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%% This BibTeX bibliography file was created using BibDesk.
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%% http://bibdesk.sourceforge.net/
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%% Created for Pierre-Francois Loos at 2019-11-03 14:37:22 +0100
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%% Created for Pierre-Francois Loos at 2019-11-03 21:54:11 +0100
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%% Saved with string encoding Unicode (UTF-8)
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@ -82,6 +82,76 @@
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@string{theo = {J. Mol. Struct. (THEOCHEM)}}
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@article{Wat96,
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Author = {John D. Watts and Rodney J. Bartlett},
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Date-Added = {2019-11-03 21:54:08 +0100},
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Date-Modified = {2019-11-03 21:54:08 +0100},
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Doi = {https://doi.org/10.1016/0009-2614(96)00708-7},
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Issn = {0009-2614},
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Journal = {Chem. Phys. Lett.},
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Number = {5},
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Pages = {581--588},
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Title = {Iterative and Non-Iterative Triple Excitation Corrections in Coupled-Cluster Methods for Excited Electronic States: the EOM-CCSDT-3 and EOM-CCSD($\tilde{T}$) Methods},
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Url = {http://www.sciencedirect.com/science/article/pii/0009261496007087},
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Volume = {258},
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Year = {1996},
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Bdsk-Url-1 = {http://www.sciencedirect.com/science/article/pii/0009261496007087},
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Bdsk-Url-2 = {https://doi.org/10.1016/0009-2614(96)00708-7}}
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@article{Chr77,
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Author = {Christiansen, P. A. and McCullough, E. A.},
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Date-Added = {2019-11-03 21:52:57 +0100},
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Date-Modified = {2019-11-03 21:52:57 +0100},
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Journal = JCP,
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Pages = {1877},
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Volume = 67,
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Year = 1977}
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@article{Noo97,
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Author = {Marcel Nooijen and Rodney J. Bartlett},
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Date-Added = {2019-11-03 21:50:58 +0100},
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Date-Modified = {2019-11-03 21:50:58 +0100},
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Doi = {10.1063/1.474000},
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Eprint = {https://doi.org/10.1063/1.474000},
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Journal = {J. Chem. Phys.},
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Number = {15},
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Pages = {6441-6448},
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Title = {A New Method for Excited States: Similarity Transformed Equation-Of-Motion Coupled-Cluster Theory},
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Url = {https://doi.org/10.1063/1.474000},
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Volume = {106},
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Year = {1997},
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Bdsk-Url-1 = {https://doi.org/10.1063/1.474000}}
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@article{Str88,
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Author = {Strinati, G.},
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Date-Added = {2019-11-03 20:52:10 +0100},
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Date-Modified = {2019-11-03 20:52:16 +0100},
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Doi = {10.1007/BF02725962},
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Issn = {1826-9850},
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Journal = {Riv. Nuovo Cimento},
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Language = {en},
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Month = dec,
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Number = {12},
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Pages = {1--86},
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Title = {Application of the {{Green}}'s Functions Method to the Study of the Optical Properties of Semiconductors},
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Volume = {11},
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Year = {1988},
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Bdsk-Url-1 = {https://dx.doi.org/10.1007/BF02725962}}
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@article{Hed65,
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Author = {Hedin, Lars},
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Date-Added = {2019-11-03 20:50:54 +0100},
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Date-Modified = {2019-11-03 20:51:00 +0100},
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Doi = {10.1103/PhysRev.139.A796},
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File = {/Users/loos/Zotero/storage/ZGMCVKPC/Hedin_1965.pdf},
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Journal = {Phys. Rev.},
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Number = {3A},
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Pages = {A796},
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Title = {New Method for Calculating the One-Particle {{Green}}'s Function with Application to the Electron-Gas Problem},
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Volume = {139},
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Year = {1965},
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Bdsk-Url-1 = {https://doi.org/10.1103/PhysRev.139.A796}}
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@article{Cur97,
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Author = {Curtiss, L. A. and Raghavachari, K. and Referm, P. C. and Pople, J. A.},
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Date-Added = {2019-11-03 14:35:45 +0100},
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@ -1,4 +1,4 @@
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\documentclass[aps,prb,reprint,noshowkeys,superscriptaddress]{revtex4-1}
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\documentclass[aip,jcp,reprint,noshowkeys,superscriptaddress]{revtex4-1}
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\usepackage{graphicx,dcolumn,bm,xcolor,microtype,multirow,amscd,amsmath,amssymb,amsfonts,physics,mhchem,longtable,pifont,wrapfig,txfonts,multirow}
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\usepackage{natbib}
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@ -70,14 +70,14 @@
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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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\begin{wrapfigure}[15]{o}[-1.25cm]{0.4\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 for small- and medium-size molecules.
