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Manuscript/GW-srDFT.bib
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@ -1,13 +1,46 @@
%% This BibTeX bibliography file was created using BibDesk.
%% http://bibdesk.sourceforge.net/
%% Created for Pierre-Francois Loos at 2019-10-12 14:32:34 +0200
%% Created for Pierre-Francois Loos at 2019-10-14 10:04:33 +0200
%% Saved with string encoding Unicode (UTF-8)
@article{Jacquemin_2016,
Author = {D. Jacquemin and I. Duchemin and X. Blase},
Date-Added = {2019-10-14 10:02:38 +0200},
Date-Modified = {2019-10-14 10:03:34 +0200},
Doi = {10.1080/00268976.2015.1119901},
Journal = {Mol. Phys.},
Pages = {957},
Title = {Assessment Of The Convergence Of Partially Self- Consistent Bse/Gw Calculations},
Volume = {114},
Year = {2016}}
@article{Marini_2009,
Author = {Andrea Marini and Conor Hogan and Myrta Gruning and Daniele Varsano },
Date-Added = {2019-10-14 09:56:13 +0200},
Date-Modified = {2019-10-14 10:04:07 +0200},
Doi = {10.1016/j.cpc.2009.02.003},
Journal = {Comp. Phys. Comm.},
Pages = {1392},
Title = {Yambo: An Ab Initio Tool For Excited State Calculations},
Volume = {180},
Year = {2009}}
@article{Deslippe_2012,
Author = {Jack Deslippe and Georgy Samsonidze and David A. Strubbe and Manish Jain and Marvin L. Cohen and Steven G. Louie},
Date-Added = {2019-10-14 09:53:28 +0200},
Date-Modified = {2019-10-14 10:04:32 +0200},
Doi = {10.1016/j.cpc.2011.12.006},
Journal = {Comput. Phys. Commun.},
Pages = {1269},
Title = {BerkeleyGW: A Massively Parallel Computer Package for the Calculation of the Quasiparticle and Optical Properties of Materials and Nanostructures},
Volume = {183},
Year = {2012}}
@article{Lewis_2019a,
Author = {Alan M. Lewis and Timothy C. Berkelbach},
Date-Added = {2019-10-12 14:31:33 +0200},
@ -17,7 +50,8 @@
Pages = {2925},
Title = {Vertex Corrections to the Polarizability Do Not Improve the GW Approximation for the Ionization Potential of Molecules},
Volume = {15},
Year = {2019}}
Year = {2019},
Bdsk-Url-1 = {https://doi.org/10.1021/acs.jctc.8b00995}}
@article{Veril_2018,
Author = {M. Veril and P. Romaniello and J. A. Berger and P. F. Loos},

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Manuscript/GW-srDFT.tex
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@ -181,9 +181,10 @@ We also compute the ionization potentials of the five canonical nucleobase (aden
\label{sec:intro}
%%%%%%%%%%%%%%%%%%%%%%%%
The purpose of many-body perturbation theory (MBPT) 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 MBPT, the \textit{screening} of the Coulomb interaction is a central quantity, and 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}
In MBPT, 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}
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} and is getting increasingly popular in molecular systems \cite{Blase_2011, Faber_2011, Bruneval_2012, Bruneval_2013, 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 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}
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}
The {\GW} approximation stems from the acclaimed Hedin's equations \cite{Hedin_1965}
\begin{subequations}
\begin{align}
@ -215,7 +216,7 @@ Within the {\GW} approximation, one bypasses the calculation of the vertex corre
\end{equation}
Depending on the degree of self-consistency one is willing to perform, there exists several types of {\GW} calculations. \cite{Loos_2018}
The simplest and most popular variant of {\GW} is perturbative {\GW}, or {\GOWO}. \cite{Hybertsen_1985a, Hybertsen_1986}
Although obviously starting-point dependent, it has been widely used in the literature to study solids, atoms and molecules. \cite{Bruneval_2012, Bruneval_2013, vanSetten_2015, vanSetten_2018}
Although obviously starting-point dependent, \cite{Bruneval_2013, Jacquemin_2016, Gui_2018} it has been widely used in the literature to study solids, atoms and molecules. \cite{Bruneval_2012, Bruneval_2013, vanSetten_2015, vanSetten_2018}
For finite systems such as atoms and molecules, partially \cite{Hybertsen_1986, Shishkin_2007, Blase_2011, Faber_2011} or fully self-consistent \cite{Caruso_2012, Caruso_2013, Caruso_2013a, Caruso_2013b} {\GW} methods have shown great promise. \cite{Ke_2011, Blase_2011, Faber_2011, Koval_2014, Hung_2016, Blase_2018, Jacquemin_2017}
Similar to other electron correlation methods, MBPT methods suffer from the usual slow convergence of energetic properties with respect to the size of the one-electron basis set.
