Antoine Marie

Pierre-François Loos

Reference Energies for Valence Ionizations and Satellite Transitions

Upon ionization of an atom or a molecule, another electron (or more) can be simultaneously excited. These concurrently generated states are called "satellites" (or shake-up transitions) as they appear in ionization spectra as higher-energy peaks with weaker intensity and larger width than the main peaks associated with single-particle ionizations. Satellites, which correspond to electronically excited states of the cationic species, are notoriously challenging to model using conventional single-reference methods due to their high excitation degree compared to the neutral reference state. This work reports 40 satellite transition energies and 58 valence ionization potentials of full configuration interaction (FCI) quality computed in small molecular systems. Following the protocol developed for the quest database [Véril, M.; et al. Wiley Interdiscip. Rev.: Comput. Mol. Sci. 2021, 11, e1517], these reference energies are computed using the configuration interaction using a perturbative selection made iteratively (CIPSI) method. In addition, the accuracy of the well-known coupled-cluster (CC) hierarchy (CC2, CCSD, CC3, CCSDT, CC4, and CCSDTQ) is gauged against these new accurate references. The performances of various approximations based on many-body Green's functions (GW , GF2, and T -matrix) for ionization potentials are also analyzed. Their limitations in correctly modeling satellite transitions are discussed.

I. INTRODUCTION

Ionization spectra, probed through techniques like UV-Vis, X-ray, synchrotron radiation, or electron impact spectroscopy, are invaluable tools in experimental chemistry for unraveling the structural intricacies of atoms, molecules, clusters, or solids. [1][2][3] Through the positions and intensities of their peaks, these spectra offer key information about the sampled system. For example, these measurements can be realized in various phases (gas, liquid, or solid) and, hence, analyzed to understand changes in electronic structure in these different phases. [4][5][6][7] Typically, within the energy range from 10 eV to 40 eV, valence-shell ionization occurs, while the core shell is probed at significantly higher energies. 8 This higherenergy region is not considered in the present study but a) Electronic mail: amarie@irsamc.ups-tlse.fr b) Electronic mail: loos@irsamc.ups-tlse.fr the concepts that we discuss below in the context of valence-shell spectra are also encountered in the case of core electron spectroscopy. Particularly, between 10 eV and 20 eV, ionization spectra of small molecules usually exhibit well-defined peaks. These sharp and intense ionization peaks are essentially single-particle processes, i.e., an electron is ejected from the molecule and measured by the detector. These first peaks are associated with outer-valence orbitals. At slightly higher energies, typically several eV, the situation is more complex as, in addition to inner-valence single-particle ionization peaks, additional broader and less intense peaks appear. These are referred to as satellites or shake-up transitions.

In molecules, satellites represent ionization events coupled with the simultaneous excitation of one or more electrons. They are thus intrinsically many-body phenomena, as one must describe at least two electrons and one hole. Satellite transitions can be seen as the equivalent of double excitations in the realm of neutral excitations. Because one must describe processes involving two electrons and two holes, double excitations pose significant challenges for theoretical methods, [9][10][11] and the same holds true for satellite transitions. Consequently, such states can hardly be described by mean-field formalisms, such as Hartree-Fock (HF) theory. Thus, properly accounting for correlation effects is crucial to describe satellite transitions. 12 In particular, a recent study has emphasized the dynamic nature of this correlation. 13 In the following the term "ionization" is employed to refer to single-particle processes, also called Koopmans' states.

Theoretical benchmarks play a pivotal role in evaluating the accuracy of approximation methods. [14][15][16][17][18][19][20][21][22][23] Concerning principal ionization potentials (IPs), which correspond to an electron detachment from the highestoccupied molecular orbital, two prominent benchmark sets are widely recognized: the extensively used GW 100 test set [24][25][26] and a set comprising 24 organic acceptor molecules. [27][28][29][30] Both sets rely on reference values obtained from coupled-cluster (CC) with singles, doubles, and perturbative triples [CCSD(T)] calculations [31][32][33] and determined by the energy difference between the neutral and cationic ground-state energies. [START_REF] Shavitt | Many-Body Methods in Chemistry and Physics: MBPT and Coupled-Cluster Theory[END_REF] Recently, Ranasinghe et al. created a comprehensive benchmark set including not only principal IPs but also outer-and inner-valence IPs of organic molecules. 35 Reference values for this set were computed using the IP version of the equation-ofmotion (IP-EOM) formalism [36][37][38][39][40] of CC theory with up to quadruple excitations (IP-EOM-CCSDTQ) for the smallest molecules. 35 Note that these benchmarks and the present work deal with vertical IPs, and we shall not address their adiabatic counterparts here.

To demonstrate its predictive capability for valence ionization spectra, an electronic structure method must precisely locate the positions of both outer-and innervalence ionization potentials, along with valence satellites. However, to the best of our knowledge, no established theoretical benchmarks exist for satellite energies in molecules. Consequently, the primary goal of this manuscript is to establish such a set of values. Finally, it is important to mention that to be fully predictive a method should be able to predict the intensities associated with of these transitions. However, benchmarking intensities is beyond the scope of this work and will be considered in a future study.

Nowadays, a plethora of methods exist to compute IPs in molecular systems. The most straightforward among them is HF where occupied orbital energies serve as approximations of the IPs (up to a minus sign) by virtue of Koopmans' theorem. [START_REF] Szabo | Modern quantum chemistry[END_REF] Similarly, within densityfunctional theory (DFT), the Kohn-Sham (KS) orbital energies can be used as approximate ionization energies. [START_REF] Perdew | [END_REF][43][44] The accurate computation of IPs within KS-DFT is still an ongoing research field with, for example, long-range corrected functionals, 45 KS potential adjustors, 46,47 doublehybrids functionals, 48 or even functionals directly optimized for IPs. 35,[49][50][51] An alternative way to compute electron detachment energies at the HF or KS-DFT levels is through the state-specific self-consistent-field (∆SCF) formalism, where one optimizes both the neutral ground state and the cationic state of interest, the IPs being computed as the difference between these two total energies. This strategy has been mainly used to compute core binding energies and is known to perform better than Koopmans' theorem thanks to orbital relaxation. [52][53][54][55][56][57][58] Mean-field methods, such as HF and KS-DFT, provide a first approximation to IPs but greater accuracy is often required. The well-known configuration interaction (CI) and CC formalisms provide two systematically improvable paths toward the exact IPs. [38][39][40]59 Within both frameworks, IPs can be obtained through a diagonalization of a given Hamiltonian matrix in the (N -1)-electron sector of the Fock space or through a state-specific formalism similar to ∆SCF. Ranasinghe et al. have shown that the mean absolute error (MAE) of IP-EOM-CCSDT with respect to CCSDTQ is only 0.03 eV for a set containing 42 IPs of small molecules. 35 Considering the same set, the cheaper IP-EOM-CCSD method has a MAE of 0.2 eV. Recently, the unitary CC formalism has also been employed within the IP-EOM formalism to compute IPs. 60 As mentioned above, the ∆SCF strategy can be extended to correlated methods which leads to the ∆CC method as an alternative to obtain IPs. 61 Once again, it has been mainly used to compute core IPs but it is also possible to determine valence IPs. [62][63][64] Selected CI (SCI) [65][66][67][68] provides yet another systematically improvable formalism for IPs. Indeed, by increasing progressively the number of determinants included in the variational space, one can in principle reach any desired accuracy, up to the full CI (FCI) limit. [69][70][71][72][73][74][75] Recently, the adaptative sampling CI algorithm 72,[76][77][78] has been used to compute accurate valence ionization spectra of small molecules. 13 In contrast to the wave function methods previously mentioned, one can also compute IPs via a more natural way based on electron propagators (or Green's functions), such as the GW approximation [START_REF] Martin | Interacting Electrons: Theory and Computational Approaches[END_REF][START_REF] Golze | [END_REF][START_REF] Marie | The GW Approximation: A Quantum Chemistry Perspective[END_REF] or the algebraic diagrammatic construction (ADC). [START_REF] Schirmer | Many-Body Methods for Atoms, Molecules and Clusters[END_REF][START_REF] Banerjee | [END_REF] The GW methodology has a myriad of variants. Its one-shot G 0 W 0 version, [84][85][86][87][88][89][90] which was first popularized in condensed matter physics, is now routinely employed to compute IPs of molecular systems and can be applied to systems with thousands of correlated electrons. [91][92][93][94][95][96][97][98][99][100][101][102][103] Other flavors of GW such as eigenvalue-only self-consistent GW (evGW ) 104-108 and quasi-particle self-consistent GW (qsGW ) [108][109][110][111][112][113] have also been benchmarked for IPs. Although the GW method is by far the most popular approach nowadays, there exist some alternatives, such as the second Born [also known as secondorder Green's function (GF2) in the quantum chemistry community] [START_REF] Szabo | Modern quantum chemistry[END_REF][114][115][116][117][118][119][120][121][122][123][124][125][126][127][128][129] or the T -matrix 130-147 approximations. However, none of them has enjoyed the popularity and performances reached by GW . [148][149][150] On the darker side, one of the main flaws of the GW approximation is its lack of systematic improvability, especially compared to the wave function methods mentioned above. Various beyond-GW schemes have been designed and gauged, but none of them seem to offer, at a reasonable cost, a systematic route toward exact IPs. 131,132,137,[151][152][153][154][155][156][157][158][159][160][161][162][163][164][165][166][167][168] The prediction of satellite peaks in molecules garnered attention in the late 20th century. In the 70's, Schirmer and coworkers applied extensively the 2ph-TDA [and the closely related ADC (3)] formalism to study the innervalence region of small molecules. 12,[169][170][171][172][173][174][175][176][177][178][179][180][181] CI methods were also employed by other groups to study this energetic region. [182][183][184][185][186][187][188][189][190][191][192][193] In both formalisms, the satellite energies are easily accessible as they correspond to higher-energy roots of the ADC and CI matrices. After relative successes for outer-valence ionizations, it was quickly realized that the inner-valence shell is much more difficult to describe due to the overlap between the inner-valence ionization and the outer-valence satellite peaks. 173 As mentioned above, the satellites present in this energy range cannot be described without taking into account electron correlation at a high level of theory. Even more troublesome, in some cases, the orbital picture (or quasiparticle approximation) completely breaks down. In other words, it becomes meaningless to assign the character of ionization or satellite to a given transition. 169,173 In the following decades, the symmetry-adapted-cluster (SAC) CI was extensively used to study the inner-valence ionization spectra of small organic molecules. [194][195][196][197][198][199][200][201][202][203][204][205] SAC-CI was shown to be able to compute satellite energies in quantitative agreement with experiments while methods based on Green's functions have been in qualitative agreement, at best. Satellites, sometimes called sidebands, have been extensively studied in the context of materials. [START_REF] Martin | Interacting Electrons: Theory and Computational Approaches[END_REF] These additional peaks, which can have different natures, are observed in photoemission spectra of metals, semiconductors, and insulators. [206][207][208][209][210][211][212][213][214][215] In "simple" metals, such as bulk sodium 206,213 or its paradigmatic version, the uniform electron gas, 25,214,[216][217][218][219][220][221] satellites are usually created by the strong coupling between electrons and plasmon excitations. It is widely recognized that GW does not properly describe satellite structures in solids, and it is required to include vertex corrections to describe these many-body effects. One of the most common schemes to study satellites in solids is the cumulant expansion, 210,219,[222][223][224] which is formally linked to electron-boson Hamiltonians. 156,[225][226][227] Nowadays, computational and theoretical progress allows us to systematically converge to exact neutral excitation energies of small molecules, 10,73,[228][229][230][231] and this holds as well for charged excitations like IPs. For example, Olsen et al. computed the exact first three IPs of water using FCI, 59 while Kamiya and Hirata went up to IP-EOM-CCSDTQ to compute highly accurate satellite energies for CO and N 2 . 39 As mentioned previously, a set of 42 IPs of CCSDTQ quality is also available now. 35 Finally, Chatterjee and Sokolov recently computed 27 valence IPs using the semistochastic heatbath SCI method 73,232,233 in order to benchmark their multi-reference implementation of ADC. 234,235 They also report FCI-quality energies for the four lowest satellite states of the carbon dimer. The present manuscript contributes to this line of research by providing 40 satellite energies of FCI quality. Additionally, 58 valence IPs are reported as well, among which 37 were not present in Chatterjee's CCSDTQ nor Sokolov's FCI benchmark set. 35 This study is part of a larger database of highly accurate vertical neutral excitation energies named quest which now includes more than 900 excitation energies. 23,230,231,[236][237][238][239][240][241][242][243] Our hope is that these new data will serve as a valuable resource for encouraging the development of novel approximate methods dedicated to computing satellite energies, building on the success of benchmarks with highly-accurate reference energies and properties.

