A mountaineering strategy to excited states:
highly-accurate oscillator strengths and dipole moments
of small molecules

Amara Chrayteh, Aymeric Blondel, Pierre-Francois Loos, Denis Jacquemin

► To cite this version:

Amara Chrayteh, Aymeric Blondel, Pierre-Francois Loos, Denis Jacquemin. A mountaineering strategy to excited states: highly-accurate oscillator strengths and dipole moments of small molecules. Journal of Chemical Theory and Computation, 2021, 17 (1), pp.416-438. 10.1021/acs.jctc.0c01111. hal-03018539

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# A Mountaineering Strategy to Excited States: Highly-Accurate Oscillator Strengths and Dipole Moments of Small Molecules

Amara Chrayteh,† Aymeric Blondel,† Pierre-François Loos,‡ and Denis Jacquemin*,†

†Université de Nantes, CNRS, CEISAM UMR 6230, F-44000 Nantes, France

‡Laboratoire de Chimie et Physique Quantiques, Université de Toulouse, CNRS, UPS, France

E-mail: Denis.Jacquemin@univ-nantes.fr

## Abstract

This work presents a series of highly-accurate excited-state properties obtained using high-order coupled-cluster (CC) calculations performed with a series of diffuse containing basis sets, as well as extensive comparisons with experimental values. Indeed, we have computed both the main ground-to-excited transition property, the oscillator strength, as well as the ground- and excited-state dipole moments, considering thirteen small molecules (hydridoboron, hydrogen chloride, water, hydrogen sulfide, boron fluoride, carbon monoxide, dinitrogen, ethylene, formaldehyde, thioformaldehyde, nitroxyl, fluorocarbene, and silylidene). We systematically include corrections up to the quintuple (CCSDTQP) in the CC expansion and extrapolate to the complete basis set limit. When comparisons with experimental measurements are possible, that is, when a number of consistent experimental data can be found, theory typically provides values falling within the experimental error bar for the excited-state properties. Besides completing our previous studies focussed on transition energies (J. Chem. Theory Comput. 14 (2018) 4360–4379, ibid. 15 (2019) 1939–1956, ibid. 16 (2020) 1711–1741, and ibid. 16 (2020) 3720–3736), this work also provides ultra-accurate dipoles and oscillator strengths that could be employed for future theoretical benchmarks.

## 1. INTRODUCTION

In the formidable quest aiming at reaching high accuracy in the modeling of electronically excited states (ESs), the primary focus has been set on vertical excitation energies,¹ defined as the difference in total energies between a given ES and its corresponding ground state (GS) at fixed geometry (typically the ground-state equilibrium geometry). The main reasons for this choice are, on the one hand, the availability of many theoretical models for computing total ES energies, and, on the other hand, the fact that obtaining accurate transition energies is generally viewed as a prerequisite for further ES investigations. Along the years, more and more accurate vertical transition energies have become available. Probably the most illustrative example is provided by the well-known Thiel set,²⁻⁴ encompassing 223 values obtained at the CC3 and CASPT2 levels, as well as our recent efforts to obtain chemically-accurate vertical energies within ±0.03 eV of the full configuration interaction (FCI) limit for more than 400 ES.⁵⁻⁸ Whilst such sets are obviously useful to benchmark lower-order methods, they remain nevertheless intrinsically limited by two factors. First, vertical transition energies remain inaccessible experimentally (in the vast majority of the cases), preventing direct comparisons with measurements. To circumvent this issue, several groups have turned their attention to 0-0 energies,⁹⁻²³ defined as the differences between the ES and GS energies determined at their respective minimum and corrected for zero-point vibrational effects, be-

cause 0-0 energies allow straightforward theory-experiment comparisons. To compute 0-0 energies, one must however determine ES geometries and vibrations, which limits the number of methods that can be applied. This explains why compromise “hybrid” protocols employing different levels of theory for the structural and energetic parameters are popular in this particular field.¹²,¹⁸,¹⁹,²¹ Interestingly, the accuracy of the underlying geometries has been shown to be rather irrelevant,¹⁴,²² indicating that benchmarks of 0-0 energies still mainly assess the quality of the (adiabatic) transition energies. Second, vertical transition energies do provide a very partial characterization of the ES as one typically needs to determine transition probabilities (oscillator strengths, *f*) as well as ES properties (such as structures and dipole moments, *μ*) to attain a better grasp of the photophysics. On a more theoretical viewpoint, it is also known that a method can provide accurate transition energies while failing to deliver accurate ES properties,²⁴⁻²⁶ indicating that benchmarks relying on transition energies as unique gauge can lead to incomplete and/or erroneous conclusions regarding the strengths and weaknesses of a given theoretical model.

In the above-defined framework, let us now summarize the efforts that have been made to define accurate reference ES properties for significant sets of compounds.

First, for ES geometries, which are not our focus here, several ensembles of structures have been reported at CASPT2,²⁷ SAC-CI,²⁸,²⁹ Mk-MRCCSD,³⁰ QMC,³¹ and
# A Mountaineering Strategy to Excited States: Highly-Accurate Oscillator Strengths and Dipole Moments of Small Molecules

Amara Chrayteh,† Aymeric Blondel,† Pierre-François Loos,‡ and Denis Jacquemin*,†

†Université de Nantes, CNRS, CEISAM UMR 6230, F-44000 Nantes, France

‡Laboratoire de Chimie et Physique Quantiques, Université de Toulouse, CNRS, UPS, France

E-mail: Denis.Jacquemin@univ-nantes.fr

## Abstract

This work presents a series of highly-accurate excited-state properties obtained using high-order coupled-cluster (CC) calculations performed with a series of diffuse containing basis sets, as well as extensive comparisons with experimental values. Indeed, we have computed both the main ground-to-excited transition property, the oscillator strength, as well as the ground- and excited-state dipole moments, considering thirteen small molecules (hydridoboron, hydrogen chloride, water, hydrogen sulfide, boron fluoride, carbon monoxide, dinitrogen, ethylene, formaldehyde, thioformaldehyde, nitroxyl, fluorocarbene, and silylidene). We systematically include corrections up to the quintuple (CCSDTQP) in the CC expansion and extrapolate to the complete basis set limit. When comparisons with experimental measurements are possible, that is, when a number of consistent experimental data can be found, theory typically provides values falling within the experimental error bar for the excited-state properties. Besides completing our previous studies focussed on transition energies (J. Chem. Theory Comput. 14 (2018) 4360–4379, ibid. 15 (2019) 1939–1956, ibid. 16 (2020) 1711–1741, and ibid. 16 (2020) 3720–3736), this work also provides ultra-accurate dipoles and oscillator strengths that could be employed for future theoretical benchmarks.

## 1. INTRODUCTION

In the formidable quest aiming at reaching high accuracy in the modeling of electronically excited states (ESs), the primary focus has been set on vertical excitation energies,¹ defined as the difference in total energies between a given ES and its corresponding ground state (GS) at fixed geometry (typically the ground-state equilibrium geometry). The main reasons for this choice are, on the one hand, the availability of many theoretical models for computing total ES energies, and, on the other hand, the fact that obtaining accurate transition energies is generally viewed as a prerequisite for further ES investigations. Along the years, more and more accurate vertical transition energies have become available. Probably the most illustrative example is provided by the well-known Thiel set,²⁻⁴ encompassing 223 values obtained at the CC3 and CASPT2 levels, as well as our recent efforts to obtain chemically-accurate vertical energies within ±0.03 eV of the full configuration interaction (FCI) limit for more than 400 ES.⁵⁻⁸ Whilst such sets are obviously useful to benchmark lower-order methods, they remain nevertheless intrinsically limited by two factors. First, vertical transition energies remain inaccessible experimentally (in the vast majority of the cases), preventing direct comparisons with measurements. To circumvent this issue, several groups have turned their attention to 0-0 energies,⁹⁻²³ defined as the differences between the ES and GS energies determined at their respective minimum and corrected for zero-point vibrational effects, be-

cause 0-0 energies allow straightforward theory-experiment comparisons. To compute 0-0 energies, one must however determine ES geometries and vibrations, which limits the number of methods that can be applied. This explains why compromise “hybrid” protocols employing different levels of theory for the structural and energetic parameters are popular in this particular field.¹²,¹⁸,¹⁹,²¹ Interestingly, the accuracy of the underlying geometries has been shown to be rather irrelevant,¹⁴,²² indicating that benchmarks of 0-0 energies still mainly assess the quality of the (adiabatic) transition energies. Second, vertical transition energies do provide a very partial characterization of the ES as one typically needs to determine transition probabilities (oscillator strengths, *f*) as well as ES properties (such as structures and dipole moments, *μ*) to attain a better grasp of the photophysics. On a more theoretical viewpoint, it is also known that a method can provide accurate transition energies while failing to deliver accurate ES properties,²⁴⁻²⁶ indicating that benchmarks relying on transition energies as unique gauge can lead to incomplete and/or erroneous conclusions regarding the strengths and weaknesses of a given theoretical model.

In the above-defined framework, let us now summarize the efforts that have been made to define accurate reference ES properties for significant sets of compounds.

First, for ES geometries, which are not our focus here, several ensembles of structures have been reported at CASPT2,²⁷ SAC-CI,²⁸,²⁹ Mk-MRCCSD,³⁰ QMC,³¹ and
CC2<sup>32</sup> levels. To the best of our knowledge, the most ex-
tensive dataset of accurate ES structural parameters remains
the one defined by some of us during the last few years: it
contains bond lengths, valence angles, and torsion angles de-
termined at the CC3 and CASPT2 levels with aug-cc-pVTZ
for several dozens of small organic compounds.<sup>22,24,33–35</sup> In
this framework, it is also worth mentioning the works of
Szalay’s group on the shape of potential energy surfaces, in
which CCSDT references are defined,<sup>25,26,36</sup> as well as the
publications of Olivucci’s group focussing on the topology
of conical intersections, in which the results of high-level
multi-reference calculations are available.<sup>37,38</sup>

Second, for transition properties, the most accurate refer-
ence values we are aware of (for a significant set of transitions)
are: i) the CC3/TZVP oscillator strengths obtained by Kan-
nar and Szalay<sup>39</sup> for the Thiel set (this work also includes
15 CCSDT/TZVP oscillator strengths) that can be compared
to CASPT2/TZVP values obtained earlier for the same set;<sup>2</sup>
ii) the numerous CC3/aug-cc-pVTZ oscillator strengths de-
termined to complete our FCI energy calculations;<sup>5,7,8</sup> and
iii) the transition dipole moments computed at the CC3 or
ADC(3) levels for 15 molecules by Robinson.<sup>40</sup> There are
also earlier sets of oscillator strengths computed at lower lev-
els of theory, e.g., CCSD,<sup>41</sup> SAC-CI,<sup>42</sup> and CC2.<sup>43,44</sup> The
typical error associated with these approaches (with respect to
FCI) remains unknown at this stage. To the best of our knowl-
edge, a significant set of FCI-quality oscillator strengths has
yet to be published.

Third, theoretical datasets for accurate ES dipoles are ap-
parently even scarcer. The most advanced sets we are aware
of contain: i) CASPT2/TZVP,<sup>2</sup> and CC2/aug-cc-pVTZ ref-
erence values<sup>43</sup> determined for Thiel's set; ii) CC2/aug-cc-
pVQZ results for small- and medium-sized molecules;<sup>45</sup> and
iii) CC2/aug-cc-pVTZ data for "real-life" organic dyes.<sup>46</sup>
Comparisons with experimental ES dipoles performed in Ref.
45 yield an error of ca. 0.2 Debye (D) for both ADC(2) and
CC2, a value that can be viewed as acceptable, but is nev-
ertheless far from chemical accuracy and not on par with
the precision provided by state-of-the-art approaches for GS
dipole moments.<sup>47</sup> Of course, one can find specific works
focussing on a small number of ES dipoles and using high
level(s) of theory (see our Result Section for references).
However, such specific studies preclude valuable statistical
conclusions.

Globally, these previous studies have demonstrated that
both *f* and *μ*<sup>ES</sup> are much more basis set sensitive than tran-
sition energies and geometries. Indeed, a specific challenge
comes from intensity-borrowing effects that can vastly change
the properties of two close-lying ES of the same symme-
try while their energies remain almost unaffected. When a
change of basis set slightly tunes the energy gap between two
ESs, it might simultaneously drastically affect the properties.
This highlights that properties are much harder to accurately
estimate than energies, not only for implementation or com-
putational reasons, but also due to more fundamental aspects.

Another important fact is that, in contrast to vertical tran-
sition energies, both oscillator strengths and ES dipole mo-
ments are accessible experimentally, so comparisons between
theoretical and experimental measurements are, in principle,

possible. However, as we pointed out previously for geome-
tries,³³ such comparisons are generally far from straightfor-
ward. For the oscillator strengths, the two main measurement
techniques are electron impact and optical spectroscopies.
As nicely summarized elsewhere,⁴⁸ the former typically re-
quires extrapolations of the cross-sections measured at vari-
ous momenta/angles (such extrapolations are not error-free),
whereas the latter can be plagued by saturation and inter-
action effects (yielding to underestimation of the actual *f*
value). In any case, the so-called “electronic” or “optical” *f*,
which is of interest here, is not directly measurable, and post-
processing of the experimental raw data (lifetimes or cross
sections) is needed to access its “experimental” value.⁴⁸,⁴⁹
The measurements of ES dipoles are also cumbersome.⁵⁰
The typical strategy is to investigate solvatofluorochromism,
that is, to measure the shift of the emission wavelength in a
series of solvents of various polarities, and to fit the results
with the Lippert-Mataga equation within an Onsager-like in-
teraction model. Such approach is obviously not a direct
gas-phase measurement and comes with significant assump-
tions, making the final error often too large to allow bench-
marking.⁵¹ Alternatively, one can also measure μ<sup>ES</sup> directly
in gas phase by studying how external electric fields tune the
position and shape of the vibronic peaks (the so-called Stark
effect).⁵⁰,⁵¹ Only the latter approach can be considered as suf-
ficiently robust for reference purposes. Yet, such Stark effect
measurements do typically provide an “adiabatic” version of
μ<sup>ES</sup>, that is, the ES dipole measured at the ES equilibrium ge-
ometry. If a non-trifling structural reorganization takes place
after absorption, such as in formaldehyde, this means that the
value of μ<sup>ES</sup> computed at the GS equilibrium structure has
again no experimental counterpart. In addition, the direction
of the ES dipole moment cannot be determined directly from
the measurements of the Stark effect.

For all these reasons, it is of interest to define a set of co-
herent near-FCI oscillator strengths and excited-state dipole
moments. Indeed, the oscillator strengths are directly related
to the transition dipole moments, whereas the dipole mo-
ments can be seen as a measure of the quality of the total (GS
and ES) densities, so that these data go beyond the “simple”
characterization of vertical transition energies. This contri-
bution therefore aims at tackling this ambitious objective for
a set of small molecules similar to the one treated in our origi-
nal *Mountaineering* paper.⁵ To this end, we take advantage of
the CC hierarchy, going from CCSD to CCSDTQP for both
properties, in combination with increasingly large atomic ba-
sis sets including one or two sets of diffuse basis functions.
Whilst such calculations provide “definite answers”, they re-
main at the limit of today’s computational capabilities and
are achievable for compact molecules only, which stands as a
clear limit of the present contribution. Nevertheless, provid-
ing highly-accurate numbers for properties directly related to
the quality of the transition and the ES density is, we believe,
useful. Very recently, Hait and Head-Gordon have used, in a
density-functional theory (DFT) context, the GS dipole mo-
ment as a metric to estimate the quality of the GS density
given by many exchange-correlation functionals.⁴⁷ As these
authors nicely stated: the dipole “is perhaps the simplest
observable that captures errors in the underlying density, (...
CC2<sup>32</sup> levels. To the best of our knowledge, the most ex-
tensive dataset of accurate ES structural parameters remains
the one defined by some of us during the last few years: it
contains bond lengths, valence angles, and torsion angles de-
termined at the CC3 and CASPT2 levels with aug-cc-pVTZ
for several dozens of small organic compounds.<sup>22,24,33–35</sup> In
this framework, it is also worth mentioning the works of
Szalay’s group on the shape of potential energy surfaces, in
which CCSDT references are defined,<sup>25,26,36</sup> as well as the
publications of Olivucci’s group focussing on the topology
of conical intersections, in which the results of high-level
multi-reference calculations are available.<sup>37,38</sup>

Second, for transition properties, the most accurate refer-
ence values we are aware of (for a significant set of transitions)
are: i) the CC3/TZVP oscillator strengths obtained by Kan-
nar and Szalay<sup>39</sup> for the Thiel set (this work also includes
15 CCSDT/TZVP oscillator strengths) that can be compared
to CASPT2/TZVP values obtained earlier for the same set;<sup>2</sup>
ii) the numerous CC3/aug-cc-pVTZ oscillator strengths de-
termined to complete our FCI energy calculations;<sup>5,7,8</sup> and
iii) the transition dipole moments computed at the CC3 or
ADC(3) levels for 15 molecules by Robinson.<sup>40</sup> There are
also earlier sets of oscillator strengths computed at lower lev-
els of theory, e.g., CCSD,<sup>41</sup> SAC-CI,<sup>42</sup> and CC2.<sup>43,44</sup> The
typical error associated with these approaches (with respect to
FCI) remains unknown at this stage. To the best of our knowl-
edge, a significant set of FCI-quality oscillator strengths has
yet to be published.

Third, theoretical datasets for accurate ES dipoles are ap-
parently even scarcer. The most advanced sets we are aware
of contain: i) CASPT2/TZVP,<sup>2</sup> and CC2/aug-cc-pVTZ ref-
erence values<sup>43</sup> determined for Thiel's set; ii) CC2/aug-cc-
pVQZ results for small- and medium-sized molecules;<sup>45</sup> and
iii) CC2/aug-cc-pVTZ data for "real-life" organic dyes.<sup>46</sup>
Comparisons with experimental ES dipoles performed in Ref.
45 yield an error of ca. 0.2 Debye (D) for both ADC(2) and
CC2, a value that can be viewed as acceptable, but is nev-
ertheless far from chemical accuracy and not on par with
the precision provided by state-of-the-art approaches for GS
dipole moments.<sup>47</sup> Of course, one can find specific works
focussing on a small number of ES dipoles and using high
level(s) of theory (see our Result Section for references).
However, such specific studies preclude valuable statistical
conclusions.

Globally, these previous studies have demonstrated that
both *f* and *μ*<sup>ES</sup> are much more basis set sensitive than tran-
sition energies and geometries. Indeed, a specific challenge
comes from intensity-borrowing effects that can vastly change
the properties of two close-lying ES of the same symme-
try while their energies remain almost unaffected. When a
change of basis set slightly tunes the energy gap between two
ESs, it might simultaneously drastically affect the properties.
This highlights that properties are much harder to accurately
estimate than energies, not only for implementation or com-
putational reasons, but also due to more fundamental aspects.

Another important fact is that, in contrast to vertical tran-
sition energies, both oscillator strengths and ES dipole mo-
ments are accessible experimentally, so comparisons between
theoretical and experimental measurements are, in principle,

possible. However, as we pointed out previously for geome-
tries,³³ such comparisons are generally far from straightfor-
ward. For the oscillator strengths, the two main measurement
techniques are electron impact and optical spectroscopies.
As nicely summarized elsewhere,⁴⁸ the former typically re-
quires extrapolations of the cross-sections measured at vari-
ous momenta/angles (such extrapolations are not error-free),
whereas the latter can be plagued by saturation and inter-
action effects (yielding to underestimation of the actual *f*
value). In any case, the so-called “electronic” or “optical” *f*,
which is of interest here, is not directly measurable, and post-
processing of the experimental raw data (lifetimes or cross
sections) is needed to access its “experimental” value.⁴⁸,⁴⁹
The measurements of ES dipoles are also cumbersome.⁵⁰
The typical strategy is to investigate solvatofluorochromism,
that is, to measure the shift of the emission wavelength in a
series of solvents of various polarities, and to fit the results
with the Lippert-Mataga equation within an Onsager-like in-
teraction model. Such approach is obviously not a direct
gas-phase measurement and comes with significant assump-
tions, making the final error often too large to allow bench-
marking.⁵¹ Alternatively, one can also measure μ<sup>ES</sup> directly
in gas phase by studying how external electric fields tune the
position and shape of the vibronic peaks (the so-called Stark
effect).⁵⁰,⁵¹ Only the latter approach can be considered as suf-
ficiently robust for reference purposes. Yet, such Stark effect
measurements do typically provide an “adiabatic” version of
μ<sup>ES</sup>, that is, the ES dipole measured at the ES equilibrium ge-
ometry. If a non-trifling structural reorganization takes place
after absorption, such as in formaldehyde, this means that the
value of μ<sup>ES</sup> computed at the GS equilibrium structure has
again no experimental counterpart. In addition, the direction
of the ES dipole moment cannot be determined directly from
the measurements of the Stark effect.

For all these reasons, it is of interest to define a set of co-
herent near-FCI oscillator strengths and excited-state dipole
moments. Indeed, the oscillator strengths are directly related
to the transition dipole moments, whereas the dipole mo-
ments can be seen as a measure of the quality of the total (GS
and ES) densities, so that these data go beyond the “simple”
characterization of vertical transition energies. This contri-
bution therefore aims at tackling this ambitious objective for
a set of small molecules similar to the one treated in our origi-
nal *Mountaineering* paper.⁵ To this end, we take advantage of
the CC hierarchy, going from CCSD to CCSDTQP for both
properties, in combination with increasingly large atomic ba-
sis sets including one or two sets of diffuse basis functions.
Whilst such calculations provide “definite answers”, they re-
main at the limit of today’s computational capabilities and
are achievable for compact molecules only, which stands as a
clear limit of the present contribution. Nevertheless, provid-
ing highly-accurate numbers for properties directly related to
the quality of the transition and the ES density is, we believe,
useful. Very recently, Hait and Head-Gordon have used, in a
density-functional theory (DFT) context, the GS dipole mo-
ment as a metric to estimate the quality of the GS density
given by many exchange-correlation functionals.⁴⁷ As these
authors nicely stated: the dipole “is perhaps the simplest
observable that captures errors in the underlying density, (...
and is) a relevant density derived quantity to examine for DFA testing and development". The extension to ESs (and time-dependent DFT) is obviously natural, but such task is difficult due to the lack of definite reference values. In this framework, we also note some very recent efforts for computing ES first-order properties with the many-body expansion FCI (MBE-FCI) approach.⁵²

## 2. COMPUTATIONAL METHODS

### 2.1 Geometries and basis sets

All our geometries are obtained at the CC3/aug-cc-pVTZ⁵³,⁵⁴ without using the frozen-core (FC) approximation (i.e, we correlate all the electrons). These geometries are given in the Supporting Information (SI). Note that several structures come from previous works,⁵,⁸ and we used this level of theory here for consistency. New optimizations have been achieved with the DALTON 2017⁵⁵ and CFOUR 2.1⁵⁶ codes applying default parameters in both cases.

As in our previous works,⁵⁻⁸ we consider the diffuse-containing Pople's 6-31+G(d) and Dunning's (d-)aug-cc-pVXZ (X = D, T, Q, and 5) atomic basis sets in all ES calculations.

In contrast with our previous studies, in which the complete basis set limit (CBS) could be obtained by a brute-force approach, *f* and *μ*ES converge, in some cases, slower than the energy with respect to the basis set size. Therefore, we have performed CBS extrapolation by applying the well-known Helgaker formula,⁵⁷

$$P_{\text{CBS}} = \frac{P_{X-1}(X-1)^3 - P_X X^3}{(X-1)^3 - X^3}, \quad (1)$$

in which *X* equals 2, 3, 4, ... for D, T, Q, ... in the Dunning series and *P* is the property under investigation. Interestingly, this approach was successfully used for GS dipole moments.⁵⁸ In practice, we performed four extrapolations using both the CCSD and CCSDT results obtained with the pairs of largest singly- and doubly-augmented basis sets accessible of at least triple-ζ size, i.e., i) CCSD/aug-cc-pVQZ and aug-cc-pV5Z; CCSDT/aug-cc-pVTZ and aug-cc-pVQZ; CCSD/d-aug-cc-pVTZ and d-aug-cc-pVQZ; and CCSDT/d-aug-cc-pVTZ and d-aug-cc-pVQZ for molecules in Sections 3.1–3.5; ii) CCSD/aug-cc-pVTZ and aug-cc-pVQZ; and CCSD/d-aug-cc-pVTZ and d-aug-cc-pVQZ for molecules in Sections 3.6–3.9. In such a way, we can estimate the extrapolation error, and provide error bars for the CBS values, although such error bar is likely underestimated for the latter set of compounds. These CCSD and/or CCSDT CBS values are next used to correct the properties obtained at higher levels (e.g., CCSDTQ) with a finite basis using the approach described below.

### 2.2 Reference calculations

We have chosen to use the MRCC (2017 and 2019) program,⁵⁹,⁶⁰ for performing our CC calculations, as this code allows to set up an arbitrary CC expansion order. We therefore use the CCSD,⁶¹⁻⁶⁵ CCSDT,⁶⁶⁻⁷⁰ CCSDTQ,⁷¹⁻⁷⁴ and CCSDTQP⁷²⁻⁷⁴ hierarchy for energies, oscillator strengths, GS and ES dipoles. All these values have been obtained within the FC approximation. The interested reader may find

discussions about the impact of this approximation and the importance of core correlation functions for transition energies in some of our previous works.²²,⁷⁵ For CCSD, we performed several test calculations with GAUSSIAN 16,⁷⁶ Q-CHEM 5.2,⁷⁷ DALTON 2017,⁵⁵ and eT 1.0⁷⁸ and we could not detect any significant discrepancy with respect to the MRCC results. At this stage, it is important to stress that all these calculations rely on the so-called linear-response (LR) formalism,⁷³,⁷⁹,⁸⁰ so that while the same transition energies would be obtained with the equation-of-motion (EOM) approach, the properties would be different. However, it is known that the two formalisms become equivalent when the CC wave function becomes exact. As we strive here to be as close as possible from the FCI limit, we trust that our theoretical best estimates (TBEs) are not significantly affected by the selection of the LR implementation. At the CCSD level, we also provide a comparison between the EOM and LR oscillator strength values obtained with various codes in Table S1 in the SI. The differences found between the EOM and LR formalisms are very small. The interested reader may also find extensive comparisons of oscillator strengths determined within the two formalisms elsewhere.⁸¹ We also note that all our oscillator strengths are given in the length gauge, the most commonly applied gauge, but again this choice is likely irrelevant when one is targeting near-exact values. The interested reader can find comparisons between CC oscillator strengths determined with the length, velocity, and mixed length-velocity gauges on small compounds elsewhere.⁸² As expected the impact of the gauge was found to decrease when increasing the order of the CC expansion, especially when triples are included. Finally, we report in the Tables below the so-called orbital-relaxed dipoles, which are more accurate than the so-called orbital-unrelaxed dipoles in which the impact of the external field on the orbitals is neglected. Details on various approaches and their implementations for correlated first-order properties can be found elsewhere.⁸⁰,⁸³,⁸⁴

Beyond the basis set extrapolation discussed above, we also define TBEs in the following. To this end, we use an incremental strategy for the transition energies and dipoles, e.g., at the aug-cc-pVTZ level,

$$P(\text{TBE}) = P(\text{Low}/\text{AVTZ}) + P(\text{High}/\text{AVDZ}) - P(\text{Low}/\text{AVDZ}), \quad (2)$$

where Low and High denote, e.g., CCSDTQ and CCSDTQP (see footnotes in the Tables for specific details). For oscillator strengths, we applied the corresponding multiplicative approach, e.g.,

$$f(\text{TBE}) = f(\text{Low}/\text{AVTZ}) \frac{f(\text{High}/\text{AVDZ})}{f(\text{Low}/\text{AVDZ})}. \quad (3)$$

Such incremental strategy that used a double-ζ result to estimate “Q” or “P” effects is commonly employed in the CC literature.⁷³,⁸⁵⁻⁸⁹

## 3. RESULTS AND DISCUSSION

Below, we discuss individual molecules going up on size progressively. Concerning literature references, we do not
and is) a relevant density derived quantity to examine for DFA testing and development". The extension to ESs (and time-dependent DFT) is obviously natural, but such task is difficult due to the lack of definite reference values. In this framework, we also note some very recent efforts for computing ES first-order properties with the many-body expansion FCI (MBE-FCI) approach.⁵²

## 2. COMPUTATIONAL METHODS

### 2.1 Geometries and basis sets

All our geometries are obtained at the CC3/aug-cc-pVTZ⁵³,⁵⁴ without using the frozen-core (FC) approximation (i.e, we correlate all the electrons). These geometries are given in the Supporting Information (SI). Note that several structures come from previous works,⁵,⁸ and we used this level of theory here for consistency. New optimizations have been achieved with the DALTON 2017⁵⁵ and CFOUR 2.1⁵⁶ codes applying default parameters in both cases.

