changes in intro

Manuscript/biblio.bib

@@ -8473,6 +8473,16 @@ year = {2016},
 note = {doi: 10.1002/wcms.1257}
 }
 
+% 
+@article{LooPraSceGinTou-ARX-19,
+   author    = {P.-F. Loos and B. Pradines and A. Scemama and E. Giner and J. Toulouse},
+   title     = {A density-based basis-set incompleteness correction for GW methods},
+   journal   = {arXiv:1910.12238},
+   volume    = {},
+   pages     = {},
+   year      = {}
+}
+
 % 
 @article{LooPraSceTouGin-JPCL-19,
    author    = {P.-F. Loos and B. Pradines and A. Scemama and J. Toulouse and E. Giner},

Manuscript/srDFT_SC.bbl

@@ -6,7 +6,7 @@
 %Control: page (0) single
 %Control: year (1) truncated
 %Control: production of eprint (0) enabled
-\begin{thebibliography}{60}%
+\begin{thebibliography}{65}%
 \makeatletter
 \providecommand \@ifxundefined [1]{%
  \@ifx{#1\undefined}
@@ -98,6 +98,17 @@
   {Kutzelnigg}},\ }\href@noop {} {\bibfield  {journal} {\bibinfo  {journal} {J.
   Chem. Phys.}\ }\textbf {\bibinfo {volume} {101}},\ \bibinfo {pages} {7738}
   (\bibinfo {year} {1994})}\BibitemShut {NoStop}%
+\bibitem [{\citenamefont {Helgaker}\ \emph {et~al.}(1997)\citenamefont
+  {Helgaker}, \citenamefont {Klopper}, \citenamefont {Koch},\ and\
+  \citenamefont {Noga}}]{HelKloKocNog-JCP-97}%
+  \BibitemOpen
+  \bibfield  {author} {\bibinfo {author} {\bibfnamefont {T.}~\bibnamefont
+  {Helgaker}}, \bibinfo {author} {\bibfnamefont {W.}~\bibnamefont {Klopper}},
+  \bibinfo {author} {\bibfnamefont {H.}~\bibnamefont {Koch}}, \ and\ \bibinfo
+  {author} {\bibfnamefont {J.}~\bibnamefont {Noga}},\ }\href@noop {} {\bibfield
+   {journal} {\bibinfo  {journal} {J. Chem. Phys.}\ }\textbf {\bibinfo {volume}
+  {106}},\ \bibinfo {pages} {9639} (\bibinfo {year} {1997})}\BibitemShut
+  {NoStop}%
 \bibitem [{\citenamefont {Halkier}\ \emph {et~al.}(1998)\citenamefont
   {Halkier}, \citenamefont {Helgaker}, \citenamefont {J{\o}rgensen},
   \citenamefont {Klopper}, \citenamefont {Koch}, \citenamefont {Olsen},\ and\
@@ -380,6 +391,23 @@
   {\bibinfo {volume} {142}},\ \bibinfo {pages} {154123} (\bibinfo {year}
   {2015})},\ \bibinfo {note} {erratum: J. Chem. Phys. {\bf 142}, 219901
   (2015)}\BibitemShut {NoStop}%
+\bibitem [{\citenamefont {Kalai}\ and\ \citenamefont
+  {Toulouse}(2018)}]{KalTou-JCP-18}%
+  \BibitemOpen
+  \bibfield  {author} {\bibinfo {author} {\bibfnamefont {C.}~\bibnamefont
+  {Kalai}}\ and\ \bibinfo {author} {\bibfnamefont {J.}~\bibnamefont
+  {Toulouse}},\ }\href {\doibase 10.1063/1.5025561} {\bibfield  {journal}
+  {\bibinfo  {journal} {J. Chem. Phys.}\ }\textbf {\bibinfo {volume} {148}},\
+  \bibinfo {pages} {164105} (\bibinfo {year} {2018})}\BibitemShut {NoStop}%
+\bibitem [{\citenamefont {Kalai}, \citenamefont {Mussard},\ and\ \citenamefont
+  {Toulouse}(2019)}]{KalMusTou-JCP-19}%
+  \BibitemOpen
+  \bibfield  {author} {\bibinfo {author} {\bibfnamefont {C.}~\bibnamefont
+  {Kalai}}, \bibinfo {author} {\bibfnamefont {B.}~\bibnamefont {Mussard}}, \
+  and\ \bibinfo {author} {\bibfnamefont {J.}~\bibnamefont {Toulouse}},\ }\href
+  {\doibase 10.1063/1.5108536} {\bibfield  {journal} {\bibinfo  {journal} {J.
+  Chem. Phys.}\ }\textbf {\bibinfo {volume} {151}},\ \bibinfo {pages} {074102}
