From 99ab17e44e43b97d80656d13caa61f106113f77d Mon Sep 17 00:00:00 2001
From: Pierre-Francois Loos <pierrefrancois.loos@gmail.com>
Date: Tue, 28 May 2019 13:53:05 +0200
Subject: [PATCH] Theory

---
 Manuscript/Ex-srDFT.bib | 402 ++++++++++++++++++++++++++++++++++++++--
 Manuscript/Ex-srDFT.tex | 146 ++++++++-------
 2 files changed, 471 insertions(+), 77 deletions(-)

diff --git a/Manuscript/Ex-srDFT.bib b/Manuscript/Ex-srDFT.bib
index 897fa4f..aeb208d 100644
--- a/Manuscript/Ex-srDFT.bib
+++ b/Manuscript/Ex-srDFT.bib
@@ -1,13 +1,391 @@
 %% This BibTeX bibliography file was created using BibDesk.
 %% http://bibdesk.sourceforge.net/
 
-%% Created for Pierre-Francois Loos at 2019-05-25 17:51:22 +0200 
+%% Created for Pierre-Francois Loos at 2019-05-28 13:36:42 +0200 
 
 
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+@article{BarDelPerMat-JMS-97,
+	Author = {Rodney J. Bartlett and Janet E. Del Bene and S.Ajith Perera and Renee-Peloquin Mattie},
+	Date-Added = {2019-05-28 13:34:13 +0200},
+	Date-Modified = {2019-05-28 13:35:03 +0200},
+	Doi = {https://doi.org/10.1016/S0166-1280(97)90277-3},
+	Issn = {0166-1280},
+	Journal = {J. Mol. Struct. (THEOCHEM)},
+	Keywords = {Ammonia, Spectra, Heat of formation, Properties, Correlation effects},
+	Pages = {157--168},
+	Title = {Ammonia: The Prototypical Lone Pair Molecule},
+	Url = {http://www.sciencedirect.com/science/article/pii/S0166128097902773},
+	Volume = {400},
+	Year = {1997},
+	Bdsk-Url-1 = {http://www.sciencedirect.com/science/article/pii/S0166128097902773},
+	Bdsk-Url-2 = {https://doi.org/10.1016/S0166-1280(97)90277-3}}
+
+@article{SchGoe-JCTC-17,
+	Author = {Schwabe, Tobias and Goerigk, Lars},
+	Date-Added = {2019-05-28 13:33:10 +0200},
+	Date-Modified = {2019-05-28 13:33:22 +0200},
+	Doi = {10.1021/acs.jctc.7b00386},
+	Eprint = {https://doi.org/10.1021/acs.jctc.7b00386},
+	Journal = {J. Chem. Theory Comput.},
+	Number = {9},
+	Pages = {4307--4323},
+	Title = {Time-Dependent Double-Hybrid Density Functionals with Spin-Component and Spin-Opposite Scaling},
+	Url = {https://doi.org/10.1021/acs.jctc.7b00386},
+	Volume = {13},
+	Year = {2017},
+	Bdsk-Url-1 = {https://doi.org/10.1021/acs.jctc.7b00386}}
+
+@article{RubSerMer-JCP-08,
+	Author = {Mercedes Rubio and Luis Serrano-Andr{\'e}s and Manuela Merch{\'a}n},
+	Date-Added = {2019-05-28 12:21:04 +0200},
+	Date-Modified = {2019-05-28 12:21:20 +0200},
+	Doi = {10.1063/1.2837827},
+	Eprint = {https://doi.org/10.1063/1.2837827},
+	Journal = {J. Chem. Phys.},
+	Number = {10},
+	Pages = {104305},
+	Title = {Excited States of the Water Molecule: Analysis of the Valence and Rydberg Character},
+	Url = {https://doi.org/10.1063/1.2837827},
+	Volume = {128},
+	Year = {2008},
+	Bdsk-Url-1 = {https://doi.org/10.1063/1.2837827}}
+
+@article{LiPal-JCP-11,
+	Author = {Xiangzhu Li and Josef Paldus},
+	Date-Added = {2019-05-28 12:20:10 +0200},
+	Date-Modified = {2019-05-28 12:20:19 +0200},
+	Doi = {10.1063/1.3595513},
+	Eprint = {https://doi.org/10.1063/1.3595513},
+	Journal = {J. Chem. Phys.},
+	Number = {21},
+	Pages = {214118},
+	Title = {Multi-Reference State-Universal Coupled-Cluster Approaches to Electronically Excited States},
+	Url = {https://doi.org/10.1063/1.3595513},
+	Volume = {134},
+	Year = {2011},
+	Bdsk-Url-1 = {https://doi.org/10.1063/1.3595513}}
+
+@article{CaiTozRei-JCP-00,
+	Author = {Zheng-Li Cai and David J. Tozer and Jeffrey R. Reimers},
+	Date-Added = {2019-05-28 12:19:14 +0200},
+	Date-Modified = {2019-05-28 12:19:44 +0200},
+	Doi = {10.1063/1.1312826},
+	Eprint = {https://doi.org/10.1063/1.1312826},
+	Journal = {J. Chem. Phys.},
+	Number = {17},
+	Pages = {7084--7096},
+	Title = {Time-Dependent Density-Functional Determination of Arbitrary Singlet and Triplet Excited-State Potential Energy Surfaces: Application to the Water Molecule},
+	Url = {https://doi.org/10.1063/1.1312826},
+	Volume = {113},
+	Year = {2000},
+	Bdsk-Url-1 = {https://doi.org/10.1063/1.1312826}}
+
+@article{PurZhaKra-JCP-09,
+	Author = {Purwanto, Wirawan and Zhang, Shiwei and Krakauer, Henry},
+	Date-Added = {2019-05-28 12:00:39 +0200},
+	Date-Modified = {2019-05-28 12:00:56 +0200},