As in our previous works,⁵⁻⁸ we consider the diffuse-containing Pople's 6-31+G(d) and Dunning's (d-)aug-cc-pVXZ (X = D, T, Q, and 5) atomic basis sets in all ES calculations.

In contrast with our previous studies, in which the complete basis set limit (CBS) could be obtained by a brute-force approach, *f* and *μ*ES converge, in some cases, slower than the energy with respect to the basis set size. Therefore, we have performed CBS extrapolation by applying the well-known Helgaker formula,⁵⁷

$$P_{\text{CBS}} = \frac{P_{X-1}(X-1)^3 - P_X X^3}{(X-1)^3 - X^3}, \quad (1)$$

in which *X* equals 2, 3, 4, ... for D, T, Q, ... in the Dunning series and *P* is the property under investigation. Interestingly, this approach was successfully used for GS dipole moments.⁵⁸ In practice, we performed four extrapolations using both the CCSD and CCSDT results obtained with the pairs of largest singly- and doubly-augmented basis sets accessible of at least triple-ζ size, i.e., i) CCSD/aug-cc-pVQZ and aug-cc-pV5Z; CCSDT/aug-cc-pVTZ and aug-cc-pVQZ; CCSD/d-aug-cc-pVTZ and d-aug-cc-pVQZ; and CCSDT/d-aug-cc-pVTZ and d-aug-cc-pVQZ for molecules in Sections 3.1–3.5; ii) CCSD/aug-cc-pVTZ and aug-cc-pVQZ; and CCSD/d-aug-cc-pVTZ and d-aug-cc-pVQZ for molecules in Sections 3.6–3.9. In such a way, we can estimate the extrapolation error, and provide error bars for the CBS values, although such error bar is likely underestimated for the latter set of compounds. These CCSD and/or CCSDT CBS values are next used to correct the properties obtained at higher levels (e.g., CCSDTQ) with a finite basis using the approach described below.

### 2.2 Reference calculations

We have chosen to use the MRCC (2017 and 2019) program,⁵⁹,⁶⁰ for performing our CC calculations, as this code allows to set up an arbitrary CC expansion order. We therefore use the CCSD,⁶¹⁻⁶⁵ CCSDT,⁶⁶⁻⁷⁰ CCSDTQ,⁷¹⁻⁷⁴ and CCSDTQP⁷²⁻⁷⁴ hierarchy for energies, oscillator strengths, GS and ES dipoles. All these values have been obtained within the FC approximation. The interested reader may find

discussions about the impact of this approximation and the importance of core correlation functions for transition energies in some of our previous works.²²,⁷⁵ For CCSD, we performed several test calculations with GAUSSIAN 16,⁷⁶ Q-CHEM 5.2,⁷⁷ DALTON 2017,⁵⁵ and eT 1.0⁷⁸ and we could not detect any significant discrepancy with respect to the MRCC results. At this stage, it is important to stress that all these calculations rely on the so-called linear-response (LR) formalism,⁷³,⁷⁹,⁸⁰ so that while the same transition energies would be obtained with the equation-of-motion (EOM) approach, the properties would be different. However, it is known that the two formalisms become equivalent when the CC wave function becomes exact. As we strive here to be as close as possible from the FCI limit, we trust that our theoretical best estimates (TBEs) are not significantly affected by the selection of the LR implementation. At the CCSD level, we also provide a comparison between the EOM and LR oscillator strength values obtained with various codes in Table S1 in the SI. The differences found between the EOM and LR formalisms are very small. The interested reader may also find extensive comparisons of oscillator strengths determined within the two formalisms elsewhere.⁸¹ We also note that all our oscillator strengths are given in the length gauge, the most commonly applied gauge, but again this choice is likely irrelevant when one is targeting near-exact values. The interested reader can find comparisons between CC oscillator strengths determined with the length, velocity, and mixed length-velocity gauges on small compounds elsewhere.⁸² As expected the impact of the gauge was found to decrease when increasing the order of the CC expansion, especially when triples are included. Finally, we report in the Tables below the so-called orbital-relaxed dipoles, which are more accurate than the so-called orbital-unrelaxed dipoles in which the impact of the external field on the orbitals is neglected. Details on various approaches and their implementations for correlated first-order properties can be found elsewhere.⁸⁰,⁸³,⁸⁴

Beyond the basis set extrapolation discussed above, we also define TBEs in the following. To this end, we use an incremental strategy for the transition energies and dipoles, e.g., at the aug-cc-pVTZ level,

$$P(\text{TBE}) = P(\text{Low}/\text{AVTZ}) + P(\text{High}/\text{AVDZ}) - P(\text{Low}/\text{AVDZ}), \quad (2)$$

where Low and High denote, e.g., CCSDTQ and CCSDTQP (see footnotes in the Tables for specific details). For oscillator strengths, we applied the corresponding multiplicative approach, e.g.,

$$f(\text{TBE}) = f(\text{Low}/\text{AVTZ}) \frac{f(\text{High}/\text{AVDZ})}{f(\text{Low}/\text{AVDZ})}. \quad (3)$$

Such incremental strategy that used a double-ζ result to estimate “Q” or “P” effects is commonly employed in the CC literature.⁷³,⁸⁵⁻⁸⁹

## 3. RESULTS AND DISCUSSION

Below, we discuss individual molecules going up on size progressively. Concerning literature references, we do not
<table><caption>Table 1: Ground-state dipole moment $\mu^{\text{GS}}$, vertical transition energies $\Delta E_{\text{vert}}$, oscillator strengths $f$, and excited-state dipole moments, $\mu_{\text{vert}}^{\text{ES}}$ and $\mu_{\text{adia}}^{\text{ES}}$, determined for BH (GS and ES geometries) and HCl (GS geometry). Transition energies are in eV and dipoles in D.</caption><thead><tr><th rowspan="2">Basis</th><th rowspan="2">Method</th><th colspan="5">BH</th><th colspan="5">HCl</th></tr><tr><th><sup>1</sup>Σ<sup>+</sup><br>μ<sup>GS</sup></th><th>ΔE<sub>vert</sub></th><th><sup>1</sup>Π (Val)<br>f</th><th>μ<sup>ES</sup><sub>vert</sub></th><th>μ<sup>ES</sup><sub>adia</sub></th><th><sup>1</sup>Σ<sup>+</sup><br>μ<sup>GS</sup></th><th>ΔE<sub>vert</sub></th><th><sup>1</sup>Π (Val)<br>f</th><th>μ<sup>ES</sup><sub>vert</sub></th></tr></thead><tbody><tr><td rowspan="5">aug-cc-pVDZ</td><td>CCSD</td><td>1.389</td><td>2.970</td><td>0.051</td><td>0.530</td><td>0.513</td><td>1.147</td><td>7.862</td><td>0.066</td><td>-2.773</td></tr><tr><td>CCSDT</td><td>1.371</td><td>2.946</td><td>0.049</td><td>0.543</td><td>0.527</td><td>1.131</td><td>7.815</td><td>0.065</td><td>-2.745</td></tr><tr><td>CCSDTQ</td><td>1.370</td><td>2.947</td><td>0.049</td><td>0.545</td><td>0.528</td><td>1.130</td><td>7.822</td><td>0.065</td><td>-2.728</td></tr><tr><td>CCSDTQP</td><td>1.370</td><td>2.947</td><td>0.049</td><td>0.545</td><td>0.528</td><td>1.130</td><td>7.823</td><td>0.065</td><td>-2.727</td></tr><tr><td>CCSD</td><td>1.433</td><td>2.928</td><td>0.050</td><td>0.550</td><td>0.534</td><td>1.097</td><td>7.906</td><td>0.056</td><td>-2.526</td></tr><tr><td rowspan="5">aug-cc-pVTZ</td><td>CCSDT</td><td>1.410</td><td>2.900</td><td>0.048</td><td>0.558</td><td>0.541</td><td>1.085</td><td>7.834</td><td>0.055</td><td>-2.515</td></tr><tr><td>CCSDTQ</td><td>1.409</td><td>2.901</td><td>0.048</td><td>0.559</td><td>0.542</td><td>1.084</td><td>7.837</td><td>0.055</td><td>-2.502</td></tr><tr><td>CCSD</td><td>1.440</td><td>2.918</td><td>0.050</td><td>0.555</td><td>0.538</td><td>1.111</td><td>7.954</td><td>0.051</td><td>-2.410</td></tr><tr><td>CCSDT</td><td>1.416</td><td>2.890</td><td>0.048</td><td>0.561</td><td>0.544</td><td>1.098</td><td>7.880</td><td>0.050</td><td>-2.410</td></tr><tr><td>CCSD</td><td>1.443</td><td>2.915</td><td>0.050</td><td>0.556</td><td>0.540</td><td>1.109</td><td>7.961</td><td>0.048</td><td>-2.336</td></tr><tr><td rowspan="5">d-aug-cc-pVDZ</td><td>CCSD</td><td>1.388</td><td>2.969</td><td>0.051</td><td>0.519</td><td>0.503</td><td>1.135</td><td>7.836</td><td>0.064</td><td>-2.697</td></tr><tr><td>CCSDT</td><td>1.370</td><td>2.945</td><td>0.049</td><td>0.533</td><td>0.516</td><td>1.119</td><td>7.787</td><td>0.063</td><td>-2.670</td></tr><tr><td>CCSD</td><td>1.432</td><td>2.927</td><td>0.049</td><td>0.550</td><td>0.534</td><td>1.096</td><td>7.894</td><td>0.055</td><td>-2.491</td></tr><tr><td>CCSDT</td><td>1.409</td><td>2.900</td><td>0.048</td><td>0.558</td><td>0.541</td><td>1.084</td><td>7.822</td><td>0.054</td><td>-2.480</td></tr><tr><td>CCSD</td><td>1.440</td><td>2.918</td><td>0.050</td><td>0.555</td><td>0.538</td><td>1.111</td><td>7.949</td><td>0.050</td><td>-2.405</td></tr><tr><td rowspan="5">d-aug-cc-pVQZ</td><td>CCSD</td><td>1.416</td><td>2.890</td><td>0.048</td><td>0.561</td><td>0.544</td><td>1.098</td><td>7.876</td><td>0.050</td><td>-2.406</td></tr><tr><td>CCSDT</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td rowspan="2">aug-cc-pVTZ</td><td>TBE<sup>a</sup></td><td>1.409</td><td>2.901</td><td>0.048</td><td>0.559</td><td>0.542</td><td>1.084</td><td>7.837</td><td>0.055</td><td>-2.501</td></tr><tr><td>TBE<sup>b</sup></td><td>1.42±0.00</td><td>2.88±0.01</td><td>0.048±0.001</td><td>0.57±0.01</td><td>0.55±0.01</td><td>1.11±0.01</td><td>7.91±0.01</td><td>0.046±0.001</td><td>-2.32±0.01</td></tr><tr><td rowspan="2">CBS</td><td>Th.</td><td>1.425<sup>c</sup></td><td>2.944<sup>d</sup></td><td></td><td></td><td></td><td>1.106<sup>f</sup></td><td>7.94<sup>g</sup></td><td>0.081<sup>g</sup></td><td></td></tr><tr><td>Exp.</td><td>1.27±0.21<sup>h</sup></td><td></td><td>0.044<sup>i</sup></td><td></td><td>0.58±0.04<sup>h</sup></td><td></td><td>8.05<sup>j</sup></td><td>0.051<sup>j</sup></td><td></td></tr><tr><td rowspan="2">Lit.</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td>0.042±0.004<sup>k</sup></td><td></td></tr><tr><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td>0.052±0.006<sup>l</sup></td><td></td></tr></tbody></table>

<sup>a</sup>Computed using CCSDTQ/aug-cc-pVTZ values and CCSDTQP/aug-cc-pVDZ corrections. Note that CCSDTQ is equivalent to FCI for BH; <sup>b</sup>See Computational Methods section; <sup>c</sup>Average between the MR-ACPF/aug-cc-pCV7Z(i) values obtained for r = 1.220 and 1.225 Å in Ref. 90;

<sup>d</sup>FCI/aug-cc-pVDZ value of Ref. 54; <sup>e</sup>CC2/aug-cc-pVQZ result from Ref. 45; <sup>f</sup>CCSD(T)/CBS value from Ref. 47; <sup>g</sup>CISDTQ/aug-cc-pCVQZ value from Ref. 91. A *f* value of 0.071 is also reported in Table II of the same work; <sup>h</sup>Stark (emission) measurements of Ref. 92; <sup>i</sup>*f*<sub>00</sub> obtained from laser-induced fluorescence in Ref. 93. There is no major contributions from other bands according to this work (see references therein for previous experimental values). A slightly older experiment (Ref. 94) reports a smaller estimate of 0.045±0.02; <sup>j</sup>Absorption values from Ref. 95 (ΔE corresponds to the maximum of absorption); <sup>k</sup>EELS value from Ref. 96; <sup>l</sup>HR-EELS value from Ref. 97.

intend to provide an exhaustive list of all previous works for each system considered here, but rather to highlight the studies and comparisons that we have found valuable for the present work.

## 3.1 BH and HCl

Let us start by a tiny compound, BH. Our results are listed in Table 1 and although the size of this molecule seems ridiculous (only 4 valence electrons), some valuable conclusions can be obtained. We note that $\mu^{\text{GS}}$ is slightly too large with CCSD (irrespective of the basis set) but the CC convergence is fast and CCSDT is obviously sufficient. Our TBE/CBS of 1.42 D for $\mu^{\text{GS}}$ is equivalent to a recent ultra-accurate estimate,⁹⁰ and also falls within the error bar of the only available experimental value we are aware of: 1.27±0.21 D.⁹² Such large error bar is explainable: the experiment relied on an analysis of the emission spectrum.⁹² It is quite obvious that the theoretical estimate is more trustworthy in this specific case, indicating that previous error analyses based on the 1.27 D value likely significantly exaggerated the ADC(2) and CC2 overestimations,⁴⁵ but underestimated the QMC error⁹⁸ for $\mu^{\text{GS}}$. For the lowest transition energy in BH, our values fit with previous FCI calculations,⁵⁴ and one again notices rather quick convergence of $\Delta E_{\text{vert}}$ with respect to both basis set size and CC order. For the oscillator strength,

there is also an astonishing stability of the values, as the considered ES is well separated from higher-lying ones of the same spatial symmetry. Our TBE is close to the most recent measurement (of $f_{00}$) we could find.⁹³ For $\mu^{\text{ES}}$, one notes a very small decrease of the amplitude between the GS and ES geometries, which is a logical consequence of the tiny geometrical relaxation (+0.018 Å), and our TBE is within the rather small experimental error bar (see bottom of Table 1). Finally, as can be seen in Table S2, these results are not affected by the FC approximation, e.g., the difference between CCSDTQ(FC)/aug-cc-pVTZ and CCSDTQ(Full)/aug-cc-pCVTZ is 0.001 for *f* and 0.002 D for the dipole moments.

For HCl, the GS dipole moment does not cause any specific challenge and our TBE is equivalent to the one recently reported by the Head-Gordon group.⁴⁷ As expected,⁵ CCS-DTQ provides converged $\Delta E_{\text{vert}}$ and this holds for both *f* and $\mu^{\text{ES}}$. When increasing the size of the basis set, one sees a significant decrease of the oscillator strength (ca. -25% from aug-cc-pVDZ to aug-cc-pV5Z) and of the ES dipole (-16% for the same basis pair), whereas the transition energy varies by 1% only. Nevertheless, the CBS extrapolations are stable for all investigated properties. For HCl, it is difficult to obtain a very accurate experimental *f* value, in part due to the mixing of the <sup>1</sup>Π and <sup>3</sup>Π states.⁹⁵ There is therefore a
<table><caption>Table 1: Ground-state dipole moment $\mu^{\text{GS}}$, vertical transition energies $\Delta E_{\text{vert}}$, oscillator strengths $f$, and excited-state dipole moments, $\mu_{\text{vert}}^{\text{ES}}$ and $\mu_{\text{adia}}^{\text{ES}}$, determined for BH (GS and ES geometries) and HCl (GS geometry). Transition energies are in eV and dipoles in D.</caption><thead><tr><th rowspan="2">Basis</th><th rowspan="2">Method</th><th colspan="5">BH</th><th colspan="5">HCl</th></tr><tr><th><sup>1</sup>Σ<sup>+</sup><br>μ<sup>GS</sup></th><th>ΔE<sub>vert</sub></th><th><sup>1</sup>Π (Val)<br>f</th><th>μ<sup>ES</sup><sub>vert</sub></th><th>μ<sup>ES</sup><sub>adia</sub></th><th><sup>1</sup>Σ<sup>+</sup><br>μ<sup>GS</sup></th><th>ΔE<sub>vert</sub></th><th><sup>1</sup>Π (Val)<br>f</th><th>μ<sup>ES</sup><sub>vert</sub></th></tr></thead><tbody><tr><td rowspan="5">aug-cc-pVDZ</td><td>CCSD</td><td>1.389</td><td>2.970</td><td>0.051</td><td>0.530</td><td>0.513</td><td>1.147</td><td>7.862</td><td>0.066</td><td>-2.773</td></tr><tr><td>CCSDT</td><td>1.371</td><td>2.946</td><td>0.049</td><td>0.543</td><td>0.527</td><td>1.131</td><td>7.815</td><td>0.065</td><td>-2.745</td></tr><tr><td>CCSDTQ</td><td>1.370</td><td>2.947</td><td>0.049</td><td>0.545</td><td>0.528</td><td>1.130</td><td>7.822</td><td>0.065</td><td>-2.728</td></tr><tr><td>CCSDTQP</td><td>1.370</td><td>2.947</td><td>0.049</td><td>0.545</td><td>0.528</td><td>1.130</td><td>7.823</td><td>0.065</td><td>-2.727</td></tr><tr><td>CCSD</td><td>1.433</td><td>2.928</td><td>0.050</td><td>0.550</td><td>0.534</td><td>1.097</td><td>7.906</td><td>0.056</td><td>-2.526</td></tr><tr><td rowspan="5">aug-cc-pVTZ</td><td>CCSDT</td><td>1.410</td><td>2.900</td><td>0.048</td><td>0.558</td><td>0.541</td><td>1.085</td><td>7.834</td><td>0.055</td><td>-2.515</td></tr><tr><td>CCSDTQ</td><td>1.409</td><td>2.901</td><td>0.048</td><td>0.559</td><td>0.542</td><td>1.084</td><td>7.837</td><td>0.055</td><td>-2.502</td></tr><tr><td>CCSD</td><td>1.440</td><td>2.918</td><td>0.050</td><td>0.555</td><td>0.538</td><td>1.111</td><td>7.954</td><td>0.051</td><td>-2.410</td></tr><tr><td>CCSDT</td><td>1.416</td><td>2.890</td><td>0.048</td><td>0.561</td><td>0.544</td><td>1.098</td><td>7.880</td><td>0.050</td><td>-2.410</td></tr><tr><td>CCSD</td><td>1.443</td><td>2.915</td><td>0.050</td><td>0.556</td><td>0.540</td><td>1.109</td><td>7.961</td><td>0.048</td><td>-2.336</td></tr><tr><td rowspan="5">d-aug-cc-pVDZ</td><td>CCSD</td><td>1.388</td><td>2.969</td><td>0.051</td><td>0.519</td><td>0.503</td><td>1.135</td><td>7.836</td><td>0.064</td><td>-2.697</td></tr><tr><td>CCSDT</td><td>1.370</td><td>2.945</td><td>0.049</td><td>0.533</td><td>0.516</td><td>1.119</td><td>7.787</td><td>0.063</td><td>-2.670</td></tr><tr><td>CCSD</td><td>1.432</td><td>2.927</td><td>0.049</td><td>0.550</td><td>0.534</td><td>1.096</td><td>7.894</td><td>0.055</td><td>-2.491</td></tr><tr><td>CCSDT</td><td>1.409</td><td>2.900</td><td>0.048</td><td>0.558</td><td>0.541</td><td>1.084</td><td>7.822</td><td>0.054</td><td>-2.480</td></tr><tr><td>CCSD</td><td>1.440</td><td>2.918</td><td>0.050</td><td>0.555</td><td>0.538</td><td>1.111</td><td>7.949</td><td>0.050</td><td>-2.405</td></tr><tr><td rowspan="5">d-aug-cc-pVQZ</td><td>CCSD</td><td>1.416</td><td>2.890</td><td>0.048</td><td>0.561</td><td>0.544</td><td>1.098</td><td>7.876</td><td>0.050</td><td>-2.406</td></tr><tr><td>CCSDT</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td rowspan="2">aug-cc-pVTZ</td><td>TBE<sup>a</sup></td><td>1.409</td><td>2.901</td><td>0.048</td><td>0.559</td><td>0.542</td><td>1.084</td><td>7.837</td><td>0.055</td><td>-2.501</td></tr><tr><td>TBE<sup>b</sup></td><td>1.42±0.00</td><td>2.88±0.01</td><td>0.048±0.001</td><td>0.57±0.01</td><td>0.55±0.01</td><td>1.11±0.01</td><td>7.91±0.01</td><td>0.046±0.001</td><td>-2.32±0.01</td></tr><tr><td rowspan="2">CBS</td><td>Th.</td><td>1.425<sup>c</sup></td><td>2.944<sup>d</sup></td><td></td><td></td><td></td><td>1.106<sup>f</sup></td><td>7.94<sup>g</sup></td><td>0.081<sup>g</sup></td><td></td></tr><tr><td>Exp.</td><td>1.27±0.21<sup>h</sup></td><td></td><td>0.044<sup>i</sup></td><td></td><td>0.58±0.04<sup>h</sup></td><td></td><td>8.05<sup>j</sup></td><td>0.051<sup>j</sup></td><td></td></tr><tr><td rowspan="2">Lit.</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td>0.042±0.004<sup>k</sup></td><td></td></tr><tr><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td>0.052±0.006<sup>l</sup></td><td></td></tr></tbody></table>

<sup>a</sup>Computed using CCSDTQ/aug-cc-pVTZ values and CCSDTQP/aug-cc-pVDZ corrections. Note that CCSDTQ is equivalent to FCI for BH; <sup>b</sup>See Computational Methods section; <sup>c</sup>Average between the MR-ACPF/aug-cc-pCV7Z(i) values obtained for r = 1.220 and 1.225 Å in Ref. 90;

<sup>d</sup>FCI/aug-cc-pVDZ value of Ref. 54; <sup>e</sup>CC2/aug-cc-pVQZ result from Ref. 45; <sup>f</sup>CCSD(T)/CBS value from Ref. 47; <sup>g</sup>CISDTQ/aug-cc-pCVQZ value from Ref. 91. A *f* value of 0.071 is also reported in Table II of the same work; <sup>h</sup>Stark (emission) measurements of Ref. 92; <sup>i</sup>*f*<sub>00</sub> obtained from laser-induced fluorescence in Ref. 93. There is no major contributions from other bands according to this work (see references therein for previous experimental values). A slightly older experiment (Ref. 94) reports a smaller estimate of 0.045±0.02; <sup>j</sup>Absorption values from Ref. 95 (ΔE corresponds to the maximum of absorption); <sup>k</sup>EELS value from Ref. 96; <sup>l</sup>HR-EELS value from Ref. 97.

intend to provide an exhaustive list of all previous works for each system considered here, but rather to highlight the studies and comparisons that we have found valuable for the present work.

## 3.1 BH and HCl

Let us start by a tiny compound, BH. Our results are listed in Table 1 and although the size of this molecule seems ridiculous (only 4 valence electrons), some valuable conclusions can be obtained. We note that $\mu^{\text{GS}}$ is slightly too large with CCSD (irrespective of the basis set) but the CC convergence is fast and CCSDT is obviously sufficient. Our TBE/CBS of 1.42 D for $\mu^{\text{GS}}$ is equivalent to a recent ultra-accurate estimate,⁹⁰ and also falls within the error bar of the only available experimental value we are aware of: 1.27±0.21 D.⁹² Such large error bar is explainable: the experiment relied on an analysis of the emission spectrum.⁹² It is quite obvious that the theoretical estimate is more trustworthy in this specific case, indicating that previous error analyses based on the 1.27 D value likely significantly exaggerated the ADC(2) and CC2 overestimations,⁴⁵ but underestimated the QMC error⁹⁸ for $\mu^{\text{GS}}$. For the lowest transition energy in BH, our values fit with previous FCI calculations,⁵⁴ and one again notices rather quick convergence of $\Delta E_{\text{vert}}$ with respect to both basis set size and CC order. For the oscillator strength,

there is also an astonishing stability of the values, as the considered ES is well separated from higher-lying ones of the same spatial symmetry. Our TBE is close to the most recent measurement (of $f_{00}$) we could find.⁹³ For $\mu^{\text{ES}}$, one notes a very small decrease of the amplitude between the GS and ES geometries, which is a logical consequence of the tiny geometrical relaxation (+0.018 Å), and our TBE is within the rather small experimental error bar (see bottom of Table 1). Finally, as can be seen in Table S2, these results are not affected by the FC approximation, e.g., the difference between CCSDTQ(FC)/aug-cc-pVTZ and CCSDTQ(Full)/aug-cc-pCVTZ is 0.001 for *f* and 0.002 D for the dipole moments.

