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%% This BibTeX bibliography file was created using BibDesk.
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%% https://bibdesk.sourceforge.io/
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@inbook{Bartlett_1986,
abstract = {A diagrammatic derivation of the coupled-cluster (CC) linear response equations for gradients is presented. MBPT approximations emerge as low-order iterations of the CC equations. In CC theory a knowledge of the change in cluster amplitudes with displacement is required, which would not be necessary if the coefficients were variationally optimum, as in the CI approach. However, it is shown that the CC linear response equations can be put in a form where there is no more difficulty in evaluating CC gradients than in the variational CI procedure. This offers a powerful approach for identifying critical points on energy surfaces and in evaluating other properties than the energy.},
address = {Dordrecht},
author = {Bartlett, Rodney J.},
booktitle = {Geometrical Derivatives of Energy Surfaces and Molecular Properties},
date-added = {2022-10-12 13:42:54 +0200},
date-modified = {2022-10-12 13:42:59 +0200},
doi = {10.1007/978-94-009-4584-5_4},
editor = {J{\o}rgensen, Poul and Simons, Jack},
isbn = {978-94-009-4584-5},
pages = {35--61},
publisher = {Springer Netherlands},
title = {Analytical Evaluation of Gradients in Coupled-Cluster and Many-Body Perturbation Theory},
url = {https://doi.org/10.1007/978-94-009-4584-5_4},
year = {1986},
bdsk-url-1 = {https://doi.org/10.1007/978-94-009-4584-5_4}}
@article{Forster_2020,
author = {F{\"o}rster, Arno and Visscher, Lucas},
date-added = {2022-10-12 10:52:33 +0200},
date-modified = {2022-10-12 10:52:55 +0200},
doi = {10.1021/acs.jctc.0c00693},
journal = {J. Chem. Theory Comput.},
number = {12},
pages = {7381-7399},
title = {Low-Order Scaling G0W0 by Pair Atomic Density Fitting},
volume = {16},
year = {2020},
bdsk-url-1 = {https://doi.org/10.1021/acs.jctc.0c00693}}
@article{Linner_2019,
author = {Linn\'er, E. and Aryasetiawan, F.},
date-added = {2022-10-12 10:39:35 +0200},
date-modified = {2022-10-12 10:39:45 +0200},
doi = {10.1103/PhysRevB.100.235106},
issue = {23},
journal = {Phys. Rev. B},
month = {Dec},
numpages = {6},
pages = {235106},
publisher = {American Physical Society},
title = {Ensemble Green's function theory for interacting electrons with degenerate ground states},
url = {https://link-aps-org-s.docadis.univ-tlse3.fr/doi/10.1103/PhysRevB.100.235106},
volume = {100},
year = {2019},
bdsk-url-1 = {https://link-aps-org-s.docadis.univ-tlse3.fr/doi/10.1103/PhysRevB.100.235106},
bdsk-url-2 = {https://doi.org/10.1103/PhysRevB.100.235106}}
@article{Brouder_2009,
author = {Brouder, Christian and Panati, Gianluca and Stoltz, Gabriel},
date-added = {2022-10-12 10:38:17 +0200},
date-modified = {2022-10-12 10:38:26 +0200},
doi = {10.1103/PhysRevLett.103.230401},
issue = {23},
journal = {Phys. Rev. Lett.},
month = {Dec},
numpages = {4},
pages = {230401},
publisher = {American Physical Society},
title = {Many-Body Green Function of Degenerate Systems},
url = {https://link.aps.org/doi/10.1103/PhysRevLett.103.230401},
volume = {103},
year = {2009},
bdsk-url-1 = {https://link.aps.org/doi/10.1103/PhysRevLett.103.230401},
bdsk-url-2 = {https://doi.org/10.1103/PhysRevLett.103.230401}}
@article{Lyakh_2012,
author = {Lyakh, Dmitry I. and Musia{\l}, Monika and Lotrich, Victor F. and Bartlett, Rodney J.},
date-added = {2022-10-12 08:56:00 +0200},
date-modified = {2022-10-12 08:56:18 +0200},
doi = {10.1021/cr2001417},
journal = {Chem. Rev.},
number = {1},
pages = {182-243},
title = {Multireference Nature of Chemistry: The Coupled-Cluster View},
volume = {112},
year = {2012},
bdsk-url-1 = {https://doi.org/10.1021/cr2001417}}
@article{Evangelista_2018,
author = {Evangelista,Francesco A.},
date-added = {2022-10-12 08:53:15 +0200},
date-modified = {2022-10-12 08:53:30 +0200},
doi = {10.1063/1.5039496},
journal = {J. Chem. Phys.},
number = {3},
pages = {030901},
title = {Perspective: Multireference coupled cluster theories of dynamical electron correlation},
volume = {149},
year = {2018},
bdsk-url-1 = {https://doi.org/10.1063/1.5039496}}
@article{McKeon_2022,
author = {McKeon,Caroline A. and Hamed,Samia M. and Bruneval,Fabien and Neaton,Jeffrey B.},
date-added = {2022-10-11 21:50:49 +0200},
date-modified = {2022-10-11 21:51:03 +0200},
doi = {10.1063/5.0097582},
journal = {J. Chem. Phys.},
number = {7},
pages = {074103},
title = {An optimally tuned range-separated hybrid starting point for ab initio GW plus Bethe--Salpeter equation calculations of molecules},
volume = {157},
year = {2022},
bdsk-url-1 = {https://doi.org/10.1063/5.0097582}}
@article{Handy_1989,
abstract = {A size-consistent set of equations for electron correlation which are limited to double substitutions, based on Brueckner orbitals, is discussed. Called BD theory, it is shown that at fifth order of perturbation theory, BD incorporates more terms than CCSD and QCISD. The simplicity of the equations leads to an elegant gradient theory. Preliminary applications are reported.},
author = {Nicholas C. Handy and John A. Pople and Martin Head-Gordon and Krishnan Raghavachari and Gary W. Trucks},
date-added = {2022-10-11 16:13:12 +0200},
date-modified = {2022-10-11 16:13:31 +0200},
doi = {https://doi.org/10.1016/0009-2614(89)85013-4},
journal = {Chem. Phys. Lett.},
number = {2},
pages = {185-192},
title = {Size-consistent Brueckner theory limited to double substitutions},
volume = {164},
year = {1989},
bdsk-url-1 = {https://www.sciencedirect.com/science/article/pii/0009261489850134},
bdsk-url-2 = {https://doi.org/10.1016/0009-2614(89)85013-4}}
@article{Musial_2003a,
author = {Musia{\l},Monika and Kucharski,Stanis{\l}aw A. and Bartlett,Rodney J.},
date-added = {2022-10-11 16:05:07 +0200},
date-modified = {2022-10-11 16:05:20 +0200},
doi = {10.1063/1.1527013},
journal = {J. Chem. Phys.},
number = {3},
pages = {1128-1136},
title = {Equation-of-motion coupled cluster method with full inclusion of the connected triple excitations for ionized states: IP-EOM-CCSDT},
volume = {118},
year = {2003},
bdsk-url-1 = {https://doi.org/10.1063/1.1527013}}
@article{Musial_2003b,
author = {Musia{\l},Monika and Bartlett,Rodney J.},
date-added = {2022-10-11 16:00:48 +0200},
date-modified = {2022-10-11 16:01:02 +0200},
doi = {10.1063/1.1584657},
journal = {J. Chem. Phys.},
number = {4},
pages = {1901-1908},
title = {Equation-of-motion coupled cluster method with full inclusion of connected triple excitations for electron-attached states: EA-EOM-CCSDT},
volume = {119},
year = {2003},
bdsk-url-1 = {https://doi.org/10.1063/1.1584657}}
@article{Nooijen_1995,
author = {Nooijen,Marcel and Bartlett,Rodney J.},
date-added = {2022-10-11 15:59:36 +0200},
date-modified = {2022-10-11 15:59:51 +0200},
doi = {10.1063/1.468592},
journal = {J. Chem. Phys.},
number = {9},
pages = {3629-3647},
title = {Equation of motion coupled cluster method for electron attachment},
volume = {102},
year = {1995},
bdsk-url-1 = {https://doi.org/10.1063/1.468592}}
@article{Stanton_1994,
author = {Stanton,John F. and Gauss,J{\"u}rgen},
date-added = {2022-10-11 15:56:56 +0200},
date-modified = {2022-10-11 15:57:11 +0200},
doi = {10.1063/1.468022},
journal = {J. Chem. Phys.},
number = {10},
pages = {8938-8944},
title = {Analytic energy derivatives for ionized states described by the equation‐of‐motion coupled cluster method},
volume = {101},
year = {1994},
bdsk-url-1 = {https://doi.org/10.1063/1.468022}}
@article{Caylak_2021,
author = {{\c C}aylak, Onur and Baumeier, Bj{\"o}rn},
date-added = {2022-10-11 13:29:42 +0200},
date-modified = {2022-10-11 13:30:08 +0200},
doi = {10.1021/acs.jctc.0c01099},
journal = {J. Chem. Theory Comput.},
number = {2},
pages = {879-888},
title = {Excited-State Geometry Optimization of Small Molecules with Many-Body Green's Functions Theory},
volume = {17},
year = {2021},
bdsk-url-1 = {https://doi.org/10.1021/acs.jctc.0c01099}}
@article{IsmailBeigi_2003,
author = {Ismail-Beigi, Sohrab and Louie, Steven G.},
date-added = {2022-10-11 13:28:50 +0200},
date-modified = {2022-10-11 13:29:10 +0200},
doi = {10.1103/PhysRevLett.90.076401},
issue = {7},
journal = {Phys. Rev. Lett.},
month = {Feb},
numpages = {4},
pages = {076401},
publisher = {American Physical Society},
title = {Excited-State Forces within a First-Principles Green's Function Formalism},
url = {https://link.aps.org/doi/10.1103/PhysRevLett.90.076401},
volume = {90},
year = {2003},
bdsk-url-1 = {https://link.aps.org/doi/10.1103/PhysRevLett.90.076401},
bdsk-url-2 = {https://doi.org/10.1103/PhysRevLett.90.076401}}
@article{Knysh_2022,
author = {Knysh,Iryna and Duchemin,Ivan and Blase,X. and Jacquemin,Denis M.},
date-added = {2022-10-11 13:26:25 +0200},
date-modified = {2022-10-11 13:26:41 +0200},
doi = {10.1063/5.0121121},
journal = {J. Chem. Phys.},
number = {ja},
pages = {null},
title = {Modelling of excited state potential energy surfaces with the Bethe−Salpeter equation formalism: The 4-(dimethylamino)benzonitrile twist},
volume = {0},
year = {0},
bdsk-url-1 = {https://doi.org/10.1063/5.0121121}}
@article{Loos_2022,
author = {Loos,Pierre-Fran{\c c}ois and Romaniello,Pina},
date-added = {2022-04-27 14:39:38 +0200},
date-modified = {2022-04-27 14:39:53 +0200},
date-added = {2022-10-11 10:48:31 +0200},
date-modified = {2022-10-11 10:48:31 +0200},
doi = {10.1063/5.0088364},
journal = {J. Chem. Phys.},
number = {16},
@ -21,6 +237,325 @@
year = {2022},
bdsk-url-1 = {https://doi.org/10.1063/5.0088364}}
@article{Chen_2017,
abstract = { Random-phase approximation (RPA) methods are rapidly emerging as cost-effective validation tools for semilocal density functional computations. We present the theoretical background of RPA in an intuitive rather than formal fashion, focusing on the physical picture of screening and simple diagrammatic analysis. A new decomposition of the RPA correlation energy into plasmonic modes leads to an appealing visualization of electron correlation in terms of charge density fluctuations. Recent developments in the areas of beyond-RPA methods, RPA correlation potentials, and efficient algorithms for RPA energy and property calculations are reviewed. The ability of RPA to approximately capture static correlation in molecules is quantified by an analysis of RPA natural occupation numbers. We illustrate the use of RPA methods in applications to small-gap systems such as open-shell d- and f-element compounds, radicals, and weakly bound complexes, where semilocal density functional results exhibit strong functional dependence. },
author = {Chen, Guo P. and Voora, Vamsee K. and Agee, Matthew M. and Balasubramani, Sree Ganesh and Furche, Filipp},
date-added = {2022-10-11 09:40:36 +0200},
date-modified = {2022-10-11 09:41:28 +0200},
doi = {10.1146/annurev-physchem-040215-112308},
journal = {Ann. Rev. Phys. Chem.},
number = {1},