+	Doi = {10.1063/1.3077920},
+	File = {/Users/loos/Zotero/storage/JDM6C32K/Purwanto et al. - 2009 - Excited state calculations using phaseless auxilia.pdf},
+	Issn = {0021-9606, 1089-7690},
+	Journal = {J. Chem. Phys.},
+	Language = {en},
+	Month = mar,
+	Number = {9},
+	Pages = {094107},
+	Shorttitle = {Excited State Calculations Using Phaseless Auxiliary-Field Quantum {{Monte Carlo}}},
+	Title = {Excited State Calculations Using Phaseless Auxiliary-Field Quantum {{Monte Carlo}}: {{Potential}} Energy Curves of Low-Lying {{C2}} Singlet States},
+	Volume = {130},
+	Year = {2009},
+	Bdsk-Url-1 = {https://doi.org/10.1063/1.3077920}}
+
+@article{Var-JCP-08,
+	Author = {Varandas, A. J. C.},
+	Date-Added = {2019-05-28 11:58:55 +0200},
+	Date-Modified = {2019-05-28 11:59:09 +0200},
+	Doi = {10.1063/1.3036115},
+	Issn = {0021-9606, 1089-7690},
+	Journal = {J. Chem. Phys.},
+	Language = {en},
+	Month = dec,
+	Number = {23},
+	Pages = {234103},
+	Shorttitle = {Extrapolation to the Complete-Basis-Set Limit and the Implications of Avoided Crossings},
+	Title = {Extrapolation to the Complete-Basis-Set Limit and the Implications of Avoided Crossings: {{The X $\Sigma$1g}}+, {{B $\Delta$1g}}, and {{B}}${'}$ {{$\Sigma$1g}}+ States of {{C2}}},
+	Volume = {129},
+	Year = {2008},
+	Bdsk-Url-1 = {https://doi.org/10.1063/1.3036115}}
+
+@article{VarRoc-PTRSMPES-18,
+	Author = {Varandas, A. J. C. and Rocha, C. M. R.},
+	Date-Added = {2019-05-28 11:58:55 +0200},
+	Date-Modified = {2019-05-28 11:59:35 +0200},
+	Doi = {10.1098/rsta.2017.0145},
+	File = {/Users/loos/Zotero/storage/VP3T2AAG/Varandas and Rocha - 2018 - iCi sub ini sub ( ini =2−4) c.pdf},
+	Issn = {1364-503X, 1471-2962},
+	Journal = {Philos. Trans. R. Soc. Math. Phys. Eng. Sci.},
+	Language = {en},
+	Month = mar,
+	Number = {2115},
+	Pages = {20170145},
+	Shorttitle = {{\emph{C}} {\textsubscript{ {\emph{n}} }} ( {\emph{n}} =2-4)},
+	Title = {{\emph{C}} {\textsubscript{ {\emph{n}} }} ( {\emph{n}} =2-4): Current Status},
+	Volume = {376},
+	Year = {2018},
+	Bdsk-Url-1 = {https://doi.org/10.1098/rsta.2017.0145}}
+
+@article{SokCha-JCP-16,
+	Author = {Sokolov, Alexander Yu. and Chan, Garnet Kin-Lic},
+	Date-Added = {2019-05-28 11:57:29 +0200},
+	Date-Modified = {2019-05-28 11:58:02 +0200},
+	Doi = {10.1063/1.4941606},
+	Issn = {0021-9606, 1089-7690},
+	Journal = {J. Chem. Phys.},
+	Language = {en},
+	Month = feb,
+	Number = {6},
+	Pages = {064102},
+	Title = {A Time-Dependent Formulation of Multi-Reference Perturbation Theory},
+	Volume = {144},
+	Year = {2016},
+	Bdsk-Url-1 = {https://doi.org/10.1063/1.4941606}}
+
+@article{Sha-JCP-15,
+	Author = {Sharma, Sandeep},
+	Date-Added = {2019-05-28 11:56:29 +0200},
+	Date-Modified = {2019-05-28 11:56:44 +0200},
+	Doi = {10.1063/1.4905237},
+	Issn = {0021-9606, 1089-7690},
+	Journal = {J. Chem. Phys.},
+	Language = {en},
+	Month = jan,
+	Number = {2},
+	Pages = {024107},
+	Title = {A General Non-{{Abelian}} Density Matrix Renormalization Group Algorithm with Application to the {{C}} {\textsubscript{2}} Dimer},
+	Volume = {142},
+	Year = {2015},
+	Bdsk-Url-1 = {https://doi.org/10.1063/1.4905237}}
+
+@article{AbrShe-JCP-04,
+	Author = {Abrams, Micah L. and Sherrill, C. David},
+	Date-Added = {2019-05-28 11:54:44 +0200},
+	Date-Modified = {2019-05-28 11:55:26 +0200},
+	Doi = {10.1063/1.1804498},
+	Issn = {0021-9606, 1089-7690},
+	Journal = {J. Chem. Phys.},
+	Language = {en},
+	Month = nov,
+	Number = {19},
+	Pages = {9211-9219},
+	Shorttitle = {Full Configuration Interaction Potential Energy Curves for the {{X 1$\Sigma$g}}+, {{B 1$\Delta$g}}, and {{B}}${'}$ {{1$\Sigma$g}}+ States of {{C2}}},
+	Title = {Full Configuration Interaction Potential Energy Curves for the {{X 1$\Sigma_g^+$}}, {{B 1$\Delta_g$}}, and {{B}}${'}$ {{1$\Sigma_g^+$}} States of {{C$_2$}}: {{A}} Challenge for Approximate Methods},
+	Volume = {121},
+	Year = {2004},
+	Bdsk-Url-1 = {https://doi.org/10.1063/1.1804498}}
+
+@article{AbrShe-CPL-05,
+	Author = {Abrams, Micah L. and Sherrill, C. David},
+	Date-Added = {2019-05-28 11:54:44 +0200},
+	Date-Modified = {2019-05-28 11:55:38 +0200},
+	Doi = {10.1016/j.cplett.2005.06.107},
