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Pierre-Francois Loos
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%% This BibTeX bibliography file was created using BibDesk.
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%% http://bibdesk.sourceforge.net/
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%% https://bibdesk.sourceforge.io/
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%% Created for Pierre-Francois Loos at 2022-02-18 09:40:08 +0100
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%% Created for Pierre-Francois Loos at 2022-02-19 13:53:00 +0100
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%% Saved with string encoding Unicode (UTF-8)
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@article{Hedin_1999,
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author = {Lars Hedin},
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date-added = {2022-02-19 13:51:59 +0100},
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date-modified = {2022-02-19 13:51:59 +0100},
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journal = {J Phys.: Cond. Mat.},
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number = {42},
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pages = {R489-R528},
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title = {On correlation effects in electron spectroscopies and the GW approximation},
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volume = {11},
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year = {1999}}
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@article{Bruneval_2006,
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author = {Bruneval, Fabien and Vast, Nathalie and Reining, Lucia},
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date-added = {2022-02-19 13:51:49 +0100},
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date-modified = {2022-02-19 13:51:49 +0100},
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doi = {10.1103/PhysRevB.74.045102},
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issue = {4},
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journal = {Phys. Rev. B},
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month = {Jul},
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numpages = {15},
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pages = {045102},
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publisher = {American Physical Society},
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title = {Effect of self-consistency on quasiparticles in solids},
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url = {https://link.aps.org/doi/10.1103/PhysRevB.74.045102},
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volume = {74},
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year = {2006},
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bdsk-url-1 = {https://link.aps.org/doi/10.1103/PhysRevB.74.045102},
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bdsk-url-2 = {https://doi.org/10.1103/PhysRevB.74.045102}}
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@article{Blase_2016,
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author = {Blase, Xavier and Boulanger, Paul and Bruneval, Fabien and Fernandez-Serra, Marivi and Duchemin, Ivan},
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date-added = {2022-02-19 13:51:38 +0100},
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date-modified = {2022-02-19 13:51:38 +0100},
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doi = {10.1063/1.4940139},
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file = {/Users/loos/Zotero/storage/LWI3LN6F/Blase_2016.pdf},
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issn = {0021-9606, 1089-7690},
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journal = {J. Chem. Phys.},
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language = {en},
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month = jan,
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number = {3},
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pages = {034109},
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title = {{{{\emph{GW}}}} and {{Bethe}}-{{Salpeter}} Study of Small Water Clusters},
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volume = {144},
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year = {2016},
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bdsk-url-1 = {https://dx.doi.org/10.1063/1.4940139}}
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@article{Evangelista_2014b,
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author = {Evangelista,Francesco A.},
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date-added = {2022-02-18 11:16:04 +0100},
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date-modified = {2022-02-18 11:16:24 +0100},
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doi = {10.1063/1.4890660},
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journal = {J. Chem. Phys.},
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number = {5},
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pages = {054109},
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title = {A driven similarity renormalization group approach to quantum many-body problems},
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volume = {141},
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year = {2014},
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bdsk-url-1 = {https://doi.org/10.1063/1.4890660}}
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@article{DiSabatino_2021,
