From fa7f0cb97acc4fbb2a44104a409393d4ab137699 Mon Sep 17 00:00:00 2001 From: Pierre-Francois Loos Date: Sat, 25 Jan 2020 21:58:47 +0100 Subject: [PATCH] first draft of theory --- BSE-PES.tex | 212 ++++++++++++++++++++++----------------------- Data/N2_GS_VDZ.pdf | Bin 55077 -> 51114 bytes 2 files changed, 102 insertions(+), 110 deletions(-) diff --git a/BSE-PES.tex b/BSE-PES.tex index c552a90..d5a55c5 100644 --- a/BSE-PES.tex +++ b/BSE-PES.tex @@ -63,7 +63,7 @@ \newcommand{\Enuc}{E^\text{nuc}} \newcommand{\Ec}{E_\text{c}} \newcommand{\EHF}{E^\text{HF}} -\newcommand{\EBSE}[1]{E_{#1}^\text{BSE}} +\newcommand{\EBSE}{E^\text{BSE}} \newcommand{\EcRPA}{E_\text{c}^\text{dRPA}} \newcommand{\EcBSE}{E_\text{c}^\text{BSE}} \newcommand{\EcsBSE}{{}^1\EcBSE} @@ -80,51 +80,41 @@ \newcommand{\eGW}[1]{\epsilon^\text{\GW}_{#1}} \newcommand{\eevGW}[1]{\epsilon^\text{\evGW}_{#1}} \newcommand{\eGnWn}[2]{\epsilon^\text{\GnWn{#2}}_{#1}} -\newcommand{\Om}[1]{\Omega_{#1}} +\newcommand{\Om}[2]{\Omega_{#1}^{#2}} % Matrix elements -\newcommand{\A}[1]{A_{#1}} -\newcommand{\B}[1]{B_{#1}} +\newcommand{\A}[2]{A_{#1}^{#2}} +\newcommand{\tA}[2]{\Tilde{A}_{#1}^{#2}} +\newcommand{\B}[2]{B_{#1}^{#2}} \renewcommand{\S}[1]{S_{#1}} -\newcommand{\ABSE}[1]{A^\text{BSE}_{#1}} -\newcommand{\BBSE}[1]{B^\text{BSE}_{#1}} -\newcommand{\ARPA}[1]{A^\text{RPA}_{#1}} -\newcommand{\BRPA}[1]{B^\text{RPA}_{#1}} +\newcommand{\ABSE}[2]{A_{#1}^{#2,\text{BSE}}} +\newcommand{\BBSE}[2]{B_{#1}^{#2,\text{BSE}}} +\newcommand{\ARPA}[2]{A_{#1}^{#2,\text{RPA}}} +\newcommand{\BRPA}[2]{B_{#1}^{#2,\text{RPA}}} \newcommand{\G}[1]{G_{#1}} \newcommand{\LBSE}[1]{L_{#1}} \newcommand{\XiBSE}[1]{\Xi_{#1}} \newcommand{\Po}[1]{P_{#1}} -\newcommand{\W}[1]{W_{#1}} +\newcommand{\W}[2]{W_{#1}^{#2}} \newcommand{\Wc}[1]{W^\text{c}_{#1}} \newcommand{\vc}[1]{v_{#1}} \newcommand{\Sig}[1]{\Sigma_{#1}} \newcommand{\SigGW}[1]{\Sigma^\text{\GW}_{#1}} \newcommand{\Z}[1]{Z_{#1}} \newcommand{\MO}[1]{\phi_{#1}} +\newcommand{\ERI}[2]{(#1|#2)} +\newcommand{\sERI}[2]{[#1|#2]} % excitation energies -\newcommand{\OmRPA}[1]{\Omega^\text{RPA}_{#1}} -\newcommand{\OmCIS}[1]{\Omega^\text{CIS}_{#1}} -\newcommand{\OmTDHF}[1]{\Omega^\text{TDHF}_{#1}} -\newcommand{\OmBSE}[1]{\Omega^\text{BSE}_{#1}} +\newcommand{\OmRPA}[2]{\Omega_{#1}^{#2,\text{RPA}}} +\newcommand{\OmRPAx}[2]{\Omega_{#1}^{#2,\text{RPAx}}} +\newcommand{\OmBSE}[2]{\Omega_{#1}^{#2,\text{BSE}}} \newcommand{\spinup}{\downarrow} \newcommand{\spindw}{\uparrow} \newcommand{\singlet}{\uparrow\downarrow} \newcommand{\triplet}{\uparrow\uparrow} -\newcommand{\Oms}[1]{{}^{1}\Omega_{#1}} -\newcommand{\OmsRPA}[1]{{}^{1}\Omega^\text{RPA}_{#1}} -\newcommand{\OmsCIS}[1]{{}^{1}\Omega^\text{CIS}_{#1}} -\newcommand{\OmsTDHF}[1]{{}^{1}\Omega^\text{TDHF}_{#1}} -\newcommand{\OmsBSE}[1]{{}^{1}\Omega^\text{BSE}_{#1}} - -\newcommand{\Omt}[1]{{}^{3}\Omega_{#1}} -\newcommand{\OmtRPA}[1]{{}^{3}\Omega^\text{RPA}_{#1}} -\newcommand{\OmtCIS}[1]{{}^{3}\Omega^\text{CIS}_{#1}} -\newcommand{\OmtTDHF}[1]{{}^{3}\Omega^\text{TDHF}_{#1}} -\newcommand{\OmtBSE}[1]{{}^{3}\Omega^\text{BSE}_{#1}} - % Matrices \newcommand{\bO}{\mathbf{0}} \newcommand{\bI}{\mathbf{1}} @@ -139,16 +129,15 @@ \newcommand{\bde}{\mathbf{\Delta\epsilon}} \newcommand{\bdeHF}{\mathbf{\Delta\epsilon}^\text{HF}} \newcommand{\bdeGW}{\mathbf{\Delta\epsilon}^{GW}} -\newcommand{\bOm}{\mathbf{\Omega}} -\newcommand{\bA}{\mathbf{A}} -\newcommand{\bAs}{{}^1\bA} -\newcommand{\bAt}{{}^3\bA} -\newcommand{\bB}{\mathbf{B}} -\newcommand{\bX}{\mathbf{X}} -\newcommand{\bY}{\mathbf{Y}} -\newcommand{\bZ}{\mathbf{Z}} +\newcommand{\bOm}[1]{\mathbf{\Omega}^{#1}} +\newcommand{\bA}[1]{\mathbf{A}^{#1}} +\newcommand{\btA}[1]{\Tilde{\mathbf{A}}^{#1}} +\newcommand{\bB}[1]{\mathbf{B}^{#1}} +\newcommand{\bX}[1]{\mathbf{X}^{#1}} +\newcommand{\bY}[1]{\mathbf{Y}^{#1}} +\newcommand{\bZ}[1]{\mathbf{Z}^{#1}} \newcommand{\bK}{\mathbf{K}} -\newcommand{\bP}{\mathbf{P}} +\newcommand{\bP}[1]{\mathbf{P}^{#1}} % units \newcommand{\IneV}[1]{#1 eV} @@ -190,8 +179,8 @@ However, we also observe, in some cases, unphysical irregularities on the ground \maketitle %%%%%%%%%%%%%%%%%%%%%%%% -\section{Introduction} -\label{sec:intro} +%\section{Introduction} +%\label{sec:intro} %%%%%%%%%%%%%%%%%%%%%%%% The features of ground- and excited-state potential energy surfaces (PES) are critical for the faithful description and a deeper understanding of photochemical and photophysical processes. \cite{Bernardi_1996,Olivucci_2010,Robb_2007} @@ -236,22 +225,22 @@ Embracing this definition, the purpose of the present study is to investigate th The location of the minima on the ground- and (singlet and triplet) excited-state PES is of particular interest. This study is a first preliminary step towards the development of analytical nuclear gradients within the BSE@{\GW} formalism.} -The paper is organized as follows. -In Sec.~\ref{sec:theo}, we introduce the equations behind the BSE formalism. -In particular, the general equations are reported in Subsec.~\ref{sec:BSE}, and the corresponding equations obtained in a finite basis in Subsec.~\ref{sec:BSE_basis}. -Subsection \ref{sec:BSE_energy} defines the BSE total energy, and various other quantities of interest for this study. -Computational details needed to reproduce the results of the present work are reported in Sec.~\ref{sec:comp_details}. -Section \ref{sec:PES} reports ground-state PES for various diatomic molecules. -Finally, we draw our conclusion in Sec.~\ref{sec:conclusion}. +%The paper is organized as follows. +%In Sec.~\ref{sec:theo}, we introduce the equations behind the BSE formalism. +%In particular, the general equations are reported in Subsec.~\ref{sec:BSE}, and the corresponding equations obtained in a finite basis in Subsec.~\ref{sec:BSE_basis}. +%Subsection \ref{sec:BSE_energy} defines the BSE total energy, and various other quantities of interest for this study. +%Computational details needed to reproduce the results of the present work are reported in Sec.~\ref{sec:comp_details}. +%Section \ref{sec:PES} reports ground-state PES for various diatomic molecules. +%Finally, we draw our conclusion in Sec.~\ref{sec:conclusion}. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -\section{Theory} -\label{sec:theo} +%\section{Theory} +%\label{sec:theo} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -\subsection{The Bethe-Salpeter equation} -\label{sec:BSE} +%\subsection{The Bethe-Salpeter equation} +%\label{sec:BSE} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% In order to compute the neutral (optical) excitations of the system and their associated oscillator strengths, the BSE expresses the two-body propagator \cite{Strinati_1988, Martin_2016} \begin{multline} @@ -272,120 +261,120 @@ which takes into account the self-consistent variation of the Hartree potential In Eq.