+	Issn = {0009-2614},
+	Journal = {Chem. Phys. Lett.},
+	Month = {Aug},
+	Number = {1-3},
+	Pages = {121--124},
+	Publisher = {Elsevier BV},
+	Title = {Important configurations in configuration interaction and coupled-cluster wave functions},
+	Url = {http://dx.doi.org/10.1016/j.cplett.2005.06.107},
+	Volume = {412},
+	Year = {2005},
+	Bdsk-Url-1 = {http://dx.doi.org/10.1016/j.cplett.2005.06.107}}
+
+@article{AngCimPas-MP-12,
+	Author = {Angeli, Celestino and Cimiraglia, Renzo and Pastore, Mariachiara},
+	Date-Added = {2019-05-28 11:53:10 +0200},
+	Date-Modified = {2019-05-28 11:53:28 +0200},
+	Doi = {10.1080/00268976.2012.689872},
+	Issn = {0026-8976, 1362-3028},
+	Journal = {Mol. Phys.},
+	Language = {en},
+	Month = dec,
+	Number = {23},
+	Pages = {2963-2968},
+	Shorttitle = {A Comparison of Various Approaches in Internally Contracted Multireference Configuration Interaction},
+	Title = {A Comparison of Various Approaches in Internally Contracted Multireference Configuration Interaction: The Carbon Dimer as a Test Case},
+	Volume = {110},
+	Year = {2012},
+	Bdsk-Url-1 = {https://doi.org/10.1080/00268976.2012.689872}}
+
+@article{SchSilSauThi-JCP-08,
+	Author = {Schreiber, M. and Silva-Junior, M. R. and Sauer, S. P. A. and Thiel, W.},
+	Date-Added = {2019-05-28 11:41:37 +0200},
+	Date-Modified = {2019-05-28 11:42:24 +0200},
+	Doi = {10.1063/1.2889385},
+	Journal = {J. Chem. Phys.},
+	Pages = {134110},
+	Title = {Benchmarks for Electronically Excited States: CASPT2, CC2, CCSD and CC3},
+	Volume = 128,
+	Year = 2008,
+	Bdsk-Url-1 = {https://doi.org/10.1063/1.2889385}}
+
+@article{SilSauSchThi-MP-10,
+	Author = {Silva-Junior, M. R. and Sauer, S. P. A. and Schreiber, M. and Thiel, W.},
+	Date-Added = {2019-05-28 11:41:37 +0200},
+	Date-Modified = {2019-05-28 11:42:05 +0200},
+	Doi = {10.1080/00268970903549047},
+	Journal = {Mol. Phys.},
+	Pages = {453--465},
+	Title = {Basis Set Effects on Coupled Cluster Benchmarks of Electronically Excited States: CC3, CCSDR(3) and CC2},
+	Volume = 108,
+	Year = 2010,
+	Bdsk-Url-1 = {https://doi.org/10.1080/00268970903549047}}
+
+@article{SilSchSauThi-JCP-10,
+	Author = {Silva-Junior, M. R. and Schreiber, M. and Sauer, S. P. A. and Thiel, W.},
+	Date-Added = {2019-05-28 11:41:37 +0200},
+	Date-Modified = {2019-05-28 11:42:45 +0200},
+	Doi = {10.1063/1.3499598},
+	Journal = {J. Chem. Phys.},
+	Pages = {174318},
+	Title = {Benchmarks of Electronically Excited States: Basis Set Effecs Benchmarks of Electronically Excited States: Basis Set Effects on CASPT2 Results},
+	Volume = 133,
+	Year = 2010,
+	Bdsk-Url-1 = {https://doi.org/10.1063/1.3499598}}
+
+@article{Ang-JCC-08,
+	Author = {C. Angeli},
+	Date-Added = {2019-05-28 11:37:39 +0200},
+	Date-Modified = {2019-05-28 11:39:34 +0200},
+	Doi = {10.1002/jcc.21155},
+	Journal = {J. Comput. Chem.},
+	Pages = {1319--1333},
+	Title = {On the Nature of the $\pi \rightarrow \pi^*$ Ionic Excited States: The V State of Ethene as a Prototype},
+	Volume = {30},
+	Year = {2008},
+	Bdsk-Url-1 = {https://doi.org/10.1002/jcc.21155}}
+
+@article{Ang-IJQC-10,
+	Author = {Angeli, Celestino},
+	Date-Added = {2019-05-28 11:37:39 +0200},
+	Date-Modified = {2019-05-28 11:38:19 +0200},
+	Doi = {10.1002/qua.22597},
+	Issn = {00207608, 1097461X},
+	Journal = {Int. J. Quantum Chem.},
+	Language = {en},
+	Pages = {2436-2447},
+	Title = {An Analysis of the Dynamic $\sigma$ Polarization in the {{V}} State of Ethene},
+	Year = {2010},
+	Bdsk-Url-1 = {https://doi.org/10.1002/qua.22597}}
+
+@article{DadSmaBooAlaFil-JCTC-12,
+	Author = {Daday, Csaba and Smart, Simon and Booth, George H. and Alavi, Ali and Filippi, Claudia},
+	Date-Added = {2019-05-28 11:37:39 +0200},
+	Date-Modified = {2019-05-28 11:39:00 +0200},
+	Doi = {10.1021/ct300486d},
+	File = {/Users/loos/Zotero/storage/APCJKTM8/Daday et al. - 2012 - Full Configuration Interaction Excitations of Ethe.pdf},
+	Issn = {1549-9618, 1549-9626},
+	Journal = {J. Chem. Theory. Comput.},
+	Language = {en},
+	Month = nov,
+	Number = {11},
+	Pages = {4441-4451},
+	Shorttitle = {Full {{Configuration Interaction Excitations}} of {{Ethene}} and {{Butadiene}}},
