diff --git a/Data/DNA_EA.dat b/Data/DNA_EA.dat index b86d1aa..9b74ef5 100644 --- a/Data/DNA_EA.dat +++ b/Data/DNA_EA.dat @@ -1,5 +1,5 @@ A 73-24-5 1.572302 -1.112845488267253E-002 -1.244641965646358E-002 0.818218 -5.905035393513733E-003 -5.850193328953127E-003 0.4735 0.1725 -0.54 C 71-30-7 1.400624 -1.104440375011201E-002 -1.232916433335030E-002 0.613380 -5.855474578578402E-003 -5.807852486273912E-003 0.2638 -0.0257 NA -G 73-40-5 1.884993 -1.139399072772152E-002 -1.290878357845295E-002 1.122363 0.0 0.0 0.7487 0.4149 NA +G 73-40-5 1.884993 -1.139399072772152E-002 -1.290878357845295E-002 1.122363 -6.138508297163354E-003 -6.157876784101071E-003 0.7487 0.4149 NA T 65-71-4 1.194189 -1.125726615899225E-002 -1.278928447622424E-002 0.388406 -5.942694859945368E-003 -5.970028247300262E-003 0.0564 -0.2136 -0.29 U 66-22-8 1.157819 -1.116178581583812E-002 -1.263984272930559E-002 0.350665 -5.928731990069141E-003 -5.951570440565138E-003 0.0124 -0.2832 -0.22 diff --git a/Manuscript/GW-srDFT.tex b/Manuscript/GW-srDFT.tex index 35594b0..c81a7c7 100644 --- a/Manuscript/GW-srDFT.tex +++ b/Manuscript/GW-srDFT.tex @@ -182,7 +182,8 @@ We also compute the ionization potentials of the five canonical nucleobases (ade \section{Introduction} \label{sec:intro} %%%%%%%%%%%%%%%%%%%%%%%% -The purpose of many-body perturbation theory (MBPT) based on Green functions is to solve the formidable many-body problem by adding the electron-electron Coulomb interaction perturbatively starting from an independent-particle model. \cite{MarReiCep-BOOK-16} In this approach, the \textit{screening} of the Coulomb interaction is an essential quantity that is responsible for a rich variety of phenomena that would be otherwise absent (such as quasiparticle satellites and lifetimes). \cite{Aryasetiawan_1998, Onida_2002, Reining_2017} \jt{Is it the screened Coulomb interaction which is responsible for these or more generally the Coulomb interaction? I would say the second.} +The purpose of many-body perturbation theory (MBPT) based on Green functions is to solve the formidable many-body problem by adding the electron-electron Coulomb interaction perturbatively starting from an independent-particle model. \cite{MarReiCep-BOOK-16} In this approach, the \textit{screening} of the Coulomb interaction is an essential quantity that is responsible for a rich variety of phenomena that would be otherwise absent (such as quasiparticle satellites and lifetimes). \cite{Aryasetiawan_1998, Onida_2002, Reining_2017} +%\jt{Is it the screened Coulomb interaction which is responsible for these or more generally the Coulomb interaction? I would say the second.} The so-called {\GW} approximation is the workhorse of MBPT and has a long and successful history in the calculation of the electronic structure of solids. \cite{Aryasetiawan_1998, Onida_2002, Reining_2017} {\GW} is getting increasingly popular in molecular systems \cite{Blase_2011, Faber_2011, Bruneval_2012, Bruneval_2015, Bruneval_2016, Bruneval_2016a, Boulanger_2014, Blase_2016, Li_2017, Hung_2016, Hung_2017, vanSetten_2015, vanSetten_2018, Ou_2016, Ou_2018, Faber_2014} thanks to efficient implementation relying on plane waves \cite{Marini_2009, Deslippe_2012, Maggio_2017} or local basis functions. \cite{Blase_2011, Blase_2018, Bruneval_2016, vanSetten_2013, Kaplan_2015, Kaplan_2016, Krause_2017, Caruso_2012, Caruso_2013, Caruso_2013a, Caruso_2013b} @@ -336,9 +337,8 @@ For example, if the reference is Hartree-Fock ($\HF$), $\Sig{\text{ref}}{\Bas}(\ In this subsection, we provide the minimal set of equations required to describe {\GOWO}. More details can be found, for example, in Refs.