II. COMPUTATIONAL DETAILS

The geometries of the molecular systems considered here have been optimized using cfour 244 following quest's protocol, 23,238 i.e. at the CC3/aug-cc-pVTZ level 245,246 without frozen-core approximation. The corresponding cartesian coordinates can be found in the Supporting Information. Throughout the paper the basis sets considered are Pople's 6-31+G* 247-253 and Dunning's aug-cc-pVXZ (where X = D, T, and Q). [254][255][256][257] A. Selected CI calculations All SCI calculations have been performed using the configuration interaction using a perturbative selection made iteratively (CIPSI) algorithm 67,228,[258][259][260][261] as implemented in quantum package. 262 For more details about the CIPSI method and its implementation, see Ref. 262. The frozen-core approximation has been enforced in all calculations using the conventions of gaussian16 263 and cfour, 244 except for Li and Be where the 1s orbital was not frozen.

We followed a two-step procedure to obtain the ionization and satellite energies, I N ν , at the SCI level. First, two single-state calculations are performed for the N -and (N -1)-electron ground states. This yields the principal IP of the system,

I N 0 = E N -1 0 -E N 0
, where E N -1 0 and E N 0 are the ground-state energies of the N -and (N -1)-electron systems, respectively. Then, a third, multi-state calculation is performed to compute the neutral excitation energies of the (N -1)-electron system,

∆E N -1 ν = E N -1 ν -E N 0 , where E N -1 ν
is the energy of the νth excited states associated with the (N -1)-electron system. Combining these three calculations, one gets

I N ν = E N -1 0 -E N 0 + ∆E N -1 ν (1)
Because single-state calculations converge faster than their multi-state counterparts, the limiting factor associated with the present CIPSI calculations are the convergence of the excitation energies ∆E N -1 ν , and this is what determines ultimately the overall accuracy of I N ν . For each system and state, the SCI variational energy has been extrapolated as a function of the second-order perturbative correction using a linear weighted fit using the last 3 to 6 CIPSI iterations. 73,74,264,[START_REF] Burton | Rationale for the Extrapolation Procedure in Selected Configuration Interaction[END_REF] The weights have been taken as the square of the inverse of the perturbative correction. The estimated FCI energy is then chosen amongst these extrapolated values obtained with a variable number of points such that the standard error associated with the extrapolated energy is minimal. Below, we report error bars associated with these extrapolated FCI values. However, it is worth remembering that these do not correspond to genuine statistical errors. The fitting procedure has been performed with mathematica using default settings. 266

B. Coupled-cluster calculations

The EOM-CC calculations have been done using cfour with the default convergence thresholds. 244 Again, the frozen-core approximation was enforced systematically. IP-EOM-CC calculations, i.e., diagonalization of the CC effective Hamiltonian in the (N -1)-electron sector of the Fock space, [36][37][38][39][40] have been performed for CCSD, 32,36,[267][268][269][270] CCSDT, 37,[271][272][273] and CCSDTQ. [274][275][276][277][278] At the CCSD level, the EOM space includes the one-hole (1h) and the two-hole-one-particle (2h1p) configurations, while the three-hole-two-particle (3h2p) and four-hole-three-particle (4h3p) configurations are further added at the CCSDT and CCSDTQ, respectively. Note that, within the CC formalism, we assume that the IP and electron affinity (EA) sectors are decoupled. [279][280][281][282] For CC2, 283,284 CC3, 245,246,[285][286][287]240,[288][289][290] diagonalization in the (N -1)-electron sector of the Fock space is not available yet. Hence, it has been carried out in the N -electron sector of the Fock space 36,37,[291][292][293][294][295] with an additional very diffuse (or bath) orbital with zero energy to obtain ionization and satellite energies. Therefore, at the CC2 level, the EOM space includes the one-hole-one-particle (1h1p) and the two-holetwo-particle (2h2p) configurations, while the three-holethree-particle (3h3p) and four-hole-four-particle (4h4p) configurations are further added at the CC3 and CC4 levels, respectively. These two schemes produce identical IPs and satellite energies but, for a given level of theory, the diagonalization in the N -electron sector is more computationally demanding due to the larger size of the EOM space (see Ref. 296 for more details). In each scheme, the desired states have been obtained thanks to the root-following Davidson algorithm implemented in cfour. The initial vectors were built using the dominant configurations of the SCI vectors.

The ∆CCSD(T) calculations have been performed with gaussian16. 263 These calculations are based on a closedshell restricted HF reference and an open-shell unrestricted HF reference for the neutral and cationic species, respectively. 149

C. Green's function calculations

Many-body Green's function calculations have been carried out with the open-source software quack. 297 In the following, we use the acronyms G 0 W 0 , G 0 F(2), and G 0 T 0 to refer to the one-shot schemes where one relies on the GW , second-Born, and T -matrix self-energies, respectively. Each approximated scheme considered in this work (G 0 W 0 , qsGW , G 0 F(2), and G 0 T 0 ) relies on HF quantities as starting point. We refer the reader to Refs. 81 and 150 for additional details about the theory and implementation of these methods. The infinitesimal broadening parameter η is set to 0.001 E h for all calculations. It is worth mentioning that we do not linearize the quasiparticle equation to obtain the quasiparticle energies. The qsGW calculations are performed with the regularized scheme based on the similarity renormalization group approach, as described in Ref. 113. A flow parameter of s = 500 is employed. All (occupied and virtual) orbitals are corrected. The spectral weight of each quasiparticle solution is reported in Supporting Information. Compared to the EOM-CC formalism discussed in Sec. II B, it is important to mention that, in the Green's function framework, the IP and EA sectors [i.e. the 1h and one-particle (1p) configurations] are actually coupled, 150,281,298 effectively creating higher-order diagrams. [START_REF] Schirmer | Many-Body Methods for Atoms, Molecules and Clusters[END_REF]298

III. RESULTS AND DISCUSSION

The present section is partitioned into subsections, each dedicated to a distinct group of related molecules. Within these subsections, we focus our attention mainly on the satellite states while IPs are addressed in Sec. III G.

Each state considered in this work is reported alongside its symmetry label, e.g. 1 2 B 1 for the principal IP of water. Furthermore, the main orbitals involved in the ionization process are specified. For example, the N -electron ground state of water has the following dominant configuration (1a 1 ) 2 (2a 1 ) 2 (1b 2 ) 2 (3a 1 ) 2 (1b 1 ) 2 , while the configuration of the (N -1)-electron ground state is (1a 1 ) 2 (2a 1 ) 2 (1b 2 ) 2 (3a 1 ) 2 (1b 1 ) 1 . Hence, we denote the principal IP as (1b 1 ) -1 to indicate that an electron has been ionized from the 1b 1 orbital. The lowest satellite of water, i.e. the 2 2 B 1 state of configuration (1a 1 ) 2 (2a 1 ) 2 (1b 2 ) 2 (3a 1 ) 1 (1b 1 ) 1 (4a 1 ) 1 , is labeled as (3a 1 ) -1 (1b 1 ) -1 (4a 1 ) 1 to signify that one electron was detached from the orbital 1b 1 and 3a 1 , one of them being subsequently promoted to the virtual orbital 4a 1 and the other ionized. In some cases, additional valence completeactive-space CI calculations have been performed using molpro to determine the symmetry of the FCI states. 299 A. 10-electron molecules: Ne, HF, H2O, NH3, and CH4

The water molecule has been extensively studied experimentally using photoionization and electron impact Water (H2O) State/Conf. Hydrogen fluoride (HF) State/Conf. spectroscopy. 204,304,305 For example, a high-resolution spectrum of liquid water is crucial as a first step for understanding the photoelectron spectra of aqueous phases. 5,6 On the other hand, its gas-phase ionization spectrum is now well understood. The experimental ionization spectrum of water is plotted in Fig. 1, serving as a representative example to illustrate the following discussion.

2 2 B1/(3a1) -1 (1b1) -1 (4a1) 1 2 2 A1/(1b1) -2 (4a1) 1 3 2 B1/(3a1) -1 (1b1) -1 (4a1)
(CH4) State/Conf. 2 2 A1/(3a1) -2 (4a1) 1 2 2 E/(3a1) -2 (3e) 1 2 2 T2/(1t2) -2 (3a1)
2 2 Σ + /(1π) -2 (4σ) 1 1 2 ∆/(1π) -2 (4σ)
The first three sharp peaks at 12.6, 14.8, and 18.7 eV are associated with electron detachments from the three outer-valence orbitals, 1b 1 , 3a 1 , and 1b 2 , respectively. 204 Then, a broader yet intense peak corresponding to the fourth ionization, (2a 1 ) -1 , is found at 32.4 eV surrounded by several close-lying satellite peaks. Additionally, there is a smaller broad satellite peak at 27.1 eV (magnified red peak in Fig. 1). Table I gathers the FCI reference values corresponding to the three lowest satellites identified in our study. The first two, which are of 2 B 1 and 2 A 1 symmetries, lie close to each other around 27.1 eV. The third satellite is of 2 B 1 symmetry and is found at slightly higher energy, approximately 29 eV. The ordering and the absolute energies of the three satellites align well with previous SAC-CI results reported by Ning et al. 204 In addition, they showed that, at the ADC(3) level, the energy of the 2 2 A 1 state is overestimated by approximately 2.7 eV while the 2 2 B 1 and 3 2 B 1 states are missing. It is also worth noting that early CI and Green's function studies had qualitatively predicted the 2 2 A 1 satellite. 12,187 Finally, we do not consider the broad peak at 32.4 eV here because it is technically out of reach for our current SCI implementation. However, it has been studied by Mejuto-Zaera and coworkers who have shown that vertex corrections are required to correctly describe this complex part of the spectrum where strong many-body effects are at play. 13 For the three satellites of water, CCSDTQ is in nearperfect agreement with FCI in all basis sets with errors inferior to 0.03 eV. CC4 is slightly worse than CCSDTQ but is still an excellent approximation given its lower computational cost and its approximate treatment of quadruples. The CCSDT satellite energies are overestimated by approximately 0.5 eV, while CC3 appears to struggle for this system. Indeed, the CC3 energies are badly underestimated with errors up to 1.5 eV, and the ordering of the first two satellites is wrongly predicted. 306 Finally, CCSD and CC2 are not considered for satellites as their poor performance (wrong by several eV) makes the assignment of these states extremely challenging.