@ -223,19 +224,18 @@ This can be tracked down to the lack of explicit electron-electron terms modelin
Pioneered by Hylleraas \cite{Hyl-ZP-29} in the 1930's and popularized in the 1990's by Kutzelnigg and coworkers \cite{Kut-TCA-85, NogKut-JCP-94, KutKlo-JCP-91} (and subsequently others \cite{KonBisVal-CR-12, HatKloKohTew-CR-12, TenNog-WIREs-12, GruHirOhnTen-JCP-17}), the so-called F12 methods overcome this slow convergence by employing geminal basis functions that closely resemble the correlation holes in electronic wave functions.
F12 methods are now routinely employed in computational chemistry and provide robust tools for electronic structure calculations where small basis sets may be used to obtain near complete basis set (CBS) limit accuracy. \cite{TewKloNeiHat-PCCP-07}
The basis-set correction presented here follow a different avenue, and relies on the range-separated density-functional theory (RS-DFT) formalism to capture, thanks to a short-range correlation functional, the missing part of the short-range correlation effects. \cite{GinPraFerAssSavTou-JCP-18}
As shown in recent studies on both ground- and excited-state properties, \cite{LooPraSceTouGin-JPCL-19, GinSceTouLoo-JCP-19} similar to F12 methods, it significantly speeds up the convergence of energetics towards the CBS limit while avoiding the usage of large auxiliary basis sets that are used in F12 methods to avoid the numerous three- and four-electron integrals. \cite{KonBisVal-CR-12, HatKloKohTew-CR-12, TenNog-WIREs-12, GruHirOhnTen-JCP-17, Barca_2018}
The basis-set correction presented here follow a different route, and relies on the range-separated density-functional theory (RS-DFT) formalism to capture, thanks to a short-range correlation functional, the missing part of the short-range correlation effects. \cite{GinPraFerAssSavTou-JCP-18}
As shown in recent studies on both ground- and excited-state properties, \cite{LooPraSceTouGin-JPCL-19, GinSceTouLoo-JCP-19} similar to F12 methods, it significantly speeds up the convergence of energetics towards the CBS limit while avoiding the usage of the large auxiliary basis sets that are used in F12 methods to avoid the numerous three- and four-electron integrals. \cite{KonBisVal-CR-12, HatKloKohTew-CR-12, TenNog-WIREs-12, GruHirOhnTen-JCP-17, Barca_2018}
Explicitly correlated F12 correction schemes have been derived for second-order Green's function methods (GF2) \cite{SzaboBook, Casida_1989, Casida_1991, Stefanucci_2013, Ortiz_2013, Phillips_2014, Phillips_2015, Rusakov_2014, Rusakov_2016, Hirata_2015, Hirata_2017, Loos_2018} by Ten-no and coworkers \cite{Ohnishi_2016, Johnson_2018} and Valeev and coworkers. \cite{Pavosevic_2017, Teke_2019}
However, to the best of our knowledge, a F12-based correction for {\GW} has not been designed yet.
In the present manuscript, we illustrate the performance of the density-based basis set correction developed in Refs.~\onlinecite{GinPraFerAssSavTou-JCP-18, LooPraSceTouGin-JPCL-19, GinSceTouLoo-JCP-19} on ionization potentials obtained within {\GOWO}.
Note that the present basis set correction can be straightforwardly applied to other properties (\textit{e.g.}, electron affinities and fundamental gap), as well as other flavors of (self-consistent) {\GW} or Green's function-based methods, such as GF2 (and its higher-order variants).
