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@ -16,7 +16,6 @@ H 0.00000000 -2.54514584 3.51352303
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cc-pVDZ 100 BF
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Cfour
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HF -192.808319784581
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MP2 -193.471694583002 -0.663374798421
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MP3 -193.509358554889360 -0.701038770308868
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@ -46,7 +45,6 @@ H 0.00000000 -2.59168789 3.47168051
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cc-pVDZ 90 BF
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Cfour
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HF -228.643312572484575
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MP2 -229.350762946309 -0.707450373824
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MP3 -229.371132765573179 -0.727820193088592
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@ -60,24 +58,6 @@ CCSDT -229.407644692870491 -0.764332120385917
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CCSDTQ -229.409966781261375 -0.76665420877679
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CCSD(T) -229.4073471 -0.7640345275
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Dalton
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HF -228.643312572485
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MP2 -229.3507629658025451
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CC2 -229.360495928343
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CCSD -229.3782852538
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CC3 -229.4090426487
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MRCC
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HF -228.6433125723933699
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MP2 -229.350762954683
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CCSD -229.37828519649 -0.734972624097
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CC3 -229.409042619076
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CCSDT -229.407644699645
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CC4 -229.410178825868
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CISD -229.260645347830
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CISDT -229.27741998
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=========
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Imidazole
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=========
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@ -94,7 +74,6 @@ H 2.08673940 -3.67001102 0.00000000
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cc-pVDZ 90 BF
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Cfour
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HF -224.835356714467
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MP2 -225.555786629729 -0.720429915262
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MP3 -225.573185257380885 -0.737828542914060
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@ -125,7 +104,6 @@ H 0.00000000 -2.56726559 -3.47837232
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cc-pVDZ 95 BF
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Cfour
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HF -208.828614693633
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MP2 -209.524336644342 -0.695721950709
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MP3 -209.549217802376802 -0.720603108743368
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@ -155,7 +133,6 @@ H 0.00000000 -2.48760051 -4.16768392
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cc-pVDZ 94 BF
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Cfour
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HF -551.321002444800
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MP2 -551.982490121638 -0.661487676839
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MP3 -552.010435926354262 -0.689433481554693
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@ -188,11 +165,11 @@ H 0.00000000 4.67337115 0.00000000
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cc-pVDZ 114 BF
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Cfour
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HF -230.722244985640
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MP2 -231.504577062867 -0.782332077228
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MP3 -231.538632128269825 -0.816387142630248
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MP4 -231.580784896961347 -0.858539911321773
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MP5 -231.576028 -0.8537830144
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CC2 -231.511657283736 -0.789412298096
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CC3 -231.581387684239189 -0.859142698599615
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CC4 -231.582831060510870 -0.860586074871286
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@ -218,11 +195,11 @@ H 0.00000000 -3.88751412 -2.35234226
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cc-pVDZ 104 BF
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Cfour
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HF -262.703010944756
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MP2 -263.537628678036 -0.834617733280
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MP3 -263.556666298770153 -0.853655354014131
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MP4 -263.605918033939417 -0.902907089183363
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MP5 -263.596846 -0.8938350552
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CC2 -263.547474392626 -0.844463447870
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CC3 -263.604508857029998 -0.901497912273981
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CC4 -263.605647787083853 -0.902636842327822
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@ -248,11 +225,11 @@ H 0.00000000 -2.39011496 -4.03967703
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cc-pVDZ 104 BF
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Cfour
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HF -262.669873445804
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MP2 -263.508614029774 -0.838740583969
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MP3 -263.527149327902 -0.857275882097803
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MP4 -263.577799236667 -0.907925790862876
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MP5 -263.5681322 -0.8982587542
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CC2 -263.518782299415 -0.848908853611
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CC3 -263.576087839704883 -0.906214393900430
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CC4 -263.577327205407130 -0.907453759605642
