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@ -1,22 +1,145 @@
%% This BibTeX bibliography file was created using BibDesk. %% This BibTeX bibliography file was created using BibDesk.
%% https://bibdesk.sourceforge.io/ %% https://bibdesk.sourceforge.io/
%% Created for Pierre-Francois Loos at 2022-10-12 13:42:59 +0200 %% Created for Pierre-Francois Loos at 2023-01-20 10:08:25 +0100
%% Saved with string encoding Unicode (UTF-8) %% Saved with string encoding Unicode (UTF-8)
@incollection{CsanakBook,
author = {Csanak, Gy and Taylor, HS and Yaris, Robert},
booktitle = {Advances in atomic and molecular physics},
date-added = {2023-01-20 10:08:22 +0100},
date-modified = {2023-01-20 10:08:22 +0100},
pages = {287--361},
publisher = {Elsevier},
title = {Green's function technique in atomic and molecular physics},
volume = {7},
year = {1971}}
@article{Roos_1995,
abstract = {A level shift technique is suggested for removal of intruder states in multiconfigurational second-order perturbation theory (CASPT2). The first-order wavefunction is first calculated with a level shift parameter large enough to remove the intruder states. The effect of the level shift on the second-order energy is removed by a back correction technique (the LS correction). It is shown that intruder states are removed with little effect on the remaining part of the correlation energy. New potential curves have been computed for the X1Σg+ and the {\'a}{\AA}̊, ΔG12 = 535(452) cm−1, D0 = 1.54(1.44) eV. The corresponding values f{\'a}{\'a}{\'a}{\'a}{\AA}{\AA}{\AA}{\AA}{\AA}{\AA}̊, ΔG12 = 667(574) cm−1, Te = 1.79(1.76) eV.},
author = {Bj{\"o}rn O. Roos and Kerstin Andersson},
date-added = {2023-01-20 09:53:48 +0100},
date-modified = {2023-01-20 09:54:17 +0100},
doi = {https://doi.org/10.1016/0009-2614(95)01010-7},
issn = {0009-2614},
journal = {Chem. Phys. Lett.},
number = {2},
pages = {215--223},
title = {Multiconfigurational Perturbation Theory with Level Shift --- the Cr$_2$ Potential Revisited},
url = {http://www.sciencedirect.com/science/article/pii/0009261495010107},
volume = {245},
year = {1995},
bdsk-url-1 = {http://www.sciencedirect.com/science/article/pii/0009261495010107},
bdsk-url-2 = {https://doi.org/10.1016/0009-2614(95)01010-7}}
@article{Andersson_1995a,
abstract = {The electronic spectrum of Cr2 has been studied using multiconfigurational second-order perturbation theory. Potential curves for 18 electronic states have been computed. By employing an atomic natural orbital (ANO) basis set of the size 8s7p6d4f and by including 3s3p correlation effects and relativistic corrections, the following results were obtained (experimental data within parentheses): First, the location of two 1Σu+ states at 2.86 eV (2.70) and 2.51 eV (2.40) above the ground state. Second, spectroscopic constants for the a′ 3Σu+ state located 1.82 eV (1.76) above the ground state are: re = 1.67 {\AA}̊ (1.65$\pm$$\pm$ 0.02) and ΔG12 = 680 cm−1 (574). Third, a plausible candidate for the metastable state detected in 1985 by Moskovits et al. is the 5Σg+ state in the ground state manifold with the vibrational frequency ωe = 148 cm−1 (79 in a matrix). The vertical transition 5Σu+ ← 5Σg+ is 2.07 eV (2.11).},
author = {K. Andersson},
date-added = {2023-01-20 09:52:39 +0100},
date-modified = {2023-01-20 09:52:46 +0100},
doi = {https://doi.org/10.1016/0009-2614(95)00328-2},
issn = {0009-2614},
journal = {Chem. Phys. Lett.},
number = {3},
pages = {212--221},
title = {The Electronic Spectrum of Cr$_2$},
url = {http://www.sciencedirect.com/science/article/pii/0009261495003282},
volume = {237},
year = {1995},
bdsk-url-1 = {http://www.sciencedirect.com/science/article/pii/0009261495003282},
bdsk-url-2 = {https://doi.org/10.1016/0009-2614(95)00328-2}}
@article{Andersson_1995b,
abstract = {A new one-particle zeroth-order Hamiltonian is proposed for perturbation theory with a complete active space self-consistent field (CASSCF) reference function. With the new partitioning of the Hamiltonian, reference functions dominated by a closed-shell configuration, on one hand, and an open-shell configuration, on the other hand, are treated in similar and balanced ways. This leads to a better description of excitation energies and dissociation energies. The new zeroth-order Hamiltonian has been tested on CH2, SiH2, NH2, CH3, N2, NO, and O2, for which full configuration interaction (FCI) results are available. Further, excitation energies and dissociation energies for the N2 molecule have been compared to corresponding multireference (MR) CI results. Finally, the dissociation energies for a large number of benchmark molecules containing first-row atoms (the ``G1'' test) have been compared to experimental data. The computed excitation energies compare very well with the corresponding FCI and MRCI values. In most cases the errors are well below 1 kcal/mol. The dissociation energies, on the other hand, are in general improved in the new treatment but have a tendency to be overestimated when compared to other more accurate methods.},
author = {Andersson, K.},
date-added = {2023-01-20 09:52:39 +0100},
date-modified = {2023-01-20 09:52:55 +0100},
doi = {10.1007/BF01113860},
journal = {Theor. Chim. Acta},
pages = {31--46},
title = {Different Forms of the Zeroth-Order Hamiltonian in Second-Order Perturbation Theory with a Complete Active Space Self-Consistent Field Reference Function},
url = {https://doi.org/10.1007/BF01113860},
volume = {91},
year = {1995},
bdsk-url-1 = {https://doi.org/10.1007/BF01113860}}
@article{Andersson_1994,
abstract = {The potential energy curve of the Cr2 ground state has been obtained by multiconfigurational second-order perturbation theory. In this study 6--8 singularities appear in the potential curve for bond distances ranging from 1.5 to 2.2 {\AA}. However, these singularities are weak enough to allow the determination of approximate spectroscopic constants. By employing a large atomic natural orbital (ANO) basis set of the size 8s7p6d4f and by including 3s, 3p correlation effects and relativistic corrections, the following values were obtained (experimental data within parentheses): equilibrium bond length re = 1.71 {\AA} (1.68 {\AA}), harmonic vibrational frequency ωe = 625 cm−1 (481 cm−1) and dissociation energy D0 = 1.54 eV (1.44$\pm$0.06 eV).},
author = {K. Andersson and B.O. Roos and Malmqvist, Per Aake. and P.-O. Widmark},
date-added = {2023-01-20 09:52:08 +0100},
date-modified = {2023-01-20 09:52:37 +0100},
doi = {https://doi.org/10.1016/0009-2614(94)01183-4},
issn = {0009-2614},
journal = {Chem. Phys. Lett.},
number = {4},
pages = {391--397},
title = {The Cr$_2$ Potential Energy Curve Studied with Multiconfigurational Second-Order Perturbation Theory},
url = {http://www.sciencedirect.com/science/article/pii/0009261494011834},
volume = {230},
year = {1994},
bdsk-url-1 = {http://www.sciencedirect.com/science/article/pii/0009261494011834},
bdsk-url-2 = {https://doi.org/10.1016/0009-2614(94)01183-4}}
@article{Frosini_2022c,
author = {M. Frosini and T. Duguet and J.-P. Ebran and B. Bally and H. Hergert and T. R. Rodr{\'{\i}}guez and R. Roth and J. M. Yao and V. Som{\`{a}}},
date-added = {2023-01-20 09:45:17 +0100},
