2019-12-06 14:06:25 +01:00
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\begin { document}
\title { Supplementary Materials for ``A Density-Based Basis-Set Incompleteness Correction for GW Methods''}
\author { Pierre-Fran\c { c} ois Loos}
\email [Corresponding author: ] { loos@irsamc.ups-tlse.fr}
\affiliation { \LCPQ }
\author { Barth\' el\' emy Pradines}
\affiliation { \LCT }
\affiliation { \ISCD }
\author { Anthony Scemama}
\affiliation { \LCPQ }
\author { Emmanuel Giner}
\affiliation { \LCT }
\author { Julien Toulouse}
\email [Corresponding author: ] { toulouse@lct.jussieu.fr}
\affiliation { \LCT }
\affiliation { \IUF }
\begin { abstract}
\end { abstract}
\maketitle
2019-12-13 22:38:13 +01:00
%Macros:
\newcommand { \basis } [0]{ \mathcal { B} }
\newcommand { \efuncbasispbe } [0]{ \bar { E} _ { \text { srPBE} } ^ { \basis } [n]}
\newcommand { \epspbeueg } [0]{ \bar { \varepsilon } ^ { \text { sr} ,\text { PBE} } _ { \text { c,md} } }
\newcommand { \epspbe } [0]{ \varepsilon ^ { \text { PBE} } _ { \text { c} } }
\newcommand { \potpbeueg } [0]{ \bar { v} _ { \text { srPBE} } ^ { \basis } }
\newcommand { \potpbe } [0]{ v^ { \text { PBE} } _ { \text { c} } }
\section { PBE-based complementary potential $ \potpbeueg $ }
Here, we provide the explicit expression of the PBE-based complementary potential in the case of closed-shell systems such as the ones studied in the present paper.
The PBE-based correlation energy functional with multideterminant reference (ECMD) has been previously reported in Ref.~\onlinecite { Loos_ 2019} and is defined by the following equation:
\begin { equation}
\label { eq:def_ pbe}
\efuncbasispbe = \int n({ \bf r} )\epspbeueg (n({ \bf r} ),s({ \bf r} ),\mu ^ { \basis } (\br { } )) d\br { } ,
\end { equation}
with,
\begin { equation}
\label { eq:def_ epsipbeueg}
\epspbeueg (n,s,\mu ) = \frac { \epspbe (n,s)} { 1+\beta (n,s)\mu ^ 3} ,
\end { equation}
where $ \epspbe ( n,s ) $ is the usual PBE correlation functional \cite { Perdew_ 1996} , $ s = \nabla n / n ^ { 4 / 3 } $ is the reduced density gradient,
\begin { equation}
\beta (n,s) = \frac { 3} { 2\sqrt { \pi } (1-\sqrt { 2} )} \frac { \epspbe (n,s)} { n_ 2^ { \text { UEG} } (n)/n} ,
\end { equation}
and
\begin { equation}
\label { eq:uegotop}
n_ 2^ { \text { UEG} } (n)=n^ 2g_ 0(r_ s)
\end { equation}
is the on-top pair density of the uniform electron gas (UEG). In Eq.~\eqref { eq:uegotop} , $ r _ s = ( 4 \pi n / 3 ) ^ { - 1 / 3 } $ the Wigner-Seitz radius and $ g _ 0 ( r _ s ) $ is the UEG on-top pair-distribution function. The parametrization of $ g _ 0 ( r _ s ) $ is given in Eq.~(46) of Ref.~\onlinecite { Gori-Giorgi_ 2006} .
The potential of this GGA ECMD complementary functional has the following form:
\begin { equation}
\begin { split}
\potpbeueg [n]
& = \fdv { \efuncbasispbe } { n}
\\
% & = \frac{\partial n \epspbeueg }{\partial n}- \nabla . \frac{\partial n \epspbeueg }{\partial \nabla n}\\
& =\epspbeueg + n \pdv { \epspbeueg } { n} - \nabla \cdot \qty ( n \pdv { \epspbeueg } { \nabla n} ).
\end { split}
\end { equation}
Hence, we have to compute two main contributions: the scalar part $ \pdv { \epspbeueg } { n } $ and the gradient part $ \pdv { \epspbeueg } { \nabla n } $ .