pages = {421-445},
title = {Random-Phase Approximation Methods},
volume = {68},
year = {2017},
bdsk-url-1 = {https://doi.org/10.1146/annurev-physchem-040215-112308}}
@article{Nooijen_2000,
author = {Nooijen,Marcel and Lotrich,Victor},
date-added = {2022-10-10 16:49:19 +0200},
date-modified = {2022-10-10 16:49:45 +0200},
doi = {10.1063/1.481828},
journal = {J. Chem. Phys.},
number = {2},
pages = {494-507},
title = {Extended similarity transformed equation-of-motion coupled cluster theory (extended-STEOM-CC): Applications to doubly excited states and transition metal compounds},
volume = {113},
year = {2000},
bdsk-url-1 = {https://doi.org/10.1063/1.481828}}
@article{Nooijen_1997c,
author = {Nooijen,Marcel and Bartlett,Rodney J.},
date-added = {2022-10-10 16:37:36 +0200},
date-modified = {2022-10-10 16:38:50 +0200},
doi = {10.1063/1.473635},
journal = {J. Chem. Phys.},
number = {15},
pages = {6449-6455},
title = {Similarity transformed equation-of-motion coupled-cluster study of ionized, electron attached, and excited states of free base porphin},
volume = {106},
year = {1997},
bdsk-url-1 = {https://doi.org/10.1063/1.473635}}
@article{Nooijen_1997b,
author = {Nooijen,Marcel and Bartlett,Rodney J.},
date-added = {2022-10-10 16:37:17 +0200},
date-modified = {2022-10-10 16:38:55 +0200},
doi = {10.1063/1.474922},
journal = {J. Chem. Phys.},
number = {17},
pages = {6812-6830},
title = {Similarity transformed equation-of-motion coupled-cluster theory: Details, examples, and comparisons},
volume = {107},
year = {1997},
bdsk-url-1 = {https://doi.org/10.1063/1.474922}}
@article{Nooijen_1997a,
author = {Nooijen,Marcel and Bartlett,Rodney J.},
date-added = {2022-10-10 16:36:53 +0200},
date-modified = {2022-10-10 16:38:27 +0200},
doi = {10.1063/1.474000},
journal = {J. Chem. Phys.},
number = {15},
pages = {6441-6448},
title = {A new method for excited states: Similarity transformed equation-of-motion coupled-cluster theory},
volume = {106},
year = {1997},
bdsk-url-1 = {https://doi.org/10.1063/1.474000}}
@book{Shavitt_2009,
address = {{Cambridge}},
author = {Shavitt, Isaiah and Bartlett, Rodney J.},
date-added = {2022-10-10 10:46:31 +0200},
date-modified = {2022-10-10 10:46:31 +0200},
doi = {10.1017/CBO9780511596834},
file = {/home/antoinem/Zotero/storage/HCDGARAQ/Shavitt and Bartlett - 2009 - Many-Body Methods in Chemistry and Physics MBPT a.pdf;/home/antoinem/Zotero/storage/3B8MK5GF/D12027E4DAF75CE8214671D842C6B80C.html},
isbn = {978-0-521-81832-2},
publisher = {{Cambridge University Press}},
series = {Cambridge {{Molecular Science}}},
title = {Many-{{Body Methods}} in {{Chemistry}} and {{Physics}}: {{MBPT}} and {{Coupled}}-{{Cluster Theory}}},
year = {2009},
bdsk-url-1 = {https://doi.org/10.1017/CBO9780511596834}}
@article{Musial_2007,
author = {Musia{\l}, Monika and Bartlett, Rodney J.},
date-added = {2022-10-10 10:46:26 +0200},
date-modified = {2022-10-10 10:46:26 +0200},
doi = {10.1063/1.2747245},
file = {/home/antoinem/Zotero/storage/TT5MN29Q/Musia{\l} and Bartlett - 2007 - Addition by subtraction in coupled cluster theory..pdf;/home/antoinem/Zotero/storage/R8DKBIVQ/1.html},
journal = {J. Chem. Phys.},
pages = {024106},
publisher = {{American Institute of Physics}},
title = {Addition by Subtraction in Coupled Cluster Theory. {{II}}. {{Equation}}-of-Motion Coupled Cluster Method for Excited, Ionized, and Electron-Attached States Based on the {{nCC}} Ground State Wave Function},
volume = {127},
year = {2007},
bdsk-url-1 = {https://doi.org/10.1063/1.2747245}}
@article{Bartlett_2007,
author = {Bartlett, Rodney J. and Musia{\l}, Monika},
date-added = {2022-10-10 10:46:26 +0200},
date-modified = {2022-10-10 10:46:26 +0200},
doi = {10.1103/RevModPhys.79.291},
journal = {Rev. Mod. Phys.},
pages = {291--352},
title = {Coupled-Cluster Theory in Quantum Chemistry},
volume = {79},
year = {2007},
bdsk-url-1 = {https://doi.org/10.1103/RevModPhys.79.291}}
@incollection{Crawford_2000,
author = {Crawford, T. Daniel and Schaefer, Henry F.},
booktitle = {Reviews in {{Computational Chemistry}}},
date-added = {2022-10-10 10:46:21 +0200},
date-modified = {2022-10-10 10:46:21 +0200},
doi = {10.1002/9780470125915.ch2},
file = {/home/antoinem/Zotero/storage/SS7HANWJ/9780470125915.html},
isbn = {978-0-470-12591-5},
pages = {33--136},
publisher = {{John Wiley \& Sons, Ltd}},
title = {An {{Introduction}} to {{Coupled Cluster Theory}} for {{Computational Chemists}}},
year = {2000},
bdsk-url-1 = {https://doi.org/10.1002/9780470125915.ch2}}
@article{Paldus_1972,
author = {Paldus, J. and \ifmmode \check{C}\else \v{C}\fi{}\'{\i}\ifmmode \check{z}\else \v{z}\fi{}ek, J. and Shavitt, I.},
date-added = {2022-10-10 10:46:17 +0200},
date-modified = {2022-10-10 10:46:17 +0200},
doi = {10.1103/PhysRevA.5.50},
issue = {1},
journal = {Phys. Rev. A},
month = {Jan},
numpages = {0},
pages = {50--67},
publisher = {American Physical Society},
title = {Correlation Problems in Atomic and Molecular Systems. IV. Extended Coupled-Pair Many-Electron Theory and Its Application to the B${\mathrm{H}}_{3}$ Molecule},
url = {https://link.aps.org/doi/10.1103/PhysRevA.5.50},
volume = {5},
year = {1972},
bdsk-url-1 = {https://link.aps.org/doi/10.1103/PhysRevA.5.50},
bdsk-url-2 = {https://doi.org/10.1103/PhysRevA.5.50}}
@article{Cizek_1966,
author = {{\v C}{\'\i}{\v z}ek, Ji{\v r}{\'\i}},
date-added = {2022-10-10 10:46:12 +0200},
date-modified = {2022-10-10 10:46:12 +0200},
doi = {10.1063/1.1727484},
file = {/home/antoinem/Zotero/storage/PR39PXU8/1.html},
journal = {J. Chem. Phys.},
pages = {4256--4266},
publisher = {{American Institute of Physics}},
title = {On the {{Correlation Problem}} in {{Atomic}} and {{Molecular Systems}}. {{Calculation}} of {{Wavefunction Components}} in {{Ursell}}-{{Type Expansion Using Quantum}}-{{Field Theoretical Methods}}},
volume = {45},
year = {1966},
bdsk-url-1 = {https://doi.org/10.1063/1.1727484}}
@article{Bohm_1953,
author = {Bohm, David and Pines, David},
date-added = {2022-10-10 10:36:26 +0200},
date-modified = {2022-10-10 10:36:34 +0200},
doi = {10.1103/PhysRev.92.609},
issue = {3},
journal = {Phys. Rev.},
month = {Nov},
numpages = {0},
pages = {609--625},
publisher = {American Physical Society},
title = {A Collective Description of Electron Interactions: III. Coulomb Interactions in a Degenerate Electron Gas},
url = {https://link.aps.org/doi/10.1103/PhysRev.92.609},
volume = {92},
year = {1953},
bdsk-url-1 = {https://link.aps.org/doi/10.1103/PhysRev.92.609},
bdsk-url-2 = {https://doi.org/10.1103/PhysRev.92.609}}
@article{Pines_1952,
author = {Pines, David and Bohm, David},
date-added = {2022-10-10 10:36:03 +0200},
date-modified = {2022-10-10 10:36:11 +0200},
doi = {10.1103/PhysRev.85.338},
issue = {2},
journal = {Phys. Rev.},
month = {Jan},
numpages = {0},
pages = {338--353},
publisher = {American Physical Society},
title = {A Collective Description of Electron Interactions: II. Collective $\mathrm{vs}$ Individual Particle Aspects of the Interactions},
url = {https://link.aps.org/doi/10.1103/PhysRev.85.338},
volume = {85},
year = {1952},
bdsk-url-1 = {https://link.aps.org/doi/10.1103/PhysRev.85.338},
bdsk-url-2 = {https://doi.org/10.1103/PhysRev.85.338}}
@article{Bohm_1951,
author = {Bohm, David and Pines, David},
date-added = {2022-10-10 10:35:34 +0200},
date-modified = {2022-10-10 10:35:41 +0200},
doi = {10.1103/PhysRev.82.625},
issue = {5},
journal = {Phys. Rev.},
month = {Jun},
numpages = {0},
pages = {625--634},
publisher = {American Physical Society},
title = {A Collective Description of Electron Interactions. I. Magnetic Interactions},
url = {https://link.aps.org/doi/10.1103/PhysRev.82.625},
volume = {82},
year = {1951},
bdsk-url-1 = {https://link.aps.org/doi/10.1103/PhysRev.82.625},
bdsk-url-2 = {https://doi.org/10.1103/PhysRev.82.625}}
@article{Rishi_2020,
author = {Rishi,Varun and Perera,Ajith and Bartlett,Rodney J.},
date-added = {2022-10-05 14:51:45 +0200},
date-modified = {2022-10-05 14:52:00 +0200},
doi = {10.1063/5.0023862},
journal = {J. Chem. Phys.},
number = {23},
pages = {234101},
title = {A route to improving RPA excitation energies through its connection to equation-of-motion coupled cluster theory},
volume = {153},
year = {2020},
bdsk-url-1 = {https://doi.org/10.1063/5.0023862}}
@article{Jansen_2010,
author = {Jansen,Georg and Liu,Ru-Fen and {\'A}ngy{\'a}n,J{\'a}nos G.},
date-added = {2022-10-05 14:50:41 +0200},
date-modified = {2022-10-05 14:50:56 +0200},
doi = {10.1063/1.3481575},
journal = {J. Chem. Phys.},
number = {15},
pages = {154106},
title = {On the equivalence of ring-coupled cluster and adiabatic connection fluctuation-dissipation theorem random phase approximation correlation energy expressions},
volume = {133},
year = {2010},
bdsk-url-1 = {https://doi.org/10.1063/1.3481575}}
@article{Freeman_1977,
author = {Freeman, David L.},
date-added = {2022-10-05 14:48:42 +0200},
date-modified = {2022-10-05 14:49:07 +0200},
doi = {10.1103/PhysRevB.15.5512},
issue = {12},
journal = {Phys. Rev. B},
month = {Jun},
numpages = {0},
pages = {5512--5521},
publisher = {American Physical Society},
title = {Coupled-cluster expansion applied to the electron gas: Inclusion of ring and exchange effects},
url = {https://link.aps.org/doi/10.1103/PhysRevB.15.5512},
volume = {15},
year = {1977},
bdsk-url-1 = {https://link.aps.org/doi/10.1103/PhysRevB.15.5512},
bdsk-url-2 = {https://doi.org/10.1103/PhysRevB.15.5512}}
@article{Emrich_1981,
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date-modified = {2022-10-05 10:58:35 +0200},
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pages = {379-396},
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volume = {351},
year = {1981},
bdsk-url-1 = {https://www.sciencedirect.com/science/article/pii/0375947481901792},
bdsk-url-2 = {https://doi.org/10.1016/0375-9474(81)90179-2}}
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title = {Full-frequency dynamical Bethe--Salpeter equation without frequency and a study of double excitations},
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bdsk-url-1 = {https://link.aps.org/doi/10.1103/PhysRevD.49.4214},
bdsk-url-2 = {https://doi.org/10.1103/PhysRevD.49.4214}}
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}
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title = {The {{In-Medium Similarity Renormalization Group}}: {{A}} Novel Ab Initio Method for Nuclei},
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}
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title = {In-Medium Similarity Renormalization Group for Open-Shell Nuclei},
author = {Tsukiyama, K. and Bogner, S. K. and Schwenk, A.},
year = {2012},
journal = {Phys. Rev. C},
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number = {6},
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author = {White,Steven R.},
date-added = {2022-02-21 15:42:00 +0100},
@ -233,8 +860,55 @@
year = {2002},
bdsk-url-1 = {https://doi.org/10.1063/1.1508370}}