+	Title = {Full {{Configuration Interaction Excitations}} of {{Ethene}} and {{Butadiene}}: {{Resolution}} of an {{Ancient Question}}},
+	Volume = {8},
+	Year = {2012},
+	Bdsk-Url-1 = {https://doi.org/10.1021/ct300486d}}
+
+@article{SerMarNebLinRoo-JCP-93,
+	Author = {{Serrano-Andr\'es}, Luis and Merch\'an, Manuela and Nebot-Gil, Ignacio and Lindh, Roland and Roos, Bj\"orn O.},
+	Date-Added = {2019-05-28 11:37:39 +0200},
+	Date-Modified = {2019-05-28 11:40:06 +0200},
+	Doi = {10.1063/1.465071},
+	Issn = {0021-9606, 1089-7690},
+	Journal = {J. Chem. Phys.},
+	Language = {en},
+	Month = feb,
+	Number = {4},
+	Pages = {3151-3162},
+	Shorttitle = {Towards an Accurate Molecular Orbital Theory for Excited States},
+	Title = {Towards an Accurate Molecular Orbital Theory for Excited States: {{Ethene}}, Butadiene, and Hexatriene},
+	Volume = {98},
+	Year = {1993},
+	Bdsk-Url-1 = {https://doi.org/10.1063/1.465071}}
+
+@article{WibOliTru-JPCA-02,
+	Author = {Wiberg, K. B. and de Oliveria, A. E. and Trucks, G.},
+	Date-Added = {2019-05-28 11:37:39 +0200},
+	Date-Modified = {2019-05-28 11:38:07 +0200},
+	Doi = {10.1021/jp014123x},
+	Journal = {J. Phys. Chem. A},
+	Pages = {4192--4199},
+	Title = {A Comparison of the Electronic Transition Energies for Ethene, Isobutene, Formaldehyde, and Acetone Calculated Using RPA, TDDFT, and EOM-CCSD. Effect of Basis Sets},
+	Volume = {106},
+	Year = 2002,
+	Bdsk-Url-1 = {https://doi.org/10.1021/jp014123x}}
+
+@article{BarPaiLis-JCP-04,
+	Author = {M. Barbatti and J. Paier and H. Lischka},
+	Date-Added = {2019-05-28 11:32:07 +0200},
+	Date-Modified = {2019-05-28 11:36:03 +0200},
+	Doi = {10.1063/1.1807378},
+	Journal = {J. Chem. Phys.},
+	Pages = {11614},
+	Title = {Photochemistry Of Ethylene: A Multireference Configuration Interaction Investigation Of The Excited-State Energy Surfaces},
+	Volume = {121},
+	Year = {20004},
+	Bdsk-Url-1 = {https://doi.org/10.1063/1.1807378}}
+
+@article{ChiHolAdaOttUmrShaZim-JPCA-18,
+	Author = {Chien, Alan D. and Holmes, Adam A. and Otten, Matthew and Umrigar, C. J. and Sharma, Sandeep and Zimmerman, Paul M.},
+	Date-Added = {2019-05-28 11:32:07 +0200},
+	Date-Modified = {2019-05-28 11:33:45 +0200},
+	Doi = {10.1021/acs.jpca.8b01554},
+	File = {/Users/loos/Zotero/storage/J96RZ7JP/Chien et al. - 2018 - Excited States of Methylene, Polyenes, and Ozone f.pdf},
+	Issn = {1089-5639, 1520-5215},
+	Journal = {J. Phys. Chem. A},
+	Language = {en},
+	Month = mar,
+	Number = {10},
+	Pages = {2714--2722},
+	Title = {Excited {{States}} of {{Methylene}}, {{Polyenes}}, and {{Ozone}} from {{Heat}}-{{Bath Configuration Interaction}}},
+	Volume = {122},
+	Year = {2018},
+	Bdsk-Url-1 = {https://doi.org/10.1021/acs.jpca.8b01554}}
+
+@article{FelPetDav-JCP-14,
+	Author = {David Feller and Kirk A. Peterson and Ernest R. Davidson},
+	Date-Added = {2019-05-28 11:32:07 +0200},
+	Date-Modified = {2019-05-28 11:34:05 +0200},
+	Doi = {10.1063/1.4894482},
+	Journal = {J. Chem. Phys.},
+	Number = {10},
+	Pages = {104302},
+	Title = {A Systematic Approach to Vertically Excited States of Ethylene Using Configuration Interaction and Coupled Cluster Techniques},
+	Volume = {141},
+	Year = {2014},
+	Bdsk-Url-1 = {http://dx.doi.org/10.1063/1.4894482}}
+
+@article{WatGwaBar-JCP-96,
+	Author = {Watts, John D. and Gwaltney, Steven R. and Bartlett, Rodney J.},
+	Date-Added = {2019-05-28 11:32:07 +0200},
+	Date-Modified = {2019-05-28 11:34:37 +0200},
+	Doi = {10.1063/1.471988},
+	Issn = {0021-9606, 1089-7690},
+	Journal = {J. Chem. Phys.},
+	Language = {en},
+	Month = oct,
+	Number = {16},
+	Pages = {6979-6988},
+	Title = {Coupled-cluster Calculations of the Excitation Energies of Ethylene, Butadiene, and Cyclopentadiene},
+	Volume = {105},
+	Year = {1996},
+	Bdsk-Url-1 = {https://doi.org/10.1063/1.471988}}
+
 @article{Koh-JCP-09,
 	Author = {A. Kohn},
 	Date-Added = {2019-05-25 17:49:32 +0200},
@@ -17,7 +395,8 @@
 	Pages = {104104},
 	Title = {A modified ansatz for explicitly correlated coupled-cluster wave functions that is suitable for response theory},
 	Volume = {130},
-	Year = {2009}}
+	Year = {2009},
+	Bdsk-Url-1 = {https://doi.org/10.1063/1.3079543}}
 