date-modified = {2023-01-20 09:45:37 +0100},
doi = {10.1140/epja/s10050-022-00694-x},
journal = {Eur. Phys. J. A},
month = apr,
number = {4},
publisher = {Springer Science and Business Media {LLC}},
title = {Multi-reference many-body perturbation theory for nuclei},
url = {https://doi.org/10.1140/epja/s10050-022-00694-x},
volume = {58},
year = {2022},
bdsk-url-1 = {https://doi.org/10.1140/epja/s10050-022-00694-x}}
@article{Frosini_2022b,
author = {M. Frosini and T. Duguet and J.-P. Ebran and B. Bally and H. Hergert and T. R. Rodr{\'{\i}}guez and R. Roth and J. M. Yao and V. Som{\`{a}}},
date-added = {2023-01-20 09:44:49 +0100},
date-modified = {2023-01-20 09:45:46 +0100},
doi = {10.1140/epja/s10050-022-00694-x},
journal = {Eur. Phys. J. A},
month = apr,
number = {4},
publisher = {Springer Science and Business Media {LLC}},
title = {Multi-reference many-body perturbation theory for nuclei},
url = {https://doi.org/10.1140/epja/s10050-022-00694-x},
volume = {58},
year = {2022},
bdsk-url-1 = {https://doi.org/10.1140/epja/s10050-022-00694-x}}
@article{Frosini_2022a,
author = {M. Frosini and T. Duguet and J.-P. Ebran and V. Som{\`{a}}},
date-added = {2023-01-20 09:44:22 +0100},
date-modified = {2023-01-20 09:45:43 +0100},
doi = {10.1140/epja/s10050-022-00692-z},
journal = {Eur. Phys. J. A},
month = apr,
number = {4},
publisher = {Springer Science and Business Media {LLC}},
title = {Multi-reference many-body perturbation theory for nuclei},
url = {https://doi.org/10.1140/epja/s10050-022-00692-z},
volume = {58},
year = {2022},
bdsk-url-1 = {https://doi.org/10.1140/epja/s10050-022-00692-z}}
@misc{Tolle_2022, @misc{Tolle_2022,
title = {Exact Relationships between the {{GW}} Approximation and Equation-of-Motion Coupled-Cluster Theories through the Quasi-Boson Formalism}, archiveprefix = {arXiv},
author = {T{\"o}lle, Johannes and Chan, Garnet Kin-Lic}, author = {T{\"o}lle, Johannes and Chan, Garnet Kin-Lic},
year = {2022}, doi = {10.48550/arXiv.2212.08982},
number = {arXiv:2212.08982}, eprint = {2212.08982},
eprint = {2212.08982}, eprinttype = {arxiv},
eprinttype = {arxiv}, number = {arXiv:2212.08982},
primaryclass = {cond-mat, physics:physics}, primaryclass = {cond-mat, physics:physics},
doi = {10.48550/arXiv.2212.08982}, title = {Exact Relationships between the {{GW}} Approximation and Equation-of-Motion Coupled-Cluster Theories through the Quasi-Boson Formalism},
archiveprefix = {arXiv} year = {2022},
} bdsk-url-1 = {https://doi.org/10.48550/arXiv.2212.08982}}
@inbook{Bartlett_1986, @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.}, 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.},
@ -748,19 +871,20 @@
bdsk-url-1 = {https://doi.org/10.1002/andp.19945060203}} bdsk-url-1 = {https://doi.org/10.1002/andp.19945060203}}
@article{Glazek_1993, @article{Glazek_1993,
title = {Renormalization of Hamiltonians}, author = {G\l{}azek, Stanis\l{}aw D. and Wilson, Kenneth G.},
author = {G\l{}azek, Stanis\l{}aw D. and Wilson, Kenneth G.}, doi = {10.1103/PhysRevD.48.5863},
journal = {Phys. Rev. D}, issue = {12},
volume = {48}, journal = {Phys. Rev. D},
issue = {12}, month = {Dec},
pages = {5863--5872}, numpages = {0},
numpages = {0}, pages = {5863--5872},
year = {1993}, publisher = {American Physical Society},
month = {Dec}, title = {Renormalization of Hamiltonians},
publisher = {American Physical Society}, url = {https://link.aps.org/doi/10.1103/PhysRevD.48.5863},
doi = {10.1103/PhysRevD.48.5863}, volume = {48},
url = {https://link.aps.org/doi/10.1103/PhysRevD.48.5863} year = {1993},
} bdsk-url-1 = {https://link.aps.org/doi/10.1103/PhysRevD.48.5863},
bdsk-url-2 = {https://doi.org/10.1103/PhysRevD.48.5863}}
@article{Glazek_1994, @article{Glazek_1994,
author = {Glazek, Stanislaw D. and Wilson, Kenneth G.}, author = {Glazek, Stanislaw D. and Wilson, Kenneth G.},
@ -781,81 +905,81 @@
bdsk-url-2 = {https://doi.org/10.1103/PhysRevD.49.4214}} bdsk-url-2 = {https://doi.org/10.1103/PhysRevD.49.4214}}
@book{Kehrein_2006, @book{Kehrein_2006,
title = {The {{Flow Equation Approach}} to {{Many-Particle Systems}}}, doi = {10.1007/3-540-34068-8},
year = {2006}, isbn = {978-3-540-34067-6},
series = {Springer {{Tracts}} in {{Modern Physics}}}, series = {Springer {{Tracts}} in {{Modern Physics}}},
volume = {217}, title = {The {{Flow Equation Approach}} to {{Many-Particle Systems}}},
doi = {10.1007/3-540-34068-8}, volume = {217},
isbn = {978-3-540-34067-6} year = {2006},
} bdsk-url-1 = {https://doi.org/10.1007/3-540-34068-8}}
@article{Bogner_2007, @article{Bogner_2007,
title = {Similarity Renormalization Group for Nucleon-Nucleon Interactions}, author = {Bogner, S. K. and Furnstahl, R. J. and Perry, R. J.},
author = {Bogner, S. K. and Furnstahl, R. J. and Perry, R. J.}, doi = {10.1103/PhysRevC.75.061001},
year = {2007}, journal = {Phys. Rev. C},
journal = {Phys. Rev. C}, number = {6},
volume = {75}, pages = {061001},
number = {6}, title = {Similarity Renormalization Group for Nucleon-Nucleon Interactions},
pages = {061001}, volume = {75},
doi = {10.1103/PhysRevC.75.061001} year = {2007},
} bdsk-url-1 = {https://doi.org/10.1103/PhysRevC.75.061001}}
@article{Hergert_2013, @article{Hergert_2013,
title = {In-Medium Similarity Renormalization Group with Chiral Two- plus Three-Nucleon Interactions}, author = {Hergert, H. and Bogner, S. K. and Binder, S. and Calci, A. and Langhammer, J. and Roth, R. and Schwenk, A.},
author = {Hergert, H. and Bogner, S. K. and Binder, S. and Calci, A. and Langhammer, J. and Roth, R. and Schwenk, A.}, doi = {10.1103/PhysRevC.87.034307},
year = {2013}, journal = {Phys. Rev. C},
journal = {Phys. Rev. C}, number = {3},
volume = {87}, pages = {034307},
number = {3}, title = {In-Medium Similarity Renormalization Group with Chiral Two- plus Three-Nucleon Interactions},
pages = {034307}, volume = {87},
doi = {10.1103/PhysRevC.87.034307} year = {2013},
} bdsk-url-1 = {https://doi.org/10.1103/PhysRevC.87.034307}}
@article{Hergert_2016, @article{Hergert_2016,
title = {In-Medium Similarity Renormalization Group for Closed and Open-Shell Nuclei}, author = {Hergert, H.},
author = {Hergert, H.}, doi = {10.1088/1402-4896/92/2/023002},
year = {2016}, issn = {1402-4896},
journal = {Phys. Scr.}, journal = {Phys. Scr.},
volume = {92}, number = {2},
number = {2}, pages = {023002},
pages = {023002}, title = {In-Medium Similarity Renormalization Group for Closed and Open-Shell Nuclei},
issn = {1402-4896}, volume = {92},
doi = {10.1088/1402-4896/92/2/023002} year = {2016},
} bdsk-url-1 = {https://doi.org/10.1088/1402-4896/92/2/023002}}
@article{Hergert_2016a, @article{Hergert_2016a,
title = {The {{In-Medium Similarity Renormalization Group}}: {{A}} Novel Ab Initio Method for Nuclei}, author = {Hergert, H. and Bogner, S. K. and Morris, T. D. and Schwenk, A. and Tsukiyama, K.},
author = {Hergert, H. and Bogner, S. K. and Morris, T. D. and Schwenk, A. and Tsukiyama, K.}, doi = {10.1016/j.physrep.2015.12.007},
year = {2016}, issn = {0370-1573},
journal = {Physics Reports}, journal = {Physics Reports},
series = {Memorial {{Volume}} in {{Honor}} of {{Gerald E}}. {{Brown}}}, pages = {165--222},
volume = {621}, series = {Memorial {{Volume}} in {{Honor}} of {{Gerald E}}. {{Brown}}},
