299 lines
22 KiB
TeX
299 lines
22 KiB
TeX
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\newcommand{\LCPQ}{Laboratoire de Chimie et Physique Quantiques, Universit\'e de Toulouse, CNRS, UPS, France}
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\newcommand{\CEISAM}{Laboratoire CEISAM - UMR CNRS 6230, Universit\'e de Nantes, 2 Rue de la Houssini\`ere, BP 92208, 44322 Nantes Cedex 3, France}
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\begin{document}
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\title{The Quest For Highly Accurate Excitation Energies}
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\author{Pierre-Fran\c{c}ois \surname{Loos}}
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\email{loos@irsamc.ups-tlse.fr}
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\affiliation{\LCPQ}
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\author{Anthony \surname{Scemama}}
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\affiliation{\LCPQ}
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\author{Denis \surname{Jacquemin}}
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\email[Corresponding author: ]{Denis.Jacquemin@univ-nantes.fr}
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\affiliation{\CEISAM}
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\begin{abstract}
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%\begin{wrapfigure}[13]{o}[-1.25cm]{0.5\linewidth}
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% \centering
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% \includegraphics[width=\linewidth]{TOC}
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%\end{wrapfigure}
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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.
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We describe
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i) the evolution of ab initio reference methods, e.g., originally CASPT2 (Roos, Serrano-Andres in the 1990's), then high-level CCn (as in the
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acclaimed Thiel benchmark series in the 2000's), and now selected CI methods thanks to their resurgence in the past five years;
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ii) how these high-level methods have allowed to assess fairly and accurately the performances of lower-order methods, e.g., ADC,
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TD-DFT and BSE;
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iii) the current potentiality of these various methods from both an expert and non-expert points of view;
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iv) what we believe could be the future developments in the field.
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\end{abstract}
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\maketitle
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%%%%%%%%%%%%%%%%%%%%
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%%% INTRODUCTION %%%
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%%%%%%%%%%%%%%%%%%%%
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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).
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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.
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The factors that makes this quest for high accuracy particularly delicate are very diverse.
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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.
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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.
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Even more problematic, experimental spectra might not be available in gas phase, and, in the worst-case scenario, wrong assignments may have occured.
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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}
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%However, they require, from a theoretical point of view, access to the optimised excited-state geometry as well as its harmonic vibration frequencies.
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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, valence, Rydberg, singlet, triplet, etc).
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Therefore, it would be highly desirable to possess a computational method or protocol providing a balanced treatment of the entire ``spectrum'' of excited states.
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And let's be honest, none of the existing methods does provide this at an affordable cost.
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What are the requirement of the ``perfect'' theoretical model?
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As mentioned above, a balanced treatment of excited states with different character is highly desirable.
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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.
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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.
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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.
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Finally, low computational scaling with respect to system size and small memory footprint cannot be disregarded.
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Although the simultaneous fulfilment of all these requirements seems elusive, it is always essential to keep these criteria in mind.
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%%% TABLE I %%%
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%\begin{squeezetable}
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\begin{table}
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\caption{Scaling of various excited state methods.
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$N$ is the number of basis functions.}
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\label{tab:method}
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\begin{ruledtabular}
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\begin{tabular}{lccc}
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\mr{2}{*}{Method} & Formal & Oscillator & Analytic \\
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& scaling & strength & gradient \\
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\hline
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CIS & $N^5$ & \cmark & \cmark \\
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CIS(D) & $N^5$ & \cmark & \cmark \\
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ADC(2) & $N^5$ & \cmark & \cmark \\
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CC2 & $N^5$ & \cmark & \cmark \\
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\\
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TD-DFT & $N^6$ & \cmark & \cmark \\
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BSE@GW & $N^6$ & \cmark & \xmark \\
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ADC(3) & $N^6$ & \cmark & \xmark \\
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EOM-CCSD & $N^6$ & \cmark & \cmark \\
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\\
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CC3 & $N^7$ & \cmark & \xmark \\
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\\
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EOM-CCSDT & $N^8$ & \xmark & \xmark \\
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EOM-CCSDTQ & $N^{10}$ & \xmark & \xmark \\
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EOM-CCSDTQP & $N^{12}$ & \xmark & \xmark \\
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\\
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CASPT2 & $N!$ & \cmark & \cmark \\
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NEVPT2 & $N!$ & \cmark & \cmark \\
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FCI & $N!$ & \xmark & \xmark \\
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\end{tabular}
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\end{ruledtabular}
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\end{table}
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%\end{squeezetable}
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%**************
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%** HISTORY **%
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%**************
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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.
