Taking into account Denis modifications
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\includegraphics[width=\linewidth]{TOC}
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\end{wrapfigure}
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We provide a personal overview of the successive steps that made possible to obtain increasingly accurate excitation energies and properties with computational chemistry tools, eventually leading to chemically accurate vertical transition energies for small- and
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medium-size molecules. First, we describe the evolution of \textit{ab initio} state-of-the-art methods use to define benchmark values, with originally Roos' CASPT2 method, and then third-order coupled cluster methods as in the renowned Thiel set of excitation
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medium-size molecules. First, we describe the evolution of \textit{ab initio} state-of-the-art methods employed to define benchmark values, with originally Roos' CASPT2 method, and then third-order coupled cluster methods as in the renowned Thiel set of excitation
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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.
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These methods have been able to routinely deliver highly accurate excitation energies for small molecules as well as medium-size molecules with a compact basis sets for both single and double excitations. Second, we describe how these high-level methods
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These methods have been able to routinely deliver highly accurate excitation energies for small molecules, as well as medium-size molecules with compact basis sets for both single and double excitations. Second, we describe how these high-level methods
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and the creation of large, diverse and accurate benchmark sets of excitation energies have allowed to assess fairly and accurately the performance of computationally lighter theoretical models (\eg, TD-DFT, BSE, ADC, EOM-CC, etc) for different types of
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excited states ($\pi \ra \pi^*$, $n \ra \pi^*$, valence, Rydberg, singlet, triplet, double excitation, etc). We conclude this \textit{Perspective} by discussing the current potentiality of these methods from both an expert and a non-expert points of view, and what we
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believe could be the future theoretical and technological developments in the field.
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@ -90,14 +90,15 @@ photochemical processes. The factors that makes this quest for high accuracy par
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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. In the case of vertical excitation energies, \ie, excitation energies at a fixed geometry, band maxima
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do not usually match theoretical values as one needs to take into account both 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, no clear
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assignments could be made. 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{Die04b,Win13,Fan14b,Loo19b}
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assignment could be made. 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{Die04b,Win13,Fan14b,Loo19b}
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Second, developing theories suite for excited states is more complex because some fundamental safety nets, \eg, bounding from below, are not available in contrast to the ground electronic states, which additionally makes the results often less accurate for excited states
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\titou{Second, developing theories suited for excited states is more complex than ground-state theories because some fundamental safety nets, \eg, bounding from below, are not available in contrast to the ground electronic states, which additionally makes the results often less accurate for excited states
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than for ground states.
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For a given accuracy, excited-state methods are usually more expensive than their ground-state counterpart.}
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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 natures ($\pi \ra \pi^*$, $n \ra \pi^*$, charge transfer, double excitation, valence, Rydberg, singlet,
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triplet, etc). Therefore, it would be highly desirable to possess a computational method (or protocol) that provides a balanced treatment of the entire ``spectrum'' of excited states. And let's be honest, none of the existing methods does provide such feat for large compounds
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at an affordable cost.
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triplet, etc). Therefore, it would be highly desirable to possess a computational method (or protocol) that provides 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 such a feat at an affordable cost for chemically-meaningful compounds.
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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)
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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.
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@ -107,12 +108,12 @@ respect to system size and small memory footprint cannot be disregarded. Althoug
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%%% TABLE I %%%
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%\begin{squeezetable}
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\begin{table}
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\caption{Formal computational scaling of various excited-state methods with respect to the number of one-electron basis functions $N$ and the accessibility of various key properties in widely available codes.
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\caption{Formal computational scaling of various excited-state methods with respect to the number of one-electron basis functions $N$ and the accessibility of various key properties in \titou{widely available} computational software packages.
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The typical error range of estimate for single excitations is also provided as a very rough indicator of the method accuracy.}
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\label{tab:method}
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\begin{ruledtabular}
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\begin{tabular}{lcccc}
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\mr{2}{*}{Method} & Formal & Oscillator & Analytic & Typical \\
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\mr{2}{*}{Method} & Formal & Oscillator & Analytical & Typical \\
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& scaling & strength & gradients & error (eV) \\
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\hline
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TD-DFT & $N^4$ & \cmark & \cmark & $0.2$--$0.4$ \\
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@ -142,8 +143,8 @@ The typical error range of estimate for single excitations is also provided as a
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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 aiming to obtain 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 method, the theory was derived much earlier than their actual implementation in electronic structure software packages and the same applies to the analytic gradients when they are available.