pages = {165--222}, title = {The {{In-Medium Similarity Renormalization Group}}: {{A}} Novel Ab Initio Method for Nuclei},
issn = {0370-1573}, volume = {621},
doi = {10.1016/j.physrep.2015.12.007} year = {2016},
} bdsk-url-1 = {https://doi.org/10.1016/j.physrep.2015.12.007}}
@article{Tsukiyama_2011, @article{Tsukiyama_2011,
title = {In-{{Medium Similarity Renormalization Group For Nuclei}}}, author = {Tsukiyama, K. and Bogner, S. K. and Schwenk, A.},
author = {Tsukiyama, K. and Bogner, S. K. and Schwenk, A.}, doi = {10.1103/PhysRevLett.106.222502},
year = {2011}, journal = {Phys. Rev. Lett.},
journal = {Phys. Rev. Lett.}, number = {22},
volume = {106}, pages = {222502},
number = {22}, title = {In-{{Medium Similarity Renormalization Group For Nuclei}}},
pages = {222502}, volume = {106},
doi = {10.1103/PhysRevLett.106.222502} year = {2011},
} bdsk-url-1 = {https://doi.org/10.1103/PhysRevLett.106.222502}}
@article{Tsukiyama_2012, @article{Tsukiyama_2012,
title = {In-Medium Similarity Renormalization Group for Open-Shell Nuclei}, author = {Tsukiyama, K. and Bogner, S. K. and Schwenk, A.},
author = {Tsukiyama, K. and Bogner, S. K. and Schwenk, A.}, doi = {10.1103/PhysRevC.85.061304},
year = {2012}, journal = {Phys. Rev. C},
journal = {Phys. Rev. C}, number = {6},
volume = {85}, pages = {061304},
number = {6}, title = {In-Medium Similarity Renormalization Group for Open-Shell Nuclei},
pages = {061304}, volume = {85},
doi = {10.1103/PhysRevC.85.061304} year = {2012},
} bdsk-url-1 = {https://doi.org/10.1103/PhysRevC.85.061304}}
@article{White_2002, @article{White_2002,
author = {White,Steven R.}, author = {White,Steven R.},
@ -870,58 +994,62 @@
year = {2002}, year = {2002},
bdsk-url-1 = {https://doi.org/10.1063/1.1508370}} bdsk-url-1 = {https://doi.org/10.1063/1.1508370}}
@article{Li_2015, @article{ChenyangLi_2015,
title = {Multireference {{Driven Similarity Renormalization Group}}: {{A Second-Order Perturbative Analysis}}}, author = {Li, Chenyang and Evangelista, Francesco A.},
author = {Li, Chenyang and Evangelista, Francesco A.}, date-modified = {2023-01-20 09:22:43 +0100},
year = {2015}, doi = {10.1021/acs.jctc.5b00134},
journal = {J. Chem. Theory Comput.}, issn = {1549-9618},
volume = {11}, journal = {J. Chem. Theory Comput.},
number = {5}, number = {5},
pages = {2097--2108}, pages = {2097--2108},
issn = {1549-9618}, title = {Multireference {{Driven Similarity Renormalization Group}}: {{A Second-Order Perturbative Analysis}}},
doi = {10.1021/acs.jctc.5b00134} volume = {11},
} year = {2015},
bdsk-url-1 = {https://doi.org/10.1021/acs.jctc.5b00134}}
@article{Li_2016, @article{ChenyangLi_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.},
author = {Li, Chenyang and Evangelista, Francesco A.}, date-modified = {2023-01-20 09:20:09 +0100},
year = {2016}, doi = {10.1063/1.4947218},
journal = {J. Chem. Phys.}, issn = {0021-9606},
volume = {144}, journal = {J. Chem. Phys.},
number = {16}, number = {16},
pages = {164114}, pages = {164114},
issn = {0021-9606}, title = {Towards Numerically Robust Multireference Theories: {{The}} Driven Similarity Renormalization Group Truncated to One- and Two-Body Operators},
doi = {10.1063/1.4947218} volume = {144},
} year = {2016},
bdsk-url-1 = {https://doi.org/10.1063/1.4947218}}
@article{Li_2017, @article{ChenyangLi_2017,
title = {Driven Similarity Renormalization Group: {{Third-order}} Multireference Perturbation Theory}, author = {Li, Chenyang and Evangelista, Francesco A.},
author = {Li, Chenyang and Evangelista, Francesco A.}, date-modified = {2023-01-20 09:21:30 +0100},
year = {2017}, doi = {10.1063/1.4979016},
journal = {J. Chem. Phys.}, issn = {0021-9606},
volume = {146}, journal = {J. Chem. Phys.},
number = {12}, number = {12},
pages = {124132}, pages = {124132},
issn = {0021-9606}, title = {Driven Similarity Renormalization Group: {{Third-order}} Multireference Perturbation Theory},
doi = {10.1063/1.4979016} volume = {146},
} year = {2017},
bdsk-url-1 = {https://doi.org/10.1063/1.4979016}}
@article{Li_2018, @article{ChenyangLi_2018,
title = {Driven Similarity Renormalization Group for Excited States: {{A}} State-Averaged Perturbation Theory}, author = {Li, Chenyang and Evangelista, Francesco A.},
author = {Li, Chenyang and Evangelista, Francesco A.}, date-modified = {2023-01-20 09:22:23 +0100},
year = {2018}, doi = {10.1063/1.5019793},
journal = {J. Chem. Phys.}, issn = {0021-9606},
volume = {148}, journal = {J. Chem. Phys.},
number = {12}, number = {12},
pages = {124106}, pages = {124106},
issn = {0021-9606}, title = {Driven Similarity Renormalization Group for Excited States: {{A}} State-Averaged Perturbation Theory},
doi = {10.1063/1.5019793} volume = {148},
} year = {2018},
bdsk-url-1 = {https://doi.org/10.1063/1.5019793}}
@article{Li_2019a, @article{ChenyangLi_2019a,
author = {Li, Chenyang and Evangelista, Francesco A.}, author = {Li, Chenyang and Evangelista, Francesco A.},
date-added = {2022-02-21 14:27:55 +0100}, date-added = {2022-02-21 14:27:55 +0100},
date-modified = {2022-02-21 14:28:27 +0100}, date-modified = {2023-01-20 09:23:34 +0100},
doi = {10.1146/annurev-physchem-042018-052416}, doi = {10.1146/annurev-physchem-042018-052416},
journal = {Annu. Rev. Phys. Chem.}, journal = {Annu. Rev. Phys. Chem.},
number = {1}, number = {1},
@ -930,55 +1058,56 @@
volume = {70}, volume = {70},
year = {2019}, year = {2019},
bdsk-url-1 = {https://doi.org/10.1146/annurev-physchem-042018-052416}} bdsk-url-1 = {https://doi.org/10.1146/annurev-physchem-042018-052416}}
@article{Roos_1995, @article{Roos_1995a,
title = {Multiconfigurational Perturbation Theory with Level Shift \textemdash{} the {{Cr2}} Potential Revisited}, author = {Roos, Bj{\"o}rn O. and Andersson, Kerstin},
author = {Roos, Bj{\"o}rn O. and Andersson, Kerstin}, date-modified = {2023-01-20 09:54:01 +0100},
year = {1995}, doi = {10.1016/0009-2614(95)01010-7},
journal = {Chemical Physics Letters}, issn = {0009-2614},
volume = {245}, journal = {Chemical Physics Letters},
number = {2}, number = {2},
pages = {215--223}, pages = {215--223},
issn = {0009-2614}, title = {Multiconfigurational Perturbation Theory with Level Shift \textemdash{} the {{Cr2}} Potential Revisited},
doi = {10.1016/0009-2614(95)01010-7} volume = {245},
} year = {1995},
bdsk-url-1 = {https://doi.org/10.1016/0009-2614(95)01010-7}}
@article{Olsen_2000, @article{Olsen_2000,
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},
author = {Olsen, Jeppe and J{\o}rgensen, Poul and Helgaker, Trygve and Christiansen, Ove}, doi = {10.1063/1.481611},
year = {2000}, issn = {0021-9606},
journal = {The Journal of Chemical Physics}, journal = {The Journal of Chemical Physics},
volume = {112}, number = {22},
number = {22}, pages = {9736--9748},
pages = {9736--9748}, title = {Divergence in {{M\o ller}}\textendash{{Plesset}} Theory: {{A}} Simple Explanation Based on a Two-State Model},
issn = {0021-9606}, volume = {112},
doi = {10.1063/1.481611} year = {2000},
} bdsk-url-1 = {https://doi.org/10.1063/1.481611}}
@article{Choe_2001, @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},
author = {Choe, Yoong-Kee and Witek, Henryk A. and Finley, James P. and Hirao, Kimihiko}, copyright = {\textcopyright{} 2001 American Institute of Physics.},
