Day 2

Manuscript/ExPerspective.tex

@@ -1,5 +1,5 @@
 \documentclass[aps,prb,reprint,noshowkeys,superscriptaddress]{revtex4-1}
-\usepackage{graphicx,dcolumn,bm,xcolor,microtype,multirow,amscd,amsmath,amssymb,amsfonts,physics,mhchem,longtable,pifont,wrapfig,txfonts}
+\usepackage{graphicx,dcolumn,bm,xcolor,microtype,multirow,amscd,amsmath,amssymb,amsfonts,physics,mhchem,longtable,pifont,wrapfig,txfonts,multirow}
 
 \usepackage{natbib}
 \usepackage[extra]{tipa}
@@ -37,6 +37,7 @@
     citecolor=blue
 }
 \newcommand{\mc}{\multicolumn}
+\newcommand{\mr}{\multirow}
 \newcommand{\ie}{\textit{i.e.}}
 \newcommand{\eg}{\textit{e.g.}}
 \newcommand{\ra}{\rightarrow}
@@ -88,84 +89,140 @@ iv) what we believe could be the future developments in the field.
 %%%%%%%%%%%%%%%%%%%%
 %%% INTRODUCTION %%%
 %%%%%%%%%%%%%%%%%%%%
-The accurate modelling of excited-state properties from \textit{ab initio} quantum chemistry methods is a challenging yet self-proclaimed ambition of the electronic structure theory community that will certainly keep us busy for (at the very least) the next few decades to come.
+The accurate modelling of excited-state properties from \textit{ab initio} quantum chemistry methods is a challenging yet self-proclaimed ambition of the electronic structure theory community that will certainly keep us busy for (at the very least) the next few decades to come (see, for example, Ref.~\onlinecite{Gon12} and references therein).
 Of particular interest is the access to precise excitation energies, \ie, the energy difference between the ground and excited states, and their intimate link with photochemical processes and photochemistry in general.
-The factors that make this quest for high accuracy particularly delicate are very diverse.
+The factors that makes this quest for high accuracy particularly delicate are very diverse.
+
 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 the geometric relaxation and the zero-point vibrational energy motion.
-Even more problematic, experimental spectra might not available in gas phase, and in the worst-case scenario, wrong assignments may occur.
-For a faithful comparison between theory and experimental, the so-called 0-0 energies are definitely a better playground.
-However, they require from a theoretical point of view access to the optimized excited-state geometry as well as its harmonic vibration frequencies.
+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 occured.
+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}
+%However, they require, from a theoretical point of view, access to the optimised excited-state geometry as well as its harmonic vibration frequencies.
 
-What makes also excited states fascinating and challenging at the same time are usually extremely close in energy and can have very different nature ($\pi \ra \pi^*$, $n \ra \pi^*$, charge transfer, double excitations, Rydberg states, etc).
-Therefore one needs --- in principle at least ---  a computational method providing a balanced theoretical treatment of all these excited states.
-And let's be honest, none of the existing methods does provide this.
+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 nature ($\pi \ra \pi^*$, $n \ra \pi^*$, charge transfer, double excitation, Rydberg, 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.
+And let's be honest, none of the existing methods does provide this at an affordable cost.
 
-If you were asking for the perfect excited-state method for Christmas, what would you (realistically) ask for?
-The requirement for the ``perfect'' theoretical model would be:
-i) balanced treatment of excited states with different character.
-ii) chemically accurate excitation energies ie error smaller than 1 kcal/mol or $0.05$ eV.
-iii) other properties (such as oscillator strength, dipole moment, optimisation for excited state geometries (analytical highly desirable)
-iv) Minimal user input requirement (black box method) and minimisation of chemical intuition in order not to bias the results.
-v) Low computational scaling with respect to system size and small memory footprint.
-In summary, each method has its own strengths and weaknesses, and none of them is able to provide accurate and reliable excitation energies in all scenarios. 
-
-Although people usually don't really like reading, reviewing or even the idea of benchmark studies, these are definitely essential for the validation of existing theoretical methods and to understand their strengths and, more importantly, their limitations.
-
-
-There is a clear need for computationally inexpensive electronic structure theory methods which can model accurately excited-state energetics and their corresponding properties. 
-Although time-dependent density-functional theory (TD-DFT) and the Bethe-Salpeter equation (BSE) formalism have emerged as powerful tools for computing excitation energies, fundamental deficiencies remain to be solved. 
-For example, the simplest and most widespread approximation in state-of-the-art electronic structure programs where TD-DFT and BSE are implemented consists in neglecting memory effects. 
-This has drastic consequences such as, for example, the complete absence of double excitations from the TD-DFT and BSE spectra.
+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 gradient, 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 bias brought by the user's 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 always essential to keep these criteria in mind.
+%In summary, each method has its own strengths and weaknesses, and none of them is able to provide accurate and reliable excitation energies in all scenarios. 
 