 @article{ShiWer-JCP-11,
 	Author = {T. Shiozaki and H.-J. Werner},
@@ -64,7 +443,8 @@
 	Pages = {607--630},
 	Title = {Multireference explicitly correlated F12 theories},
 	Volume = {111},
-	Year = {2013}}
+	Year = {2013},
+	Bdsk-Url-1 = {https://doi.org/10.1080/00268976.2013.779393}}
 
 @article{NeiHatKlo-JCP-06,
 	Author = {C. Neiss and C. Hattig and W. Klopper},
@@ -75,7 +455,8 @@
 	Pages = {064111},
 	Title = {Extensions of r12 corrections to CC2-R12 for excited states},
 	Volume = {125},
-	Year = {2006}}
+	Year = {2006},
+	Bdsk-Url-1 = {https://doi.org/10.1063/1.2335443%F4%8F%B0%83}}
 
 @article{HanKoh-JCP-09,
 	Author = {M. Hanauer and A. Kohn},
@@ -86,7 +467,8 @@
 	Pages = {124118},
 	Title = {Response properties with explicitly correlated coupled-cluster methods using a Slater-type correlation factor and cusp conditions},
 	Volume = {131},
-	Year = {2009}}
+	Year = {2009},
+	Bdsk-Url-1 = {https://doi.org/10.1063/1.3238237%F4%8F%B0%83}}
 
 @article{FliHatKlo-JCP-06,
 	Author = {H. Fliegl and C. Hattig and W. Klopper},
@@ -97,7 +479,8 @@
 	Pages = {044112},
 	Title = {Coupled-cluster response theory with linear-r12 corrections: The CC2-R12 model for excitation energies},
 	Volume = {124},
-	Year = {2006}}
+	Year = {2006},
+	Bdsk-Url-1 = {https://doi.org/10.1063/1.2161183%F4%8F%B0%83}}
 
 @article{LooPraSceTouGin-JPCL-19,
 	Author = {P. F. Loos and B. Pradines and A. Scemama and J. Toulouse and E. Giner},
@@ -8989,13 +9372,6 @@
 	Volume = {17},
 	Year = {1969}}
 
-@article{PurZhaKra-JCP-09,
-	Author = {W. Purwanto and S. Zhang and H. Krakauer},
-	Journal = {J. Chem. Phys.},
-	Pages = {094107},
-	Volume = {130},
-	Year = {2009}}
-
 @misc{Qmc-PROG-XX,
 	Note = {QMCMOL, a quantum Monte Carlo program written by R. Assaraf, F. Colonna, X. Krokidis, P. Reinhardt and coworkers.},
 	Url = {http://www.lct.jussieu.fr/pagesequipe/qmcmol/qmcmol/},
diff --git a/Manuscript/Ex-srDFT.tex b/Manuscript/Ex-srDFT.tex
index 93434b7..a759103 100644
--- a/Manuscript/Ex-srDFT.tex
+++ b/Manuscript/Ex-srDFT.tex
@@ -83,7 +83,9 @@
 \newcommand{\n}[2]{n_{#1}^{#2}}
 \newcommand{\Ec}{E_\text{c}}
 \newcommand{\E}[2]{E_{#1}^{#2}}
+\newcommand{\DE}[2]{\Delta E_{#1}^{#2}}
 \newcommand{\bE}[2]{\Bar{E}_{#1}^{#2}}
+\newcommand{\DbE}[2]{\Delta \Bar{E}_{#1}^{#2}}
 \newcommand{\bEc}[1]{\Bar{E}_\text{c,md}^{#1}}
 \newcommand{\e}[2]{\varepsilon_{#1}^{#2}}
 \newcommand{\be}[2]{\Bar{\varepsilon}_{#1}^{#2}}
@@ -218,6 +220,45 @@ Contrary to our recent study on atomization and correlation energies, \cite{LooP
 \label{sec:theory}
 %%%%%%%%%%%%%%%%%%%%%%%%
 
+The present basis set correction assumes that we have, in a given (finite) basis set $\Bas$, the ground-state and the $k$th excited-state energies, $\E{0}{\Bas}$ and $\E{k}{\Bas}$, their one-electron densities, $\n{k}{\Bas}$ and $\n{0}{\Bas}$, as well as their opposite-spin on-top pair densities, $\n{2,0}{\Bas}(\br{},\br{})$ and $\n{2,k}{\Bas}(\br{},\br{})$,
+Therefore, the complete basis set (CBS) energy of the ground and excited states may be approximated as
+\begin{align}
+	\label{eq:ECBS}
+	\E{0}{\CBS} & \approx \E{0}{\Bas} + \bE{}{\Bas}[\n{0}{\Bas}],
+	&
+	\E{k}{\CBS} & \approx \E{k}{\Bas} + \bE{}{\Bas}[\n{k}{\Bas}],
+\end{align}
+where 
+\begin{equation}
+	\label{eq:E_funcbasis}
+	 \bE{}{\Bas}[\n{}{}]
+	= \min_{\wf{}{} \rightsquigarrow \n{}{}}  \mel*{\wf{}{}}{\hT + \hWee{}}{\wf{}{}} 
+	- \min_{\wf{}{\Bas} \rightsquigarrow \n{}{}}  \mel*{\wf{}{\Bas}}{\hT + \hWee{}}{\wf{}{\Bas}}
+\end{equation}
+is the basis-dependent complementary density functional, 
+\begin{align}
+	\hT & = - \frac{1}{2} \sum_{i}^{\Ne} \nabla_i^2,
+	&
+	\hWee{} & = \sum_{i<j}^{\Ne} r_{ij}^{-1},
+\end{align}
+are the kinetic and interelectronic repulsion operators, respectively, and $\wf{}{\Bas}$ and $\wf{}{}$ are two general $\Ne$-electron normalized wave functions belonging to the Hilbert space spanned by $\Bas$ and the complete basis set (CBS), respectively.
+The notation $\wf{}{} \rightsquigarrow \n{}{}$ states that $\wf{}{}$ yield the density $\n{}{}$.
+
+Hence, the CBS excitation energy reads
+\begin{equation}
+	\DE{k}{\CBS} = \E{k}{\CBS} - \E{0}{\CBS} = \DE{k}{\Bas} + \DbE{}{\Bas}[\n{0}{\Bas},\n{k}{\Bas}],
+\end{equation}
+where
+\begin{equation}
+	\label{eq:DEB}
+	\DE{k}{\Bas} = \E{k}{\Bas} - \E{0}{\Bas}
+\end{equation}
+is the excitation energy in $\Bas$ and
+\begin{equation}
+	\label{eq:DbE}
+	\DbE{}{\Bas}[\n{0}{\Bas},\n{k}{\Bas}] = \bE{}{\Bas}[\n{k}{\Bas}] - \bE{}{\Bas}[\n{0}{\Bas}]
+\end{equation}
+its basis set correction.
 
