revision theory
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@ -56,6 +56,8 @@
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\newcommand{\cD}{\mathcal{D}}
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\newcommand{\Ne}{N}
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\newcommand{\Nbas}{N_\text{bas}}
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\newcommand{\Nocc}{N_\text{occ}}
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\newcommand{\Nvirt}{N_\text{virt}}
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% energies
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@ -365,9 +367,9 @@ Within the {\GW} approximation, the correlation part of the self-energy reads
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\Sig{\text{c},p}{\Bas}(\omega)
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& = \mel*{\MO{p}}{\Sig{\text{c}}{\Bas}(\omega)}{\MO{p}}
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\\
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& = 2 \sum_{i}^\text{occ} \sum_{m} \frac{[pi|m]^2}{\omega - \e{i} + \Om{m} - i \eta}
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& = 2 \sum_{i}^{\Nocc} \sum_{m} \frac{[pi|m]^2}{\omega - \e{i} + \Om{m} - i \eta}
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\\
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& + 2 \sum_{a}^\text{virt} \sum_{m} \frac{[pa|m]^2}{\omega - \e{a} - \Om{m} + i \eta},
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& + 2 \sum_{a}^{\Nvirt} \sum_{m} \frac{[pa|m]^2}{\omega - \e{a} - \Om{m} + i \eta},
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\end{split}
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\end{equation}
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where $m$ labels excited states and $\eta$ is a positive infinitesimal.
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@ -431,20 +433,36 @@ In particular, the ionization potential (IP) and electron affinity (EA) are extr
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where $\eGOWO{\HOMO}$ and $\eGOWO{\LUMO}$ are the HOMO and LUMO orbital energies, respectively.
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%%%%%%%%%%%%%%%%%%%%%%%%
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\subsection{Short-range correlation functionals}
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\label{sec:srDFT}
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\subsection{Basis set correction}
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\label{sec:BSC}
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%%%%%%%%%%%%%%%%%%%%%%%%
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The fundamental idea behind the present basis set correction is to recognize that the two-electron interaction $\abs*{\br{} - \br{}'}^{-1}$ projected in a finite basis $\Bas$ is a finite, non-divergent quantity at $\abs*{\br{} - \br{}'} = 0$.
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We can therefore define an effective two-electron interaction which equals the Coulomb operator in an incomplete basis, \ie,
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\titou{The fundamental idea behind the present basis set correction is to recognize that the singular two-electron Coulomb interaction $\abs*{\br{} - \br{}'}^{-1}$ projected in a finite basis $\Bas$ is a finite, non-divergent quantity at $\abs*{\br{} - \br{}'} = 0$, which ``resembles'' the long-range interaction operator $\abs*{\br{} - \br{}'}^{-1} \erf(\rsmu{}{} \abs*{\br{} - \br{}'})$ used within RS-DFT. \cite{Giner_2018}
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We can therefore define an effective two-electron operator which equals its Coulomb counterpart in an incomplete basis, \ie,
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\begin{equation}
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\iint \frac{\n{2}{\Bas}(\br{},\br{}')}{\abs*{\br{} - \br{}'}} d\br{} d\br{}'
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=
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\iint \n{2}{\Bas}(\br{},\br{}') W(\br{},\br{}') d\br{} d\br{}'.
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\iint \n{2}{\Bas}(\br{},\br{}') W^{\Bas}(\br{},\br{}') d\br{} d\br{}'.
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\end{equation}
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A convenient choice is, for example,
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Because the value of $W^{\Bas}(\br{},\br{}')$ at coalescence, $W^{\Bas}(\br{},\br{})$, is necessarily finite in a finite basis $\Bas$, one can map this operator to a non-divergent, long-range interaction of the form
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\begin{equation}
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\label{eq:Wapprox}
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W^{\Bas}(\br{},\br{}')
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\approx \frac{1}{2} \qty{
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\frac{\erf[\rsmu{}{\Bas}(\br{}) \abs*{\br{} - \br{}'}]}{\abs*{\br{} - \br{}'}}
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+ \frac{\erf[\rsmu{}{\Bas}(\br{}') \abs*{\br{} - \br{}'}]}{\abs*{\br{} - \br{}'}}
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}.
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\end{equation}
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The information about the finiteness of the basis set has been transferred to the range-separation \textit{function} $\rsmu{}{\Bas}(\br{})$, and its value can be set by ensuring that the two sides of Eq.~\eqref{eq:Wapprox} are strictly equal at $\abs*{\br{} - \br{}'} = 0$.