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abstract = {Using the simple (symmetric) Hubbard dimer, we analyze some important features of the GW approximation. We show that the problem of the existence of multiple quasiparticle solutions in the (perturbative) one-shot GW method and its partially self-consistent version is solved by full self-consistency. We also analyze the neutral excitation spectrum using the Bethe-Salpeter equation (BSE) formalism within the standard GW approximation and find, in particular, that 1) some neutral excitation energies become complex when the electron-electron interaction U increases, which can be traced back to the approximate nature of the GW quasiparticle energies; 2) the BSE formalism yields accurate correlation energies over a wide range of U when the trace (or plasmon) formula is employed; 3) the trace formula is sensitive to the occurrence of complex excitation energies (especially singlet), while the expression obtained from the adiabatic-connection fluctuation-dissipation theorem (ACFDT) is more stable (yet less accurate); 4) the trace formula has the correct behavior for weak (i.e., small U) interaction, unlike the ACFDT expression.},
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author = {Di Sabatino, S. and Loos, P.-F. and Romaniello, P.},
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@ -14686,21 +14745,6 @@
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year = {2014},
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bdsk-url-1 = {https://dx.doi.org/10.1088/1367-2630/16/11/119601}}
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@article{Blase_2016,
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author = {Blase, Xavier and Boulanger, Paul and Bruneval, Fabien and Fernandez-Serra, Marivi and Duchemin, Ivan},
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doi = {10.1063/1.4940139},
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file = {/Users/loos/Zotero/storage/LWI3LN6F/Blase_2016.pdf},
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issn = {0021-9606, 1089-7690},
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journal = {J. Chem. Phys.},
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language = {en},
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month = jan,
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number = {3},
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pages = {034109},
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title = {{{{\emph{GW}}}} and {{Bethe}}-{{Salpeter}} Study of Small Water Clusters},
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volume = {144},
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year = {2016},
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bdsk-url-1 = {https://dx.doi.org/10.1063/1.4940139}}
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@article{Bruneval_2009,
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author = {Bruneval, Fabien},
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doi = {10.1103/PhysRevLett.103.176403},
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@ -47,12 +47,6 @@
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\newcommand{\KS}{\text{KS}}
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\newcommand{\HF}{\text{HF}}
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\newcommand{\RPA}{\text{RPA}}
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\newcommand{\ppRPA}{\text{pp-RPA}}
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\newcommand{\BSE}{\text{BSE}}
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\newcommand{\dBSE}{\text{dBSE}}
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\newcommand{\stat}{\text{stat}}
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\newcommand{\dyn}{\text{dyn}}
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\newcommand{\TDA}{\text{TDA}}
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%
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\newcommand{\Norb}{N}
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@ -75,6 +69,7 @@
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% Matrix elements
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\newcommand{\Sig}[2]{\Sigma_{#1}^{#2}}
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\newcommand{\SigC}[1]{\Sigma^\text{c}_{#1}}
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\newcommand{\rSigC}[1]{\widetilde{\Sigma}^\text{c}_{#1}}
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\newcommand{\SigX}[1]{\Sigma^\text{x}_{#1}}
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\newcommand{\SigXC}[1]{\Sigma^\text{xc}_{#1}}
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\newcommand{\MO}[1]{\phi_{#1}}
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@ -129,7 +124,7 @@
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\begin{abstract}
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By upfolding the frequency-dependent $GW$ quasiparticle equation, we explain the appearance of multiple solutions and unphysical discontinuities in various physical quantities computed within the $GW$ approximation.
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By considering the $GW$ self-energy as an effective Hamiltonian, the appearance of these multiple solutions and discontinuities can be directly related to the intruder states problem.
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By considering the $GW$ self-energy as an effective Hamiltonian, the appearance of these multiple solutions and discontinuities can be directly related to the intruder state problem.
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A simple and efficient regularization procedure is proposed to avoid such issues.