@article{Li_2015,
title = {Multireference {{Driven Similarity Renormalization Group}}: {{A Second-Order Perturbative Analysis}}},
author = {Li, Chenyang and Evangelista, Francesco A.},
year = {2015},
journal = {J. Chem. Theory Comput.},
volume = {11},
number = {5},
pages = {2097--2108},
issn = {1549-9618},
doi = {10.1021/acs.jctc.5b00134}
}
@article{Li_2016,
title = {Towards Numerically Robust Multireference Theories: {{The}} Driven Similarity Renormalization Group Truncated to One- and Two-Body Operators},
author = {Li, Chenyang and Evangelista, Francesco A.},
year = {2016},
journal = {J. Chem. Phys.},
volume = {144},
number = {16},
pages = {164114},
issn = {0021-9606},
doi = {10.1063/1.4947218}
}
@article{Li_2017,
title = {Driven Similarity Renormalization Group: {{Third-order}} Multireference Perturbation Theory},
author = {Li, Chenyang and Evangelista, Francesco A.},
year = {2017},
journal = {J. Chem. Phys.},
volume = {146},
number = {12},
pages = {124132},
issn = {0021-9606},
doi = {10.1063/1.4979016}
}
@article{Li_2018,
title = {Driven Similarity Renormalization Group for Excited States: {{A}} State-Averaged Perturbation Theory},
author = {Li, Chenyang and Evangelista, Francesco A.},
year = {2018},
journal = {J. Chem. Phys.},
volume = {148},
number = {12},
pages = {124106},
issn = {0021-9606},
doi = {10.1063/1.5019793}
}
@article{Li_2019a,
abstract = { The driven similarity renormalization group (DSRG) provides an alternative way to address the intruder state problem in quantum chemistry. In this review, we discuss recent developments of multireference methods based on the DSRG. We provide a pedagogical introduction to the DSRG and its various extensions and discuss its formal properties in great detail. In addition, we report several illustrative applications of the DSRG to molecular systems. },
author = {Li, Chenyang and Evangelista, Francesco A.},
date-added = {2022-02-21 14:27:55 +0100},
date-modified = {2022-02-21 14:28:27 +0100},
@ -246,6 +920,55 @@
volume = {70},
year = {2019},
bdsk-url-1 = {https://doi.org/10.1146/annurev-physchem-042018-052416}}
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title = {Multiconfigurational Perturbation Theory with Level Shift \textemdash{} the {{Cr2}} Potential Revisited},
author = {Roos, Bj{\"o}rn O. and Andersson, Kerstin},
year = {1995},
journal = {Chemical Physics Letters},
volume = {245},
number = {2},
pages = {215--223},
issn = {0009-2614},
doi = {10.1016/0009-2614(95)01010-7}
}
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title = {Divergence in {{M\o ller}}\textendash{{Plesset}} Theory: {{A}} Simple Explanation Based on a Two-State Model},
author = {Olsen, Jeppe and J{\o}rgensen, Poul and Helgaker, Trygve and Christiansen, Ove},
year = {2000},
journal = {The Journal of Chemical Physics},
volume = {112},
number = {22},
pages = {9736--9748},
issn = {0021-9606},
doi = {10.1063/1.481611}
}
@article{Choe_2001,
title = {Identifying and Removing Intruder States in Multireference {{Mo}}/Ller\textendash{{Plesset}} Perturbation Theory},
author = {Choe, Yoong-Kee and Witek, Henryk A. and Finley, James P. and Hirao, Kimihiko},
year = {2001},
journal = {The Journal of Chemical Physics},
volume = {114},
number = {9},
pages = {3913},
issn = {0021-9606},
doi = {10.1063/1.1345510},
copyright = {\textcopyright{} 2001 American Institute of Physics.}
}
@article{Battaglia_2022,
title = {Regularized {{CASPT2}}: An {{Intruder-State-Free Approach}}},
author = {Battaglia, Stefano and Frans{\'e}n, Lina and Fdez. Galv{\'a}n, Ignacio and Lindh, Roland},
year = {2022},
journal = {Journal of Chemical Theory and Computation},
volume = {18},
number = {8},
pages = {4814--4825},
issn = {1549-9618},
doi = {10.1021/acs.jctc.2c00368}
}
@article{Forsberg_1997,
abstract = {In multiconfigurational perturbation theory, so-called intruders may cause singularities in the potential energy functions, at geometries where an energy denominator becomes zero. When the singularities are weak, they may be successfully removed by level shift techniques. When applied to excited states, a small shift merely moves the singularity. A large shift may cause new divergencies, and too large shifts are unacceptable since the potential function is affected in regions further away from the singularities. This Letter presents an alternative which may be regarded as an imaginary shift. The singularities are not moved, but disappear completely. They are replaced by a small distortion of the potential function. Applications to the N2 ground state, its A3/gEu+ state, and the Cr2 ground state show that the distortion caused by this procedure is small.},
@ -454,10 +1177,10 @@
year = {2014},
bdsk-url-1 = {https://doi.org/10.1063/1.4865816}}
@article{Bintrim_2021a,
@article{Bintrim_2021,
author = {Bintrim,Sylvia J. and Berkelbach,Timothy C.},
date-added = {2021-11-03 22:50:51 +0100},
date-modified = {2021-11-03 22:51:09 +0100},
date-modified = {2022-10-04 16:48:56 +0200},
doi = {10.1063/5.0035141},
journal = {J. Chem. Phys.},
number = {4},
@ -467,16 +1190,6 @@
year = {2021},
bdsk-url-1 = {https://doi.org/10.1063/5.0035141}}
@misc{Bintrim_2021b,
archiveprefix = {arXiv},
author = {Sylvia J. Bintrim and Timothy C. Berkelbach},
date-added = {2021-11-03 22:49:47 +0100},
date-modified = {2021-11-03 22:50:18 +0100},
eprint = {2110.03850},
primaryclass = {cond-mat.mtrl-sci},
title = {Full-frequency dynamical Bethe-Salpeter equation without frequency and a study of double excitations},
year = {2021}}
@article{Loos_2021,
author = {Loos, Pierre-Fran{\c c}ois and Comin, Massimiliano and Blase, Xavier and Jacquemin, Denis},
date-added = {2021-11-03 16:57:23 +0100},
@ -10882,23 +11595,6 @@
year = {1999},
bdsk-url-1 = {https://doi.org/10.1016/S1386-1425(98)00261-3}}
@article{Nooijen_2000,
author = {Nooijen, Marcel},
date-added = {2020-01-01 21:36:51 +0100},
date-modified = {2020-01-01 21:36:52 +0100},
doi = {10.1021/jp993983z},
issn = {1089-5639, 1520-5215},
journal = {J. Phys. Chem. A},
language = {en},
month = may,
number = {19},
pages = {4553-4561},
shorttitle = {Electronic {{Excitation Spectrum}} of {\emph{s}} -{{Tetrazine}}},
title = {Electronic {{Excitation Spectrum}} of {\emph{s}} -{{Tetrazine}}: {{An Extended}}-{{STEOM}}-{{CCSD Study}}},
volume = {104},
year = {2000},
bdsk-url-1 = {https://doi.org/10.1021/jp993983z}}
@article{Noro_2000,
author = {Noro, Takeshi and Sekiya, Masahiro and Koga, Toshikatsu and Matsuyama, Hisashi},
date-added = {2020-01-01 21:36:51 +0100},
@ -15261,7 +15957,20 @@
title = {Valence {{Electron Photoemission Spectrum}} of {{Semiconductors}}: {{{\emph{Ab Initio}}}} {{Description}} of {{Multiple Satellites}}},
volume = {107},
year = {2011},
bdsk-url-1 = {https://dx.doi.org/10.1103/PhysRevLett.107.166401}}
bdsk-url-1 = {https://dx.doi.org/10.1103/PhysRevLett.107.166401}
}
@article{Ismail-Beigi_2017,
title = {Justifying Quasiparticle Self-Consistent Schemes via Gradient Optimization in {{Baym}}\textendash{{Kadanoff}} Theory},
author = {{Ismail-Beigi}, Sohrab},
year = {2017},
journal = {Journal of Physics: Condensed Matter},
volume = {29},
number = {38},
pages = {385501},
issn = {0953-8984},
doi = {10.1088/1361-648X/aa7803}
}
@article{vanSchilfgaarde_2006,
author = {{van Schilfgaarde}, M. and Kotani, Takao and Faleev, S.},

785
Manuscript/SRGGW.tex
View File

@ -1,5 +1,5 @@
\documentclass[aip,jcp,reprint,noshowkeys,superscriptaddress]{revtex4-1}
\usepackage{graphicx,dcolumn,bm,xcolor,microtype,multirow,amscd,amsmath,amssymb,amsfonts,physics,longtable,wrapfig,txfonts}
\usepackage{graphicx,dcolumn,bm,xcolor,microtype,multirow,amscd,amsmath,amssymb,amsfonts,physics,longtable,wrapfig,txfonts,bbold}
\usepackage[version=4]{mhchem}
\usepackage[utf8]{inputenc}
@ -10,117 +10,20 @@
colorlinks=true,
citecolor=blue,
breaklinks=true
]{hyperref}
]{hyperref}
\urlstyle{same}
\newcommand{\ie}{\textit{i.e.}}
\newcommand{\eg}{\textit{e.g.}}
\newcommand{\alert}[1]{\textcolor{red}{#1}}
\usepackage[normalem]{ulem}
\newcommand{\titou}[1]{\textcolor{red}{#1}}
\newcommand{\trashPFL}[1]{\textcolor{r\ed}{\sout{#1}}}
\newcommand{\trashPFL}[1]{\textcolor{\red}{\sout{#1}}}
\newcommand{\PFL}[1]{\titou{(\underline{\bf PFL}: #1)}}
\newcommand{\mc}{\multicolumn}
\newcommand{\fnm}{\footnotemark}
\newcommand{\fnt}{\footnotetext}
\newcommand{\tabc}[1]{\multicolumn{1}{c}{#1}}
\newcommand{\QP}{\textsc{quantum package}}
\newcommand{\T}[1]{#1^{\intercal}}
% coordinates
\newcommand{\br}{\boldsymbol{r}}
\newcommand{\bx}{\boldsymbol{x}}
\newcommand{\dbr}{d\br}
\newcommand{\dbx}{d\bx}
% methods
\newcommand{\GW}{\text{$GW$}}
\newcommand{\evGW}{ev$GW$}
\newcommand{\qsGW}{qs$GW$}
\newcommand{\GOWO}{$G_0W_0$}
\newcommand{\Hxc}{\text{Hxc}}
\newcommand{\xc}{\text{xc}}
\newcommand{\Ha}{\text{H}}
\newcommand{\co}{\text{c}}
\newcommand{\x}{\text{x}}
\newcommand{\KS}{\text{KS}}
\newcommand{\HF}{\text{HF}}
\newcommand{\RPA}{\text{RPA}}
%
\newcommand{\Ne}{N}
\newcommand{\Norb}{K}
\newcommand{\Nocc}{O}
\newcommand{\Nvir}{V}
% operators
\newcommand{\hH}{\Hat{H}}
\newcommand{\hS}{\Hat{S}}
% energies
\newcommand{\Enuc}{E^\text{nuc}}
\newcommand{\Ec}[1]{E_\text{c}^{#1}}
\newcommand{\EHF}{E^\text{HF}}
% orbital energies
\newcommand{\eps}{\epsilon}
\newcommand{\reps}{\Tilde{\epsilon}}
\newcommand{\Om}{\Omega}
% Matrix elements
\newcommand{\SigC}{\Sigma^\text{c}}
\newcommand{\rSigC}{\Tilde{\Sigma}^\text{c}}
\newcommand{\MO}[1]{\phi_{#1}}
\newcommand{\SO}[1]{\psi_{#1}}
\newcommand{\ERI}[2]{(#1|#2)}
\newcommand{\rbra}[1]{(#1|}
\newcommand{\rket}[1]{|#1)}
% Matrices
\newcommand{\bO}{\boldsymbol{0}}
\newcommand{\bI}{\boldsymbol{1}}
\newcommand{\bH}{\boldsymbol{H}}
\newcommand{\bSigC}{\boldsymbol{\Sigma}^{\text{c}}}
\newcommand{\be}{\boldsymbol{\epsilon}}
\newcommand{\bOm}{\boldsymbol{\Omega}}
\newcommand{\bA}{\boldsymbol{A}}
\newcommand{\bB}{\boldsymbol{B}}
\newcommand{\bC}{\boldsymbol{C}}
\newcommand{\bD}{\boldsymbol{D}}
\newcommand{\bU}{\boldsymbol{U}}
\newcommand{\bV}{\boldsymbol{V}}
\newcommand{\bW}{\boldsymbol{W}}
\newcommand{\bX}{\boldsymbol{X}}
\newcommand{\bY}{\boldsymbol{Y}}
\newcommand{\bZ}{\boldsymbol{Z}}
\newcommand{\bc}{\boldsymbol{c}}
% orbitals, gaps, etc
\newcommand{\IP}{I}
\newcommand{\EA}{A}
\newcommand{\HOMO}{\text{HOMO}}
\newcommand{\LUMO}{\text{LUMO}}
\newcommand{\Eg}{E_\text{g}}
\newcommand{\EgFun}{\Eg^\text{fund}}
\newcommand{\EgOpt}{\Eg^\text{opt}}
\newcommand{\EB}{E_B}
% shortcuts for greek letters
\newcommand{\si}{\sigma}
\newcommand{\la}{\lambda}
\newcommand{\RHH}{R_{\ce{H-H}}}
\newcommand{\ii}{\mathrm{i}}
\newcommand{\ant}[1]{\textcolor{green}{#1}}
% addresses
\newcommand{\LCPQ}{Laboratoire de Chimie et Physique Quantiques (UMR 5626), Universit\'e de Toulouse, CNRS, UPS, France}
\begin{document}
\title{A Similarity Renormalization Group Approach to $GW$}
\title{A Similarity Renormalization Group Approach To Many-Body Perturbation Theory}
\author{Antoine \surname{Marie}}
\email{amarie@irsamc.ups-tlse.fr}
@ -141,290 +44,446 @@ Here comes the abstract.
