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181
Manuscript/ExPerspective.bib
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@ -1526,3 +1526,184 @@
Volume = {94},
Year = {1990},
Bdsk-Url-1 = {http://dx.doi.org/10.1021/j100377a012}}
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title = {{Analytic nuclear gradients of the algebraic-diagrammatic construction scheme for the polarization propagator up to third order of perturbation theory}},
journal = {J. Chem. Phys.},
volume = {150},
number = {17},
pages = {174110},
year = {2019},
month = {May},
issn = {1089-7690},
publisher = {American Institute of Physics},
doi = {10.1063/1.5085117}
}
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title = {{Coupled Cluster Theory on Graphics Processing Units I. The Coupled Cluster Doubles Method}},
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}

39
Manuscript/ExPerspective.tex
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@ -111,7 +111,7 @@ And let's be honest, none of the existing methods does provide this at an afford
What are the requirement of the ``perfect'' theoretical model?
As mentioned above, a balanced treatment of excited states with different character is highly desirable.
Moreover, chemically accurate excitation energies (\ie, with error smaller than $1$ kcal/mol or $0.043$ eV) would be also beneficial in order to provide a quantitative chemical picture.
Moreover, chemically accurate excitation energies (\ie, with error smaller than $1$~kcal/mol or $0.043$~eV) would be also beneficial in order to provide a quantitative chemical picture.
The access to other properties, such as oscillator strengths, dipole moments and analytical energy gradients, is also an asset if one wants to compare with experimental data.
Let us not forget about the requirements of minimal user input and minimal chemical intuition (\ie, black box method preferable) in order to minimize the potential bias brought by the user appreciation of the problem complexity.
Finally, low computational scaling with respect to system size and small memory footprint cannot be disregarded.
@ -137,17 +137,17 @@ The typical error range or estimate for single excitations is also provided as a
ADC(2) & $N^5$ & \cmark & \cmark & $0.1$--$0.2$ \\
CC2 & $N^5$ & \cmark & \cmark & $0.1$--$0.2$ \\
\\
ADC(3) & $N^6$ & \cmark & \xmark & $0.1$--$0.2$ \\
ADC(3) & $N^6$ & \cmark & \cmark & $0.1$--$0.2$ \\
EOM-CCSD & $N^6$ & \cmark & \cmark & $\sim 0.10$ \\
\\
CC3 & $N^7$ & \cmark & \xmark & $\sim 0.04$ \\
CC3 & $N^7$ & \cmark & \cmark & $\sim 0.04$ \\
\\
EOM-CCSDT & $N^8$ & \xmark & \xmark & $\sim 0.03$ \\
EOM-CCSDTQ & $N^{10}$ & \xmark & \xmark & $\sim 0.01$ \\
\\
CASPT2 or NEVPT2 & $N!$ & \cmark & \cmark & $0.1$--$0.2$ \\
SCI & $N!$ & \xmark & \xmark & $\sim 0.03$ \\
FCI & $N!$ & \cmark & \xmark & $0.0$ \\
FCI & $N!$ & \cmark & \cmark & $0.0$ \\
\end{tabular}
\end{ruledtabular}
\end{table}
@ -157,7 +157,7 @@ The typical error range or estimate for single excitations is also provided as a
%** HISTORY **%
%**************
Before detailing some key past and present contributions towards the obtention of highly accurate excitation energies, we start by giving a historical overview of the various excited-state \textit{ab initio} methods that have emerged in the last fifty years.
Interestingly, for pretty much every single methods, the theory was derived much earlier than their actual implementation in electronic structure software packages (same applies to the analytic gradients when available).
Interestingly, for pretty much every single method, the theory was derived much earlier than their actual implementation in electronic structure software packages (same applies to the analytic gradients when available).
Here, we only mention methods that, we think, ended up becoming mainstream.
%%%%%%%%%%%%%%%%%%%%%
@ -166,9 +166,8 @@ Here, we only mention methods that, we think, ended up becoming mainstream.