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Before detailing some key past and present contributions aiming at obtaining 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 method, the theory was derived much earlier than their actual implementation in electronic structure software packages and the same applies to the analytical gradients when they are available.
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%%%%%%%%%%%%%%%%%%%%%
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%%% POPLE'S GROUP %%%
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@ -159,12 +160,12 @@ This second-order correction greatly reduces the magnitude of the error compared
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%%%%%%%%%%%%%%%%%%%
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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.
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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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Although it took more than ten years to obtain analytical 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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Nonetheless, it is common knowledge that CASPT2 has the clear tendency of underestimating vertical excitation energies in organic molecules.
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Driven by Angeli 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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For example, NEVPT2 is known to be intruder state free and size consistent.
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The limited applicability of these multiconfigurational methods is mainly due to the need of carefully defining an active space based on the desired transition(s) in order to obtain meaningful results, as well as their factorial computational growth with the number of active electrons and orbitals.
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We also point out that some emergent approaches, like DMRG (density matrix renormalization group), \cite{Bai19} also offer a new path of development for these multiconfigurational theories.
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We also point out that some emergent approaches, like DMRG (density matrix renormalization group), \cite{Bai19} also offer a new path for the development of these multiconfigurational theories.
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With a typical minimal valence active space tailored for the desired transitions, the usual error in CASPT2 or NEVPT2 calculations is $0.1$--$0.2$ eV.
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%%%%%%%%%%%%%
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@ -174,8 +175,8 @@ The advent of time-dependent density-functional theory (TD-DFT) \cite{Run84,Cas9
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For low-lying valence excited states, TD-DFT calculations relying on hybrid exchange-correlation functionals have a typical error of $0.2$--$0.4$ eV.
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However, a large number of shortcomings were quickly discovered. \cite{Toz98,Toz99,Dre04,Mai04,Dre05,Lev06,Eli11}
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In the present context, one of the most annoying feature of TD-DFT --- in its most standard (adiabatic) approximation --- is its inability to describe, even qualitatively, charge-transfer states, \cite{Toz99,Dre04} Rydberg states, \cite{Toz98} and double excitations. \cite{Mai04,Lev06,Eli11}
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One of the main related issues 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{Goe19,Sue19} it is difficult to
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select a functional adequate for all families of transitions, \cite{Lau13} despite the development of new more robust approaches, including the so-called double-hybrids. \cite{Goe10a,Bre16,Sch17}
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One of the main related issues 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{Goe19,Sue19}
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More specifically, despite the development of new, more robust approaches (including the so-called double hybrids \cite{Goe10a,Bre16,Sch17}), it is still difficult (not to say impossible) to select a functional adequate for all families of transitions. \cite{Lau13}
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Moreover, the difficulty of making TD-DFT systematically improvable obviously hampers its applicability.
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Despite all of this, TD-DFT remains nowadays the most employed excited-state method in the electronic structure community (and beyond).
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@ -192,7 +193,7 @@ For the sake of brevity, we drop the EOM acronym in the rest of this \textit{Per
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The 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}
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As a $N^7$ method (where $N$ is the number of basis functions), CC3 has a particularly interesting accuracy/cost ratio with errors usually below the chemical accuracy threshold. \cite{Hat05c,Loo18a,Loo18b,Loo19a}
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The series CC2, CCSD, CC3, CCSDT defines a hierarchy of models with $N^5$, $N^6$, $N^7$ and $N^8$ scaling, respectively.
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It is also noteworthy that CCSDT and CC3 are also able to pinpoint the presence of double excitations, a feature that is absent from both CCSD and CC2. \cite{Loo19c}
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It is also noteworthy that CCSDT and CC3 are also able to \titou{pinpoint} the presence of double excitations, a feature that is absent from both CCSD and CC2. \cite{Loo19c}
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%%%%%%%%%%%%%%%%%%%
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%%% ADC METHODS %%%
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@ -201,14 +202,14 @@ It is also important to mention the recent rejuvenation of the second- and third
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This renaissance was certainly initiated by the enormous amount of work invested by Dreuw's group in order to provide a fast and efficient implementation of these methods, \cite{Dre15} including the analytical gradients, \cite{Reh19} as well as other interesting variants.