For HCl, the GS dipole moment does not cause any specific challenge and our TBE is equivalent to the one recently reported by the Head-Gordon group.⁴⁷ As expected,⁵ CCS-DTQ provides converged $\Delta E_{\text{vert}}$ and this holds for both *f* and $\mu^{\text{ES}}$. When increasing the size of the basis set, one sees a significant decrease of the oscillator strength (ca. -25% from aug-cc-pVDZ to aug-cc-pV5Z) and of the ES dipole (-16% for the same basis pair), whereas the transition energy varies by 1% only. Nevertheless, the CBS extrapolations are stable for all investigated properties. For HCl, it is difficult to obtain a very accurate experimental *f* value, in part due to the mixing of the <sup>1</sup>Π and <sup>3</sup>Π states.⁹⁵ There is therefore a
broad range of measured values (see bottom of Table 1 as well as Table 2 in Ref. 97), and our TBE/CBS of 0.046 is compatible with the two most recent experiments. In contrast, we could not find any experimental $\mu^{\text{ES}}$ estimate, which is a logical consequence of the dissociative character of the lowest singlet ES of HCl.

## 3.2 H₂O and H₂S

The results obtained for water and hydrogen sulfide are listed in Tables 2 and 3, respectively. The CCSDT/aug-cc-pVTZ estimate of the ground-state dipole of water is already within 0.01 D of both previous CASPT2⁹⁹ and experimental values.¹⁰¹ For all transition energies, as discussed in our earlier work,⁵ the calculations are converged with CCSDTQ, but one needs a rather large basis set (especially in terms of diffuse functions) to be chemically accurate, which is quite usual for Rydberg transitions in small compounds. Nevertheless, CCSDT/aug-cc-pVTZ delivers $\Delta E_{\text{vert}}$ with an error of 1–3% only as compared to the most accurate estimates (see bottom of Table 2) for all three transitions. For the lowest $B_1$ excitation, the oscillator strength $f$ varies rather mildly with the selected level of theory and basis set, although one notices a general decreasing trend when improving the method. Our TBE of 0.052±0.001 falls in the error bar of a recent experiment,¹⁰³ and is only slightly larger than the previous most accurate TBE we are aware of.⁴⁸ More exhaustive lists of additional theoretical and experimental values can be found in Table IV of Ref. 103 and Table 6 of Ref. 48 for the lowest transition. The reported $f$ values in these tables are in the range 0.041–0.060. The magnitude of $\mu^{\text{ES}}$ for this $B_1$ ES increases significantly with the basis set size, and a difference of -0.21 D exists between our aug-cc-pVTZ and CBS TBEs. The latter compares well with an earlier CASPT2 value.⁹⁹ This trend is even exacerbated for the $\mu^{\text{ES}}$ value associated with the $A_2$ state that changes by -0.56 D from aug-cc-pVTZ to CBS, whereas no significant changes can be noticed between CCSDT and CCSDTQ. For the lowest $A_1$ ES, there is a significant mixing with a close-lying state of the same symmetry, and the addition of a second set of diffuse is mandatory to obtain reasonable estimates of the oscillator strength (which is much too large with aug-cc-pVTZ). Our TBE/CBS $f$ value of 0.062 could still be slightly too large, but the experimental values range from 0.041 to 0.073 (see Table 3 in Ref. 100). In contrast to what we found for the two lower-lying ES, $\mu^{\text{ES}}$ for the $A_1$ ES is not very sensitive to the basis set size, with a difference of -0.09 D only between the aug-cc-pVTZ and CBS TBEs. We can already conclude, from the data of Table 2, that the basis set required to reach accurate estimates not only differs from one ES to another, but might also be very different, for a given ES, in oscillator strengths and ES dipoles. On a brighter note, the improvements brought by the Q and P excitations are rather limited, meaning that CCSDT seems already sufficient (for water at least).

As can be seen in Table 3, the ground-state dipole moment of hydrogen sulfide remains almost unchanged when increasing the CC order or the size of the basis set: all estimates fall in a tight window: 1.00±0.03 D. Our TBE/CBS of 0.99 D is very close from the experimental value of 0.974±0.005 D, although we use a theoretical geometry. For the lowest $A_2$

transition, the excitation energies are almost converged with CCSDT, but a rather large basis set is required, like in water. Our TBE/CBS of 6.10±0.03 eV agrees with previous high-order estimates.⁵,⁹⁹,¹⁰⁵ To the best of our knowledge there are no accurate experimental estimates for this dark state. While the methodological effects remain firmly under control for the ES dipole, CCSDT being again sufficient, the basis set effects are huge: at the CCSDT level, we have $\mu^{\text{ES}} = 0.13$ D with aug-cc-pVDZ but more than four times larger (0.65 D) with d-aug-cc-pVQZ. Our TBE/CBS of 0.72±0.02 D is slightly larger than the best previous estimate we have found (0.65 D).⁹⁹ The transition energy to the second ES, of $B_1$ symmetry, is also basis set sensitive, although in that case larger bases yield larger (and not smaller) transition energies, as shown in Table 3. Our TBE/CBS of 6.33±0.01 eV is close to previous estimates,⁵,⁹⁹,¹⁰⁵ and also consistent with measurements,¹⁰⁷ although we recall that such comparisons should be made with care. The dipole moment of the $B_1$ state is very large and relatively insensitive to the basis set as compared to its $A_2$ counterpart. Our TBE/CBS of -1.74±0.02 D is very close to an earlier CASPT2 estimate.⁹⁹ Finally, the computed oscillator strength is rather methodologically insensitive and our best estimate of 0.060 is within 0.010 of the two most recent measurements we could found,¹⁰⁸,¹⁰⁹ and compares favorably with two (rather old) theoretical estimates: 0.081¹¹⁰ and 0.075.¹¹¹

## 3.3 BF

For this diatomic, the lowest Π ES behaves rather nicely, and one notices in Table 4 that we could reach very stable estimates for all investigated properties, CCSDT/aug-cc-pVTZ delivering already sufficiently accurate values. It is noteworthy, that for any given basis set, CCSD underestimates $\mu^{\text{GS}}$ ($\mu^{\text{ES}}$) by ca. 4% (25%), highlighting the difficulty posed by ES for “simple” methods. The experimental value of $\mu^{\text{GS}}$ was measured to be 0.5±0.2 D,¹¹² a value that was suggested to be too low more than five scores ago.¹¹³ Indeed, the present TBE/CBS of 0.80 D is significantly above the upper limit of the experimental error bar, but in good agreement with an earlier MRCI+Q value (0.84 D) obtained with a very large basis set.¹¹⁴ Our best estimate for $\Delta E_{\text{vert}}$, 6.39 eV, is slightly larger than a rather old CIPSI value of 6.329 eV,¹¹⁵ whereas unfortunately, Ref. 114 lists adiabatic energies only. For the oscillator strength, the TBE of Table 4 is larger than previous experimental¹¹⁶ and theoretical¹¹³ data, but the uniformity of our estimates gives confidence in their quality. Eventually, we predict a significant drop in polarity when going from the GS to the ES, with a $\mu^{\text{ES}}$ value of ca. 0.27 D. This value is much larger than the MRCI+Q $\mu^{\text{ES}}$ reported by Magoulas and coworkers (0.01 D),¹¹⁴ but the latter is obtained on the ES rather than the GS geometry.

## 3.4 CO

For C=O, we considered six different excited states of various nature, and our results are listed in Table 5. For the three lowest transitions of valence character, one notices that CCSDT is again sufficient, with very small corrections brought by the Q and P excitations, e.g., the Q-induced changes attain ca. ±0.01–0.02 D only for $\mu^{\text{ES}}$. For these three transitions, the basis set effects are also firmly under control for all prop-
broad range of measured values (see bottom of Table 1 as well as Table 2 in Ref. 97), and our TBE/CBS of 0.046 is compatible with the two most recent experiments. In contrast, we could not find any experimental $\mu^{\text{ES}}$ estimate, which is a logical consequence of the dissociative character of the lowest singlet ES of HCl.

### 3.2 H₂O and H₂S

The results obtained for water and hydrogen sulfide are listed in Tables 2 and 3, respectively. The CCSDT/aug-cc-pVTZ estimate of the ground-state dipole of water is already within 0.01 D of both previous CASPT2⁹⁹ and experimental values.¹⁰¹ For all transition energies, as discussed in our earlier work,⁵ the calculations are converged with CCSDTQ, but one needs a rather large basis set (especially in terms of diffuse functions) to be chemically accurate, which is quite usual for Rydberg transitions in small compounds. Nevertheless, CCSDT/aug-cc-pVTZ delivers $\Delta E_{\text{vert}}$ with an error of 1–3% only as compared to the most accurate estimates (see bottom of Table 2) for all three transitions. For the lowest $B_1$ excitation, the oscillator strength $f$ varies rather mildly with the selected level of theory and basis set, although one notices a general decreasing trend when improving the method. Our TBE of 0.052±0.001 falls in the error bar of a recent experiment,¹⁰³ and is only slightly larger than the previous most accurate TBE we are aware of.⁴⁸ More exhaustive lists of additional theoretical and experimental values can be found in Table IV of Ref. 103 and Table 6 of Ref. 48 for the lowest transition. The reported $f$ values in these tables are in the range 0.041–0.060. The magnitude of $\mu^{\text{ES}}$ for this $B_1$ ES increases significantly with the basis set size, and a difference of -0.21 D exists between our aug-cc-pVTZ and CBS TBEs. The latter compares well with an earlier CASPT2 value.⁹⁹ This trend is even exacerbated for the $\mu^{\text{ES}}$ value associated with the $A_2$ state that changes by -0.56 D from aug-cc-pVTZ to CBS, whereas no significant changes can be noticed between CCSDT and CCSDTQ. For the lowest $A_1$ ES, there is a significant mixing with a close-lying state of the same symmetry, and the addition of a second set of diffuse is mandatory to obtain reasonable estimates of the oscillator strength (which is much too large with aug-cc-pVTZ). Our TBE/CBS $f$ value of 0.062 could still be slightly too large, but the experimental values range from 0.041 to 0.073 (see Table 3 in Ref. 100). In contrast to what we found for the two lower-lying ES, $\mu^{\text{ES}}$ for the $A_1$ ES is not very sensitive to the basis set size, with a difference of -0.09 D only between the aug-cc-pVTZ and CBS TBEs. We can already conclude, from the data of Table 2, that the basis set required to reach accurate estimates not only differs from one ES to another, but might also be very different, for a given ES, in oscillator strengths and ES dipoles. On a brighter note, the improvements brought by the Q and P excitations are rather limited, meaning that CCSDT seems already sufficient (for water at least).

As can be seen in Table 3, the ground-state dipole moment of hydrogen sulfide remains almost unchanged when increasing the CC order or the size of the basis set: all estimates fall in a tight window: 1.00±0.03 D. Our TBE/CBS of 0.99 D is very close from the experimental value of 0.974±0.005 D, although we use a theoretical geometry. For the lowest $A_2$

transition, the excitation energies are almost converged with CCSDT, but a rather large basis set is required, like in water. Our TBE/CBS of 6.10±0.03 eV agrees with previous high-order estimates.⁵,⁹⁹,¹⁰⁵ To the best of our knowledge there are no accurate experimental estimates for this dark state. While the methodological effects remain firmly under control for the ES dipole, CCSDT being again sufficient, the basis set effects are huge: at the CCSDT level, we have $\mu^{\text{ES}} = 0.13$ D with aug-cc-pVDZ but more than four times larger (0.65 D) with d-aug-cc-pVQZ. Our TBE/CBS of 0.72±0.02 D is slightly larger than the best previous estimate we have found (0.65 D).⁹⁹ The transition energy to the second ES, of $B_1$ symmetry, is also basis set sensitive, although in that case larger bases yield larger (and not smaller) transition energies, as shown in Table 3. Our TBE/CBS of 6.33±0.01 eV is close to previous estimates,⁵,⁹⁹,¹⁰⁵ and also consistent with measurements,¹⁰⁷ although we recall that such comparisons should be made with care. The dipole moment of the $B_1$ state is very large and relatively insensitive to the basis set as compared to its $A_2$ counterpart. Our TBE/CBS of -1.74±0.02 D is very close to an earlier CASPT2 estimate.⁹⁹ Finally, the computed oscillator strength is rather methodologically insensitive and our best estimate of 0.060 is within 0.010 of the two most recent measurements we could found,¹⁰⁸,¹⁰⁹ and compares favorably with two (rather old) theoretical estimates: 0.081¹¹⁰ and 0.075.¹¹¹

### 3.3 BF

For this diatomic, the lowest Π ES behaves rather nicely, and one notices in Table 4 that we could reach very stable estimates for all investigated properties, CCSDT/aug-cc-pVTZ delivering already sufficiently accurate values. It is noteworthy, that for any given basis set, CCSD underestimates $\mu^{\text{GS}}$ ($\mu^{\text{ES}}$) by ca. 4% (25%), highlighting the difficulty posed by ES for “simple” methods. The experimental value of $\mu^{\text{GS}}$ was measured to be 0.5±0.2 D,¹¹² a value that was suggested to be too low more than five scores ago.¹¹³ Indeed, the present TBE/CBS of 0.80 D is significantly above the upper limit of the experimental error bar, but in good agreement with an earlier MRCI+Q value (0.84 D) obtained with a very large basis set.¹¹⁴ Our best estimate for $\Delta E_{\text{vert}}$, 6.39 eV, is slightly larger than a rather old CIPSI value of 6.329 eV,¹¹⁵ whereas unfortunately, Ref. 114 lists adiabatic energies only. For the oscillator strength, the TBE of Table 4 is larger than previous experimental¹¹⁶ and theoretical¹¹³ data, but the uniformity of our estimates gives confidence in their quality. Eventually, we predict a significant drop in polarity when going from the GS to the ES, with a $\mu^{\text{ES}}$ value of ca. 0.27 D. This value is much larger than the MRCI+Q $\mu^{\text{ES}}$ reported by Magoulas and coworkers (0.01 D),¹¹⁴ but the latter is obtained on the ES rather than the GS geometry.

### 3.4 CO

For C=O, we considered six different excited states of various nature, and our results are listed in Table 5. For the three lowest transitions of valence character, one notices that CCSDT is again sufficient, with very small corrections brought by the Q and P excitations, e.g., the Q-induced changes attain ca. ±0.01–0.02 D only for $\mu^{\text{ES}}$. For these three transitions, the basis set effects are also firmly under control for all prop-
<table><caption>Table 2: Ground-state dipole moment $\mu^{GS}$, vertical transition energies $\Delta E_{\text{vert}}$, oscillator strengths $f$, and excited state dipole moments $\mu_{\text{vert}}^{\text{ES}}$ determined for H$_2$O (GS geometry). See caption of Table 1 for details.</caption><thead><tr><th rowspan="2">Basis</th><th rowspan="2">Method</th><th colspan="2">1<sup>A</sup><sub>1</sub></th><th colspan="2">1<sup>B</sup><sub>1</sub> (Ryd, n → 3s)</th><th colspan="2">1<sup>A</sup><sub>2</sub> (Ryd, n → 3p)</th><th colspan="2">1<sup>A</sup><sub>1</sub> (Ryd, n → 3s)</th></tr><tr><th>μ<sup>GS</sup></th><th>ΔE<sub>vert</sub></th><th>f</th><th>μ<sup>ES</sup><sub>vert</sub></th><th>ΔE<sub>vert</sub></th><th>μ<sup>ES</sup><sub>vert</sub></th><th>ΔE<sub>vert</sub></th><th>f</th><th>μ<sup>ES</sup><sub>vert</sub></th></tr></thead><tbody><tr><td rowspan="6">aug-cc-pVDZ</td><td>CCSD</td><td>1.870</td><td>7.447</td><td>0.057</td><td>-1.404</td><td>9.213</td><td>-0.936</td><td>9.861</td><td>0.103</td><td>-1.095</td></tr><tr><td>CCSDT</td><td>1.849</td><td>7.497</td><td>0.058</td><td>-1.420</td><td>9.279</td><td>-0.978</td><td>9.903</td><td>0.104</td><td>-1.149</td></tr><tr><td>CCSDTQ</td><td>1.848</td><td>7.529</td><td>0.058</td><td>-1.415</td><td>9.313</td><td>-0.974</td><td>9.937</td><td>0.105</td><td>-1.143</td></tr><tr><td>CCSDTQP</td><td>1.848</td><td>7.522</td><td>0.058</td><td>-1.414</td><td>9.318</td><td>-0.972</td><td>9.941</td><td>0.105</td><td>-1.141</td></tr><tr><td>CCSD</td><td>1.864</td><td>7.597</td><td>0.053</td><td>-1.549</td><td>9.361</td><td>-1.056</td><td>9.957</td><td>0.098</td><td>-1.163</td></tr><tr><td>CCSDT</td><td>1.842</td><td>7.591</td><td>0.054</td><td>-1.565</td><td>9.368</td><td>-1.110</td><td>9.949</td><td>0.100</td><td>-1.221</td></tr><tr><td rowspan="6">aug-cc-pVTZ</td><td>CCSD</td><td>1.840</td><td>7.600</td><td>0.052</td><td>-1.655</td><td>9.422</td><td>-1.237</td><td>10.004</td><td>0.095</td><td>-1.226</td></tr><tr><td>CCSDT</td><td>1.873</td><td>7.660</td><td>0.052</td><td>-1.655</td><td>9.422</td><td>-1.237</td><td>10.004</td><td>0.095</td><td>-1.226</td></tr><tr><td>CCSDTQ</td><td>1.850</td><td>7.637</td><td>0.053</td><td>-1.667</td><td>9.410</td><td>-1.294</td><td>9.980</td><td>0.097</td><td>-1.286</td></tr><tr><td>CCSD</td><td>1.876</td><td>7.683</td><td>0.051</td><td>-1.726</td><td>9.444</td><td>-1.309</td><td>10.010</td><td>0.089</td><td>-1.229</td></tr><tr><td>CCSDT</td><td>1.840</td><td>7.614</td><td>0.054</td><td>-1.558</td><td>9.405</td><td>-1.106</td><td>9.985</td><td>0.100</td><td>-1.213</td></tr><tr><td>CCSDTQP</td><td>1.86±0.01</td><td>7.71±0.02</td><td>0.052±0.001</td><td>-1.77±0.04</td><td>9.49±0.02</td><td>-1.67±0.01</td><td>9.99±0.01</td><td>0.062±0.002</td><td>-1.30±0.03</td></tr><tr><td rowspan="4">d-aug-cc-pVDZ</td><td>CCSD</td><td>1.861</td><td>7.429</td><td>0.052</td><td>-1.759</td><td>9.179</td><td>-1.658</td><td>9.731</td><td>0.050</td><td>-0.884</td></tr><tr><td>CCSDT</td><td>1.841</td><td>7.479</td><td>0.053</td><td>-1.754</td><td>9.244</td><td>-1.703</td><td>9.792</td><td>0.057</td><td>-1.021</td></tr><tr><td>CCSDTQ</td><td>1.865</td><td>7.592</td><td>0.051</td><td>-1.764</td><td>9.348</td><td>-1.626</td><td>9.869</td><td>0.057</td><td>-1.154</td></tr><tr><td>CCSDTQP</td><td>1.843</td><td>7.586</td><td>0.052</td><td>-1.767</td><td>9.353</td><td>-1.681</td><td>9.872</td><td>0.062</td><td>-1.232</td></tr><tr><td rowspan="6">aug-cc-pVQZ</td><td>CCSD</td><td>1.874</td><td>7.659</td><td>0.051</td><td>-1.765</td><td>9.416</td><td>-1.622</td><td>9.932</td><td>0.058</td><td>-1.228</td></tr><tr><td>CCSDT</td><td>1.851</td><td>7.636</td><td>0.052</td><td>-1.770</td><td>9.403</td><td>-1.678</td><td>9.917</td><td>0.063</td><td>-1.295</td></tr><tr><td>CCSDTQP</td><td>1.840</td><td>7.614</td><td>0.054</td><td>-1.558</td><td>9.405</td><td>-1.106</td><td>9.985</td><td>0.100</td><td>-1.213</td></tr><tr><td>CCSDTQP</td><td>1.86±0.01</td><td>7.71±0.02</td><td>0.052±0.001</td><td>-1.77±0.04</td><td>9.49±0.02</td><td>-1.67±0.01</td><td>9.99±0.01</td><td>0.062±0.002</td><td>-1.30±0.03</td></tr><tr><td>CCSDT</td><td>1.853<sup>c</sup></td><td>7.66<sup>c</sup></td><td>0.054<sup>d</sup></td><td>-1.787<sup>c</sup></td><td>9.42<sup>c</sup></td><td>-1.682<sup>c</sup></td><td>9.97<sup>e</sup></td><td>0.100<sup>d</sup></td><td></td></tr><tr><td>CCSDTQP</td><td>1.86±0.01</td><td>7.70<sup>e</sup></td><td>0.049<sup>f</sup></td><td>-1.787<sup>c</sup></td><td>9.47<sup>e</sup></td><td>-1.682<sup>c</sup></td><td>9.97<sup>e</sup></td><td>0.100<sup>d</sup></td><td></td></tr><tr><td rowspan="4">aug-cc-pVTZ</td><td>TBE<sup>a</sup></td><td>1.853<sup>c</sup></td><td>7.66<sup>c</sup></td><td>0.054<sup>d</sup></td><td>-1.787<sup>c</sup></td><td>9.42<sup>c</sup></td><td>-1.682<sup>c</sup></td><td>9.97<sup>e</sup></td><td>0.100<sup>d</sup></td><td></td></tr><tr><td>TBE<sup>b</sup></td><td>1.853<sup>c</sup></td><td>7.66<sup>c</sup></td><td>0.054<sup>d</sup></td><td>-1.787<sup>c</sup></td><td>9.42<sup>c</sup></td><td>-1.682<sup>c</sup></td><td>9.97<sup>e</sup></td><td>0.100<sup>d</sup></td><td></td></tr><tr><td>Lit.</td><td>Th.</td><td>1.853<sup>c</sup></td><td>7.66<sup>c</sup></td><td>0.054<sup>d</sup></td><td>-1.787<sup>c</sup></td><td>9.42<sup>c</sup></td><td>-1.682<sup>c</sup></td><td>9.97<sup>e</sup></td><td>0.100<sup>d</sup></td><td></td></tr><tr><td>Exp.</td><td>Th.</td><td>1.853<sup>c</sup></td><td>7.66<sup>c</sup></td><td>0.054<sup>d</sup></td><td>-1.787<sup>c</sup></td><td>9.42<sup>c</sup></td><td>-1.682<sup>c</sup></td><td>9.97<sup>e</sup></td><td>0.100<sup>d</sup></td><td></td></tr></tbody></table>

a,b See corresponding footnotes in Table 1; c CASPT2/d-aug-cc-pVQZ values from Ref. 99; d LR-CC3/aug-cc-pVTZ value from Ref. 5; e Basis set corrected exFCI/aug-cc-pVQZ values from Ref. 5; f 2FVCAS/MR-CI/CBS values (f in length gauge) from Ref. 48; g 2FVCAS/MR-CI/d-aug-cc-pV5Z values (f in length gauge) from Ref. 100; h Average of the three (very close) experimental values reported in Table 1 of Ref. 101; i Energy loss experiment from Ref. 102; j Electron impact from Ref. 103; k Electron impact from Ref. 104 integrated by Borges in Ref. 100.