~\eqref{eq:BSE}, $\LBSE{0}(1,2,1',2') = - i \G{}(1,2')\G{}(2,1')$, and $(1)=(\br{1},\sigma_1,t_1)$ is a composite index gathering space, spin and time variables. In the {\GW} approximation, \cite{Hedin_1965,Aryasetiawan_1998,Onida_2002,Martin_2016,Reining_2017} we have \begin{equation} - \SigGW{\xc}(1,2) = i \G{}(1,2) \W{}(1^+,2), + \SigGW{\xc}(1,2) = i \G{}(1,2) \W{}{}(1^+,2), \end{equation} -where $\W{}$ is the screened Coulomb operator, and hence the BSE reduces to +where $\W{}{}$ is the screened Coulomb operator, and hence the BSE reduces to \begin{equation} - \XiBSE{}(3,5,4,6) = \delta(3,4) \delta(5,6) \vc{}(3,6) - \delta(3,6) \delta(4,5) \W{}(3,4), + \XiBSE{}(3,5,4,6) = \delta(3,4) \delta(5,6) \vc{}(3,6) - \delta(3,6) \delta(4,5) \W{}{}(3,4), \end{equation} -where, as commonly done, we have neglected the term $\delta \W{}/\delta \G{}$ in the functional derivative of the self-energy. \cite{Hanke_1980,Strinati_1984,Strinati_1982} -Finally, the static approximation is enforced, \ie, $\W{}(1,2) = \W{}(\{\br{1}, \sigma_1, t_1\},\{\br{2}, \sigma_2, t_2\}) \delta(t_1-t_2)$, which corresponds to restricting $\W{}$ to its static limit, \ie, $\W{}(1,2) = \W{}(\{\br{1},\sigma_1\},\{\br{2},\sigma_2\}; \omega=0)$. +where, as commonly done, we have neglected the term $\delta \W{}{}/\delta \G{}$ in the functional derivative of the self-energy. \cite{Hanke_1980,Strinati_1984,Strinati_1982} +Finally, the static approximation is enforced, \ie, $\W{}{}(1,2) = \W{}{}(\{\br{1}, \sigma_1, t_1\},\{\br{2}, \sigma_2, t_2\}) \delta(t_1-t_2)$, which corresponds to restricting $\W{}{}$ to its static limit, \ie, $\W{}{}(1,2) = \W{}{}(\{\br{1},\sigma_1\},\{\br{2},\sigma_2\}; \omega=0)$. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -\subsection{BSE in a finite basis} -\label{sec:BSE_basis} +%\subsection{BSE in a finite basis} +%\label{sec:BSE_basis} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -For a closed-shell system, in order to compute the singlet BSE excitation energies within the static approximation in a finite basis, one must solve the following linear response problem \cite{Casida,Dreuw_2005,Martin_2016} +For a closed-shell system, in order to compute the singlet BSE excitation energies (within the static approximation) of the physical system (\ie, $\IS = 1$) in a finite basis, one must solve the following linear response problem \cite{Casida,Dreuw_2005,Martin_2016} \begin{equation} \label{eq:LR} \begin{pmatrix} - \bA & \bB \\ - -\bB & -\bA \\ + \bA{\IS} & \bB{\IS} \\ + -\bB{\IS} & -\bA{\IS} \\ \end{pmatrix} \begin{pmatrix} - \bX_m \\ - \bY_m \\ + \bX{\IS}_m \\ + \bY{\IS}_m \\ \end{pmatrix} = - \Om{m} + \Om{m}{\IS} \begin{pmatrix} - \bX_m \\ - \bY_m \\ + \bX{\IS}_m \\ + \bY{\IS}_m \\ \end{pmatrix}, \end{equation} -where $\Om{m}$ is the $m$th excitation energy with eigenvector $\T{(\bX_m \, \bY_m)}$, and we have assumed real-valued spatial orbitals $\{\MO{p}(\br{})\}_{1 \le p \le \Norb}$. -The matrices $\bA$, $\bB$, $\bX$, and $\bY$ are all of size $\Nocc \Nvir \times \Nocc \Nvir$ where $\Nocc$ and $\Nvir$ are the number of occupied and virtual orbitals (\ie, $\Norb = \Nocc + \Nvir$), respectively. +where $\Om{m}{\IS}$ is the $m$th excitation energy with eigenvector $\T{(\bX{\IS}_m \, \bY{\IS}_m)}$ at interpolation strength $\IS$, and we have assumed real-valued spatial orbitals $\{\MO{p}(\br{})\}_{1 \le p \le \Norb}$. +The matrices $\bA{\IS}$, $\bB{\IS}$, $\bX{\IS}$, and $\bY{\IS}$ are all of size $\Nocc \Nvir \times \Nocc \Nvir$ where $\Nocc$ and $\Nvir$ are the number of occupied and virtual orbitals (\ie, $\Norb = \Nocc + \Nvir$), respectively. In the following, the index $m$ labels the $\Nocc \Nvir$ single excitations, $i$ and $j$ are occupied orbitals, $a$ and $b$ are unoccupied orbitals, while $p$, $q$, $r$, and $s$ indicate arbitrary orbitals. -In the absence of instabilities (\ie, $\bA - \bB$ is positive-definite), \cite{Dreuw_2005} Eq.~\eqref{eq:LR} is usually transformed into an Hermitian eigenvalue problem of smaller dimension +In the absence of instabilities (\ie, $\bA{\IS} - \bB{\IS}$ is positive-definite), \cite{Dreuw_2005} Eq.~\eqref{eq:LR} is usually transformed into an Hermitian eigenvalue problem of smaller dimension \begin{equation} \label{eq:small-LR} - (\bA - \bB)^{1/2} (\bA + \bB) (\bA - \bB)^{1/2} \bZ = \bOm^2 \bZ, + (\bA{\IS} - \bB{\IS})^{1/2} (\bA{\IS} + \bB{\IS}) (\bA{\IS} - \bB{\IS})^{1/2} \bZ{\IS} = (\bOm{\IS})^2 \bZ{\IS}, \end{equation} where the excitation amplitudes are \begin{subequations} \begin{align} - \bX + \bY = \bOm^{-1/2} (\bA - \bB)^{1/2} \bZ, + \bX{\IS} + \bY{\IS} = (\bOm{\IS})^{-1/2} (\bA{\IS} - \bB{\IS})^{1/2} \bZ{\IS}, \\ - \bX - \bY = \bOm^{1/2} (\bA - \bB)^{-1/2} \bZ. + \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} & = \delta_{ij} \delta_{ab} (\eGW{a} - \eGW{i}) + \W{ia,bj}(\omega = 0) - (ia|jb), + \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} & = \W{ia,jb}(\omega = 0) - (ib|ja) , + \BBSE{ia,jb}{\IS} & = \W{ia,jb}{\IS}(\omega = 0) - \lambda \ERI{ib}{ja} , \end{align} \end{subequations} where $\eGW{p}$ are the {\GW} quasiparticle energies, \begin{multline} \label{eq:W} - \W{ia,jb}(\omega) = 2 (ia|jb) + \W{ia,jb}{\IS}(\omega) = 2 \ERI{ia}{jb} \\ - + \sum_m^{\Nocc \Nvir} [ia|m] [jb|m] \qty(\frac{1}{\omega - \OmRPA{m} + i \eta} + \frac{1}{\omega + \OmRPA{m} - i \eta}) + + \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, +are the elements of the screened Coulomb operator $\W{}{\IS}$, \begin{equation} - [pq|m] = \sum_i^{\Nocc} \sum_a^{\Nvir} (pq|ia) (\bX_m+\bY_m)_{ia} + \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, \begin{equation} - (pq|rs) = \iint \frac{\MO{p}(\br{}) \MO{q}(\br{}) \MO{r}(\br{}') \MO{s}(\br{}')}{\abs*{\br{} - \br{}'}} \dbr{} \dbr{}', + \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. -In Eq.~\eqref{eq:W}, $\eta$ is a positive infinitesimal, and $\OmRPA{m}$ are the direct (\ie, without exchange) RPA neutral excitation energies computed during the {\GW} calculation by solving the linear eigenvalue problem \eqref{eq:LR} with matrix elements +are the bare two-electron integrals, and $\delta_{pq}$ is the Kronecker delta. \cite{NISTbook} +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_BSE} - \ARPA{ia,jb} & = \delta_{ij} \delta_{ab} (\eHF{a} - \eHF{i}) + (ia|bj), + \ARPA{ia,jb}{\IS} & = \delta_{ij} \delta_{ab} (\eHF{a} - \eHF{i}) + \IS \ERI{ia}{bj}, \\ - \BRPA{ia,jb} & = (ia|jb), + \BRPA{ia,jb}{\IS} & = \IS \ERI{ia}{jb}, \end{align} \end{subequations} -and $\eHF{p}$ are the HF orbital energies. +where $\eHF{p}$ are the HF orbital energies. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -\subsection{Ground-state BSE energy} -\label{sec:BSE_energy} +%\subsection{Ground-state BSE energy} +%\label{sec:BSE_energy} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -The key quantity to define in the present context is the total ground-state BSE energy. +The key quantity to define in the present context is the total ground-state BSE energy $\EBSE$. Although this choice is not unique, \cite{Holzer_2018} we propose to define it as \begin{equation} \label{eq:EtotBSE} - \EBSE{m} = \Enuc + \EHF + \EcBSE + \EBSE = \Enuc + \EHF + \EcBSE \end{equation} where $\Enuc$ and $\EHF$ are the nuclear repulsion energy and ground-state HF energy (respectively), and \begin{equation} \label{eq:EcBSE} - \EcBSE = \frac{1}{2} \int_0^1 \Tr(\bK \bP_\IS) d\IS + \EcBSE = \frac{1}{2} \int_0^1 \Tr(\bK \bP{\IS}) d\IS \end{equation} is the ground-state BSE correlation energy computed in the adiabatic connection framework, where \begin{equation} \bK = \begin{pmatrix} - \bA' & \bB \\ - \bB & \bA' \\ + \btA{\IS=1} & \bB{\IS=1} \\ + \bB{\IS=1} & \btA{\IS=1} \\ \end{pmatrix} \end{equation} -is the interaction kernel \cite{Angyan_2011, Holzer_2018} [with $\A{ia,jb}' = (ia|bj)$], +is the interaction kernel \cite{Angyan_2011, Holzer_2018} [with $\tA{ia,jb}{\IS} = \IS \ERI{ia}{bj}$], \begin{equation} -\bP_\lambda = + \bP{\IS} = \begin{pmatrix} - \bY_\IS \T{\bY}_\IS & \bY_\IS \T{\bX}_\IS \\ - \bX_\IS \T{\bY}_\IS & \bX_\IS \T{\bX}_\IS \\ + \bY{\IS} \T{(\bY{\IS})} & \bY{\IS} \T{(\bX{\IS})} \\ + \bX{\IS} \T{(\bY{\IS})} & \bX{\IS} \T{(\bX{\IS})} \\ \end{pmatrix} - \begin{pmatrix} @@ -393,18 +382,21 @@ is the interaction kernel \cite{Angyan_2011, Holzer_2018} [with $\A{ia,jb}' = (i \bO & \bI \\ \end{pmatrix} \end{equation} -is the correlation part of the two-electron density matrix at interaction strength $\IS$, $\Tr$ denotes the matrix trace. -Note that, it is unnecessary to compute the triplet contribution as it is strictly zero. -Note that the present formulation is different from the plasmon formulation. \cite{Schuck_Book, Gell-Mann_1957, Rowe_1968, Sawada_1957b, Li_2020} -Note that, at the dRPA level, the plasmon and adiabatic connection formulations are equivalent. \cite{Sawada_1957b, Fukuta_1964, Furche_2008} -However, this is not the case at the BSE level. +is the correlation part of the two-electron density matrix at interaction strength $\IS$, $\Tr$ denotes the matrix trace and $\T{}$ the matrix transpose. +Note that the present definition of the BSE correlation energy [see Eq.~\eqref{eq:EcBSE}] has be labeled ``XBS'' for ``extended Bethe Salpeter'' by Holzer et al. \cite{Holzer_2018} + +Several important comments are in order here. +For spin-restricted closed-shell molecular systems around their equilibrium geometry (such as the ones studied here), it is rare to encounter singlet instabilities. +However, they may appear in the presence of strong correlation (\eg, when the bond is stretch). +In such a case, this hampers the use of Eq.