 %Let us assume that we have reasonable approximations of the FCI energy and density of a $\Ne$-electron system in an incomplete basis set $\Bas$, say the CCSD(T) energy $\E{\CCSDT}{\Bas}$ and the Hartree-Fock (HF) density $\n{\HF}{\Bas}$. 
 %According to Eq.~(15) of Ref.~\citenum{GinPraFerAssSavTou-JCP-18}, the exact ground-state energy $\E{}{}$ may be approximated as
@@ -333,75 +374,47 @@ Contrary to our recent study on atomization and correlation energies, \cite{LooP
 %Indeed, the two functionals coincide if $\wf{}{\Bas} = \wf{}{\rsmu{}{}}$.
 %Therefore, we approximate $\bE{}{\Bas}[\n{}{}]$ by ECMD functionals evaluated with the range-separation function $\rsmu{}{\Bas}(\br{})$. 
 
-%The local-density approximation (LDA) of the ECMD complementary functional is defined as 
-%\begin{equation}
-%	\label{eq:def_lda_tot}
-%	\bE{\LDA}{\Bas}[\n{}{},\rsmu{}{\Bas}] = \int \n{}{}(\br{}) \be{\text{c,md}}{\sr,\LDA}\qty(\n{}{}(\br{}),\zeta(\br{}),\rsmu{}{\Bas}(\br{})) \dbr{},
-%\end{equation}
-%where $\zeta = (\n{\uparrow}{} - \n{\downarrow}{})/\n{}{}$ is the spin polarization and $\be{\text{c,md}}{\sr,\LDA}(\n{}{},\zeta,\rsmu{}{})$ is the ECMD short-range correlation energy per electron of the uniform electron gas (UEG) \cite{LooGil-WIRES-16} parameterized in Ref.~\citenum{PazMorGorBac-PRB-06}.
-%The short-range LDA correlation functional relies on the transferability of the physics of the UEG which is certainly valid for large $\mu$ but is known to over correlate for small $\mu$. 
-%In order to correct such a defect, inspired by the recent functional proposed by some of the authors~\cite{FerGinTou-JCP-18}, we propose here a new Perdew-Burke-Ernzerhof (PBE)-based ECMD functional 
-%Inspired by the recent functional proposed by some of the authors~\cite{FerGinTou-JCP-18}, we propose here a new Perdew-Burke-Ernzerhof (PBE)-based ECMD functional 
-%\begin{equation}
-%	\label{eq:def_pbe_tot}
-%	\bE{\PBE}{\Bas}[\n{}{},\rsmu{}{\Bas}] =
-%	\int \n{}{}(\br{}) \be{\text{c,md}}{\sr,\PBE}\qty(\n{}{}(\br{}),s(\br{}),\zeta(\br{}),\rsmu{}{\Bas}(\br{})) \dbr{},
-%\end{equation}
-%where \titou{$\zeta = (\n{\uparrow}{} - \n{\downarrow}{})/\n{}{}$ is the spin polarization and} $s=\abs{\nabla \n{}{}}/\n{}{4/3}$ is the reduced density gradient. 
-%$\be{\text{c,md}}{\sr,\PBE}\qty(\n{}{},s,\zeta,\rsmu{}{})$ interpolates between the usual PBE correlation functional, \cite{PerBurErn-PRL-96} $\e{\text{c}}{\PBE}(\n{}{},s,\zeta)$, at $\rsmu{}{}=0$ and the exact large-$\rsmu{}{}$ behavior, \cite{TouColSav-PRA-04, GorSav-PRA-06, PazMorGorBac-PRB-06} yielding
-%\begin{subequations}
-%\begin{gather}
-%	\label{eq:epsilon_cmdpbe}
-%	\be{\text{c,md}}{\sr,\PBE}(\n{}{},s,\zeta,\rsmu{}{}) = \frac{\e{\text{c}}{\PBE}(\n{}{},s,\zeta)}{1 + \beta(\n{}{},s,\zeta) \rsmu{}{3} },
-%	\\
-%	\label{eq:beta_cmdpbe}
-%	\beta(\n{}{},s,\zeta) = \frac{3}{2\sqrt{\pi} (1 - \sqrt{2} )} \frac{\e{\text{c}}{\PBE}(\n{}{},s,\zeta)}{\n{2}{\UEG}(\n{}{},\zeta)}.
-%\end{gather}
-%\end{subequations}
-%The difference between the ECMD functional defined in Ref.~\citenum{FerGinTou-JCP-18} and the present expression \eqref{eq:epsilon_cmdpbe}-\eqref{eq:beta_cmdpbe} is that we approximate here the on-top pair density by its \titou{uniform electron gas \cite{LooGil-WIRES-16} (UEG)} version, i.e.~$\n{2}{\Bas}(\br{},\br{})  \approx \n{2}{\UEG}(\n{}{}(\br{}),\zeta(\br{}))$, where $\n{2}{\UEG}(\n{}{},\zeta) \approx \n{}{2} (1-\zeta^2) g_0(n)$ with the parametrization of the UEG on-top pair-distribution function $g_0(n)$ given in Eq.~(46) of Ref.~\citenum{GorSav-PRA-06}. 
-%This represents a major computational saving without loss of accuracy for weakly correlated systems as we eschew the computation of $\n{2}{\Bas}(\br{},\br{})$.
-%The complementary functional $\bE{}{\Bas}[\n{\HF}{\Bas}]$ is approximated by $\bE{\PBE}{\Bas}[\n{\HF}{\Bas},\rsmu{}{\Bas}]$ where $\rsmu{}{\Bas}(\br{})$ is given by Eq.~\eqref{eq:mu_of_r}.
 