<table><caption>Table 3: Ground-state dipole moment $\mu^{GS}$, vertical transition energies $\Delta E_{\text{vert}}$, oscillator strengths $f$, and excited state dipole moments $\mu_{\text{vert}}^{\text{ES}}$ determined for H$_2$S (GS geometry). See caption of Table 1 for details.</caption><thead><tr><th rowspan="2">Basis</th><th rowspan="2">Method</th><th colspan="2">1<sup>A</sup><sub>1</sub></th><th colspan="2">1<sup>A</sup><sub>2</sub> (Ryd, n → 4p)</th><th colspan="2">1<sup>B</sup><sub>1</sub> (Ryd, n → 4s)</th></tr><tr><th>μ<sup>GS</sup></th><th>ΔE<sub>vert</sub></th><th>μ<sup>ES</sup></th><th>ΔE<sub>vert</sub></th><th>f</th><th>μ<sup>ES</sup></th></tr></thead><tbody><tr><td rowspan="6">aug-cc-pVDZ</td><td>CCSD</td><td>1.031</td><td>6.343</td><td>0.113</td><td>6.141</td><td>0.068</td><td>-1.983</td></tr><tr><td>CCSDT</td><td>1.016</td><td>6.286</td><td>0.131</td><td>6.098</td><td>0.067</td><td>-1.946</td></tr><tr><td>CCSDTQ</td><td>1.015</td><td>6.286</td><td>0.137</td><td>6.103</td><td>0.067</td><td>-1.934</td></tr><tr><td>CCSDTQP</td><td>1.015</td><td>6.286</td><td>0.137</td><td>6.103</td><td>0.067</td><td>-1.933</td></tr><tr><td>CCSD</td><td>0.990</td><td>6.246</td><td>0.503</td><td>6.295</td><td>0.064</td><td>-1.893</td></tr><tr><td>CCSDT</td><td>0.978</td><td>6.185</td><td>0.496</td><td>6.237</td><td>0.063</td><td>-1.875</td></tr><tr><td rowspan="6">aug-cc-pVTZ</td><td>CCSD</td><td>0.977</td><td>6.181</td><td>0.498</td><td>6.238</td><td>0.063</td><td>-1.866</td></tr><tr><td>CCSDT</td><td>1.001</td><td>6.212</td><td>0.650</td><td>6.349</td><td>0.062</td><td>-1.822</td></tr><tr><td>CCSDTQ</td><td>0.990</td><td>6.153</td><td>0.636</td><td>6.288</td><td>0.061</td><td>-1.815</td></tr><tr><td>CCSD</td><td>0.998</td><td>6.177</td><td>0.691</td><td>6.368</td><td>0.061</td><td>-1.794</td></tr><tr><td>CCSDT</td><td>1.017</td><td>6.297</td><td>0.445</td><td>6.130</td><td>0.065</td><td>-1.811</td></tr><tr><td>CCSDTQP</td><td>1.002</td><td>6.241</td><td>0.458</td><td>6.086</td><td>0.065</td><td>-1.776</td></tr><tr><td rowspan="6">d-aug-cc-pVDZ</td><td>CCSD</td><td>0.989</td><td>6.228</td><td>0.597</td><td>6.292</td><td>0.062</td><td>-1.774</td></tr><tr><td>CCSDT</td><td>0.977</td><td>6.167</td><td>0.587</td><td>6.234</td><td>0.062</td><td>-1.761</td></tr><tr><td>CCSDTQP</td><td>1.001</td><td>6.206</td><td>0.668</td><td>6.347</td><td>0.061</td><td>-1.758</td></tr><tr><td>CCSD</td><td>0.989</td><td>6.147</td><td>0.653</td><td>6.286</td><td>0.061</td><td>-1.755</td></tr><tr><td>CCSDTQP</td><td>0.977</td><td>6.181</td><td>0.498</td><td>6.238</td><td>0.063</td><td>-1.865</td></tr><tr><td>CCSDT</td><td>0.99±0.01</td><td>6.10±0.03</td><td>0.72±0.01</td><td>6.33±0.01</td><td>0.060±0.001</td><td>-1.74±0.02</td></tr><tr><td rowspan="4">aug-cc-pVTZ</td><td>TBE<sup>a</sup></td><td>0.989<sup>c</sup></td><td>6.12<sup>c</sup>;6.10<sup>d</sup>;6.10<sup>e</sup></td><td>0.653<sup>c</sup></td><td>6.27<sup>c</sup>;6.29<sup>d</sup>;6.33<sup>e</sup></td><td>0.063<sup>f</sup></td><td>-1.733<sup>c</sup></td></tr><tr><td>TBE<sup>b</sup></td><td>0.974±0.005<sup>g</sup></td><td></td><td></td><td>6.326<sup>h</sup></td><td>0.0542<sup>i</sup>;0.0547<sup>j</sup></td><td></td></tr><tr><td>Lit.</td><td>Th.</td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>Exp.</td><td>Th.</td><td></td><td></td><td></td><td></td><td></td></tr></tbody></table>

a,b See corresponding footnotes in Table 1; c CASPT2/d-aug-cc-pVQZ values from Ref. 99; d Basis set corrected exFCI/aug-cc-pVQZ values from Ref. 5; e SS-RASPT2/ANO-RCC-VTZP+diffuse values from Ref. 105; f LR-CC3/aug-cc-pVTZ value from Ref. 5; g From Ref. 106; h From Ref. 107 (see Table 9 of this work); i From Ref. 108, obtained by integrating the experimental absorption spectrum in the 5.2–7.7 eV region; j From Ref. 109, obtained by integrating the experimental absorption spectrum in the 5.2–7.7 eV region.
<table><caption>Table 4: Ground-state dipole moment $\mu^{\text{GS}}$, vertical transition energies $\Delta E_{\text{vert}}$, oscillator strengths $f$, and excited state dipole moments $\mu^{\text{ES}}$ determined for BF (GS geometry). See caption of Table 1 for details.</caption><thead><tr><th rowspan="2">Basis</th><th rowspan="2">Method</th><th colspan="2"><sup>1</sup>A<sub>1</sub></th><th colspan="2"><sup>1</sup>Π(Val, σ → π*)</th></tr><tr><th>μ<sup>GS</sup></th><th>ΔE<sub>vert</sub></th><th>f</th><th>μ<sup>ES</sup></th></tr></thead><tbody><tr><td rowspan="4">aug-cc-pVDZ</td><td>CCSD</td><td>0.832</td><td>6.534</td><td>0.479</td><td>0.240</td></tr><tr><td>CCSDT</td><td>0.861</td><td>6.491</td><td>0.475</td><td>0.311</td></tr><tr><td>CCSDTQ</td><td>0.861</td><td>6.486</td><td>0.474</td><td>0.316</td></tr><tr><td>CCSDTQP</td><td>0.860</td><td>6.485</td><td>0.474</td><td>0.316</td></tr><tr><td rowspan="4">aug-cc-pVTZ</td><td>CCSD</td><td>0.794</td><td>6.464</td><td>0.475</td><td>0.222</td></tr><tr><td>CCSDT</td><td>0.824</td><td>6.423</td><td>0.469</td><td>0.293</td></tr><tr><td>CCSDTQ</td><td>0.824</td><td>6.417</td><td>0.468</td><td>0.300</td></tr><tr><td>CCSDTQ</td><td>0.812</td><td>6.411</td><td>0.468</td><td>0.279</td></tr><tr><td>aug-cc-pVQZ</td><td>CCSD</td><td>0.783</td><td>6.449</td><td>0.475</td><td>0.207</td></tr><tr><td rowspan="2">aug-cc-pV5Z</td><td>CCSD</td><td>0.782</td><td>6.443</td><td>0.475</td><td>0.202</td></tr><tr><td>CCSD</td><td>0.822</td><td>6.521</td><td>0.478</td><td>0.262</td></tr><tr><td>d-aug-cc-pVDZ</td><td>CCSD</td><td>0.850</td><td>6.477</td><td>0.474</td><td>0.330</td></tr><tr><td>d-aug-cc-pVTZ</td><td>CCSD</td><td>0.794</td><td>6.459</td><td>0.476</td><td>0.237</td></tr><tr><td>d-aug-cc-pVQZ</td><td>CCSD</td><td>0.824</td><td>6.419</td><td>0.469</td><td>0.308</td></tr><tr><td rowspan="2">aug-cc-pVTZ</td><td>CCSD</td><td>0.784</td><td>6.448</td><td>0.475</td><td>0.213</td></tr><tr><td>CCSDT</td><td>0.813</td><td>6.409</td><td>0.468</td><td>0.285</td></tr><tr><td rowspan="2">aug-cc-pVTZ</td><td>TBE<sup>a</sup></td><td>0.824</td><td>6.417</td><td>0.468</td><td>0.299</td></tr><tr><td>TBE<sup>b</sup></td><td>0.80±0.01</td><td>6.39±0.01</td><td>0.467±0.001</td><td>0.27±0.01</td></tr><tr><td rowspan="2">CBS</td><td>Th.</td><td>0.84<sup>c</sup></td><td>6.329<sup>d</sup></td><td>0.30<sup>e</sup></td><td>0.01<sup>c</sup></td></tr><tr><td>Exp.</td><td>0.5±0.2<sup>f</sup></td><td></td><td>0.40<sup>g</sup></td><td></td></tr><tr><td>Lit.</td><td></td><td></td><td></td><td></td><td></td></tr></tbody></table>

a,b See corresponding footnotes in Table 1; c MRCI+Q/aug-cc-pV6Z values from Ref. 114, the ES dipole was obtained on the corresponding ES geometry; d CIPSI/(11s7p4d3f)/[6s4p4d3f] estimate from Ref. 115; e Sum of the MRCI *f* value determined for GS *v''* = 0, from Table 5 of Ref. 113; *f* From Ref. 112; *g* *f*<sub>00</sub> value from Ref. 116.

erties, and, both *f* and $\mu^{\text{ES}}$ are not significantly affected by the second set of diffuse orbitals and *aug-cc-pVQZ* is likely sufficiently large to obtain accurate estimates. As a consequence, it is rather straightforward to get stable CBS extrapolations. The moderate impact brought by the quadruples seems to pertain for the three higher-lying Rydberg ESs of carbon monoxide, except for the dipole moment of the second $\Sigma^+$ transition for which $\mu^{\text{ES}}$ increases by +0.111 D when going from CCSDT/aug-cc-pVTZ to CCSDTQ/aug-cc-pVTZ. As expected, much larger basis sets are needed to obtain converged properties for the Rydberg ESs than for their valence counterparts. Indeed, when going from *aug-cc-pVTZ* to *d-aug-cc-pVQZ*, the CCSDT ES dipoles of the three lowest Rydberg states drastically change from -3.89 to -1.81 D, from +6.09 to +4.34 D, and from +0.14 to -1.44 D, respectively. In the same time, the CCSDT value of *f* for the two significantly dipole-allowed transitions decrease by -36% when considering the same basis set pair. Obviously such behavior makes the error bar obtained for our TBE/CBS estimates non-negligible (see below). In contrast, and as already noted above, the transition energies of these Rydberg ESs can be effectively estimated with high accuracy.

When comparing with previously reported experimental or theoretical estimates, one should keep in mind that the properties of carbon monoxide are strongly affected by the bond length,¹¹⁸ and that we used a theoretical geometry with a slightly too elongated double bond (2.142 bohr versus 2.132 bohr experimentally). Our TBE/CBS for $\mu^{\text{GS}}$ is 0.10 D, which

is slightly too low as compared to both the experimental measurements¹¹⁶ and the theoretical calculations performed at the experimental geometry.⁴⁷,¹²³ For the lowest and well-studied Π ES, the present 8.46±0.01 eV $\Delta E_{\text{vert}}$ value is close to our recent FCI estimate (8.48 eV)⁵ as well as to the experimental value (8.51 eV) deduced with the help of the measured spectroscopic constants and the reconstruction of the potential energy surfaces.¹²⁴ The computed value of *f* = 0.165 is in very reasonable agreement with the estimates obtained by dipole (e,e) and (γ,γ) spectroscopies: 0.181¹¹⁹ and 0.194,¹²¹ respectively. We refer the interested readers to Tables 3 and 4 of Ref. 119, Table 6 of Ref. 125, and Table II of Ref. 126 for more complete lists of experimental and theoretical estimates of the oscillator strength. There is a quite significant elongation of the double bond in the lowest ES. As a consequence the value of $\mu^{\text{ES}}$ determined on the GS (-0.19±0.01 D) and ES (-0.39±0.01 D) structures do differ significantly. The two Stark effect measurements we are aware of yield $\mu^{\text{ES}}$ = -0.15 ± 0.05 D (analysis of the 0-0 band)¹²⁰ and $\mu^{\text{ES}}$ = -0.34 ± 0.01 D (two-photon LIF),¹²² which are also somehow inconsistent, yet of the same order of magnitude as our GS and ES values, respectively. For the two other valence transitions, of $\Sigma^-$ and $\Delta$ symmetries, the TBE/CBS for $\Delta E_{\text{vert}}$ show small errors (1–2 %) as compared to the “experimental” values.⁴⁹ We could not find any experimental estimate of the ES dipoles for these states, likely because they are dark, and we believe that the data listed in Table 5 stand as the most accurate ES dipole values proposed to date. They indicate that the dipole moments of these two ESs are parallel to the one of the GS but have much larger amplitudes. For the three Rydberg transitions considered, our best estimates of the transition energies are fully compatible with the measurements. The lowest Rydberg ES has a very low oscillator strength, a result consistent with previous theoretical⁵,⁴⁹ and experimental¹¹⁹ investigations. More interestingly, its dipole moment is large and negative. The two available measurements return dipoles of -1.60±0.15¹²⁰ and -1.95±0.03 D,¹²² whereas the strong basis set effects bring uncertainty to our TBE/CBS (-2.49±0.59 D). The lower bound (-1.90 D) fits the latest experimental value quite well. We have also computed the ES dipole of this state at its equilibrium geometry, but the changes are very limited (see Table S4 in the SI). The second Rydberg $\Sigma^+$ transition is much more dipole allowed than the first, and our TBE/CBS of 0.146±0.016 for the oscillator strength is consistent with the most recent measurement (0.136).¹²¹ The dipole moment of this ES is large, positive, and rather unaffected by structural relaxation effects (Table S4 in the SI), our extrapolated value of 4.94±0.47 D agreeing well with the two experimental estimates of 4.52±0.35¹²⁰ and 4.50±0.07 D.¹²² Eventually, for the highest ES of CO considered herein, the extrapolated *f* value perfectly fits the measurements.¹¹⁹,¹²¹ We provide, as far as we know, the first estimate of its dipole moment, which is rather small and negative (see bottom of Table 5).

### 3.5 N₂

The understanding of the nature of the ES in the highly symmetric N₂ molecule is certainly challenging from both an experimental and a theoretical point of view.¹²⁷,¹²⁸,¹³⁰ Of particular relevance for the present work is the nature and
<table><caption>Table 4: Ground-state dipole moment $\mu^{\text{GS}}$, vertical transition energies $\Delta E_{\text{vert}}$, oscillator strengths $f$, and excited state dipole moments $\mu^{\text{ES}}$ determined for BF (GS geometry). See caption of Table 1 for details.</caption><thead><tr><th rowspan="2">Basis</th><th rowspan="2">Method</th><th colspan="2"><sup>1</sup>A<sub>1</sub></th><th colspan="2"><sup>1</sup>Π(Val, σ → π*)</th></tr><tr><th>μ<sup>GS</sup></th><th>ΔE<sub>vert</sub></th><th>f</th><th>μ<sup>ES</sup></th></tr></thead><tbody><tr><td rowspan="4">aug-cc-pVDZ</td><td>CCSD</td><td>0.832</td><td>6.534</td><td>0.479</td><td>0.240</td></tr><tr><td>CCSDT</td><td>0.861</td><td>6.491</td><td>0.475</td><td>0.311</td></tr><tr><td>CCSDTQ</td><td>0.861</td><td>6.486</td><td>0.474</td><td>0.316</td></tr><tr><td>CCSDTQP</td><td>0.860</td><td>6.485</td><td>0.474</td><td>0.316</td></tr><tr><td rowspan="4">aug-cc-pVTZ</td><td>CCSD</td><td>0.794</td><td>6.464</td><td>0.475</td><td>0.222</td></tr><tr><td>CCSDT</td><td>0.824</td><td>6.423</td><td>0.469</td><td>0.293</td></tr><tr><td>CCSDTQ</td><td>0.824</td><td>6.417</td><td>0.468</td><td>0.300</td></tr><tr><td>CCSDTQ</td><td>0.812</td><td>6.411</td><td>0.468</td><td>0.279</td></tr><tr><td>aug-cc-pVQZ</td><td>CCSD</td><td>0.783</td><td>6.449</td><td>0.475</td><td>0.207</td></tr><tr><td rowspan="2">aug-cc-pV5Z</td><td>CCSD</td><td>0.782</td><td>6.443</td><td>0.475</td><td>0.202</td></tr><tr><td>CCSD</td><td>0.822</td><td>6.521</td><td>0.478</td><td>0.262</td></tr><tr><td rowspan="2">d-aug-cc-pVDZ</td><td>CCSD</td><td>0.850</td><td>6.477</td><td>0.474</td><td>0.330</td></tr><tr><td>CCSDT</td><td>0.794</td><td>6.459</td><td>0.476</td><td>0.237</td></tr><tr><td rowspan="2">d-aug-cc-pVTZ</td><td>CCSD</td><td>0.824</td><td>6.419</td><td>0.469</td><td>0.308</td></tr><tr><td>CCSDT</td><td>0.784</td><td>6.448</td><td>0.475</td><td>0.213</td></tr><tr><td rowspan="2">d-aug-cc-pVQZ</td><td>CCSD</td><td>0.813</td><td>6.409</td><td>0.468</td><td>0.285</td></tr><tr><td>CCSDT</td><td>0.824</td><td>6.417</td><td>0.468</td><td>0.299</td></tr><tr><td rowspan="2">aug-cc-pVTZ</td><td>TBE<sup>a</sup></td><td>0.80±0.01</td><td>6.39±0.01</td><td>0.467±0.001</td><td>0.27±0.01</td></tr><tr><td>TBE<sup>b</sup></td><td>0.84<sup>c</sup></td><td>6.329<sup>d</sup></td><td>0.30<sup>e</sup></td><td>0.01<sup>c</sup></td></tr><tr><td>Lit.</td><td>Th.</td><td>0.5±0.2<sup>f</sup></td><td></td><td>0.40<sup>g</sup></td><td></td></tr><tr><td></td><td>Exp.</td><td></td><td></td><td></td><td></td></tr></tbody></table>

<sup>a,b</sup>See corresponding footnotes in Table 1; <sup>c</sup>MRCI+Q/aug-cc-pV6Z values from Ref. 114, the ES dipole was obtained on the corresponding ES geometry; <sup>d</sup>CIPSI/(11s7p4d3f)/[6s4p4d3f] estimate from Ref. 115; <sup>e</sup>Sum of the MRCI *f* value determined for GS *ν''* = 0, from Table 5 of Ref. 113; <sup>f</sup>From Ref. 112; <sup>g</sup>*f*<sub>00</sub> value from Ref. 116.

erties, and, both *f* and $\mu^{\text{ES}}$ are not significantly affected by the second set of diffuse orbitals and *aug-cc-pVQZ* is likely sufficiently large to obtain accurate estimates. As a consequence, it is rather straightforward to get stable CBS extrapolations. The moderate impact brought by the quadruples seems to pertain for the three higher-lying Rydberg ESs of carbon monoxide, except for the dipole moment of the second $\Sigma^+$ transition, for which $\mu^{\text{ES}}$ increases by +0.111 D when going from CCSDT/aug-cc-pVTZ to CCSDTQ/aug-cc-pVTZ. As expected, much larger basis sets are needed to obtain converged properties for the Rydberg ESs than for their valence counterparts. Indeed, when going from *aug-cc-pVTZ* to *d-aug-cc-pVQZ*, the CCSDT ES dipoles of the three lowest Rydberg states drastically change from -3.89 to -1.81 D, from +6.09 to +4.34 D, and from +0.14 to -1.44 D, respectively. In the same time, the CCSDT value of *f* for the two significantly dipole-allowed transitions decrease by -36% when considering the same basis set pair. Obviously such behavior makes the error bar obtained for our TBE/CBS estimates non-negligible (see below). In contrast, and as already noted above, the transition energies of these Rydberg ESs can be effectively estimated with high accuracy.

When comparing with previously reported experimental or theoretical estimates, one should keep in mind that the properties of carbon monoxide are strongly affected by the bond length,<sup>118</sup> and that we used a theoretical geometry with a slightly too elongated double bond (2.142 bohr versus 2.132 bohr experimentally). Our TBE/CBS for $\mu^{\text{GS}}$ is 0.10 D, which

is slightly too low as compared to both the experimental measurements<sup>116</sup> and the theoretical calculations performed at the experimental geometry.<sup>47,123</sup> For the lowest and well-studied $\Pi$ ES, the present 8.46±0.01 eV $\Delta E_{\text{vert}}$ value is close to our recent FCI estimate (8.48 eV)<sup>5</sup> as well as to the experimental value (8.51 eV) deduced with the help of the measured spectroscopic constants and the reconstruction of the potential energy surfaces.<sup>124</sup> The computed value of *f* = 0.165 is in very reasonable agreement with the estimates obtained by dipole (e,e) and (γ,γ) spectroscopies: 0.181<sup>119</sup> and 0.194,<sup>121</sup> respectively. We refer the interested readers to Tables 3 and 4 of Ref. 119, Table 6 of Ref. 125, and Table II of Ref. 126 for more complete lists of experimental and theoretical estimates of the oscillator strength. There is a quite significant elongation of the double bond in the lowest ES. As a consequence the value of $\mu^{\text{ES}}$ determined on the GS (-0.19±0.01 D) and ES (-0.39±0.01 D) structures do differ significantly. The two Stark effect measurements we are aware of yield $\mu^{\text{ES}} = -0.15 \pm 0.05$ D (analysis of the 0-0 band)<sup>120</sup> and $\mu^{\text{ES}} = -0.34 \pm 0.01$ D (two-photon LIF),<sup>122</sup> which are also somehow inconsistent, yet of the same order of magnitude as our GS and ES values, respectively. For the two other valence transitions, of $\Sigma^-$ and $\Delta$ symmetries, the TBE/CBS for $\Delta E_{\text{vert}}$ show small errors (1–2 %) as compared to the “experimental” values.<sup>49</sup> We could not find any experimental estimate of the ES dipoles for these states, likely because they are dark, and we believe that the data listed in Table 5 stand as the most accurate ES dipole values proposed to date. They indicate that the dipole moments of these two ESs are parallel to the one of the GS but have much larger amplitudes. For the three Rydberg transitions considered, our best estimates of the transition energies are fully compatible with the measurements. The lowest Rydberg ES has a very low oscillator strength, a result consistent with previous theoretical<sup>5,49</sup> and experimental<sup>119</sup> investigations. More interestingly, its dipole moment is large and negative. The two available measurements return dipoles of -1.60±0.15<sup>120</sup> and -1.95±0.03 D,<sup>122</sup> whereas the strong basis set effects bring uncertainty to our TBE/CBS (-2.49±0.59 D). The lower bound (-1.90 D) fits the latest experimental value quite well. We have also computed the ES dipole of this state at its equilibrium geometry, but the changes are very limited (see Table S4 in the SI). The second Rydberg $\Sigma^+$ transition is much more dipole allowed than the first, and our TBE/CBS of 0.146±0.016 for the oscillator strength is consistent with the most recent measurement (0.136).<sup>121</sup> The dipole moment of this ES is large, positive, and rather unaffected by structural relaxation effects (Table S4 in the SI), our extrapolated value of 4.94±0.47 D agreeing well with the two experimental estimates of 4.52±0.35<sup>120</sup> and 4.50±0.07 D.<sup>122</sup> Eventually, for the highest ES of CO considered herein, the extrapolated *f* value perfectly fits the measurements.<sup>119,121</sup> We provide, as far as we know, the first estimate of its dipole moment, which is rather small and negative (see bottom of Table 5).

### 3.5 N<sub>2</sub>

The understanding of the nature of the ES in the highly symmetric N<sub>2</sub> molecule is certainly challenging from both an experimental and a theoretical point of view.<sup>127,128,130</sup> Of particular relevance for the present work is the nature and
<table><caption>Table 5: Ground-state dipole moment $\mu^{GS}$, vertical transition energies $\Delta E_{\text{vert}}$, oscillator strengths $f$, and excited state dipole moments, $\mu_{\text{vert}}^{\text{ES}}$ and $\mu_{\text{adia}}^{\text{ES}}$, determined for various transitions in CO at its GS geometry. The transition energies are in eV and the dipoles in D.</caption><thead><tr><th rowspan="2">Basis</th><th rowspan="2">Method</th><th colspan="2">1Σ<sup>+</sup></th><th colspan="3">1Π(Val, n → π<sup>*</sup>)</th><th colspan="2">1Σ<sup>-</sup>(Val, π → π<sup>*</sup>)</th><th colspan="2">1Δ(Val, π → π<sup>*</sup>)</th></tr><tr><th>μ<sup>GS</sup></th><th>ΔE<sub>vert</sub></th><th>f</th><th>μ<sup>ES</sup><sub>vert</sub></th><th>μ<sup>ES</sup><sub>adia</sub></th><th>ΔE<sub>vert</sub></th><th>μ<sup>ES</sup><sub>vert</sub></th><th>ΔE<sub>vert</sub></th><th>μ<sup>ES</sup><sub>vert</sub></th></tr></thead><tbody><tr><td rowspan="4">aug-cc-pVDZ</td><td>CCSD</td><td>0.078</td><td>8.671</td><td>0.167</td><td>-0.157</td><td>-0.438</td><td>10.096</td><td>1.574</td><td>10.210</td><td>1.405</td></tr><tr><td>CCSDT</td><td>0.121</td><td>8.574</td><td>0.173</td><td>-0.079</td><td>-0.331</td><td>10.062</td><td>1.597</td><td>10.178</td><td>1.432</td></tr><tr><td>CCSDTQ</td><td>0.130</td><td>8.563</td><td>0.174</td><td>-0.072</td><td>-0.317</td><td>10.057</td><td>1.613</td><td>10.169</td><td>1.446</td></tr><tr><td>CCSDTQP</td><td>0.132</td><td>8.561</td><td>0.175</td><td>-0.069</td><td>-0.312</td><td>10.057</td><td>1.615</td><td>10.168</td><td>1.448</td></tr><tr><td rowspan="4">aug-cc-pVTZ</td><td>CCSD</td><td>0.051</td><td>8.587</td><td>0.161</td><td>-0.227</td><td>-0.478</td><td>9.986</td><td>1.577</td><td>10.123</td><td>1.413</td></tr><tr><td>CCSDT</td><td>0.104</td><td>8.492</td><td>0.164</td><td>-0.137</td><td>-0.358</td><td>9.940</td><td>1.595</td><td>10.076</td><td>1.432</td></tr><tr><td>CCSDTQ</td><td>0.113</td><td>8.480</td><td>0.166</td><td>-0.129</td><td>-0.344</td><td>9.932</td><td>1.612</td><td>10.066</td><td>1.449</td></tr><tr><td>CCSDTQP</td><td>0.094</td><td>8.480</td><td>0.163</td><td>-0.169</td><td>-0.385</td><td>9.940</td><td>1.586</td><td>10.073</td><td>1.421</td></tr><tr><td rowspan="2">aug-cc-pVQZ</td><td>CCSD</td><td>0.039</td><td>8.574</td><td>0.160</td><td>-0.264</td><td>-0.511</td><td>9.992</td><td>1.567</td><td>10.127</td><td>1.402</td></tr><tr><td>CCSDT</td><td>0.094</td><td>8.480</td><td>0.163</td><td>-0.199</td><td>-0.339</td><td>9.932</td><td>1.614</td><td>10.065</td><td>1.450</td></tr><tr><td rowspan="2">aug-cc-pV5Z</td><td>CCSD</td><td>0.037</td><td>8.571</td><td>0.160</td><td>-0.280</td><td>-0.523</td><td>9.932</td><td>1.614</td><td>10.065</td><td>1.450</td></tr><tr><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td rowspan="2">d-aug-cc-pVDZ</td><td>CCSD</td><td>0.085</td><td>8.663</td><td>0.167</td><td>-0.149</td><td>-0.442</td><td>10.087</td><td>1.561</td><td>10.199</td><td>1.386</td></tr><tr><td>CCSDT</td><td>0.128</td><td>8.565</td><td>0.173</td><td>-0.069</td><td>-0.334</td><td>10.053</td><td>1.584</td><td>10.167</td><td>1.413</td></tr><tr><td rowspan="2">d-aug-cc-pVTZ</td><td>CCSD</td><td>0.053</td><td>8.582</td><td>0.160</td><td>-0.232</td><td>-0.483</td><td>9.983</td><td>1.568</td><td>10.120</td><td>1.401</td></tr><tr><td>CCSDT</td><td>0.106</td><td>8.487</td><td>0.164</td><td>-0.141</td><td>-0.364</td><td>9.937</td><td>1.586</td><td>10.073</td><td>1.421</td></tr><tr><td rowspan="2">d-aug-cc-pVQZ</td><td>CCSD</td><td>0.040</td><td>8.572</td><td>0.160</td><td>-0.269</td><td>-0.513</td><td>9.992</td><td>1.563</td><td>10.126</td><td>1.397</td></tr><tr><td>CCSDT</td><td>0.094</td><td>8.478</td><td>0.163</td><td>-0.173</td><td>-0.389</td><td>9.940</td><td>1.581</td><td>10.073</td><td>1.416</td></tr><tr><td rowspan="2">aug-cc-pVTZ</td><td>TBE<sup>a</sup></td><td>0.115</td><td>8.478</td><td>0.166</td><td>-0.126</td><td>-0.339</td><td>9.932</td><td>1.614</td><td>10.065</td><td>1.450</td></tr><tr><td>CBS</td><td>0.097±0.002</td><td>8.46±0.01</td><td>0.165±0.001</td><td>-0.19±0.01</td><td>-0.39±0.01</td><td>9.94±0.01</td><td>1.60±0.01</td><td>10.07±0.01</td><td>1.43±0.01</td></tr><tr><td rowspan="3">Lit.</td><td>Th.</td><td>0.1172<sup>c</sup></td><td>8.541<sup>d</sup></td><td>0.121<sup>e</sup></td><td>-0.135<sup>f</sup></td><td>-0.05<sup>h</sup>;-0.19<sup>i</sup></td><td>10.045<sup>d</sup></td><td></td><td>10.182<sup>d</sup></td><td></td></tr><tr><td></td><td>0.091<sup>f</sup></td><td>8.48<sup>g</sup></td><td>0.168<sup>g</sup></td><td></td><td></td><td>9.98<sup>g</sup></td><td></td><td>10.10<sup>g</sup></td><td></td></tr><tr><td>Exp.</td><td>0.122<sup>j</sup></td><td>8.51<sup>k</sup></td><td>0.181<sup>l</sup></td><td></td><td>-0.15±0.05<sup>m</sup></td><td>9.88<sup>k</sup></td><td></td><td>10.23<sup>k</sup></td><td></td></tr><tr><td></td><td></td><td></td><td>0.194<sup>n</sup></td><td></td><td>-0.335±0.013<sup>o</sup></td><td></td><td></td><td></td><td></td></tr><tr><th rowspan="2">Basis</th><th rowspan="2">Method</th><th colspan="3">1Σ<sup>+</sup> (Ryd)</th><th colspan="3">1Σ<sup>+</sup> (Ryd)</th><th colspan="3">1Π(Ryd)</th></tr><tr><th>ΔE<sub>vert</sub></th><th>f</th><th>μ<sup>ES</sup><sub>vert</sub></th><th>ΔE<sub>vert</sub></th><th>f</th><th>μ<sup>ES</sup><sub>vert</sub></th><th>ΔE<sub>vert</sub></th><th>f</th><th>μ<sup>ES</sup><sub>vert</sub></th></tr><tr><td rowspan="4">aug-cc-pVDZ</td><td>CCSD</td><td>11.171</td><td>0.003</td><td>-4.134</td><td>11.710</td><td>0.248</td><td>6.323</td><td>11.973</td><td>0.132</td><td>0.192</td></tr><tr><td>CCSDT</td><td>10.944</td><td>0.001</td><td>-4.519</td><td>11.518</td><td>0.240</td><td>6.699</td><td>11.767</td><td>0.124</td><td>0.149</td></tr><tr><td>CCSDTQ</td><td>10.926</td><td>0.000</td><td>-4.572</td><td>11.510</td><td>0.238</td><td>6.768</td><td>11.758</td><td>0.122</td><td>0.143</td></tr><tr><td>CCSDTQP</td><td>10.919</td><td>0.000</td><td>-4.592</td><td>11.506</td><td>0.238</td><td>6.792</td><td>11.753</td><td>0.121</td><td>0.141</td></tr><tr><td rowspan="4">aug-cc-pVTZ</td><td>CCSD</td><td>11.222</td><td>0.008</td><td>-3.376</td><td>11.751</td><td>0.208</td><td>5.600</td><td>11.960</td><td>0.115</td><td>0.195</td></tr><tr><td>CCSDT</td><td>10.987</td><td>0.004</td><td>-3.894</td><td>11.540</td><td>0.203</td><td>6.094</td><td>11.737</td><td>0.110</td><td>0.138</td></tr><tr><td>CCSDTQ</td><td>10.963</td><td>0.003</td><td>-3.994</td><td>11.523</td><td>0.202</td><td>6.205</td><td>11.720</td><td>0.108</td><td>0.126</td></tr><tr><td>CCSDTQP</td><td>11.190</td><td>0.010</td><td>-2.792</td><td>11.733</td><td>0.183</td><td>5.097</td><td>11.916</td><td>0.103</td><td>0.084</td></tr><tr><td rowspan="2">aug-cc-pVQZ</td><td>CCSD</td><td>10.954</td><td>0.006</td><td>-3.350</td><td>11.514</td><td>0.180</td><td>5.619</td><td>11.687</td><td>0.098</td><td>0.007</td></tr><tr><td>CCSDT</td><td>11.133</td><td>0.012</td><td>-2.077</td><td>11.691</td><td>0.160</td><td>4.614</td><td>11.851</td><td>0.091</td><td>-0.158</td></tr><tr><td rowspan="2">d-aug-cc-pVDZ</td><td>CCSD</td><td>10.795</td><td>0.009</td><td>-1.462</td><td>11.393</td><td>0.134</td><td>3.498</td><td>11.535</td><td>0.067</td><td>-1.957</td></tr><tr><td>CCSDT</td><td>10.569</td><td>0.006</td><td>-1.981</td><td>11.175</td><td>0.129</td><td>3.927</td><td>11.313</td><td>0.061</td><td>-2.146</td></tr><tr><td rowspan="2">d-aug-cc-pVTZ</td><td>CCSD</td><td>10.960</td><td>0.010</td><td>-1.337</td><td>11.567</td><td>0.134</td><td>3.814</td><td>11.700</td><td>0.074</td><td>-1.449</td></tr><tr><td>CCSDT</td><td>10.726</td><td>0.007</td><td>-1.855</td><td>11.340</td><td>0.130</td><td>4.216</td><td>11.468</td><td>0.069</td><td>-1.648</td></tr><tr><td rowspan="2">d-aug-cc-pVQZ</td><td>CCSD</td><td>11.010</td><td>0.011</td><td>-1.298</td><td>11.617</td><td>0.134</td><td>3.946</td><td>11.749</td><td>0.075</td><td>-1.233</td></tr><tr><td>CCSDT</td><td>10.774</td><td>0.007</td><td>-1.810</td><td>11.388</td><td>0.130</td><td>4.337</td><td>11.516</td><td>0.070</td><td>-1.437</td></tr><tr><td rowspan="2">aug-cc-pVTZ</td><td>TBE<sup>a</sup></td><td>10.956</td><td>0.003</td><td>-4.014</td><td>11.519</td><td>0.202</td><td>6.229</td><td>11.715</td><td>0.107</td><td>0.124</td></tr><tr><td>CBS</td><td>10.84±0.06</td><td>0.005±0.001</td><td>-2.49±0.59</td><td>11.43±0.04</td><td>0.146±0.016</td><td>4.94±0.47</td><td>11.58±0.05</td><td>0.078±0.009</td><td>-0.70±0.60</td></tr><tr><td rowspan="3">Lit.</td><td>Th.</td><td>10.983<sup>d</sup></td><td>0.029<sup>e</sup></td><td>-2.79<sup>i</sup></td><td></td><td>0.133<sup>e</sup></td><td>5.34<sup>i</sup></td><td></td><td>0.106<sup>g</sup></td><td></td></tr><tr><td></td><td>10.80<sup>g</sup></td><td>0.003<sup>g</sup></td><td></td><td>11.42<sup>g</sup></td><td>0.200<sup>g</sup></td><td>11.55<sup>g</sup></td><td></td><td></td><td></td></tr><tr><td>Exp.</td><td>10.78<sup>k</sup></td><td>0.009<sup>l</sup></td><td>-1.60±0.15<sup>m</sup></td><td>11.40<sup>k</sup></td><td>0.121<sup>l</sup></td><td>4.52±0.35<sup>m</sup></td><td>11.53<sup>k</sup></td><td>0.074<sup>l</sup></td><td></td></tr><tr><td></td><td></td><td></td><td>-1.95±0.03<sup>o</sup></td><td></td><td>0.136<sup>n</sup></td><td>4.50±0.07<sup>o</sup></td><td></td><td>0.074<sup>n</sup></td><td></td></tr></tbody></table>