~\eqref{eq:EcBSE}. +Triplet instabilities are much more common. +However, contrary to the plasmon formulation (an alternative expression of the BSE correlation energy), \cite{Schuck_Book, Gell-Mann_1957, Rowe_1968, Sawada_1957b, Li_2020} the triplet excitations do not contribute in the adiabatic connection formulation, which is an indisputable advantage of the ACFDT appraoch. +Although at the RPA level, the plasmon and adiabatic connection formulations are equivalent, \cite{Sawada_1957b, Fukuta_1964, Furche_2008} this is not the case at the BSE level. -One of the indisputable advantage of the adiabatic connection formulation is that the triplet does not contribute. -Therefore, the triplet instabilities, some of the triplet excitation energies are negative, and must be discarded as the resonant-only part of the BSE excitonic Hamiltonian has to be considered. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -\section{Computational details} -\label{sec:comp_details} +%\section{Computational details} +%\label{sec:comp_details} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% All the preliminary {\GW} calculations performed to obtain the screened Coulomb operator and the quasiparticle energies have been done using a (restricted) Hartree-Fock (HF) starting point, which is a very adequate choice in the case of the (small) systems that we have considered here. Perturbative {\GW} (or {\GOWO}) \cite{Hybertsen_1985a, Hybertsen_1986} calculations are employed as starting point to compute the BSE neutral excitations. @@ -421,8 +413,8 @@ Because Eq.~\eqref{eq:EcBSE} requires the entire BSE excitation spectrum (both s This step is, by far, the computational bottleneck in our current implementation. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -\section{Potential energy surfaces} -\label{sec:PES} +%\section{Potential energy surfaces} +%\label{sec:PES} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%% TABLE I %%% @@ -512,23 +504,23 @@ Additional graphs for other basis sets and within the frozen-core approximation \end{figure*} %%% %%% %%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -\subsection{Hydrogen molecule} -\label{sec:H2} +%\subsection{Hydrogen molecule} +%\label{sec:H2} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -\subsection{Lithium hydride and lithium fluoride} -\label{sec:LiX} +%\subsection{Lithium hydride and lithium fluoride} +%\label{sec:LiX} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%% -\subsection{The isoelectronic sequence: \ce{N2}, \ce{CO}, and \ce{BF}} -\label{sec:isoN2} +%\subsection{The isoelectronic sequence: \ce{N2}, \ce{CO}, and \ce{BF}} +%\label{sec:isoN2} %%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%% -\section{Conclusion} -\label{sec:conclusion} +%\section{Conclusion} +%\label{sec:conclusion} %%%%%%%%%%%%%%%%%%%%%%%% \titou{A nice conclusion here saying that what we have done is pretty awesome.} diff --git a/Data/N2_GS_VDZ.pdf b/Data/N2_GS_VDZ.pdf index d8ffaef1462ff26c3ee5e1cf94714d6db1479d01..7c132eab5526f03ab2a7fca3dcf4ccd06b2e965f 100644 GIT binary patch delta 38690 zcmV)7K*ztOt^=x%1CUaGk{-6tUr|fhz|Pmg z-uG7Ce!lfUDT*}Gmh zyH*~r17no_*4ul3?|r_ml<{_6?{)Cs_j|qWx2wLN9sMe`y!UauZTIu) zQB3zm&iP)Bi+>q>OTFHYcD$8#zF#{gv9x`D^qxJx-;cX)WVJcw^P#i6UkBgQ`uuo* zD+6EPi`tl9e=Ap?ZPk13_iHw+%;sFPIoo6%^?knUt?uK0{pQ@4<9I*HoSkyLpYt!^ zp!N4#oy()>TNn>a;A~Hyk8E$Hm-l+sw>q2H8^*a{h>r1A`+TSzL#&u+E7qjzA8_j=T~Hm7Dz6&SM( 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