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These Green's function one-electron propagator techniques represent interesting alternatives thanks to their reduced scaling compared to their CC equivalents.
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In that regard, ADC(2) is particularly attractive with an error around $0.1$--$0.2$ eV.
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However, we have recently observed that ADC(3) generally overcorrects the ADC(2) excitation energies and is less accurate than CC3 \cite{Tro02,Loo18a,Loo20}.
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However, we have recently observed that ADC(3) generally overcorrects the ADC(2) excitation energies and is significantly less accurate than CC3. \cite{Tro02,Loo18a,Loo20}
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%%%%%%%%%%%%%%
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%%% BSE@GW %%%
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%%%%%%%%%%%%%%
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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 \emph{GW} calculation \cite{Hed65}).
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BSE has gained momentum in the past few years and is a serious candidate as a computationally inexpensive electronic structure theory method that can model accurately excited states (with a typical error of $0.1$--$0.3$ eV) and some of the corresponding properties. \cite{Jac17b,Bla18}
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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 \emph{GW} calculation, BSE@\emph{GW} has been shown to be weakly dependent on its starting point, that is on the functional selected for the underlying DFT calculation. \cite{Jac16,Gui18}
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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 \emph{GW} calculation, BSE@\emph{GW} has been shown to be weakly dependent on its starting point (\ie, on the functional selected for the underlying DFT calculation). \cite{Jac16,Gui18}
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However, due to the adiabatic (\ie, static) approximation, doubly excited states are completely absent from the BSE spectrum.
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%%%%%%%%%%%%%%%%%%%
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@ -219,7 +220,7 @@ SCI methods rely on the same principle as the usual CI approach, except that det
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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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The main advantage of SCI methods is that no a priori assumption is made on the type of electron correlation.
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Therefore, at the price of a brute force calculation, a SCI calculation is less biased by the user appreciation of the problem's complexity.
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One of the strength of one of the implementation, based on the CIPSI (configuration interaction using a perturbative selection made iteratively) algorithm developed by Huron, Rancurel, and Malrieu in 1973 \cite{Hur73} is its parallel efficiency which makes possible to run on thousands of cores. \cite{Gar19}
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One of the strength of one of the implementation, based on the CIPSI (configuration interaction using a perturbative selection made iteratively) algorithm developed by Huron, Rancurel, and Malrieu in 1973 \cite{Hur73} is its parallel efficiency which makes possible to run on thousands of CPU cores. \cite{Gar19}
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Thanks to these tremendous features, SCI methods deliver 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 originating from the extrapolation procedure. \cite{Gar19}
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However, although the \textit{``exponential wall''} is pushed back, this type of method is only applicable to molecules with a small number of heavy atoms and/or relatively compact basis sets.
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@ -275,7 +276,7 @@ These TBEs were quickly refined with the larger \emph{aug}-cc-pVTZ basis set, \c
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As a direct evidence of the actual 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 the Introduction of Ref.~\onlinecite{Loo18a} and references therein).
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Theoretical improvements of Thiel's set were slow but steady, highlighting further its quality.
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In 2013, Watson et al.\cite{Wat13} computed CCSDT-3/TZVP (an iterative approximation of the triples of CCSDT \cite{Wat96}) excitation energies for the Thiel set.
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In 2013, Watson \textit{et al.} \cite{Wat13} computed CCSDT-3/TZVP (an iterative approximation of the triples of CCSDT \cite{Wat96}) 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 model was the most 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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@ -286,16 +287,16 @@ These two studies clearly demonstrate and motivate the need for higher accuracy
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%%% JACQUEMIN'S SET %%%
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%%%%%%%%%%%%%%%%%%%%%%%
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Recently, we made, what we think, is a significant contribution to this quest for highly accurate excitation energies. \cite{Loo18a}
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More specifically, we studied 18 small molecules (1 to 3 non-hydrogen atoms) with sizes ranging from one to three non-hydrogen atoms.