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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.
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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 in small basis sets for single and double excitations.
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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 in compact basis sets for single and double excitations.
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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.
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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.
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\end{abstract}
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@ -88,7 +88,7 @@ We conclude this \textit{Perspective} by discussing the current potentiality of
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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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Of particular interest is the access to precise excitation energies, \ie, the energy difference between 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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@ -97,71 +97,72 @@ Even more problematic, experimental spectra might not be available in gas phase,
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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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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).
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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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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.
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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 potential bias brought by the user 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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Although the simultaneous fulfilment of all these requirements seems elusive, it is 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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%\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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Interestingly, for pretty much every single method, 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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CIS lacks electron correlation and therefore grossly overestimates excitation energies and wrongly orders excited states.
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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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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.
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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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@ -175,7 +176,7 @@ The advent of time-dependent density-functional theory (TD-DFT) \cite{Run84,Dre0
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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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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}
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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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@ -185,33 +186,34 @@ Thanks to the development of coupled cluster (CC) response theory, \cite{Koc90}
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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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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 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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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}
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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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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}
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%%%%%%%%%%%%%%%%%%%
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%%% ADC METHODS %%%
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%%%%%%%%%%%%%%%%%%%
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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, 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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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.
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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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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}
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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 (which is usually performed on top of a GW calculation).
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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 GW calculation \cite{Hed65}).
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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}
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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 self-consistently GW calculation, is weakly dependent on the starting point. \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 GW calculation, BSE@GW has been shown to be weakly dependent on its starting point. \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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%%% SCI METHODS %%%
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%%%%%%%%%%%%%%%%%%%
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In the past five years, we have witnessed a resurgence of selected CI (SCI) methods \cite{Gin13,Gin15} thanks to the development and implementation of new and fast algorithms to select cleverly determinants in the full CI (FCI) space (see Refs.~\onlinecite{Gar18,Gar19} and references therein).
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Importantly in the context of the present \textit{Perspective} article, we have witnessed a resurgence of selected CI (SCI) methods \cite{Ben69,Whi69,Hur73} in the past five years \cite{Gin13,Gin15} thanks to the development and implementation of new, fast and efficient algorithms to select cleverly determinants in the full CI (FCI) space (see Refs.~\onlinecite{Gar18,Gar19} and references therein).
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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 or energy.
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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 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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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 a ver large number of cores.
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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 complexity.
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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.
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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}
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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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%\cite{Caf16,Gar17b,Hol16,Sha17,Hol17,Chi18,Sce18,Sce18b,Loo18a,Gar18}.
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@ -219,14 +221,13 @@ However, although the \textit{``exponential wall''} is pushed back, this type of
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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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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, ultimately leading to an unbalanced description of excited states with distinct natures.
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||||
%*****************
|
||||
%** BENCHMARKS ***
|
||||
%*****************
|
||||
Although sometimes decried, benchmark set of molecules and their corresponding reference data are definitely essential for the validation of existing theoretical methods and to understand their strengths and, more importantly, their limitations.
|
||||
These sets have started to emerge at the end of the 1990's for ground-state properties with the acclaimed G2 test set designed by Pople's group. \cite{Cur97}
|
||||
Although sometimes decried, benchmark sets of molecules and their corresponding reference data are definitely essential for the validation of existing theoretical models and to bring to light and subsequently understand their strengths and, more importantly, their limitations.
