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Manuscript/ExPerspective.bib

@@ -1,7 +1,7 @@
 %% This BibTeX bibliography file was created using BibDesk.
 %% http://bibdesk.sourceforge.net/
 
-%% Created for Pierre-Francois Loos at 2019-11-03 21:54:11 +0100 
+%% Created for Pierre-Francois Loos at 2019-11-06 15:17:19 +0100 
 
 
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Manuscript/ExPerspective.tex

@@ -74,11 +74,11 @@
 	\centering
   \includegraphics[width=\linewidth]{TOC}
 \end{wrapfigure}
-We provide a global overview of the successive steps that made possible to obtain increasingly accurate excited-state energies and properties, eventually leading to chemically-accurate excitation energies for small- and medium-size molecules.
+We provide a global overview of the successive steps that made possible to obtain increasingly accurate excitation energies and properties, eventually leading to chemically-accurate vertical transition energies for small- and medium-size molecules.
 First, we describe the evolution of \textit{ab initio} state-of-the-art methods, with originally Roos' CASPT2 method, and then third-order coupled cluster methods as in the renowned Thiel set of excitation energies described in a remarkable series of papers in the 2000's.
 More recently, this quest for highly accurate excitation energies was reinitiated thanks to the resurgence of selected configuration interaction (SCI) methods and their efficient parallel implementation.
 These methods have been able to routinely deliver highly-accurate excitation energies for small molecules as well as medium-size molecules in compact basis sets for single and double excitations.
-Second, we describe how these high-level methods and the creation of large, broad and accurate benchmark sets of excitation energies have allowed to assess fairly and accurately the performances of lower-order methods (\eg, TD-DFT, BSE, ADC, EOM-CC, etc) for different types of excited states.
+Second, we describe how these high-level methods and the creation of large, broad and accurate benchmark sets of excitation energies have allowed to assess fairly and accurately the performances of yet accurate, lower-order methods (\eg, TD-DFT, BSE, ADC, EOM-CC, etc) for different types of excited states.
 We conclude this \textit{Perspective} by discussing the current potentiality of these methods from both an expert and a non-expert point of view, and what we believe could be the future theoretical and technological developments in the field.
 \end{abstract}
 
@@ -94,7 +94,7 @@ The factors that makes this quest for high accuracy particularly delicate are ve
 First of all (and maybe surprisingly), it is, in most cases, tricky to obtain reliable and accurate experimental data that one can straightforwardly compare to theoretical values.
 In the case of vertical excitation energies, \ie, excitation energies at a fixed geometry, band maxima does not usually match theoretical values as one needs to take into account geometric relaxation and zero-point vibrational energy motion.
 Even more problematic, experimental spectra might not be available in gas phase, and, in the worst-case scenario, wrong assignments may have occurred.
-For a more faithful comparison between theory and experiment, although more computationally demanding, the so-called 0-0 energies are definitely a safer playground. \cite{Loo19b}
+For a more faithful comparison between theory and experiment, although more computationally demanding, the so-called 0-0 energies are definitely a safer playground. \cite{Loo18b,Loo19a,Loo19b}
 
 Another feature that makes excited states particularly fascinating and challenging is that they can be both extremely close in energy from each other and have very different natures ($\pi \ra \pi^*$, $n \ra \pi^*$, charge transfer, double excitation, valence, Rydberg, singlet, triplet, etc).
 Therefore, it would be highly desirable to possess a computational method or protocol providing a balanced treatment of the entire ``spectrum'' of excited states.
@@ -102,43 +102,42 @@ And let's be honest, none of the existing methods does provide this at an afford
 