year = {2001}, doi = {10.1063/1.1345510},
journal = {The Journal of Chemical Physics}, issn = {0021-9606},
volume = {114}, journal = {The Journal of Chemical Physics},
number = {9}, number = {9},
pages = {3913}, pages = {3913},
issn = {0021-9606}, title = {Identifying and Removing Intruder States in Multireference {{Mo}}/Ller\textendash{{Plesset}} Perturbation Theory},
doi = {10.1063/1.1345510}, volume = {114},
copyright = {\textcopyright{} 2001 American Institute of Physics.} year = {2001},
} bdsk-url-1 = {https://doi.org/10.1063/1.1345510}}
@article{Battaglia_2022, @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},
author = {Battaglia, Stefano and Frans{\'e}n, Lina and Fdez. Galv{\'a}n, Ignacio and Lindh, Roland}, doi = {10.1021/acs.jctc.2c00368},
year = {2022}, issn = {1549-9618},
journal = {Journal of Chemical Theory and Computation}, journal = {Journal of Chemical Theory and Computation},
volume = {18}, number = {8},
number = {8}, pages = {4814--4825},
pages = {4814--4825}, title = {Regularized {{CASPT2}}: An {{Intruder-State-Free Approach}}},
issn = {1549-9618}, volume = {18},
doi = {10.1021/acs.jctc.2c00368} year = {2022},
} bdsk-url-1 = {https://doi.org/10.1021/acs.jctc.2c00368}}
@article{Forsberg_1997, @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.}, 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.},
@ -4372,10 +4501,10 @@
bdsk-url-1 = {https://link.aps.org/doi/10.1103/PhysRevB.51.6868}, bdsk-url-1 = {https://link.aps.org/doi/10.1103/PhysRevB.51.6868},
bdsk-url-2 = {https://doi.org/10.1103/PhysRevB.51.6868}} bdsk-url-2 = {https://doi.org/10.1103/PhysRevB.51.6868}}
@article{Blase_2011, @article{Blase_2011a,
author = {Blase,X. and Attaccalite,C.}, author = {Blase,X. and Attaccalite,C.},
date-added = {2020-05-18 21:40:28 +0200}, date-added = {2020-05-18 21:40:28 +0200},
date-modified = {2020-05-21 08:47:22 +0200}, date-modified = {2023-01-20 09:20:43 +0100},
doi = {10.1063/1.3655352}, doi = {10.1063/1.3655352},
journal = {Appl. Phys. Lett.}, journal = {Appl. Phys. Lett.},
number = {17}, number = {17},
@ -4975,39 +5104,39 @@
year = {2010}, year = {2010},
bdsk-url-1 = {https://link.aps.org/doi/10.1103/PhysRevB.81.115105}, bdsk-url-1 = {https://link.aps.org/doi/10.1103/PhysRevB.81.115105},
bdsk-url-2 = {https://doi.org/10.1103/PhysRevB.81.115105}} bdsk-url-2 = {https://doi.org/10.1103/PhysRevB.81.115105}}
@article{Godby_1986, @article{Godby_1986,
title = {Accurate {{Exchange-Correlation Potential}} for {{Silicon}} and {{Its Discontinuity}} on {{Addition}} of an {{Electron}}}, author = {Godby, R. W. and Schl{\"u}ter, M. and Sham, L. J.},
author = {Godby, R. W. and Schl{\"u}ter, M. and Sham, L. J.}, doi = {10.1103/PhysRevLett.56.2415},
year = {1986}, journal = {Phys. Rev. Lett.},
journal = {Phys. Rev. Lett.}, number = {22},
volume = {56}, pages = {2415--2418},
number = {22}, title = {Accurate {{Exchange-Correlation Potential}} for {{Silicon}} and {{Its Discontinuity}} on {{Addition}} of an {{Electron}}},
pages = {2415--2418}, volume = {56},
doi = {10.1103/PhysRevLett.56.2415} year = {1986},
} bdsk-url-1 = {https://doi.org/10.1103/PhysRevLett.56.2415}}
@article{Godby_1987, @article{Godby_1987,
title = {Trends in Self-Energy Operators and Their Corresponding Exchange-Correlation Potentials}, author = {Godby, R. W. and Schl{\"u}ter, M. and Sham, L. J.},
author = {Godby, R. W. and Schl{\"u}ter, M. and Sham, L. J.}, doi = {10.1103/PhysRevB.36.6497},
year = {1987}, journal = {Phys. Rev. B},
journal = {Phys. Rev. B}, number = {12},
volume = {36}, pages = {6497--6500},
number = {12}, title = {Trends in Self-Energy Operators and Their Corresponding Exchange-Correlation Potentials},
pages = {6497--6500}, volume = {36},
doi = {10.1103/PhysRevB.36.6497} year = {1987},
} bdsk-url-1 = {https://doi.org/10.1103/PhysRevB.36.6497}}
@article{Godby_1987a, @article{Godby_1987a,
title = {Quasiparticle Energies in {{GaAs}} and {{AlAs}}}, author = {Godby, R. W. and Schl{\"u}ter, M. and Sham, L. J.},
author = {Godby, R. W. and Schl{\"u}ter, M. and Sham, L. J.}, doi = {10.1103/PhysRevB.35.4170},
year = {1987}, journal = {Phys. Rev. B},
journal = {Phys. Rev. B}, number = {8},
volume = {35}, pages = {4170--4171},
number = {8}, title = {Quasiparticle Energies in {{GaAs}} and {{AlAs}}},
pages = {4170--4171}, volume = {35},
doi = {10.1103/PhysRevB.35.4170} year = {1987},
} bdsk-url-1 = {https://doi.org/10.1103/PhysRevB.35.4170}}
@article{Godby_1988, @article{Godby_1988,
author = {Godby, R. W. and Schl\"uter, M. and Sham, L. J.}, author = {Godby, R. W. and Schl\"uter, M. and Sham, L. J.},
@ -6172,16 +6301,17 @@
bdsk-url-1 = {http://link.aps.org/doi/10.1103/PhysRevLett.45.290}, bdsk-url-1 = {http://link.aps.org/doi/10.1103/PhysRevLett.45.290},
bdsk-url-2 = {http://dx.doi.org/10.1103/PhysRevLett.45.290}} bdsk-url-2 = {http://dx.doi.org/10.1103/PhysRevLett.45.290}}
@article{Strinati_1982, @article{Strinati_1982a,
title = {Dynamical Aspects of Correlation Corrections in a Covalent Crystal}, author = {Strinati, G. and Mattausch, H. J. and Hanke, W.},
author = {Strinati, G. and Mattausch, H. J. and Hanke, W.}, date-modified = {2023-01-20 09:26:16 +0100},
year = {1982}, doi = {10.1103/PhysRevB.25.2867},
journal = {Physical Review B}, journal = {Physical Review B},
volume = {25}, number = {4},
number = {4}, pages = {2867--2888},
pages = {2867--2888}, title = {Dynamical Aspects of Correlation Corrections in a Covalent Crystal},
doi = {10.1103/PhysRevB.25.2867} volume = {25},
} year = {1982},
bdsk-url-1 = {https://doi.org/10.1103/PhysRevB.25.2867}}
@article{Strinati_1988, @article{Strinati_1988,
author = {Strinati, G.}, author = {Strinati, G.},
@ -6664,10 +6794,10 @@
title = {Electronic excitation energies of molecular systems from the Bethe-Salpeter equation: Example of the H$_2$ molecule}, title = {Electronic excitation energies of molecular systems from the Bethe-Salpeter equation: Example of the H$_2$ molecule},
year = {2013}} year = {2013}}
@article{Strinati_1982, @article{Strinati_1982b,
author = {G. Strinati}, author = {G. Strinati},
date-added = {2020-01-03 21:01:54 +0100}, date-added = {2020-01-03 21:01:54 +0100},
date-modified = {2020-01-03 21:02:37 +0100}, date-modified = {2023-01-20 09:26:19 +0100},
doi = {10.1103/PhysRevLett.49.1519}, doi = {10.1103/PhysRevLett.49.1519},
journal = {Phys. Rev. Lett.}, journal = {Phys. Rev. Lett.},
pages = {1519}, pages = {1519},
@ -15979,126 +16109,100 @@
title = {Valence {{Electron Photoemission Spectrum}} of {{Semiconductors}}: {{{\emph{Ab Initio}}}} {{Description}} of {{Multiple Satellites}}}, title = {Valence {{Electron Photoemission Spectrum}} of {{Semiconductors}}: {{{\emph{Ab Initio}}}} {{Description}} of {{Multiple Satellites}}},
volume = {107}, volume = {107},
year = {2011}, 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, @article{Ismail-Beigi_2017,
title = {Justifying Quasiparticle Self-Consistent Schemes via Gradient Optimization in {{Baym}}\textendash{{Kadanoff}} Theory}, author = {{Ismail-Beigi}, Sohrab},
author = {{Ismail-Beigi}, Sohrab}, doi = {10.1088/1361-648X/aa7803},
year = {2017}, issn = {0953-8984},
journal = {Journal of Physics: Condensed Matter}, journal = {Journal of Physics: Condensed Matter},