 %%% TABLE I %%%
+%\begin{squeezetable}
 \begin{table}
 \caption{Scaling of various excited state methods.
 $N$ is the number of basis functions.}
 \label{tab:method}
 \begin{ruledtabular}
-\begin{tabular}{lcccl}
-	Method			&	Scaling		&	Oscillator 	&	Analytic 		&	Software	\\
-					&				&	strength	&	gradient		&		\\
+\begin{tabular}{lccc}
+	\mr{2}{*}{Method}	&	Formal		&	Oscillator 	&	Analytic 	\\
+						&	scaling		&	strength	&	gradient	\\
 	\hline
-	TD-DFT			&	$N^4$		&	\cmark		&	\cmark			&	GAUSSIAN	\\
-	BSE@GW			&	$N^6$		&	\cmark		&	\xmark			&	FIESTA		\\
-	CIS(D)			&	$N^7$		&	\cmark		&	\cmark			&	DALTON		\\
-	CIS(D$_\infty$)	&	$N^7$		&	\cmark		&	\cmark			&	MRCC		\\
-	CC2				&	$N^6$		&	\cmark		&	\cmark			&	DALTON		\\
-	CC3				&	$N^7$		&	\xmark		&	\xmark			&	DALTON		\\
-	ADC(2)			&	$N^7$		&	\cmark		&	\cmark			&	Q-CHEM		\\
-	ADC(3)			&	$N^7$		&	\cmark		&	\xmark			&	Q-CHEM		\\
-	EOM-CCSD		&	$N^7$		&	\cmark		&	\cmark			&	DALTON		\\
-	STEOM-CCSD		&	$N^7$		&	\cmark		&	\xmark			&	ORCA		\\
-	EOM-CCSDR(3)	&	$N^7$		&	\cmark		&	\cmark			&	DALTON		\\
-	EOM-CCSDT-3		&	$N^7$		&	\xmark		&	\xmark			&	CFOUR		\\
-	EOM-CCSDT		&	$N^8$		&	\xmark		&	\xmark			&	CFOUR		\\
-	EOM-CCSDTQ		&	$N^{10}$	&	\xmark		&	\xmark			&	MRCC		\\
-	EOM-CCSDTQP		&	$N^{12}$	&	\xmark		&	\xmark			&	MRCC		\\
-	CASPT2			&	$e^N$		&	\cmark		&	\cmark			&	MOLPRO		\\
-	NEVPT2			&	$e^N$		&	\xmark		&	\xmark			&	MOLPRO		\\
-	sCI				&	$e^N$		&	\xmark		&	\xmark			&	QP2			\\
+	CIS					&	$N^5$		&	\cmark		&	\cmark		\\
+	CIS(D)				&	$N^5$		&	\cmark		&	\cmark		\\
+	ADC(2)				&	$N^5$		&	\cmark		&	\cmark		\\
+	CC2					&	$N^5$		&	\cmark		&	\cmark		\\
+	\\
+	TD-DFT				&	$N^6$		&	\cmark		&	\cmark		\\
+	BSE@GW				&	$N^6$		&	\cmark		&	\xmark		\\
+	ADC(3)				&	$N^6$		&	\cmark		&	\xmark		\\
+	EOM-CCSD			&	$N^6$		&	\cmark		&	\cmark		\\
+	\\
+	CC3					&	$N^7$		&	\cmark		&	\xmark		\\
+	\\
+	EOM-CCSDT			&	$N^8$		&	\xmark		&	\xmark		\\
+	EOM-CCSDTQ			&	$N^{10}$	&	\xmark		&	\xmark		\\
+	EOM-CCSDTQP			&	$N^{12}$	&	\xmark		&	\xmark		\\
+	\\
+	CASPT2				&	$N!$		&	\cmark		&	\cmark		\\
+	NEVPT2				&	$N!$		&	\cmark		&	\cmark		\\
+	FCI					&	$N!$		&	\xmark		&	\xmark		\\
 \end{tabular}
 \end{ruledtabular}
 \end{table}
+%\end{squeezetable}
+
+%%%%%%%%%%%%%%%
+%%% 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.
+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} 
+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} 
+
 %%%%%%%%%%%%%%%%%%%
 %%% ROOS' GROUP %%%
 %%%%%%%%%%%%%%%%%%%
-Roos' group probably kick started the whole thing thanks to their CASPT2 method.
+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 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, the creation of the second-order $n$-electron valence state perturbation theory (NEVPT2) method \cite{Ang01} 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.
+
+%%%%%%%%%%%%%
+%%% TDDFT %%%
+%%%%%%%%%%%%%
+The advent of time-dependent density-functional theory (TD-DFT) \cite{Run84} 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.
+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}
+Moreover, the difficulty of making TD-DFT systematically improvable obviously hampers its applicability.
+One of the main problem is the selection of the exchange-correlation functional and the variation of the results one can obtain with different choices.
+Despite all of this, TD-DFT is still nowadays the most employed excited-state method in the electronic structure community.
+ 
+%%%%%%%%%%%%%%%%%%
+%%% CC METHODS %%%
+%%%%%%%%%%%%%%%%%%
+Thanks to the development of coupled cluster (CC) response theory, \cite{Koc90} and the hugh growth of computer power, EOM-CCSD \cite{Sta93}became mainstream in the 2000's.
+EOM-CCSD gradient were also quickly available. \cite{Sta95}
+Higher orders are possible but extremely expensive. \cite{Nog87, Kuc91}
+This was quickly followed by the CC2 \cite{Chr95} and CC3 \cite{Chr95b} methods.
+
+
+%%%%%%%%%%%%%%
+%%% BSE@GW %%%
+%%%%%%%%%%%%%%
+In that regard, the Bethe-Salpeter equation (BSE) formalism is a real plus.
+
+There is a clear need for computationally inexpensive electronic structure theory methods which can model accurately excited-state energetics and their corresponding properties. 
+Although  and TD-DFT the BSE formalism have emerged as powerful tools for computing excitation energies, fundamental deficiencies remain to be solved. 
+For example, the simplest and most widespread approximation in state-of-the-art electronic structure programs where TD-DFT and BSE are implemented consists in neglecting memory effects. 
+This has drastic consequences such as, for example, the complete absence of double excitations from the TD-DFT and BSE spectra.
 
