija,p}^{\text{2h1p}}$ and $t_{iab,p}^{\text{2p1h}} $, it means that one must solve the following amplitude equations \begin{subequations} \begin{align} \begin{split} @@ -553,6 +556,7 @@ with \Delta_{iab,p}^{\text{2p1h}} & = \e{a}{} + \e{b}{} - \e{i}{} - \e{p}{} \end{align} \end{subequations} +One can then employed the usual iterative procedure to solve these non-linear equations by updating the amplitudes via \begin{subequations} \begin{align} t_{ija,p}^{\text{2h1p}} & \leftarrow t_{ija,p}^{\text{2h1p}} - \frac{r_{ija,p}^{\text{2h1p}}}{\Delta_{ij}^{ap}} @@ -568,80 +572,6 @@ The quasiparticle energies $\e{p}{GW}$ are thus provided by the eigenvalues of $ is the correlation part of the $GW$ self-energy. The $G_0W_0$ quasiparticle energies can be easily obtained via the procedure described in Ref.~\onlinecite{Monino_2022}. -%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -%\section{Coupled-cluster theory} -%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -%The Schr\"odinger equation is -%\begin{equation} -% \hH \Psi = E \Psi -%\end{equation} -%where the bare Hamiltonian reads -%\begin{equation} -% \hH = \sum_{pq} h_{p}^{q} \cre{p} \ani{q} + \frac{1}{4} \sum_{pqrs} v_{pq}^{rs} \cre{p} \cre{q} \ani{s} \ani{r} -%\end{equation} -%where $h_{p}^{q} = \mel{p}{\hh}{q}$ are the core Hamiltonian elements and $v_{pq}^{rs} = \ERI{pq}{rs}$ are antisymmetrized two-electron integrals. -%The creation and annihilation operators fulfils the following anticommutation relationships: -%\begin{align} -% \cre{p} \cre{q} + \cre{q} \cre{p} & = 0 -% & -% \ani{p} \ani{q} + \ani{q} \ani{p} & = 0 -% & -% \cre{p} \ani{q} + \ani{p} \cre{q} & = \delta_{pq} -%\end{align} -%In terms of normal-ordered (with respect to the reference wave function $\Psi_0$) quantities (which are indicated with curly braces), the Schr\"odinger equation reads -%\begin{equation} -% \hHN \Psi = \Delta E \Psi -%\end{equation} -%where $\Delta E = E - E_0 = E - \mel{\Psi_0}{\hH}{\Psi_0}$ is the correlation energy and the normal-ordered Hamiltonian $\hHN = \hH - E_0$ is -%\begin{equation} -% \hHN = \hFN + \hVN = \sum_{pq} f_{p}^{q} \NO{\cre{p}\ani{q}} + \frac{1}{4} \sum_{pqrs} v_{pq}^{rs} \NO{\cre{p}\cre{q}\ani{s}\ani{r}} -%\end{equation} -%with $f_{p}^{q} = \mel{p}{\hf}{q}$ the elements of the Fock operator. -% and where the creation and annihilation operators are written compactly as -%\begin{equation} -% \ca{pq\cdots}{rs\cdots} = \cre{p} \cre{q} \cdots \ani{s} \ani{r} -%\end{equation} - -%Within coupled-cluster theory, thanks to the following exponential ansatz of the wave function -%\begin{equation} -% \Psi = e^{\hT} \Psi_0 -%\end{equation} -%and the introducing the following similarity-transformed Hamiltonian -%\begin{equation} -% \bH = e^{-\hT} \hHN e^{\hT} -%\end{equation} -%one recasts the Schr\"odinger equation as -%\begin{equation} -% \bH \Psi_0 = \Delta E \Psi_0 -%\end{equation} -%where $e^{\hT}$ is the wave operator and $\hT$ is the excitation operator. -%The Schr\"odinger equation is recasted as -% -%Matrix elements involving $\bH$ can be efficiently and elegantly evaluated thanks to Wick's theorem and the so-called Campbell-Baker-Hausdorff (BCH) formula which leads to a linear combination of nested communtators -%\begin{equation} -%\begin{split} -% \bH -% & = \hHN + \comm{\hHN}{\hT{}{}} -% + \frac{1}{2!} \comm{\comm{\hHN}{\hT{}{}}}{\hT{}{}} -% \\ -% & + \frac{1}{3!} \comm{\comm{\comm{\hHN}{\hT{}{}}}{\hT{}{}}}{\hT{}{}} -% + \frac{1}{4!} \comm{\comm{\comm{\hHN}{\hT{}{}}}{\hT{}{}}}{\hT{}{}} -%\end{split} -%\end{equation} -%which naturally truncate at the fourth term due to the two-body nature of the bare Hamiltonian. -% -%In coupled-cluster theory, it is usual to truncate the excitation operator at the $n$th order, \ie, -%\begin{equation} -% \hT = \sum_{k=1}^{n} \hT_k -%\end{equation} -%where the normal-ordered $k$-fold ``neutral'' excitation operator -%\begin{equation} -% \hT_{k} = \frac{1}{(k!)^2} \sum_{ij\cdots} \sum_{ab\cdots} t_{ij\cdots}^{ab\cdots} -% \NO{\cre{a}\cre{b}\cdots}{\ani{i}\ani{j}\cdots} -%\end{equation} -%creates, when applied to $\ket*{\Psi_0}$, $k$-fold excited $\Ne$-electron determinants of the form $\ket*{\Psi_{ij\cdots}^{ab\cdots}}$. -%It is worth mentioning here that the excitation operators do commute with each other, \ie, $\comm{\hT_{k_1}}{\hT_{k_2}} = 0$ but do not commute with $\hHN$, \ie, $\comm{\hT_k}{\hHN} \neq 0$. - %====================== %\subsection{EE-EOM-CCD} %======================