volume = {29}, number = {38},
number = {38}, pages = {385501},
pages = {385501}, title = {Justifying Quasiparticle Self-Consistent Schemes via Gradient Optimization in {{Baym}}\textendash{{Kadanoff}} Theory},
issn = {0953-8984}, volume = {29},
doi = {10.1088/1361-648X/aa7803} year = {2017},
} bdsk-url-1 = {https://doi.org/10.1088/1361-648X/aa7803}}
@article{Bruneval_2013, @article{Bruneval_2013,
title = {Benchmarking the {{Starting Points}} of the {{GW Approximation}} for {{Molecules}}}, author = {Bruneval, Fabien and Marques, Miguel A. L.},
author = {Bruneval, Fabien and Marques, Miguel A. L.}, doi = {10.1021/ct300835h},
year = {2013}, issn = {1549-9618},
journal = {Journal of Chemical Theory and Computation}, journal = {Journal of Chemical Theory and Computation},
volume = {9}, number = {1},
number = {1}, pages = {324--329},
pages = {324--329}, title = {Benchmarking the {{Starting Points}} of the {{GW Approximation}} for {{Molecules}}},
issn = {1549-9618}, volume = {9},
doi = {10.1021/ct300835h} year = {2013},
} bdsk-url-1 = {https://doi.org/10.1021/ct300835h}}
@article{Caruso_2016, @article{Blase_2011b,
title = {Benchmark of {{GW Approaches}} for the {{GW100 Test Set}}}, author = {Blase, X. and Attaccalite, C. and Olevano, V.},
author = {Caruso, Fabio and Dauth, Matthias and {van Setten}, Michiel J. and Rinke, Patrick}, date-modified = {2023-01-20 09:20:58 +0100},
year = {2016}, doi = {10.1103/PhysRevB.83.115103},
journal = {Journal of Chemical Theory and Computation}, journal = {Phys. Rev. B},
volume = {12}, number = {11},
number = {10}, pages = {115103},
pages = {5076--5087}, title = {First-Principles \$\textbackslash mathit\{\vphantom\}{{GW}}\vphantom\{\}\$ Calculations for Fullerenes, Porphyrins, Phtalocyanine, and Other Molecules of Interest for Organic Photovoltaic Applications},
issn = {1549-9618}, volume = {83},
doi = {10.1021/acs.jctc.6b00774} year = {2011},
} bdsk-url-1 = {https://doi.org/10.1103/PhysRevB.83.115103}}
@article{Blase_2011,
title = {First-Principles \$\textbackslash mathit\{\vphantom\}{{GW}}\vphantom\{\}\$ Calculations for Fullerenes, Porphyrins, Phtalocyanine, and Other Molecules of Interest for Organic Photovoltaic Applications},
author = {Blase, X. and Attaccalite, C. and Olevano, V.},
year = {2011},
journal = {Physical Review B},
volume = {83},
number = {11},
pages = {115103},
doi = {10.1103/PhysRevB.83.115103}
}
@article{Shishkin_2007, @article{Shishkin_2007,
title = {Self-Consistent \${{GW}}\$ Calculations for Semiconductors and Insulators}, author = {Shishkin, M. and Kresse, G.},
author = {Shishkin, M. and Kresse, G.}, doi = {10.1103/PhysRevB.75.235102},
year = {2007}, journal = {Physical Review B},
journal = {Physical Review B}, number = {23},
volume = {75}, pages = {235102},
number = {23}, title = {Self-Consistent \${{GW}}\$ Calculations for Semiconductors and Insulators},
pages = {235102}, volume = {75},
doi = {10.1103/PhysRevB.75.235102} year = {2007},
} bdsk-url-1 = {https://doi.org/10.1103/PhysRevB.75.235102}}
@article{Wilhelm_2016, @article{Wilhelm_2016,
title = {{{GW}} in the {{Gaussian}} and {{Plane Waves Scheme}} with {{Application}} to {{Linear Acenes}}}, author = {Wilhelm, Jan and Del Ben, Mauro and Hutter, J{\"u}rg},
author = {Wilhelm, Jan and Del Ben, Mauro and Hutter, J{\"u}rg}, doi = {10.1021/acs.jctc.6b00380},
year = {2016}, issn = {1549-9618},
journal = {Journal of Chemical Theory and Computation}, journal = {Journal of Chemical Theory and Computation},
volume = {12}, number = {8},
number = {8}, pages = {3623--3635},
pages = {3623--3635}, title = {{{GW}} in the {{Gaussian}} and {{Plane Waves Scheme}} with {{Application}} to {{Linear Acenes}}},
issn = {1549-9618}, volume = {12},
doi = {10.1021/acs.jctc.6b00380} year = {2016},
} bdsk-url-1 = {https://doi.org/10.1021/acs.jctc.6b00380}}
@article{Gallandi_2015, @article{Gallandi_2015,
title = {Long-{{Range Corrected DFT Meets GW}}: {{Vibrationally Resolved Photoelectron Spectra}} from {{First Principles}}}, author = {Gallandi, Lukas and K{\"o}rzd{\"o}rfer, Thomas},
author = {Gallandi, Lukas and K{\"o}rzd{\"o}rfer, Thomas}, doi = {10.1021/acs.jctc.5b00820},
year = {2015}, issn = {1549-9618},
journal = {Journal of Chemical Theory and Computation}, journal = {Journal of Chemical Theory and Computation},
volume = {11}, number = {11},
number = {11}, pages = {5391--5400},
pages = {5391--5400}, title = {Long-{{Range Corrected DFT Meets GW}}: {{Vibrationally Resolved Photoelectron Spectra}} from {{First Principles}}},
issn = {1549-9618}, volume = {11},
doi = {10.1021/acs.jctc.5b00820} year = {2015},
} bdsk-url-1 = {https://doi.org/10.1021/acs.jctc.5b00820}}
@article{Gallandi_2016,
title = {Accurate {{Ionization Potentials}} and {{Electron Affinities}} of {{Acceptor Molecules II}}: {{Non-Empirically Tuned Long-Range Corrected Hybrid Functionals}}},
author = {Gallandi, Lukas and Marom, Noa and Rinke, Patrick and K{\"o}rzd{\"o}rfer, Thomas},
year = {2016},
journal = {Journal of Chemical Theory and Computation},
volume = {12},
number = {2},
pages = {605--614},
issn = {1549-9618},
doi = {10.1021/acs.jctc.5b00873}
}
@article{Korzdorfer_2012, @article{Korzdorfer_2012,
title = {Strategy for Finding a Reliable Starting Point for \$\{\vphantom\}{{G}}\vphantom\{\}\_\{0\}\{\vphantom\}{{W}}\vphantom\{\}\_\{0\}\$ Demonstrated for Molecules}, author = {K{\"o}rzd{\"o}rfer, Thomas and Marom, Noa},
author = {K{\"o}rzd{\"o}rfer, Thomas and Marom, Noa}, doi = {10.1103/PhysRevB.86.041110},
year = {2012}, journal = {Physical Review B},
journal = {Physical Review B}, number = {4},
volume = {86}, pages = {041110},
number = {4}, title = {Strategy for Finding a Reliable Starting Point for \$\{\vphantom\}{{G}}\vphantom\{\}\_\{0\}\{\vphantom\}{{W}}\vphantom\{\}\_\{0\}\$ Demonstrated for Molecules},
pages = {041110}, volume = {86},
doi = {10.1103/PhysRevB.86.041110} year = {2012},
} bdsk-url-1 = {https://doi.org/10.1103/PhysRevB.86.041110}}
@article{Marom_2012, @article{Marom_2012,
title = {Benchmark of \${{GW}}\$ Methods for Azabenzenes}, author = {Marom, Noa and Caruso, Fabio and Ren, Xinguo and Hofmann, Oliver T. and K{\"o}rzd{\"o}rfer, Thomas and Chelikowsky, James R. and Rubio, Angel and Scheffler, Matthias and Rinke, Patrick},
author = {Marom, Noa and Caruso, Fabio and Ren, Xinguo and Hofmann, Oliver T. and K{\"o}rzd{\"o}rfer, Thomas and Chelikowsky, James R. and Rubio, Angel and Scheffler, Matthias and Rinke, Patrick}, doi = {10.1103/PhysRevB.86.245127},
year = {2012}, journal = {Physical Review B},
journal = {Physical Review B}, number = {24},
volume = {86}, pages = {245127},
number = {24}, title = {Benchmark of \${{GW}}\$ Methods for Azabenzenes},
pages = {245127}, volume = {86},
doi = {10.1103/PhysRevB.86.245127} year = {2012},
} bdsk-url-1 = {https://doi.org/10.1103/PhysRevB.86.245127}}
@article{vanSchilfgaarde_2006, @article{vanSchilfgaarde_2006,
author = {{van Schilfgaarde}, M. and Kotani, Takao and Faleev, S.}, author = {{van Schilfgaarde}, M. and Kotani, Takao and Faleev, S.},

150
Manuscript/SRGGW.tex
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@ -1,17 +1,52 @@
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@ -49,46 +84,49 @@ Here comes the abstract.
