diff --git a/BF_GS_VQZ.pdf b/BF_GS_VQZ.pdf index f62aa68..cb2c24f 100644 Binary files a/BF_GS_VQZ.pdf and b/BF_GS_VQZ.pdf differ diff --git a/BSE-PES.tex b/BSE-PES.tex index b57ec81..b1cfbfa 100644 --- a/BSE-PES.tex +++ b/BSE-PES.tex @@ -108,6 +108,7 @@ %% bold in Table \newcommand{\bb}[1]{\textbf{#1}} \newcommand{\rb}[1]{\textbf{\textcolor{red}{#1}}} +\newcommand{\gb}[1]{\textbf{\textcolor{darkgreen}{#1}}} % excitation energies \newcommand{\OmRPA}[2]{\Omega_{#1}^{#2,\text{RPA}}} @@ -303,15 +304,15 @@ 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} -With the Mulliken notation for the bare two-electron integrals +With Mulliken's notation of 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 $\IS$ +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{}{\IS}(\br{},\br{}') \MO{r}(\br{}') \MO{s}(\br{}') \dbr{} \dbr{}', \end{equation} -the $(\bA{\IS},\bB{\IS})$ BSE matrix elements read: +the BSE matrix elements read \begin{subequations} \begin{align} \label{eq:LR_BSE-A} @@ -328,20 +329,19 @@ In the standard BSE approach, the screened Coulomb potential $\W{}{\IS}$ is buil \label{eq:wrpa} \begin{align} \W{}{\IS}(\br{},\br{}') - & = \int \dbr{1} \frac{\epsilon_{\IS}^{-1}(\br{},\br{1}; \omega=0)}{\abs{\br{1} - \br{}'}}, + & = \int \frac{\epsilon_{\IS}^{-1}(\br{},\br{}''; \omega=0)}{\abs*{\br{}' - \br{}''}} \dbr{}'' , \\ \epsilon_{\IS}(\br{},\br{}'; \omega) - & = \delta(\br{}-\br{}') - \IS \int \dbr{1} \frac{\chi_{0}(\br{},\br{1}; \omega)}{\abs{\br{1} - \br{}'}}, + & = \delta(\br{}-\br{}') - \IS \int \frac{\chi_{0}(\br{},\br{}''; \omega)}{\abs*{\br{}' - \br{}''}} \dbr{}'' , \end{align} \end{subequations} -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 $W$.} +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 spatial 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} \\ \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 $\IS$ read \begin{equation} \sERI{pq}{m} = \sum_i^{\Nocc} \sum_a^{\Nvir} \ERI{pq}{ia} (\bX{\IS}_m + \bY{\IS}_m)_{ia}. @@ -359,9 +359,10 @@ In Eq.~\eqref{eq:W}, $\eta$ is a positive infinitesimal, and $\OmRPA{m}{\IS}$ a \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 +The relationship between the BSE formalism and the well-known RPAx (\ie, RPA with exchange) approach can be obtained by switching off the screening %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: +so that $\W{}{\IS}$ reduces to the bare Coulomb potential $\vc{}$. +In this 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} @@ -412,10 +413,10 @@ 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 been 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 named ``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. +However, as commonly done within RPA and RPAx, \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 Eqs.~\eqref{eq:LR_RPA-A} and \eqref{eq:LR_RPA-B}, and their RPAx analogs in Eqs.~\eqref{eq:LR_RPAx-A} and \eqref{eq:LR_RPAx-B}. @@ -468,6 +469,7 @@ Additional graphs for other basis sets can be found in the {\SI}. \caption{ Equilibrium distances (in bohr) of the ground state of diatomic molecules obtained at various levels of theory and basis sets. The reference CC3 and corresponding BSE@{\GOWO}@HF data are highlighted in bold black and bold red for visual convenience, respectively. +The values in parenthesis have been obtained via a Morse fit of the PES. } \label{tab:Req} @@ -490,17 +492,17 @@ The reference CC3 and corresponding BSE@{\GOWO}@HF data are highlighted in bold & cc-pVTZ & 1.393 & 3.004 & 2.968 & 2.405 & 2.095 & 2.144 & 2.383 & 2.636 \\ & cc-pVQZ & 1.391 & 3.008 & 