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%\bigskip
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%\begin{center}
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@ -144,24 +139,53 @@ A simple and efficient regularization procedure is proposed to avoid such issues
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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The $GW$ approximation of many-body perturbation theory \cite{Hedin_1965,Martin_2016} allows to compute accurate charged excitation (\ie, ionization potentials and electron affinities) in solids and molecules. \cite{Aryasetiawan_1998,Onida_2002,Reining_2017,Golze_2019}
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Its popularity in the molecular structure community is rapidly increasing \cite{Ke_2011,Bruneval_2012,Bruneval_2013,Bruneval_2015,Blase_2016,Bruneval_2016, Bruneval_2016a,Koval_2014,Hung_2016,Blase_2018,Boulanger_2014,Li_2017,Hung_2016,Hung_2017,vanSetten_2015,vanSetten_2018,vanSetten_2015, Maggio_2017,vanSetten_2018,Richard_2016,Gallandi_2016,Knight_2016,Dolgounitcheva_2016,Bruneval_2015,Krause_2015,Govoni_2018,Caruso_2016} thanks to its relatively low cost and somehow surprising accuracy for weakly-correlated systems. \cite{Bruneval_2021}
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Its popularity in the molecular structure community is rapidly growing \cite{Ke_2011,Bruneval_2012,Bruneval_2013,Bruneval_2015,Blase_2016,Bruneval_2016, Bruneval_2016a,Koval_2014,Hung_2016,Blase_2018,Boulanger_2014,Li_2017,Hung_2016,Hung_2017,vanSetten_2015,vanSetten_2018,vanSetten_2015, Maggio_2017,vanSetten_2018,Richard_2016,Gallandi_2016,Knight_2016,Dolgounitcheva_2016,Bruneval_2015,Krause_2015,Govoni_2018,Caruso_2016} thanks to its relatively low cost and somehow surprising accuracy for weakly-correlated systems. \cite{Bruneval_2021}
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The idea behind the $GW$ approximation (or Green's function-based methods in general) is to recast the many-body problem into a set of non-linear one-body equations. The introduction of the self-energy $\Sigma$ links the non-interacting Green's function $G_0$ to its fully-interacting version $G$ via the following Dyson equation:
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\begin{equation}
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G = G_0 + G_0 \Sigma G
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\end{equation}
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Electron correlation is then explicitly incorporated into one-body quantities via a sequence of self-consistent steps known as Hedin's equations. \cite{Hedin_1965}
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%which connect $G$, the irreducible vertex function $\Gamma$, the irreducible polarizability $P$, the dynamically-screened Coulomb interaction $W$, and $\Sigma$ through a set of five equations.
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%\begin{subequations}
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%\begin{align}
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% \label{eq:G}
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% & G(12) = G_0(12) + \int G_\text{H}(13) \Sigma(34) G(42) d(34),
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% \\
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% \label{eq:Gamma}
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% & \Gamma(123) = \delta(12) \delta(13)
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% \notag
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% \\
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% & \qquad \qquad + \int \fdv{\Sigma(12)}{G(45)} G(46) G(75) \Gamma(673) d(4567),
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% \\
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% \label{eq:P}
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% & P(12) = - i \int G(13) \Gamma(324) G(41) d(34),
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% \\
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% \label{eq:W}
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% & W(12) = v(12) + \int v(13) P(34) W(42) d(34),
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% \\
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% \label{eq:Sig}
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% & \Sigma(12) = i \int G(13) W(14) \Gamma(324) d(34),
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%\end{align}
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%\end{subequations}
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%where $v$ is the bare Coulomb interaction, $\delta(12)$ is Dirac's delta function and $(1)$ is a composite coordinate gathering spin, space and time variables $(\sigma_1,\boldsymbol{r}_1,t_1)$.
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In recent studies, \cite{Loos_2018b,Veril_2018,Berger_2021,DiSabatino_2021} we discovered that one can observe (unphysical) irregularities and/or discontinuities in the energy surfaces of several key quantities (ionization potential, electron affinity, fundamental gap, total and correlation energies, as well as vertical excitation energies) even in the weakly correlated regime.
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These issues were discovered in Ref.~\onlinecite{Loos_2018b} while studying a model two-electron system \cite{Seidl_2007,Loos_2009a,Loos_2009c} and they were further investigated in Ref.~\onlinecite{Veril_2018}, where we provided additional evidences and explanations of these undesirable features in real molecular systems.