The remainder of this section is concerned with four molecules isoelectronic to water, namely CH 4 , NH 3 , HF, and Ne. For each of these molecules, Table I provides FCI reference values for the IPs corresponding to the two outermost valence orbitals. In addition, two satellite energies are reported for hydrogen fluoride and ammonia while one satellite is presented for methane. Experimental values for the IPs of these four molecules have been measured multiple times and are reported in Table I. 4,7,189,191,192,300,302,[307][308][309] Yencha and coworkers measured the inner-valence photoelectron spectrum of HF. It displays a well-defined peak around 33 eV which appears in close agreement with the FCI energies for the 2 2 Σ + state. 309 In addition, a doubly-degenerate satellite of 2 ∆ symmetry has also been computed. In the various NH 3 ionization spectra reported in the literature, there is no satellite peak around 24 eV which may correspond to the 2 2 A 1 and 2 2 E FCI states. 189,201,300 Nevertheless, the FCI energies align well with the SAC-CI energies of Ishida and coworkers who predicted that these two satellite states have very low intensity. 201 The first satellite observed in the inner-valence region of the photoionization spectrum of CH 4 is a very weak and broad peak at 29.2 eV. 301 This peak is also measured at 28.56 eV using electron momentum spectroscopy experiments. 192 The energy of the first satellite calculated at the FCI level, and associated with the (2t 2 ) -2 (3a 1 ) 1 process, compares well with the experimental data. It is worth noting that CC3 behaves similarly in HF and H 2 O, yet it appears to be a much better approximation for the satellite states of NH 3 and CH 4 .

Among the 12 IPs computed for this series of molecules, 11 of them have a weight larger than 0.85 on the 1h dominant configuration (in the aug-cc-pVDZ basis set). Only the 3 2 A 1 state of ammonia has a quite smaller weight (0.58) on the corresponding 1h determinant. This exemplifies the breakdown of the orbital picture in the inner-valence ionization spectrum, 12,[169][170][171][172][173][174][175][176][177][178][179][180][181] which signature is a significant weight on both 1h and 2h1p configurations, hence preventing us from assigning the solution as a clear IP or satellite. The performance of the various approximations for IPs will be statistically gauged in Sec. III G. Thus far, we have exclusively assessed the performance of different rungs of the CC hierarchy. Although shakeup transition energies can also be computed within the Green's function framework, the task is notably more challenging, especially when compared to the more straightforward nature of IP-EOM-CC. This complexity arises from the fact that satellites, existing as non-linear solutions of the quasiparticle equation, [START_REF] Martin | Interacting Electrons: Theory and Computational Approaches[END_REF] prove much more difficult to converge using Newton-Raphson algorithms than the quasiparticle solutions, which are representative of typical IPs. Fortunately, an alternative and equivalent pathway exists where one solves a larger linear eigenvalue problem instead of solving the non-linear quasiparticle equation. [START_REF] Schirmer | Many-Body Methods for Atoms, Molecules and Clusters[END_REF]125,150,282,[310][311][312][313][314] In such a case, satellites are obtained as higher-energy roots via diagonalization of the so-called "upfolded" matrix built in the basis of the 2h1p and two-particle-one-hole (2p1h) configurations in addition to the 1h and 1p configurations.

The satellite energies of H 2 O and NH 3 computed with G 0 W 0 , G 0 F(2), and G 0 T 0 are presented in Table II. qsGW is not considered here as its static approximation naturally discards all the satellite solutions. The third column shows the diagonal elements of the upfolded matrix associated with the 2h1p configurations, while the fourth column displays the associated eigenvalues. One can immediately observe that the eigenvalues do not improve upon the diagonal elements. This is due to the lack of higher-order (such as 3h2p) configurations that are essential to correlate satellites. This parallels the description of double excitations which require at least triple excitations (i.e. 3h3p configurations) in addition to the 2h2p configurations to correlate doubly-excited states. [9][10][11] Regarding the two satellites of ammonia, the T -matrix zeroth-order elements are utterly inaccurate. This discrepancy arises because, at the T -matrix level, satellite energies are described as the sum of a Koopmans' electron attachment energy (1p configuration) and a double electron detachment energy [two-hole (2h) configuration] stemming from the particleparticle random-phase approximation. 150,[315][316][317][318] On the other hand, G 0 W 0 and G 0 F(2) offer decent estimates of these satellite energies.

The remaining three rows of Table II contain the energies corresponding to the three satellites of water discussed previously. The 2 2 A 1 state is the easiest to identify as its 2h1p dominant configuration clearly corresponds to the (1b 1 ) -2 (4a 1 ) 1 process. The eigenvectors associated with the 2 2 B 1 and 3 2 B 1 states, which correspond to the (3a 1 ) -1 (1b 1 ) -1 (4a 1 ) 1 and (1b 1 ) -1 (3a 1 ) -1 (4a 1 ) 1 processes respectively, are nearly degenerate and highly entangled. This is thus harder, if not impossible, to assign these states.

Because of these assignment problems, the G 0 W 0 , G 0 F( 2), and G 0 T 0 satellite energies have not been computed for the other molecules considered in this study. To alleviate this issue, there is a notable appeal for a self-energy approximation including vertex corrections capable of effectively addressing satellite states. As mentioned previously, Green's-function-based methods such as the 2ph-TDA 171 and ADC(3) 180, 319,320 (first named extended 2ph-TDA 179 ) have shown success in qualitatively modeling the inner-valence region of experimental spectra. 12,[169][170][171][172][173][174][175][176][177][178][179]181,321 Sokolov's recent multi-reference ADC(2) scheme 322 is also a promising avenue. In particular, it has shown potential in describing the satellites of the carbon dimer (see below). 234,235 While a detailed quantitative analysis of these approaches on the present benchmark set would be interesting, it lies beyond the scope of this study.

C. 14-electron molecules: N2, CO, and BF

The nitrogen and carbon monoxide molecules have been extensively studied both experimentally 324-330 and theoretically. 39,169,180,182,183,185,197,[331][332][333] Their ionization spectra are similar as they exhibit three sharp and intense peaks, corresponding to Koopmans' states, below 20 eV. Their respective fourth IP, corresponding to electron detachment from the orbital 2σ g for N 2 and 3σ for CO, lies above 30 eV. Several peaks can be found below these ionizations, i.e. between 20 and 30 eV. 324,327,328 These correspond to satellite states associated with the three outer-valence orbitals. Note that Schirmer et al. have shown (using the 2ph-TDA method 171 ) that the quasiparticle approximation breaks down in the region of the fourth ionizations of CO and N 2 . 169,180 However, as shown below, the peaks between 20 and 30 eV have a well-defined satellite character.

Baltzer et al. produced, using He(II) photoelectron spectroscopy, accurate experimental values for the outervalence IPs (see Table III) and the first satellite peaks of N 2 . 325 In particular, they reported a value of 25.514 eV Carbon monoxide (CO) State/Conf. Dinitrogen (N2) State/Conf. for an intense satellite peak, as well as 24.788 eV for a very weak peak. These peaks were assigned 2 Σ + u and 2 Π g symmetry, respectively, based on CI calculations. Note that the 2 Π g satellite peak is more intense when measured by resonance Auger spectroscopy. 324 We report FCI values for both satellites as well as a slightly higher third one with 2 Σ - u symmetry. This latter state is not observed experimentally but plays an important role nonetheless as it is involved in the dissociation pathways between the 2 Σ + u and 4 Π u states. 331 Similar to its isoelectronic N 2 molecule, CO exhibits shake-up peaks between the (4σ) -1 and (3σ) -1 ionizations. Using monochromatized X-ray excited photoelectron spectroscopy, Svensson et al. observed an intense 2 Σ + satellite peak with energy 23.7 eV as well as a weak 2 Π satellite at 22.7 eV. 324 This is in agreement with older He(II) photoelectron spectroscopy experiments done by Asbrink and coworkers. 328 FCI values for both satellites, as well as for the higher-energy 1 2 ∆ state, are reported in Table II.

2 2 Π/(5σ) -2 (2π) 1 2 2 Σ + /(1π) -1 (5σ) -1 (2π) 1 1 2 ∆/(1π) -1 (5σ) -1 (2π)
1 2 Πg/(3σg) -2 (4πg) 1 1 2 Σ + u /(3σg) -1 (3πu) -1 (4πg) 1 1 2 Σ - u /(3σg) -1 (3πu) -1 (4πg)
The performance of CC schemes for these six satellites is similar to what we have observed for the 10-electron series. Yet, it is interesting to note that CCSDTQ seems to struggle slightly more with the 2 2 Π satellite of CO and the 1 2 Π g and 1 2 Σ - u states of N 2 . The boron fluoride molecule is isoelectronic to CO and N 2 but its ionization spectrum is much harder to obtain experimentally because BF is a quite non-volatile compound, meaning that the measurements have to be done at high temperatures. 323 Yet, Hildenbrand and coworkers managed to measure its principal IP using electron impact spectroscopic and they reported a value of 11.06 eV. The 1 2 Σ + FCI state is in good agreement with this value. Table II also displays two additional IPs and one satellite. The order of the 2 Π states in BF is reversed with respect to its isoelectronic species: the (5σ) -2 (2π) 1 satellite state has a lower energy than the (1π) -1 ionization. In this case, CCSDTQ accurately describes the satellite of Π symmetry.

D. 12-electron molecules: LiF, BeO, BN, and C2

We now direct our attention toward the 12-electron isoelectronic molecules: LiF, BeO, BN, and C 2 . These four molecules are quite challenging for theoretical methods as, except for LiF, their ground states have a strong multi-reference character. [335][336][337][338][339][340][341][START_REF] Ammar | Can GW Handle Multireference Systems?[END_REF] For example, BN and BeO are among the eight molecules of the GW 100 set having multiple solution issues at the GW level. 24,25 (C 2 is not considered in the GW 100 set but would certainly fall in the same category.) Another noteworthy observation about these molecules is that their lowest unoccupied molecular orbital has a negative energy, which means that their respective anions are stable.

LiF is a relatively non-volatile molecule, and as a result, experimental data became accessible during the second phase of the development of ultraviolet photoelectron spectroscopy. 334 In addition, lithium fluoride vapor is not solely composed of monomers but also includes dimers, trimers, or even tetramers, posing challenges for more precise measurements of the Koopmans states of LiF. Berkowitz et al. measured the first two IPs using He(I) photoelectron spectroscopy : 11.50, 11.67, and 11.94 eV for the 1 2 Π 3/2,3/2 , 1 2 Π 3/2,1/2 , and 1 2 Σ + states, respectively. In our study, the spin-orbit coupling is not accounted for.

Therefore, we report a single value for the 1 2 Π state, while experimentally two distinct ionization energies are measured.