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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 methods, SCI calculations and large diffuse basis sets, we were able to compute a list of 110 highly accurate vertical excitation energies for excited states of various natures (valence, Rydberg, $n \ra \pi^*$, $\pi \ra \pi^*$, singlet, triplet and doubly excited) based on accurate CC3/\emph{aug}-cc-pVTZ geometries.
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In the following, we label this set of TBEs as {\SetA}.
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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 (see Fig.~\ref{fig:Set1}): CIS(D), ADC(2), CC2, STEOM-CCSD, \cite{Noo97} CCSD, CCSDR(3), \cite{Chr77} and CCSDT-3. \cite{Wat96}
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Importantly, it allowed us to benchmark a series of popular excited-state wave function methods accounting for double and triple excitations (see Fig.~\ref{fig:Set1}): CIS(D), ADC(2), ADC(3), CC2, STEOM-CCSD, \cite{Noo97} CCSD, CCSDR(3), \cite{Chr77} CCSDT-3, \cite{Wat96} CC3, CCSDT, and CCSDTQ.
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Our main conclusion was that CC3 is extremely accurate (with a mean absolute error of only $\sim 0.03$ eV), and that, although less accurate than CC3, CCSDT-3 could be used as a reliable reference for benchmark studies.
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Quite surprisingly, ADC(3) was found to have a clear tendency to overcorrect its second-order version ADC(2).
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In a second study, \cite{Loo19c} using a similar combination of theoretical models (but mostly extrapolated SCI energies), we provided accurate reference excitation energies for transitions involving a substantial amount of double excitations using a series of increasingly large diffuse-containing atomic basis sets (up to \emph{aug}-cc-pVQZ when technically feasible).
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This set gathers 20 vertical transitions from 14 small- and medium-sized molecules, a set we label as {\SetB} in the remaining of this \emph{Perspective}.
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An important addition to this second study was the computation of double excitations with various flavors of multiconfigurational methods (CASSCF, CASPT2, and NEVPT2) in addition to high-order CC methods including perturbative or iterative triple corrections (see Fig.~\ref{fig:Set2}).
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An important addition to this second study was the computation of double excitations with various flavors of multiconfigurational methods (CASSCF, CASPT2, and NEVPT2) in addition to high-order CC methods including, at least, perturbative or iterative triple corrections (see Fig.~\ref{fig:Set2}).
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Our results clearly evidence that the error in CC methods is intimately related to the amount of double-excitation character in the vertical transition.
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For ``pure'' double excitations (\ie, for transitions which do not mix with single excitations), the error in CC3 and CCSDT can easily reach $1$ and $0.5$ eV, respectively, while it goes down to a few tenths of an electronvolt for more common transitions (such as in \emph{trans}-butadiene and benzene) involving a significant amount of singles.\cite{Shu17,Bar18b,Bar18a}
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The quality of the excitation energies obtained with multiconfigurational methods was harder to predict as the overall accuracy of these methods is highly dependent on both the system and the selected active space.
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@ -327,7 +328,7 @@ Each approach was applied in combination with diffuse-containing atomic basis se
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For all the transitions of the {\SetC} set, we reported at least CC3/\emph{aug}-cc-pVQZ transition energies as well as CC3/\emph{aug}-cc-pVTZ oscillator strengths for each dipole-allowed transition.
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Pursuing our previous benchmarking efforts, \cite{Loo18a,Loo19c} we confirmed that CC3 almost systematically delivers transition energies in agreement with higher-level theoretical models ($\pm 0.04$ eV) except for transitions presenting a dominant double excitation character (see Fig.~\ref{fig:Set3}).
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This settles down, at least for now, the debate by demonstrating the superiority of CC3 (in terms of accuracy) compared to methods like CCSDT-3 or ADC(3).
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Moreover, thanks to the exhaustive and detailed comparisons made in Ref.~\onlinecite{Loo20}, we could safely conclude that CC3 also regularly outperforms CASPT2 (that underestimates) and NEVPT2 (that overestimates) as long as the corresponding transition does not show any strong multiple excitation character.