<footnote>a,b See corresponding footnotes in Table 1; cCCSD(T)/CBS value from Ref. 47; dCCSDT/PVTZ+ values from Ref. 68; eSOPPA (with “all corrections”) from Ref. 49; fFCI/cc-pVDZ from Ref. 117; gFrom Ref. 5: the values od ΔE<sub>vert</sub> are basis set corrected exFCI/aug-cc-pVQZ values whereas f are LR-CC3/aug-cc-pVTZ values. Note that a factor of two linked to the degeneracy was incorrectly omitted in this earlier work for the oscillator strengths of the two Π transitions; hCC2/AVQZ result from Ref. 45; iFrom Ref. 118; jFrom Ref. 116; kVertical values estimated in Ref. 49 on the basis of the experimental spectroscopic data of Ref. 116; lDipole (e,e) spectroscopy of Ref. 119. For the higher ES, we give the contributions given for the two vibrational quanta; mStark measurements of the 0-0 bands from Ref. 120. Note that the value for the lowest ES is considered as an upper limit in this work. The sign of the dipole is assumed to be consistent with theory; nDipole (γ,γ) measurements from Ref. 121, summing over the different vibrational contributions; oTwo-photon laser induced fluorescence spectroscopy measurement of Stark effect from Ref. 122. The sign of the dipole is assumed to be consistent with theory.</footnote>
<table><caption>Table 6: Vertical transition energies $\Delta E_{\text{vert}}$ and oscillator strengths $f$ determined for $N_2$ (GS geometry). See caption of Table 1 for details.</caption><thead><tr><th rowspan="2">Basis</th><th rowspan="2">Method</th><th colspan="2">¹Π<sub>u</sub> (Ryd)</th><th colspan="2">¹Σ<sup>+</sup><sub>u</sub> (Ryd)</th><th colspan="2">¹Π<sub>u</sub> (Ryd)</th><th colspan="2">¹Π<sub>u</sub> (Ryd)</th></tr><tr><th>ΔE<sub>vert</sub></th><th>f</th><th>ΔE<sub>vert</sub></th><th>f</th><th>ΔE<sub>vert</sub></th><th>f</th><th>ΔE<sub>vert</sub></th><th>f</th></tr></thead><tbody><tr><td rowspan="4">aug-cc-pVDZ</td><td>CCSD</td><td>13.451</td><td>0.531</td><td>13.250</td><td>0.311</td><td>13.765</td><td>0.014</td><td>14.497</td><td>0.148</td></tr><tr><td>CCSDT</td><td>13.174</td><td>0.469</td><td>13.131</td><td>0.334</td><td>13.591</td><td>0.020</td><td>14.228</td><td>0.163</td></tr><tr><td>CCSDTQ</td><td>13.131</td><td>0.458</td><td>13.109</td><td>0.337</td><td>13.560</td><td>0.027</td><td>14.221</td><td>0.164</td></tr><tr><td>CCSDTQP</td><td>13.127</td><td>0.457</td><td>13.107</td><td>0.338</td><td>13.558</td><td>0.028</td><td>14.216</td><td>0.164</td></tr><tr><td rowspan="4">aug-cc-pVTZ</td><td>CCSD</td><td>13.422</td><td>0.439</td><td>13.264</td><td>0.263</td><td>13.674</td><td>0.053</td><td>14.307</td><td>0.136</td></tr><tr><td>CCSDT</td><td>13.140</td><td>0.435</td><td>13.118</td><td>0.281</td><td>13.455</td><td>0.008</td><td>14.034</td><td>0.148</td></tr><tr><td>CCSDTQ</td><td>13.095</td><td>0.424</td><td>13.090</td><td>0.285</td><td>13.419</td><td>0.014</td><td>14.014</td><td>0.149</td></tr><tr><td>CCSDTQP</td><td>13.354</td><td>0.357</td><td>13.242</td><td>0.242</td><td>13.638</td><td>0.113</td><td>14.216</td><td>0.134</td></tr><tr><td rowspan="2">aug-cc-pVQZ</td><td>CCSD</td><td>13.108</td><td>0.422</td><td>13.088</td><td>0.258</td><td>13.372</td><td>0.000</td><td>13.935</td><td>0.145</td></tr><tr><td>CCSDT</td><td>13.235</td><td>0.266</td><td>13.195</td><td>0.219</td><td>13.621</td><td>0.181</td><td>14.100</td><td>0.126</td></tr><tr><td rowspan="2">aug-cc-pV5Z</td><td>CCSD</td><td>13.037</td><td>0.356</td><td>13.039</td><td></td><td>13.306</td><td>0.044</td><td>13.816</td><td>0.138</td></tr><tr><td>CCSDT</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td rowspan="4">d-aug-cc-pVDZ</td><td>CCSD</td><td>12.784</td><td>0.170</td><td>12.822</td><td>0.172</td><td>13.640</td><td>0.101</td><td>13.537</td><td>0.234</td></tr><tr><td>CCSDT</td><td>12.669</td><td>0.198</td><td>12.712</td><td>0.183</td><td>13.262</td><td>0.256</td><td>13.300</td><td>0.014</td></tr><tr><td>CCSD</td><td>12.978</td><td>0.178</td><td>13.026</td><td>0.181</td><td>13.599</td><td>0.314</td><td>13.682</td><td>0.007</td></tr><tr><td>CCSDT</td><td>12.827</td><td>0.224</td><td>12.885</td><td>0.193</td><td>13.257</td><td>0.168</td><td>13.396</td><td>0.073</td></tr><tr><td rowspan="2">d-aug-cc-pVQZ</td><td>CCSD</td><td>13.039</td><td>0.181</td><td>13.090</td><td>0.184</td><td>13.601</td><td>0.276</td><td>13.730</td><td>0.043</td></tr><tr><td>CCSDT</td><td>12.876</td><td>0.234</td><td>12.939</td><td>0.196</td><td>13.256</td><td>0.460</td><td>13.443</td><td>0.082</td></tr><tr><td rowspan="2">aug-cc-pVTZ</td><td>TBE<sup>a</sup></td><td>13.090</td><td>0.423</td><td>13.088</td><td>0.286</td><td>13.417</td><td>0.015</td><td>14.009</td><td>0.148</td></tr><tr><td>TBE<sup>b</sup></td><td>12.83±0.08</td><td>0.22±0.06<sup>c</sup></td><td>12.96±0.01</td><td>0.207±0.005</td><td>13.27±0.07</td><td><sup>d</sup></td><td>13.57±0.12</td><td>0.10±0.03<sup>c</sup></td></tr><tr><td rowspan="4">CBS</td><td>Lit.</td><td>12.73<sup>e</sup></td><td>0.458<sup>e</sup></td><td>12.95<sup>e</sup></td><td>0.296<sup>e</sup></td><td>13.27<sup>e</sup></td><td>0.000<sup>e</sup></td><td></td><td></td></tr><tr><td>Th.</td><td></td><td>0.091<sup>f</sup>; 0.063<sup>g</sup></td><td></td><td>0.65<sup>f</sup>; 0.221<sup>g</sup></td><td></td><td>0.32<sup>f</sup>; 0.091<sup>g</sup></td><td></td><td>0.083<sup>g</sup></td></tr><tr><td rowspan="2">Exp.</td><td>12.78<sup>h</sup></td><td>0.243<sup>i</sup></td><td>12.96<sup>h</sup></td><td>0.279<sup>i</sup></td><td>13.10<sup>h</sup></td><td>0.145<sup>i</sup></td><td></td><td>0.080<sup>i</sup></td></tr><tr><td>12.90<sup>j</sup></td><td></td><td>12.98<sup>j</sup></td><td>0.223<sup>k</sup></td><td>13.24<sup>j</sup></td><td></td><td></td><td>13.63<sup>j</sup></td></tr></tbody></table>

a,b See corresponding footnotes in Table 1; c The extrapolation is very challenging due to the mixing, tentative values; d Too unstable to report any reasonable CBS estimate; e From Ref. 5: energies are basis set corrected exFCI/aug-cc-pVQZ values and the oscillator strengths are LR-CC3/aug-cc-pVTZ values. Note that a factor of two linked to the degeneracy was incorrectly omitted in this earlier work for the $Π_u$ transitions; f RPA values in length gauge from Ref. 127; g SOPA values from Ref. 128; h Experimental vertical values given in Ref. 127 deduced from spectroscopic data of Ref. 116; i Integrated intensities from electron scattering of Ref. 129. j Experimental vertical values given in Ref. 130 deduced from spectroscopic data of Ref. 116; k Integrated electron impact induced emission intensities of Ref. 131.

relative ordering of the three lowest $^1\Pi_u$ states. Here, we have classified them following the nature of the underlying MOs,¹³² and characterized all of them as Rydberg, although alternative yet reasonable assignments can be found in the literature (e.g., see Tables I and II in Ref. 128). Our results are given in Table 6. The convergence with respect to the CC excitation order is rather quick for both energies and oscillator strengths, with nevertheless non-negligible contributions from the quadruples for the transition energies, a likely consequence of the presence of a triple bond. If basis set effects follow the expected trends for the energies of these high-lying ES (i.e., a decrease of $\Delta E_{\text{vert}}$ when increasing the basis set size, and a significant impact of the second set of diffuse functions), the changes in the oscillator strength of the $^1\Pi_u$ states when enlarging the basis set are dramatic. As an illustration, the oscillator strength of the second $^1\Pi_u$ transition is 0.000 at the CCSDT/aug-cc-pVQZ level but 0.460 at the CCSDT/d-aug-cc-pVQZ level. If the extreme sensitivity of $N_2$'s oscillator strengths to the computational setup is known for years (e.g., see Table 6 in Ref. 127), it remains a very striking example of the state mixing nightmare. As a consequence, while we could obtain solid TBE/aug-cc-pVTZ values in all cases, the extrapolation to CBS of the $f$ values of the $^1\Pi_u$ transitions is clearly problematic. In comparison, while the basis set effects are far from being negligible for the $^1\Sigma_u^+$ ES, it is definitely possible to establish robust CBS $f$ values for this transition.

For the vertical excitation energies, our TBE/CBS of 12.83,

13.27, and 13.57 eV for the three lowest $^1\Pi_u$ transitions and of 12.96 eV for the lowest $^1\Sigma_u^+$ transition are all in reasonable agreement with current state-of-the-art values. Indeed, as can be seen from the bottom of Table 6, these values are close to the experimental vertical data deduced elsewhere,¹³⁰ as well as to our recent FCI/CBS estimates.⁵ When turning to the oscillator strengths, there is quite a diversity in the experimentally-measured values (see Table 8 of Ref. 129 for integrated values, and Table 1 of Ref. 133 for individual vibrronic contributions from various earlier works). For the $^1\Sigma_u^+$ transition, several estimates are available and they show a wide range on both the theoretical and experimental sides (see Table 3 of Ref. 134), and our TBE/CBS $f$ value of 0.207 is reasonably in line with the most recent electron impact value that we have found (0.223),¹³¹ yet significantly smaller than a value obtained by electron scattering (0.279).¹²⁹ For the first and third $^1\Pi_u$ transitions, our estimates of the oscillator strength come with large error bars (0.22±0.06 and 0.10±0.03, respectively), which nevertheless cover the most recent experimental values (0.243 and 0.080),¹²⁹ an outcome that we found satisfying. As stated above, no reasonable estimate could be obtained for the oscillator strengths of the remaining transition. It is likely that high-level multi-reference calculations would be welcome in this specific case.

### 3.6 Ethylene

The ESs of this model $\pi$-conjugated hydrocarbon have puzzled chemists for years, in particular the relative ordering and nature of the two lowest singlet ESs considered here (Table
<table><caption>Table 6: Vertical transition energies $\Delta E_{\text{vert}}$ and oscillator strengths $f$ determined for $N_2$ (GS geometry). See caption of Table 1 for details.</caption><thead><tr><th rowspan="2">Basis</th><th rowspan="2">Method</th><th colspan="2"><sup>1</sup>Π<sub>u</sub> (Ryd)</th><th colspan="2"><sup>1</sup>Σ<sup>+</sup><sub>u</sub> (Ryd)</th><th colspan="2"><sup>1</sup>Π<sub>u</sub> (Ryd)</th><th colspan="2"><sup>1</sup>Π<sub>u</sub> (Ryd)</th></tr><tr><th>ΔE<sub>vert</sub></th><th>f</th><th>ΔE<sub>vert</sub></th><th>f</th><th>ΔE<sub>vert</sub></th><th>f</th><th>ΔE<sub>vert</sub></th><th>f</th></tr></thead><tbody><tr><td rowspan="4">aug-cc-pVDZ</td><td>CCSD</td><td>13.451</td><td>0.531</td><td>13.250</td><td>0.311</td><td>13.765</td><td>0.014</td><td>14.497</td><td>0.148</td></tr><tr><td>CCSDT</td><td>13.174</td><td>0.469</td><td>13.131</td><td>0.334</td><td>13.591</td><td>0.020</td><td>14.228</td><td>0.163</td></tr><tr><td>CCSDTQ</td><td>13.131</td><td>0.458</td><td>13.109</td><td>0.337</td><td>13.560</td><td>0.027</td><td>14.221</td><td>0.164</td></tr><tr><td>CCSDTQP</td><td>13.127</td><td>0.457</td><td>13.107</td><td>0.338</td><td>13.558</td><td>0.028</td><td>14.216</td><td>0.164</td></tr><tr><td rowspan="4">aug-cc-pVTZ</td><td>CCSD</td><td>13.422</td><td>0.439</td><td>13.264</td><td>0.263</td><td>13.674</td><td>0.053</td><td>14.307</td><td>0.136</td></tr><tr><td>CCSDT</td><td>13.140</td><td>0.435</td><td>13.118</td><td>0.281</td><td>13.455</td><td>0.008</td><td>14.034</td><td>0.148</td></tr><tr><td>CCSDTQ</td><td>13.095</td><td>0.424</td><td>13.090</td><td>0.285</td><td>13.419</td><td>0.014</td><td>14.014</td><td>0.149</td></tr><tr><td>CCSDTQP</td><td>13.354</td><td>0.357</td><td>13.242</td><td>0.242</td><td>13.638</td><td>0.113</td><td>14.216</td><td>0.134</td></tr><tr><td rowspan="2">aug-cc-pVQZ</td><td>CCSD</td><td>13.108</td><td>0.422</td><td>13.088</td><td>0.258</td><td>13.372</td><td>0.000</td><td>13.935</td><td>0.145</td></tr><tr><td>CCSDT</td><td>13.235</td><td>0.266</td><td>13.195</td><td>0.219</td><td>13.621</td><td>0.181</td><td>14.100</td><td>0.126</td></tr><tr><td rowspan="2">aug-cc-pV5Z</td><td>CCSD</td><td>13.037</td><td>0.356</td><td>13.039</td><td></td><td>13.306</td><td>0.044</td><td>13.816</td><td>0.138</td></tr><tr><td>CCSDT</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td rowspan="4">d-aug-cc-pVDZ</td><td>CCSD</td><td>12.784</td><td>0.170</td><td>12.822</td><td>0.172</td><td>13.640</td><td>0.101</td><td>13.537</td><td>0.234</td></tr><tr><td>CCSDT</td><td>12.669</td><td>0.198</td><td>12.712</td><td>0.183</td><td>13.262</td><td>0.256</td><td>13.300</td><td>0.014</td></tr><tr><td>CCSD</td><td>12.978</td><td>0.178</td><td>13.026</td><td>0.181</td><td>13.599</td><td>0.314</td><td>13.682</td><td>0.007</td></tr><tr><td>CCSDT</td><td>12.827</td><td>0.224</td><td>12.885</td><td>0.193</td><td>13.257</td><td>0.168</td><td>13.396</td><td>0.073</td></tr><tr><td rowspan="2">d-aug-cc-pVQZ</td><td>CCSD</td><td>13.039</td><td>0.181</td><td>13.090</td><td>0.184</td><td>13.601</td><td>0.276</td><td>13.730</td><td>0.043</td></tr><tr><td>CCSDT</td><td>12.876</td><td>0.234</td><td>12.939</td><td>0.196</td><td>13.256</td><td>0.460</td><td>13.443</td><td>0.082</td></tr><tr><td rowspan="2">aug-cc-pVTZ</td><td>TBE<sup>a</sup></td><td>13.090</td><td>0.423</td><td>13.088</td><td>0.286</td><td>13.417</td><td>0.015</td><td>14.009</td><td>0.148</td></tr><tr><td>TBE<sup>b</sup></td><td>12.83±0.08</td><td>0.22±0.06<sup>c</sup></td><td>12.96±0.01</td><td>0.207±0.005</td><td>13.27±0.07</td><td><sup>d</sup></td><td>13.57±0.12</td><td>0.10±0.03<sup>c</sup></td></tr><tr><td rowspan="4">CBS</td><td>Lit.</td><td>12.73<sup>e</sup></td><td>0.458<sup>e</sup></td><td>12.95<sup>e</sup></td><td>0.296<sup>e</sup></td><td>13.27<sup>e</sup></td><td>0.000<sup>e</sup></td><td></td><td></td></tr><tr><td>Th.</td><td></td><td>0.091<sup>f</sup>; 0.063<sup>g</sup></td><td></td><td>0.65<sup>f</sup>; 0.221<sup>g</sup></td><td></td><td>0.32<sup>f</sup>; 0.091<sup>g</sup></td><td></td><td>0.083<sup>g</sup></td></tr><tr><td rowspan="2">Exp.</td><td>12.78<sup>h</sup></td><td>0.243<sup>i</sup></td><td>12.96<sup>h</sup></td><td>0.279<sup>i</sup></td><td>13.10<sup>h</sup></td><td>0.145<sup>i</sup></td><td></td><td>0.080<sup>i</sup></td></tr><tr><td>12.90<sup>j</sup></td><td></td><td>12.98<sup>j</sup></td><td>0.223<sup>k</sup></td><td>13.24<sup>j</sup></td><td></td><td>13.63<sup>j</sup></td><td></td><td></td></tr></tbody></table>

a,b See corresponding footnotes in Table 1; c The extrapolation is very challenging due to the mixing, tentative values; d Too unstable to report any reasonable CBS estimate; e From Ref. 5: energies are basis set corrected exFCI/aug-cc-pVQZ values and the oscillator strengths are LR-CC3/aug-cc-pVTZ values. Note that a factor of two linked to the degeneracy was incorrectly omitted in this earlier work for the $\Pi_u$ transitions; f RPA values in length gauge from Ref. 127; g SOPA values from Ref. 128; h Experimental vertical values given in Ref. 127 deduced from spectroscopic data of Ref. 116; i Integrated intensities from electron scattering of Ref. 129. j Experimental vertical values given in Ref. 130 deduced from spectroscopic data of Ref. 116; k Integrated electron impact induced emission intensities of Ref. 131.

relative ordering of the three lowest $^1\Pi_u$ states. Here, we have classified them following the nature of the underlying MOs,¹³² and characterized all of them as Rydberg, although alternative yet reasonable assignments can be found in the literature (e.g., see Tables I and II in Ref. 128). Our results are given in Table 6. The convergence with respect to the CC excitation order is rather quick for both energies and oscillator strengths, with nevertheless non-negligible contributions from the quadruples for the transition energies, a likely consequence of the presence of a triple bond. If basis set effects follow the expected trends for the energies of these high-lying ES (i.e., a decrease of $\Delta E_{\text{vert}}$ when increasing the basis set size, and a significant impact of the second set of diffuse functions), the changes in the oscillator strength of the $^1\Pi_u$ states when enlarging the basis set are dramatic. As an illustration, the oscillator strength of the second $^1\Pi_u$ transition is 0.000 at the CCSDT/aug-cc-pVQZ level but 0.460 at the CCSDT/d-aug-cc-pVQZ level. If the extreme sensitivity of $N_2$'s oscillator strengths to the computational setup is known for years (e.g., see Table 6 in Ref. 127), it remains a very striking example of the state mixing nightmare. As a consequence, while we could obtain solid TBE/aug-cc-pVTZ values in all cases, the extrapolation to CBS of the $f$ values of the $^1\Pi_u$ transitions is clearly problematic. In comparison, while the basis set effects are far from being negligible for the $^1\Sigma_u^+$ ES, it is definitely possible to establish robust CBS $f$ values for this transition.