\maketitle
%%%%%%%%%%%%%%%%%%%%%%
%=================================================================%
\section{Introduction}
%%%%%%%%%%%%%%%%%%%%%%
Here comes the introduction.
\label{sec:intro}
%=================================================================%
%%%%%%%%%%%%%%%%%
\section{Undressing $GW$ one determinant at a time}
%%%%%%%%%%%%%%%%%
In the case of {\GOWO}, the quasiparticle equation reads
\begin{equation}
\label{eq:qp_eq}
\eps_p + \SigC_p(\omega) - \omega = 0
\end{equation}
where $\eps_p$ is the one-electron energy of the HF spatial orbital $\MO{p}(\br)$ and the correlation part of the frequency-dependent self-energy is
\begin{equation}
\label{eq:SigC}
\SigC_p(\omega)
= \sum_{im} \frac{2\ERI{pi}{m}^2}{\omega - \eps_i + \Om_m{}}
+ \sum_{am} \frac{2\ERI{pa}{m}^2}{\omega - \eps_a - \Om_m{}}
\end{equation}
where
\begin{equation}
\ERI{pq}{m} = \sum_{ia} \ERI{pq}{ia} X_{ia,m}
\end{equation}
are the screened two-electron repulsion integrals where $\Om_m$ and $\bX_m$ are the $m$th RPA eigenvalue and eigenvector obtained by solving the following (linear) RPA eigenvalue system within the Tamm-Dancoff approximation:
\begin{equation}
\bA^{\RPA} \cdot \bX_m = \Om_m \bX_m
\end{equation}
with
\begin{equation}
A_{ia,jb}^{} = (\eps_a - \eps_i) \delta_{ij} \delta_{ab} + \ERI{ia}{bj}
\end{equation}
and
\begin{equation}
\ERI{pq}{ia} = \iint \MO{p}(\br_1) \MO{q}(\br_1) \frac{1}{\abs{\br_1 - \br_2}} \MO{i}(\br_2) \MO{a}(\br_2) d\br_1 \dbr_2
\end{equation}
One-body Green's functions provide a natural and elegant way to access charged excitations energies of a physical system. \cite{Martin_2016}
The one-body non-linear Hedin's equations give a recipe to obtain the exact interacting one-body Green's function and therefore the exact ionization potentials and electron affinities. \cite{Hedin_1965}
Unfortunately, fully solving the Hedin's equations is out of reach and one must resort to approximations.
In particular, the GW approximation, \cite{Hedin_1965} which has first been mainly used in the context of solids \ant{ref?} and is now widely used for molecules as well \ant{ref?}, provides fairly accurate results for weakly correlated systems\cite{Hung_2017,vanSetten_2015,vanSetten_2018,Caruso_2016,Korbel_2014,Bruneval_2021} at a low computational cost. \cite{Foerster_2011,Liu_2016,Wilhelm_2018,Forster_2021,Duchemin_2021}
The spectral weight of a solution $\eps_{p,\si}^{\GW}$ (where $\si$ numbers the solution for a given orbital $p$) is given by
The $GW$ approximation is an approximation for the self-energy $\Sigma$ which role is to relate the exact interacting Green's function $G$ to a non-interacting reference one $G_0$ through the Dyson equation
\begin{equation}
\label{eq:Z}
0 \le Z_{p,\si} = \qty[ 1 - \eval{\pdv{\SigC_p(\omega)}{\omega}}_{\omega = \eps_{p,\si}^{\GW}} ]^{-1} \le 1
\end{equation}
with the following sum rules:
\begin{align}
\sum_{\si} Z_{p,\si} & = 1
&
\sum_{\si} Z_{p,\si} \eps_{p,\si}^{\GW} & = \eps_{p}
\end{align}
Here, we $p,q,r$ indicate arbitrary (\ie, occupied or unoccupied) orbitals, $i,j,k,l$ are occupied orbitals, while $a,b,c,d$ are unoccupied (virtual) orbitals.
As shown recently, the quasiparticle equation \eqref{eq:qp_eq} can be recast as a linear eigensystem with exactly the same solution:
\begin{equation}
\bH_p \cdot \bc_{p,\si} = \eps_{p,\si}^{\GW} \bc_{p,\si}
\label{eq:dyson}
G = G_0 + G_0\Sigma G.
\end{equation}
with
\begin{equation}
\label{eq:Hp}
\bH_p =
\begin{pmatrix}
\eps_{p} & \bV_p^{\text{2h1p}} & \bV_p^{\text{2p1h}}
\\
\T{(\bV_p^{\text{2h1p}})} & \bC^{\text{2h1p}} & \bO
\\
\T{(\bV_p^{\text{2p1h}})} & \bO & \bC^{\text{2p1h}}
\end{pmatrix}
\end{equation}
and where the expressions of the 2h1p and 2p1h blocks reads
\begin{subequations}
\begin{align}
\label{eq:C2h1p}
C^\text{2h1p}_{IJA,KCL} & = \qty[ \qty( \eps_I + \eps_J - \eps_A) \delta_{JL} \delta_{AC} - 2 \ERI{JA}{CL} ] \delta_{IK}
\\
\label{eq:C2p1h}
C^\text{2p1h}_{IAB,KCD} & = \qty[ \qty( \eps_A + \eps_B - \eps_I) \delta_{IK} \delta_{AC} + 2 \ERI{AI}{KC} ] \delta_{BD}
\end{align}
\end{subequations}
with the following expressions for the coupling blocks:
\begin{subequations}
\begin{align}
\label{eq:V2h1p}
V^\text{2h1p}_{p,KLC} & = \sqrt{2} \ERI{pK}{CL}
\\
\label{eq:V2p1h}
V^\text{2p1h}_{p,KCD} & = \sqrt{2} \ERI{pD}{KC}
\end{align}
\end{subequations}
Here, we use lower case letters for the electronic configurations belonging to the reference model state and upper case letters for the external determinants (\ie, the perturbers).
The self-energy encapsulates all the exchange-correlation effects which are not taken in account in the reference system.
Approximating $\Sigma$ as the first order truncation of its perturbation expansion in terms of the screened interaction $W$ gives the so-called $GW$ approximation. \cite{Hedin_1965, Martin_2016}
Alternatively one could choose to define $\Sigma$ as the $n$-th order expansion in terms of the bare Coulomb interaction leading to the GF($n$) class of approximations. \cite{Hirata_2015,Hirata_2017}
The GF(2) approximation is also known has the second Born approximation. \ant{ref ?}
By solving the secular equation
\begin{equation}
\det[ \bH_p - \omega \bI ] = 0
\end{equation}
we recover the dynamical expression of the self-energy \eqref{eq:SigC}, \ie,
\begin{multline}
\SigC_p(\omega)
= \bV_p^{\text{2h1p}} \cdot \qty(\omega \bI - \bC^{\text{2h1p}} )^{-1} \cdot \T{\qty(\bV_p^{\text{2h1p}})}
\\
+ \bV_p^{\text{2p1h}} \cdot \qty(\omega \bI - \bC^{\text{2p1h}} )^{-1} \cdot \T{\qty(\bV_p^{\text{2p1h}})}
\end{multline}
with
\begin{equation}
\label{eq:Z_proj}
Z_{p,\si} = \qty[ c_{p,\si,1} ]^{2}
\end{equation}
Despite a wide range of successes, many-body perturbation theory is not flawless.
It has been shown that a variety of physical quantities such as charged and neutral excitations energies or correlation and total energies computed within many-body perturbation theory exhibits some discontinuities. \cite{Veril_2018,Loos_2018b}
Even more worrying these discontinuities can happen in the weakly correlated regime where GW is thought to be valid.
These discontinuities are due to the transfer of spectral weight between two solutions of the quasi-particle equation. \cite{Monino_2022}
This is another occurrence of the infamous intruder-state problem. \cite{Roos_1995,Olsen_2000,Choe_2001} \ant{more ref}
In addition, systems for which two quasi-particle solutions have a similar spectral weight are known to be particularly difficult to converge for partially self-consistent GW. \cite{Forster_2021}
In the presence of intruder states, it might be interesting to move additional electronic configurations in the reference space.
Let us label as $p$ the reference 1h ($p = i$) or 1p ($p = a$) determinant and $qia$ the additional 2h1p ($qia = ija$) or 2p1h ($qia = iab$) that one wants to consider explicitly in the model space.
Equation \label{eq:Hp} can then be written exactly as
\begin{equation}
\label{eq:Hp_qia}
\bH_{p,qia} =
\begin{pmatrix}
\eps_p & V_{p,qia} & \bV_p^{\text{2h1p}} & \bV_p^{\text{2p1h}}
\\
V_{qia,p} & \eps_{qia} & \bC_{qia}^{\text{2h1p}} & \bC_{qia}^{\text{2p1h}}
\\
\T{(\bV_p^{\text{2h1p}})} & \T{(\bC_{qia}^{\text{2h1p}})} & \bC^{\text{2h1p}} & \bO
\\
\T{(\bV_p^{\text{2p1h}})} & \T{(\bC_{qia}^{\text{2p1h}})} & \bO & \bC^{\text{2p1h}}
\end{pmatrix}
\end{equation}
with new blocks defined as
\begin{subequations}
\begin{gather}
\eps_{qia} = \text{sgn}(\eps_q - \mu) \qty[ \qty(\eps_q + \eps_a - \eps_i ) + 2 \ERI{ia}{ia} ]
\\
C_{qia,KLC}^\text{2h1p} = - 2 \ERI{ia}{CL} \delta_{qK}
\\
C_{qia,KCD}^\text{2p1h} = + 2 \ERI{ia}{KC} \delta_{qD}
\\
V_{p,qia} = V_{qia,p} = \sqrt{2} \ERI{pq}{ia}
\end{gather}
\end{subequations}
where $\text{sgn}$ is the sign function and $\mu$ is the chemical potential.
In a recent study, Monino and Loos showed that these discontinuities could be removed by introducing a regularizer inspired by the similarity renormalisation group (SRG) in the quasi-particle equation. \cite{Monino_2022}
Encouraged by this result, this work will investigate the application of the SRG formalism to many-body perturbation theory in its $GW$ and GF(2) variants.
The SRG has been developed independently by Wegner \cite{Wegner_1994} and Glazek and Wilson \cite{Glazek_1993,Glazek_1994} in the context of condensed matter systems and light-front quantum field theories, respectively.
This formalism has been been introduced in quantum chemistry by White \cite{White_2002} before being explored in more details by Evangelista and his co-workers in the context of multi-reference electron correlation theories. \cite{Evangelista_2014b,Li_2015, Li_2016, Li_2017, Li_2018, Li_2019a}
The SRG has also been successful in the context of nuclear theory, \cite{Bogner_2007,Tsukiyama_2011,Tsukiyama_2012,Hergert_2013,Hergert_2016} see Ref.\onlinecite{Hergert_2016a} for a recent review in this field. \ant{Maybe search for recent papers of T. Duguet as well.}
The expressions of $\bC^{\text{2h1p}}$, $\bC^{\text{2p1h}}$, $\bV_p^{\text{2h1p}}$, and $\bV_p^{\text{2p1h}}$ remain identical to the ones given in Eqs.~\eqref{eq:C2h1p}, \eqref{eq:C2p1h}, \eqref{eq:V2h1p}, and \eqref{eq:V2p1h} but one has to remove the contribution from the 2h1p or 2p1h configuration $qia$.
While $\eps_p$ represents the relative energy (with respect to the $N$-electron HF reference determinant) of the 1h or 1p configuration, $\eps_{qia} \equiv C_{qia,qia}$ is the relative energy of the 2h1p or 2p1h configuration with respect to the $N$-electron HF reference determinant.
Therefore, when $\eps_p$ and $\eps_{qia}$ becomes of similar magnitude, one might want to move the 2h1p or 2p1h configuration from the external to the internal space in order to avoid intruder state problems.
The SRG transformation aims at decoupling a reference space from an external space while folding information about the coupling in the reference space.
This is often during such decoupling that intruder states appear. \ant{ref}
Yet, SRG is particularly well-suited to avoid them because the speed to which each external configurations is decoupled is proportional to the energy difference between each external configurations and the reference space.
Because by definition intruder states have energies really close to the reference energies therefore they will be the last decoupled.