Moreover, we are currently investigating the performances of the present approach for linear response theory, in order to speed up the convergence of excitation energies obtained within the random-phase approximation (RPA) \cite{Casida_1995, Dreuw_2005} and Bethe-Salpeter equation (BSE) formalism. \cite{Strinati_1988, Leng_2016, Blase_2018}
The paper is organized as follows.
In Sec.~\ref{sec:theory}, we provide details about the theory behind the present basis set correction and its adaptation to {\GW} methods.
Results are reported and discussed in Sec.~\ref{sec:results}.
Results for a large collection of molecular systems are reported and discussed in Sec.~\ref{sec:results}.
Finally, we draw our conclusions in Sec.~\ref{sec:conclusion}.
Unless otherwise stated, atomic units are used throughout.
@ -248,7 +248,7 @@ Unless otherwise stated, atomic units are used throughout.
\subsection{MBPT with DFT basis set correction}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Following Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}, we start by defining, for a $\Ne$-electron system with nuclei-electron potential $\vne(\b{r})$, the approximate ground-state energy for one-electron densities $\n{}{}$ which are ``representable'' in a finite basis set $\Bas$
Following Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}, we start by defining, for a $\Ne$-electron system with nuclei-electron potential $\vne(\br{})$, the approximate ground-state energy for one-electron densities $\n{}{}$ which are ``representable'' in a finite basis set $\Bas$
\begin{equation}
\E{0}{\Bas} = \min_{\n{}{} \in \cD^\Bas} \qty{ \F{}{}[n] + \int \vne(\br{}) \n{}{}(\br{}) \dbr{} },
\label{eq:E0B}
@ -303,7 +303,7 @@ The stationary condition from Eq.~\eqref{eq:E0BGB} gives the following Dyson equ
(\G{}{\Bas})^{-1} = (\G{0}{\Bas})^{-1}- \Sig{\Hxc}{\Bas}[\G{}{\Bas}]- \bSig{}{\Bas}[\n{\G{}{\Bas}}{}],
\label{eq:Dyson}
\end{equation}
where $(\G{0}{\Bas})^{-1}$ is the basis projection of the inverse non-interacting Green's function with potential $\vne(\b{r})$
where $(\G{0}{\Bas})^{-1}$ is the basis projection of the inverse non-interacting Green's function with potential $\vne(\br{})$
\begin{equation}
(\G{0}{})^{-1}(\br{},\br{}',\omega)= \qty(\omega + \frac{\nabla_{\br{}}^2}{2} + \vne(\br{}) + \lambda) \delta(\br{}-\br{}')
\end{equation}
@ -470,9 +470,9 @@ The mean absolute deviation (MAD), root-mean-square deviation (RMSD), and maximu
\ce{SH2} & 10.10 & 10.49 & 10.65 & 10.72 & 10.44 & 10.67 & 10.76 & 10.78 & 10.45 & 10.66 & 10.74 & 10.77 & 10.78 \\
\ce{F2} & 15.93 & 16.30 & 16.51 & 16.61 & 16.42 & 16.56 & 16.67 & 16.71 & 16.58 & 16.61 & 16.67 & 16.71 & 16.69 \\
\hline
MAD & 0.66 & 0.30 & 0.13 & 0.06 & 0.33 & 0.13 & 0.04 & 0.01 & 0.27 & 0.12 & 0.04 & 0.01 \\
RMSD & 0.71 & 0.32 & 0.14 & 0.06 & 0.37 & 0.14 & 0.04 & 0.01 & 0.30 & 0.13 & 0.05 & 0.01 \\
MAX & 1.08 & 0.46 & 0.22 & 0.10 & 0.65 & 0.22 & 0.07 & 0.03 & 0.54 & 0.20 & 0.08 & 0.03 \\
MAD & 0.66 & 0.30 & 0.13 & 0.06 & 0.33 & 0.13 & 0.04 & 0.01 & 0.27 & 0.12 & 0.04 & 0.01 \\
RMSD & 0.71 & 0.32 & 0.14 & 0.06 & 0.37 & 0.14 & 0.04 & 0.01 & 0.30 & 0.13 & 0.05 & 0.01 \\
MAX & 1.08 & 0.46 & 0.22 & 0.10 & 0.65 & 0.22 & 0.07 & 0.03 & 0.54 & 0.20 & 0.08 & 0.03 \\
\end{tabular}
\end{ruledtabular}
\end{table*}
@ -549,7 +549,6 @@ Moreover, the infinitesimal $\eta$ in Eq.~\eqref{eq:SigC} has been set to zero.