 What are the requirement of the ``perfect'' theoretical model?
 As mentioned above, a balanced treatment of excited states with different character is highly desirable.
-Moreover, chemically-accurate excitation energies (\ie, with error smaller than $1$ kcal/mol or $0.043$ eV) would be also beneficial in order to provide a quantitative picture.
-The access to other properties, such as oscillator strength, dipole moment and analytical energy gradients, is also an asset if one wants to compare with experimental data.
-Let us not forget about minimal user input and chemical intuition requirements (\ie, black box method preferable) in order to minimise the potential bias brought by the user appreciation of the problem complexity.
+Moreover, chemically-accurate excitation energies (\ie, with error smaller than $1$ kcal/mol or $0.043$ eV) would be also beneficial in order to provide a quantitative chemical picture.
+The access to other properties, such as oscillator strengths, dipole moments and analytical energy gradients, is also an asset if one wants to compare with experimental data.
+Let us not forget about the requirements of minimal user input and chemical intuition (\ie, black box method preferable) in order to minimise the potential bias brought by the user appreciation of the problem complexity.
 Finally, low computational scaling with respect to system size and small memory footprint cannot be disregarded.
 Although the simultaneous fulfilment of all these requirements seems elusive, it is essential to keep these criteria in mind.
+Table \ref{tab:method} is here for fulfil such a purpose.
 
 %%% TABLE I %%%
 \begin{squeezetable}
 \begin{table}
-\caption{Scaling of various excited state methods.
-$N$ is the number of basis functions.}
+\caption{Scaling of various excited state methods with the number of basis functions $N$ and the availability of various key properties.}
 \label{tab:method}
 \begin{ruledtabular}
-\begin{tabular}{lccc}
-	\mr{2}{*}{Method}	&	Formal		&	Oscillator 	&	Analytic 	\\
-						&	scaling		&	strength	&	gradient	\\
+\begin{tabular}{lcccc}
+	\mr{2}{*}{Method}	&	Formal		&	Oscillator 	&	Analytic 		&	Typical 		\\
+						&	scaling		&	strength	&	gradients		&	error (eV)		\\
 	\hline
-	TD-DFT				&	$N^4$		&	\cmark		&	\cmark		\\
-	BSE@GW				&	$N^4$		&	\cmark		&	\xmark		\\
+	TD-DFT				&	$N^4$		&	\cmark		&	\cmark			&	$0.2$-$0.4$		\\
+	BSE@GW				&	$N^4$		&	\cmark		&	\xmark			&	$0.1$-$0.3$		\\
 	\\
-	CIS					&	$N^5$		&	\cmark		&	\cmark		\\
-	CIS(D)				&	$N^5$		&	\cmark		&	\cmark		\\
-	ADC(2)				&	$N^5$		&	\cmark		&	\cmark		\\
-	CC2					&	$N^5$		&	\cmark		&	\cmark		\\
+	CIS					&	$N^5$		&	\cmark		&	\cmark			&	$1.0$-$1.5$		\\
+	CIS(D)				&	$N^5$		&	\xmark		&	\cmark			&	$0.2$-$0.3$		\\
+	ADC(2)				&	$N^5$		&	\cmark		&	\cmark			&	$0.1$-$0.2$		\\
+	CC2					&	$N^5$		&	\cmark		&	\cmark			&	$0.1$-$0.2$		\\
 	\\
-	ADC(3)				&	$N^6$		&	\cmark		&	\xmark		\\
-	EOM-CCSD			&	$N^6$		&	\cmark		&	\cmark		\\
+	ADC(3)				&	$N^6$		&	\cmark		&	\xmark			&	$0.1$-$0.2$		\\
+	EOM-CCSD			&	$N^6$		&	\cmark		&	\cmark			&	$\sim 0.10$		\\
 	\\
-	CC3					&	$N^7$		&	\cmark		&	\xmark		\\
+	CC3					&	$N^7$		&	\cmark		&	\xmark			&	$\sim 0.04$		\\
 	\\
-	EOM-CCSDT			&	$N^8$		&	\xmark		&	\xmark		\\
-	EOM-CCSDTQ			&	$N^{10}$	&	\xmark		&	\xmark		\\
-	EOM-CCSDTQP			&	$N^{12}$	&	\xmark		&	\xmark		\\
+	EOM-CCSDT			&	$N^8$		&	\xmark		&	\xmark			&	$\sim 0.03$		\\
+	EOM-CCSDTQ			&	$N^{10}$	&	\xmark		&	\xmark			&	$\sim 0.01$		\\
 	\\
-	CASPT2				&	$N!$		&	\cmark		&	\cmark		\\
-	NEVPT2				&	$N!$		&	\cmark		&	\cmark		\\
-	SCI or FCI			&	$N!$		&	\xmark		&	\xmark		\\
+	CASPT2	or NEVPT2	&	$N!$		&	\cmark		&	\cmark			&	$0.1$-$0.2$		\\
+	SCI					&	$N!$		&	\cmark		&	\cmark			&	$\sim 0.03$		\\
+	FCI					&	$N!$		&	\cmark		&	\xmark			&	$0.0$			\\
 \end{tabular}
 \end{ruledtabular}
 \end{table}
@@ -148,59 +147,67 @@ $N$ is the number of basis functions.}
 %** HISTORY **%
 %**************
 Before detailing some key past and present contributions towards the obtention of highly-accurate excitation energies, we start by giving a historical overview of the various excited-state \textit{ab initio} methods that have emerged in the last fifty years.
-Interestingly, for pretty much every single method, the theory was derived much earlier than their actual implementation in electronic structure software packages.
+Interestingly, for pretty much every single method, the theory was derived much earlier than their actual implementation in electronic structure software packages (same applies to the analytic gradients when available).
 Here, we only mention methods that, we think, ended up becoming mainstream.
 