+  (\bibinfo {year} {2019})}\BibitemShut {NoStop}%
 \bibitem [{\citenamefont {Leininger}\ \emph {et~al.}(1997)\citenamefont
   {Leininger}, \citenamefont {Stoll}, \citenamefont {Werner},\ and\
   \citenamefont {Savin}}]{LeiStoWerSav-CPL-97}%
@@ -442,6 +470,14 @@
   {pages} {084103} (\bibinfo {year} {2019})},\ \Eprint
   {http://arxiv.org/abs/https://doi.org/10.1063/1.5082638}
   {https://doi.org/10.1063/1.5082638} \BibitemShut {NoStop}%
+\bibitem [{\citenamefont {Mussard}\ and\ \citenamefont
+  {Toulouse}(2017)}]{MusTou-MP-17}%
+  \BibitemOpen
+  \bibfield  {author} {\bibinfo {author} {\bibfnamefont {B.}~\bibnamefont
+  {Mussard}}\ and\ \bibinfo {author} {\bibfnamefont {J.}~\bibnamefont
+  {Toulouse}},\ }\href@noop {} {\bibfield  {journal} {\bibinfo  {journal} {Mol.
+  Phys.}\ }\textbf {\bibinfo {volume} {115}},\ \bibinfo {pages} {161} (\bibinfo
+  {year} {2017})}\BibitemShut {NoStop}%
 \bibitem [{\citenamefont {Giner}\ \emph {et~al.}(2018)\citenamefont {Giner},
   \citenamefont {Pradines}, \citenamefont {Fert\'e}, \citenamefont {Assaraf},
   \citenamefont {Savin},\ and\ \citenamefont
@@ -485,15 +521,25 @@
   {pages} {144118} (\bibinfo {year} {2019})},\ \Eprint
   {http://arxiv.org/abs/https://doi.org/10.1063/1.5122976}
   {https://doi.org/10.1063/1.5122976} \BibitemShut {NoStop}%
+\bibitem [{\citenamefont {Loos}\ \emph {et~al.}()\citenamefont {Loos},
+  \citenamefont {Pradines}, \citenamefont {Scemama}, \citenamefont {Giner},\
+  and\ \citenamefont {Toulouse}}]{LooPraSceGinTou-ARX-19}%
+  \BibitemOpen
+  \bibfield  {author} {\bibinfo {author} {\bibfnamefont {P.-F.}\ \bibnamefont
+  {Loos}}, \bibinfo {author} {\bibfnamefont {B.}~\bibnamefont {Pradines}},
+  \bibinfo {author} {\bibfnamefont {A.}~\bibnamefont {Scemama}}, \bibinfo
+  {author} {\bibfnamefont {E.}~\bibnamefont {Giner}}, \ and\ \bibinfo {author}
+  {\bibfnamefont {J.}~\bibnamefont {Toulouse}},\ }\href@noop {} {\bibinfo
+  {journal} {arXiv:1910.12238}\ }\BibitemShut {NoStop}%
 \bibitem [{\citenamefont {Toulouse}, \citenamefont {Gori-Giorgi},\ and\
   \citenamefont {Savin}(2005)}]{TouGorSav-TCA-05}%
   \BibitemOpen
-  \bibfield  {author} {\bibinfo {author} {\bibfnamefont {J.}~\bibnamefont
+\bibfield  {journal} {  }\bibfield  {author} {\bibinfo {author} {\bibfnamefont
-  {Toulouse}}, \bibinfo {author} {\bibfnamefont {P.}~\bibnamefont
+  {J.}~\bibnamefont {Toulouse}}, \bibinfo {author} {\bibfnamefont
-  {Gori-Giorgi}}, \ and\ \bibinfo {author} {\bibfnamefont {A.}~\bibnamefont
+  {P.}~\bibnamefont {Gori-Giorgi}}, \ and\ \bibinfo {author} {\bibfnamefont
-  {Savin}},\ }\href@noop {} {\bibfield  {journal} {\bibinfo  {journal} {Theor.
+  {A.}~\bibnamefont {Savin}},\ }\href@noop {} {\bibfield  {journal} {\bibinfo
-  Chem. Acc.}\ }\textbf {\bibinfo {volume} {114}},\ \bibinfo {pages} {305}
+  {journal} {Theor. Chem. Acc.}\ }\textbf {\bibinfo {volume} {114}},\ \bibinfo
-  (\bibinfo {year} {2005})}\BibitemShut {NoStop}%
+  {pages} {305} (\bibinfo {year} {2005})}\BibitemShut {NoStop}%
 \bibitem [{\citenamefont {Perdew}, \citenamefont {Burke},\ and\ \citenamefont
   {Ernzerhof}(1996)}]{PerBurErn-PRL-96}%
   \BibitemOpen