\label{sec:intro} \label{sec:intro}
%=================================================================% %=================================================================%
One-body Green's functions provide a natural and elegant way to access the charged excitation energies of a physical system. \cite{Martin_2016,Golze_2019} One-body Green's functions provide a natural and elegant way to access the charged excitation energies of a physical system. \cite{CsanakBook,FetterBook,Martin_2016,Golze_2019}
The 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} The non-linear Hedin's equations provide 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 Hedin's equations is out of reach and one must resort to approximations. Unfortunately, fully solving 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 \cite{Strinati_1980,Strinati_1982,Hybertsen_1985,Hybertsen_1986,Godby_1986,Godby_1987,Godby_1987a,Godby_1988,Blase_1995} 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} In particular, the $GW$ approximation, \cite{Hedin_1965,Aryasetiawan_1998,Onida_2002,Reining_2017,Golze_2019,Bruneval_2021} which has been first introduced in the context of solids \cite{Strinati_1980,Strinati_1982a,Strinati_1982b,Hybertsen_1985,Hybertsen_1986,Godby_1986,Godby_1987,Godby_1987a,Godby_1988,Blase_1995} and is now widely used for molecules as well, \cite{Rohlfing_1999a,Horst_1999,Puschnig_2002,Tiago_2003,Rocca_2010,Boulanger_2014,Jacquemin_2015a,Bruneval_2015,Jacquemin_2015b,Hirose_2015,Jacquemin_2017a,Jacquemin_2017b,Rangel_2017,Krause_2017,Gui_2018,Blase_2018,Liu_2020,Li_2017,Li_2019,Li_2020,Li_2021,Blase_2020,Holzer_2018a,Holzer_2018b,Loos_2020e,Loos_2021,McKeon_2022} provides fairly accurate charged excitation energies 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_2019,Duchemin_2020,Duchemin_2021}
The $GW$ method approximates the self-energy $\Sigma$ which relates the exact interacting Green's function $G$ to a non-interacting reference one $G_S$ through the Dyson equation The $GW$ method approximates the self-energy $\Sigma$ which relates the exact interacting Green's function $G$ to a non-interacting reference one $G_S$ through a Dyson equation
\begin{equation} \begin{equation}
\label{eq:dyson} \label{eq:dyson}
G = G_S + G_S\Sigma G. G = G_S + G_S\Sigma G.
\end{equation} \end{equation}
The self-energy encapsulates all the Hartree-exchange-correlation effects which are not taken into account in the reference system. The self-energy encapsulates all the Hartree-exchange-correlation effects which are not taken into account in the reference system.
%Throughout this manuscript the references are chosen to be the Hartree-Fock (HF) ones so that the self-energy only account for the missing correlation. %Throughout this manuscript the references are chosen to be the Hartree-Fock (HF) ones so that the self-energy only account for the missing correlation.
Approximating $\Sigma$ as the first order term of its perturbation expansion with respect to the screened interaction $W$ gives the so-called $GW$ approximation. \cite{Hedin_1965, Martin_2016} Approximating $\Sigma$ as the first-order term of its perturbation expansion with respect to the screened interaction $W$ yields 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} 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{SzaboBook,Ortiz_2013,Hirata_2015,Hirata_2017}
The GF(2) approximation is also known as the second Born approximation. \ant{ref ?} The GF(2) approximation \cite{Casida_1989,Casida_1991,Phillips_2014,Phillips_2015,Rusakov_2014,Rusakov_2016,Backhouse_2021,Backhouse_2020b,Backhouse_2020a,Pokhilko_2021a,Pokhilko_2021b,Pokhilko_2022} is also known as the second Born approximation in condensed matter physics. \cite{Stefanucci_2013}
Despite a wide range of successes, many-body perturbation theory is not flawless. Despite a wide range of successes, many-body perturbation theory is not flawless. \cite{vanSetten_2015,Maggio_2017,Duchemin_2020}
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 exhibit some discontinuities. \cite{Veril_2018,Loos_2018b} 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 exhibit some discontinuities. \cite{Veril_2018,Loos_2018b,Loos_2020e,Berger_2021,DiSabatino_2021}
Even more worrying these discontinuities can happen in the weakly correlated regime where $GW$ is thought to be valid. Even more worrying these discontinuities can happen in the weakly correlated regime where $GW$ is thought to be valid.
These discontinuities are due to a transfer of spectral weight between two solutions of the quasi-particle equation. \cite{Monino_2022} These discontinuities are due to a 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} This is another occurrence of the infamous intruder-state problem. \cite{Andersson_1994,Andersson_1995a,Roos_1995,Forsberg_1997,Olsen_2000,Choe_2001}
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 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 a recent study, Monino and Loos showed that these discontinuities could be removed by the introduction of a regularizer inspired by the similarity renormalization group (SRG) in the quasi-particle equation. \cite{Monino_2022} In a recent study, Monino and Loos showed that these discontinuities could be removed by the introduction of a regularizer inspired by the similarity renormalization 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. Encouraged by the recent successes of regularization schemes in many-body quantum chemistry methods, as in single- and multi-reference perturbation theory, \cite{Lee_2018a,Shee_2021,Evangelista_2014b,ChenyangLi_2019a,Battaglia_2022} 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. 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 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} This formalism has been introduced in quantum chemistry by White \cite{White_2002} before being explored in more detail by Evangelista and his co-workers in the context of multi-reference electron correlation theories. \cite{Evangelista_2014b,ChenyangLi_2015, ChenyangLi_2016,ChenyangLi_2017,ChenyangLi_2018,ChenyangLi_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 SRG has also been successful in the context of nuclear structure theory, where it was first developed as a mature computational tool thanks to the work of several research groups.
\cite{Bogner_2007,Tsukiyama_2011,Tsukiyama_2012,Hergert_2013,Hergert_2016,Frosini_2022a,Frosini_2022b,Frosini_2022c}
See Ref.~\onlinecite{Hergert_2016a} for a recent review in this field.
The SRG transformation aims at decoupling a reference space from an external space while folding information about the coupling in the reference space. The SRG transformation aims at decoupling an internal (or reference) space from an external space while incorporating information about their coupling in the reference space.
This is often during such decoupling that intruder states appear. \ant{ref} This process can often result in the appearance of intruder states. \cite{Evangelista_2014b,ChenyangLi_2019a}
However, SRG is particularly well-suited to avoid them because the decoupling of each external configuration is inversely proportional to its energy difference with the reference space. However, SRG is particularly well-suited to avoid them because the decoupling of each external configuration is inversely proportional to its energy difference with the reference space.
Because intruder states have energies really close to the reference energies they will be the last ones decoupled. By definition, intruder states have energies that are close to the reference energy, and therefore are the last to be decoupled.
Therefore the SRG continuous transformation can be stopped once every external configurations except the intruder ones have been decoupled. By stopping the SRG transformation once all external configurations except the intruder states have been decoupled,
This provides a way to fold in information about the coupling in the reference space while avoiding intruder states. correlation effects between the internal and external spaces can be incorporated (or folded) without the presence of intruder 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. The goal of this manuscript is to determine if the SRG formalism can effectively address the issue of intruder states in many-body perturbation theory, as it has in other areas of electronic and nuclear structure theory.
We begin by reviewing the $GW$ formalism in Sec.~\ref{sec:gw} and then briefly review the SRG formalism in Sec.~\ref{sec:srg}. The manuscript is organized as follows.
Section~\ref{sec:theoretical} is concluded by a perturbative analysis of the SRG formalism applied to $GW$ (see Sec.~\ref{sec:srggw}). We begin by reviewing the $GW$ approximation in Sec.~\ref{sec:gw} and then briefly review the SRG formalism in Sec.~\ref{sec:srg}.
The computational details of our implementation are provided in Sec.~\ref{sec:comp_det} before turning to the results section. Section~\ref{sec:theoretical} is concluded by a perturbative analysis of SRG applied to $GW$ (see Sec.~\ref{sec:srggw}).
The computational details are provided in Sec.~\ref{sec:comp_det} before turning to the results section.
This section starts by This section starts by
%=================================================================% %=================================================================%
@ -107,12 +145,12 @@ The central equation of many-body perturbation theory based on Hedin's equations
\left[ \bF + \bSig(\omega = \epsilon_p) \right] \psi_p = \epsilon_p \psi_p, \left[ \bF + \bSig(\omega = \epsilon_p) \right] \psi_p = \epsilon_p \psi_p,
\end{equation} \end{equation}
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. 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 represented by $\bF$. The self-energy can be physically understood as a dynamical \titou{screening} correction to the Hartree-Fock (HF) problem represented by $\bF$.
Similarly to the HF case, this equation needs to be solved self-consistently. Similarly to the HF case, this equation needs to be solved self-consistently.
Note that $\bSig$ is dynamical, \ie it depends on both the eigenvalues $\epsilon_p$ and eigenvectors $\psi_p$ while $\bF$ depends only on the eigenvectors. Note that $\bSig$ is dynamical, \ie it depends on both the eigenvalues $\epsilon_p$ and eigenvectors $\psi_p$ while $\bF$ depends only on the eigenvectors.
Because of this $\omega$ dependence, fully solving this equation is a rather complicated task, hence several approximate solving schemes has been developed. Because of this $\omega$ 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. 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.~\eqref{eq:quasipart_eq} are neglected and the self-consistency is abandoned.