To the best of our knowledge, no gas phase experimental values are available for the three remaining species (see Ref. 343 for a study in solid phase). Nonetheless, they are an interesting playground for theoretical methods due to their multi-reference character. We start by discussing BeO as it has the less pronounced multi-reference character out of these three molecules. Table IV displays the excitation energies of the two lowest Koopmans states and the first two satellites. These four states have the same dominant configurations and ordering as the ones of lithium fluoride. However, the satellite states of BeO are much lower in energy than those of LiF. The 2 2 Π state of BeO is interesting as it exhibits the largest error of this benchmark set at the CCSDTQ level. At the CC4 level, the 2 2 Π and 1 2 Σ -states are drastically underestimated. This is also the case for the two satellite states of LiF, these four states having, by far, the largest CC4 errors of this benchmark set. They are also hugely underestimated at the CC3 level and, as for CC2 and CCSD, we have not reported these energies as they are not meaningful. Unfortunately, at this stage, we have no clear explanation for the failure of CC3 and CC4 in LiF and BeO.

BN and C 2 have the strongest multi-reference character among these four molecules. [START_REF] Ammar | Can GW Handle Multireference Systems?[END_REF] The ordering of their state differs from the one of LiF and BeO as their lowest satellite states are below their second IP. Furthermore, the ordering of the satellites is also different than the two previous molecules. The first satellite of boron nitride has the same dominant configuration as in the latter two molecules but the second satellite is of 1 2 ∆ symmetry with a (1π) -2 (5σ) 1 dominant process. This satellite is also found in C 2 but even lower in the energy spectrum as the 1 2 ∆ g state is the lowest-energy satellite of the carbon dimer. Table IV reports two additional FCI satellite transition energies of C 2 . Note that the satellite transition energies of BeO, BN, and C 2 are the lowest of the present set. The three satellite states of the carbon dimer have already been studied by Chatterjee and Sokolov. 234,235 In particular, they have shown that ADC(3) performs poorly and does not even predict enough satellite states. On the other hand, their extension of ADC(2) using a multi-determinantal reference 322 can predict each state and be in quantitative agreement with FCI. 234,235 E. Third-row molecules: CS, Ar, HCl, H2S, PH3, SiH4, and LiCl

The molecules examined in this subsection have been obtained by substituting a second-row atom with its thirdrow analog in some of the molecules discussed above. These molecules with more diffuse density have their ionization shifted towards zero with respect to their secondrow counterparts (see Tables V, VI, and VII). Consequently, the breakdown of the orbital picture occurs at Lithium fluoride (LiF) Beryllium oxide (BeO) State/Conf. The first molecule considered in this subsection is carbon sulfide. In 1972, two independent studies measured its photoelectron spectrum up to 20 eV. 346,347 One can clearly distinguish four well-defined peaks in this energy range. While the assignment of the two lowest peaks is straightforward, i.e. IPs associated with the two outermost orbitals, the interpretation of the other two remained Carbon sulfide (CS) State/Conf. Lithium chloride (LiCl) Fluorine (F2) State/Conf. elusive for several years. Thanks to theoretical studies performed several years later, it became clear that the third peak is due to a multi-particle process while the fourth one is associated with an electron detachment from the orbital 6σ. 172,234,235,306 Note that, as explained by Schirmer et al.,172 one has to be particularly careful when labeling the third peak as a satellite because its FCI vector has a coefficient of 0.49 on the 1h configuration (6σ) -1 and of 0.40 on the 2h1p configuration (2π) -1 (7σ) -1 (3π) 1 (in 6-31+G* basis set). This is yet another example of a strong configuration mixing.

1 2 Σ -/(1π) -2 (5σ) 1 2 2 Π/(4σ) -1 (1π) -1 (5σ) 1 1 2 Σ -/(1π) -2 (5σ) 1 Exp. CC3 × × × × × × × × × × × × CCSDT
2 2 Π/(4σ) -1 (1π) -1 (5σ) 1 1 2 Σ -/(1π) -2 (5σ) 1 1 2 ∆/(1π) -2 (5σ) 1 Exp. CC3 × × × × 13
2 2 Σ + /(2π) -1 (7σ) -1 (3π) 1 3 2 Σ + /(2π) -1 (7σ) -1 (3π) 1 1 2 ∆/(2π) -1 (7σ) -1 (3π)
2 2 Σ + /(2π) -2 (7σ) 1 2 2 Π/(6σ) -1 (2π) -1 (7σ) 1 2 2 Σ + g /(1πu) -1 (1πg) -1 (3σu)
Despite this, the 2 2 Σ + state is classified as a satellite in Table V. Indeed, higher in energy there is another FCI solution of 2 Σ + symmetry with an even larger weight, 0.63, on the 1h determinant (6σ) -1 and coefficients smaller than 0.31 on the 2h1p determinants. In addition, the third peak has a pronounced vibrational structure while the fourth peak is sharp like the (7σ) -1 one. 347 This is why the higher 2 Σ + state is classified as the third single ionization in Table V. Two other satellite states, 3 2 Σ + and 1 2 ∆, that are not visible on the experimental spectrum, are reported in Table V.

Next, we consider lithium chloride and compare it with its second-row analog, lithium fluoride. The experimental challenges outlined earlier for LiF are similar for LiCl. Experimental values have first been reported independently by two groups in 1979, 334,[START_REF] Potts | Faraday Discuss[END_REF] and revised values, measured by He(I) spectroscopy, have been published recently, 344 and are reported in Table V. The FCI results predict two close-lying satellites around 20 eV. Unfortunately, the experimental studies mentioned above do not probe this energy range. The ADC(3) calculations of Tomasello et al. also predict two satellite lines around 21.5 eV. 349 Finally, we examine the 18-electron isoelectronic hydrides as analogs to the 10-electron series discussed in Subsec. III A. Historically, it was quickly realized that the satellite structure of H 2 S is significantly more complex than the one of H 2 O. 174 This intricate structure can be observed in the electron momentum spectrum of French et al. 350 They recorded a first very weak satellite at 19.63 eV which is in agreement with earlier measurements 351 as well as photoelectron spectrum measured using synchrotron radiation. 352 Several years later, extensive SAC-CI results have been reported and show qualitative agreement with experiments. 200 (See also earlier calculations from Refs. 193 and 195.) However, the FCI results (see Table VII) exhibit some significant difference with the SAC-CI results of Ehara et al. because the 2 2 A 1 satellite has a lower energy than the 2 2 B 2 state. In addition, the FCI transition energy associated with the 2 2 B 1 state is 2 eV lower than the one computed in Ref. 200. The 2 2 A 1 state is known to be the one observed at 19.63 eV in Ref. 350 and is sometimes referred to as a shakedown state as it "borrows" intensity from the higher-lying (4a 1 ) -1 ionization. 200,352,353 Our FCI estimate for this state is 18.745 eV while the SAC-CI energy of Ehara and collaborators is 20.00 eV. 200 In this specific scenario, calculating the adiabatic transition energy related to this state would undoubtedly provide a more faithful comparison with the experimental result.

For PH 3 , there is one satellite of symmetry E that is analog to the two 2 2 E state of NH 3 . However, in the case of phosphine, there is no analog for the 2 2 A 1 satellite of ammonia. This is in agreement with the SAC-CI results of Ishida et al. who also found a single satellite below the (4a 1 ) -1 ionization threshold. 201 Two FCI states, with symmetry 2 Σ + and 2 ∆, are reported for HCl. These states are analog to the HF satellites reported in Table I although they have significantly lower energies in HCl. The satellite structure between 20 eV and the (4σ) -1 ionization around 26 eV is notably intricate. 354 Additionally, this structure is characterized by weak signals and some of its features were even not observed in previous studies performed at a lower level of theory. [355][356][357] The assignment of the various peaks in this energy range is beyond the scope of this work. Yet, one can mention that the first FCI satellite is in qualitative agreement with the first satellite peak measured by synchrotron radiation spectroscopy at 21.57 eV. 354 F. Miscellaneous molecules: F2, CO2, CH2O, and BH3

In this last subset, a few miscellaneous molecules are considered. First, the F 2 molecule is of interest as it is isoelectronic to SiH 4 , PH 3 , H 2 S, HCl and Ar. Due to the absence of third-row atoms in F 2 , Cederbaum et al. observed that there is no breakdown of the orbital picture, as observed in the 18-electron hydride series. 170 Three IPs and one satellite transition energy of fluorine are reported in Table V. As documented in Ref. 170, these four states have a clear dominant configuration in their corresponding FCI vectors. The 2 2 Σ + g satellite is not observed in photoionization 302,358 or in electron impact spectra. 345 However, a satellite state with the same symmetry and similar energy has been computed using ADC( 4) and multi-reference CI. 345 Carbon dioxide and formaldehyde are two small organic molecules that have been widely studied both experimentally and theoretically. Experimental studies have shown that CO 2 first measurable satellite peak is around 22 eV, 188,205,326,359 while four ionization peaks are observed below 20 eV. Tian's experimental values for these IPs, measured by electron momentum spectroscopy, are reported alongside our FCI estimates in Table VIII. 205 These IPs have already been computed at various levels of theory such as CI, 188 ADC, 234,235,194,205 CC, 35 and even FCI. 234,235 The spectrum of formaldehyde is slightly harder to interpret. The electron momentum spectrum displays a shark peak at 10.9 eV as well as a broad band between 12 and 18 eV. 190 On the other hand, one can observe four different peaks below 17 eV in the corresponding photoionization spectra. 177,360 These peaks clearly correspond to ionizations from the four outermost orbitals (see Table VIII) and have already been computed using both wave function and Green's function methods. 35,38,177,234,235,332 Hochlaf et al. also mention a very weak band around 18 eV assigned as a satellite of 2 B 2 symmetry, 360 which can also be observed in Ref. 177. This is in nice agreement with the 3 2 B 2 FCI satellite state. There is an additional 2 2 B 1 satellite state with slightly lower energy than the previous one.

Finally, a small boron hydride is considered as we have seen above that boron-containing molecules such as BN are quite challenging. To the best of our knowledge, experimental results on BH 3 are quite scarce but the principal IP has been measured using mass spectroscopy in the 60's. 361,362 Finally, after discussing each molecule individually, the statistics over the whole set are reported and discussed in this subsection. Figure 2 displays the mean sign error (MSE) and MAE of the various methods considered in this study with respect to the new FCI references. These statistical errors have been computed for the 58 IPs in the aug-cc-pVTZ basis set. Several other statistical descriptors are also reported in Table VI.

The CC hierarchy (CCSD, CCSDT, and CCSDTQ) behaves as expected, i.e. being more and more accurate as the rank of the excitation is increased. Chemical accuracy (i.e., error below 0.043 eV as represented by the horizontal green line in the lower panel Fig. 2) is reached at the CCSDT level. The least expensive CC2 method does not perform well for IPs as already observed previously. [364][365][366] This has been attributed to the same underlying issue observed in CC2 for Rydberg 367 and charge-transfer excited states. 368 Figure 2 also shows that CC3 and CC4 are good approximations, for IPs, of their respective parents, CCSDT and CCSDTQ. This could have been expected for CC3 as it is known to be a good approximation of CCSDT for Rydberg excited states. 367 In addition, these four methods have very small MSEs and do not, on average, underestimate (as CC2) or overestimate (as CCSD) the IPs. Therefore, implementations of IP-EOM-CC3 and IP-EOM-CC4 would be valuable to lower the cost of the present implementation based on EE-EOM. CC3 and CC4 could be certainly employed as reference methods for larger molecular systems. 240,290,369 For the sake of completeness, we also report in Table VI the statistical descriptors for the propagator methods. However, their trends are now well-known. 113,142,149,150 The G 0 T 0 MAE is very close to the G 0 W 0 one whereas the second-Born approximation exhibits significantly poorer performance. 142,149,150 The self-consistent qsGW slightly mitigates the error compared to the one-shot GW version. It is also interesting to note that GF2 results are very close to those of CC2. This could have been expected as GF2 is equivalent to ADC(2) [START_REF] Schirmer | Many-Body Methods for Atoms, Molecules and Clusters[END_REF]124,150 and the latter is closely related to the CC2 approximation. 370 Finally, the principal IP of the 23 molecules considered so far have been computed at the ∆CCSD(T) in the augcc-pVTZ basis set (see Supporting Information). This method has been used as the reference for the GW 100 dataset and it is interesting to benchmark it now that we have access to FCI references. 25,61,149,371 The last line of Table VI reports the corresponding statistical descriptor. In particular, its MAE and MSE of 0.021 eV and 0.016 eV, respectively, show that the state-specific ∆CCSD(T) method can indeed be employed as a reference.