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@ -279,11 +256,11 @@ H 0.00000000 -3.88059079 2.40341581
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cc-pVDZ 109 BF
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Cfour
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HF -246.715184707013
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MP2 -247.522656534580 -0.807471827567
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MP3 -247.549220047586402 -0.834035340573813
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MP4 -247.595108304867409 -0.879923597854822
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MP5 -247.5880598 -0.872875093
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CC2 -247.531513026372 -0.816328319360
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CC3 -247.594784382246985 -0.879599675234389
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CC4 -247.596006679647360 -0.880821972634781
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@ -309,11 +286,11 @@ H 0.00000000 -4.05149341 -2.16351748
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cc-pVDZ 104 BF
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Cfour
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HF -262.713652796899
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MP2 -263.543710290177 -0.830057493278
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MP3 -263.563265009893314 -0.849612212994749
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MP4 -263.612919430489910 -0.899266633591386
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MP5
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CC2 -263.554961634033 -0.841308837134
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CC3 -263.612018090419497 -0.898365293520950
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CC4 -263.612916665641080 -0.899263868742531
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@ -336,11 +313,11 @@ H 0.00000000 0.00000000 -4.41850901
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cc-pVDZ 94 BF
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Cfour
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HF -294.615743073901
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MP2 -295.511666063031 -0.895922989130
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MP3 -295.515200031440543 -0.899456957539510
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MP4 -295.574313404387055 -0.958570330486010
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MP5 -295.5600125 -0.9442694261
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CC2 -295.524712564571 -0.908969490669
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CC3 -295.570596292645462 -0.954853218744419
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CC4 -295.571623462749471 -0.955880388848436
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@ -365,11 +342,11 @@ H 0.00000000 0.00000000 4.48366420
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cc-pVDZ 99 BF
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Cfour
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HF -278.717319389695
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MP2 -279.567838672171 -0.850519282476
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MP3 -279.580912529411535 -0.863593139716035
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MP4 -279.634037190758932 -0.916717801063422
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MP5 -279.6227554 -0.9054360103
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CC2 -279.581725808334 -0.864406418639
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CC3 -279.632903200419094 -0.915583810723581
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CC4 -279.633395016566340 -0.916075626870853
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@ -378,5 +355,3 @@ CCSDT -279.630035882850507 -0.912716493155011
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CCSDTQ
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CCSD(T) -279.6304634 -0.9131440103
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@ -50,6 +50,13 @@
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% Abstract
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\begin{abstract}
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We report (frozen-core) full configuration interaction (FCI) energies in finite Hilbert spaces for various five- and six-membered rings.
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In the continuity of our recent work on the benzene molecule [\href{https://doi.org/10.1063/5.0027617}{J. Chem. Phys. \textbf{153}, 176101 (2020)}], itself motivated by the blind challenge of Eriksen \textit{et al.} [\href{https://doi.org/10.1021/acs.jpclett.0c02621}{J. Phys. Chem. Lett. \textbf{11}, 8922 (2020)}] on the same system, we report reference frozen-core correlation energies for twelve cyclic molecules (cyclopentadiene, furan, imidazole, pyrrole, thiophene, benzene, pyrazine, pyridazine, pyridine, pyrimidine, tetrazine, and triazine) in the standard correlation-consistent double-$\zeta$ Dunning basis set (cc-pVDZ).
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This corresponds to Hilbert spaces with sizes ranging from $10^{20}$ (for thiophene) to $10^{36}$ (for benzene).
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Our estimates are based on localized-orbital-based selected configuration interaction (SCI) calculations performed with the \textit{Configuration Interaction using a Perturbative Selection made Iteratively} (CIPSI) algorithm.
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The performance and convergence properties of several series of methods are investigated.
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In particular, we study the convergence properties of ii) the M{\o}ller-Plesset perturbation series up to fifth-order (MP2, MP3, MP4, and MP5), ii) the iterative approximate single-reference coupled-cluster series CC2, CC3, and CC4, and ii) the single-reference coupled-cluster series CCSD, CCSDT, and CCSDTQ.
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The performance of the ground-state gold standard CCSD(T) is also investigated.
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\end{abstract}
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% Title
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@ -57,8 +64,28 @@
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\section{Introduction}
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\begin{figure*}
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\includegraphics[width=\linewidth]{mol}
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\caption{
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Five-membered rings (top) and six-membered rings (bottom) considered in this study.