-%As most WFT calculations are performed within the frozen-core (FC) approximation, it is important to define an effective interaction within a subset of MOs. 
-%We then naturally split the basis set as $\Bas = \Cor \bigcup \BasFC$ (where $\Cor$ and $\BasFC$ are the sets of core and active MOs, respectively) and define the FC version of the effective interaction as
-% \begin{equation}
-% 	\label{eq:WFC}
-%	\WFC{}{\Bas}(\br{1},\br{2})  = 
-%    \begin{cases}
-%	\fFC{}{\Bas}(\br{1},\br{2})/\nFC{2}{\Bas}(\br{1},\br{2}),  & \text{if $\nFC{2}{\Bas}(\br{1},\br{2}) \ne 0$},
-%	\\
-%	\infty,		&	\text{otherwise,}
-%    \end{cases}
-% \end{equation}
-%with 
-%\begin{subequations}
-%\begin{gather}
-%	\label{eq:fbasisval}
-%	\fFC{}{\Bas}(\br{1},\br{2})
-%	=  \sum_{pq \in \Bas} \sum_{rstu \in \BasFC}  \SO{p}{1} \SO{q}{2} \V{pq}{rs} \Gam{rs}{tu} \SO{t}{1} \SO{u}{2},
-%	\\
-%	\nFC{2}{\Bas}(\br{1},\br{2})
-%	= \sum_{pqrs \in \BasFC}  \SO{p}{1} \SO{q}{2} \Gam{pq}{rs} \SO{r}{1} \SO{s}{2}, 
-%\end{gather}
-%\end{subequations}
-%and the corresponding FC range-separation function $\rsmuFC{}{\Bas}(\br{}) = (\sqrt{\pi}/2) \WFC{}{\Bas}(\br{},\br{})$.
-%It is noteworthy that, within the present definition, $\WFC{}{\Bas}(\br{1},\br{2})$ still tends to the regular Coulomb interaction as $\Bas \to \CBS$.
-%Defining $\nFC{\HF}{\Bas}$ as the FC (i.e.~valence-only) $\HF$ one-electron density in $\Bas$, the FC contribution of the complementary functional is then approximated by $\bE{\PBE}{\Bas}[\nFC{\HF}{\Bas},\rsmuFC{}{\Bas}]$.
 
-%The most computationally intensive task of the present approach is the evaluation of $\W{}{\Bas}(\br{},\br{})$ at each quadrature grid point. 
-%In the general case (i.e.~$\wf{}{\Bas}$ is a multi-determinant expansion), we compute this embarrassingly parallel step in $\order*{\Ng \Nb^4}$ computational cost with a memory requirement of $\order*{ \Ng \Nb^2}$, where $\Nb$ is the number of basis functions in $\Bas$.
-%The computational cost can be reduced to $\order*{ \Ng \Ne^2 \Nb^2}$ with no memory footprint when $\wf{}{\Bas}$ is a single Slater determinant.
-%As shown in Ref.~\citenum{GinPraFerAssSavTou-JCP-18}, this choice for $\wf{}{\Bas}$ already provides, for weakly correlated systems, a quantitative representation of the incompleteness of $\Bas$.
-%Hence, we will stick to this choice throughout the present study.
-%In our current implementation, the computational bottleneck is the four-index transformation to get the two-electron integrals in the MO basis which appear in Eqs.~\eqref{eq:n2basis} and \eqref{eq:fbasis}. 
-%Nevertheless, this step usually has to be performed for most correlated WFT calculations. 
+
 
 %To conclude this section, we point out that, thanks to the definitions \eqref{eq:def_weebasis} and \eqref{eq:mu_of_r} as well as the properties \eqref{eq:lim_W} and \eqref{eq:large_mu_ecmd}, independently of the DFT functional, the present basis-set correction
 %i) can be applied to any WFT method that provides an energy and a density, 
 %ii) does not correct one-electron systems, and
 %iii) vanishes in the CBS limit, hence guaranteeing an unaltered CBS limit for a given WFT method. 
 
+%%%%%%%%%%%%%%%%%%%%%%%%
+\subsection{Short-range correlation functionals}
+\label{sec:func}
+%%%%%%%%%%%%%%%%%%%%%%%%
 
+The local-density approximation (LDA) of the ECMD complementary functional is defined as 
+\begin{equation}
+	\label{eq:def_lda_tot}
+	\bE{\LDA}{\Bas}[\n{}{},\rsmu{}{\Bas}] = \int \n{}{}(\br{}) \be{\text{c,md}}{\sr,\LDA}\qty(\n{}{}(\br{}),\zeta(\br{}),\rsmu{}{\Bas}(\br{})) \dbr{},
+\end{equation}
+where $\zeta = (\n{\uparrow}{} - \n{\downarrow}{})/\n{}{}$ is the spin polarization and $\be{\text{c,md}}{\sr,\LDA}(\n{}{},\zeta,\rsmu{}{})$ is the ECMD short-range correlation energy per electron of the uniform electron gas (UEG) \cite{LooGil-WIRES-16} parameterized in Ref.~\citenum{PazMorGorBac-PRB-06}.
+
+To go beyond the LDA and cure its over correlation at small $\mu$, some of the authors recently proposed a Perdew-Burke-Ernzerhof (PBE)-based ECMD functional \cite{FerGinTou-JCP-18}, 
+\begin{equation}
+	\label{eq:def_pbe_tot}
+	\bE{\PBE}{\Bas}[\n{}{},\rsmu{}{\Bas}] =
+	\int \n{}{}(\br{}) \be{\text{c,md}}{\sr,\PBE}\qty(\n{}{}(\br{}),s(\br{}),\zeta(\br{}),\rsmu{}{\Bas}(\br{})) \dbr{},
+\end{equation}
+where $s = \abs{\nabla \n{}{}}/\n{}{4/3}$ is the reduced density gradient. 
+$\be{\text{c,md}}{\sr,\PBE}\qty(\n{}{},s,\zeta,\rsmu{}{})$ interpolates between the usual PBE correlation functional, \cite{PerBurErn-PRL-96} $\e{\text{c}}{\PBE}(\n{}{},s,\zeta)$, at $\rsmu{}{}=0$ and the exact large-$\rsmu{}{}$ behavior, \cite{TouColSav-PRA-04, GorSav-PRA-06, PazMorGorBac-PRB-06} yielding
+\begin{subequations}
+\begin{gather}
+	\label{eq:epsilon_cmdpbe}
+	\be{\text{c,md}}{\sr,\PBE}(\n{}{},s,\zeta,\rsmu{}{}) = \frac{\e{\text{c}}{\PBE}(\n{}{},s,\zeta)}{1 + \beta(\n{}{},s,\zeta) \rsmu{}{3} },
+	\\
+	\label{eq:beta_cmdpbe}
+	\beta(\n{}{},s,\zeta) = \frac{3}{2\sqrt{\pi} (1 - \sqrt{2} )} \frac{\e{\text{c}}{\PBE}(\n{}{},s,\zeta)}{\n{2}{\Bas}(\br{},\br{})}.
+\end{gather}
+\end{subequations}
+We will refer to this functional as the ``on top'' PBE (PBEot) ECMD functional.
+More recently, we have also proposed a simplified version of the PBEot functional where we replaced the on-top pair density by its UEG version, i.e.~$\n{2}{\Bas}(\br{},\br{})  \approx \n{2}{\UEG}(\n{}{}(\br{}),\zeta(\br{}))$, where $\n{2}{\UEG}(\n{}{},\zeta) \approx \n{}{2} (1-\zeta^2) g_0(n)$ with the parametrization of the UEG on-top pair-distribution function $g_0(n)$ given in Eq.~(46) of Ref.~\citenum{GorSav-PRA-06}. 
+This computationally-lighter functional will be refer to as PBE.
 