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Knowing that $\lim_{r \to 0} \erf(\mu r)/r = 2\mu/\sqrt{\pi}$, this yields
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\begin{equation}
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\rsmu{}{\Bas}(\br{}) = \frac{\sqrt{\pi}}{2} W^{\Bas}(\br{},\br{}).
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\end{equation}
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}
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\titou{A convenient choice for the on-top value of the two-electron effective operator is, for example,
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\begin{equation}
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\label{eq:W}
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W(\br{},\br{}') =
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W^{\Bas}(\br{},\br{}) =
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\begin{cases}
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f^{\Bas}(\br{})/\n{2}{\Bas}(\br{}), & \n{2}{\Bas}(\br{}) \neq 0, \\
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\infty, & \text{otherwise}, \\
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@ -452,23 +470,23 @@ A convenient choice is, for example,
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\end{equation}
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where, in the case of a single-determinant method (such as HF and KS-DFT),
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\begin{equation}
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f(\br{}) = \sum_{pq}^{\Nbas} \sum_{ij}^\text{occ} \MO{p}(\br{}) \MO{q}(\br{}) \braket{pq}{ij} \MO{i}(\br{}) \MO{j}(\br{})
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f^{\Bas}(\br{}) = \sum_{pq}^{\Nbas} \sum_{ij}^{\Nocc} \MO{p}(\br{}) \MO{q}(\br{}) (pi|qj) \MO{i}(\br{}) \MO{j}(\br{}),
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\end{equation}
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and
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\begin{equation}
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\n{2}{\Bas}(\br{}) = \frac{[\n{}{\Bas}(\br{})]^2}{4}
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\n{2}{\Bas}(\br{}) = \frac{1}{4} [\n{}{\Bas}(\br{})]^2
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\end{equation}
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is the opposite-spin on-top pair density.
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The effective operator $W(\br{},\br{}')$ has some interesting properties.
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is the opposite-spin on-top pair density in a closed-shell system.
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}
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\titou{The effective operator $W^{\Bas}(\br{},\br{}')$ has some interesting properties.
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For example, we have
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\begin{equation}
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\lim_{\Bas \to \CBS} W(\br{},\br{}') = \abs*{\br{} - \br{}'}^{-1}
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\lim_{\Bas \to \CBS} W^{\Bas}(\br{},\br{}') = \abs*{\br{} - \br{}'}^{-1},
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\end{equation}
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which means that, in the limit of a complete basis, one recovers the genuine (divergent) Coulomb operator.
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Consequently, the magnitude of the correction tends to zero (see Eq.~\eqref{eq:limSig}).
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Note also that the divergence condition of $W(\br{},\br{}')$ in Eq.~\eqref{eq:W} evidences that one-electron systems are free of correction.
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Because the value of $W(\br{},\br{}')$ at coalescence, $W(\br{},\br{})$, is necessarily finite in a finite basis $\Bas$, one can map its value to a non-divergent, long-range interaction of the form $\abs*{\br{} - \br{}'}^{-1} \erf(\rsmu{}{} \abs*{\br{} - \br{}'})$.
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which means that, in the limit of a complete basis, one recovers the genuine (divergent) Coulomb operator, and that the magnitude of the energetic correction tends to zero [see Eq.~\eqref{eq:limSig}].
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Note also that the divergence condition of $W^{\Bas}(\br{},\br{}')$ in Eq.~\eqref{eq:W} ensures that one-electron systems are free of correction.
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}
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%%%%%%%%%%%%%%%%%%%%%%%%
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\subsection{Short-range correlation functionals}
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@ -478,7 +496,8 @@ Because the value of $W(\br{},\br{}')$ at coalescence, $W(\br{},\br{})$, is nece
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The frequency-independent local self-energy $\bSig{}{\Bas}[\n{}{}](\br{},\br{}') = \bpot{}{\Bas}[\n{}{}](\br{}) \delta(\br{}-\br{}')$ originates from the functional derivative of complementary basis-correction density functionals $\bpot{}{\Bas}[\n{}{}](\br{}) = \delta \bE{}{\Bas}[\n{}{}] / \delta \n{}{}(\br{})$.
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Here, we employ two types of complementary, short-range correlation functionals $\bE{}{\Bas}[\n{}{}]$: a short-range local-density approximation ($\srLDA$) functional with multideterminant reference \cite{Toulouse_2005, Paziani_2006} and a short-range Perdew-Burke-Ernzerhof ($\srPBE$) correlation functional \cite{Ferte_2019, Loos_2019} which interpolates between the usual PBE functional \cite{Perdew_1996} at $\mu = 0$ and the exact large-$\mu$ behavior \cite{Toulouse_2004, Gori-Giorgi_2006, Paziani_2006} using the on-top pair density from the uniform-electron gas. \cite{Loos_2019}
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Additionally to the one-electron density calculated from the HF or KS orbitals, these RS-DFT functionals require a range-separation function $\rsmu{}{\Bas}(\br{})$ which automatically adapts to the spatial inhomogeneity of the basis-set incompleteness error and is computed using the HF or KS opposite-spin pair-density matrix in the basis set $\Bas$.