The first mainstream \textit{ab initio} method for excited states was probably CIS (configuration interaction with singles) which has been around since the 1970's. \cite{Ben71}
CIS lacks electron correlation and therefore grossly overestimates excitation energies and wrongly orders excited states.
It is not unusual to have errors of the order of $1$ eV which precludes the usage of CIS as a quantitative quantum chemistry method.
Twenty years later, CIS(D) which adds a second-order perturbative correction to CIS was developed and implemented thanks to the efforts of Head-Gordon and coworkers. \cite{Hea94}
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,Ish95}
This second-order correction significantly reduces the magnitude of the error compared to CIS, with a typical error range of $0.2$--$0.3$ eV.
Unfortunately, to the best of our knowledge, analytic nuclear gradients are not available for CIS(D).
%%%%%%%%%%%%%%%%%%%
%%% ROOS' GROUP %%%
@ -177,10 +176,10 @@ In the early 1990's, the complete-active-space self-consistent field (CASSCF) me
This was a real breakthrough.
Although it took more than ten years to obtain analytic nuclear gradients, \cite{Cel03} CASPT2 was probably the first method that could provide quantitative results for molecular excited states of genuine photochemical interest. \cite{Roo96}
Nonetheless, it is common knowledge that CASPT2 has the strong tendency of underestimating vertical excitation energies in organic molecules.
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.
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.
For example, NEVPT2 is known to be intruder state free and size consistent.
The limited applicability of these so-called multiconfigurational methods is mainly due to the necessity of defining an active space based on the desired transition(s), as well as their factorial computational growth with the number of active electrons and orbitals.
With a typical minimal valence active space tailored for the desired transitions, the typical error in CASPT2 or NEVPT2 calculations is $0.1$--$0.2$ eV.
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.
%%%%%%%%%%%%%
%%% TDDFT %%%
@ -190,7 +189,7 @@ For low-lying excited states, TD-DFT calculations relying on hybrid exchange-cor
However, a large number of shortcomings were quickly discovered. \cite{Toz98,Toz99,Dre04,Mai04,Dre05,Lev06,Eli11}
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}
Moreover, the difficulty of making TD-DFT systematically improvable obviously hampers its applicability.
One of the main issue is the selection of the exchange-correlation functional from an ever-growing zoo of functionals and the variation of the excitation energies that one can observe with different functionals. \cite{Sue19}
One of the main 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{Sue19}
Despite all of this, TD-DFT is still nowadays the most employed excited-state method in the electronic structure community (and beyond).
%%%%%%%%%%%%%%%%%%
@ -199,7 +198,7 @@ Despite all of this, TD-DFT is still nowadays the most employed excited-state me
Thanks to the development of coupled cluster (CC) response theory, \cite{Koc90} and the huge growth of computational ressources, equation-of-motion coupled cluster with singles and doubles (EOM-CCSD) \cite{Sta93} became mainstream in the 2000's.
EOM-CCSD gradients were also quickly available. \cite{Sta95}
With EOM-CCSD, it is not unusual to have errors as small as $0.1$ eV, and a typical overestimation of the vertical transition energies.
Its third-order version, EOM-CCSDT, was also implemented and provides, at a significant higher cost, high accuracy for single excitations. \cite{Nog87}
Its third-order version, EOM-CCSDT, was also implemented and provides, at a significantly higher cost, high accuracy for single excitations. \cite{Nog87}
Although extremely expensive and tedious to implement, higher orders are also technically possible for small systems thanks to automatically generated code. \cite{Kuc91,Hir04}
The EOM-CC family of methods was quickly followed by an approximated and computationally lighter family with, in front line, the second-order CC2 model \cite{Chr95} and its third-order extension, CC3. \cite{Chr95b}
@ -212,7 +211,7 @@ For the sake of brevity, we drop the EOM acronym in the rest of this \textit{Per
%%% ADC METHODS %%%
%%%%%%%%%%%%%%%%%%%
It is also important to mention the recent rejuvenation 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.
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 [including the ADC(2) analytical gradients] as well as other interesting variants. \cite{Dre15}
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.
These Green's function one-electron propagator techniques represent interesting alternatives thanks to their reduced scaling compared to their CC equivalents.