 %%%%%%%%%%%%%%%%%%%%%
 %%% THIEL'S GROUP %%%
 %%%%%%%%%%%%%%%%%%%%%
-A major contribution originates from the Thiel's group
+A major contribution originates from the Thiel's group \cite{Sch08,Sil08,Sau09,Sil10b,Sil10c} with in particular the introduction of the Thiel's set  gathering, for the first time, a large number of excitation energies of different natures. \cite{Sch08}
 
+Although people usually don't really like reading, reviewing or even the idea of benchmark studies, these are definitely essential for the validation of existing theoretical methods and to understand their strengths and, more importantly, their limitations.
+
+%%%%%%%%%%%%%%%%%%%
+%%% ADC METHODS %%%
+%%%%%%%%%%%%%%%%%%%
+ADC(2) \cite{Dre15}, ADC(3) \cite{Dre15},  
 %%%%%%%%%%%%%%%%%%%
 %%% SCI METHODS %%%
 %%%%%%%%%%%%%%%%%%%
+
+Alternatively to CC and multiconfigurational methods, one can also compute transition energies for various types of excited states using selected configuration interaction (sCI) methods \cite{Ben69,Whi69,Hur73} which have recently demonstrated their ability to reach near full CI (FCI) quality energies for small molecules \cite{Gin13,Caf14,Gin15,Caf16,Gar17b,Hol16,Sha17,Hol17,Chi18,Sce18a,Sce18b,Loo18b,Gar18}.
+The idea behind such methods is to avoid the exponential increase of the size of the CI expansion by retaining the most energetically relevant determinants only, thanks to the use of a second-order energetic criterion to select perturbatively determinants in the FCI space.
+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.
+
+
 In the past five years, we have witnessed a resurgence of selected CI (sCI) methods thanks to the development and implementation of new and fast algorithm to select cleverly determinants in the FCI space.
 sCI methods rely on the same principle as the usual CI approach, except that determinants are not chosen a priori based on occupation or excitation criteria but selected among the entire set of determinants based on their estimated contribution to the FCI wave function. 
 Indeed, it has been noticed long ago that, even inside a predefined subspace of determinants, only a small number of them significantly contributes.
-Therefore, an on-the-fly selection of determinants is a rather natural idea that has been proposed in the late 1960s by Bender and Davidson19 as well as Whitten and Hackmeyer.s
+Therefore, an on-the-fly selection of determinants is a rather natural idea that has been proposed in the late 1960s by Bender and Davidson19 as well as Whitten and Hackmeyer's
 sCI methods are still very much under active development. 
 The main advantage of sCI methods is that no a priori assumption is made on the type of electronic correlation. 
 Therefore, at the price of a brute force calculation, a sCI calculation is less biased by the user's appreciation of the problem's complexity.