Figure 3 shows the distribution of the errors associated with the satellite transitions computed with CC methods including at least triple excitations, namely, CC3, CCSDT, CC4, and CCSDTQ. The corresponding statistical descriptors are reported in Table VII. The MAE of CCSDTQ is 0.040 eV, i.e. just below chemical accuracy, while CC4 and its approximate treatment of quadruples achieve a 0.087 eV MAE. Interestingly, while CC4 absolute errors are, on average, larger than CCSDTQ, its MSE is closer to zero. CC3 and CCSDT have MAEs of 0.749 eV and 0.528 eV, respectively, and once again CC3 and CC4 have lower MSEs than their parent methods (0.232 eV and 0.528 eV). Hence, methods accounting for triple excitations, even fully, should be used with care for satellites. It is also interesting to note that these

IV. CONCLUSION

We have reported 40 FCI satellite transition energies computed in 23 small molecules. These energies have been calculated with increasingly large basis sets ranging from Pople's 6-31+G* to Dunning's aug-cc-pVXZ (where X = D, T, and Q). In addition, 58 FCI reference values for outer-and inner-valence IPs of the same molecular set have been presented. This work is the tenth layer of reference values of the quest database 23,238 and the first one to include charged excitations.

Various CC methods have been employed to compute IPs (CC2, CCSD, CC3, CCSDT, CC4, and CCSDTQ) and satellite transition energies (CC3, CCSDT, CC4, and CCSDTQ), and their performances have been assessed using the FCI reference values. It has been shown that CC3 and CC4 are faithful approximations of CCSDT and CCSDTQ for IPs, respectively, while the CC2 approximate treatment of double excitations induces large errors with respect to CCSD. For the satellites, our study reveals that chemical accuracy is reached only at the CCSDTQ level, highlighting the intricate and complex correlation effects involved in such states and their overall challenging nature for computational methods.

The performance of various propagator methods (G 0 W 0 , G 0 F(2), G 0 T 0 , and qsGW ) have also been gauged. The poor performance of these methods for satellite transition energies has been discussed in detail. These results call for the development of new methods capable of describing such states. Studying the convergence of the ADC hierarchy using these new benchmark values is a possible outlook. Finally, assessing methods designed in the condensed matter community, such as the cumulant Green's function, on these small molecular systems would certainly be interesting. Work along this line is presently underway.

One obvious perspective that needs to be addressed is the extension to transition intensities, which are of crucial importance for direct comparisons with experimental spectra. An approximate electronic structure method should not only aim to accurately describe the excitedstate energy but also the transition intensities associated with it. Within the present SCI formalism, computing intensities is not straightforward but this is feasible, as demonstrated in Refs. 13, 373, and 374, and is planned for future investigation. file for each molecule. This json file contains all the IPs and satellite transition energies of a given molecule as well as the FCI incertitudes and the spectral weight associated with the G 0 W 0 , G 0 F(2), and G 0 T 0 quasiparticle energies.

2 FIG. 1 .

 21 FIG. 1. Gaussian fit of the experimental ionization spectrum of water in gas phase measured by Ning and coworkers. The fitting parameters can be found in Ref. 204. The red peak at 27.1 eV has been magnified by a factor 10.

FIG. 3 .

 3 FIG.3. Distribution of the errors with respect to FCI of the various methods considered in this work. These errors are computed for the 36 satellites of this set in the aug-cc-pVTZ basis set. The satellites of LiF and BeO have been excluded (see main text). Note the different scale of the horizontal axis in the leftmost plot.

TABLE I .

 I Valence ionizations and satellite transition energies (in eV) of the 10-electron series for various methods and basis sets. AVXZ stands for aug-cc-pVXZ (where X = D, T, and Q). Selected experimental values are also reported.

			Basis				Basis			Basis
	Methods	6-31+G *	AVDZ	AVTZ	AVQZ	6-31+G *	AVDZ	AVTZ	AVQZ	6-31+G *	AVDZ	AVTZ	AVQZ
	Mol .						Water (H2O)
	State/Conf.		1 2 B1/(1b1) -1			1 2 A1/(3a1) -1			1 2 B2/(1b2) -1
	Exp.		12.6 204			14.8 204			18.7 204
	CC2	11.159	11.345	11.541	11.620	13.513	13.645	13.791	13.863	18.035	18.039	18.145	18.211
	CCSD	12.170	12.386	12.594	12.675	14.502	14.677	14.825	14.895	18.861	18.888	18.972	19.032
	CC3	12.287	12.519	12.661	12.722	14.621	14.811	14.899	14.949	18.950	18.993	19.023	19.065
	CCSDT	12.276	12.491	12.629	12.689	14.601	14.776	14.861	14.910	18.919	18.951	18.981	19.022
	CC4	12.307	12.543	12.683	12.741	14.635	14.832	14.920	14.968	18.952	18.999	19.030	19.070
	CCSDTQ	12.304	12.534	12.673	×	14.631	14.822	14.907	×	18.947	18.990	19.018	×
	FCI	12.309	12.540	12.679	12.737	14.636	14.829	14.915	14.962	18.950	18.995	19.024	19.063
	G0W0	12.312	12.485	12.884	13.080	14.625	14.781	15.106	15.285	18.818	18.865	19.129	19.290
	qsGW	12.379	12.640	12.879	12.982	14.696	14.932	15.107	15.197	18.965	19.069	19.188	19.271
	G0F(2)	11.110	11.279	11.555	11.675	13.507	13.626	13.837	13.945	17.983	17.978	18.141	18.236
	G0T0	11.967	12.095	12.357	×	14.240	14.336	14.532	×	18.459	18.429	18.572	×
	Mol.						Ammonia (NH3)
	State/Conf.		1 2 A1/(3a1) -1			1 2 E/(1eg) -1			3 2 A1/(2a1) -1
	Exp.		10.93 300			16.6 300
	CC2	9.779	9.986	10.168	10.234	15.794	15.828	15.960	16.019	27.646	27.381	27.365	×
	CCSD	10.434	10.677	10.862	10.923	16.403	16.473	16.588	16.639	27.745	27.696	27.855	27.915
	CC3	10.447	10.746	10.888	10.935	16.407	16.520	16.592	16.629	27.252	27.114	27.191	×
	CCSDT	10.449	10.734	10.876	10.922	16.399	16.500	16.573	16.609	26.773	26.724	26.899	×
	CC4	10.461	10.761	10.901	×	16.417	16.533	16.603	×	26.669	26.621	26.746	×
	CCSDTQ	10.461	10.760	10.899	×	16.415	16.529	16.598	×	26.698	26.645	26.768	×
	FCI	10.463	10.762	10.901	10.945	16.418	16.534	16.603	16.640	26.683	26.659	26.779	26.833(1)
	G0W0	10.675	10.837	11.201	11.362	16.527	16.578	16.867	17.007	28.241	28.117	28.427	28.463
	qsGW	10.520	10.870	11.094	11.176	16.468	16.655	16.805	16.878	28.029	27.962	27.980	28.151
	G0F(2)	9.841	9.994	10.244	10.345	15.817	15.814	16.002	16.088	27.589	27.638	27.729	×
	G0T0	10.399	10.497	10.716	×	16.217	16.170	16.330	×	28.860	28.738	28.860	×
	Mol.				Methane (CH4)				Hydrogen fluoride (HF)
	State/Conf.		1 2 T2/(1t2) -1			1 2 A1/(2a1) -1			1 2 Π/(1π) -1
	Exp.		14.5 301			23.0 301			16.19 302
	CC2	13.787	13.888	14.028	14.079	23.289	23.227	23.311	23.352	14.431	14.559	14.725	14.813
	CCSD	14.102	14.258	14.387	14.428	23.238	23.247	23.383	23.426	15.688	15.837	16.021	16.117
	CC3	14.060	14.270	14.365	14.395	23.034	23.035	23.138	23.173	15.917	16.036	16.126	16.194
	CCSDT	14.068	14.269	14.365	14.395	23.040	23.035	23.135	23.171	15.885	15.992	16.077	16.145
	CC4	14.072	14.284	14.376	×	23.039	23.050	23.142	×	15.947	16.068	16.161	16.227
	CCSDTQ	14.073	14.284	14.376	×	23.042	23.052	23.143	×	15.935	16.051	16.140	16.205
	FCI	14.073	14.285	14.377	14.407	23.043	23.056	23.146	23.148(10)	15.941	16.059	16.149	16.214
	G0W0	14.338	14.466	14.753	14.872	23.647	23.626	23.875	23.988	15.679	15.868	16.237	16.453
	qsGW	14.142	14.446	14.621	14.686	23.248	23.426	23.550	23.605	16.001	16.144	16.349	16.469
	G0F(2)	13.861	13.913	14.102	14.176	23.377	23.257	23.385	23.447	14.280	14.437	14.685	14.815
	G0T0	14.117	14.117	14.275	×	24.107	24.051	24.163	×	15.334	15.466	15.721	×
	Mol.	Hydrogen fluoride (HF)					Neon (Ne)
	State/Conf.		1 2 Σ + /(3σ) -1			1 2 P/(2p) -1			1 2 S/(2s) -1
	Exp.		19.90 302			21.57 303			48.46 303
	CC2	18.740	18.814	18.908	18.982	19.874	20.017	20.144	20.236	47.483	47.265	47.187	47.207
	CCSD	19.777	19.861	19.946	20.021	21.030	21.168	21.326	21.432	48.735	48.363	48.426	48.494
	CC3	19.980	20.040	20.050	20.100	21.353	21.417	21.449	21.522	48.652	48.263	48.145	48.168
	CCSDT	19.933	19.989	19.995	20.045	21.304	21.367	21.398	21.473	48.725	48.330	48.229	48.270
	CC4	19.986	20.051	20.065	20.114	21.375	21.434	21.473	21.546	48.829	48.424	48.316	48.349
	CCSDTQ	19.974	20.036	20.046	20.094	21.362	21.421	21.455	21.527	48.811	48.406	48.293	48.326
	FCI	19.979	20.043	20.054	20.102	21.365	21.426	21.461	21.533	48.822	48.417	48.306	48.340
	G0W0	19.662	19.812	20.074	20.259	20.859	21.104	21.432	21.655	47.851	47.785	47.950	48.085
	qsGW	19.984	20.084	20.203	20.304	21.361	21.435	21.592	21.729	47.844	47.652	47.560	47.566
	G0F(2)	18.644	18.744	18.899	19.007	19.642	19.851	20.066	20.202	47.246	47.082	47.055	47.096
	G0T0	19.312	19.402	19.551	×	20.671	20.847	21.085	×	48.966	48.851	48.886	×
	Mol.

TABLE II .

