diff --git a/BSE-PES.bib b/BSE-PES.bib index 386b692..a7b4962 100644 --- a/BSE-PES.bib +++ b/BSE-PES.bib @@ -13049,3 +13049,23 @@ Year = {2016}, Bdsk-Url-1 = {https://link.aps.org/doi/10.1103/PhysRevB.93.235113}, Bdsk-Url-2 = {https://doi.org/10.1103/PhysRevB.93.235113}} + +%%%% Xavier + +@misc{complexw, + note = {In the case of complex molecular orbitals, see Ref.~\citenum{Holzer_2019} for a correct use of complex conjugation in the spectral representation of $W$. } +} + +@article{Holzer_2019, +author = {Holzer,Christof and Teale,Andrew M. and Hampe,Florian and Stopkowicz,Stella and Helgaker,Trygve and Klopper,Wim }, +title = {GW quasiparticle energies of atoms in strong magnetic fields}, +journal = { J. Chem. Phys. }, +volume = {150}, +number = {21}, +pages = {214112}, +year = {2019}, +doi = {10.1063/1.5093396}, +URL = { https://doi.org/10.1063/1.5093396}, +eprint = { https://doi.org/10.1063/1.5093396} +} + diff --git a/BSE-PES.tex b/BSE-PES.tex index fa04162..360c7e2 100644 --- a/BSE-PES.tex +++ b/BSE-PES.tex @@ -303,42 +303,71 @@ where the excitation amplitudes are \bX{\IS} - \bY{\IS} = (\bOm{\IS})^{1/2} (\bA{\IS} - \bB{\IS})^{-1/2} \bZ{\IS}. \end{align} \end{subequations} -In the case of BSE, the specific expression of the matrix elements are -\begin{subequations} -\begin{align} - \label{eq:LR_BSE} - \ABSE{ia,jb}{\IS} & = \delta_{ij} \delta_{ab} (\eGW{a} - \eGW{i}) + \IS \W{ia,bj}{\IS}(\omega = 0) - \lambda \ERI{ia}{jb}, - \\ - \BBSE{ia,jb}{\IS} & = \W{ia,jb}{\IS}(\omega = 0) - \lambda \ERI{ia}{bj} , -\end{align} -\end{subequations} -where $\eGW{p}$ are the $GW$ quasiparticle energies, -\begin{multline} -\label{eq:W} - \W{ia,jb}{\IS}(\omega) = 2 \ERI{ia}{jb} - \\ - + \sum_m^{\Nocc \Nvir} \sERI{ia}{m} \sERI{jb}{m} \qty(\frac{1}{\omega - \OmRPA{m}{\IS} + i \eta} + \frac{1}{\omega + \OmRPA{m}{\IS} - i \eta}) -\end{multline} -are the elements of the screened Coulomb operator, -\begin{equation} - \sERI{pq}{m} = \sum_i^{\Nocc} \sum_a^{\Nvir} \ERI{pq}{ia} (\bX{\IS}_m + \bY{\IS}_m)_{ia} -\end{equation} -are the screened two-electron integrals, +With the Mulliken notation for the bare two-electron integrals \begin{equation} \ERI{pq}{rs} = \iint \frac{\MO{p}(\br{}) \MO{q}(\br{}) \MO{r}(\br{}') \MO{s}(\br{}')}{\abs*{\br{} - \br{}'}} \dbr{} \dbr{}', \end{equation} -are the bare two-electron integrals, and $\delta_{pq}$ is the Kronecker delta. \cite{NISTbook} + and the corresponding (static) screened Coulomb potential matrix elements at coupling strength $\lambda$ +\begin{equation} + \W{pq,rs}{\IS} = \iint \MO{p}(\br{}) \MO{q}(\br{}) W^{\lambda}(\br{},\br{}') \MO{r}(\br{}') \MO{s}(\br{}') \dbr{} \dbr{}', +\end{equation} +the $(\bA{\IS},\bB{\IS})$ BSE matrix elements read: +\begin{subequations} +\label{eq:LR_BSE} +\begin{align} + \ABSE{ia,jb}{\IS} & = \delta_{ij} \delta_{ab} (\eGW{a} - \eGW{i}) + \IS \left[ \ERI{ia}{jb} - \W{ij,ab}{\IS} \right], + \\ + \BBSE{ia,jb}{\IS} & = \lambda \left[ \ERI{ia}{bj} - \W{ib,aj}{\IS} \right], +\end{align} +\end{subequations} +where $\eGW{p}$ are the $GW$ quasiparticle energies. +%In the standard BSE implementation, the screened Coulomb potential $W^{\lambda}$ is taken to be static $(\omega \rightarrow 0)$. +In the standard BSE approach, the screened Coulomb potential $W^{\lambda}$ is built within the direct RPA scheme: +\begin{subequations} +\label{eq:wrpa} +\begin{align} + W^{\lambda}({\bf r},{\bf r}') &= \int d{\bf r}_1 \; \frac{\epsilon_{\lambda}^{-1}({\bf r},{\bf r}_1; \omega=0) } { |{\bf r}_1-{\bf r}' | }, \\ + \epsilon_{\lambda}({\bf r},{\bf r}'; \omega) &= \delta({\bf r}-{\bf r}') - \lambda \int d{\bf r}_1 \; \frac{ \chi_{0}({\bf r},{\bf r}_1; \omega) }{ |{\bf r}_1 - {\bf r}'| }, +\end{align} +\end{subequations} +with $\epsilon_{\lambda}$ the dielectric function at coupling constant $\lambda$ and $\chi_{0}$ the non-interacting polarizability. In the occupied-to-virtual molecular orbitals product basis, the spectral representation of $W^{\lambda}$ can be written as follows in the case of real molecular orbitals: \cite{complexw} +\begin{multline} +\label{eq:W} + \W{ij,ab}{\IS}(\omega) = \textcolor{red}{\sout{2}} \ERI{ij}{ab} + \sum_m^{\Nocc \Nvir} \sERI{ij}{m} \sERI{ab}{m} + \\ + \times \qty(\frac{1}{\omega - \OmRPA{m}{\IS} + i \eta} \textcolor{red}{-} \frac{1}{\omega + \OmRPA{m}{\IS} - i \eta}) +\end{multline} +where the \xavier{ \sout{screened two-electron integrals} spectral weights} $\sERI{pq}{m}$ at coupling strength $\lambda$ read: +\begin{equation} + \sERI{pq}{m} = \sum_i^{\Nocc} \sum_a^{\Nvir} \ERI{pq}{ia} (\bX{\IS}_m + \bY{\IS}_m)_{ia} +\end{equation} + In Eq.~\eqref{eq:W}, $\eta$ is a positive infinitesimal, and $\OmRPA{m}{\IS}$ are the direct (\ie, without exchange) RPA neutral excitation energies computed by solving the linear eigenvalue problem \eqref{eq:LR} with the following matrix elements \begin{subequations} \begin{align} \label{eq:LR_RPA} - \ARPA{ia,jb}{\IS} & = \delta_{ij} \delta_{ab} (\eHF{a} - \eHF{i}) + 2 \IS \ERI{ia}{bj}, + \ARPA{ia,jb}{\IS} & = \delta_{ij} \delta_{ab} (\eHF{a} - \eHF{i}) + \IS \ERI{ia}{jb}, \\ - \BRPA{ia,jb}{\IS} & = 2 \IS \ERI{ia}{jb}, + \BRPA{ia,jb}{\IS} & = \IS \ERI{ia}{bj}, \end{align} \end{subequations} where $\eHF{p}$ are the HF orbital energies. +The relation between the BSE formalism and the well-known RPAx approach can be obtained by switching off the screening +%namely setting $\epsilon_{\lambda}({\bf r},{\bf r}'; \omega) = \delta({\bf r}-{\bf r}')$ +so that $W^{\lambda}$ reduces to the bare Coulomb potential. In that limit, the $GW$ quasiparticle energies reduce to the Hartree-Fock eigenvalues, and Eqs.