For the vertical excitation energies, our TBE/CBS of 12.83,

13.27, and 13.57 eV for the three lowest $^1\Pi_u$ transitions and of 12.96 eV for the lowest $^1\Sigma_u^+$ transition are all in reasonable agreement with current state-of-the-art values. Indeed, as can be seen from the bottom of Table 6, these values are close to the experimental vertical data deduced elsewhere,¹³⁰ as well as to our recent FCI/CBS estimates.⁵ When turning to the oscillator strengths, there is quite a diversity in the experimentally-measured values (see Table 8 of Ref. 129 for integrated values, and Table 1 of Ref. 133 for individual vibrational contributions from various earlier works). For the $^1\Sigma_u^+$ transition, several estimates are available and they show a wide range on both the theoretical and experimental sides (see Table 3 of Ref. 134), and our TBE/CBS $f$ value of 0.207 is reasonably in line with the most recent electron impact value that we have found (0.223),¹³¹ yet significantly smaller than a value obtained by electron scattering (0.279).¹²⁹ For the first and third $^1\Pi_u$ transitions, our estimates of the oscillator strength come with large error bars (0.22±0.06 and 0.10±0.03, respectively), which nevertheless cover the most recent experimental values (0.243 and 0.080),¹²⁹ an outcome that we found satisfying. As stated above, no reasonable estimate could be obtained for the oscillator strengths of the remaining transition. It is likely that high-level multi-reference calculations would be welcome in this specific case.

### 3.6 Ethylene

The ESs of this model $\pi$-conjugated hydrocarbon have puzzled chemists for years, in particular the relative ordering and nature of the two lowest singlet ESs considered here (Table
7). As can be seen, the oscillator strength of the lowest Rydberg state of $B_{3u}$ symmetry is much too large with 6-31+G(d) but rapidly converges with the Dunning series, the addition of a second set of diffuse functions playing no significant role. In fact the *f* values for this transition converges faster when climbing the methodology ladder than the $\Delta E_{\text{vert}}$ values. For the valence $\pi \rightarrow \pi^* B_{1u}$ transition, CCSD seems to slightly overestimate the oscillator strength and the convergence with basis set size is nearly reached with aug-cc-pVTZ. Our best estimates are 7.42 eV (*f*=0.076) and 7.90 eV (*f*=0.338), for the Rydberg and valence transitions, respectively.

<table><caption>Table 7: Vertical transition energies $\Delta E_{\text{vert}}$ and oscillator strengths *f* of ethylene (GS geometry). See caption of Table 1 for details.</caption><thead><tr><th rowspan="2">Basis</th><th rowspan="2">Method</th><th colspan="2"><sup>1</sup>B<sub>3u</sub> (Ryd)</th><th colspan="2"><sup>1</sup>B<sub>1u</sub> (Val)</th></tr><tr><th>ΔE<sub>vert</sub></th><th><i>f</i></th><th>ΔE<sub>vert</sub></th><th><i>f</i></th></tr></thead><tbody><tr><td rowspan="4">6-31+G(d)</td><td>CCSD</td><td>7.814</td><td>0.152</td><td>8.275</td><td>0.380</td></tr><tr><td>CCSDT</td><td>7.725</td><td>0.151</td><td>8.152</td><td>0.365</td></tr><tr><td>CCSDTQ</td><td>7.722</td><td>0.150</td><td>8.137</td><td>0.364</td></tr><tr><td>CCSDTQP</td><td>7.722</td><td>0.150</td><td>8.135</td><td>0.364</td></tr><tr><td rowspan="3">aug-cc-pVDZ</td><td>CCSD</td><td>7.323</td><td>0.080</td><td>8.035</td><td>0.365</td></tr><tr><td>CCSDT</td><td>7.294</td><td>0.080</td><td>7.944</td><td>0.352</td></tr><tr><td>CCSDTQ</td><td>7.303</td><td>0.080</td><td>7.932</td><td>0.351</td></tr><tr><td rowspan="2">aug-cc-pVTZ</td><td>CCSD</td><td>7.417</td><td>0.078</td><td>8.020</td><td>0.362</td></tr><tr><td>CCSDT</td><td>7.365</td><td>0.078</td><td>7.918</td><td>0.346</td></tr><tr><td>aug-cc-pVQZ</td><td>CCSD</td><td>7.451</td><td>0.078</td><td>8.023</td><td>0.360</td></tr><tr><td>d-aug-cc-pVDZ</td><td>CCSD</td><td>7.301</td><td>0.078</td><td>8.008</td><td>0.345</td></tr><tr><td>d-aug-cc-pVTZ</td><td>CCSD</td><td>7.273</td><td>0.078</td><td>7.920</td><td>0.336</td></tr><tr><td>d-aug-cc-pVQZ</td><td>CCSD</td><td>7.409</td><td>0.077</td><td>8.009</td><td>0.353</td></tr><tr><td>d-aug-cc-pVQZ</td><td>CCSD</td><td>7.357</td><td>0.077</td><td>7.908</td><td>0.339</td></tr><tr><td>aug-cc-pVTZ</td><td>TBE<sup>a</sup></td><td>7.374</td><td>0.078</td><td>7.905</td><td>0.345</td></tr><tr><td>CBS</td><td>TBE<sup>b</sup></td><td>7.42±0.02</td><td>0.076±0.001</td><td>7.90±0.01</td><td>0.338±0.005</td></tr><tr><td rowspan="3">Lit.</td><td>Th.</td><td>7.45<sup>c</sup></td><td>0.069<sup>c</sup></td><td>8.00<sup>c</sup></td><td>0.333<sup>c</sup></td></tr><tr><td></td><td>7.43<sup>d</sup></td><td>0.078<sup>d</sup></td><td>7.92<sup>d</sup></td><td>0.348<sup>d</sup></td></tr><tr><td>Exp.</td><td>7.11<sup>e</sup></td><td>~0.04<sup>f</sup></td><td>7.60<sup>e</sup></td><td>0.348<sup>f</sup></td></tr></tbody></table>

On the theoretical side, the most advanced theoretical study of ethylene's ES likely remains the 2014 investigation of Feller et al.⁸⁸ With their best composite theory, these authors reported transition energies of 7.45 eV and 8.00 eV with respective *f* values of 0.069 and 0.333 (see footnotes of Table 7 for details). One can also compare to the values of Ref. 5: FCI/CBS estimates of 7.43 and 7.92 eV with CC3/AVTZ *f* values of 0.078 and 0.346, respectively. Older works report oscillator strengths of 0.389 for the valence state at the CCSDT/TZVP level,³⁹ and of 0.078 and 0.358 with CCSD.¹³⁹ The experimental measurements of the oscillator strength do not allow to attribute values to individual transitions due to strong overlapping.¹⁴⁰ The generally used experimental reference values are 0.34 or 0.29 for the valence transition, both estimates being obtained from measurements of the vacuum absorption spectrum performed in 1953¹³⁸ and

1955,¹⁴¹ respectively. However, a more recent dipole (e,e) spectroscopy study suggests that the originally measured oscillator strengths in the 7.4–8.0 eV regions are probably too low by ca. 10–15%.¹⁴⁰ Our TBE of 0.338 is therefore reasonably in line with the current experimental knowledge. For the Rydberg ES, the only experimental estimate we are aware of has been reported in 1969 as a “total *f* perhaps about 0.04”.¹³⁷ Given the consistency of all theoretical estimates, it seems rather reasonable to state that our current TBE of 0.076 is significantly more trustworthy. In ethylene, in contrast to N₂, theory has clearly the edge because the considered transitions have different spatial symmetries.

## 3.7 Formaldehyde and thioformaldehyde

The ESs of formaldehyde have been extensively studied before with almost all possible theoretical approaches,²,⁵,⁴⁵,⁷⁴,¹⁴²⁻¹⁵⁵ and we have considered here two valence and three Rydberg states. Our results are collected in Table 8 for these fives ESs. The hallmark $n \rightarrow \pi^*$ transition behaves nicely from a theoretical point of view, and the convergences of both $\Delta E_{\text{vert}}$ and $\mu^{\text{ES}}$ with respect to the basis set size are very quick, with a negligible effect of the second set of diffuse functions. Likewise, the corrections brought by the Q and P terms in the CC expansion are rather negligible, and CCSDT/aug-cc-pVTZ values are trustworthy. The higher-lying $\pi \rightarrow \pi^*$ valence ES is significantly more challenging as, on the one hand, the second set of diffuse functions significantly increases the transition energy (by ca. +0.2 eV), decreases the oscillator strength (by roughly 10%), and greatly amplifies the ES dipole (by a factor of 2 or 3), whereas, on the other hand, going from CCSDT to CCSDTQ yields a non-negligible drop of the computed dipole. Extrapolation to the CBS limit is therefore uneasy for the latter property. The oscillator strengths of the three Rydberg ESs are all relatively small, but their absolute and relative amplitudes are fairly independent on the selected level of theory and basis sets, though double augmentation induces a small decrease of the magnitude of the oscillator strength. The value of $\mu^{\text{ES}}$ for the lowest-lying Rydberg ES cannot be adequately described with the selected Pople basis set, but are easy to extrapolate using Dunning’s series. For the second (third) Rydberg ES, all tested approaches agree on the rather small (moderate) amplitude for $\mu^{\text{ES}}$, but the basis set effects are quite drastic. For instance, considering the higher-lying Rydberg ES, the CCSDT $\mu^{\text{ES}}$ value is -0.06 D with 6-31+G(d), -2.19 D with aug-cc-pVDZ, and -0.37 D with d-aug-cc-pVTZ. Clearly it is challenging to get a definitive CBS estimate.

A very accurate measurement of $\mu^{\text{GS}}$ for formaldehyde is available at 2.3321±0.0005 D (molecular beam electric resonance spectroscopy),¹⁵⁹ and our theoretical TBE of 2.41±0.01 D seems slightly too large, but is in very good agreement with previous CCSD(T)/CBS (2.393 D)⁴⁷ and CCSD(T)/aug-cc-pVQZ (2.382 D) estimates.¹⁶⁶ As early as 2004, Hirata proposed a CCSDT/aug-cc-pVDZ value of $\mu^{\text{GS}}$ at 2.33 D,⁷⁴ right on the experimental spot, but an experimental geometry was used and the orbital relaxation effects neglected, which might have induced a very slight error compensation. For the hallmark lowest $n \rightarrow \pi^*$ transition, one can find several experimental estimates of $\mu^{\text{ES}}$:

<footnote><sup>a</sup>CCSDT/aug-cc-pVTZ values corrected for Q effects using aug-cc-pVDZ and for P effects using 6-31+G(d); <sup>b</sup>See Computational section; <sup>c</sup>From Ref. 88: best composite theory for energies, icCAS(12/15)-CI/VDZ+ for the oscillator strengths of the Rydberg transition and icINO(12/16)-CI/VDZ+ extrapolated to FCI for the oscillator strengths of the valence transition; <sup>d</sup>FCI/CBS (for transition energies) and CC3/aug-cc-pVTZ (for *f*) values from Ref. 5; <sup>e</sup>Experimental values collected in Ref. 135 (see the discussions in Refs. 2, 88, and 136); <sup>f</sup>From Ref. 137 (see text); <sup>g</sup>Vacuum absorption from Ref. 138.</footnote>
7). As can be seen, the oscillator strength of the lowest Rydberg state of $B_{3u}$ symmetry is much too large with 6-31+G(d) but rapidly converges with the Dunning series, the addition of a second set of diffuse functions playing no significant role. In fact the *f* values for this transition converges faster when climbing the methodology ladder than the $\Delta E_{\text{vert}}$ values. For the valence $\pi \rightarrow \pi^* B_{1u}$ transition, CCSD seems to slightly overestimate the oscillator strength and the convergence with basis set size is nearly reached with aug-cc-pVTZ. Our best estimates are 7.42 eV (*f*=0.076) and 7.90 eV (*f*=0.338), for the Rydberg and valence transitions, respectively.

<table><caption>Table 7: Vertical transition energies $\Delta E_{\text{vert}}$ and oscillator strengths *f* of ethylene (GS geometry). See caption of Table 1 for details.</caption><thead><tr><th rowspan="2">Basis</th><th rowspan="2">Method</th><th colspan="2"><sup>1</sup>B<sub>3u</sub> (Ryd)</th><th colspan="2"><sup>1</sup>B<sub>1u</sub> (Val)</th></tr><tr><th>ΔE<sub>vert</sub></th><th><i>f</i></th><th>ΔE<sub>vert</sub></th><th><i>f</i></th></tr></thead><tbody><tr><td rowspan="4">6-31+G(d)</td><td>CCSD</td><td>7.814</td><td>0.152</td><td>8.275</td><td>0.380</td></tr><tr><td>CCSDT</td><td>7.725</td><td>0.151</td><td>8.152</td><td>0.365</td></tr><tr><td>CCSDTQ</td><td>7.722</td><td>0.150</td><td>8.137</td><td>0.364</td></tr><tr><td>CCSDTQP</td><td>7.722</td><td>0.150</td><td>8.135</td><td>0.364</td></tr><tr><td rowspan="3">aug-cc-pVDZ</td><td>CCSD</td><td>7.323</td><td>0.080</td><td>8.035</td><td>0.365</td></tr><tr><td>CCSDT</td><td>7.294</td><td>0.080</td><td>7.944</td><td>0.352</td></tr><tr><td>CCSDTQ</td><td>7.303</td><td>0.080</td><td>7.932</td><td>0.351</td></tr><tr><td rowspan="2">aug-cc-pVTZ</td><td>CCSD</td><td>7.417</td><td>0.078</td><td>8.020</td><td>0.362</td></tr><tr><td>CCSDT</td><td>7.365</td><td>0.078</td><td>7.918</td><td>0.346</td></tr><tr><td>aug-cc-pVQZ</td><td>CCSD</td><td>7.451</td><td>0.078</td><td>8.023</td><td>0.360</td></tr><tr><td>d-aug-cc-pVDZ</td><td>CCSD</td><td>7.301</td><td>0.078</td><td>8.008</td><td>0.345</td></tr><tr><td>d-aug-cc-pVTZ</td><td>CCSD</td><td>7.273</td><td>0.078</td><td>7.920</td><td>0.336</td></tr><tr><td>d-aug-cc-pVTZ</td><td>CCSD</td><td>7.409</td><td>0.077</td><td>8.009</td><td>0.353</td></tr><tr><td>d-aug-cc-pVQZ</td><td>CCSD</td><td>7.357</td><td>0.077</td><td>7.908</td><td>0.339</td></tr><tr><td>aug-cc-pVTZ</td><td>TBE<sup>a</sup></td><td>7.374</td><td>0.078</td><td>7.905</td><td>0.345</td></tr><tr><td>CBS</td><td>TBE<sup>b</sup></td><td>7.42±0.02</td><td>0.076±0.001</td><td>7.90±0.01</td><td>0.338±0.005</td></tr><tr><td rowspan="3">Lit.</td><td>Th.</td><td>7.45<sup>c</sup></td><td>0.069<sup>c</sup></td><td>8.00<sup>c</sup></td><td>0.333<sup>c</sup></td></tr><tr><td></td><td>7.43<sup>d</sup></td><td>0.078<sup>d</sup></td><td>7.92<sup>d</sup></td><td>0.348<sup>d</sup></td></tr><tr><td>Exp.</td><td>7.11<sup>e</sup></td><td>~0.04<sup>f</sup></td><td>7.60<sup>e</sup></td><td>0.348<sup>g</sup></td></tr></tbody></table>

On the theoretical side, the most advanced theoretical study of ethylene's ES likely remains the 2014 investigation of Feller *et al.*<sup>88</sup> With their best composite theory, these authors reported transition energies of 7.45 eV and 8.00 eV with respective *f* values of 0.069 and 0.333 (see footnotes of Table 7 for details). One can also compare to the values of Ref. 5: FCI/CBS estimates of 7.43 and 7.92 eV with CC3/AVTZ *f* values of 0.078 and 0.346, respectively. Older works report oscillator strengths of 0.389 for the valence state at the CCSDT/TZVP level,<sup>39</sup> and of 0.078 and 0.358 with CCSD.<sup>139</sup> The experimental measurements of the oscillator strength do not allow to attribute values to individual transitions due to strong overlapping.<sup>140</sup> The generally used experimental reference values are 0.34 or 0.29 for the valence transition, both estimates being obtained from measurements of the vacuum absorption spectrum performed in 1953<sup>138</sup> and

1955,<sup>141</sup> respectively. However, a more recent dipole (e,e) spectroscopy study suggests that the originally measured oscillator strengths in the 7.4–8.0 eV regions are probably too low by ca. 10–15%.<sup>140</sup> Our TBE of 0.338 is therefore reasonably in line with the current experimental knowledge. For the Rydberg ES, the only experimental estimate we are aware of has been reported in 1969 as a “total *f* perhaps about 0.04”.<sup>137</sup> Given the consistency of all theoretical estimates, it seems rather reasonable to state that our current TBE of 0.076 is significantly more trustworthy. In ethylene, in contrast to N<sub>2</sub>, theory has clearly the edge because the considered transitions have different spatial symmetries.

## 3.7 Formaldehyde and thioformaldehyde

The ESs of formaldehyde have been extensively studied before with almost all possible theoretical approaches,<sup>2,5,45,74,142–155</sup> and we have considered here two valence and three Rydberg states. Our results are collected in Table 8 for these fives ESs. The hallmark $n \rightarrow \pi^*$ transition behaves nicely from a theoretical point of view, and the convergences of both $\Delta E_{\text{vert}}$ and $\mu^{\text{ES}}$ with respect to the basis set size are very quick, with a negligible effect of the second set of diffuse functions. Likewise, the corrections brought by the Q and P terms in the CC expansion are rather negligible, and CCSDT/aug-cc-pVTZ values are trustworthy. The higher-lying $\pi \rightarrow \pi^*$ valence ES is significantly more challenging as, on the one hand, the second set of diffuse functions significantly increases the transition energy (by ca. +0.2 eV), decreases the oscillator strength (by roughly 10%), and greatly amplifies the ES dipole (by a factor of 2 or 3), whereas, on the other hand, going from CCSDT to CCSDTQ yields a non-negligible drop of the computed dipole. Extrapolation to the CBS limit is therefore uneasy for the latter property. The oscillator strengths of the three Rydberg ESs are all relatively small, but their absolute and relative amplitudes are fairly independent on the selected level of theory and basis sets, though double augmentation induces a small decrease of the magnitude of the oscillator strength. The value of $\mu^{\text{ES}}$ for the lowest-lying Rydberg ES cannot be adequately described with the selected Pople basis set, but are easy to extrapolate using Dunning’s series. For the second (third) Rydberg ES, all tested approaches agree on the rather small (moderate) amplitude for $\mu^{\text{ES}}$, but the basis set effects are quite drastic. For instance, considering the higher-lying Rydberg ES, the CCSDT $\mu^{\text{ES}}$ value is -0.06 D with 6-31+G(d), -2.19 D with aug-cc-pVDZ, and -0.37 D with d-aug-cc-pVTZ. Clearly it is challenging to get a definitive CBS estimate.

A very accurate measurement of $\mu^{\text{GS}}$ for formaldehyde is available at 2.3321±0.0005 D (molecular beam electric resonance spectroscopy),<sup>159</sup> and our theoretical TBE of 2.41±0.01 D seems slightly too large, but is in very good agreement with previous CCSD(T)/CBS (2.393 D)<sup>47</sup> and CCSD(T)/aug-cc-pVQZ (2.382 D) estimates.<sup>166</sup> As early as 2004, Hirata proposed a CCSDT/aug-cc-pVDZ value of $\mu^{\text{GS}}$ at 2.33 D,<sup>74</sup> right on the experimental spot, but an experimental geometry was used and the orbital relaxation effects neglected, which might have induced a very slight error compensation. For the hallmark lowest $n \rightarrow \pi^*$ transition, one can find several experimental estimates of $\mu^{\text{ES}}$:

<footnote><sup>a</sup>CCSDT/aug-cc-pVTZ values corrected for Q effects using aug-cc-pVDZ and for P effects using 6-31+G(d); <sup>b</sup>See Computational section; <sup>c</sup>From Ref. 88: best composite theory for energies, icCAS(12/15)-CI/VDZ+ for the oscillator strengths of the Rydberg transition and icINO(12/16)-CI/VDZ+ extrapolated to FCI for the oscillator strengths of the valence transition; <sup>d</sup>FCI/CBS (for transition energies) and CC3/aug-cc-pVTZ (for *f*) values from Ref. 5; <sup>e</sup>Experimental values collected in Ref. 135 (see the discussions in Refs. 2, 88, and 136); <sup>f</sup>From Ref. 137 (see text); <sup>g</sup>Vacuum absorption from Ref. 138.</footnote>
<table><caption>Table 8: Ground-state dipole moment $\mu^{\text{GS}}$, vertical transition energies $\Delta E_{\text{vert}}$, oscillator strengths $f$, and excited state dipole moments, $\mu_{\text{vert}}^{\text{ES}}$ and $\mu_{\text{adia}}^{\text{ES}}$, determined of formaldehyde. See caption of Table 1 for details.</caption><thead><tr><th rowspan="2">Basis</th><th rowspan="2">Method</th><th colspan="3"><sup>1</sup>A<sub>1</sub></th><th colspan="3"><sup>1</sup>A<sub>2</sub> (Val, n → π*)</th><th colspan="3"><sup>1</sup>B<sub>2</sub> (Ryd, n → 3s)</th></tr><tr><th>μ<sup>GS</sup></th><th>ΔE<sub>vert</sub></th><th>μ<sup>ES</sup><sub>vert</sub></th><th>μ<sup>ES</sup><sub>adia</sub></th><th>ΔE<sub>vert</sub></th><th>f</th><th>μ<sup>ES</sup><sub>vert</sub></th><th>μ<sup>ES</sup><sub>adia</sub></th></tr></thead><tbody><tr><td rowspan="4">6-31+G(d)</td><td>CCSD</td><td>2.584</td><td>4.031</td><td>1.710</td><td>1.870</td><td>7.238</td><td>0.017</td><td>-0.901</td><td>-0.479</td></tr><tr><td>CCSDT</td><td>2.529</td><td>4.012</td><td>1.649</td><td>1.790</td><td>7.232</td><td>0.021</td><td>-1.459</td><td>-0.990</td></tr><tr><td>CCSDTQ</td><td>2.518</td><td>4.022</td><td>1.629</td><td>1.751</td><td>7.279</td><td>0.020</td><td>-1.364</td><td>-0.906</td></tr><tr><td>CCSDTQP</td><td>2.517</td><td>4.023</td><td>1.627</td><td>1.746</td><td>7.287</td><td>0.020</td><td>-1.342</td><td>-0.885</td></tr><tr><td rowspan="4">aug-cc-pVDZ</td><td>CCSD</td><td>2.427</td><td>4.020</td><td>1.397</td><td>1.602</td><td>7.043</td><td>0.018</td><td>-2.078</td><td>-1.634</td></tr><tr><td>CCSDT</td><td>2.368</td><td>3.986</td><td>1.337</td><td>1.524</td><td>7.040</td><td>0.020</td><td>-2.457</td><td>-1.976</td></tr><tr><td>CCSDTQ</td><td>2.356</td><td>3.997</td><td>1.319</td><td>1.486</td><td>7.091</td><td>0.020</td><td>-2.403</td><td>-1.932</td></tr><tr><td>CCSD</td><td>2.457</td><td>4.013</td><td>1.416</td><td>1.620</td><td>7.231</td><td>0.018</td><td>-1.929</td><td>-1.476</td></tr><tr><td rowspan="2">aug-cc-pVTZ</td><td>CCSD</td><td>2.389</td><td>3.954</td><td>1.346</td><td>1.534</td><td>7.165</td><td>0.020</td><td>-2.379</td><td>-1.873</td></tr><tr><td>CCSDT</td><td>2.475</td><td>4.024</td><td>1.436</td><td>1.635</td><td>7.296</td><td>0.018</td><td>-1.861</td><td>-1.406</td></tr><tr><td rowspan="4">d-aug-cc-pVDZ</td><td>CCSD</td><td>2.417</td><td>4.012</td><td>1.386</td><td>1.596</td><td>7.027</td><td>0.016</td><td>-1.839</td><td>-1.410</td></tr><tr><td>CCSDT</td><td>2.359</td><td>3.978</td><td>1.327</td><td>1.518</td><td>7.024</td><td>0.019</td><td>-2.282</td><td>-1.803</td></tr><tr><td>CCSD</td><td>2.454</td><td>4.011</td><td>1.418</td><td>1.619</td><td>7.224</td><td>0.017</td><td>-1.751</td><td>-1.309</td></tr><tr><td>CCSDTQ</td><td>2.387</td><td>3.952</td><td>1.347</td><td>1.533</td><td>7.158</td><td>0.020</td><td>-2.248</td><td>-1.743</td></tr><tr><td rowspan="2">d-aug-cc-pVQZ</td><td>CCSD</td><td>2.475</td><td>4.023</td><td>1.439</td><td>1.636</td><td>7.192</td><td>0.018</td><td>-1.747</td><td>-1.299</td></tr><tr><td>CCSDT</td><td>2.375</td><td>3.966</td><td>1.325</td><td>1.491</td><td>7.225</td><td>0.020</td><td>-2.302</td><td>-1.809</td></tr><tr><td rowspan="2">aug-cc-pVTZ</td><td>TBE<sup>a</sup></td><td>2.41±0.01</td><td>3.99±0.01</td><td>1.36±0.01</td><td>1.52±0.01</td><td>7.34±0.01</td><td>0.020±0.001</td><td>-2.15±0.03</td><td>-1.66±0.03</td></tr><tr><td>TBE<sup>b</sup></td><td>2.393<sup>c</sup></td><td>3.98<sup>d</sup></td><td>1.33<sup>e</sup></td><td>1.48<sup>e</sup></td><td>7.12<sup>d</sup></td><td>0.018<sup>d</sup>;0.025<sup>e</sup></td><td>-2.52<sup>e</sup></td><td>-3.45<sup>f</sup></td></tr><tr><td rowspan="2">Lit.</td><td>Th.</td><td>2.333<sup>e</sup></td><td>3.97<sup>g</sup></td><td></td><td>1.73<sup>h</sup></td><td>7.30<sup>g</sup></td><td>0.021<sup>g</sup>;0.018<sup>i</sup></td><td></td><td></td></tr><tr><td>Exp.</td><td>2.332<sup>k</sup></td><td>4.07<sup>l</sup></td><td></td><td>1.53±0.11<sup>m</sup></td><td>7.11<sup>k</sup></td><td>0.028<sup>n</sup></td><td></td><td>-0.33±0.16<sup>o</sup></td></tr><tr><td colspan="2"></td><td colspan="3"><sup>1</sup>B<sub>2</sub> (Ryd, n → 3p)</td><td colspan="3"><sup>1</sup>A<sub>1</sub> (Ryd, n → 3p)</td><td colspan="3"><sup>1</sup>A<sub>1</sub> (Val, π → π*)</td></tr><tr><th>Basis</th><th>Method</th><th>ΔE<sub>vert</sub></th><th>f</th><th>μ<sup>ES</sup><sub>vert</sub></th><th>ΔE<sub>vert</sub></th><th>f</th><th>μ<sup>ES</sup><sub>vert</sub></th><th>ΔE<sub>vert</sub></th><th>f</th><th>μ<sup>ES</sup><sub>vert</sub></th></tr><tr><td rowspan="4">6-31+G(d)</td><td>CCSD</td><td>7.994</td><td>0.042</td><td>0.199</td><td>8.282</td><td>0.060</td><td>0.082</td><td>10.042</td><td>0.183</td><td>2.057</td></tr><tr><td>CCSDT</td><td>8.007</td><td>0.040</td><td>0.625</td><td>8.295</td><td>0.058</td><td>-0.062</td><td>9.829</td><td>0.163</td><td>1.737</td></tr><tr><td>CCSDTQ</td><td>8.045</td><td>0.040</td><td>0.504</td><td>8.341</td><td>0.058</td><td>-0.076</td><td>9.779</td><td>0.159</td><td>1.670</td></tr><tr><td>CCSDTQP</td><td>8.051</td><td>0.041</td><td>0.483</td><td>8.350</td><td>0.058</td><td>-0.074</td><td>9.776</td><td>0.158</td><td>1.663</td></tr><tr><td rowspan="4">aug-cc-pVDZ</td><td>CCSD</td><td>7.993</td><td>0.044</td><td>0.414</td><td>8.052</td><td>0.058</td><td>-2.058</td><td>9.752</td><td>0.157</td><td>1.470</td></tr><tr><td>CCSDT</td><td>8.002</td><td>0.042</td><td>0.631</td><td>8.068</td><td>0.057</td><td>-2.190</td><td>9.588</td><td>0.147</td><td>1.356</td></tr><tr><td>CCSDTQ</td><td>8.045</td><td>0.042</td><td>0.557</td><td>8.119</td><td>0.057</td><td>-2.202</td><td>9.544</td><td>0.143</td><td>1.292</td></tr><tr><td>CCSD</td><td>8.120</td><td>0.040</td><td>0.319</td><td>8.210</td><td>0.054</td><td>-1.438</td><td>9.670</td><td>0.139</td><td>1.503</td></tr><tr><td rowspan="2">aug-cc-pVTZ</td><td>CCSD</td><td>8.070</td><td>0.038</td><td>0.549</td><td>8.164</td><td>0.052</td><td>-1.617</td><td>9.488</td><td>0.131</td><td>1.372</td></tr><tr><td>CCSDT</td><td>8.153</td><td>0.039</td><td>0.292</td><td>8.267</td><td>0.053</td><td>-1.145</td><td>9.647</td><td>0.125</td><td>1.388</td></tr><tr><td rowspan="4">d-aug-cc-pVDZ</td><td>CCSD</td><td>7.834</td><td>0.036</td><td>0.062</td><td>7.962</td><td>0.049</td><td>-0.019</td><td>9.885</td><td>0.126</td><td>3.672</td></tr><tr><td>CCSDT</td><td>7.846</td><td>0.035</td><td>0.280</td><td>7.983</td><td>0.049</td><td>-0.301</td><td>9.748</td><td>0.119</td><td>3.553</td></tr><tr><td>CCSD</td><td>8.026</td><td>0.035</td><td>-0.104</td><td>8.169</td><td>0.050</td><td>-0.044</td><td>9.862</td><td>0.124</td><td>3.783</td></tr><tr><td>CCSDTQ</td><td>7.978</td><td>0.033</td><td>0.117</td><td>8.126</td><td>0.049</td><td>-0.369</td><td>9.719</td><td>0.108</td><td>4.347</td></tr><tr><td rowspan="2">d-aug-cc-pVQZ</td><td>CCSD</td><td>8.095</td><td>0.035</td><td>-0.115</td><td>8.240</td><td>0.050</td><td>-0.055</td><td>9.869</td><td>0.123</td><td>3.918</td></tr><tr><td>CCSDT</td><td>8.119</td><td>0.039</td><td>0.454</td><td>8.224</td><td>0.052</td><td>-1.628</td><td>9.441</td><td>0.127</td><td>1.301</td></tr><tr><td rowspan="2">aug-cc-pVTZ</td><td>TBE<sup>a</sup></td><td>8.16±0.02</td><td>0.035±0.002</td><td>0.21±0.20</td><td>8.28±0.04</td><td>0.050±0.001</td><td>-0.69±0.43</td><td>9.52±0.12</td><td>0.107±0.002</td><td>2.46±1.36</td></tr><tr><td>TBE<sup>b</sup></td><td>7.94<sup>d</sup></td><td>0.040<sup>d</sup>;0.041<sup>e</sup></td><td>0.85<sup>e</sup></td><td>8.16<sup>d</sup></td><td>0.043<sup>d</sup>;0.058<sup>e</sup></td><td>-2.16<sup>e</sup></td><td>9.83<sup>d</sup></td><td>0.100<sup>d</sup></td><td></td></tr><tr><td rowspan="2">Lit.</td><td>Th.</td><td>8.14<sup>g</sup></td><td>0.037<sup>g</sup>;0.035<sup>i</sup></td><td></td><td>8.27<sup>g</sup></td><td>0.052<sup>g</sup>;0.050<sup>i</sup></td><td></td><td>9.26<sup>g</sup></td><td>0.135<sup>g</sup>;0.093<sup>i</sup></td><td></td></tr><tr><td>Exp.</td><td>7.97<sup>l</sup></td><td>0.017<sup>n</sup></td><td></td><td>8.14<sup>l</sup></td><td>0.032<sup>n</sup></td><td></td><td></td><td></td><td></td></tr></tbody></table>