\subsection { Scalar contribution}
For the scalar contribution, we simply differenciate Eq.~\eqref { eq:def_ epsipbeueg} with respect to the density:
\begin { equation}
\pdv { \epspbeueg } { n}
= \frac { \potpbe } { 1+\beta \mu ^ 3}
- \frac { \epspbe \mu ^ 3} { (1+\beta \mu ^ 3)^ 2} \pdv { \beta } { n} ,
\end { equation}
where $ \potpbe = \pdv { \epspbe } { n } $ and
\begin { equation}
\pdv { \beta } { n}
= \frac { 3} { 2\sqrt { \pi } (1-\sqrt { 2} )}
\Bigg [ \frac { \potpbe } { n_ 2^ { \text { UEG} } /n}
- \frac { \epspbe } { (n_ 2^ { \text { UEG} } /n)^ 2} \frac { \partial (n_ 2^ { \text { UEG} } /n)} { \partial n} \Bigg ].
\end { equation}
The only remaining missing part is the derivative of $ n _ 2 ^ { \text { UEG } } / n $ with respect to the density:
\begin { equation}
\pdv { (n_ 2^ { \text { UEG} } /n)} { n} = \pdv { [n g_ 0(r_ s)]} { n} = g_ 0(r_ s)+ n \pdv { g_ 0(r_ s)} { n} .
\end { equation}
with
\begin { equation}
\pdv { g_ 0(r_ s)} { n} = \pdv { r_ s} { n} \pdv { g_ 0(r_ s)} { r_ s} = -(6 n^ { 2} \sqrt { \pi } )^ { -2/3} \pdv { g_ 0(r_ s)} { r_ s} .
\end { equation}
The derivative with respect to $ r _ s $ can be expressed as
\begin { equation}
\begin { aligned}
\pdv { g_ 0(r_ s)} { r_ s}
& = \frac { e^ { -F\, r_ s} } { 2} \big [ (-B + 2 C r_ s + 3 D r_ s^ 2 + 4 E r_ s^ 3)
\\
& - F (1 - B r_ s + C r_ s^ 2 + D r_ s^ 3 + E r_ s^ 4) \big ],
\end { aligned}
\end { equation}
with
\begin { align}
C & = 0.0819306, \\
F & = 0.752411, \\
D & = -0.0127713,\\
E & =0.00185898,\\
B & = 0.7317 - F.
\end { align}
\subsection { Gradient contribution}
For the gradient part, we also used the chain rule:
\begin { equation}
\pdv { \epspbeueg } { \nabla n} = \pdv { \epspbeueg } { \epspbe } \pdv { \epspbe } { \nabla n} .
\end { equation}
The term $ \pdv { \epspbe } { \nabla n } $ is already known (\textbf { ref??} ), and the partial derivative of $ \epspbeueg $ with respect to $ \epspbe $ is
\begin { equation}
\pdv { \epspbeueg } { \epspbe }
= \frac { 1} { 1+\beta \mu ^ 3}
- \frac { \epspbe \mu ^ 3} { (1+\beta \mu ^ 3)^ 2} \pdv { \beta } { \epspbe } ,
\end { equation}
where
\begin { equation}
\pdv { \beta } { \epspbe } = \frac { 3} { 2\sqrt { \pi } (1-\sqrt { 2} )} \frac { 1} { n_ 2^ { \text { UEG} } /n} .
\end { equation}
2019-12-06 14:06:25 +01:00
\begin { figure*}
\includegraphics [width=\linewidth] { IP_ G0W0HF}
\caption {
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.
The thick black line represents the CBS value obtained by extrapolation with the three largest basis sets.
\label { fig:IP_ G0W0HF}
}
\end { figure*}
\begin { figure*}
\includegraphics [width=\linewidth] { IP_ G0W0PBE0}
\caption {
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.
The thick black line represents the CBS value obtained by extrapolation with the three largest basis sets.
\label { fig:IP_ G0W0HF}
}
\end { figure*}
\bibliography { ../GW-srDFT,../GW-srDFT-control}
\end { document}