In this case, there are $K$ quasi-particle equations that read In this case, there are $K$ quasi-particle equations that read
tr\begin{equation} tr\begin{equation}
\label{eq:G0W0} \label{eq:G0W0}
@ -126,18 +164,18 @@ These solutions can be characterized by their spectral weight defined as the ren
0 \leq Z_{p,s} = \left[ 1 - \pdv{\Sigma_{p}(\omega)}{\omega}\bigg|_{\omega=\epsilon_{p,s}} \right]^{-1} \leq 1. 0 \leq Z_{p,s} = \left[ 1 - \pdv{\Sigma_{p}(\omega)}{\omega}\bigg|_{\omega=\epsilon_{p,s}} \right]^{-1} \leq 1.
\end{equation} \end{equation}
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. 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. However, in some cases, Eq.~\eqref{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 states. 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 states.
One obvious flaw of the one-shot scheme mentioned above is its starting point dependence. 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. Indeed, in Eq.~\eqref{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 optimize the starting point to obtain the best one-shot energies possible. \cite{Korzdorfer_2012,Marom_2012,Bruneval_2013,Gallandi_2015,Caruso_2016, Gallandi_2016} Therefore, one could try to optimize the starting point to obtain the best one-shot energies possible. \cite{Korzdorfer_2012,Marom_2012,Bruneval_2013,Gallandi_2015,Caruso_2016, Gallandi_2016}
Alternatively, one could solve this equation self-consistently leading to the eigenvalue-only self-consistent scheme. \cite{Shishkin_2007,Blase_2011,Marom_2012,Kaplan_2016,Wilhelm_2016} Alternatively, one could solve this equation self-consistently leading to the eigenvalue-only self-consistent scheme. \cite{Shishkin_2007,Blase_2011b,Marom_2012,Kaplan_2016,Wilhelm_2016}
To do so the energy of the quasi-particle solution of the previous iteration is used to build Eq.~(\ref{eq:G0W0}) and then this equation is solved for $\omega$ again until convergence is reached. To do so the energy of the quasi-particle solution of the previous iteration is used to build Eq.~\eqref{eq:G0W0} and then this equation is solved 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. 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. \cite{Marom_2012} 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. \cite{Marom_2012}
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 update both the energies and the molecular orbitals, one needs to take into account the off-diagonal elements in Eq.~\eqref{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 $\bSig^{\qs}$. 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 $\bF + \bSig^{\qs}$. The algorithm to solve the qs problem is totally analog to the HF case with $\bF$ replaced by $\bF + \bSig^{\qs}$.
Various choices for $\bSig^\qs$ are possible but the most popular one is the following Hermitian approximation Various choices for $\bSig^\qs$ are possible but the most popular one is the following Hermitian approximation
@ -161,8 +199,8 @@ But it would be more rigorous, and more instructive, to obtain this regularizer
This is the aim of this work. 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. 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. However, to do so one needs to identify the coupling terms in Eq.~\eqref{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. The way around this problem is to transform Eq.~\eqref{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 $GW$ in the Tamm-Dancoff approximation (TDA) case but the same derivation could be done for the non-TDA $GW$ and GF(2) self-energies. From now on, we will restrict ourselves to the $GW$ in the Tamm-Dancoff approximation (TDA) case but the same derivation could be done for the non-TDA $GW$ and GF(2) self-energies.
The corresponding formula are given in Appendix~\ref{sec:nonTDA} and \ref{sec:GF2}, respectively. The corresponding formula are given in Appendix~\ref{sec:nonTDA} and \ref{sec:GF2}, respectively.
The upfolded $GW$ quasi-particle equation is the following The upfolded $GW$ quasi-particle equation is the following
@ -209,7 +247,7 @@ The usual $GW$ non-linear equation
\label{eq:GWnonlin} \label{eq:GWnonlin}
\left( \bF + \bSig(\omega) \right) \bX = \omega \bX, \left( \bF + \bSig(\omega) \right) \bX = \omega \bX,
\end{equation} \end{equation}
can be obtained by applying L\"odwin partitioning technique to Eq.~(\ref{eq:GWlin}) \cite{Lowdin_1963,Bintrim_2021} which gives the following the expression for the self-energy can be obtained by applying L\"odwin partitioning technique to Eq.~\eqref{eq:GWlin} \cite{Lowdin_1963,Bintrim_2021} which gives the following the expression for the self-energy
\begin{align} \begin{align}
\bSig(\omega) &= \bV^{\hhp} \left(\omega \mathbb{1} - \bC^{\hhp}\right)^{-1} (\bV^{\hhp})^{\mathsf{T}} \\ \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 &+ \bV^{\pph} \left(\omega \mathbb{1} - \bC^{\pph})^{-1} (\bV^{\pph}\right)^{\mathsf{T}}, \notag
@ -234,9 +272,9 @@ with
\begin{equation} \begin{equation}
A^\dRPA_{ij,ab} = (\epsilon_i - \epsilon_a) \delta_{ij}\delta_{ab} + \eri{ib}{aj}. A^\dRPA_{ij,ab} = (\epsilon_i - \epsilon_a) \delta_{ij}\delta_{ab} + \eri{ib}{aj}.
\end{equation} \end{equation}
$\boldsymbol{\Omega}$ is the diagonal matrix of eigenvalues and its elements $\Omega_v$ appear in Eq.~(\ref{eq:GW_selfenergy}). $\boldsymbol{\Omega}$ is the diagonal matrix of eigenvalues and its elements $\Omega_v$ appear in Eq.~\eqref{eq:GW_selfenergy}.
Equations~(\ref{eq:GWlin}) and~(\ref{eq:GWnonlin}) have exactly the same solutions but one is linear and the other not. Equations \eqref{eq:GWlin} and \eqref{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 $\order{K^3}$ while it is $\order{K}$ in the latter one. The price to pay for this linearity is that the size of the matrix in the former equation is $\order{K^3}$ while it is $\order{K}$ in the latter one.
We refer to Ref.~\onlinecite{Bintrim_2021} for a detailed discussion of the up/downfolding processes of the $GW$ equations (see also Chapter 8 of Ref.~\onlinecite{Schirmer_2018} for the GF(2) case). We refer to Ref.~\onlinecite{Bintrim_2021} for a detailed discussion of the up/downfolding processes of the $GW$ equations (see also Chapter 8 of Ref.~\onlinecite{Schirmer_2018} for the GF(2) case).
@ -255,7 +293,7 @@ Therefore, the transformed Hamiltonian
\bH(s) = \bU(s) \, \bH \, \bU^\dag(s), \bH(s) = \bU(s) \, \bH \, \bU^\dag(s),
\end{equation} \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. 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}) with respect to $s$. An evolution equation for $\bH(s)$ can be easily obtained by deriving Eq.~\eqref{eq:SRG_Ham} with respect to $s$.
This gives the flow equation This gives the flow equation
\begin{equation} \begin{equation}
\label{eq:flowEquation} \label{eq:flowEquation}
@ -302,7 +340,7 @@ For finite values of $s$, we have the following perturbation expansion of the Ha
\bH(s) = \bH^{(0)}(s) + \lambda ~ \bH^{(1)}(s) + \lambda^2 \bH^{(2)}(s) + \dots \bH(s) = \bH^{(0)}(s) + \lambda ~ \bH^{(1)}(s) + \lambda^2 \bH^{(2)}(s) + \dots
\end{equation} \end{equation}
Hence, the generator $\boldsymbol{\eta}(s)$ admits a perturbation expansion as well. 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. Then, one can collect order by order the terms in Eq.~\eqref{eq:flowEquation} and solve analytically the low-order differential equations.
%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%
\subsection{Renormalized GW} \subsection{Renormalized GW}
@ -329,7 +367,7 @@ As hinted at the end of section~\ref{sec:gw}, the diagonal and off-diagonal part
\end{pmatrix} \end{pmatrix}
\end{align} \end{align}
where we have omitted the $s$ dependence of the matrix elements for the sake of brevity. where we have omitted the $s$ dependence of the matrix elements for the sake of brevity.
Then, the aim is to solve order by order the flow equation [see Eq.~(\ref{eq:flowEquation})] knowing that the initial conditions are Then, the aim is to solve order by order the flow equation [see Eq.~\eqref{eq:flowEquation}] knowing that the initial conditions are
\begin{align} \begin{align}
\bHd{0}(0) &= \begin{pmatrix} \bHd{0}(0) &= \begin{pmatrix}
\bF{}{} & \bO \\ \bF{}{} & \bO \\
@ -349,11 +387,11 @@ Then, the aim is to solve order by order the flow equation [see Eq.~(\ref{eq:flo
\end{pmatrix} \notag \end{pmatrix} \notag
\end{align} \end{align}
where we have defined the matrices $\bC$ and $\bV$ that collects the 2h1p and 2p1h channels for the sake of conciseness. where we have defined the matrices $\bC$ and $\bV$ that collects the 2h1p and 2p1h channels for the sake of conciseness.