<footnote>a,b See corresponding footnotes in Table 7; c CCSD(T)/CBS value from Ref. 47; d MR-AQCC-LRT calculations from Ref. 156; e CCSDT/aug-cc-pVDZ results from Ref. 74; f MRDCI value from Ref. 157; g exFCI/aug-cc-pVTZ transition energies corrected for basis set effects up to d-aug-cc-pVFZ and LR-CC3/aug-cc-pVTZ for f from Ref. 5; h CC2/aug-cc-pVQZ figure from Ref. 45; i EOM-CCSD from Ref. 158; j CASPT2/TZVP values from Ref. 2; k Electric resonance spectroscopy from Ref. 159; l Various experimental sources collected in Ref. 135; m Stark effect measurement on lineshapes from Ref. 160; n EELS values from Ref. 161; o Values measured from polarized electrochromism reported in Refs. 162,163; p Stark effect from quantum beat spectroscopy from Ref. 164; q Dipole (e,e) spectroscopy from Ref. 165.</footnote>
<table><caption>Table 8: Ground-state dipole moment $\mu^{\text{GS}}$, vertical transition energies $\Delta E_{\text{vert}}$, oscillator strengths $f$, and excited state dipole moments, $\mu_{\text{vert}}^{\text{ES}}$ and $\mu_{\text{adia}}^{\text{ES}}$, determined of formaldehyde. See caption of Table 1 for details.</caption><thead><tr><th rowspan="2">Basis</th><th rowspan="2">Method</th><th colspan="3"><sup>1</sup>A<sub>1</sub></th><th colspan="3"><sup>1</sup>A<sub>2</sub> (Val, n → π<sup>*</sup>)</th><th colspan="3"><sup>1</sup>B<sub>2</sub> (Ryd, n → 3s)</th></tr><tr><th>μ<sup>GS</sup></th><th>ΔE<sub>vert</sub></th><th>μ<sup>ES</sup><sub>vert</sub></th><th>μ<sup>ES</sup><sub>adia</sub></th><th>ΔE<sub>vert</sub></th><th>f</th><th>μ<sup>ES</sup><sub>vert</sub></th><th>μ<sup>ES</sup><sub>adia</sub></th></tr></thead><tbody><tr><td rowspan="5">6-31+G(d)</td><td>CCSD</td><td>2.584</td><td>4.031</td><td>1.710</td><td>1.870</td><td>7.238</td><td>0.017</td><td>-0.901</td><td>-0.479</td></tr><tr><td>CCSDT</td><td>2.529</td><td>4.012</td><td>1.649</td><td>1.790</td><td>7.232</td><td>0.021</td><td>-1.459</td><td>-0.990</td></tr><tr><td>CCSDTQ</td><td>2.518</td><td>4.022</td><td>1.629</td><td>1.751</td><td>7.279</td><td>0.020</td><td>-1.364</td><td>-0.906</td></tr><tr><td>CCSDTQP</td><td>2.517</td><td>4.023</td><td>1.627</td><td>1.746</td><td>7.287</td><td>0.020</td><td>-1.342</td><td>-0.885</td></tr><tr><td>aug-cc-pVDZ</td><td>CCSD</td><td>2.427</td><td>4.020</td><td>1.397</td><td>1.602</td><td>7.043</td><td>0.018</td><td>-2.078</td><td>-1.634</td></tr><tr><td rowspan="5">aug-cc-pVTZ</td><td>CCSDT</td><td>2.368</td><td>3.986</td><td>1.337</td><td>1.524</td><td>7.040</td><td>0.020</td><td>-2.457</td><td>-1.976</td></tr><tr><td>CCSDTQ</td><td>2.356</td><td>3.997</td><td>1.319</td><td>1.486</td><td>7.091</td><td>0.020</td><td>-2.403</td><td>-1.932</td></tr><tr><td>CCSD</td><td>2.457</td><td>4.013</td><td>1.416</td><td>1.620</td><td>7.231</td><td>0.018</td><td>-1.929</td><td>-1.476</td></tr><tr><td>CCSDT</td><td>2.389</td><td>3.954</td><td>1.346</td><td>1.534</td><td>7.165</td><td>0.020</td><td>-2.379</td><td>-1.873</td></tr><tr><td>aug-cc-pVQZ</td><td>CCSD</td><td>2.475</td><td>4.024</td><td>1.436</td><td>1.635</td><td>7.296</td><td>0.018</td><td>-1.861</td><td>-1.406</td></tr><tr><td rowspan="5">d-aug-cc-pVDZ</td><td>CCSD</td><td>2.417</td><td>4.012</td><td>1.386</td><td>1.596</td><td>7.027</td><td>0.016</td><td>-1.839</td><td>-1.410</td></tr><tr><td>CCSDT</td><td>2.359</td><td>3.978</td><td>1.327</td><td>1.518</td><td>7.024</td><td>0.019</td><td>-2.282</td><td>-1.803</td></tr><tr><td>aug-cc-pVTZ</td><td>CCSD</td><td>2.454</td><td>4.011</td><td>1.418</td><td>1.619</td><td>7.224</td><td>0.017</td><td>-1.751</td><td>-1.309</td></tr><tr><td>CCSDTQ</td><td>2.387</td><td>3.952</td><td>1.347</td><td>1.533</td><td>7.158</td><td>0.020</td><td>-2.248</td><td>-1.743</td></tr><tr><td>aug-cc-pVQZ</td><td>CCSD</td><td>2.475</td><td>4.023</td><td>1.439</td><td>1.636</td><td>7.192</td><td>0.018</td><td>-1.747</td><td>-1.299</td></tr><tr><td rowspan="3">aug-cc-pVTZ</td><td>TBE<sup>a</sup></td><td>2.375</td><td>3.966</td><td>1.325</td><td>1.491</td><td>7.225</td><td>0.020</td><td>-2.302</td><td>-1.809</td></tr><tr><td>CBS</td><td>2.41±0.01</td><td>3.99±0.01</td><td>1.36±0.01</td><td>1.52±0.01</td><td>7.34±0.01</td><td>0.020±0.001</td><td>-2.15±0.03</td><td>-1.66±0.03</td></tr><tr><td>Lit.</td><td>Th.</td><td>2.393<sup>c</sup></td><td>3.98<sup>d</sup></td><td>1.33<sup>e</sup></td><td>1.48<sup>e</sup></td><td>7.12<sup>d</sup></td><td>0.018<sup>d</sup>;0.025<sup>e</sup></td><td>-2.52<sup>e</sup></td><td>-3.45<sup>f</sup></td></tr><tr><td rowspan="3">aug-cc-pVQZ</td><td></td><td>2.333<sup>e</sup></td><td>3.97<sup>g</sup></td><td></td><td>1.73<sup>h</sup></td><td>7.30<sup>g</sup></td><td>0.021<sup>g</sup>;0.018<sup>i</sup></td><td></td><td></td></tr><tr><td></td><td>2.44<sup>j</sup></td><td>3.98<sup>j</sup></td><td></td><td>1.46<sup>j</sup></td><td></td><td></td><td></td><td></td></tr><tr><td>Exp.</td><td>2.332<sup>k</sup></td><td>4.07<sup>l</sup></td><td></td><td>1.53±0.11<sup>m</sup></td><td>7.11<sup>k</sup></td><td>0.028<sup>n</sup></td><td></td><td>-0.33±0.16<sup>o</sup></td></tr><tr><td></td><td></td><td></td><td></td><td>1.47<sup>p</sup></td><td></td><td>0.041<sup>q</sup></td><td></td><td></td><td></td></tr><tr><th rowspan="2">Basis</th><th rowspan="2">Method</th><th colspan="3"><sup>1</sup>B<sub>2</sub> (Ryd, n → 3p)</th><th colspan="3"><sup>1</sup>A<sub>1</sub> (Ryd, n → 3p)</th><th colspan="3"><sup>1</sup>A<sub>1</sub> (Val, π → π<sup>*</sup>)</th></tr><tr><th>ΔE<sub>vert</sub></th><th>f</th><th>μ<sup>ES</sup><sub>vert</sub></th><th>ΔE<sub>vert</sub></th><th>f</th><th>μ<sup>ES</sup><sub>vert</sub></th><th>ΔE<sub>vert</sub></th><th>f</th><th>μ<sup>ES</sup><sub>vert</sub></th></tr><tr><td rowspan="5">6-31+G(d)</td><td>CCSD</td><td>7.994</td><td>0.042</td><td>0.199</td><td>8.282</td><td>0.060</td><td>0.082</td><td>10.042</td><td>0.183</td><td>2.057</td></tr><tr><td>CCSDT</td><td>8.007</td><td>0.040</td><td>0.625</td><td>8.295</td><td>0.058</td><td>-0.062</td><td>9.829</td><td>0.163</td><td>1.737</td></tr><tr><td>CCSDTQ</td><td>8.045</td><td>0.040</td><td>0.504</td><td>8.341</td><td>0.058</td><td>-0.076</td><td>9.779</td><td>0.159</td><td>1.670</td></tr><tr><td>CCSDTQP</td><td>8.051</td><td>0.041</td><td>0.483</td><td>8.350</td><td>0.058</td><td>-0.074</td><td>9.776</td><td>0.158</td><td>1.663</td></tr><tr><td>aug-cc-pVDZ</td><td>CCSD</td><td>7.993</td><td>0.044</td><td>0.414</td><td>8.052</td><td>0.058</td><td>-2.058</td><td>9.752</td><td>0.157</td><td>1.470</td></tr><tr><td rowspan="5">aug-cc-pVTZ</td><td>CCSD</td><td>8.002</td><td>0.042</td><td>0.631</td><td>8.068</td><td>0.057</td><td>-2.190</td><td>9.588</td><td>0.147</td><td>1.356</td></tr><tr><td>CCSDT</td><td>8.045</td><td>0.042</td><td>0.557</td><td>8.119</td><td>0.057</td><td>-2.202</td><td>9.544</td><td>0.143</td><td>1.292</td></tr><tr><td>CCSDTQ</td><td>8.120</td><td>0.040</td><td>0.319</td><td>8.210</td><td>0.054</td><td>-1.438</td><td>9.670</td><td>0.139</td><td>1.503</td></tr><tr><td>CCSD</td><td>8.070</td><td>0.038</td><td>0.549</td><td>8.164</td><td>0.052</td><td>-1.617</td><td>9.488</td><td>0.131</td><td>1.372</td></tr><tr><td>aug-cc-pVQZ</td><td>CCSD</td><td>8.153</td><td>0.039</td><td>0.292</td><td>8.267</td><td>0.053</td><td>-1.145</td><td>9.647</td><td>0.125</td><td>1.388</td></tr><tr><td rowspan="5">d-aug-cc-pVDZ</td><td>CCSD</td><td>7.834</td><td>0.036</td><td>0.062</td><td>7.962</td><td>0.049</td><td>-0.019</td><td>9.885</td><td>0.126</td><td>3.672</td></tr><tr><td>CCSDT</td><td>7.846</td><td>0.035</td><td>0.280</td><td>7.983</td><td>0.049</td><td>-0.301</td><td>9.748</td><td>0.119</td><td>3.553</td></tr><tr><td>aug-cc-pVTZ</td><td>CCSD</td><td>8.026</td><td>0.035</td><td>-0.104</td><td>8.169</td><td>0.050</td><td>-0.044</td><td>9.862</td><td>0.124</td><td>3.783</td></tr><tr><td>CCSDTQ</td><td>8.798</td><td>0.033</td><td>0.117</td><td>8.126</td><td>0.049</td><td>-0.369</td><td>9.719</td><td>0.108</td><td>4.347</td></tr><tr><td>aug-cc-pVQZ</td><td>CCSD</td><td>8.095</td><td>0.035</td><td>-0.115</td><td>8.240</td><td>0.050</td><td>-0.055</td><td>9.869</td><td>0.123</td><td>3.918</td></tr><tr><td rowspan="3">aug-cc-pVTZ</td><td>TBE<sup>a</sup></td><td>8.119</td><td>0.039</td><td>0.454</td><td>8.224</td><td>0.052</td><td>-1.628</td><td>9.441</td><td>0.127</td><td>1.301</td></tr><tr><td>CBS</td><td>8.16±0.02</td><td>0.035±0.002</td><td>0.21±0.20</td><td>8.28±0.04</td><td>0.050±0.001</td><td>-0.69±0.43</td><td>9.52±0.12</td><td>0.107±0.002</td><td>2.46±1.36</td></tr><tr><td>Lit.</td><td>Th.</td><td>7.94<sup>d</sup></td><td>0.040<sup>d</sup>;0.041<sup>e</sup></td><td>0.85<sup>e</sup></td><td>8.16<sup>d</sup></td><td>0.043<sup>d</sup>;0.058<sup>e</sup></td><td>-2.16<sup>e</sup></td><td>9.83<sup>d</sup></td><td>0.100<sup>d</sup></td><td></td></tr><tr><td rowspan="3">aug-cc-pVQZ</td><td></td><td>8.14<sup>g</sup></td><td>0.037<sup>g</sup>;0.035<sup>i</sup></td><td></td><td>8.27<sup>g</sup></td><td>0.052<sup>g</sup>;0.050<sup>i</sup></td><td></td><td>9.26<sup>g</sup></td><td>0.135<sup>g</sup>;0.093<sup>i</sup></td><td></td></tr><tr><td></td><td>7.97<sup>l</sup></td><td>0.017<sup>n</sup></td><td></td><td>8.14<sup>l</sup></td><td>0.032<sup>n</sup></td><td></td><td></td><td></td><td></td></tr><tr><td>Exp.</td><td></td><td>0.018<sup>q</sup></td><td></td><td></td><td>0.061<sup>q</sup></td><td></td><td></td><td></td><td></td><td></td></tr></tbody></table>

<footnote>a,b See corresponding footnotes in Table 7; c CCSD(T)/CBS value from Ref. 47; d MR-AQCC-LRT calculations from Ref. 156; e CCSDT/aug-cc-pVDZ results from Ref. 74; f MRDCI value from Ref. 157; g exFCI/aug-cc-pVTZ transition energies corrected for basis set effects up to d-aug-cc-pVFZ and LR-CC3/aug-cc-pVTZ for f from Ref. 5; h CC2/aug-cc-pVQZ figure from Ref. 45; i EOM-CCSD from Ref. 158; j CASPT2/TZVP values from Ref. 2; k Electric resonance spectroscopy from Ref. 159; l Various experimental sources collected in Ref. 135; m Stark effect measurement on lineshapes from Ref. 160; n EELS values from Ref. 161; o Values measured from polarized electrochromism reported in Refs. 162,163; p Stark effect from quantum beat spectroscopy from Ref. 164; q Dipole (e,e) spectroscopy from Ref. 165.</footnote>
<table><caption>Table 9: Ground-state dipole moment $\mu^{GS}$, vertical transition energies $\Delta E_{\text{vert}}$, oscillator strengths $f$, and excited state dipole moments, $\mu_{\text{vert}}^{\text{ES}}$ and $\mu_{\text{adia}}^{\text{ES}}$, determined of thioformaldehyde. See caption of Table 1 for details.</caption><thead><tr><th rowspan="2">Basis</th><th rowspan="2">Method</th><th colspan="2"><sup>1</sup>A<sub>1</sub></th><th colspan="2"><sup>1</sup>A<sub>2</sub> (Val, n → π*)</th><th colspan="2"><sup>1</sup>B<sub>2</sub> (Ryd, n → 4s)</th><th colspan="3"><sup>1</sup>A<sub>1</sub> (Val, π → π*)</th></tr><tr><th>μ<sup>GS</sup></th><th>ΔE<sub>vert</sub></th><th>μ<sup>ES</sup><sub>vert</sub></th><th>μ<sup>ES</sup><sub>adia</sub></th><th>ΔE<sub>vert</sub></th><th>f</th><th>μ<sup>ES</sup><sub>vert</sub></th><th>ΔE<sub>vert</sub></th><th>f</th><th>μ<sup>ES</sup><sub>vert</sub></th></tr></thead><tbody><tr><td rowspan="4">6-31+G(d)</td><td>CCSD</td><td>1.747</td><td>2.302</td><td>0.933</td><td>0.968</td><td>5.937</td><td>0.019</td><td>-3.378</td><td>-3.070</td><td>6.961</td><td>0.261</td><td>1.964</td></tr><tr><td>CCSDT</td><td>1.733</td><td>2.244</td><td>0.948</td><td>0.990</td><td>5.875</td><td>0.018</td><td>-3.570</td><td>-3.257</td><td>6.790</td><td>0.223</td><td>1.745</td></tr><tr><td>CCSDTQ</td><td>1.720</td><td>2.246</td><td>0.919</td><td>0.947</td><td>5.890</td><td>0.019</td><td>-3.563</td><td>-3.253</td><td>6.713</td><td>0.191</td><td>1.334</td></tr><tr><td>CCSDTQP</td><td>1.719</td><td>2.247</td><td>0.917</td><td>0.943</td><td>5.893</td><td>0.019</td><td>-3.562</td><td>-3.252</td><td>6.708</td><td>0.189</td><td>1.305</td></tr><tr><td rowspan="4">aug-cc-pVDZ</td><td>CCSD</td><td>1.742</td><td>2.325</td><td>0.851</td><td>0.873</td><td>5.841</td><td>0.012</td><td>-3.938</td><td>-3.693</td><td>6.749</td><td>0.251</td><td>1.875</td></tr><tr><td>CCSDT</td><td>1.716</td><td>2.253</td><td>0.870</td><td>0.898</td><td>5.796</td><td>0.011</td><td>-4.140</td><td>-3.885</td><td>6.597</td><td>0.182</td><td>1.753</td></tr><tr><td>CCSDTQ</td><td>1.704</td><td>2.255</td><td>0.848</td><td>0.864</td><td>5.817</td><td>0.011</td><td>-4.134</td><td>-3.885</td><td>6.512</td><td>0.152</td><td>1.264</td></tr><tr><td>CCSDTQ</td><td>1.737</td><td>2.291</td><td>0.848</td><td>0.870</td><td>5.970</td><td>0.014</td><td>-3.374</td><td>-3.149</td><td>6.633</td><td>0.206</td><td>2.379</td></tr><tr><td rowspan="2">aug-cc-pVTZ</td><td>CCSD</td><td>1.706</td><td>2.207</td><td>0.865</td><td>0.890</td><td>5.900</td><td>0.012</td><td>-3.665</td><td>-3.422</td><td>6.467</td><td>0.163</td><td>1.698</td></tr><tr><td>CCSDT</td><td>1.759</td><td>2.296</td><td>0.869</td><td>0.890</td><td>6.018</td><td>0.014</td><td>-3.162</td><td>-2.944</td><td>6.607</td><td>0.200</td><td>2.385</td></tr><tr><td rowspan="4">d-aug-cc-pVDZ</td><td>CCSD</td><td>1.735</td><td>2.324</td><td>0.858</td><td>0.877</td><td>5.804</td><td>0.015</td><td>-3.258</td><td>-3.055</td><td>6.678</td><td>0.165</td><td>-0.038</td></tr><tr><td>CCSDT</td><td>1.709</td><td>2.252</td><td>0.875</td><td>0.901</td><td>5.761</td><td>0.014</td><td>-3.495</td><td>-3.281</td><td>6.577</td><td>0.191</td><td>1.749</td></tr><tr><td>CCSD</td><td>1.737</td><td>2.291</td><td>0.851</td><td>0.872</td><td>5.958</td><td>0.015</td><td>-3.050</td><td>-2.847</td><td>6.627</td><td>0.206</td><td>2.376</td></tr><tr><td>CCSDT</td><td>1.707</td><td>2.207</td><td>0.867</td><td>0.892</td><td>5.888</td><td>0.013</td><td>-3.379</td><td>-3.154</td><td>6.463</td><td>0.162</td><td>2.193</td></tr><tr><td rowspan="2">d-aug-cc-pVQZ</td><td>CCSD</td><td>1.759</td><td>2.296</td><td>0.871</td><td>0.892</td><td>6.012</td><td>0.014</td><td>-2.999</td><td>-2.794</td><td>6.604</td><td>0.199</td><td>2.383</td></tr><tr><td>aug-cc-pVTZ</td><td>TBE<sup>a</sup></td><td>1.694</td><td>2.210</td><td>0.840</td><td>0.851</td><td>5.923</td><td>0.013</td><td>-3.658</td><td>-3.421</td><td>6.377</td><td>0.135</td><td>1.179</td></tr><tr><td rowspan="2">CBS</td><td>TBE<sup>b</sup></td><td>1.73±0.01</td><td>2.22±0.01</td><td>0.088±0.01</td><td>0.089±0.01</td><td>6.01±0.01</td><td>0.013±0.001</td><td>-3.27±0.03</td><td>-3.05±0.02</td><td>6.33±0.01</td><td>0.13±0.01</td><td>1.20±0.01</td></tr><tr><td>Lit.</td><td>1.700<sup>c</sup><br/>1.72<sup>e</sup></td><td>2.20<sup>d</sup></td><td></td><td>0.96<sup>e</sup></td><td>5.99<sup>d</sup></td><td>0.012<sup>f</sup></td><td></td><td></td><td>6.34<sup>g</sup></td><td>0.178<sup>f</sup></td><td></td></tr><tr><td rowspan="2">Exp.</td><td></td><td>1.649<sup>h</sup></td><td>2.033<sup>i</sup></td><td></td><td>0.850±0.002<sup>j</sup></td><td>5.841<sup>i</sup></td><td></td><td></td><td>-2.2±0.3<sup>k</sup></td><td>6.60<sup>i</sup></td><td></td><td></td></tr><tr><td></td><td>1.647<sup>l</sup></td><td></td><td></td><td>0.815±0.020<sup>m</sup></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr></tbody></table>

a,b See corresponding footnotes in Table 7; c CCSD(T)/aug-cc-pVQZ from Ref. 166; d exFCI/aug-cc-pVTZ transition energies corrected for basis set effects up to d-aug-cc-pVQZ from Ref. 5; e CC2/aug-cc-pVQZ from Ref. 45; f LR-CC3/aug-cc-pVTZ f from Ref. 5; g CCSDTQ/aug-cc-pVDZ transition energy corrected for basis set effects up to d-aug-cc-pVQZ from Ref. 5; h Molecular beam electric resonance value from Ref. 159; i 0-0 energies listed in Table 13 of Ref. 167; j Microwave-optical double resonance measurements of Stark effect from Ref. 168; k Stark effect on the absorption spectrum from Ref. 169; l Stark effect measurement on the microwave spectra from Ref. 170; m Intermodulated fluorescence of Stark effect from Ref. 171.