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We refer the interested reader to Refs.~\onlinecite{Giner_2018, Loos_2019, Giner_2019} where our procedure is thoroughly detailed and the explicit expressions of these two short-range correlation functionals are provided.
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We refer the interested reader to Refs.~\onlinecite{Giner_2018, Loos_2019, Giner_2019} where our procedure is thoroughly detailed.
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\titou{The explicit expressions of these two short-range correlation functionals, as well as their functional derivative, are provided in the {\SI}.}
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The basis set corrected {\GOWO} quasiparticle energies are thus given by
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@ -739,8 +758,8 @@ We hope to report on this in the near future.
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%%%%%%%%%%%%%%%%%%%%%%%%
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\section*{Supporting Information}
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%%%%%%%%%%%%%%%%%%%%%%%%
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See {\SI} for additional graphs reporting the convergence of the ionization potentials of the GW20 subset with respect to the size of the basis set.
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\titou{The numerical data of Tables \ref{tab:GW20_HF} and \ref{tab:GW20_PBE0} are provided in txt and json format.}
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See {\SI} for \titou{the explicit expression of the short-range correlation functionals (and their functional derivatives)}, additional graphs reporting the convergence of the ionization potentials of the GW20 subset with respect to the size of the basis set, \titou{and
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the numerical data of Tables \ref{tab:GW20_HF} and \ref{tab:GW20_PBE0} (provided in txt and json formats).}
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%%%%%%%%%%%%%%%%%%%%%%%%
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\begin{acknowledgements}
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205
JCTC_revision/SI/GW-srDFT-SI.tex
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205
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@ -0,0 +1,205 @@
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\documentclass[aip,jcp,reprint,noshowkeys]{revtex4-1}
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\usepackage{graphicx,dcolumn,bm,xcolor,microtype,multirow,amscd,amsmath,amssymb,amsfonts,physics,mhchem,longtable,wrapfig}
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\usepackage{natbib}
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\usepackage[extra]{tipa}
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\bibliographystyle{achemso}
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\AtBeginDocument{\nocite{achemso-control}}
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\usepackage{mathpazo,libertine}
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\usepackage{hyperref}
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\hypersetup{
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colorlinks=true,
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linkcolor=blue,
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filecolor=blue,
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urlcolor=blue,
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citecolor=blue
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}
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\urlstyle{same}
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\newcommand{\alert}[1]{\textcolor{red}{#1}}
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\definecolor{darkgreen}{HTML}{009900}
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\usepackage[normalem]{ulem}
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\newcommand{\titou}[1]{\textcolor{red}{#1}}
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\newcommand{\jt}[1]{\textcolor{purple}{#1}}
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\newcommand{\manu}[1]{\textcolor{darkgreen}{#1}}
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\newcommand{\toto}[1]{\textcolor{brown}{#1}}
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\newcommand{\trashPFL}[1]{\textcolor{red}{\sout{#1}}}
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\newcommand{\trashJT}[1]{\textcolor{purple}{\sout{#1}}}
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\newcommand{\trashMG}[1]{\textcolor{darkgreen}{\sout{#1}}}
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\newcommand{\trashAS}[1]{\textcolor{brown}{\sout{#1}}}
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\newcommand{\MG}[1]{\manu{(\underline{\bf MG}: #1)}}
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\newcommand{\JT}[1]{\juju{(\underline{\bf JT}: #1)}}
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\newcommand{\PFL}[1]{\titou{(\underline{\bf PFL}: #1)}}
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\newcommand{\AS}[1]{\toto{(\underline{\bf TOTO}: #1)}}
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\usepackage{hyperref}
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\hypersetup{
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colorlinks=true,
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linkcolor=blue,
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filecolor=blue,
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urlcolor=blue,
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citecolor=blue
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}
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\newcommand{\mc}{\multicolumn}
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\newcommand{\fnm}{\footnotemark}
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\newcommand{\fnt}{\footnotetext}
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\newcommand{\tabc}[1]{\multicolumn{1}{c}{#1}}
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\newcommand{\SI}{\textcolor{blue}{supporting information}}
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\newcommand{\QP}{\textsc{quantum package}}
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% methods