 II Satellite transition energies of ammonia and water computed with Green's function methods in the aug-cc-pVDZ basis set. The FCI values are reported for comparison purposes.

	Molecule State Method Diag. element Eigenvalue FCI
			G0F(2)	24.324	24.328
	NH3	2 2 A1 G0W0	24.408	24.410	23.829
			G0T0	40.441	40.444
			G0F(2)	24.977	24.977
	NH3	2 2 E	G0W0	24.997	24.997	25.655
			G0T0	41.094	41.094
			G0F(2)	30.759	30.759
	H2O	2 2 B1 G0W0	30.846	30.846	27.065
			G0T0	×	×
			G0F(2)	28.683	28.683
	H2O	2 2 A1 G0W0	28.770	28.770	27.131
			G0T0	×	×
			G0F(2)	30.759	30.781
	H2O	3 2 B1 G0W0	30.863	30.867	28.754
			G0T0	×	×
	B. Satellite in Green's functions methods

TABLE III .

 III Valence ionizations and satellite transition energies (in eV) of the 14-electron series for various methods and basis sets. AVXZ stands for aug-cc-pVXZ (where X = D, T, and Q). Selected experimental values are also reported.

				Basis			Basis				Basis
	Methods 6-31+G * AVDZ	AVTZ	AVQZ	6-31+G *	AVDZ	AVTZ	AVQZ	6-31+G * AVDZ AVTZ	AVQZ
	Mol.						Boron fluoride (BF)
	State/Conf.		1 2 Σ + /(5σ) -1			2 2 Π/(1π) -1			2 2 Σ + /(4σ) -1
	Exp.		11.06 323
	CC2	10.751	10.824	10.944	10.987	17.136	17.274	17.385	17.467	19.767	19.957 19.962 20.041
	CCSD	11.080	11.154	11.250	11.279	17.920	18.028	18.172	18.246	21.017	21.208 21.253 21.342
	CC3	10.914	11.004	11.100	11.127	18.606	18.722	18.743	18.794	20.745	20.910 20.918 20.984
	CCSDT	10.969	11.057	11.157	11.185	18.474	18.584	18.622	18.677	20.708	20.866 20.875 20.946
	CC4	10.965	11.053	11.148	11.175	18.475	18.593	18.626	18.679	20.770	20.920 20.917 20.981
	CCSDTQ	10.967	11.054	11.150	×	18.453	18.569	18.599	×	20.734	20.882 20.874	×
	FCI	10.966	11.054	11.149	11.175	18.466	18.581	18.612	18.664	20.765	20.913 20.906 20.970(1)
	G0W0	11.053	11.117	11.325	11.420	18.237	18.456	18.743	18.949	21.142	21.402 21.567 21.778
	qsGW	10.862	10.989	11.167	11.240	18.513	18.662	18.784	18.899	21.389	21.583 21.585 21.701
	G0F(2)	10.859	10.915	11.052	11.114	16.958	17.132	17.328	17.454	19.654	19.878 19.955 20.079
	G0T0	10.821	10.856	10.955	×	17.947	18.105	18.304	×	20.739	20.955 21.041	×
	Mol.						Carbon monoxide (CO)
	State/Conf.		1 2 Σ + /(5σ) -1			1 2 Π/(1π) -1			2 2 Σ + /(4σ) -1
	Exp.		14.01 324			17.0 324			19.7 324
	CC2	13.550	13.584	13.748	13.809	16.289	16.349	16.505	16.581	18.175	18.316 18.400 18.464
	CCSD	13.948	13.998	14.190	14.246	16.793	16.865	17.024	17.095	19.501	19.657 19.790 19.867
	CC3	13.614	13.697	13.863	13.912	16.826	16.902	17.018	17.075	19.512	19.664 19.744 19.807
	CCSDT	13.693	13.770	13.952	14.005	16.762	16.838	16.960	17.016	19.347	19.498 19.583 19.647
	CC4	13.678	13.760	13.933	13.984	16.751	16.835	16.955	17.009	19.410	19.566 19.653 19.715
	CCSDTQ	13.679	13.761	13.935	×	16.755	16.837	16.958	×	19.376	19.532 19.616	×
	FCI	13.670	13.752	13.925	13.975	16.762	16.845	16.966	17.017(2)	19.393	19.550 19.637 19.699(1)
	G0W0	14.461	14.467	14.777	14.915	16.677	16.762	17.083	17.264	19.869	20.045 20.300 20.485
	qsGW	13.980	14.080	14.318	14.416	16.836	16.932	17.124	17.231	19.899	20.071 20.191 20.298
	G0F(2)	13.856	13.857	14.067	14.154	16.134	16.204	16.422	16.534	18.165	18.317 18.460 18.564
	G0T0	14.163	14.143	14.324	×	16.422	16.470	16.666	×	19.333	19.481 19.613	×
	Mol.						Dinitrogen (N2)
	State/Conf. Exp.		1 2 Σ + g /(3σg) -1 15.580 325			1 2 Πu/(1πu) -1 16.926 325			1 2 Σ + u /(2σu) -1 18.751 325
	CC2	14.613	14.649	14.814	14.877	16.932	16.943	17.104	17.178	17.803	17.862 17.991	18.037
	CCSD	15.382	15.424	15.641	15.709	17.065	17.087	17.228	17.287	18.654	18.721 18.931	18.991
	CC3	15.282	15.349	15.519	15.574	16.669	16.719	16.837	16.885	18.598	18.680 18.849	18.899
	CCSDT	15.270	15.333	15.517	15.574	16.765	16.812	16.950	17.001	18.502	18.585 18.763	18.816
	CC4	15.220	15.293	15.471	15.526	16.770	16.821	16.940	16.987	18.403	18.493 18.669	18.720
	CCSDTQ	15.237	15.309	15.487	×	16.764	16.815	16.936	×	18.429	18.519 18.696	×
	FCI	15.235	15.308	15.486	15.541	16.759	16.811	16.933	16.981	18.427	18.516 18.692	18.742
	G0W0	15.959	15.984	16.350	16.519	16.781	16.790	17.093	17.259	19.515	19.558 19.862	20.000
	qsGW	15.575	15.663	15.914	16.020	16.640	16.706	16.903	17.006	19.125	19.221 19.425	19.513
	G0F(2)	14.824	14.845	15.080	15.181	16.956	16.952	17.158	17.261	17.974	18.020 18.201	18.274
	G0T0	15.494	15.502	15.722	×	16.673	16.653	16.820	×	18.993	19.021 19.190	×
	Mol.		Boron Fluoride (BF)
	State/Conf.		1 2 Π/(5σ) -2 (2π) 1
	Exp.
	CC3	17.494	17.541	17.607	17.637
	CCSDT	17.410	17.462	17.532	17.567
	CC4	17.293	17.345	17.393	17.419
	CCSDTQ	17.303	17.355	17.405	×
	FCI	17.297	17.346	17.392	17.417
	Mol.

TABLE IV .

 IV Valence ionizations and satellite transition energies (in eV) of the 12-electron series for various methods and basis sets. AVXZ stands for aug-cc-pVXZ (where X = D, T, and Q). Selected experimental values are also reported.

			Basis			Basis				Basis
	Methods 6-31+G * AVDZ AVTZ	AVQZ	6-31+G * AVDZ AVTZ	AVQZ	6-31+G * AVDZ	AVTZ	AVQZ
	Mol.			Lithium fluoride (LiF)				Beryllium oxide (BeO)
	State/Conf.		1 2 Π/(1π) -1			1 2 Σ + /(4σ) -1			1 2 Π/(1π) -1
	Exp.		11.50,11.67 334			11.94 334
	CC2	9.481	9.588	9.804	9.895	9.801	9.923 10.109 10.209	9.615	9.712	9.818	9.894
	CCSD	11.078	11.193 11.398	11.493	11.566	11.701 11.874 11.979	9.708	9.808	9.875	9.941
	CC3	11.142	11.222 11.375	11.449	11.588	11.682 11.802 11.883	9.976	10.096	10.243	10.320
	CCSDT	11.165	11.247 11.379	11.452	11.641	11.738 11.834 11.914	9.750	9.849	9.864	9.916
	CC4	11.270	11.351 11.496	11.567	11.764	11.863 11.971 12.049	9.749	9.850	9.867	9.922
	CCSDTQ	11.234	11.315 11.453	×	11.719	11.816 11.917	×	9.831	9.930	9.939	×
	FCI	11.246	11.328 11.468	11.538	11.735	11.833 11.933 12.018(2)	9.863	9.962	9.972	10.018
	G0W0	10.797	10.979 11.384	11.594	11.339	11.549 11.915 12.139	9.356	9.489	9.727	9.907
	qsGW	11.249	11.330 11.575	11.699	11.809	11.926 12.119 12.255	9.991	10.079	10.168	10.251
	G0F(2)	9.294	9.445	9.729	9.857	9.644	9.812 10.063 10.200	7.957	8.072	8.195	8.302
	G0T0	10.512	10.644 10.923	×	10.968	11.120 11.366	×	8.885	8.975	9.108	×
	Mol.		Beryllium oxide (BeO)				Boron nitride (BN)
	State/Conf.		1 2 Σ + /(4σ) -1			1 2 Π/(1π) -1			1 2 Σ + /(4σ) -1
	Exp.
	CC2	10.523	10.620 10.667 10.735	10.734	10.792 10.927	10.991	12.842	12.870 12.979	13.018
	CCSD	10.861	10.987 11.006 11.082	11.776	11.850 11.971	12.018	13.571	13.624 13.697	13.720
	CC3	11.128	11.269 11.370 11.454	11.825	11.941 12.057	12.101	13.641	13.718 13.806	13.828
	CCSDT	10.830	10.962 10.916 10.975	11.778	11.871 11.980	12.019	13.642	13.716 13.790	13.808
	CC4	10.825	10.959 10.915 10.977	11.681	11.797 11.902	11.940	13.534	13.626 13.700	13.718
	CCSDTQ	10.923	11.056 11.007	×	11.754	11.860 11.966	×	13.580	13.667 13.745	×
	FCI	10.970	11.103 11.056 11.115(2)	11.767	11.875 11.980	12.019	13.571	13.660 13.729(11) 13.710(70)
	G0W0	10.628	10.798 10.996 11.200	11.423	11.447 11.752	11.907	13.154	13.171 13.447	13.590
	qsGW	11.083	11.199 11.241 11.341	11.597	11.711 11.898	11.987	13.376	13.490 13.621	13.693
	G0F(2)	8.499	8.648	8.720	8.837	10.817	10.857 11.031	11.122	12.207	12.195 12.382	12.454
	G0T0	9.930	10.064 10.155	×	10.988	10.996 11.159	×	12.499	12.512 12.657	×
	Mol.		Carbon dimer (C2)
	State/Conf.		1 2 Πu/(2πu) -1
	Exp.
	CC2	12.742	12.779 12.951	13.023
	CCSD	12.770	12.830 12.978	13.030
	CC3	11.930	12.058 12.177	12.215
	CCSDT	12.289	12.391 12.540	12.585
	CC4	12.231	12.347 12.472	12.511
	CCSDTQ	12.225	12.340 12.471	×
	FCI	12.205	12.323 12.463	12.497
	G0W0	12.621	12.613 12.928	13.082
	qsGW	12.202	12.344 12.561	12.656
	G0F(2)	12.882	12.870 13.078	13.175
	G0T0	12.482	12.454 12.625	×
	Mol.						Carbon dimer (C2)
	State/Conf.		1 2 ∆g/(2πu) -2 (3σg) 1		1 2 Σ -g /(2πu) -2 (3σg) 1			1 2 Σ + g /(2πu) -2 (3σg) 1
	Exp.
	CC3	14.644	14.713 14.815	14.850	14.846	14.957 15.087	15.123	15.360	15.353	15.435	15.460
	CCSDT	14.494	14.568 14.680	14.729	14.721	14.833 14.998	15.051	15.086	15.072	15.194	15.246
	CC4	13.920	14.007 14.052	14.076	14.196	14.308 14.359	14.388	14.304	14.331	14.413	14.439
	CCSDTQ	13.879	13.969 14.041	×	14.182	14.200 14.310	×	14.209	14.316	14.423	×
	FCI	13.798	13.889 13.944	13.963	14.084	14.099 14.167	14.193	14.108	14.244	14.337	14.359(1)
	Mol.

