diff --git a/BF_GS_VQZ.pdf b/BF_GS_VQZ.pdf new file mode 100644 index 0000000..f62aa68 Binary files /dev/null and b/BF_GS_VQZ.pdf differ diff --git a/BF_GS_VTZ.pdf b/BF_GS_VTZ.pdf deleted file mode 100644 index b614af2..0000000 Binary files a/BF_GS_VTZ.pdf and /dev/null differ diff --git a/BSE-PES.bib b/BSE-PES.bib index a7b4962..d6dd4f5 100644 --- a/BSE-PES.bib +++ b/BSE-PES.bib @@ -1,7 +1,7 @@ %% This BibTeX bibliography file was created using BibDesk. %% http://bibdesk.sourceforge.net/ -%% Created for Pierre-Francois Loos at 2020-01-28 17:57:57 +0100 +%% Created for Pierre-Francois Loos at 2020-02-03 16:35:45 +0100 %% Saved with string encoding Unicode (UTF-8) @@ -13050,22 +13050,15 @@ 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} -} - + Author = {Holzer,Christof and Teale,Andrew M. and Hampe,Florian and Stopkowicz,Stella and Helgaker,Trygve and Klopper,Wim}, + Doi = {10.1063/1.5093396}, + Eprint = {https://doi.org/10.1063/1.5093396}, + Journal = {J. Chem. Phys.}, + Number = {21}, + Pages = {214112}, + Title = {GW quasiparticle energies of atoms in strong magnetic fields}, + Url = {https://doi.org/10.1063/1.5093396}, + Volume = {150}, + Year = {2019}, + Bdsk-Url-1 = {https://doi.org/10.1063/1.5093396}} diff --git a/BSE-PES.tex b/BSE-PES.tex index 85e3c6b..b57ec81 100644 --- a/BSE-PES.tex +++ b/BSE-PES.tex @@ -307,9 +307,9 @@ 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} - and the corresponding (static) screened Coulomb potential matrix elements at coupling strength $\lambda$ + and the corresponding (static) screened Coulomb potential matrix elements at coupling strength $\IS$ \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{}', + \W{pq,rs}{\IS} = \iint \MO{p}(\br{}) \MO{q}(\br{}) \W{}{\IS}(\br{},\br{}') \MO{r}(\br{}') \MO{s}(\br{}') \dbr{} \dbr{}', \end{equation} the $(\bA{\IS},\bB{\IS})$ BSE matrix elements read: \begin{subequations} @@ -318,20 +318,23 @@ the $(\bA{\IS},\bB{\IS})$ BSE matrix elements read: \ABSE{ia,jb}{\IS} & = \delta_{ij} \delta_{ab} (\eGW{a} - \eGW{i}) + \IS \qty[ 2 \ERI{ia}{jb} - \W{ij,ab}{\IS} ], \\ \label{eq:LR_BSE-B} - \BBSE{ia,jb}{\IS} & = \lambda \qty[ 2 \ERI{ia}{bj} - \W{ib,aj}{\IS} ], + \BBSE{ia,jb}{\IS} & = \IS \qty[ 2 \ERI{ia}{bj} - \W{ib,aj}{\IS} ], \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: +%In the standard BSE implementation, the screened Coulomb potential $\W{}{\IS}$ is taken to be static $(\omega \rightarrow 0)$. +In the standard BSE approach, the screened Coulomb potential $\W{}{\IS}$ 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}'| }, + \W{}{\IS}(\br{},\br{}') + & = \int \dbr{1} \frac{\epsilon_{\IS}^{-1}(\br{},\br{1}; \omega=0)}{\abs{\br{1} - \br{}'}}, + \\ + \epsilon_{\IS}(\br{},\br{}'; \omega) + & = \delta(\br{}-\br{}') - \IS \int \dbr{1} \frac{\chi_{0}(\br{},\br{1}; \omega)}{\abs{\br{1} - \br{}'}}, \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} +with $\epsilon_{\IS}$ the dielectric function at coupling constant $\IS$ and $\chi_{0}$ the non-interacting polarizability. In the occupied-to-virtual molecular orbitals product basis, the spectral representation of $\W{}{\IS}$ can be written as follows in the case of real molecular orbitals \footnote{In the case of complex molecular orbitals, see Ref.~\onlinecite{Holzer_2019} for a correct use of complex conjugation in the spectral representation of $W$.