|
||||
These sets have started to emerge at the end of the 1990's for ground-state properties with the acclaimed G2 test set designed by the Pople group. \cite{Cur97}
|
||||
For excited states, things started moving a little later but some major contributions were able to put things back on track pretty quickly.
|
||||
|
||||
%%%%%%%%%%%%%%%%%%%
|
||||
@ -241,8 +242,8 @@ These TBEs were sooner refined with the larger aug-cc-pVTZ basis set, \cite{Sil1
|
||||
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).
|
||||
|
||||
Theoretical improvements of Thiel's set were slow but steady, highlighting further their quality.
|
||||
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.
|
||||
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.
|
||||
In 2013, Watson et al.\cite{Wat13} computed EOM-CCSDT-3/TZVP (an iterative approximation of the triples of EOM-CCSDT \cite{Wat96}) excitation energies for the Thiel set.
|
||||
Their quality were very similar to the CC3 values reported in Ref.~\onlinecite{Sau09} and the authors could not appreciate which model was more accurate.
|
||||
Similarly, Dreuw and coworkers performed ADC(3) calculations on Thiel's set and arrived at the same kind of conclusion: \cite{Har14}
|
||||
\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''}.
|
||||
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.
|
||||
@ -252,22 +253,21 @@ Finally, let us mention the work of Kannar and Szalay who reported EOM-CCSDT exc
|
||||
%%%%%%%%%%%%%%%%%%%%%%%
|
||||
Recently, we also made, what we think, is a significant contribution to the quest for highly-accurate excitation energies. \cite{Loo18a}
|
||||
More specifically, we studied 18 small molecules with sizes ranging from one to three non-hydrogen atoms.
|
||||
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.
|
||||
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].
|
||||
For such systems, using a combination of high-order CC, 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 characters (valence, Rydberg, $n \ra \pi^*$, $\pi \ra \pi^*$, singlet, triplet and doubly-excited) based on accurate CC3/aug-cc-pVTZ geometries.
|
||||
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, \cite{Noo97} CCSD, CCSDR(3), \cite{Chr77} and CCSDT-3 \cite{Wat96}].
|
||||
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).
|
||||
|
||||
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).
|
||||
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).
|
||||
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.
|
||||
Our results clearly evidenced that the error in CC methods is intimately related to the amount of double-excitation character of the transition.
|
||||
%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
|
||||
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 excitation using a series of increasingly large diffuse-containing atomic basis sets (up to aug-cc-pVTZ when technically feasible).
|
||||
Our set gathers 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).
|
||||
An important addition to this second study was the computation of excitation energies with multiconfigurational methods (CASSCF, CASPT2, and NEVPT2) in addition to high-order CC methods including perturbative and iterative triple corrections.
|
||||
Our results clearly evidenced that the error in CC methods is intimately related to the amount of double-excitation character in the vertical transition.
|
||||
For ``pure'' double excitations (\ie, for transitions which do not mix with single excitations), the error in CC3 and EOM-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 trans-butadiene) involving a significant amount of singles.
|
||||
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.
|
||||
|
||||
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}
|
||||
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.
|
||||
All approaches being combined with diffuse-containing atomic basis sets.
|
||||
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.
|
||||
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.
|
||||
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).
|
||||
In our latest study, \cite{Loo20} in order to provide more general conclusions, we generated highly-accurate vertical transition energies for larger compounds with a set composed by 27 molecules encompassing from 4 to 6 non-hydrogen atoms for a total of \alert{238} vertical transition energies of various natures.
|
||||
To obtain these energies, we used EOM-CC methods up to the highest possible order (CC3, CCSDT, and CCSDTQ), SCI calculations up to several million determinants, as well as the most robust multiconfigurational method, NEVPT2, each approach being combined with diffuse-containing atomic basis sets.
|
||||
For all these transitions, we reported at least CC3/aug-cc-pVQZ transition energies and as well as CC3/aug-cc-pVTZ oscillator strengths for all dipole-allowed transitions.
|
||||
In agreement with our two previous studies, \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 predominant double excitation character.
|
||||
|
||||
%%%%%%%%%%%%%%%%%
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%%% COMPUTERS %%%
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Loading…
Reference in New Issue
Block a user