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In particular, we show that each branch of the self-energy $\Sigma$ is associated with a distinct quasiparticle solution, and that each switch between solutions implies a significant discontinuity in the quasiparticle energy due to the transfer of weight between two solutions of the quasiparticle equation. \cite{Veril_2018}
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It was shown that these problems could be alleviated by using a static COHSEX self-energy \cite{Berger_2021} and by considering a fully self-consistent scheme. \cite{DiSabatino_2021}
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However, none of these solutions is completely satisfying as a static approximation of the self-energy can induce significant loss of accuracy and fully self-consistent calculations can be quite challenging terms of implementation and cost.
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In particular, we showed that each branch of the self-energy $\Sigma$ is associated with a distinct quasiparticle solution, and that each switch between solutions implies a significant discontinuity in the quasiparticle energy due to the transfer of weight between two solutions of the quasiparticle equation. \cite{Veril_2018}
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It was shown that these problems could be alleviated by using a static Coulomb-hole plus screened-exchange (COHSEX) \cite{Hedin_1965,Hybertsen_1986,Hedin_1999,Bruneval_2006} self-energy \cite{Berger_2021} or by considering a fully self-consistent scheme. \cite{DiSabatino_2021}
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However, none of these solutions is completely satisfying as a static approximation of the self-energy can induce significant loss in accuracy and fully self-consistent calculations can be quite challenging in terms of implementation and cost.
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In the present article, we provide further physical insights into the origin of these discontinuities by highlighting, in particular, the role of intruder states.
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These new insights allow us to provide a cheap and efficient regularization scheme in order to avoid these issues.
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In the present article, via an upfolding process of the non-linear $GW$ equation, \cite{Bintrim_2021a} we provide further physical insights into the origin of these discontinuities by highlighting, in particular, the role of intruder states.
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Inspired by regularized electronic structure theories, \cite{Lee_2018a,Evangelista_2014b} these new insights allow us to provide a cheap and efficient regularization scheme in order to avoid these issues.
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We consider {\GOWO} \cite{Strinati_1980,Hybertsen_1985a,Hybertsen_1986,Godby_1988,Linden_1988,Northrup_1991,Blase_1994,Rohlfing_1995,Shishkin_2007} for the sake of simplicity but the same analysis can be performed in the case of (partially) self-consistent schemes.\cite{Hybertsen_1986,Shishkin_2007,Blase_2011,Faber_2011,Rangel_2016,Gui_2018,Faleev_2004,vanSchilfgaarde_2006,Kotani_2007,Ke_2011,Kaplan_2016}
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Here, we consider the one-shot {\GOWO} \cite{Strinati_1980,Hybertsen_1985a,Hybertsen_1986,Godby_1988,Linden_1988,Northrup_1991,Blase_1994,Rohlfing_1995,Shishkin_2007} for the sake of simplicity but the same analysis can be performed in the case of (partially) self-consistent schemes.\cite{Hybertsen_1986,Shishkin_2007,Blase_2011,Faber_2011,Rangel_2016,Gui_2018,Faleev_2004,vanSchilfgaarde_2006,Kotani_2007,Ke_2011,Kaplan_2016}
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Moreover, we consider a restricted Hartree-Fock (HF) starting point but it can be straightforwardly extended to a Kohn-Sham (KS) starting point.
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\section{Downfold: The non-linear $GW$ problem}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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Here, for the sake of simplicity, we consider a restricted Hartree-Fock (HF) starting point but the present analysis can be straightforwardly extended to a Kohn-Sham (KS) starting point.