Therefore the SRG continuous transformation can be stopped once every external configurations except the intruder ones have been decoupled.
Doing so, it gives a way to fold in information in the reference space while avoiding intruder states.
Downfolding Eq.~\eqref{eq:Hp_qia} yields the following frequency-dependent self-energy matrix
\begin{equation}
\label{eq:Hp}
\bSigC_{p,qia}(\omega) =
\begin{pmatrix}
\eps_p + \SigC_p(\omega) & V_{p,qia} + \SigC_{p,qia}(\omega)
\\
V_{qia,p} + \SigC_{qia,p}(\omega) & \eps_{qia} + \SigC_{qia}(\omega)
\\
\end{pmatrix}
\end{equation}
with the dynamical self-energies
\begin{subequations}
\begin{gather}
\begin{split}
\SigC_p(\omega)
& = \bV_p^{\text{2h1p}} \cdot \qty(\omega \bI - \bC^{\text{2h1p}} )^{-1} \cdot \T{\qty(\bV_p^{\text{2h1p}})}
\\
& + \bV_p^{\text{2p1h}} \cdot \qty(\omega \bI - \bC^{\text{2p1h}} )^{-1} \cdot \T{\qty(\bV_p^{\text{2p1h}})}
\end{split}
\\
\begin{split}
\SigC_{qia}(\omega)
& = \bC_{qia}^{\text{2h1p}} \cdot \qty(\omega \bI - \bC^{\text{2h1p}} )^{-1} \cdot \T{\qty(\bC_{qia}^{\text{2h1p}})}
\\
& + \bC_{qia}^{\text{2p1h}} \cdot \qty(\omega \bI - \bC^{\text{2p1h}} )^{-1} \cdot \T{\qty(\bC_{qia}^{\text{2p1h}})}
\end{split}
\\
\begin{split}
\SigC_{p,qia}(\omega)
& = \bV_p^{\text{2h1p}} \cdot \qty(\omega \bI - \bC^{\text{2h1p}} )^{-1} \cdot \T{\qty(\bC_{qia}^{\text{2h1p}})}
\\
& + \bV_p^{\text{2p1h}} \cdot \qty(\omega \bI - \bC^{\text{2p1h}} )^{-1} \cdot \T{\qty(\bC_{qia}^{\text{2p1h}})}
\end{split}
\\
\begin{split}
\SigC_{qia,p}(\omega)
& = \bC_{qia}^{\text{2h1p}} \cdot \qty(\omega \bI - \bC^{\text{2h1p}} )^{-1} \cdot \T{\qty(\bV_p^{\text{2h1p}})}
\\
& + \bC_{qia}^{\text{2p1h}} \cdot \qty(\omega \bI - \bC^{\text{2p1h}} )^{-1} \cdot \T{\qty(\bV_p^{\text{2p1h}})}
\end{split}
\end{gather}
\end{subequations}
Of course, the present procedure can be generalized to any number of states.
The aim of this manuscript is to investigate whether SRG can treat the intruder-state problem in many-body perturbation theory as successfully as it has been in other fields.
We begin by reviewing the $GW$ formalism in Sec.~\ref{sec:gw} and then briefly review the SRG formalism in Sec.~\ref{sec:srg}.
Section~\ref{sec:theoretical} is concluded by a perturbative analysis of the SRG formalism applied to $GW$ (see Sec.~\ref{sec:srggw}).
The computational details of our implementation are provided in Sec.~\ref{sec:comp_det} before turning to the results section.
This section starts by
Solving
\begin{equation}
\bH_{p,qia} \cdot \bc_{p,qia,\si} = \eps_{p,qia,\si}^{\GW} \bc_{p,qia,\si}
\end{equation}
Because both the 1h or 1p configuration $p$ and the 2h1p or 2p1h configuration $qia$ are in the internal space, we have a new definition of the spectral weight:
\begin{equation}
\label{eq:Z_proj}
Z_{p,qia,\si} = \qty[ c_{p,qia,\si,1} ]^{2} + \qty[ c_{p,qia,\si,2} ]^{2}
\end{equation}
Without doubt, the present procedure has similarities with the dressed time-dependent density-functional theory method developed by Maitra and coworkers, \cite{Cave_2004,Maitra_2004} where one doubly-excited configuration is included in the space of single excitations, hence resulting in a dynamical kernel.
\\
%=================================================================%
\section{Theoretical background}
\label{sec:theoretical}
%=================================================================%
%%%%%%%%%%%%%%%%%%%%%%
\section{Similarity renormalization group of the $GW$ equations}
%%%%%%%%%%%%%%%%%%%%%%
\subsection{Many-body perturbation theory in the GW approximation}
\label{sec:gw}
%%%%%%%%%%%%%%%%%%%%%%
Following the similarity renormalization group (SRG) formalism, we perform a unitary transformation of the linear $GW$ equations
Within approximate many-body perturbation theory based on Hedin's equations the central equation is the so-called quasi-particle equation
\begin{equation}
\bH_p(s) = \bU(s) \, \bH_p \, \bU^\dag(s)
\label{eq:quasipart_eq}
\left[ \bF + \bSig(\omega = \epsilon_p) \right] \psi_p = \epsilon_p \psi_p,
\end{equation}
where the so-called flow parameter, $ 0 \le s < \infty$, is a time-like parameter that controls the extent of the transformation.
The purpose of this transformation is to partially decouple the internal and external spaces, or, more precisely in this case, the 1h or 1p sector from the 2h1p and 2p1h sectors, hence avoiding intruder state issues.
By construction, if $s=0$, then $\bU(s) = \bI$, \ie, $\bH_p(s=0) = \bH_p$, while, in the limit $s\to\infty$, $\bH_p(s)$ becomes diagonal.
The SRG flow equation is
\begin{equation}
\dv{\bH_p(s)}{s} = \comm{\boldsymbol{\eta}_p(s)}{\bH_p(s)}
\end{equation}
where the flow generator
\begin{equation*}
\boldsymbol{\eta}_p(s) = \dv{\bU(s)}{s} \bU^\dag(s) = - \boldsymbol{\eta}_p^\dag(s)
\end{equation*}
is an anti-hermitian operator.
where $\bF$ is the Fock matrix, \cite{SzaboBook} and $\bSig(\omega)$ is the self-energy, both are $K \times K$ matrices with $K$ the number of one-body basis functions.
The self-energy can be physically understood as a dynamical screening correction to the Hartree-Fock (HF) problem.
Because $\bSig$ is dynamical it depends on both the eigenvalues $\epsilon_p$ and eigenvectors $\psi_p$ while $\bF$ depends only on the eigenvectors.
Therefore, similarly to the HF case, this equation needs to be solved self-consistently.
We consider Wegner's canonical generator
However, because of the $\omega$ dependence, fully solving this equation is a rather complicated task, hence several approximate solving schemes has been developed.
The most popular one is probably the one-shot scheme, known as $G_0W_0$ if the self-energy is the $GW$ one, in which the off-diagonal elements of Eq.~(\ref{eq:quasipart_eq}) are neglected and the self-consistency is abandoned.
In this case, there are $K$ quasi-particle equations which read as
\begin{equation}
\boldsymbol{\eta}_p^\text{W}(s) = \comm{\bH_p^\text{d}(s)}{\bH_p(s)} = \comm{\bH_p^\text{d}(s)}{\bH_p^\text{od}(s)}
\label{eq:G0W0}
\epsilon_p^{\HF} + \Sigma_{p}(\omega) - \omega = 0,
\end{equation}
where one partitions $\bH_p(s)$ [see Eq.~\eqref{eq:Hp}] into its diagonal $\bH_p^\text{d}(s)$ and off-diagonal $\bH_p^\text{od}(s)$ parts, \ie,
where $\Sigma_{p}(\omega)$ are the diagonal elements of $\bSig$ and $\epsilon_p^{\HF}$ are the HF orbital energies.
The previous equation is non-linear with respect to $\omega$ and therefore can have multiple solutions $\epsilon_{p,s}$ for a given $p$.
These solutions can be characterised by their spectral weight defined as the renormalisation factor $Z_{p,s}$
\begin{equation}
\bH_{p}(s) = \underbrace{\bH_p^\text{d}(s)}_{\text{diagonal}} + \underbrace{\bH_p^\text{od}(s)}_{\text{off-diagonal}}
\label{eq:renorm_factor}
0 \leq Z_{p,s} = \left[ 1 - \pdv{\Sigma_{p}(\omega)}{\omega}\bigg|_{\omega=\epsilon_{p,s}} \right]^{-1} \leq 1.
\end{equation}
where we have $\lim_{s\to\infty} \bH_p^\text{od}(s) = \bO$.
The solution with the largest weight is referred to as the quasi-particle solution while the others are known as satellites or shake-up solutions.
However, in some cases Eq.~(\ref{eq:G0W0}) can have two (or more) solutions with similar weights and the quasi-particle solution is not well-defined.
In fact, these cases are related to the discontinuities and convergence problems discussed earlier because the additional solutions with large weights are the previously mentioned intruder state.
Let us now perform a perturbative analysis of the SRG equation.
For $s=0$, the partition the initial problem as
One obvious flaw of the one-shot scheme mentioned above is its starting point dependence.
Indeed, in Eq.~(\ref{eq:G0W0}) we chose to use the HF orbital energies but this is arbitrary and one could have chosen Kohn-Sham orbitals for example.
Therefore, one could try to optimise the starting point to obtain the best one-shot energies possible. \ant{add ref}
Alternatively, one could solve this equation self-consistently leading to the eigenvalue only self-consistent scheme. \cite{Kaplan_2016}
To do so one use the energy of the quasi-particle solution of the previous iteration to build Eq.~(\ref{eq:G0W0}) and then solves for $\omega$ again until convergence is reached.
However, if the quasi-particle solution is not well-defined, self-consistency can be quite difficult, if not impossible, to reach.
Even if self-consistency has been reached, the starting point dependence has not been totally removed because the results still depend on the starting molecular orbitals.
To update both the energies and the molecular orbitals, one needs to take into account the off-diagonal elements in Eq.~(\ref{eq:quasipart_eq}).
To take into account the effect of off-diagonal elements without fully solving the quasi-particle equation, one can resort to the quasi-particle self-consistent (qs) scheme in which $\bSig(\omega)$ is replaced by a static approximation $\bSig^{\qs}$.
The algorithm to solve the qs problem is totally analog to the HF case with $\bF$ replaced by $\bF + \bSig^{\qs}$.
Various choice for $\bSig^\qs$ are possible but the most used one is the following hermitian one
\begin{equation}
\bH_p(0) = \bH_p^{(0)}(0) + \la \bH_p^{(1)}(0)
\label{eq:sym_qsgw}
\Sigma_{pq}^\qs = \frac{1}{2}\Re\left(\Sigma_{pq}(\epsilon_p) + \Sigma_{pq}(\epsilon_q) \right).
\end{equation}
This form has first been introduced by Faleev and co-workers \cite{Faleev_2004,vanSchilfgaarde_2006,Kotani_2007} before being derived as the effective Hamiltonian that minimizes the length of the gradient of the Klein functional for non-interacting Green's function. \cite{Ismail-Beigi_2017}
One of the main results of this manuscript is the derivation from first principles of an alternative static hermitian form, this will be done in next section.
In this case as well self-consistency is hard to reach in cases where multiple solutions have large spectral weights.
Multiple solutions arise due to the $\omega$ dependence of the self-energy.
Therefore, by suppressing this dependence the static qs approximation relies on the fact that there is one well-defined quasi-particle solution.
If it is not the case, the qs scheme will oscillates between the solutions with large weights. \cite{Forster_2021}
Therefore convergence problems arise when a shake-up configuration has an energy similar to the associated quasi-particle solution, this satellite is the so-called intruder state.
The intruder state problem can be dealt with by introducing \textit{ad hoc} regularizers.
The $\ii eta$ term that is usually added in the denominator of the self-energy (see below) is the usual imaginary-shift regularizer used in various other theories flawed by intruder states. \cite{Battaglia_2022} \ant{more ref...}
Various other regularizers are possible and in particular one of us has shown that a regularizer inspired by the SRG had some advantages over the imaginary shift one in the $GW$ case. \cite{Monino_2022}
But it would be more rigorous to obtain this regularizer from first principles by applying the SRG formalism to many-body perturbation theory.
This is the aim of this work.
Therefore if we apply it, the SRG would gradually remove the coupling between the quasi-particle and the satellites resulting in a renormalized quasi-particle.
However, to do so one needs to identify the coupling terms in Eq.~(\ref{eq:quasipart_eq}) which is not straightforward.
The way around this problem is to transform Eq.~(\ref{eq:quasipart_eq}) to its upfolded version and the coupling terms will elegantly appear in the process.