} \begin{multline} \label{eq:W} \W{ij,ab}{\IS}(\omega) = \ERI{ij}{ab} + 2 \sum_m^{\Nocc \Nvir} \sERI{ij}{m} \sERI{ab}{m} @@ -339,7 +342,7 @@ with $\epsilon_{\lambda}$ the dielectric function at coupling constant $\lambda$ \times \qty(\frac{1}{\omega - \OmRPA{m}{\IS} + i \eta} - \frac{1}{\omega + \OmRPA{m}{\IS} - i \eta}), \end{multline} -where the spectral weights at coupling strength $\lambda$ read +where the spectral weights at coupling strength $\IS$ read \begin{equation} \sERI{pq}{m} = \sum_i^{\Nocc} \sum_a^{\Nvir} \ERI{pq}{ia} (\bX{\IS}_m + \bY{\IS}_m)_{ia}. \end{equation} @@ -357,8 +360,8 @@ In Eq.~\eqref{eq:W}, $\eta$ is a positive infinitesimal, and $\OmRPA{m}{\IS}$ a 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.~\eqref{eq:LR_BSE-A} and \eqref{eq:LR_BSE-B} to the RPAx equations: +%namely setting $\epsilon_{\IS}({\bf r},{\bf r}'; \omega) = \delta({\bf r}-{\bf r}')$ +so that $\W{}{\IS}$ reduces to the bare Coulomb potential. In that limit, the $GW$ quasiparticle energies reduce to the Hartree-Fock eigenvalues, and Eqs.~\eqref{eq:LR_BSE-A} and \eqref{eq:LR_BSE-B} to the RPAx equations: \begin{subequations} \begin{align} \label{eq:LR_RPAx-A} @@ -368,7 +371,7 @@ so that $W^{\lambda}$ reduces to the bare Coulomb potential. In that limit, t \BRPAx{ia,jb}{\IS} & = \IS \qty[ 2 \ERI{ia}{bj} - \ERI{ib}{aj} ]. \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. +%This allows to understand that the strength parameter $\IS$ enters twice in the $\IS W^{\IS}$ contribution, one time to renormalize the screening efficiency, and a second time to renormalize the direct electron-hole interaction. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %\subsection{Ground-state BSE energy} @@ -582,9 +585,9 @@ In that case again, the performance of BSE@{\GOWO}@HF are outstanding, as shown \begin{figure*} \includegraphics[height=0.26\linewidth]{N2_GS_VQZ} \includegraphics[height=0.26\linewidth]{CO_GS_VQZ} - \includegraphics[height=0.26\linewidth]{BF_GS_VTZ} + \includegraphics[height=0.26\linewidth]{BF_GS_VQZ} \caption{ -Ground-state PES of the isoelectronic series \ce{N2} (left), \ce{CO} (center), and \ce{BF} (right) around their respective equilibrium geometry obtained at various levels of theory with the \titou{cc-pVQZ} basis set. +Ground-state PES of the isoelectronic series \ce{N2} (left), \ce{CO} (center), and \ce{BF} (right) around their respective equilibrium geometry obtained at various levels of theory with the cc-pVQZ basis set. Additional graphs for other basis sets and within the frozen-core approximation can be found in the {\SI}. \label{fig:PES-N2-CO-BF} } diff --git a/CO_GS_VQZ.pdf b/CO_GS_VQZ.pdf index dd129ed..b27e08b 100644 Binary files a/CO_GS_VQZ.pdf and b/CO_GS_VQZ.pdf differ diff --git a/F2_GS_VQZ.pdf b/F2_GS_VQZ.pdf index 6982878..01a8eb1 100644 Binary files a/F2_GS_VQZ.pdf and b/F2_GS_VQZ.pdf differ diff --git a/H2_GS_VQZ.pdf b/H2_GS_VQZ.pdf index f0678ab..be05396 100644 Binary files a/H2_GS_VQZ.pdf and b/H2_GS_VQZ.pdf differ diff --git a/HCl_GS_VQZ.pdf b/HCl_GS_VQZ.pdf index 80f4051..69371df 100644 Binary files a/HCl_GS_VQZ.pdf and b/HCl_GS_VQZ.pdf differ diff --git a/LiF_GS_VQZ.pdf b/LiF_GS_VQZ.pdf index 1c87260..6445ea2 100644 Binary files a/LiF_GS_VQZ.pdf and b/LiF_GS_VQZ.pdf differ diff --git a/LiH_GS_VQZ.pdf b/LiH_GS_VQZ.pdf index 3aae5ac..82869a5 100644 Binary files a/LiH_GS_VQZ.pdf and b/LiH_GS_VQZ.pdf differ diff --git a/N2_GS_VQZ.pdf b/N2_GS_VQZ.pdf index 20dffc3..26e3f5f 100644 Binary files a/N2_GS_VQZ.pdf and b/N2_GS_VQZ.pdf differ