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Within the {\GOWO} approximation, in order to obtain the quasiparticle energies and the corresponding satellites, one solve, for each spatial orbital $p$, the following (non-linear) quasiparticle equation
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\begin{equation}
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\label{eq:qp_eq}
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@ -170,8 +194,8 @@ Within the {\GOWO} approximation, in order to obtain the quasiparticle energies
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where $\eps{p}{\HF}$ is the $p$th HF orbital energy and the correlation part of the {\GOWO} self-energy reads
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\begin{equation}
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\SigC{p}(\omega)
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= \sum_{im} \frac{2\ERI{pi}{m}^2}{\omega - \eps{i}{\HF} + \Om{m}{\RPA} - i \eta}
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+ \sum_{am} \frac{2\ERI{pa}{m}^2}{\omega - \eps{a}{\HF} - \Om{m}{\RPA} + i \eta}
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= \sum_{im} \frac{2\ERI{pi}{m}^2}{\omega - \eps{i}{\HF} + \Om{m}{\RPA}}
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+ \sum_{am} \frac{2\ERI{pa}{m}^2}{\omega - \eps{a}{\HF} - \Om{m}{\RPA}}
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\end{equation}
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Within the Tamm-Dancoff approximation, the screened two-electron integrals are given by
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\begin{equation}
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@ -195,7 +219,7 @@ As a non-linear equation, Eq.~\eqref{eq:qp_eq} has many solutions $\eps{p,s}{\GW
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0 \le Z_{p,s} = \qty[ 1 - \eval{\pdv{\SigC{p}(\omega)}{\omega}}_{\omega = \eps{p,s}{\GW}} ]^{-1} \le 1
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\end{equation}
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In a well-behaved case, one of the solution (the so-called quasiparticle) $\eps{p}{\GW} \equiv \eps{p,s=0}{\GW}$ has a large weight $Z_{p} \equiv Z_{p,s=0}$.
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Note that we have the following important conservation rules
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Note that we have the following important conservation rules \cite{Martin_1959,Baym_1961,Baym_1962}
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\begin{align}
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\sum_{s} Z_{p,s} & = 1
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&
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@ -254,9 +278,10 @@ and comparing it with Eq.~\eqref{eq:qp_eq} by setting
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\\
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+ \bV{p}{\text{2p1h}} \cdot \qty(\omega \bI - \bC{}{\text{2p1h}} )^{-1} \cdot \T{\qty(\bV{p}{\text{2p1h}})}
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\end{multline}
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where $\bI$ is the identity matrix.
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The main difference between the two approaches is that, by diagonalizing Eq.~\eqref{eq:Hp}, one has directly access to the eigenvectors associated with each quasiparticle and satellites.
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One can see this downfolding process as the construction of an frequency-dependent effective Hamiltonian where the reference (zeroth-order) space is composed by a single determinant of the 1h or 1p sector and the external (first-order) space by all the singly-excited states built from to this reference electronic configuration.
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One can see this downfolding process as the construction of a frequency-dependent effective Hamiltonian where the reference (zeroth-order) space is composed by a single determinant of the 1h or 1p sector and the external (first-order) space by all the 2h1p and 2p1h configurations.
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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@ -293,6 +318,26 @@ Of course, the way of regularizing the self-energy is not unique but here we con
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This helps greatly convergence for (partially) self-consistent $GW$ methods.
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The regularized self-energy is
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\begin{equation}
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\begin{split}
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\rSigC{p}(\omega;s)
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& = \sum_{im} 2\ERI{pi}{m}^2 f_s(\omega - \eps{i}{\HF} + \Om{m}{\RPA})
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\\
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& + \sum_{am} 2\ERI{pa}{m}^2 f_s(\omega - \eps{a}{\HF} - \Om{m}{\RPA})
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\end{split}
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\end{equation}
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where various choices for the ``regularizer'' $f_s$ are possible.
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Our investigation have shown that
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\begin{equation}
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f_s(\Delta) = \frac{1-e^{-2s\Delta^2}}{\Delta}
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\end{equation}
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is a very convenient form and has been derived from the flow equation within driven-similarity renormalization group \cite{Evangelista_2014}
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Of course, by construction, we have
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\begin{equation}
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\lim_{s\to\infty} \rSigC{p}(\omega;s) = \SigC{p}(\omega)
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\end{equation}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\section{Concluding remarks}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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||||
}, Open ]]
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||||
}
|
||||
]
|
||||
|
||||
Loading…
Reference in New Issue
Block a user