From now on, we will restrict ourselves to the $GW$ case but the same derivation could be done for the GF(2) and $GT$ self-energy and the corresponding formula are given in Appendix~\ref{sec:GF2}. \ant{do we really give GT equations?}
The upfolded $GW$ quasi-particle equation is the following
\begin{equation}
\label{eq:GWlin}
\begin{pmatrix}
\bF & \bV^{\text{2h1p}} & \bV^{\text{2p1h}} \\
(\bV^{\text{2h1p}})^{\mathrm{T}} & \bC^{\text{2h1p}} & \bO \\
(\bV^{\text{2p1h}})^{\mathrm{T}} & \bO & \bC^{\text{2p1h}} \\
\end{pmatrix}
\cdot
\begin{pmatrix}
\bX \\
\bY^{\text{2h1p}} \\
\bY^{\text{2p1h}} \\
\end{pmatrix}
=
\begin{pmatrix}
\bX \\
\bY^{\text{2h1p}} \\
\bY^{\text{2p1h}} \\
\end{pmatrix}
\cdot
\boldsymbol{\epsilon},
\end{equation}
where $\boldsymbol{\epsilon}$ is a diagonal matrix collecting the quasi-particle and satellite energies, the 2h1p and 2p1h matrix elements are
\begin{subequations}
\begin{align}
C^\text{2h1p}_{ija,klc} & = \qty[ \qty( \epsilon_{i}^{\GW} + \epsilon_{j}^{\GW} - \epsilon_{a}^{\GW}) \delta_{jl} \delta_{ac} - \eri{jc}{al} ] \delta_{ik} ,
\\
C^\text{2p1h}_{iab,kcd} & = \qty[ \qty( \epsilon_{a}^{\GW} + \epsilon_{b}^{\GW} - \epsilon_{i}^{\GW}) \delta_{ik} \delta_{ac} + \eri{ak}{ic} ] \delta_{bd},
\end{align}
\end{subequations}
and the corresponding coupling blocks read
\begin{align}
V^\text{2h1p}_{p,klc} & = \eri{pc}{kl},
&
V^\text{2p1h}_{p,kcd} & = \eri{pk}{dc}.
\end{align}
The $GW$ non-linear equation can be obtained by applying L\"odwin partitioning technique to Eq.~(\ref{eq:GWlin}) \cite{Lowdin_1963,Bintrim_2021}
\begin{equation}
\label{eq:GWnonlin}
\left( \bF + \bSig(\omega) \right) \bX = \omega \bX,
\end{equation}
with
\begin{gather}
\bH_p^{(0)}(0) =
\begin{pmatrix}
\eps_{p} & \bO & \bO
\\
\bO & \bC^{\text{2h1p}} & \bO
\\
\bO & \bO & \bC^{\text{2p1h}}
\end{pmatrix}
\\
\bH_p^{(1)}(0) =
\begin{pmatrix}
0 & \bV_p^{\text{2h1p}} & \bV_p^{\text{2p1h}}
\\
\T{(\bV_p^{\text{2h1p}})} & \bO & \bO
\\
\T{(\bV_p^{\text{2p1h}})} & \bO & \bO
\end{pmatrix}
\end{gather}
where $\la$ is the usual parameter that controls the magnitude of the perturbation.
This partitioning is reminiscent from Epstein-Nest perturbation theory.
We then expand both $\bH_p(s)$ and $\eps_{p,\si}(s)$ as power series in $\la$, such that
\begin{subequations}
\begin{align}
\bH_p(s) & = \bH_p^{(0)}(s) + \la \bH_p^{(1)}(s) + \la^2 \bH_p^{(2)}(s) + \cdots
\\
\eps_{p,\si}(s) & = \eps_{p,\si}^{(0)}(s) + \la \eps_{p,\si}^{(1)}(s) + \la^2 \eps_{p,\si}^{(2)}(s) + \cdots
\bSig(\omega) &= \bV^{\hhp} \left(\omega \mathbb{1} - \bC^{\hhp}\right)^{-1} (\bV^{\hhp})^{\mathsf{T}} \\
&+ \bV^{\pph} \left(\omega \mathbb{1} - \bC^{\pph})^{-1} (\bV^{\pph}\right)^{\mathsf{T}}, \notag
\end{align}
which can be further developed to give the usual
\begin{equation}
\label{eq:GW_selfenergy}
\Sigma_{pq}(\omega)
= \sum_{iv} \frac{W_{p,(i,v)} W_{q,(i,v)}}{\omega - \epsilon_i + \Omega_{v} - \ii \eta}
+ \sum_{av} \frac{W_{p,(a,v)}W_{q,(a,v)}}{\omega - \epsilon_a - \Omega_{v} + \ii \eta},
\end{equation}
with the screened integral defined as
\begin{equation}
\label{eq:GW_sERI}
W_{p,(q,v)} = \sum_{ia}\eri{pi}{qa}\qty( \bX_{v} + \bY_{v})_{ia},
\end{equation}
where $\bX$ and $\bY$ are the matrix of eigenvectors of the direct particle-hole RPA problem defined as
\begin{equation}
\begin{pmatrix}
\bA & \bB \\
- \bB & \bA \\
\end{pmatrix}
\begin{pmatrix}
\bX \\
\bY \\
\end{pmatrix} = \boldsymbol{\Omega}
\begin{pmatrix}
\bX \\
\bY \\
\end{pmatrix},
\end{equation}
with
\begin{align}
A^\dRPA_{ij,ab} &= (\epsilon_i - \epsilon_a) \delta_{ij}\delta_{ab} + \eri{ib}{aj}, \\
B^\dRPA_{ij,ab} &= \eri{ij}{ab}.
\end{align}
$\boldsymbol{\Omega}$ is the diagonal matrix of eigenvalues and its elements $\Omega_v$ appear in Eq.~(\ref{eq:GW_selfenergy}).
Equations~(\ref{eq:GWlin}) and~(\ref{eq:GWnonlin}) have exactly the same solutions but one is linear and the other not.
The price to pay for this linearity is that the size of the matrix in the former equation is $\mathcal{O}(K^3)$ while it is $\mathcal{O}(K)$ in the latter one.
We refer to Ref.~\onlinecite{Bintrim_2021} for a detailed discussion of the up/downfolding process of the $GW$ equations (see also Chapter 8 of Ref.~\onlinecite{Schirmer_2018} for the GF(2) case).
As can be readily seen in Eq.~\eqref{eq:GWlin}, the blocks $V^\text{2h1p}$ and $ V^\text{2p1h}$ are coupling the 1h and 1p configuration to the dressed 2h1p and 2p1h configurations.
Therefore, these blocks will be the target of our SRG transformation but before going in more details we will review the SRG formalism.
%%%%%%%%%%%%%%%%%%%%%%
\subsection{The similarity renormalization group}
\label{sec:srg}
%%%%%%%%%%%%%%%%%%%%%%
The similarity renormalization group method aims at continuously transforming an Hamiltonian to a diagonal form, or more often to a block-diagonal form.
Therefore, the transformed Hamiltonian
\begin{equation}
\label{eq:SRG_Ham}
\bH(s) = \bU(s) \, \bH \, \bU^\dag(s),
\end{equation}
depends on a flow parameter $s$, such that $\bH(s=0)$ is the initial untransformed Hamiltonian and $\bH(s=\infty)$ is the (block)-diagonal Hamiltonian.
An evolution equation for $\bH(s)$ can be easily obtained by deriving Eq~(\ref{eq:SRG_Ham}) and this gives the flow equation
\begin{equation}
\label{eq:flowEquation}
\dv{\bH(s)}{s} = \comm{\boldsymbol{\eta}(s)}{\bH(s)},
\end{equation}
where $\boldsymbol{\eta}(s)$, the flow generator, is defined as
\begin{equation}
\boldsymbol{\eta}(s) = \dv{\bU(s)}{s} \bU^\dag(s) = - \boldsymbol{\eta}^\dag(s).
\end{equation}
To solve this equation at a cost inferior to the one of diagonalizing the initial Hamiltonian, one needs to introduce approximation for $\boldsymbol{\eta}(s)$.
Before defining such an approximation, we need to define what are the blocks to suppress in order to obtain a block-diagonal Hamiltonian.
Therefore, the Hamiltonian is separated in two parts as
\begin{equation}
\bH(s) = \underbrace{\bH^\text{d}(s)}_{\text{diagonal}} + \underbrace{\bH^\text{od}(s)}_{\text{off-diagonal}}.
\end{equation}
By definition, we have the following condition on $\bH^\text{od}$
\begin{equation}
\bH^\text{od}(s=\infty) = \boldsymbol{0}.
\end{equation}
In this work, we will use Wegner's canonical generator which is defined as \cite{Wegner_1994}
\begin{equation}
\boldsymbol{\eta}^\text{W}(s) = \comm{\bH^\text{d}(s)}{\bH(s)} = \comm{\bH^\text{d}(s)}{\bH^\text{od}(s)}.
\end{equation}
If this generator is used, the following condition is verified \cite{Kehrein_2006}
\begin{equation}
\label{eq:derivative_trace}
\dv{s}\text{Tr}\left[ \bH^\text{od}(s)^2 \right] \leq 0
\end{equation}
which implies that the matrix elements of the off-diagonal part will decrease in a monotonic way.
Even more, the coupling coefficients associated with the highest energy determinants are removed first as will be evidenced by the perturbative analysis after.
The main flaw of this generator is that it generates a stiff set of ODE which is therefore difficult to solve numerically. \ant{ref}
However, here we will not tackle the full SRG problem but only consider analytical low-order perturbative expressions so we will not be affected by this problem. \cite{Evangelista_2014, Hergert_2016}
Let us now perform the perturbative analysis of the SRG equations.
For $s=0$, the initial problem is
\begin{equation}
\bH(0) = \bH^\text{d}(0) + \lambda ~ \bH^\text{od}(0)
\end{equation}
where $\lambda$ is the usual perturbation parameter and the off-diagonal part of the Hamiltonian has been defined as the perturber.
For finite values of $s$, we have the following perturbation expansion of the Hamiltonian
\begin{equation}
\label{eq:perturbation_expansionH}
\bH(s) = \bH^{(0)}(s) + \lambda ~ \bH^{(1)}(s) + \lambda^2 \bH^{(2)}(s) + \dots
\end{equation}
Hence, the generator $\boldsymbol{\eta}(s)$ admits a perturbation expansion as well.
Then, one can collect order by order the terms in Eq.~(\ref{eq:flowEquation}) and solve analytically the low-order differential equations.
%%%%%%%%%%%%%%%%%%%%%%
\subsection{Renormalised GW}
\label{sec:srggw}
%%%%%%%%%%%%%%%%%%%%%%
Finally, the SRG formalism exposed above will be applied to $GW$.
First, one needs to defined the diagonal and off-diagonal parts of the $GW$ effective Hamiltonian.
As hinted at the end of section~\ref{sec:gw}, the diagonal and off-diagonal parts will be defined as
\begin{align}
\label{eq:diag_and_offdiag}
\bH^\text{d}(s) &=
\begin{pmatrix}
\bF & \bO & \bO \\
\bO & \bC^{\text{2h1p}} & \bO \\
\bO & \bO & \bC^{\text{2p1h}} \\
\end{pmatrix}
&
\bH^\text{od}(s) &=
\begin{pmatrix}
\bO & \bV^{\text{2h1p}} & \bV^{\text{2p1h}} \\
(\bV^{\text{2h1p}})^{\mathrm{T}} & \bO & \bO \\
(\bV^{\text{2p1h}})^{\mathrm{T}} & \bO & \bO \\
\end{pmatrix}
\end{align}
where we have omitted the $s$ dependence of the matrix elements for the sake of brevity.
Then, the aim of this section is to solve analytically the flow equation [see Eq.~(\ref{eq:flowEquation})] order by order knowing that the initial conditions are
\begin{align}
\bHd{0}(0) &= \begin{pmatrix}
\bF{}{} & \bO \\
\bO & \bC{}{}
\end{pmatrix} &
\bHod{0}(0) &=\begin{pmatrix}
\bO & \bO \\
\bO & \bO
\end{pmatrix} \notag \\
\bHd{1}(0) &=\begin{pmatrix}
\bO & \bO \\
\bO & \bO
\end{pmatrix} &
\bHod{1}(0) &= \begin{pmatrix}
\bO & \bV{}{} \\
\bV{}{\dagger} & \bO \notag
\end{pmatrix} \notag
\end{align}
where we have defined $\bC$ and $\bV$ that collects the 2h1p and 2p1h channels for the sake of conciseness.
Then, the perturbative expansions can be inserted in Eq.~(\ref{eq:GWlin}) before downfolding to obtain a renormalised quasi-particle equation.
In particular, in this manuscript the focus will be on the second-order renormalized quasi-particle equation.