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\newcommand{\evGW}{ev$GW$}
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\newcommand{\qsGW}{qs$GW$}
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\newcommand{\GOWO}{$G_0W_0$}
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\newcommand{\GW}{$GW$}
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\newcommand{\GnWn}[1]{$G_{#1}W_{#1}$}
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% operators
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\newcommand{\hH}{\Hat{H}}
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% energies
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\newcommand{\Ec}{E_\text{c}}
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\newcommand{\EHF}{E_\text{HF}}
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\newcommand{\EKS}{E_\text{KS}}
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\newcommand{\EcK}{E_\text{c}^\text{Klein}}
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\newcommand{\EcRPA}{E_\text{c}^\text{RPA}}
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\newcommand{\EcGM}{E_\text{c}^\text{GM}}
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\newcommand{\EcMP}{E_c^\text{MP2}}
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\newcommand{\Egap}{E_\text{gap}}
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\newcommand{\IP}{\text{IP}}
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\newcommand{\EA}{\text{EA}}
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\newcommand{\RH}{R_{\ce{H2}}}
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\newcommand{\RF}{R_{\ce{F2}}}
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\newcommand{\RBeO}{R_{\ce{BeO}}}
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% orbital energies
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\newcommand{\nDIIS}{N^\text{DIIS}}
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\newcommand{\maxDIIS}{N_\text{max}^\text{DIIS}}
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\newcommand{\nSat}[1]{N_{#1}^\text{sat}}
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\newcommand{\eSat}[2]{\epsilon_{#1,#2}}
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\newcommand{\e}[1]{\epsilon_{#1}}
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\newcommand{\eHF}[1]{\epsilon^\text{HF}_{#1}}
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\newcommand{\teHF}[1]{\Tilde{\epsilon}^\text{HF}_{#1}}
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\newcommand{\eKS}[1]{\epsilon^\text{KS}_{#1}}
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\newcommand{\eQP}[1]{\epsilon^\text{QP}_{#1}}
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\newcommand{\eGOWO}[1]{\epsilon^\text{\GOWO}_{#1}}
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\newcommand{\eGW}[1]{\epsilon^\text{\GW}_{#1}}
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\newcommand{\eGnWn}[2]{\epsilon^\text{\GnWn{#2}}_{#1}}
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\newcommand{\de}[1]{\Delta\epsilon_{#1}}
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\newcommand{\deHF}[1]{\Delta\epsilon^\text{HF}_{#1}}
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\newcommand{\Om}[1]{\Omega_{#1}}
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\newcommand{\eHOMO}{\epsilon_\text{HOMO}}
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\newcommand{\eLUMO}{\epsilon_\text{LUMO}}
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\newcommand{\HOMO}{\text{HOMO}}
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\newcommand{\LUMO}{\text{LUMO}}
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% Matrix elements
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\newcommand{\A}[1]{A_{#1}}
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\newcommand{\B}[1]{B_{#1}}
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\newcommand{\tA}{\Tilde{A}}
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\newcommand{\tB}{\Tilde{B}}
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\renewcommand{\S}[1]{S_{#1}}
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\newcommand{\G}[1]{G_{#1}}
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\newcommand{\Po}[1]{P_{#1}}
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\newcommand{\W}[1]{W_{#1}}
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\newcommand{\Wc}[1]{W^\text{c}_{#1}}
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\newcommand{\vc}[1]{v_{#1}}
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\newcommand{\SigX}[1]{\Sigma^\text{x}_{#1}}
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\newcommand{\SigC}[1]{\Sigma^\text{c}_{#1}}
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\newcommand{\tSigC}[1]{\Tilde{\Sigma}^\text{c}_{#1}}
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\newcommand{\SigCp}[1]{\Sigma^\text{p}_{#1}}
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\newcommand{\SigCh}[1]{\Sigma^\text{h}_{#1}}
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\newcommand{\SigGW}[1]{\Sigma^\text{\GW}_{#1}}
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\newcommand{\Z}[1]{Z_{#1}}
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% Matrices
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\newcommand{\bG}{\boldsymbol{G}}
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\newcommand{\bW}{\boldsymbol{W}}
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\newcommand{\bvc}{\boldsymbol{v}}
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\newcommand{\bSig}{\boldsymbol{\Sigma}}
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\newcommand{\bSigX}{\boldsymbol{\Sigma}^\text{x}}
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\newcommand{\bSigC}{\boldsymbol{\Sigma}^\text{c}}
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\newcommand{\bSigGW}{\boldsymbol{\Sigma}^\text{\GW}}
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\newcommand{\be}{\boldsymbol{\epsilon}}
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\newcommand{\bDelta}{\boldsymbol{\Delta}}
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\newcommand{\beHF}{\boldsymbol{\epsilon}^\text{HF}}
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\newcommand{\beGW}{\boldsymbol{\epsilon}^\text{\GW}}