Manuscript/srDFT_SC.tex

@@ -264,7 +264,7 @@
 \author{Emmanuel Giner}
 \email{emmanuel.giner@lct.jussieu.fr}
 \affiliation{\LCT}
-\author{Bath\'elemy Pradines}
+\author{Barth\'el\'emy Pradines}
 \affiliation{\LCT}
 \affiliation{\ISCD}
 \author{Anthony Scemama}
@@ -293,14 +293,7 @@ The general goal of quantum chemistry is to provide reliable theoretical tools t
 
 The difficulty of obtaining a reliable theoretical description of a given chemical system can be roughly categorized by the strength of the electronic correlation appearing in its electronic structure. The so-called weakly correlated systems, such as closed-shell organic molecules near their equilibrium geometries, are typically dominated by correlation effects which do not affect the qualitative mean-field picture of the system. These weak correlation effects can be either short range(near the electron-electron coalescence point) or long range (London dispersion interactions). The theoretical description of weakly correlated systems is one of the most concrete achievement of quantum chemistry, and the main remaining issue for these systems is to push the limit of the size of the chemical systems that can be treated.  The case of the so-called strongly correlated systems, which are ubiquitous in chemistry, is much more problematic as they exhibit a much more complex electronic structure. For example, transition metal complexes, low-spin open-shell systems, covalent bond breaking situations have all in common that they cannot be even qualitatively described by a single electronic configuration. It is now clear that the usual semilocal density-functional approximations of KS DFT fail to accurately describe these situations and WFT is king for the treatment of strongly correlated systems. 
 
-In practice WFT uses a finite one-particle basis set (here referred as $\basis$) to project the Schroedinger equation whose exact solution becomes clear: the full configuration interaction (FCI) which consists in a linear algebra problem whose dimension scales exponentially with the system size. 
+In practice WFT uses a finite one-particle basis set (here denoted as $\basis$) to project the Schr\"odinger equation. The exact solution within the basis set is then provided by full configuration interaction (FCI) which consists in a linear-algebra problem with a dimension scaling exponentially with the system size. Due to this exponential growth of the FCI computational cost, introducing approximations is necessary, with at least two difficulties for strongly correlated systems: i) the qualitative description of the wave function is determined by a primary set of electronic configurations (whose size can scale exponentially in many cases) among which near degeneracies and/or strong interactions appear in the Hamiltonian matrix; ii) the quantitative description of the system requires to also account for weak correlation effects which involve many other electronic configurations with typically much smaller weights in the wave function. Addressing these two objectives is a rather complicated task for a given approximate WFT method, especially if one adds the requirement of satisfying formal properties, such as spin-multiplet degeneracy and size consistency.
-Because of the exponential growth of the FCI, many approximations have appeared and in that regard the complexity of the strong correlation problem is, at least, two-fold: 
-i)  the qualitative description of the wave function is determined by a primary set of electronic configurations (whose size can scale exponentially in many cases) among which near degeneracies and/or strong interactions appear in the Hamiltonian matrix, 
-ii) the quantitative description of the systems must take into account weak correlation effects which requires to take into account many 
-other electronic configurations with typically much smaller weights in the wave function.  
-Fulfilling these two objectives is a rather complicated task for a given approximated approach, specially if one adds the requirement of satisfying formal properties, such $S_z$ invariance or additivity of the computed energy in the case of non interacting fragments. 
-
-%energy degeneracy of spin-multiplet components
 
 %To tackle this complicated problem, many methods have been proposed and an exhaustive review of the zoology of methods for strong correlation goes beyond the scope and purpose of this article. 
 