\section{Results and Discussion}
\label{sec:results}
%%%%%%%%%%%%%%%%%%%%%%%%
In this section, we study a subset of atoms and molecules from the GW100 test set. \cite{vanSetten_2015}
In particular, we study the 20 smallest molecules of the GW100 set, a subset that we label as GW20.
This subset has been recently considered by Lewis and Berkelbach to study the effect of vertex corrections to $\W{}{}$ on IPs of molecules. \cite{Lewis_2019a}
@ -559,7 +558,6 @@ Later in this section, we also study the five canonical nucleobases (adenine, cy
\subsection{GW20}
\label{sec:GW20}
%%%%%%%%%%%%%%%%%%%%%%%%
The IPs of the GW20 obtained at the {\GOWO}@{\HF} and {\GOWO}@{\PBEO} levels with increasingly larger Dunning's basis sets cc-pVXZ (X $=$ D, T, Q and 5) are reported in Tables \ref{tab:GW20_HF} and \ref{tab:GW20_PBE0}, respectively.
The corresponding statistical deviations (with respect to the CBS values) are also reported: mean absolute deviation (MAD), root-mean-square deviation (RMSD), and maximum deviation (MAX).
These reference CBS values have been obtained with the usual X$^{-3}$ extrapolation procedure using the three largest basis sets. \cite{Bruneval_2012}
@ -589,7 +587,6 @@ For example, we evidenced in Ref.~\onlinecite{LooPraSceTouGin-JPCL-19} that quin
Here, the overall gain seems to be less important.
The potential reasons for this could be: i) potential-based DFT correction are usually less accurate than the ones based directly on energies, and ii) because the present scheme only corrects the basis set incompleteness error originating from the electron-electron cusp, some incompleteness remains at the HF or KS level.
%%% TABLE III %%%
\begin{table*}
\caption{
@ -672,7 +669,6 @@ The CCSD(T)/def2-TZVPP and experimental results are reported for comparison purp
\subsection{Nucleobases}
\label{sec:DNA}
%%%%%%%%%%%%%%%%%%%%%%%%
In order to check the transferability of the present observations to larger systems, we have computed the values of the IPs of the five canonical nucleobases at the {\GOWO}@PBE level of theory with a different family of basis sets.
The numerical values are reported in Table \ref{tab:DNA_IP}, and their error with respect to the {\GOWO}@PBE/def2-TQZVP extrapolated values (obtained via extrapolation of the def2-TZVP and def2-QZVP results) are shown in Fig.~\ref{fig:DNA_IP}.
The CCSD(T)/def2-TZVPP computed by Krause \textit{et al.} \cite{Krause_2015} as well as the experimental results extracted from Ref.~\onlinecite{vanSetten_2015} are reported for comparison purposes.
@ -692,7 +688,7 @@ These findings have been observed for different {\GW} starting points (HF, PBE o
As mentioned earlier, the present basis set correction can be straightforwardly applied to other properties of interest such as electron affinities or fundamental gap.
It is also applicable to other flavors of {\GW} such as the partially self-consistent {\evGW} or {\qsGW} methods.
We are currently investigating the performances of the present approach within linear response theory in order to speed up the convergence of excitation energies obtained within the RPA and BSE formalisms.
We are currently investigating the performances of the present approach within linear response theory in order to speed up the convergence of excitation energies obtained within the RPA and Bethe-Salpeter equation (BSE) \cite{Strinati_1988, Leng_2016, Blase_2018} formalisms.
We hope to report on this in the near future.
%%%%%%%%%%%%%%%%%%%%%%%%