%///////////////////////////%
\subsubsection{Zero-th order matrix elements}
%///////////////////////////%
There is only one zero-th order term in the right hand side of the flow equation
\begin{equation}
\dv{\bH^{(0)}}{s} = \comm{\comm{\bHd{0}}{\bHod{0}}}{\bH^{(0)}},
\end{equation}
and performing the block matrix products gives the following system of equations
\begin{subequations}
\begin{align}
\dv{\bF^{(0)}}{s} &= \bO \\
\dv{\bC^{(0)}}{s} &= \bO \\
\dv{\bV^{(0),\dagger}}{s} &= 2 \bC^{(0)}\bV^{(0),\dagger}\bF^{(0)} - \bV^{(0),\dagger}(\bF^{(0)})^2 - (\bC^{(0)})^2\bV^{(0),\dagger} \\
\dv{\bV^{(0)}}{s} &= 2 \bF^{(0)}\bV^{(0)}\bC^{(0)} - (\bF^{(0)})^2\bV^{(0)} - \bV^{(0)}(\bC^{(0)})^2
\end{align}
\end{subequations}
The last equation can be solved by introducing $\bU$ the matrix that diagonalizes $\bC^{(0)} = \bU \bD^{(0)} \bU^{-1}$ such that the differential equation for $\bV^{(0)}$ becomes
\begin{equation}
\dv{\bW^{(0)}}{s} = 2 \bF^{(0)}\bW^{(0)} \bD^{(0)} - (\bF^{(0)})^2\bW^{(0)} - \bW^{(0)} (\bD^{(0)})^2
\end{equation}
where $\bW^{(0)}= \bV^{(0)} \bU$.
Note that the matrix $\bU$ is also used in the downfolding process of Eq.~(\ref{eq:GWlin}). \cite{Bintrim_2021}
Due to the diagonal structure of $\bF^{(0)}$ and $\bD^{(0)}$, these equations can be easily solved and give
\begin{equation}
W_{p,(q,v)}^{(0)}(s) = W_{p,(q,v)}^{(0)}(0) e^{- (F_{pp}^{(0)} - D_{(q,v),(q,v)}^{(0)})^2 s}
\end{equation}
Due to the initial conditions $\bV^{(0)}(0) = \bO$, we have $\bW^{(0)}(s)=\bO$ and therefore $\bV^{(0)}(s)=\bO=\bV^{(0)}(0) $.
The two first equations of the system are trivial and finally we have
\begin{equation}
\bH^{(0)}(s) = \bH^{(0)}(0)
\end{equation}
which shows that the zero-th order matrix elements are independent of $s$.
%///////////////////////////%
\subsubsection{First order matrix elements}
%///////////////////////////%
Knowing that $\bHod{0}(s)=\bO$, the first order flow equation is
\begin{equation}
\dv{\bH^{(1)}}{s} = \comm{\comm{\bHd{0}}{\bHod{1}}}{\bHd{0}}
\end{equation}
which gives the same system of equations as in the previous subsection except that $\bV^{(0)}$ and $\bV^{(0),\dagger}$ should be replaced by $\bV^{(1)}$ and $\bV^{(1),\dagger}$.
Once again the two first equations are easily solved
\begin{align}
\bF^{(1)}(s) &= \bF^{(1)}(0) = \bO & \bC^{(1)}(s) &= \bC^{(1)}(0) = \bO.
\end{align}
and the first order coupling elements are given by (up to a multiplication by $\bU^{-1}$)
\begin{equation}
W_{p,(q,v)}^{(1)}(s) = W_{p,(q,v)}^{(1)}(0) e^{- (F_{pp}^{(0)} - D_{(q,v),(q,v)}^{(0)})^2 s}
\end{equation}
Note that at $s=0$ the elements $W_{p,(q,v)}^{(1)}(0)$ are equal to the two-electron screened integrals defined in Eq.~(\ref{eq:GW_sERI}) and that for $s\to\infty$ they go to zero.
Therefore, $W_{p,(q,v)}^{(1)}(s)$ are renormalized two-electrons screened integrals.
Note the close similarity of the first-order element expressions with the ones of Evangelista in Ref.~\onlinecite{Evangelista_2014b} obtained in a second quantization formalism (see also Ref.~\onlinecite{Hergert_2016}).
%///////////////////////////%
\subsubsection{Second-order matrix elements}
% ///////////////////////////%
The second-order renormalised quasi-particle equation is given by
\begin{equation}
\label{eq:GW_renorm}
\left( \widetilde{\bF}(s) + \widetilde{\bSig}(\omega; s) \right) \bX = \omega \bX
\end{equation}
with
\begin{align}
\widetilde{\bF}(s) &= \bF^{(0)}+\bF^{(2)}(s)\\
\widetilde{\bSig}(\omega; s) &= \bV^{(1)}(s) \left(\omega \mathbb{1} - \bC^{(0)}\right)^{-1} (\bV^{(1)}(s))^{\dagger}
\end{align}
As can be readily seen above, $\bF^{(2)}$ is the only second-order block of the effective Hamiltonian contributing to the second-order SRG quasi-particle equation.
Collecting every second-order terms and performing the block matrix products results in the following differential equation for $\bF^{(2)}$
\begin{equation}
\label{eq:diffeqF2}
\dv{\bF^{(2)}}{s} = \bF^{(0)}\bV^{(1)}\bV^{(1),\dagger} + \bV^{(1)}\bV^{(1),\dagger}\bF^{(0)} - 2 \bV^{(1)}\bC^{(0)}\bV^{(1),\dagger} .
\end{equation}
This can be solved by simple integration along with the initial condition $\bF^{(2)}=\bO$ to give
\begin{align}
F_{pq}^{(2)}(s) &= \sum_{r,v} \frac{\epsilon_{p} + \epsilon_{q} - 2 (\epsilon_r \pm \Omega_v)}{(\epsilon_p - \epsilon_r \pm \Omega_v)^2 + (\epsilon_q - \epsilon_r \pm \Omega_v)^2} W_{p,(r,v)} \notag \\
&\times W^{\dagger}_{(r,v),q}\left(1 - e^{-(\epsilon_p - \epsilon_r \pm \Omega_v)^2s} e^{-(\epsilon_q - \epsilon_r \pm \Omega_v)^2s}\right)
\end{align}
At $s=0$, this second-order correction is null and for $s\to\infty$ it tends towards the following static limit
\begin{equation}
\label{eq:static_F2}
F_{pq}^{(2)}(\infty) = \sum_{r,v} \frac{\left(\epsilon_{p} + \epsilon_{q} - 2 (\epsilon_r \pm \Omega_v)\right) W_{p,(r,v)} W_{q,(r,v)}}{(\epsilon_p - \epsilon_r \pm \Omega_v)^2 + (\epsilon_q - \epsilon_r \pm \Omega_v)^2}
\end{equation}
Therefore, the SRG flows gradually transforms the dynamic degrees of freedom in $\bSig(\omega)$ in static ones ,starting from the ones that have the largest denominators in Eq.~(\ref{eq:static_F2}).
Interestingly, the static limit, \ie $s\to\infty$ limit, of Eq.~(\ref{eq:GW_renorm}) defines an alternative qsGW approximation to the one defined by Eq.~(\ref{eq:sym_qsgw}).
Yet, both are closely related as they share the same diagonal terms.
Also, note that the hermiticity is naturally enforced in the SRG static approximation as opposed to the symmetrized case.
However, as will be discussed in more details in the results section, the convergence of the qs$GW$ scheme using $\tilde{\bF}(\infty)$ is very poor.
This is similar to the symmetric case when the imaginary shift $\ii \eta$ is set to zero.
Therefore, we will define the SRG-qs$GW$ as
\begin{align}
\label{eq:SRG_qsGW}
\Sigma_{pq}^{\text{SRG}}(s) &= \epsilon_p \delta_{pq} + \sum_{r,v} \frac{\left(\epsilon_{p} + \epsilon_{q} - 2 (\epsilon_r \pm \Omega_v)\right) W_{p,(r,v)} W_{q,(r,v)}}{(\epsilon_p - \epsilon_r \pm \Omega_v)^2 + (\epsilon_q - \epsilon_r \pm \Omega_v)^2} \notag \\
&\times \left(1 - e^{-(\epsilon_p - \epsilon_r \pm \Omega_v)^2s} e^{-(\epsilon_q - \epsilon_r \pm \Omega_v)^2s}\right)
\end{align}
which depends on the parameter $s$ analogously to the $eta$ in the usual case.
The fact that the $s\to\infty$ static limit does not well converge when used in a qs$GW$ calculation could have been predicted because in this limit even the intruder states have been included in $\tilde{\bF}$.
Therefore, we should use a value of $s$ large enough to include almost every states but small enough to avoid intruder states.
To conclude this section, we will discuss the case of discontinuities.
Indeed, we have previously said that intruder states are responsible for both the poor convergence of qs$GW$ and discontinuities in physical quantities at the $\GOWO$ level.
So is it possible to remove discontinuities by using the SRG machinery developed above?
In fact, not directly because discontinuities are due to intruder states in the dynamic part while we have seen just above that a finite value of $s$ is well-designed to avoid the intruder states in the static part.
However, doing a change of variable such that
\begin{align}
e^{-s} &= 1-e^{-t} & 1 - e^{-s} &= e^{-t}
\end{align}
hence choosing a finite value of $t$ is well-designed to avoid discontinuities in the dynamic.
In fact, the dynamic part after the change of variable is closely related to the SRG-inspired regularizer introduced by Monino and Loos in Ref.~\onlinecite{Monino_2022}.
%=================================================================%
\section{Computational details}
\label{sec:comp_det}
%=================================================================%
%=================================================================%
\section{Results}
\label{sec:results}
%=================================================================%
\begin{gather}
\bH_p^{(0)}(s) =
\begin{pmatrix}
\eps_{p}(s) & \bO & \bO
\\
\bO & \bC^{\text{2h1p}}(s) & \bO
\\
\bO & \bO & \bC^{\text{2p1h}}(s)
\end{pmatrix}
\\
\bH_p^{(1)}(s) =
\begin{pmatrix}
0 & \bV_p^{\text{2h1p}}(s) & \bV_p^{\text{2p1h}}(s)
\\
\T{(\bV_p^{\text{2h1p}}(s))} & \bO & \bO
\\
\T{(\bV_p^{\text{2p1h}}(s))} & \bO & \bO
\end{pmatrix}
\end{gather}
%%%%%%%%%%%%%%%%%%%%%%
\section{Conclusion}
@ -432,8 +491,7 @@ We then expand both $\bH_p(s)$ and $\eps_{p,\si}(s)$ as power series in $\la$, s
Here comes the conclusion.
%%%%%%%%%%%%%%%%%%%%%%%%
\acknowledgements{
This project has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (Grant agreement No.~863481).}
\acknowledgements{This project has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (Grant agreement No.~863481).}
%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
@ -442,7 +500,14 @@ This project has received funding from the European Research Council (ERC) under
The data that supports the findings of this study are available within the article.% and its supplementary material.
%%%%%%%%%%%%%%%%%%%%%%%%
\bibliography{MRGW}
\bibliography{SRGGW}
%%%%%%%%%%%%%%%%%%%%%%%%
\appendix
%%%%%%%%%%%%%%%%%%%%%%
\section{GF(2) equations}
\label{sec:GF2}
%%%%%%%%%%%%%%%%%%%%%%
\end{document}

172
Notes/Notes.tex
View File

@ -727,7 +727,20 @@ and can be solved by integration of the right side using the previous analytic f
\begin{align}
\text{Part A}: &(\dv{\bW^{(3)}}{s})_{pR} = (2 \bF{}{(0)}\bW^{(1)}\bD^{(2)} + 2 \bF{}{(2)}\bW^{(1)}\bD^{(0)})_{pR} \notag \\
&= 2\sum_{qS} \epsilon_p\delta_{pq} W_{qS}^{(1)}D_{SR}^{(2)} + F_{pq}^{(2)}W_{qS}^{(1)}D_S\delta_{SR}
&= 2\sum_{qS} \epsilon_p\delta_{pq} W_{qS}^{(1)}D_{SR}^{(2)} + F_{pq}^{(2)}W_{qS}^{(1)}D_S\delta_{SR} \notag \\
&= 2\sum_S \epsilon_pW_{pS}^{(1)}(0) e^{-\Delta_{pS}^2 s} D_{SR}^{(2)} \notag \\
&+ 2\sum_q D_S W_{qR}^{(1)}(0) e^{-\Delta_{qR}^2 s} F_{pq}^{(2)} \notag \\
&=2\sum_{qS} \epsilon_pW_{pS}^{(1)}(0) e^{-\Delta_{pS}^2 s} \frac{-\Delta_{qS}-\Delta_{qR}}{\Delta_{qS}^2+\Delta_{qR}^2} \notag \\
&\times W^{(1)}_{qS}(0)W^{(1)}_{qR}(0)(1-e^{-(\Delta_{qS}^2+\Delta_{qR}^2)s})) \notag \\
&+ 2\sum_{qS} D_S W_{qR}^{(1)}(0) e^{-\Delta_{qR}^2 s} \frac{\Delta_{pS}+\Delta_{qS}}{\Delta_{pS}^2+\Delta_{qS}^2}\notag \\
&\times W^{(1)}_{pS}(0)W^{(1)}_{qS}(0) (1 - e^{-(\Delta_{pS}^2+\Delta_{qS}^2)s}) \notag
\end{align}
\begin{align}
& (\bW^{(3)}(s))_{pR} = \sum_{qS} (2D_S - \epsilon_p - \epsilon_q) \frac{\Delta_{pS}+\Delta_{qS}}{\Delta_{pS}^2+\Delta_{qS}^2}W^{(1)}_{pS}(0)W^{(1)}_{qS}(0) W^{(1)}_{qR}(0) \notag \\
&\times \left(-\frac{e^{-\Delta_{qR}^2s} }{\Delta_{qR}^2} + \frac{e^{-(\Delta_{pS}^2+\Delta_{qS}^2+\Delta_{qR}^2)s}}{\Delta_{pS}^2+\Delta_{qS}^2+\Delta_{qR}^2}\right) \notag \\
&+ \sum_{Sq} (D_R+D_S - 2\epsilon_p) \frac{\Delta_{qS}+\Delta_{qR}}{\Delta_{qS}^2+\Delta_{qR}^2} W^{(1)}_{pS}(0)W^{(1)}_{qS}(0) W^{(1)}_{qR}(0) \notag \\
&\times \left(-\frac{e^{-\Delta_{pS}^2s} }{\Delta_{pS}^2} + \frac{e^{-(\Delta_{pS}^2+\Delta_{qS}^2+\Delta_{qR}^2)s}}{\Delta_{pS}^2+\Delta_{qS}^2+\Delta_{qR}^2}\right) \notag
\end{align}
%///////////////////////////%
@ -874,46 +887,129 @@ Finally, we discuss the renormalized correlation self-energy introduced in this
\label{fig:fig_1}
\end{figure}
%=================================================================%
\section{An alternative partitioning designed for discontinuities}
\label{sec:discontinuities}
\section{Preliminary results}
\label{sec:results}
%=================================================================%
As we have seen before the SRG scheme studied so far has been designed to renormalize the quasiparticle in the ``right way'', \ie by handling correctly the divergent denominators in the static energy expression.