@@ -320,19 +313,13 @@ Fulfilling these two objectives is a rather complicated task for a given approxi
 %Among the SCI algorithms, the CI perturbatively selected iteratively (CIPSI) can be considered as a pioneer. The main idea of the CIPSI and other related SCI algorithms is to iteratively select the most important Slater determinants thanks to perturbation theory in order to build a MRCI zeroth-order wave function which automatically concentrate the strongly interacting part of the wave function. On top of this MRCI zeroth-order wave function, a rather simple MRPT approach is used to recover the missing weak correlation and the process is iterated until reaching a given convergence criterion. It is important to notice that in the SCI algorithms, neither the SCI or the MRPT are size extensive \textit{per se}, but the extensivity property is almost recovered by approaching the FCI limit. 
 %When the SCI are affordable, their clear advantage are that they provide near FCI wave functions and energies, whatever the level of knowledge of the user on the specific physical/chemical problem considered. The drawback of SCI is certainly their \textit{intrinsic} exponential scaling due to their linear parametrisation. Nevertheless, such an exponential scaling is lowered by the smart selection of the zeroth-order wave function together with the MRPT calculation. 
 
-Besides the difficulties of accurately describing the electronic structure within a given basis set, a crucial component of the limitations of applicability of WFT concerns the slow convergence of the energies and properties with respect to the quality of the basis set. 
+Beside the difficulties of accurately describing the molecular electronic structure within a given basis set, a crucial limitation of WFT methods is the slow convergence of the energies and properties with respect to the size of the basis set. As initially shown by the seminal work of Hylleraas\cite{Hyl-ZP-29} and further developed by Kutzelnigg and coworkers~\textit{et al.}\cite{Kut-TCA-85,KutKlo-JCP-91, NogKut-JCP-94}, the main convergence problem originates from the divergence of the Coulomb electron-electron interaction at the coalescence point, which induces a discontinuity in the first derivative of the exact wave function (the so-called electron-electron cusp). Describing such a discontinuity with an incomplete one-electron basis set is impossible and, as a consequence, the convergence of the computed energies and properties can be strongly affected. To attenuate this problem, extrapolation techniques have been developed, either based on a partial-wave expansion analysis~\cite{HelKloKocNog-JCP-97,HalHelJorKloKocOlsWil-CPL-98}, or more recently based on perturbative arguments\cite{IrmHulGru-arxiv-19}. A more rigorous approach to tackle the basis-set convergence problem is provided by the so-called R12 and F12 explicitly correlated methods\cite{Ten-TCA-12,TenNog-WIREs-12,HatKloKohTew-CR-12, KonBisVal-CR-12, GruHirOhnTen-JCP-17, MaWer-WIREs-18} which introduce a geminal function explicitly depending on the interelectronic distances ensuring the correct cusp condition in the wave function, and lead to a much faster convergence of the correlation energies than usual WFT methods. For instance, using the explicitly correlated version of coupled cluster with singles, doubles, and perturbative triples (CCSD(T)) in a triple-$\zeta$ quality basis set is equivalent to using a quintuple-$\zeta$ quality basis set with the usual CCSD(T) method\cite{TewKloNeiHat-PCCP-07}, although a computational overhead is introduced by the auxiliary basis set needed to compute the three- and four-electron integrals involved in F12 theory. In addition to the computational cost, a possible drawback of F12 theory is its rather complex formalism which requires non-trivial developments for adapting it to a new method. For strongly correlated systems, several multi-reference methods have been extended to explicitly correlation (see for instance Ref.~\onlinecite{Ten-CPL-07,ShiWer-JCP-10,TorKniWer-JCP-11,DemStanMatTenPitNog-PCCP-12,GuoSivValNee-JCP-17}), including approaches based on the so-called universal F12 theory which are potentially applicable to any electronic-structure computational methods~\cite{TorVal-JCP-09,KonVal-JCP-11,HauMaoMukKlo-CPL-12,BooCleAlaTew-JCP-12}. 
-As initially shown by the seminal work of Hylleraas\cite{Hyl-ZP-29} and further developed by Kutzelnigg \textit{et. al.}\cite{Kut-TCA-85,KutKlo-JCP-91, NogKut-JCP-94}, the main convergence problem originates from the divergence of the coulomb interaction at the electron coalescence point, which induces a discontinuity in the first-derivative of the exact wave function (the so-called electron-electron cusp). 
-Describing such a discontinuity with an incomplete basis set is impossible and as a consequence, the convergence of the computed energies and properties can be strongly affected. To attenuate this problem, extrapolation techniques has been developed, either based on the Hylleraas's expansion of the coulomb operator\cite{HalHelJorKloKocOlsWil-CPL-98}, or more recently based on perturbative arguments\cite{IrmHulGru-arxiv-19}. A more rigorous approach to tackle the basis set convergence problem has been proposed by the so-called R12 and F12 methods\cite{Ten-TCA-12,TenNog-WIREs-12,HatKloKohTew-CR-12, KonBisVal-CR-12, GruHirOhnTen-JCP-17, MaWer-WIREs-18} which introduce a function explicitly depending on the interelectronic coordinates ensuring the correct cusp condition in the wave function, and the resulting correlation energies converge much faster than the usual WFT. For instance, using the explicitly correlated version of coupled cluster with single, double and perturbative triple substitution (CCSD(T)) in a triple-$\zeta$ quality basis set is equivalent to a quintuple-$\zeta$ quality of the usual CCSD(T) method\cite{TewKloNeiHat-PCCP-07}, although inherent computational overhead are introduced by the auxiliary basis sets needed to resolve the rather complex three- and four-electron integrals involved in the F12 theory. In addition to the computational cost, a possible drawback of F12 theory is its rather complex formalism which requires a quite involved mathematical development to adapt to a new theory. In the context of strong correlation, several multi-reference methods have been extended to explicitly correlation (see for instance Ref. \cite{Ten-CPL-07,ShiWer-JCP-10,TorKniWer-JCP-11,DemStanMatTenPitNog-PCCP-12,GuoSivValNee-JCP-17}), including approaches based on so-called universal F12 which are potentially applicable to any electronic structure approaches\cite{TorVal-JCP-09,KonVal-JCP-11,HauMaoMukKlo-CPL-12,BooCleAlaTew-JCP-12}. 
 