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Interestingly, for pretty much every single methods, the theory was derived much earlier than their actual implementation in electronic structure software packages.
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Here, we only mention methods that, we think, ended up becoming mainstream.
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%%%%%%%%%%%%%%%%%%%%%
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%%% POPLE'S GROUP %%%
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%%%%%%%%%%%%%%%%%%%%%
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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}
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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}
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%%%%%%%%%%%%%%%%%%%
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%%% ROOS' GROUP %%%
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%%%%%%%%%%%%%%%%%%%
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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.
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This was a real breakthrough.
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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}
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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.
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In particular, NEVPT2 is known to be intruder state free.
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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.
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%%%%%%%%%%%%%
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%%% TDDFT %%%
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%%%%%%%%%%%%%
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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.
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However, a large number of shortcomings were quickly discovered. \cite{Dre05}
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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}
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Moreover, the difficulty of making TD-DFT systematically improvable obviously hampers its applicability.
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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 one can obtain with different choices. \cite{Sue19}
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Despite all of this, TD-DFT is still nowadays the most employed excited-state method in the electronic structure community.
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%%%%%%%%%%%%%%%%%%
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%%% CC METHODS %%%
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%%%%%%%%%%%%%%%%%%
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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.
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EOM-CCSD gradients were also quickly available. \cite{Sta95}
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Its third-order version, EOM-CCSDT, was also implemented and provides, at a significant higher cost, high accuracy for single excitations. \cite{Nog87}
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Thanks to the introduction of triples, EOM-CCSDT also provides qualitative results for double excitations, a feature that is completely absent from EOM-CCSD. \cite{Loo19c}
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Although extremely expensive and tedious to implement, higher orders are also technically possible for small systems thanks to automatically-generated code. \cite{Kuc91, Hir04, }
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The EOM-CC family of methods was quickly followed by a slightly computationally lighter family with in front line the second-order CC2 method \cite{Chr95} and its third-order extension CC3 \cite{Chr95b} with formal computational scaling of $N^5$ and $N^7$ compared to $N^6$ and $N^8$ for EOM-CCSD and EOM-CCSDT, respectively.
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%%%%%%%%%%%%%%%%%%%
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%%% ADC METHODS %%%
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%%%%%%%%%%%%%%%%%%%
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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, represent interesting alternatives thanks to their reduced scaling compared to their CC equivalents.
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Moreover, Dreuw's group has put an enormous amount of work to provide a fast and efficient implementation. \cite{Dre15}
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%%%%%%%%%%%%%%
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%%% BSE@GW %%%
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%%%%%%%%%%%%%%
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More recentky Finally, let us mention the Bethe-Salpeter equation (BSE) formalism (which is usually performed on top of a GW calculation).
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There is a clear need for computationally inexpensive electronic structure theory methods which can model accurately excited-state energetics and their corresponding properties.
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Although and TD-DFT the BSE formalism have emerged as powerful tools for computing excitation energies, fundamental deficiencies remain to be solved.
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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.
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This has drastic consequences such as, for example, the complete absence of double excitations from the TD-DFT and BSE spectra.
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%%%%%%%%%%%%%%%%%%%
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%%% SCI METHODS %%%
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%%%%%%%%%%%%%%%%%%%
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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,Gin15,Caf16,Gar17b,Hol16,Sha17,Hol17,Chi18,Sce18,Sce18b,Loo18a,Gar18}.