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Moreover, thanks to the exhaustive and detailed comparisons made in Ref.~\onlinecite{Loo20}, we could safely conclude that CC3 also regularly outperforms CASPT2 (which underestimates excitation energies) and NEVPT2 (which overestimates excitation energies) as long as the corresponding transition does not show any strong multiple excitation character.
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Our current efforts are now focussing on expanding and merging these sets to create an complete test set of highly accurate excitations energies.
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In particular, we are currently generating reference excitations energies for radicals as well as more ``exotic'' molecules containing heavier atoms (such as \ce{Cl}, \ce{F}, \ce{P}, and \ce{Si}). \cite{Loo20b}
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@ -338,10 +339,11 @@ It would likely stimulate further theoretical developments in excited-state meth
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%%%%%%%%%%%%%%%%%%
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%%% Properties
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%%%%%%%%%%%%%%%%%%
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Besides all previously described works aiming at reaching chemically accurate vertical transition energies, it should be pointed out that an increasing effort is now set to determine highly-trustable excited-state properties as well. This includes first the 0-0 energies, \cite{Die04b,Hat05c,Goe10a,Sen11b,Win13,Fan14b,Loo18b,Loo19a,Loo19b}
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that, as mentioned above, as they offer well-grounded comparisons with experiment. However, as these 0-0 energies are not very sensitive to the used geometry, \cite{Sen11b,Win13,Loo19a} they are not very indicative of the quality of the underlying structures. This is why, one can find several sets of excited-state geometries
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determined with various wavefunction approaches, \cite{Pag03,Gua13,Bou13,Tun16,Bud17} a few using very refined models, \cite{Gua13,Bud17} as well as evaluations of the accuracy of the gradients at the FC point. \cite{Taj18,Taj19} The interested researchers can also find several investigations proposing sets of
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reference oscillator strengths, \cite{Sil10c,Har14,Kan14,Loo18a,Loo20b} but other more complex properties, such as two-photon cross-sections and vibrations, have been mostly determined at lower levels of theory hinting that the story is far from its end.
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\titou{Besides all previously described works aiming at reaching chemically accurate vertical transition energies, it should be pointed out that an increasing effort is now set to determine highly-trustable excited-state properties as well.
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This includes first the 0-0 energies, \cite{Die04b,Hat05c,Goe10a,Sen11b,Win13,Fan14b,Loo18b,Loo19a,Loo19b} that, as mentioned above, as they offer well-grounded comparisons with experiment.
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However, as these 0-0 energies are not very sensitive to the used geometry, \cite{Sen11b,Win13,Loo19a} they are not very indicative of the quality of the underlying structures.
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This is why, one can find several sets of excited-state geometries determined with various wavefunction approaches, \cite{Pag03,Gua13,Bou13,Tun16,Bud17} a few using very refined models, \cite{Gua13,Bud17} as well as evaluations of the accuracy of the gradients at the FC point. \cite{Taj18,Taj19}
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The interested researchers can also find several investigations proposing sets of reference oscillator strengths, \cite{Sil10c,Har14,Kan14,Loo18a,Loo20b} but other more complex properties, such as two-photon cross-sections and vibrations, have been mostly determined at lower levels of theory hinting that the story is far from its end.}
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%%%%%%%%%%%%%%%%%%
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%%% CONCLUSION %%%
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@ -349,7 +351,7 @@ reference oscillator strengths, \cite{Sil10c,Har14,Kan14,Loo18a,Loo20b} but othe
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As concluding remarks, we would like to highlight once again the major contribution brought by Roos' and Thiel's groups in an effort to define benchmark values for excited states.
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Following their footsteps, we have recently proposed a larger, even more accurate set of vertical transitions energies for various types of excited states (including double excitations). \cite{Loo18a,Loo19c,Loo20}
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This was made possible thanks to a technological renaissance of SCI methods which can now routinely produce near-FCI excitation energies for small- and medium-size organic molecules. \cite{Chi18,Gar18,Gar19}
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We hope that new technological advances will enable us to push further our quest to highly accurate excitation energies, and, importantly, of excited-state properties as well, in years to come.
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We hope that new technological advances will enable us to push further in years to come our quest to highly accurate excitation energies, and, importantly, of excited-state properties.
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%%%%%%%%%%%%%%%%%%%%%%%%
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%%% ACKNOWLEDGEMENTS %%%
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