-An alternative point of view to improve the convergence towards the CBS limit is to leave the short-range correlation effects to DFT and to use WFT to deal only with the long-range and/or strong-correlation effects. A rigorous approach to mix DFT and WFT is the range-separated DFT (RSDFT) formalism (see Ref.~\onlinecite{TouColSav-PRA-04} and references therein) which rely on a splitting of the coulomb interaction in terms of the interelectronic distance thanks to a range-separation parameter $\mu$. The advantage of such approach is at least two-folds: i) the DFT part deals only with the short-range part of the coulomb interaction, and therefore the usual semi-local approximations to the unknown exchange-correlation functional are more suited to that correlation regime, ii) as the WFT part deals with a smooth non divergent interaction, the exact wave function has no cusp\cite{GorSav-PRA-06} and therefore the basis set convergence is much faster\cite{FraMusLupTou-JCP-15}. 
+An alternative way to improve the convergence towards the complete-basis-set (CBS) limit is to treat the short-range correlation effects within DFT and to use WFT methods to deal only with the long-range and/or strong-correlation effects. A rigorous approach achieving this mixing of DFT and WFT is range-separated DFT (RSDFT) (see Ref.~\onlinecite{TouColSav-PRA-04} and references therein) which relies on a splitting of the Coulomb electron-electron interaction in terms of the interelectronic distance thanks to a range-separation parameter $\mu$. The advantage of this approach is at least two-fold: i) the DFT part deals primarily with the short-range part of the Coulomb interaction, and consequently the usual semilocal density-functional approximations are more accurate than for standard KS DFT; ii) the WFT part deals only with a smooth non-divergent interaction, and consequently the wave function has no electron-electron cusp\cite{GorSav-PRA-06} and the basis-set convergence is much faster\cite{FraMusLupTou-JCP-15}. A number of approximate RSDFT schemes have been developed involving single-reference \cite{AngGerSavTou-PRA-05, GolWerSto-PCCP-05, TouGerJanSavAng-PRL-09,JanHenScu-JCP-09, TouZhuSavJanAng-JCP-11, MusReiAngTou-JCP-15,KalTou-JCP-18,KalMusTou-JCP-19} or multi-reference \cite{LeiStoWerSav-CPL-97, FroTouJen-JCP-07, FroCimJen-PRA-10, HedKneKieJenRei-JCP-15, HedTouJen-JCP-18, FerGinTou-JCP-18} WFT methods. Nevertheless, there are still some open issues in RSDFT, such as remaining fractional-charge and fractional-spin errors in the short-range density functionals~\cite{MusTou-MP-17} or the dependence of the quality of the results on the value of the range-separation parameter $\mu$.
-Therefore, a number of approximate RS-DFT schemes have been developed within single-reference \cite{AngGerSavTou-PRA-05, GolWerSto-PCCP-05, TouGerJanSavAng-PRL-09,JanHenScu-JCP-09, TouZhuSavJanAng-JCP-11, MusReiAngTou-JCP-15} or multi-reference \cite{LeiStoWerSav-CPL-97, FroTouJen-JCP-07, FroCimJen-PRA-10, HedKneKieJenRei-JCP-15, HedTouJen-JCP-18, FerGinTou-JCP-18} WFT approaches. 
+% which can be seen as an empirical parameter. 
-Nevertheless, there are still some open issues in RSDFT, such remaining self-interaction errors or   
-the dependence of the quality of the results on the value of the range separation $\mu$ which can be seen as an empirical parameter. 
 