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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.
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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.
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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.
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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.
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Indeed, it has been noticed long ago that, even inside a predefined subspace of determinants, only a small number of them significantly contributes.
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Therefore, an on-the-fly selection of determinants is a rather natural idea that has been proposed in the late 1960's by Bender and Davidson as well as Whitten and Hackmeyer's SCI methods are still very much under active development.
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The main advantage of SCI methods is that no a priori assumption is made on the type of electronic correlation.
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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.
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The approach that we have implemented in QUANTUM PACKAGE is based on the CIPSI algorithm developed by Huron, Rancurel, and Malrieu in 1973.
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One of the strength of our implementation is its parallel efficiency which makes it possible to run on a ver large number of cores.
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%%%%%%%%%%%%%%%
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%%% SUMMARY %%%
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%%%%%%%%%%%%%%%
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In summary, each method has its own strengths and weaknesses, and none of them is able to provide accurate and reliable excitation energies for all classes of electronic states.
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This ultimately leads to an unbalanced description of different excited states.
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%*****************
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%** BENCHMARKS ***
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%*****************
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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.
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%%%%%%%%%%%%%%%%%%%
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%%% THIEL'S SET %%%
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%%%%%%%%%%%%%%%%%%%
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A major contribution was provided by the group of Walter Thiel \cite{Sch08,Sil08,Sau09,Sil10b,Sil10c} with the introduction of the so-called Thiel set of excitation energies. \cite{Sch08}
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For the first time, this set was large, broad and accurate enough to be used as a proper benchmarking set for excited states.
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More specifically, it gathers a large number of excitation energies consisting of 28 medium-size organic molecules with a total of 223 excited states (152 singlet and 71 triplet states).
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In their first study they performed CC2, EOM-CCSD, CC3 and MS-CASPT2 calculations (in the TZVP basis) in order to provide (based on additional high-level literature data) a list of best theoretical estimates (TBEs) for all these transitions.
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Their main conclusion was that \textit{``CC3 and CASPT2 excitation energies are in excellent agreement for states which are dominated by single excitations''}.
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These TBEs were sooner refined with the larger aug-cc-pVTZ basis set, \cite{Sil10b} highlighting the importance of diffuse functions (especially for Rydberg states).
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As a direct evidence of the value of reference data, these TBEs were quickly picked up to benchmark various computationally effective methods from semi-empirical to state-of-the-art \textit{ab initio} methods (see Ref.~\onlinecite{Loo18a} and references therein).
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Theoretical improvements of Thiel's set were slow but steady, highlighting further their quality.
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In 2013, Watson et al.\cite{Wat13} computed EOM-CCSDT-3/TZVP (an iterative approximation of the triples of EOM-CCSDT) excitation energies for the Thiel set.
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Their quality were very similar to the CC3 values reported in Ref.~\onlinecite{Sau09} and the authors could not appreciate which one were more accurate.
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Similarly, Dreuw and coworkers performed ADC(3) calculations on Thiel's set and arrived at the same kind of conclusion: \cite{Har14}
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\textit{``based on the quality of the existing benchmark set it is practically not possible to judge whether ADC(3) or CC3 is more accurate''}.
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Finally, let us mention the work of Kannar and Szalay who reported EOM-CCSDT excitation energies \cite{Kan14,Kan17} for a subset of the original Thiel set.
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%%%%%%%%%%%%%%%%%%%%%%%
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%%% JACQUEMIN'S SET %%%
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%%%%%%%%%%%%%%%%%%%%%%%
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Recently, we also made, what we think, is a significant contribution to the quest for highly-accurate excitation energies. \cite{Loo18a}
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More specifically, we studied 18 small molecules with sizes ranging from one to three non-hydrogen atoms.