 %%%%%%%%%%%%%%%%%%%%%
 %%% POPLE'S GROUP %%%
 %%%%%%%%%%%%%%%%%%%%%
-The first mainstream \textit{ab initio} method for excited states was probably CIS (configuration interaction with singles) which has been around since the 70's. \cite{Ben71} 
+The first mainstream \textit{ab initio} method for excited states was probably CIS (configuration interaction with singles) which has been around since the 1970's. \cite{Ben71} 
 CIS lacks electron correlation and therefore grossly overestimates excitation energies and wrongly orders excited states.
+It is not unusual to have errors of the order of $1$ eV which precludes the usage of CIS as a quantitative quantum chemistry method.
 Twenty years later, CIS(D) which adds a second-order perturbative correction to CIS was developed and implemented thanks to the efforts of Head-Gordon and coworkers. \cite{Hea94} 
+This second-order correction significantly reduces the magnitude of the error associated with CIS excitation energies, with a typical error range of $0.2$-$0.3$ eV.
  
 %%%%%%%%%%%%%%%%%%%
 %%% ROOS' GROUP %%%
 %%%%%%%%%%%%%%%%%%%
-In the early 90's, the complete-active-space self-consistent field (CASSCF) method \cite{And90} and its second-order perturbation-corrected variant CASPT2 \cite{And92} (both originally developed in Roos' group) appeared.
+In the early 1990's, the complete-active-space self-consistent field (CASSCF) method \cite{And90} and its second-order perturbation-corrected variant CASPT2 \cite{And92} (both originally developed in Roos' group) appeared.
 This was a real breakthrough.
 Although it took more than ten years to obtain analytic nuclear gradients, \cite{Cel03} CASPT2 was probably the first method that could provide quantitative results for molecular excited states of genuine photochemical interest. \cite{Roo96}
 Driven by Celestino and Malrieu, \cite{Ang01} the creation of the second-order $n$-electron valence state perturbation theory (NEVPT2) method several years later was able to cure some of the main theoretical deficiencies of CASPT2.
-In particular, NEVPT2 is known to be intruder state free.
-The limited applicability of these so-called multiconfigurational methods is mainly due to the necessity of defining an active space, as well as their factorial computational growth with the number of active electrons and orbitals.
+For example, NEVPT2 is known to be intruder state free and size consistent.
+The limited applicability of these so-called multiconfigurational methods is mainly due to the necessity of defining an active space based on the desired transition(s), as well as their factorial computational growth with the number of active electrons and orbitals.
+With a typical minimal valence active space, the typical error range of CASPT2 or NEVPT2 is $0.1$-$0.2$ eV.
 