~\citenum{vanSetten_2013, Kaplan_2016, Bruneval_2016}. -For sake of generality, we consider a $\KS$ reference. +For sake of generality, we consider closed-shell systems with a $\KS$ reference. The one-electron energies $\e{p}$ and their corresponding (real-valued) orbitals $\MO{p}(\br{})$ (which defines the basis set $\Bas$) are then $\KS$ energies and orbitals. -\jt{equations are for closed-shell systems} Within the {\GW} approximation, the correlation part of the self-energy reads \begin{equation} @@ -347,15 +347,15 @@ Within the {\GW} approximation, the correlation part of the self-energy reads \Sig{\text{c},p}{\Bas}(\omega) & = \mel*{\MO{p}}{\Sig{\text{c}}{\Bas}(\omega)}{\MO{p}} \\ - & = 2 \sum_{i}^\text{occ} \sum_{x} \frac{[pi|x]^2}{\omega - \e{i} + \Om{x} - i \eta} + & = 2 \sum_{i}^\text{occ} \sum_{m} \frac{[pi|m]^2}{\omega - \e{i} + \Om{m} - i \eta} \\ - & + 2 \sum_{a}^\text{virt} \sum_{x} \frac{[pa|x]^2}{\omega - \e{a} - \Om{x} + i \eta}, + & + 2 \sum_{a}^\text{virt} \sum_{m} \frac{[pa|m]^2}{\omega - \e{a} - \Om{m} + i \eta}, \end{split} \end{equation} -\jt{x should be defined. If it is just an index for the eigenvector/eigenvalue, maybe we should call it $m$ instead?} where $\eta$ is a positive infinitesimal. +where $m$ corresponds to a sum over the single excitations and $\eta$ is a positive infinitesimal. The screened two-electron integrals \begin{equation} - [pq|x] = \sum_{ia} (pq|ia) (\bX+\bY)_{ia}^{x} + [pq|m] = \sum_{ia} (pq|ia) (\bX+\bY)_{ia}^{m} \end{equation} are obtained via the contraction of the bare two-electron integrals \cite{Gill_1994} \begin{equation} @@ -382,11 +382,10 @@ and the transition densities $(\bX+\bY)_{ia}^{x}$ originating from a (direct) ra with \begin{align} \label{eq:RPA} - A_{ia,jb} & = \delta_{ij} \delta_{ab} (\e{a} - \e{i}) + 2 (ia|jb), + A_{ia,jb} & = \delta_{ij} \delta_{ab} (\e{a} - \e{i}) + 2 (ib|aj), & - B_{ia,jb} & = 2 (ia|bj), + B_{ia,jb} & = 2 (ia|jb), \end{align} -\jt{the integral in the B term could be written as in the A term since we use real-valued orbitals} and $\delta_{pq}$ is the Kronecker delta. \cite{NISTbook} Equation \eqref{eq:LR} also provides the RPA neutral excitation energies $\Om{x}$ which represent the poles of the screened Coulomb potential $\W{}{}$. @@ -395,7 +394,6 @@ The {\GOWO} quasiparticle energies $\eGOWO{p}$ are provided by the solution of t \label{eq:QP-G0W0} \omega = \e{p} - \Pot{\xc,p}{\Bas} + \Sig{\text{x},p}{\Bas} + \Re[\Sig{\text{c},p}{\Bas}(\omega)]. \end{equation} -\jt{an equation involving $Z_p$ seems to be missing!} with the largest renormalization weight (or factor) \begin{equation} \label{eq:Z} @@ -586,7 +584,7 @@ In a nutshell, the present basis set correction provides cc-pVQZ quality results Besides, it allows to reach chemical accuracy with the quadruple-$\zeta$ basis set, an accuracy that could not be reached even with the cc-pV5Z basis set for the conventional calculations. Very similar conclusions are drawn at the {\GOWO}@{\PBEO} level (see Table \ref{tab:GW20_PBE0}) with a slightly faster convergence to the CBS limit. -For example, at the {\GOWO}@PBE0+srLDA/cc-pVQZ level, the MAD is only $0.02$ eV with a maximum error as small as $0.09$ eV. \jt{these numbers do not exactly correspond to the ones in the Table II. Is it because of round off?} +For example, at the {\GOWO}@PBE0+srLDA/cc-pVQZ level, the MAD is only $0.02$ eV with a maximum error as small as $0.09$ eV. It is worth pointing out that, for ground-state properties such as atomization and correlation energies, the density-based correction brought a more significant basis set reduction. For example, we evidenced in Ref.~\onlinecite{LooPraSceTouGin-JPCL-19} that quintuple-$\zeta$ quality atomization and correlation energies are recovered with triple-$\zeta$ basis sets. diff --git a/Notebooks/GW100.nb b/Notebooks/GW100.nb index 226f480..203090f 100644 --- a/Notebooks/GW100.nb +++ b/Notebooks/GW100.nb @@ -10,10 +10,10 @@ NotebookFileLineBreakTest NotebookFileLineBreakTest NotebookDataPosition[ 158, 7] -NotebookDataLength[ 13702938, 242784] -NotebookOptionsPosition[ 13686389, 242463] -NotebookOutlinePosition[ 13686881, 242483] -CellTagsIndexPosition[ 13686838, 242480] +NotebookDataLength[ 13517839, 238315] +NotebookOptionsPosition[ 13500665, 237982] +NotebookOutlinePosition[ 13501141, 238001] +CellTagsIndexPosition[ 13501098, 237998] WindowFrame->Normal*) (* Beginning of Notebook Content *) @@ -31,7 +31,7 @@ Cell[BoxData[ InitializationCell->True, CellChangeTimes->{{3.7208031947801647`*^9, 3.7208032000677156`*^9}, { 3.7208034541742477`*^9, 3.720803455246439*^9}}, - 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