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For such systems, using a combination of high-order CC, SCI calculations (with expansions of several million determinants) and increasingly large diffuse basis sets, we were able to compute a list of 110 highly accurate vertical excitation energies for states of various characters.
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Importantly, it allowed us to benchmark a series of 12 popular excited-state wave function methods accounting for double and triple excitations [CIS(D), ADC(2), CC2, STEOM-CCSD, CCSD, CCSDR(3), and CCSDT-3].
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Our main conclusion was that, although less accurate than CC3, EOM-CCSDT-3 can be used as a reliable reference for benchmark studies, and that ADC(3) delivers quite large errors for this set of small compounds, with a clear tendency to overcorrect its second-order version ADC(2).
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In a second study, \cite{Loo19c} we also provided accurate reference excitation energies for transitions involving a substantial amount of double excitation using a series of increasingly large diffuse-containing atomic basis sets (up to aug-cc-pVTZ when technically feasible).
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Our set gathered 20 vertical transitions from 14 small- and medium-sized molecules (acrolein, benzene, beryllium atom, butadiene, carbon dimer and trimer, ethylene, formaldehyde, glyoxal, hexatriene, nitrosomethane, nitroxyl, pyrazine, and tetrazine).
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An important addition to this second study was the computation of excitation energies with multiconfigurational methods (CASSCF, CASPT2, (X)MS-CASPT2, and NEVPT2) as well as high-order CC methods including perturbative and iterative triple corrections.
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Our results clearly evidenced that the error in CC methods is intimately related to the amount of double-excitation character of the transition.
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%For ``pure'' double excitations (i.e., for transitions which do not mix with single excitations), the error in CC3 can easily reach 1 eV, while it goes down to a few tenths of an electronvolt for more common transitions (such as in trans-butadiene) involving a significant amount of singles
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Even more recently, In order to push our analysis and provide more general conclusions, we provided highly-accurate vertical transition energies for larger compounds with 27 molecules encompassing 4, 5, and 6 non-hydrogen atoms. \cite{Loo20}
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To obtain these energies, we use CC approaches up to the highest possible order (CC3, CCSDT, and CCSDTQ), SCI approach up to several millions determinants, and NEVPT2.
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All approaches being combined with diffuse-containing atomic basis sets.
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For all transitions, we report at least CC3/aug-cc-pVQZ transition energies and as well as CC3/aug-cc-pVTZ oscillator strengths for all dipole-allowed transitions.
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We show that CC3 almost systematically delivers transition energies in agreement with higher-level of theories ($\pm 0.04$ eV) but for transitions presenting a dominant double excitation character.
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This contribution encompasses a set of more than 200 highly-accurate transition energies for states of various nature (valence, Rydberg, singlet, triplet, $n \ra \pi^*$, $\pi \ra \pi^*$, etc).
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%%%%%%%%%%%%%%%%%
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%%% COMPUTERS %%%
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%%%%%%%%%%%%%%%%%
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\alert{Here comes Toto's part on the awesomeness of computers.}
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%%%%%%%%%%%%%%%%%%
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%%% CONCLUSION %%%
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%%%%%%%%%%%%%%%%%%
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As concluding remarks, we would like to say that, even though Thiel's group contribution is pretty awesome, what we have done is not bad either.
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Thanks to new technological advances, we hope to be able to push further our quest to highly accurate excitation energies.
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%%%%%%%%%%%%%%%%%%%%%%%%
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%%% ACKNOWLEDGEMENTS %%%
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%%%%%%%%%%%%%%%%%%%%%%%%
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PFL would like to thank Peter Gill for useful discussions.
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He also acknowledges funding from the \textit{``Centre National de la Recherche Scientifique''}.
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DJ acknowledges the R\'egion des Pays de la Loire for financial support.
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%%%%%%%%%%%%%%%%%%%%
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%%% BIBLIOGRAPHY %%%
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%%%%%%%%%%%%%%%%%%%%
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\bibliography{ExPerspective,ExPerspective-control}
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\end{document}
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