TABLE V .

 V Valence ionizations and satellite transition energies (in eV) of the third-row molecules for various methods and basis sets. AVXZ stands for aug-cc-pVXZ (where X = D, T, and Q). Selected experimental values are also reported.

			Basis				Basis			Basis
	Methods 6-31+G * AVDZ AVTZ	AVQZ	6-31+G * AVDZ	AVTZ	AVQZ	6-31+G *	AVDZ	AVTZ	AVQZ
	Mol.					Carbon sulfide (CS)
	State/Conf.		1 2 Σ + /(7σ) -1			1 2 Π/(2π) -1			2 Σ + /(6σ) -1
	Exp.
	CC2	10.627	10.745 10.847	10.900	12.791	12.897 13.014	13.083	16.698	16.817	16.945	17.005
	CCSD	11.245	11.402 11.553	11.609	12.726	12.883 13.000	13.059	16.854	16.997	17.220	17.288
	CC3	10.949	11.186 11.325	11.377	12.553	12.766 12.880	12.937	18.201	18.290	18.389	18.422
	CCSDT	10.966	11.190 11.346	11.404	12.596	12.799 12.918	12.974	17.915	18.023	18.134	18.179
	CC4	10.914	11.161 11.310	11.368	12.542	12.764 12.878	12.934	17.764	17.881	17.959	17.994
	CCSDTQ	10.920	11.166 11.316	×	12.548	12.768 12.885	×	17.749	17.865	17.947	×
	FCI	10.899	11.151 11.300	11.355	12.545	12.768 12.882(1)	12.936	17.723	17.844	17.920	17.958(2)
	G0W0	12.092	12.119 12.378	12.523	12.602	12.679 12.907	13.063	17.666	17.713	17.976	18.119
	qsGW	11.369	11.589 11.775	11.880	12.498	12.705 12.852	12.958	17.323	17.505	17.688	17.788
	G0F(2)	11.109	11.152 11.292	11.371	12.696	12.774 12.923	13.018	16.704	16.779	16.942	17.026
	G0T0	11.595	11.594 11.713	×	12.479	12.505 12.609	×	17.422	17.455	17.591	×
	Mol.

TABLE VI .

 VI These two research groups reported quite different values of 12.32 eV 361 and 11.4 eV. 362 The FCI values, presented in Table VIII, is closer to the first one, corroborating the findings of Tian et al., who computed similar values using propagator-based methods as well as CCSD(T). 363 Finally, two FCI satellite transition energies of BH 3 are reported in Table VIII.FIG. 2. Mean signed error (MSE) [upper panel] and mean absolute error (MSE) [lower panel] with respect to FCI of the various methods considered in this work. These errors are computed for the 58 IPs of this set in the aug-cc-pVTZ basis set. Mean absolute error (MAE), mean signed error (MSE), root mean square error (RMSE), standard deviation error (SDE), minimum and maximum errors (in eV) with respect to FCI of the various methods considered in this work. These descriptors are computed for the 58 IPs of this set in the aug-cc-pVTZ basis set. The ∆CCSD(T) statistical descriptors correspond only to the 23 principal IPs.

	0.4
	0.2
	0.0
	-0.2
	-0.4
	-0.6
	0.8
	0.6
	0.4
	0.2
	0.0
	Methods	MAE MSE RMSE SDE	Min	Max
	CC2	0.769 -0.594 0.940 0.735 -2.207 1.565
	CCSD	0.175	0.097 0.280 0.265 -0.700 1.075
	CC3	0.070	0.001 0.125 0.126 -0.395 0.469
	CCSDT	0.041 -0.010 0.057 0.057 -0.140 0.214
	CC4	0.015	0.005 0.027 0.027 -0.078 0.118
	CCSDTQ	0.010 -0.005 0.013 0.012 -0.049 0.027
	G0W0	0.470	0.399 0.664 0.535 -0.504 2.053
	qsGW	0.391	0.268 0.559 0.494 -1.348 1.747
	G0F(2)	0.807 -0.550 0.987 0.827 -2.336 1.623
	G0T0	0.485	0.007 0.752 0.758 -1.169 2.959
	∆CCSD(T) 0.021	0.016 0.037 0.035 -0.020 0.120
	G. Global statistics

TABLE VII .

 VII Mean absolute error (MAE), mean signed error (MSE), root mean square error (RMSE), standard deviation error (SDE), minimum and maximum errors (in eV) with respect to FCI of the various methods considered in this work. These descriptors are computed for the 36 satellites of this set in the aug-cc-pVTZ basis set. The satellites of LiF and BeO have been excluded (see main text).

	Methods	MAE MSE RMSE SDE	Min	Max
	CC3	0.749 0.232	0.830	0.809 -1.805 1.098
	CCSDT	0.528 0.528	0.582	0.249	0.104 1.195
	CC4	0.087 0.010	0.118	0.119 -0.264 0.333
	CCSDTQ 0.040 0.036	0.055	0.043 -0.026 0.143
	MAEs align very well with those computed for double
	excitations, as reported in Ref. 10.

TABLE VIII .

 VIII Valence ionizations and satellite transition energies (in eV) of the 18-electron series for various methods and basis sets. AVXZ stands for aug-cc-pVXZ (where X = D, T, and Q). Selected experimental values are also reported.

			Basis				Basis			Basis
	Methods	6-31+G *	AVDZ	AVTZ	AVQZ	6-31+G *	AVDZ	AVTZ	AVQZ	6-31+G *	AVDZ	AVTZ	AVQZ
	Mol.						(H2S)
	State/Conf.		1 2 B1/(2b1) -1			1 2 A1/(5a1) -1			1 2 B2/(2b2) -1
	Exp.		10.5 350			13.1 350			15.6 350
	CC2	9.699	9.953	10.156	10.228	12.833	13.004	13.155	13.220	15.393	15.368	15.464	15.521
	CCSD	9.908	10.208	10.421	10.490	13.057	13.280	13.440	13.497	15.540	15.591	15.684	15.734
	CC3	9.846	10.189	10.388	10.454	12.985	13.259	13.407	13.459	15.475	15.567	15.635	15.680
	CCSDT	9.853	10.190	10.390	10.455	12.985	13.253	13.399	13.450	15.460	15.552	15.619	15.664
	CC4	9.855	10.199	10.394	10.458	12.990	13.265	13.407	13.456	15.458	15.560	15.624	15.668
	CCSDTQ	9.855	10.199	10.393	×	12.990	13.265	13.406	×	15.457	15.558	15.622	×
	FCI	9.855	10.199	10.393	10.456	12.992	13.268	13.411	13.460	15.459	15.562	15.627	15.672
	G0W0	10.034	10.172	10.500	10.660	13.212	13.339	13.614	13.758	15.684	15.698	15.928	16.058
	qsGW	9.890	10.201	10.445	10.549	13.048	13.337	13.530	13.619	15.524	15.651	15.785	15.868
	G0F(2)	9.747	9.940	10.180	10.282	12.887	13.005	13.188	13.278	15.420	15.364	15.494	15.569
	G0T0	9.927	10.011	10.188	×	13.068	13.112	13.250	×	15.588	15.477	15.582	×
	Mol.						(PH3)
	State/Conf.		1 2 A1/(5a1) -1			1 2 E/(2eg) -1			2 2 A1/(4a1) -1
	Exp.		10.85 201			16.4 201			27.6 201
	CC2	10.030	10.219	10.392	10.444	13.463	13.493	13.638	13.689	21.308	21.046	21.079	21.106
	CCSD	10.198	10.448	10.623	10.662	13.521	13.637	13.784	13.825	20.144	20.221	20.361	20.396
	CC3	10.130	10.431	10.599	10.634	13.464	13.615	13.744	13.779	19.528	19.643	19.739	19.763
	CCSDT	10.129	10.431	10.599	10.634	13.454	13.606	13.734	13.769	19.317	19.487	19.583	19.610
	CC4	10.127	10.436	10.600	×	13.452	13.611	13.737	×	19.263	19.446	19.520	×
	CCSDTQ	10.127	10.437	10.600	×	13.451	13.611	13.737	×	19.259	19.442	19.511	×
	FCI	10.126	10.436	10.596	10.628	13.456	13.615	13.745	13.784	19.261	19.445	19.514	19.537
	G0W0	10.365	10.497	10.787	10.911	13.717	13.831	14.099	14.214	21.505	21.378	21.567	21.673
	qsGW	10.190	10.506	10.725	10.802	13.545	13.761	13.954	14.027	21.128	21.155	21.261	21.322
	G0F(2)	10.095	10.223	10.428	10.508	13.495	13.503	13.675	13.741	21.354	21.067	21.137	21.184
	G0T0	10.186	10.236	10.387	×	13.665	13.619	13.759	×	22.142	21.976	22.040	×
	Mol.					(SiH4)					(HCl)
	State/Conf.		1 2 T2/(2t2) -1			1 2 A1/(3a1) -1			1 2 Π/(1π) -1
	Exp.		12.8 372			18.2 372			12.745/12.830 354
	CC2	12.515	12.653	12.802	12.848	18.704	18.759	18.823	18.846	11.990	12.230	12.401	12.491
	CCSD	12.477	12.701	12.844	12.877	18.266	18.379	18.483	18.505	12.250	12.538	12.712	12.812
	CC3	12.417	12.681	12.806	12.832	18.109	18.236	18.316	18.331	12.186	12.529	12.672	12.771
	CCSDT	12.407	12.673	12.794	12.820	18.046	18.172	18.247	18.262	12.190	12.524	12.667	12.764
	CC4	12.404	12.676	12.795	×	18.033	18.171	18.241	×	12.199	12.539	12.678	12.773
	CCSDTQ	12.404	12.677	12.795	×	18.030	18.170	18.238	×	12.199	12.538	12.676	×
	FCI	12.403	12.676	12.793	12.818	18.031	18.173	18.240	18.258(2)	12.199	12.539	12.676	12.770
	G0W0	12.716	12.960	13.214	13.312	18.701	18.824	19.007	19.097	12.343	12.488	12.778	12.967
	qsGW	12.541	12.906	13.107	13.170	18.422	18.700	18.828	18.879	12.224	12.539	12.723	12.855
	G0F(2)	12.550	12.662	12.830	12.886	18.742	18.768	18.855	18.899	12.022	12.200	12.415	12.535
	G0T0	12.712	12.762	12.891	×	19.119	19.156	19.204	×	12.273	12.347	12.507	×
	Mol.		(HCl)						Argon (Ar)
	State/Conf.		1 2 Σ + /(5σ) -1			1 2 P/(3p) -1			1 2 S/(3s) -1
	Exp.		16.270 354			15.8 303			29.24 303
	CC2	16.245	16.329	16.438	16.512	15.055	15.236	15.379	15.486	31.123	30.459	30.439	30.437
	CCSD	16.467	16.599	16.708	16.782	15.308	15.534	15.672	15.802	30.353	29.838	29.881	29.907
	CC3	16.395	16.577	16.658	16.730	15.230	15.525	15.614	15.745	30.255	29.437	29.238	29.227
	CCSDT	16.388	16.564	16.644	16.715	15.228	15.514	15.606	15.735	30.261	29.445	29.214	29.216
	CC4	16.394	16.577	16.653	16.723	15.238	15.531	15.616	15.743	30.263	29.442	29.186	29.179
	CCSDTQ	16.393	16.576	16.651	×	15.237	15.529	15.614	15.740	30.261	29.442	29.182	29.175
	FCI	16.396	16.579	16.657	16.728	15.237	15.529	15.613	15.739	30.265	29.449	29.188	29.182
	G0W0	16.574	16.635	16.872	17.031	15.333	15.458	15.711	15.926	31.226	30.759	31.089	31.224
	qsGW	16.417	16.619	16.752	16.861	15.232	15.507	15.633	15.794	30.858	30.576	30.681	30.751
	G0F(2)	16.286	16.325	16.469	16.564	15.072	15.196	15.387	15.523	31.130	30.399	30.426	30.427
	G0T0	16.461	16.436	16.548	×	15.326	15.368	15.511	×	32.255	32.007	32.147	×
	Mol.						(H2S)
	State/Conf.		2 2 A1/(2b1) -2 (6a1) 1			2 2 B2/(2b1) -2 (3b2) 1		2 2 B1/(5a1) -1 (2b1) -1 (6a1) 1
	Exp.		19.63 350
	CC3	19.182	19.350	19.377	×	19.973	20.137	20.317	20.355	20.456	20.620	20.702	×
	CCSDT	18.761	19.018	19.043	×	19.675	19.961	20.136	20.190	19.948	20.185	20.269	×
	CC4	18.607	18.848	18.801	×	19.485	19.777	19.892	×	19.775	19.995	20.019	×
	CCSDTQ	18.582	18.827	18.772	×	19.467	19.765	19.868	×	19.744	19.969	19.986	×
	FCI	18.575	18.819	18.755	18.745	19.462	19.759	19.853	19.889	19.741	19.965	19.974	×
	Mol.		PH3						HCl
	State/Conf.		2 2 E				2 2 Σ + /(2π) -1 (6σ) 1			1 2 ∆/(2π) -1 (6σ) 1
	Exp.
	CC3	19.513	19.613	19.630	19.606	22.798	22.997	23.197	23.262	23.556	23.732	23.795	23.829
	CCSDT	19.072	19.217	19.255	19.245	22.277	22.676	22.885	22.982	23.188	23.532	23.532	23.590
	CC4	18.940	19.084	19.080	×	21.932	22.328	22.439	×	22.918	23.273	23.185	×
	CCSDTQ	18.907	19.055	19.046	×	21.889	22.303	22.400	×	22.885	23.257	23.155	×
	FCI	18.897	19.047	19.025	18.997	21.878	22.293	22.377	22.463	22.881	23.254	23.142	23.185