In that regard, ADC(2) is particularly attractive with an error generally around $0.1$--$0.2$ eV.
However, we have recently observed that ADC(3) generally overcorrects the ADC(2) excitation energies. \cite{Loo18a,Loo20}
@ -232,10 +231,10 @@ In the past five years, \cite{Gin13,Gin15} we have witnessed a resurgence of the
SCI methods rely on the same principle as the usual CI approach, except that determinants are not chosen \textit{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.
Indeed, it has been noticed long ago that, even inside a predefined subspace of determinants, only a small number of them significantly contributes.
The main advantage of SCI methods is that no a priori assumption is made on the type of electron correlation.
Therefore, at the price of a brute force calculation, a SCI calculation is less biased by the user appreciation of the problem complexity.
Therefore, at the price of a brute force calculation, a SCI calculation is less biased by the user appreciation of the problem's complexity.
One of the strength of our 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}
Thanks to these tremendous features, SCI methods delivers near FCI quality excitation energies for both singly and doubly excited states, \cite{Hol17,Chi18,Loo18a,Loo19c} with an error of roughly $0.03$ eV (mostly due to the extrapolation procedure \cite{Gar19}).
However, although the \textit{``exponential wall''} is pushed back, this type of methods is only applicable to molecules with a small number of heavy atoms with relatively compact basis sets.
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 due to the extrapolation procedure \cite{Gar19}).
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 with relatively compact basis sets.
%%%%%%%%%%%%%%%
%%% SUMMARY %%%
@ -314,7 +313,7 @@ Interestingly, CASPT2 and NEVPT2 were found to be more accurate for transition w
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.
This set, labeled as {\SetC} and still based on CC3/aug-cc-pVTZ geometries, is constituted by a reasonably good balance of singlet, triplet, valence, and Rydberg states.
To obtain this new, larger set of TBEs, we employed CC methods up to the highest possible order (CC3, CCSDT, and CCSDTQ), very large SCI calculations (with tens of millions of determinants), as well as the most robust multiconfigurational method, NEVPT2.
To obtain this new, larger set of TBEs, we employed CC methods up to the highest possible order (CC3, CCSDT, and CCSDTQ), very large SCI calculations (with a hundred million of determinants), as well as the most robust multiconfigurational method, NEVPT2.
Each approach was applied in combination with diffuse-containing atomic basis sets.
Because the SCI energy converges obviously slower for these larger systems, the extrapolated SCI values were employed as a``safely net'' to demonstrate the overall consistency of our CC-based protocol rather than straight out-of-the-box reference values.
For all the transitions of the {\SetC} set, we reported at least CC3/aug-cc-pVQZ transition energies as well as CC3/aug-cc-pVTZ oscillator strengths for each dipole-allowed transition.
@ -332,7 +331,13 @@ It would surely stimulate further theoretical developments in excited-state meth
%%%%%%%%%%%%%%%%%
%%% COMPUTERS %%%
%%%%%%%%%%%%%%%%%
\alert{Here comes Toto's part on the awesomeness of computers.}
\alert{
To keep on with Moore's ``Law'' in the early 2000's, computer manufacturers had no other choice than to propose multi-core chips to avoid an explosion of the energy requirements.
Doubling the number of floating-point operations per second (flops/s) by doubling the number of CPU cores only requires to double the required energy, while doubling the frequency multiplies the required energy by a factor close to 8.
This bifuraction in hardware design implied a \emph{Change of paradigm}\cite{Sut05} in the implementation and design of computational algorithms, which needed to express a large degree of parallelism to benefit from a significant acceleration.
Fifteen years later, the community has made the effort of redesigning the methods with parallel-friendly algorithms,\cite{Val10,Cle10,Gar17b,Pen16,Kri13,Sce13} but the next generation of supercomputers is going to generalize the presence of accelerators (graphical processing units, GPUs), leading to a new software crisis.
Fortunately, some authors have already prepared this transition.\cite{Dep11,Kim18,Sny15,Ufi08,Kal17}
}
%%%%%%%%%%%%%%%%%%
%%% CONCLUSION %%%

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