 
 %%%%%%%%%%%%%%%%%%%%%%%%
@@ -421,7 +434,12 @@ The geometries have been extracted from Refs.~\onlinecite{LooSceBloGarCafJac-JCT
 They are also reported in the {\SI}.
 Frozen-core calculations are systematically performed and defined as such: a \ce{He} core is frozen from \ce{Li} to \ce{Ne}, while a \ce{Ne} core is frozen from \ce{Na} to \ce{Ar}.
 The FC density-based correction is used consistently with the FC approximation in WFT methods.
+We refer the interested reader to Ref.~\onlinecite{LooPraSceTouGin-JPCL-19} for detailed explanations on how the previous equations have to be modified within the FC approximation.
 
+The most computationally intensive task of the present approach is the evaluation of $\W{}{\Bas}(\br{},\br{})$ at each quadrature grid point. 
+In the general case (i.e.~$\wf{}{\Bas}$ is a multi-determinant expansion), we compute this embarrassingly parallel step in $\order*{\Ng \Nb^4}$ computational cost with a memory requirement of $\order*{ \Ng \Nb^2}$, where $\Nb$ is the number of basis functions in $\Bas$.
+In our current implementation, the computational bottleneck is the four-index transformation to get the two-electron integrals in the MO basis which appear in Eqs.~\eqref{eq:n2basis} and \eqref{eq:fbasis}. 
+Nevertheless, this step usually has to be performed for most correlated WFT calculations. 
 
 %%%%%%%%%%%%%%%%%%%%%%%%
 \section{Results}
@@ -433,7 +451,7 @@ The FC density-based correction is used consistently with the FC approximation i
 \label{sec:CH2}
 %=======================
 
-As a first test of the present basis set correction, we consider the adiabatic transitions of methylene which have been thourhoughly studied in the literature with high-level ab initio methods.
+As a first test of the present basis set correction, we consider the adiabatic transitions of methylene which have been thourhoughly studied in the literature with high-level ab initio methods. \cite{AbrShe-JCP-04, AbrShe-CPL-05, ZimTouZhaMusUmr-JCP-09, ChiHolAdaOttUmrShaZim-JPCA-18}
 
 %%% TABLE 1 %%%
 	\begin{squeezetable}
@@ -551,7 +569,7 @@ As a first test of the present basis set correction, we consider the adiabatic t
 \label{sec:H2O-NH3}
 %=======================
 
-Water and ammonia are two interesting molecules with Rydberge excited states which are highly sensitive to the radial completeness of the one-electron basis set.
+Water \cite{CaiTozRei-JCP-00, RubSerMer-JCP-08, LiPal-JCP-11, LooSceBloGarCafJac-JCTC-18, SceBenJacCafLoo-JCP-18} and ammonia \cite{SchGoe-JCTC-17, BarDelPerMat-JMS-97, LooSceBloGarCafJac-JCTC-18} are two interesting molecules with Rydberg excited states which are highly sensitive to the radial completeness of the one-electron basis set.
 
 %%% TABLE 2 %%%
 	\begin{squeezetable}
@@ -801,7 +819,7 @@ Water and ammonia are two interesting molecules with Rydberge excited states whi
 \subsection{Doubly-Excited States of the Carbon Dimer}
 \label{sec:C2}
 %=======================
-It is also interesting to study doubly-excited states.
+It is also interesting to study doubly-excited states. \cite{AbrShe-JCP-04, AbrShe-CPL-05, Var-JCP-08, PurZhaKra-JCP-09, AngCimPas-MP-12, BooCleThoAla-JCP-11, Sha-JCP-15, SokCha-JCP-16, VarRoc-PTRSMPES-18}
 In the carbon dimer, these valence states are of $(\pi,\pi) \ra (\si,\si)$ character and they have been recently studied with state-of-the-art methods. \cite{LooBogSceCafJAc-JCTC-19}
 
 %%% FIG 4 %%%
@@ -818,7 +836,7 @@ In the carbon dimer, these valence states are of $(\pi,\pi) \ra (\si,\si)$ chara
 \label{sec:C2H4}
 %=======================
 
-Ethylene is an interesting molecules as it contains both valence and Rydberg excited states.
+Ethylene is an interesting molecules as it contains both valence and Rydberg excited states. \cite{SerMarNebLinRoo-JCP-93, WatGwaBar-JCP-96, WibOliTru-JPCA-02, BarPaiLis-JCP-04, Ang-JCC-08, SchSilSauThi-JCP-08, SilSchSauThi-JCP-10, SilSauSchThi-MP-10, Ang-IJQC-10, DadSmaBooAlaFil-JCTC-12, FelPetDav-JCP-14, ChiHolAdaOttUmrShaZim-JPCA-18}
 
 %\begin{figure}
 %	\includegraphics[width=\linewidth]{C2H2}