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\newcommand{\beGnWn}[1]{\boldsymbol{\epsilon}^\text{\GnWn{#1}}}
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\newcommand{\bdeGnWn}[1]{\Delta\boldsymbol{\epsilon}^\text{\GnWn{#1}}}
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\newcommand{\bde}{\boldsymbol{\Delta\epsilon}}
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\newcommand{\bdeHF}{\boldsymbol{\Delta\epsilon}^\text{HF}}
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\newcommand{\bdeGW}{\boldsymbol{\Delta\epsilon}^\text{GW}}
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\newcommand{\bOm}{\boldsymbol{\Omega}}
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\newcommand{\bA}{\boldsymbol{A}}
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\newcommand{\bB}{\boldsymbol{B}}
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\newcommand{\bX}{\boldsymbol{X}}
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\newcommand{\bY}{\boldsymbol{Y}}
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\newcommand{\bZ}{\boldsymbol{Z}}
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\newcommand{\fc}{f_\text{c}}
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\newcommand{\Vc}{V_\text{c}}
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\newcommand{\MO}[1]{\phi_{#1}}
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% coordinates
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\newcommand{\br}[1]{\mathbf{r}_{#1}}
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\renewcommand{\b}[1]{\mathbf{#1}}
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\renewcommand{\d}{\text{d}}
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\newcommand{\dbr}[1]{d\br{#1}}
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\renewcommand{\bra}[1]{\ensuremath{\langle #1 \vert}}
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\renewcommand{\ket}[1]{\ensuremath{\vert #1 \rangle}}
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\renewcommand{\braket}[2]{\ensuremath{\langle #1 \vert #2 \rangle}}
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\newcommand{\ISCD}{Institut des Sciences du Calcul et des Donn\'ees, Sorbonne Universit\'e, Paris, France}
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\newcommand{\LCPQ}{Laboratoire de Chimie et Physique Quantiques (UMR 5626), Universit\'e de Toulouse, CNRS, UPS, France}
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\newcommand{\LCT}{Laboratoire de Chimie Th\'eorique (UMR 7616), Sorbonne Universit\'e, CNRS, Paris, France}
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\newcommand{\IUF}{Institut Universitaire de France, Paris, France}
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\begin{document}
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\title{Supplementary Materials for ``A Density-Based Basis-Set Incompleteness Correction for GW Methods''}
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\author{Pierre-Fran\c{c}ois Loos}
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\email[Corresponding author: ]{loos@irsamc.ups-tlse.fr}
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\affiliation{\LCPQ}
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\author{Barth\'el\'emy Pradines}
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\affiliation{\LCT}
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\affiliation{\ISCD}
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\author{Anthony Scemama}
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\affiliation{\LCPQ}
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\author{Emmanuel Giner}
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\affiliation{\LCT}
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\author{Julien Toulouse}
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\email[Corresponding author: ]{toulouse@lct.jussieu.fr}
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\affiliation{\LCT}
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\affiliation{\IUF}
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\begin{abstract}
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\end{abstract}
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\maketitle
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\begin{figure*}
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\includegraphics[width=\linewidth]{IP_G0W0HF}
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\caption{
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IPs (in eV) computed at the {\GOWO}@HF (black circles), {\GOWO}@HF+srLDA (red squares), and {\GOWO}@HF+srPBE (blue diamonds) levels of theory with increasingly large Dunning's basis sets (cc-pVDZ, cc-pVTZ, cc-pVQZ, and cc-pV5Z) for the 20 smallest molecules of the GW100 set.
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The thick black line represents the CBS value obtained by extrapolation with the three largest basis sets.
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\label{fig:IP_G0W0HF}
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}
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\end{figure*}
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\begin{figure*}
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\includegraphics[width=\linewidth]{IP_G0W0PBE0}
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\caption{
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IPs (in eV) computed at the {\GOWO}@PBE0 (black circles), {\GOWO}@PBE0+srLDA (red squares), and {\GOWO}@PBE0+srPBE (blue diamonds) levels of theory with increasingly large Dunning's basis sets (cc-pVDZ, cc-pVTZ, cc-pVQZ, and cc-pV5Z) for the 20 smallest molecules of the GW100 set.
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The thick black line represents the CBS value obtained by extrapolation with the three largest basis sets.
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\label{fig:IP_G0W0HF}
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}
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\end{figure*}
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\bibliography{../GW-srDFT,../GW-srDFT-control}
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\end{document}
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Reference in New Issue
Block a user