 %%%%%%%%%%%%%
 %%% TDDFT %%%
 %%%%%%%%%%%%%
 The advent of time-dependent density-functional theory (TD-DFT) \cite{Run84,Dre05} was a real shock for the community as TD-DFT was able to provide accurate excitation energies at a much lower cost than its predecessors in a very black-box way.
-However, a large number of shortcomings were quickly discovered. \cite{Dre05}
-One of the most annoying feature of TD-DFT in the present context is its inability to describe, even qualitatively, charge-transfer states, \cite{Toz99} Rydberg states, \cite{Toz98} and double excitations. \cite{Lev06}
+For low-lying excited states, TD-DFT calculations relying on hybrid exchange-correlation functionals have a typical error of $0.2$-$0.4$ eV.
+However, a large number of shortcomings were quickly discovered. \cite{Toz98,Toz99,Dre04,Dre05,Lev06}
+In the present context, one of the most annoying feature of TD-DFT --- in its most standard form --- is its inability to describe, even qualitatively, charge-transfer states, \cite{Toz99,Dre04} Rydberg states, \cite{Toz98} and double excitations. \cite{Lev06}
 Moreover, the difficulty of making TD-DFT systematically improvable obviously hampers its applicability.
 One of the main issue is the selection of the exchange-correlation functional from an ever-growing zoo of functionals and the variation of the excitation energies that one can observe with different functionals. \cite{Sue19}
-Despite all of this, TD-DFT is still nowadays the most employed excited-state method in the electronic structure community.
+Despite all of this, TD-DFT is still nowadays the most employed excited-state method in the electronic structure community (and beyond).
  
 %%%%%%%%%%%%%%%%%%
 %%% CC METHODS %%%
 %%%%%%%%%%%%%%%%%%
 Thanks to the development of coupled cluster (CC) response theory, \cite{Koc90} and the huge growth of computational ressources, equation-of-motion coupled cluster with singles and doubles (EOM-CCSD) \cite{Sta93} became mainstream in the 2000's.
 EOM-CCSD gradients were also quickly available. \cite{Sta95}
+With EOM-CCSD, it is not unusual to have errors as small as $0.1$ eV. 
 Its third-order version, EOM-CCSDT, was also implemented and provides, at a significant higher cost, high accuracy for single excitations. \cite{Nog87}
 Although extremely expensive and tedious to implement, higher orders are also technically possible for small systems thanks to automatically-generated code. \cite{Kuc91,Hir04}
 The EOM-CC family of methods was quickly followed by an approximated and computationally lighter family with, in front line, the second-order CC2 model \cite{Chr95} and its third-order extension, CC3. \cite{Chr95b}
-The series CC2, CCSD, CC3, CCSDT defines a hierarchy of models with $N^5$, $N^6$, $N^7$ and $N^8$ scaling, respectively.
+As a $N^7$ method, CC3 has a particularly interesting accuracy/cost ratio with errors usually below chemical accuracy.
+The series CC2, CCSD, CC3, CCSDT defines a hierarchy of models with $N^5$, $N^6$, $N^7$ and $N^8$ scaling, respectively, where $N$ is the number of basis functions.
 Thanks to the introduction of triples, EOM-CCSDT and CC3 also provide qualitative results for double excitations, a feature that is completely absent from EOM-CCSD and CC2. \cite{Loo19c} 
 
 %%%%%%%%%%%%%%%%%%%
 %%% ADC METHODS %%%
 %%%%%%%%%%%%%%%%%%%
-It is also important to mention the recent advent of the second- and third-order algebraic diagrammatic construction [ADC(2) \cite{Sch82} and ADC(3) \cite{Tro99,Har14}] which scale as $N^5$ and $N^6$ respectively.
-These Green's function one-electron propagator techniques represent interesting alternatives thanks to their reduced scaling compared to their CC equivalents.
+It is also important to mention the recent advent of the second- and third-order algebraic diagrammatic construction [ADC(2) \cite{Sch82} and ADC(3) \cite{Tro99,Har14}] which scale as $N^5$ and $N^6$, respectively.
 Moreover, Dreuw's group has put an enormous amount of work to provide a fast and efficient implementation of these methods as well as other interesting variants. \cite{Dre15} 
+These Green's function one-electron propagator techniques represent interesting alternatives thanks to their reduced scaling compared to their CC equivalents.
+In that regard, ADC(2) is particularly attractive with an error generally around $0.1$-$0.2$ eV.
+However, we have recently observed that ADC(3) might generally overcorrect the ADC(2) excitation energies. \cite{Loo18a,Loo20}
 