~\ref{eq:LR_BSE} to the RPAx equations: +\begin{subequations} +\begin{align} + \label{eq:LR_RPAx} + \ARPAx{ia,jb}{\IS} & = \delta_{ij} \delta_{ab} (\eHF{a} - \eHF{i}) + \IS \left[ \ERI{ia}{jb} - \ERI{ij}{ab} \right], + \\ + \BRPAx{ia,jb}{\IS} & = \IS \left[ \ERI{ia}{bj} - \ERI{ib}{aj} \right]. +\end{align} +\end{subequations} +%This allows to understand that the strength parameter $\lambda$ enters twice in the $\lambda W^{\lambda}$ contribution, one time to renormalize the screening efficiency, and a second time to renormalize the direct electron-hole interaction. + + + %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %\subsection{Ground-state BSE energy} %\label{sec:BSE_energy} @@ -378,21 +407,13 @@ is the interaction kernel \cite{Angyan_2011, Holzer_2018} [with $\tA{ia,jb}{\IS} \end{pmatrix} \end{equation} is the correlation part of the two-electron density matrix at interaction strength $\IS$, and $\Tr$ denotes the matrix trace. -Note that the present definition of the BSE correlation energy [see Eq.~\eqref{eq:EcBSE}], which we refer to as BSE@$GW$@HF here, has be labeled ``XBS'' for ``extended Bethe Salpeter'' by Holzer \textit{et al.} \cite{Holzer_2018} +Note that the present definition of the BSE correlation energy [see Eq.~\eqref{eq:EcBSE}], which we refer to as BSE@$GW$@HF here, has been labeled ``XBS'' for ``extended Bethe Salpeter'' by Holzer \textit{et al.} \cite{Holzer_2018} Contrary to DFT where the electron density is fixed along the adiabatic path, in the present formalism, the density is not maintained as $\IS$ varies. Therefore, an additional contribution to Eq.~\eqref{eq:EcBSE} originating from the variation of the Green's function along the adiabatic connection should be added. However, as commonly done within RPA and RPAx (\ie, RPA with exchange), \cite{Toulouse_2009, Toulouse_2010, Holzer_2018} we shall neglect it in the present study. Equation \eqref{eq:EcBSE} can also be straightforwardly applied to RPA and RPAx, the only difference being the expressions of $\bA{\IS}$ and $\bB{\IS}$ used to obtain the eigenvectors $\bX{\IS}$ and $\bY{\IS}$ entering the definition of $\bP{\IS}$ [see Eq.~\eqref{eq:2DM}]. -For RPA, these expressions have been provided in Eq.~\eqref{eq:LR_RPA}, and their RPAx analogs read -\begin{subequations} -\begin{align} - \label{eq:LR_RPAx} - \ARPAx{ia,jb}{\IS} & = \delta_{ij} \delta_{ab} (\eHF{a} - \eHF{i}) + 2 \IS \ERI{ia}{bj} - \IS \ERI{ia}{jb}, - \\ - \BRPAx{ia,jb}{\IS} & = 2 \IS \ERI{ia}{jb} - \IS \ERI{ia}{bj}. -\end{align} -\end{subequations} +For RPA, these expressions have been provided in Eq.~\eqref{eq:LR_RPA}, and their RPAx analogs in Eq.~\eqref{eq:LR_RPAx}. In the following, we will refer to these two types of calculations as RPA@HF and RPAx@HF, respectively. Finally, we will also consider the RPA@$GW$@HF scheme which consists in replacing the HF orbital energies in Eq.~\eqref{eq:LR_RPA} by the $GW$ quasiparticles energies.