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\label{fig:mol}}
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\end{figure*}
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\section{Computational details}
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We follow our usual procedure \cite{Scemama_2018,Scemama_2018b,Scemama_2019,Loos_2018a,Loos_2019,Loos_2020a,Loos_2020b,Loos_2020c} by performing a preliminary SCI calculation using Hartree-Fock orbitals in order to generate a SCI wave function with at least $10^7$ determinants.
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Natural orbitals are then computed based on this wave function, and a second run is performed with localized orbitals.
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This has the advantage to produce a smoother and faster convergence of the SCI energy toward the FCI limit by taking benefit of the local character of electron correlation.\cite{Angeli_2003,Angeli_2009,BenAmor_2011,Suaud_2017,Chien_2018,Eriksen_2020}
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The Boys-Foster localization procedure \cite{Boys_1960} that we apply to the natural orbitals is performed in several orbital windows: i) core, ii) valence $\sigma$, iii) valence $\pi$, iv) valence $\pi^*$, v) valence $\sigma^*$, vi) the higher-lying $\sigma$ orbitals, and vii) the higher-lying $\pi$ orbitals.
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Like Pipek-Mezey, \cite{Pipek_1989} this choice of orbital windows allows to preserve a strict $\sigma$-$\pi$ separation in planar systems like benzene.
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The total SCI energy is defined as the sum of the variational energy $E_\text{var.}$ (computed via diagonalization of the CI matrix in the reference space) and a second-order perturbative correction $E_\text{(r)PT2}$ which takes into account the external determinants, \ie, the determinants which do not belong to the variational space but are linked to the reference space via a nonzero matrix element.
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The magnitude of $E_\text{(r)PT2}$ provides a qualitative idea of the ``distance'' to the FCI limit.
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We then linearly extrapolate the total SCI energy to $E_\text{(r)PT2} = 0$ (which effectively corresponds to the FCI limit).
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Note that, unlike excited-state calculations where it is important to enforce that the wave functions are eigenfunctions of the $\Hat{S}^2$ spin operator, \cite{Applencourt_2018} the present wave functions do not fulfil this property as we aim for the lowest possible energy of a singlet state.
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We have found that $\expval*{\Hat{S}^2}$ is, nonetheless, very close to zero for each system.
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\section{Results and discussion}
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\begin{table*}
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@ -109,6 +136,7 @@
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MP2 & $-231.5046$ & $-782.3$ & $-263.5376$ & $-834.6$ & $-263.5086$ & $-838.7$ & $-247.5227$ & $-807.5$ & $-263.5437$ & $-830.1$ & $-295.5117$ & $-895.9$ & $-279.5678$ & $-850.5$\\
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MP3 & $-231.5386$ & $-816.4$ & $-263.5567$ & $-853.7$ & $-263.5271$ & $-857.3$ & $-247.5492$ & $-834.0$ & $-263.5633$ & $-849.6$ & $-295.5152$ & $-899.5$ & $-279.5809$ & $-863.6$ \\
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MP4 & $-231.5808$ & $-858.5$ & $-263.6059$ & $-902.9$ & $-263.5778$ & $-907.9$ & $-247.5951$ & $-879.9$ & $-263.6129$ & $-899.3$ & $-295.5743$ & $-958.6$ & $-279.6340$ & $-916.7$ \\
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MP5 & $-231.5760$ & $-853.8$ & $-263.5968$ & $-893.8$ & $-263.5681$ & $-898.3$ & $-247.5881$ & $-872.9$ & & & $-295.5600$ & $-944.3$ & $-279.6228$ & $-905.4$ \\
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\hline
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CC2 & $-231.5117$ & $-789.4$ & $-263.5475$ & $-844.5$ & $-263.5188$ & $-848.9$ & $-247.5315$ & $-816.3$ & $-263.5550$ & $-841.3$ & $-295.5247$ & $-909.0$ & $-279.5817$ & $-864.4$ \\
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CC3 & $-231.5814$ & $-859.1$ & $-263.6045$ & $-901.5$ & $-263.5761$ & $-906.2$ & $-247.5948$ & $-879.6$ & $-263.6120$ & $-898.4$ & $-295.5706$ & $-954.9$ & $-279.6329$ & $-915.6$ \\
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