 %%%%%%%%%%%%%%
 %%% BSE@GW %%%
 %%%%%%%%%%%%%%
 Finally, let us mention the many-body Green's function Bethe-Salpeter equation (BSE) formalism \cite{Str88} (which is usually performed on top of a GW calculation \cite{Hed65}).
-BSE has gained momentum in the past few years and is a good candidate as a computationally inexpensive electronic structure theory method that can model accurately excited-state energetics and their corresponding properties. \cite{Jac17b,Bla18}
+BSE has gained momentum in the past few years and is a good candidate as a computationally inexpensive electronic structure theory method that can model accurately excited-state energetics (with a typical error of $0.1$-$0.3$ eV) and their corresponding properties. \cite{Jac17b,Bla18}
 One of the main advantage of BSE compared to TD-DFT (with a similar computational cost) is that it allows a faithful description of charge-transfer states and, when performed on top of a (partially) self-consistently GW calculation, BSE@GW has been shown to be weakly dependent on its starting point. \cite{Jac16,Gui18}
 However, due to the adiabatic (\ie, static) approximation, doubly-excited states are completely absent from the BSE spectrum.
 
@@ -213,7 +220,7 @@ Indeed, it has been noticed long ago that, even inside a predefined subspace of
 The main advantage of SCI methods is that no a priori assumption is made on the type of electron correlation. 
 Therefore, at the price of a brute force calculation, a SCI calculation is less biased by the user appreciation of the problem complexity.
 One of the strength of our implementation (based on the CIPSI algorithm developed by Huron, Rancurel, and Malrieu in 1973 \cite{Hur73}) is its parallel efficiency which makes possible to run on thousands of cores.
-Thanks to these tremendous features, SCI methods delivers near FCI quality excitation energies for both singly- and doubly-excited states. \cite{Hol17,Chi18,Loo18a,Loo19c}
+Thanks to these tremendous features, SCI methods delivers near FCI quality excitation energies for both singly- and doubly-excited states, \cite{Hol17,Chi18,Loo18a,Loo19c} with an error of roughly $0.03$ eV (mostly due to the extrapolation procedure). \cite{Gar19}
 However, although the \textit{``exponential wall''} is pushed back, this type of methods is only applicable to molecules with a small number of heavy atoms with relatively compact basis sets.
 
 %%%%%%%%%%%%%%%
@@ -232,7 +239,8 @@ In summary, each method has its own strengths and weaknesses, and none of them i
 %%% FIG 2 %%%
 \begin{figure}
 	\includegraphics[width=\linewidth]{Set2}
-	\caption{Mean absolute error (in eV) with respect to exFCI excitation energies from the Jacquemin set \#2 (as described in Ref.~\onlinecite{Loo19c}) for various methods taking into account at least triple excitations.}
+	\caption{Mean absolute error (in eV) (with respect to exFCI) excitation energies from the Jacquemin set \#2 (as described in Ref.~\onlinecite{Loo19c}) for various methods taking into account at least triple excitations.
+	$\%T_1$ corresponds to the percentage of single excitations in the transition.}
 	\label{fig:Set2}
 \end{figure}
 %%% %%% %%%
@@ -244,6 +252,7 @@ In summary, each method has its own strengths and weaknesses, and none of them i
 	\label{fig:Set3}
 \end{figure*}
 %%% %%% %%%
+
 %*****************
 %** BENCHMARKS ***
 %*****************