TABLE IX .

 IX Valence ionizations and satellite transition energies (in eV) of the remaining molecules for various methods and basis sets. AVXZ stands for aug-cc-pVXZ (where X = D, T, and Q). Selected experimental values are also reported.

			Basis				Basis			Basis
	Methods 6-31+G *	AVDZ	AVTZ	AVQZ	6-31+G * AVDZ	AVTZ	AVQZ	6-31+G *	AVDZ	AVTZ	AVQZ
	Mol.						Carbon dioxide (CO2)
	State/Conf. Exp.		1 2 Πg/(1πg) -1 13.8 205			1 2 Πu/(1πu) -1 17.6 205			1 2 Σ + u /(3σu) -1 18.1 205
	CC2	12.747	12.812	13.015	13.103	16.508	16.630 16.751	16.829	16.866	16.889	17.034	17.107
	CCSD	13.490	13.561	13.787	13.879	17.767	17.809 17.996	18.080	17.885	18.006	18.171	18.259
	CC3	13.456	13.528	13.696	13.773	17.275	17.335 17.463	17.531	17.708	17.829	17.928	17.999
	CCSDT	13.474	13.539	13.716	13.794	17.351	17.406 17.553	17.628	17.708	17.821	17.930	18.005
	CC4	13.488	13.562	13.733	×	17.339	17.396 17.521	×	17.783	17.904	18.009	×
	CCSDTQ	13.491	13.563	13.734	×	17.341	17.396 17.522	×	17.750	17.869	17.972	×
	FCI	13.496	13.567	13.733	13.823	17.337	17.391 17.513(7) 17.618(6) 17.766(1) 17.889(1) 17.996(2)	×
	G0W0	13.743	13.837	14.221	14.425	18.184	18.209 18.519	18.697	18.335	18.474	18.791	18.994
	qsGW	13.712	13.806	14.054	14.182	17.950	18.008 18.200	18.315	18.120	18.264	18.442	18.566
	G0F(2)	12.783	12.854	13.129	13.254	16.993	17.012 17.224	17.332	16.714	16.840	17.035	17.153
	G0T0	13.284	13.334	13.586	×	17.700	17.696 17.890	×	17.693	17.816	17.995	×
	Mol.		Carbon dioxide (CO2)
	State/Conf. Exp.		1 2 Σ + g /(4σg) -1 19.4 205
	CC2 17.746	17.864	17.980	18.048
	CCSD 19.237	19.351	19.523	19.605
	CC3 18.959	19.073	19.184	19.250
	CCSDT 18.968	19.075	19.198	19.268
	CC4 19.047	19.158	19.274	×
	CCSDTQ 19.008	19.119	19.232	×
	FCI 19.024(1) 19.137(2) 19.254(3)	×
	G0W0 19.730	19.856	20.150	20.334
	qsGW 19.474	19.604	19.776	19.888
	G0F(2) 17.949	18.069	18.245	18.348
	G0T0 19.083	19.193	19.353	×
	Mol.					Formaldehyde (CH2O)
	State/Conf.		1 2 B2/(2b2) -1			1 2 B1/(1b1) -1			1 2 A1/(5a1) -1
	Exp.		10.9 177			14.5 177			16.1 177
	CC2	9.444	9.553	9.753	9.828	13.854	13.903 14.053	14.127	14.596	14.705	14.820	14.893
	CCSD	10.486	10.625	10.848	10.924	14.410	14.480 14.606	14.667	15.833	15.964	16.105	16.181
	CC3	10.555	10.710	10.873	10.932	14.377	14.482 14.562	14.609	15.832	15.985	16.063	16.119
	CCSDT	10.533	10.680	10.840	10.898	14.381	14.478 14.562	14.609	15.811	15.953	16.030	16.086
	CC4	10.578	10.736	10.897	×	14.395	14.504 14.578	×	15.879	16.039	16.114	×
	CCSDTQ	10.574	10.729	10.887	×	14.391	14.498 14.572	×	15.864	16.018	16.090	×
	FCI	10.582	10.739	10.899	10.954	14.395	14.502 14.578(1) 14.618(9)	15.876	16.032	16.106(1) 16.161(4)
	G0W0	10.835	10.996	11.380	11.556	14.232	14.283 14.587	14.759	16.170	16.292	16.602	16.794
	qsGW	10.780	10.990	11.241	11.342	14.395	14.543 14.693	14.780	16.118	16.313	16.459	16.557
	G0F(2)	9.605	9.712	9.981	10.088	13.740	13.772 13.985	14.093	14.747	14.856	15.040	15.153
	G0T0	10.345	10.407	10.655	×	13.879	13.896 14.082	×	15.625	15.716	15.888	×
	Mol.		Formaldehyde (CH2O)					Borane (BH3)
	State/Conf. Exp.		2 2 B2/(1b2) -1 17.0 177			1 2 E ′ /(1e ′ ) -1			1 2 A ′ 1 /(2a ′ 1 ) -1
	CC2	16.704	16.738	16.866	16.921	12.942	13.081	13.231	13.276	18.210	18.337	18.427	18.465
	CCSD	17.256	17.338	17.494	17.553	12.995	13.204	13.342	13.375	18.100	18.300	18.398	18.428
	CC3	16.898	16.992	17.102	17.148	12.941	13.191	13.309	13.334	18.018	18.249	18.326	18.349
	CCSDT	16.938	17.036	17.156	17.207	12.938	13.189	13.304	13.330	18.002	18.232	18.307	18.331
	CC4	16.905	17.005	17.107	×	12.938	13.193	13.306	13.331	18.001	18.236	18.308	18.331
	CCSDTQ	16.906	17.005	17.108	×	12.938	13.193	13.307	×	18.001	18.237	18.309	×
	FCI	16.901	17.002	17.107	17.156(5)	12.938	13.194	13.307	13.332	18.001	18.238	18.310	18.332
	G0W0	17.651	17.714	18.015	18.162	13.202	13.395	13.678	13.780	18.320	18.470	18.685	18.780
	qsGW	17.463	17.635	17.821	17.909	13.047	13.371	13.584	13.651	18.111	18.410	18.544	18.601
	G0F(2)	16.460	16.463	16.656	16.742	12.989	13.093	13.278	13.341	18.255	18.342	18.459	18.515
	G0T0	17.317	17.298	17.473	×	13.130	13.179	13.327	×	18.536	18.598	18.683	×
	Mol.			Formaldehyde (CH2O)
	State/Conf.		2 2 B1/(2b2) -1 (2b1) 1		3 2 B2/(1b1) -1 (2b2) -1 (2b1) 1
	Exp.
	CC3	16.564	16.632	16.800	16.847	18.826	18.991 19.154	19.208
	CCSDT	16.684	16.786	17.004	17.083	18.761	18.956 19.153	19.233
	CC4	16.348	16.407	16.495	×	18.503	18.663 18.751	×
	CCSDTQ	16.363	16.425	16.522	×	18.515	18.679 18.772	×
	FCI	16.345	16.409	16.498(1)	×	18.512	18.678 18.763(1)	×
	Mol.				Borane (BH3)
	State/Conf.		1 2 B ′ 1 /(1e ′ ) -2 (1b1) 1			2 2 E ′ /(1e ′ ) -2 (1b1) 1
	Exp.
	CC3	19.485	19.803	19.889	19.894	20.026	20.212	20.324	20.346
	CCSDT	18.896	19.228	19.331	19.355	19.482	19.669	19.789	19.824
	CC4	18.790	19.130	19.209	19.226	19.403	19.602	19.704	19.732
	CCSDTQ	18.756	19.100	19.178	×	19.379	19.584	19.684	×
	FCI	18.754	19.099	19.176	19.194	19.377	19.583	19.685	19.714

ACKNOWLEDGMENTS

This project has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (Grant agreement No. 863481). This work used the HPC resources from CALMIP (Toulouse) under allocations 2023-18005 and 2024-18005. The authors thank Abdallah Ammar, Fábris Kossoski, Yann Damour, Alexander Sokolov, Devin Matthews, Anthony Scemama, and Denis Jacquemin for helpful comments and/or insightful discussions.

SUPPORTING INFORMATION

The Supporting Information includes the geometry of the 23 molecules considered in this study as well as a json