Once the analytical low-order perturbative expansions are known they can be inserted in Eq.~(\ref{eq:GWlin}) before downfolding to obtain a renormalized quasi-particle equation. Once the analytical low-order perturbative expansions are known they can be inserted in Eq.~\eqref{eq:GWlin} before downfolding to obtain a renormalized quasi-particle equation.
In particular, in this manuscript the focus will be on the second-order renormalized 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} \subsubsection{Zeroth-order matrix elements}
%///////////////////////////% %///////////////////////////%
There is only one zeroth order term in the right-hand side of the flow equation There is only one zeroth order term in the right-hand side of the flow equation
@ -383,7 +421,7 @@ The matrix elements of $\bU$ and $\bD^{(0)}$ are
D_{(p,v),(q,w)}^{(0)} &= \left(\epsilon_p + \text{sign}(\epsilon_p-\epsilon_F)\Omega_v\right)\delta_{pq}\delta_{vw} D_{(p,v),(q,w)}^{(0)} &= \left(\epsilon_p + \text{sign}(\epsilon_p-\epsilon_F)\Omega_v\right)\delta_{pq}\delta_{vw}
\end{align} \end{align}
where $\epsilon_F$ is the Fermi level. where $\epsilon_F$ is the Fermi level.
Note that the matrix $\bU$ is also used in the downfolding process of Eq.~(\ref{eq:GWlin}). \cite{Bintrim_2021} Note that the matrix $\bU$ is also used in the downfolding process of Eq.~\eqref{eq:GWlin}. \cite{Bintrim_2021}
Thanks to the diagonal structure of $\bF^{(0)}$ and $\bD^{(0)}$, Eq.~\eqref{eq:eqdiffW0} can be easily solved and give Thanks to the diagonal structure of $\bF^{(0)}$ and $\bD^{(0)}$, Eq.~\eqref{eq:eqdiffW0} can be easily solved and give
\begin{equation} \begin{equation}
@ -397,7 +435,7 @@ Therefore, the zeroth order Hamiltonian is
\ie it is independent of $s$. \ie it is independent of $s$.
%///////////////////////////% %///////////////////////////%
\subsubsection{First order matrix elements} \subsubsection{First-order matrix elements}
%///////////////////////////% %///////////////////////////%
Knowing that $\bHod{0}(s)=\bO$, the first order flow equation is Knowing that $\bHod{0}(s)=\bO$, the first order flow equation is
@ -415,7 +453,7 @@ and the first order coupling elements are given by (up to a multiplication by $\
W_{p,(q,v)}^{(1)}(s) &= W_{p,(q,v)}^{(1)}(0) e^{- (F_{pp}^{(0)} - D_{(q,v),(q,v)}^{(0)})^2 s} \\ W_{p,(q,v)}^{(1)}(s) &= W_{p,(q,v)}^{(1)}(0) e^{- (F_{pp}^{(0)} - D_{(q,v),(q,v)}^{(0)})^2 s} \\
&= W_{p,(q,v)}^{(1)}(0) e^{- (\epsilon_p - \epsilon_q - \text{sign}(\epsilon_q-\epsilon_F)\Omega_v)^2 s} \notag &= W_{p,(q,v)}^{(1)}(0) e^{- (\epsilon_p - \epsilon_q - \text{sign}(\epsilon_q-\epsilon_F)\Omega_v)^2 s} \notag
\end{align} \end{align}
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}) while for $s\to\infty$ they go to zero. At $s=0$ the elements $W_{p,(q,v)}^{(1)}(0)$ are equal to the two-electron screened integrals defined in Eq.~\eqref{eq:GW_sERI} while for $s\to\infty$ they go to zero.
Therefore, $W_{p,(q,v)}^{(1)}(s)$ are renormalized two-electrons screened integrals. 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}). 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}).
@ -454,9 +492,9 @@ At $s=0$, this second-order correction is null while for $s\to\infty$ it tends t
\end{equation} \end{equation}
Note that in the $s\to\infty$ limit the dynamic part of the self-energy [see Eq.~\eqref{eq:srg_sigma}] tends to zero. Note that in the $s\to\infty$ limit the dynamic part of the self-energy [see Eq.~\eqref{eq:srg_sigma}] tends to zero.
Therefore, the SRG flow transforms the dynamic part of $\bSig(\omega)$ into a static correction. Therefore, the SRG flow transforms the dynamic part of $\bSig(\omega)$ into a static correction.
This transformation is done gradually starting from the states that have the largest denominators in Eq.~(\ref{eq:static_F2}). This transformation is done gradually starting from the states that have the largest denominators in Eq.~\eqref{eq:static_F2}.
Interestingly, the static limit, \ie $s\to\infty$ limit, of Eq.~(\ref{eq:GW_renorm}) defines an alternative qs$GW$ approximation to the one defined by Eq.~(\ref{eq:sym_qsgw}) which matrix elements read as Interestingly, the static limit, \ie $s\to\infty$ limit, of Eq.~\eqref{eq:GW_renorm} defines an alternative qs$GW$ approximation to the one defined by Eq.~\eqref{eq:sym_qsgw} which matrix elements read as
\begin{equation} \begin{equation}
\label{eq:static_F2} \label{eq:static_F2}
\Sigma_{pq}^{\text{qs}GW}(\eta) = \sum_{r,v} \left( \frac{\Delta_{prv}}{\Delta_{prv}^2 + \eta^2} +\frac{\Delta_{qrv}}{\Delta_{qrv}^2 + \eta^2} \right) W_{p,(r,v)} W^{\dagger}_{(r,v),q}. \Sigma_{pq}^{\text{qs}GW}(\eta) = \sum_{r,v} \left( \frac{\Delta_{prv}}{\Delta_{prv}^2 + \eta^2} +\frac{\Delta_{qrv}}{\Delta_{qrv}^2 + \eta^2} \right) W_{p,(r,v)} W^{\dagger}_{(r,v),q}.
@ -529,7 +567,7 @@ The data that supports the findings of this study are available within the artic
\label{sec:nonTDA} \label{sec:nonTDA}
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The $GW$ self-energy without TDA is the same as in Eq.~(\ref{eq:GW_selfenergy}) but the screened integrals are now defined as The $GW$ self-energy without TDA is the same as in Eq.~\eqref{eq:GW_selfenergy} but the screened integrals are now defined as
\begin{equation} \begin{equation}
\label{eq:GWnonTDA_sERI} \label{eq:GWnonTDA_sERI}
W_{p,(q,v)} = \sum_{ia}\eri{pi}{qa}\qty( \bX_{v} + \bY_{v})_{ia}, W_{p,(q,v)} = \sum_{ia}\eri{pi}{qa}\qty( \bX_{v} + \bY_{v})_{ia},
@ -581,7 +619,7 @@ However, because we will eventually downfold again the upfolded matrix, we can u
\cdot \cdot
\boldsymbol{\epsilon}, \boldsymbol{\epsilon},
\end{equation} \end{equation}
which already depends on the screened integrals and therefore require the knowledge of the eigenvectors of the dRPA problem defined in Eq.~(\ref{eq:full_dRPA}). which already depends on the screened integrals and therefore require the knowledge of the eigenvectors of the dRPA problem defined in Eq.~\eqref{eq:full_dRPA}.
where $\boldsymbol{\epsilon}$ is a diagonal matrix collecting the quasi-particle and satellite energies, the 2h1p and 2p1h matrix elements are where $\boldsymbol{\epsilon}$ is a diagonal matrix collecting the quasi-particle and satellite energies, the 2h1p and 2p1h matrix elements are
@ -598,7 +636,7 @@ and the corresponding coupling blocks read
W^\text{2p1h}_{p,kcd} & = \sum_{ia}\eri{pi}{ca} \qty( \bX_{kd} + \bY_{kd})_{ia} W^\text{2p1h}_{p,kcd} & = \sum_{ia}\eri{pi}{ca} \qty( \bX_{kd} + \bY_{kd})_{ia}
\end{align} \end{align}
Using the SRG on this matrix instead of Eq.~(\ref{eq:GWlin}) gives the same expression for $\bW^{(1)}$, $\bF^{(2)}$ and $\bSig^{\text{SRG}}$ but now the screened integrals are the one of Eq.~\eqref{eq:GWnonTDA_sERI} and the eigenvalues $\Omega$ and eigenvectors $\bX$ and $\bY$ are the ones of the full RPA problem defined in Eq.~\eqref{eq:full_dRPA}. Using the SRG on this matrix instead of Eq.~\eqref{eq:GWlin} gives the same expression for $\bW^{(1)}$, $\bF^{(2)}$ and $\bSig^{\text{SRG}}$ but now the screened integrals are the one of Eq.~\eqref{eq:GWnonTDA_sERI} and the eigenvalues $\Omega$ and eigenvectors $\bX$ and $\bY$ are the ones of the full RPA problem defined in Eq.~\eqref{eq:full_dRPA}.
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\section{GF(2) equations} \section{GF(2) equations}