-Following this path, a very recent solution to the basis set convergence problem has been proposed by some of the preset authors\cite{GinPraFerAssSavTou-JCP-18} where they proposed to use RSDFT to take into account only the correlation effects outside a given basis set. The key idea in such a work is to realize that a wave function developed in an incomplete basis set is cusp-less could also come from a Hamiltonian with a non divergent electron-electron interaction. Therefore, the authors proposed a mapping with RSDFT through the introduction of an effective non-divergent interaction representing the usual coulomb interaction projected in an incomplete basis set. First applications to weakly correlated molecular systems have been successfully carried recently\cite{LooPraSceTouGin-JCPL-19} together with the first attempt to generalize this approach to excited states\cite{GinSceTouLoo-JCP-19}. 
+Building on the development of RSDFT, a possible solution to the basis-set convergence problem has been recently proposed by some of the present authors~\cite{GinPraFerAssSavTou-JCP-18} where RSDFT functionals are used to recover only the correlation effects outside a given basis set. The key point here is to realize that a wave function developed in an incomplete basis set is cuspless and could also come from a Hamiltonian with a non divergent electron-electron interaction. Therefore, a mapping with RSDFT can be introduced through the introduction of an effective non-divergent interaction representing the usual Coulomb electron-electron interaction projected in an incomplete basis set. First applications to weakly correlated molecular systems have been successfully carried out~\cite{LooPraSceTouGin-JCPL-19}, together with extensions of this approach to the calculations of excitation energies~\cite{GinSceTouLoo-JCP-19} and ionization potentials~\cite{LooPraSceGinTou-ARX-19}. The goal of the present work is to further develop this approach for the description of strongly correlated systems. The paper is organized as follows. In Section \ref{sec:theory} we recall the mathematical framework of the basis-set correction and we present the extension for strongly correlated systems. In particular, we focus on imposition of two important formal properties: size-consistency and spin-multiplet degeneracy.
-The goal of the present work is to push the development of this new theory toward the description of strongly correlated systems. 
+Then, in Section \ref{sec:results} we apply the method to the calculation of the potential energy curves of the C$_2$, N$_2$, O$_2$, F$_2$, and H$_{10}$ molecules up to the dissociation limit, representing prototypes of strongly correlated systems. Finally, we conclude in Section \ref{sec:conclusion}. 
-The paper is organized as follows: in section \ref{sec:theory} we recall the mathematical framework of the basis set correction and we expose the extension for strongly correlated systems. Within the present development, two important formal properties are imposed: the extensivity of the correlation energies together with the $S_z$ independence of the results. 
-Then in section \ref{sec:results} we discuss the potential energy surfaces (PES) of the C$_2$, N$_2$, O$_2$, F$_2$ and H$_{10}$ molecules up to full dissociation as a prototype of strongly correlated problems. Finally, we conclude in section \ref{sec:conclusion}. 
 
 %%%%%%%%%%%%%%%%%%%%%%%%
 \section{Theory}