2.970 & 2.395 & 2.091 & 2.137 & 2.382 & 2.634 \\ BSE@{\GOWO}@HF & cc-pVDZ & 1.437 & 3.042 & 3.000 & 2.454 & 2.107 & 2.153 & 2.407 & 2.700 \\ - & cc-pVTZ & 1.404 & 3.023 & & 2.410 & 2.068 & 2.116 & & \\ - & cc-pVQZ &\rb{1.399} &\rb{3.017} &\rb{} &\rb{} &\rb{} &\rb{} &\rb{} &\rb{} \\ + & cc-pVTZ & 1.404 & 3.023 & (2.982) & 2.410 & 2.068 & 2.116 & (2.389) & (2.647) \\ + & cc-pVQZ &\rb{1.399} &\rb{3.017} &\rb{(2.974)}&\rb{(2.408)}&\rb{(2.070)}&\rb{(2.130)}&\rb{(2.383)}&\rb{(2.640)} \\ RPA@{\GOWO}@HF & cc-pVDZ & 1.426 & 3.019 & 2.994 & 2.436 & 2.083 & 2.144 & 2.403 & 2.691 \\ - & cc-pVTZ & 1.388 & 3.013 & & 2.408 & 2.065 & 2.114 & & \\ - & cc-pVQZ & 1.382 & 3.013 & & & & & & \\ + & cc-pVTZ & 1.388 & 3.013 & (2.965) & 2.408 & 2.065 & 2.114 & (2.370) & (2.584) \\ + & cc-pVQZ & 1.382 & 3.013 & (2.965) & (2.389) & (2.045) & (2.110) & (2.367) & (2.571) \\ RPAx@HF & cc-pVDZ & 1.428 & 3.040 & 2.998 & 2.424 & 2.077 & 2.130 & 2.417 & 2.611 \\ - & cc-pVTZ & 1.395 & 3.003 & 2.971 & 2.400 & 2.046 & 2.110 & 2.368 & 2.568 \\ - & cc-pVQZ & 1.394 & 3.011 & & & & & & \\ + & cc-pVTZ & 1.395 & 3.003 &\gb{2.971} & 2.400 & 2.046 & 2.110 & 2.368 & 2.568 \\ + & cc-pVQZ & 1.394 & 3.011 & (2.943) & (2.393) & (2.041) & (2.105) & (2.367) & (2.563) \\ RPA@HF & cc-pVDZ & 1.431 & 3.021 & 2.999 & 2.424 & 2.083 & 2.134 & 2.416 & 2.623 \\ & cc-pVTZ & 1.388 & 2.978 & 2.939 & 2.396 & 2.045 & 2.110 & 2.362 & 2.579 \\ - & cc-pVQZ & 1.386 & 2.994 & & & & & & \\ + & cc-pVQZ & 1.386 & 2.994 & (2.946) & (2.385) & (2.042) & (2.104) & (2.365) & (2.571) \\ % FROZEN CORE VERSION % Method & Basis & \ce{H2} & \ce{LiH}& \ce{LiF}& \ce{N2} & \ce{CO} & \ce{BF} & \ce{F2} & \ce{HCl}\\ % \hline @@ -551,7 +553,7 @@ Here again, the BSE@{\GOWO}@HF equilibrium bond length (obtained with cc-pVQZ) i \includegraphics[width=0.49\linewidth]{LiH_GS_VQZ} \caption{ Ground-state PES of \ce{H2} (left) and \ce{LiH} (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}. +%Additional graphs for other basis sets and within the frozen-core approximation can be found in the {\SI}. \label{fig:PES-H2-LiH} } \end{figure*} @@ -572,7 +574,7 @@ Note that these irregularities would be genuine discontinuities in the case of { \includegraphics[height=0.35\linewidth]{HCl_GS_VQZ} \caption{ Ground-state PES of \ce{LiF} (left) and \ce{HCl} (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}. +%Additional graphs for other basis sets and within the frozen-core approximation can be found in the {\SI}. \label{fig:PES-LiF-HCl} } \end{figure*} @@ -588,7 +590,7 @@ In that case again, the performance of BSE@{\GOWO}@HF are outstanding, as shown \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 cc-pVQZ basis set. -Additional graphs for other basis sets and within the frozen-core approximation can be found in the {\SI}. +%Additional graphs for other basis sets and within the frozen-core approximation can be found in the {\SI}. \label{fig:PES-N2-CO-BF} } \end{figure*} @@ -603,7 +605,7 @@ However, BSE@{\GOWO}@HF is the closest to the CC3 curve \includegraphics[width=\linewidth]{F2_GS_VQZ} \caption{ Ground-state PES of \ce{F2} around its 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}. +%Additional graphs for other basis sets and within the frozen-core approximation can be found in the {\SI}. \label{fig:PES-F2} } \end{figure} diff --git a/LiF_GS_VQZ.pdf b/LiF_GS_VQZ.pdf index 6445ea2..be3be23 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 82869a5..4bde22f 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 26e3f5f..37d3b9d 100644 Binary files a/N2_GS_VQZ.pdf and b/N2_GS_VQZ.pdf differ