However, doing so we do note handle correctly the renormalization of the self-energy.
The aim of this section is to find a partitioning that do it the other way around with respect to the previous one.
The idea to obtain this is to start from the full Hamiltonian and use a perturber that remove the coupling, this gives
\begin{equation}
\bH(0) =
\begin{pmatrix}
\bF{}{} & \bV{}{}\\
\bV{}{\dagger} & \bC{}{}
\end{pmatrix}
+ \lambda
\begin{pmatrix}
\bO & -\bV{}{} \\
-\bV{}{\dagger} & \bO
\end{pmatrix}
= \bH_{\text{d}}(0) + (1-\lambda)\bH_{\text{od}}(0)
\end{equation}
We define $\lambda' = 1 - \lambda$, hence we can expand it like this
\begin{align}
\bH(s) & = \bH'^{(0)}(s) + \lambda' \bH'^{(1)}(s) + \lambda'^2 \bH'^{(2)}(s) + \cdots
\\
\bF{}{}(s) &= \bF{}{'(0)}(s) +\lambda' \bF{}{'(1)}(s) + \lambda'^2 \bF{}{'(2)}(s) + \cdots
\\
\bC{}{}(s) & = \bC{}{'(0)}(s) + \lambda' \bC{}{'(1)}(s) + \lambda'^2 \bC{}{'(2)}(s) + \cdots
\\
\bV{}{}(s) & = \bV{}{'(0)}(s) + \lambda' \bV{}{'(1)}(s) + \lambda'^2 \bV{}{'(2)}(s) + \cdots
\end{align}
We can use the expansion in terms of $\lambda$ and transform them to $\lambda'$ and then identify with the expressions above, for example for $\bF{}{}$
\begin{align}
\bF{}{}(s) & = \bF{}{(0)}(s) + (1 - \lambda') \bF{}{(1)}(s) + (1 - \lambda')^2 \bF{}{(2)}(s) + \cdots \\
&= \qty( \bF{}{(0)}(s) + \bF{}{(1)}(s) + \bF{}{(2)}(s) + \cdots) \notag \\
&+ \lambda'\qty(-\bF{}{(1)}(s) - 2 \bF{}{(2)}(s) + \cdots) \notag \\
&+ \lambda'^2 \qty(\bF{}{(2)}(s) + \cdots) \notag
\end{align}
%%%%%%%%%%%%%%%%%%%%%%
\subsection{$\eta = 0$}
%%%%%%%%%%%%%%%%%%%%%%
\begin{widetext}
\begin{table}
\caption{Symmetrized static form DIIS = 1}
\begin{tabular}{rrrrr}
& He & Ne & H2 & Ar \\
NbIter & 2.0000 & 13.0000 & 115.0000 & 6.0000 \\
HOMO & -24.3588 & -21.0664 & -16.1493 & -15.3200 \\
EHF & -2.8552 & -128.4885 & -1.1280 & -526.8000 \\
RPA & -0.0460 & -0.2271 & -0.0453 & -0.1775 \\
\end{tabular}
\end{table}
\begin{table}
\caption{SRG static for DIIS = 1}
\begin{tabular}{rrrrrrr}
& He & Ne & H2 & Ar & LiF & HCl \\
NbIter & 2.0000 & 13.0000 & 51.0000 & 4.0000 & 207.0000 & 163.0000 \\
HOMO & -24.3516 & -20.9991 & -16.2258 & -15.3139 & -10.9319 & -12.3324 \\
EHF & -2.8552 & -128.4887 & -1.1286 & -526.8002 & -106.9429 & -460.0891 \\
RPA & -0.0460 & -0.2271 & -0.0459 & -0.1774 & -0.2481 & -0.1935 \\
\end{tabular}
\end{table}
\begin{table}
\caption{Symmetrized static form DIIS = 3}
\begin{tabular}{rrrrrrrrrr}
& He & Ne & H2 & LiH & HF & Ar & LiF & F2 \\
NbIter & 2.0000 & 4.0000 & 11.0000 & 58.0000 & 227.0000 & 4.0000 & 234.0000 & 132.0000 \\
HOMO & -24.3588 & -21.0662 & -16.1493 & -7.9603 & -15.5864 & -15.3198 & -11.0257 & -15.6825 \\
EHF & -2.8552 & -128.4885 & -1.1280 & -7.9834 & -100.0101 & -526.7999 & -106.9414 & -198.6606 \\
RPA & -0.0460 & -0.2271 & -0.0453 & -0.0424 & -0.2447 & -0.1775 & -0.2493 & -0.4709 \\
\end{tabular}
\end{table}
\begin{table}
\caption{SRG static form DIIS = 3}
\begin{tabular}{rrrrrrrrrrr}
& He & Ne & H2 & LiH & HF & Ar & LiF & HCl & BF & F2 \\
NbIter & 2.0000 & 4.0000 & 38.0000 & 248.0000 & 91.0000 & 4.0000 & 28.0000 & 141.0000 & 140.0000 & 52.0000 \\
HOMO & -24.3516 & -20.9990 & -16.2258 & -7.8883 & -15.6498 & -15.3138 & -10.9314 & -12.3324 & -10.8823 & -15.7551 \\
EHF & -2.8552 & -128.4887 & -1.1286 & -7.9833 & -100.0190 & -526.8000 & -106.9429 & -460.0891 & -124.1050 & -198.6830 \\
RPA & -0.0460 & -0.2271 & -0.0459 & -0.0425 & -0.2436 & -0.1774 & -0.2484 & -0.1936 & -0.3130 & -0.4684 \\
\end{tabular}
\end{table}
\end{widetext}
%%%%%%%%%%%%%%%%%%%%%%
\subsection{$\eta = 0.01$}
%%%%%%%%%%%%%%%%%%%%%%
\begin{widetext}
\begin{table}
\caption{Symmetrized static form DIIS = 1}
\begin{tabular}{rrrrrrr}
& He & Ne & H2 & LiH & Ar & LiF \\
NbIter & 2.0000 & 13.0000 & 60.0000 & 22.0000 & 6.0000 & 29.0000 \\
HOMO & -24.3588 & -21.0664 & -16.1499 & -7.9504 & -15.3200 & -11.0513 \\
EHF & -2.8552 & -128.4885 & -1.1280 & -7.9834 & -526.8000 & -106.9418 \\
RPA & -0.0460 & -0.2271 & -0.0453 & -0.0424 & -0.1775 & -0.2493 \\
\end{tabular}
\end{table}
\begin{table}
\caption{SRG static for DIIS = 1}
\begin{tabular}{rrrrrrrrr}
& He & Ne & H2 & LiH & Ar & LiF & HCl & CH4 \\
NbIter & 2.0000 & 13.0000 & 47.0000 & 14.0000 & 4.0000 & 240.0000 & 19.0000 & 172.0000 \\
HOMO & -24.3516 & -20.9991 & -16.2258 & -7.8880 & -15.3139 & -10.9312 & -12.3324 & -14.3818 \\
EHF & -2.8552 & -128.4887 & -1.1286 & -7.9833 & -526.8002 & -106.9428 & -460.0891 & -40.1982 \\
RPA & -0.0460 & -0.2271 & -0.0459 & -0.0423 & -0.1774 & -0.2485 & -0.1935 & -0.2289 \\
\end{tabular}
\end{table}
\begin{table}[h!]
\caption{Symmetrized static form DIIS = 3}
\begin{tabular}{rrrrrrrrrrr}
& He & Ne & H2 & Li2 & LiH & HF & Ar & H2O & LiF & HCl \\
NbIter & 2.0000 & 4.0000 & 9.0000 & 12.0000 & 20.0000 & 18.0000 & 4.0000 & 33.0000 & 12.0000 & 34.0000 \\
HOMO & -24.3588 & -21.0662 & -16.1499 & -5.3026 & -7.8579 & -15.5895 & -15.3198 & -12.1571 & -11.0512 & -12.3535 \\
EHF & -2.8552 & -128.4885 & -1.1280 & -14.8685 & -7.9826 & -100.0089 & -526.7999 & -76.0202 & -106.9418 & -460.0862 \\
RPA & -0.0460 & -0.2271 & -0.0453 & -0.0382 & -0.0429 & -0.2454 & -0.1775 & -0.2507 & -0.2493 & -0.1946 \\
\hline
& BeO & CO & N2 & & BH3 & NH3 & BF & BN & SH2 \\
NbIter & 27.0000 & 18.0000 & 18.0000 & & 15.0000 & 33.0000 & 17.0000 & 26.0000 & 13.0000 \\
HOMO & -10.1480 & -14.1411 & -15.4355 & & -13.3479 & -10.5387 & -10.6645 & -11.6532 & -10.0743 \\
EHF & -89.3969 & -112.6821 & -108.9450 & & -26.3889 & -56.1865 & -124.0993 & -78.8715 & -398.6901 \\
RPA & -0.2867 & -0.3626 & -0.3538 & & -0.1504 & -0.2446 & -0.3107 & -0.3085 & -0.2056 \\
\end{tabular}
\end{table}
\begin{table}[h!]
\caption{SRG static form DIIS = 3}
\begin{tabular}{rrrrrrrrrrrrrrrrrrrrr}
Mol & He & Ne & H2 & Li2 & LiH & HF & Ar & H2O & LiF & HCl \\
NbIter & 2.0000 & 4.0000 & 8.0000 & 9.0000 & 12.0000 & 23.0000 & 4.0000 & 10.0000 & 7.0000 & 17.0000 \\
HOMO & -24.3516 & -20.9990 & -16.2258 & -5.2716 & -7.8879 & -15.6498 & -15.3138 & -12.1953 & -10.9314 & -12.3324 \\
EHF & -2.8552 & -128.4887 & -1.1286 & -14.8692 & -7.9833 & -100.0190 & -526.8000 & -76.0262 & -106.9429 & -460.0891 \\
RPA & -0.0460 & -0.2271 & -0.0459 & -0.0383 & -0.0423 & -0.2436 & -0.1774 & -0.2492 & -0.2485 & -0.1935 \\
\hline
Mol & BeO & CO & N2 & CH4 & BH3 & NH3 & BF & BN & SH2 & F2 \\
NbIter & 13.0000 & 12.0000 & 53.0000 & 39.0000 & 113.0000 & 35.0000 & 8.0000 & 80.0000 & 8.0000 & 19.0000 \\
HOMO & -10.0058 & -14.1362 & -15.4820 & -14.3811 & -13.3194 & -10.5563 & -10.8822 & -11.5763 & -10.0370 & -15.7552 \\
EHF & -89.4032 & -112.6875 & -108.9521 & -40.1982 & -26.3904 & -56.1949 & -124.1050 & -78.8792 & -398.6937 & -198.6830 \\
RPA & -0.2848 & -0.3614 & -0.3508 & -0.2288 & -0.1503 & -0.2429 & -0.3121 & -0.3125 & -0.2043 & -0.4684 \\
\end{tabular}
\end{table}
\end{widetext}
%=================================================================%
\section{Towards second quantized effective Hamiltonians for MBPT?}
\label{sec:second_quant_mbpt}