1.48±0.07,¹⁷² 1.56±0.07,¹⁷³ 1.4±0.1 D,¹⁷⁴ 1.53±0.11D,¹⁶⁰ and 1.47 D.¹⁶⁴ Somehow surprisingly, the most recent value obtained by Stark quantum beat spectroscopy has hardly been considered as reference in theoretical works than the maximal measured value of 1.56 D. On the theory side, one can highlight two significant earlier contributions (on the ES geometry): 1.48 D (CCSDT/aug-cc-pVDZ)⁷⁴ and 1.73 D (CC2/aug-cc-pVQZ).⁴⁵ We somehow reconcile these earlier results by using both large basis sets and high CC levels and considering both geometries, leading to a TBE/CBS of 1.52±0.01 D for the adiabatic value, right at the center of the experimental cloud. It is noteworthy that the geometrical relaxation induces a non-negligible increase of the magnitude of the dipole moment for the $A_2$ (¹A''') ES. We indeed found a TBE/CBS value of 1.36±0.02 D for the GS geometry. For this $\mu_{\text{vert}}^{\text{ES}}$, earlier estimates include the 1.46 D (CASPT2/TZVP)² and 1.38 D (CC2/aug-cc-pVTZ).⁴³ For the lowest $B_2$ Rydberg transition, we are aware of one experiment only (polarized electrochromism), leading to an ES dipole of -0.33±0.16 D.¹⁶²,¹⁶³ While theory does confirm the sign change, it returns a much larger amplitude for the dipole with -2.15±0.03 D (GS structure) or -1.66±0.03 D (ES geometry). At the CCSDT/aug-cc-pVDZ level, Hirata reported -2.52 D (vertical),⁷⁴ likely the best previous estimate. This significant discrepancy between theory and experiment was previously attributed to an (experimental) mixing between the two lowest-lying $B_2$ transitions.¹⁵⁷ It seems reasonable to state that theory has the edge in this

case. For the higher-lying ES, no Stark effect measurement are available, and our values are very likely more accurate than the previous ones reported at the CCSDT level but with a rather small basis set.⁷⁴ Nevertheless, the CBS extrapolation is uneasy and large error bars are obtained for all these high-lying ESs. The oscillator strengths of the three lowest Rydberg transitions, $B_2(n \rightarrow 3s)$, $B_2(n \rightarrow 3p)$, and $A_1(n \rightarrow 3s)$ have been respectively measured as 0.038, 0.017±0.02, and 0.038± 0.04 (absorption spectroscopy),¹⁷⁵ 0.028, 0.017, and 0.032 (EELS),¹⁶¹ 0.032, 0.019, and 0.036 (absorption),¹⁷⁶ and 0.041, 0.028, and 0.061 [dipole (e,e) spectroscopy].¹⁶⁵ Although the orders of magnitude are consistent with the present calculations, the theoretical values do not follow the same ranking as they yield *f* values of 0.020, 0.035, and 0.050. Such discrepancy has been attributed by other groups to the difficulty of assigning individual vibronic bands to a specific electronic transition in the experimental spectra.⁷⁴,¹⁷⁷ Interestingly, our current values are agreeing very well with previous theoretical estimates, that returned 0.018, 0.040, and 0.043 (MR-AQCC-LRT),¹⁵⁶ 0.025, 0.041, and 0.058 (CCSDT),⁷⁴ 0.018, 0.035, and 0.050 (EOM-CCSD),¹⁵⁸ and 0.021, 0.037, and 0.052 (CC3).⁵ For the brighter $\pi \rightarrow \pi^*$ transition, we are not aware of experimental *f* values, but theoretical values reported in previous works are of the order of 0.1: 0.100,¹⁵⁶ 0.093,¹⁵⁸ and 0.135,⁵ and our current TBE of 0.107±0.002 lies in the middle of these earlier data. In the original Thiel benchmark, the next $A_1$ ES with a larger *f* was actually considered.²
<table><caption>Table 9: Ground-state dipole moment $\mu^{\text{GS}}$, vertical transition energies $\Delta E_{\text{vert}}$, oscillator strengths $f$, and excited state dipole moments, $\mu_{\text{vert}}^{\text{ES}}$ and $\mu_{\text{adia}}^{\text{ES}}$, determined of thioformaldehyde. See caption of Table 1 for details.</caption><thead><tr><th rowspan="2">Basis</th><th rowspan="2">Method</th><th colspan="3"><sup>1</sup>A<sub>1</sub></th><th colspan="3"><sup>1</sup>B<sub>2</sub>(Ryd, n → 4s)</th><th colspan="3"><sup>1</sup>A<sub>1</sub>(Val, π → π*)</th></tr><tr><th>μ<sup>GS</sup></th><th>ΔE<sub>vert</sub></th><th>μ<sup>ES</sup><sub>vert</sub></th><th>μ<sup>ES</sup><sub>adia</sub></th><th>ΔE<sub>vert</sub></th><th>f</th><th>μ<sup>ES</sup><sub>vert</sub></th><th>ΔE<sub>vert</sub></th><th>f</th><th>μ<sup>ES</sup><sub>vert</sub></th></tr></thead><tbody><tr><td rowspan="4">6-31+G(d)</td><td>CCSD</td><td>1.747</td><td>2.302</td><td>0.933</td><td>0.968</td><td>5.937</td><td>0.019</td><td>-3.378</td><td>6.961</td><td>0.261</td><td>1.964</td></tr><tr><td>CCSDT</td><td>1.733</td><td>2.244</td><td>0.948</td><td>0.990</td><td>5.875</td><td>0.018</td><td>-3.570</td><td>6.790</td><td>0.223</td><td>1.745</td></tr><tr><td>CCSDTQ</td><td>1.720</td><td>2.246</td><td>0.919</td><td>0.947</td><td>5.890</td><td>0.019</td><td>-3.563</td><td>6.713</td><td>0.191</td><td>1.334</td></tr><tr><td>CCSDTQP</td><td>1.719</td><td>2.247</td><td>0.917</td><td>0.943</td><td>5.893</td><td>0.019</td><td>-3.562</td><td>6.708</td><td>0.189</td><td>1.305</td></tr><tr><td rowspan="4">aug-cc-pVDZ</td><td>CCSD</td><td>1.742</td><td>2.325</td><td>0.851</td><td>0.873</td><td>5.841</td><td>0.012</td><td>-3.938</td><td>6.749</td><td>0.251</td><td>1.875</td></tr><tr><td>CCSDT</td><td>1.716</td><td>2.253</td><td>0.870</td><td>0.898</td><td>5.796</td><td>0.011</td><td>-4.140</td><td>6.597</td><td>0.182</td><td>1.753</td></tr><tr><td>CCSDTQ</td><td>1.704</td><td>2.255</td><td>0.848</td><td>0.864</td><td>5.817</td><td>0.011</td><td>-4.134</td><td>6.512</td><td>0.152</td><td>1.264</td></tr><tr><td>CCSD</td><td>1.737</td><td>2.291</td><td>0.848</td><td>0.870</td><td>5.970</td><td>0.014</td><td>-3.374</td><td>6.633</td><td>0.206</td><td>2.379</td></tr><tr><td rowspan="2">aug-cc-pVTZ</td><td>CCSD</td><td>1.706</td><td>2.207</td><td>0.865</td><td>0.890</td><td>5.900</td><td>0.012</td><td>-3.665</td><td>6.467</td><td>0.163</td><td>1.698</td></tr><tr><td>CCSDT</td><td>1.759</td><td>2.296</td><td>0.869</td><td>0.890</td><td>6.018</td><td>0.014</td><td>-3.162</td><td>6.607</td><td>0.200</td><td>2.385</td></tr><tr><td rowspan="4">d-aug-cc-pVDZ</td><td>CCSD</td><td>1.735</td><td>2.324</td><td>0.858</td><td>0.877</td><td>5.804</td><td>0.015</td><td>-3.258</td><td>6.678</td><td>0.165</td><td>-0.038</td></tr><tr><td>CCSDT</td><td>1.709</td><td>2.252</td><td>0.875</td><td>0.901</td><td>5.761</td><td>0.014</td><td>-3.495</td><td>6.577</td><td>0.191</td><td>1.749</td></tr><tr><td>CCSD</td><td>1.737</td><td>2.291</td><td>0.851</td><td>0.872</td><td>5.958</td><td>0.015</td><td>-3.050</td><td>6.627</td><td>0.206</td><td>2.376</td></tr><tr><td>CCSDT</td><td>1.707</td><td>2.207</td><td>0.867</td><td>0.892</td><td>5.888</td><td>0.013</td><td>-3.379</td><td>6.463</td><td>0.162</td><td>2.193</td></tr><tr><td rowspan="2">d-aug-cc-pVQZ</td><td>CCSD</td><td>1.759</td><td>2.296</td><td>0.871</td><td>0.892</td><td>6.012</td><td>0.014</td><td>-2.999</td><td>6.604</td><td>0.199</td><td>2.383</td></tr><tr><td>TBE<sup>a</sup></td><td>1.694</td><td>2.210</td><td>0.840</td><td>0.851</td><td>5.923</td><td>0.013</td><td>-3.658</td><td>6.377</td><td>0.135</td><td>1.179</td></tr><tr><td rowspan="2">aug-cc-pVTZ</td><td>CBS</td><td>1.73±0.01</td><td>2.22±0.01</td><td>0.088±0.01</td><td>0.089±0.01</td><td>6.01±0.01</td><td>0.013±0.001</td><td>-3.27±0.03</td><td>6.33±0.01</td><td>0.13±0.01</td><td>1.20±0.01</td></tr><tr><td>Lit.</td><td>1.700<sup>c</sup><br/>1.72<sup>e</sup></td><td>2.20<sup>d</sup></td><td></td><td>0.96<sup>e</sup></td><td>5.99<sup>d</sup></td><td>0.012<sup>f</sup></td><td></td><td>6.34<sup>g</sup></td><td>0.178<sup>f</sup></td><td></td></tr><tr><td rowspan="2">Exp.</td><td></td><td>1.649<sup>h</sup></td><td>2.033<sup>i</sup></td><td></td><td>0.850±0.002<sup>j</sup></td><td>5.841<sup>i</sup></td><td></td><td>-2.2±0.3<sup>k</sup></td><td>6.60<sup>i</sup></td><td></td><td></td></tr><tr><td></td><td>1.647<sup>l</sup></td><td></td><td></td><td>0.815±0.020<sup>m</sup></td><td></td><td></td><td></td><td></td><td></td><td></td></tr></tbody></table>

<sup>a,b</sup>See corresponding footnotes in Table 7; <sup>c</sup>CCSD(T)/aug-cc-pVQZ from Ref. 166; <sup>d</sup>exFCI/aug-cc-pVTZ transition energies corrected for basis set effects up to d-aug-cc-pVQZ from Ref. 5; <sup>e</sup>CC2/aug-cc-pVQZ from Ref. 45; <sup>f</sup>LR-CC3/aug-cc-pVTZ from Ref. 5; <sup>g</sup>CCSDTQ/aug-cc-pVDZ transition energy corrected for basis set effects up to d-aug-cc-pVQZ from Ref. 5; <sup>h</sup>Molecular beam electric resonance value from Ref. 159; <sup>i</sup>0-0 energies listed in Table 13 of Ref. 167; <sup>j</sup>Microwave-optical double resonance measurements of Stark effect from Ref. 168; <sup>k</sup>Stark effect on the absorption spectrum from Ref. 169; <sup>l</sup>Stark effect measurement on the microwave spectra from Ref. 170; <sup>m</sup>Intermodulated fluorescence of Stark effect from Ref. 171.

1.48±0.07,¹⁷² 1.56±0.07,¹⁷³ 1.4±0.1 D,¹⁷⁴ 1.53±0.11D,¹⁶⁰ and 1.47 D.¹⁶⁴ Somehow surprisingly, the most recent value obtained by Stark quantum beat spectroscopy has hardly been considered as reference in theoretical works than the maximal measured value of 1.56 D. On the theory side, one can highlight two significant earlier contributions (on the ES geometry): 1.48 D (CCSDT/aug-cc-pVDZ)⁷⁴ and 1.73 D (CC2/aug-cc-pVQZ).⁴⁵ We somehow reconcile these earlier results by using both large basis sets and high CC levels and considering both geometries, leading to a TBE/CBS of 1.52±0.01 D for the adiabatic value, right at the center of the experimental cloud. It is noteworthy that the geometrical relaxation induces a non-negligible increase of the magnitude of the dipole moment for the $A_2$ (¹A''') ES. We indeed found a TBE/CBS value of 1.36±0.02 D for the GS geometry. For this $\mu_{\text{vert}}^{\text{ES}}$, earlier estimates include the 1.46 D (CASPT2/TZVP)² and 1.38 D (CC2/aug-cc-pVTZ).⁴³ For the lowest $B_2$ Rydberg transition, we are aware of one experiment only (polarized electrochromism), leading to an ES dipole of -0.33±0.16 D.¹⁶²,¹⁶³ While theory does confirm the sign change, it returns a much larger amplitude for the dipole with -2.15±0.03 D (GS structure) or -1.66±0.03 D (ES geometry). At the CCSDT/aug-cc-pVDZ level, Hirata reported -2.52 D (vertical),⁷⁴ likely the best previous estimate. This significant discrepancy between theory and experiment was previously attributed to an (experimental) mixing between the two lowest-lying $B_2$ transitions.¹⁵⁷ It seems reasonable to state that theory has the edge in this

case. For the higher-lying ES, no Stark effect measurement are available, and our values are very likely more accurate than the previous ones reported at the CCSDT level but with a rather small basis set.⁷⁴ Nevertheless, the CBS extrapolation is uneasy and large error bars are obtained for all these high-lying ESs. The oscillator strengths of the three lowest Rydberg transitions, $B_2(n \to 3s)$, $B_2(n \to 3p)$, and $A_1(n \to 3s)$ have been respectively measured as 0.038, 0.017±0.02, and 0.038± 0.04 (absorption spectroscopy),¹⁷⁵ 0.028, 0.017, and 0.032 (EELS),¹⁶¹ 0.032, 0.019, and 0.036 (absorption),¹⁷⁶ and 0.041, 0.028, and 0.061 [dipole (e,e) spectroscopy].¹⁶⁵ Although the orders of magnitude are consistent with the present calculations, the theoretical values do not follow the same ranking as they yield $f$ values of 0.020, 0.035, and 0.050. Such discrepancy has been attributed by other groups to the difficulty of assigning individual vibronic bands to a specific electronic transition in the experimental spectra.⁷⁴,¹⁷⁷ Interestingly, our current values are agreeing very well with previous theoretical estimates, that returned 0.018, 0.040, and 0.043 (MR-AQCC-LRT),¹⁵⁶ 0.025, 0.041, and 0.058 (CCSDT),⁷⁴ 0.018, 0.035, and 0.050 (EOM-CCSD),¹⁵⁸ and 0.021, 0.037, and 0.052 (CC3).⁵ For the brighter $\pi \to \pi^*$ transition, we are not aware of experimental $f$ values, but theoretical values reported in previous works are of the order of 0.1: 0.100,¹⁵⁶ 0.093,¹⁵⁸ and 0.135,⁵ and our current TBE of 0.107±0.002 lies in the middle of these earlier data. In the original Thiel benchmark, the next $A_1$ ES with a larger $f$ was actually considered.²
In thioformaldehyde (Table 9), one notes relatively stable $\Delta E_{\text{vert}}$ and $\mu^{\text{ES}}$ for the lowest dipole-forbidden $A_2$ transition: the convergence is rather fast with respect to both CC expansion and basis set size, so that we can safely report accurate TBE/CBS for both the GS and ES structures. The change of ES dipole between the two geometries is limited as well, contrasting with formaldehyde. This is because there is no puckering effect in thioformaldehyde's $A_2$ ES: the true minimum belongs to the $C_{2v}$ point group.33,178,179 For the first Rydberg transition ($B_2$) the impact of the basis set size is logically more pronounced with, e.g., a +0.76 D change between the CCSDT/aug-cc-pVTZ and CCSDT/d-aug-cc-pVTZ dipoles, making the TBE/CBS extrapolation uncertainty larger than for the $A_2$ ES. The difference between the values of $\mu^{\text{ES}}$ determined at the GS and ES equilibrium geometries are also larger for the Rydberg excitation than for the lowest transition, despite the planarity of all geometries. When selecting Dunning's basis sets, the weak oscillator strength of the $B_2$ transition always falls in a rather tight window (0.011–0.015). The valence $A_1(\pi \rightarrow \pi^*)$ transition is clearly no cakewalk: not only the enlargement of the basis set yields significant changes of $\mu^{\text{ES}}$ (e.g., +0.49 D from CCSDT/aug-cc-pVTZ to CCSDT/d-aug-cc-pVTZ), but, in addition, the impact of the quadruples in the CC expansion becomes significant: the Q term induces a drop of $\mu^{\text{ES}}$ by ca. -0.40 D and a decrease of $f$ by ca. 15%. For this transition CCSD is clearly insufficient to obtain accurate ES properties. A chemical understanding of the underlying reasons for this large Q effect in the $A_1$ ES of thioformaldehyde would likely require an in-depth analysis of the various densities determined at various levels of theory, which is beyond our scope.

The value of $\mu^{\text{GS}}$ in thioformaldehyde was measured very accurately: 1.6491 ± 0.0004 D,159 and our TBE of 1.73±0.01 D is slightly higher. Same comment applies to a previous CCSD(T)/aug-cc-pVQZ value of (1.700 D)166 and a CC2/aug-cc-pVQZ value of 1.72 D.45 Like in formaldehyde, one can find a series of Stark measurements relying on various spectroscopic techniques for the lowest $^1A_2$ ES. Quite a range of magnitudes have been reported for $\mu^{\text{ES}}$: 0.79±0.04 D (absorption spectroscopy),180 0.838±0.008 D (laser-induced fluorescence excitation),181 0.850±0.002 D (microwave-optical double resonance),168 and 0.815±0.020 D (intermodulated fluorescence).171 Obviously, the error bars of these measurements are not overlapping, but the latter work warns that values between 0.77 and 0.93 D can be obtained.171 Our TBEs of 0.88±0.01 D (GS geometry) and 0.89±0.01 D (ES geometry) are therefore obviously compatible with the experimental measures. The only previous wave function-based TBE we are aware of is the CC2/aug-cc-pVQZ value of 0.96 D estimated by Hellweg (ES geometry),45 that appears approximately 0.10 D too large. For the second ES (of Rydberg nature), we know only one measurement of the Stark effect (on the absorption spectrum) that led to a $\mu^{\text{ES}}$ value of -2.2 ± 0.3 D.169 Theory clearly confirms the flip of the dipole as compared to the GS, but our TBEs are significantly larger than this experimental value, irrespective of the considered geometry: -3.27 (GS geometry) and -3.05 D (ES geometry). Given the significant basis set dependence of $\mu^{\text{ES}}$ of this state, one clearly needs to be cautious but it is nevertheless

likely that the experimental value of -2.2 D is too low. To the best of our knowledge, there is no previous published value of $\mu^{\text{ES}}$ for the $A_1$ ES. Concerning the oscillator strengths, the previous TBEs are likely our CC3/aug-cc-pVTZ values: 0.012 ($B_1$) and 0.178 ($A_1$),5 which are consistent with the new values listed in Table 9. On the “experimental side”, an estimate of 0.38 was proposed for the valence transition,182 but it is based on an empirical ratio of 10 compared to an earlier estimate of the oscillator strength for the corresponding Rydberg ES of formaldehyde.175 We trust that our current TBEs are significantly more accurate.

## 3.8 Nitroxyl and fluorocarbene

Table 10 provides the dipole moments and transition energies of the lowest ES of HNO. Although this transition is not strictly forbidden by symmetry, all methods return very low $f$ values (< 0.001). Thus, we have not bothered reporting the values of the oscillator strength.

<table><caption>Table 10: Ground-state dipole moment $\mu^{\text{GS}}$, vertical transition energies $\Delta E_{\text{vert}}$, and excited state dipole moments, $\mu_{\text{vert}}^{\text{ES}}$ and $\mu_{\text{adia}}^{\text{ES}}$, determined of nitroxyl. We report the norm of the dipoles. See caption of Table 1 for details.</caption><thead><tr><th rowspan="2">Basis</th><th rowspan="2">Method</th><th colspan="2">$^1A'$</th><th colspan="2">$^1A''$ (Val, $n \rightarrow \pi^*$)</th></tr><tr><th>$\mu^{\text{GS}}$</th><th>$\Delta E_{\text{vert}}$</th><th>$\mu_{\text{vert}}^{\text{ES}}$</th><th>$\mu_{\text{adia}}^{\text{ES}}$</th></tr></thead><tbody><tr><td rowspan="4">6-31+G(d)</td><td>CCSD</td><td>1.902</td><td>1.802</td><td>1.982</td><td>2.111</td></tr><tr><td>CCSDT</td><td>1.876</td><td>1.797</td><td>1.948</td><td>2.076</td></tr><tr><td>CCSDTQ</td><td>1.869</td><td>1.799</td><td>1.938</td><td>2.063</td></tr><tr><td>CCSDTQP</td><td>1.868</td><td>1.800</td><td>1.927</td><td>2.062</td></tr><tr><td rowspan="4">aug-cc-pVDZ</td><td>CCSD</td><td>1.701</td><td>1.779</td><td>1.719</td><td>1.840</td></tr><tr><td>CCSDT</td><td>1.667</td><td>1.767</td><td>1.681</td><td>1.799</td></tr><tr><td>CCSDTQ</td><td>1.658</td><td>1.770</td><td>1.670</td><td>1.785</td></tr><tr><td>CCSD</td><td>1.722</td><td>1.756</td><td>1.727</td><td>1.850</td></tr><tr><td rowspan="2">aug-cc-pVTZ</td><td>CCSDT</td><td>1.683</td><td>1.737</td><td>1.688</td><td>1.807</td></tr><tr><td>CCSD</td><td>1.735</td><td>1.753</td><td>1.735</td><td>1.859</td></tr><tr><td rowspan="2">aug-cc-pVDZ</td><td>CCSD</td><td>1.695</td><td>1.778</td><td>1.709</td><td>1.831</td></tr><tr><td>CCSDT</td><td>1.661</td><td>1.766</td><td>1.671</td><td>1.790</td></tr><tr><td rowspan="2">aug-cc-pVTZ</td><td>CCSD</td><td>1.720</td><td>1.755</td><td>1.724</td><td>1.847</td></tr><tr><td>CCSDT</td><td>1.681</td><td>1.737</td><td>1.685</td><td>1.804</td></tr><tr><td rowspan="2">aug-cc-pVQZ</td><td>CCSD</td><td>1.735</td><td>1.753</td><td>1.679</td><td>1.795</td></tr><tr><td></td><td></td><td></td><td></td><td></td></tr><tr><td rowspan="2">aug-cc-pVTZ</td><td>TBE<sup>a</sup></td><td>1.674</td><td>1.740</td><td>1.676</td><td>1.791</td></tr><tr><td>TBE<sup>b</sup></td><td>1.69±0.01</td><td>1.73±0.01</td><td>1.69±0.01</td><td>1.80±0.01</td></tr><tr><td rowspan="2">CBS</td><td>Th.</td><td>1.654<sup>c</sup></td><td>1.74<sup>d</sup></td><td></td><td></td></tr><tr><td>Exp.</td><td>1.67±0.03<sup>e</sup><br>1.62±0.02<sup>h</sup></td><td>1.63<sup>f</sup></td><td>1.69±0.01<sup>g</sup></td><td></td></tr></tbody></table>

a,b See corresponding footnotes in Table 7; cCCSD(T)/CBS value from Ref. 47; dUnpublished exFCI/aug-cc-pVTZ value (from our groups); eFrom microwave spectroscopy (Ref. 183); f 0-0 energy from Ref. 184 and references therein. gMicrowave optical double resonance value from Ref. 185 hFrom Stark effects measurements (Ref. 186).

As can be seen in Table 10, the convergences with respect to the CC excitation order and basis size are rather fast: quadruples tune the dipole par ~-0.01 D only, and basis set extension beyond triple-$\zeta$ is unnecessary. In other words CCSDT/aug-cc-pVTZ provides very accurate estimates and the CBS extrapolations come with small error bars. One notes that the geometrical relaxation of the ES increases the predicted dipole by ca. +0.11 D for all methods. Our TBE/CBS for $\mu^{\text{GS}}$, 1.69±0.01 D, is slightly above the Hai and Head-Gordon value (1.654 D), but the two available experiments
In thioformaldehyde (Table 9), one notes relatively stable $\Delta E_{\text{vert}}$ and $\mu^{\text{ES}}$ for the lowest dipole-forbidden $A_2$ transition: the convergence is rather fast with respect to both CC expansion and basis set size, so that we can safely report accurate TBE/CBS for both the GS and ES structures. The change of ES dipole between the two geometries is limited as well, contrasting with formaldehyde. This is because there is no puckering effect in thioformaldehyde's $A_2$ ES: the true minimum belongs to the $C_{2v}$ point group.³³,¹⁷⁸,¹⁷⁹ For the first Rydberg transition ($B_2$) the impact of the basis set size is logically more pronounced with, e.g., a +0.76 D change between the CCSDT/aug-cc-pVTZ and CCSDT/d-aug-cc-pVTZ dipoles, making the TBE/CBS extrapolation uncertainty larger than for the $A_2$ ES. The difference between the values of $\mu^{\text{ES}}$ determined at the GS and ES equilibrium geometries are also larger for the Rydberg excitation than for the lowest transition, despite the planarity of all geometries. When selecting Dunning's basis sets, the weak oscillator strength of the $B_2$ transition always falls in a rather tight window (0.011–0.015). The valence $A_1(\pi \rightarrow \pi^*)$ transition is clearly no cakewalk: not only the enlargement of the basis set yields significant changes of $\mu^{\text{ES}}$ (e.g., +0.49 D from CCSDT/aug-cc-pVTZ to CCSDT/d-aug-cc-pVTZ), but, in addition, the impact of the quadruples in the CC expansion becomes significant: the Q term induces a drop of $\mu^{\text{ES}}$ by ca. -0.40 D and a decrease of $f$ by ca. 15%. For this transition CCSD is clearly insufficient to obtain accurate ES properties. A chemical understanding of the underlying reasons for this large Q effect in the $A_1$ ES of thioformaldehyde would likely require an in-depth analysis of the various densities determined at various levels of theory, which is beyond our scope.

The value of $\mu^{\text{GS}}$ in thioformaldehyde was measured very accurately: 1.6491 ± 0.0004 D,¹⁵⁹ and our TBE of 1.73±0.01 D is slightly higher. Same comment applies to a previous CCSD(T)/aug-cc-pVQZ value of (1.700 D)¹⁶⁶ and a CC2/aug-cc-pVQZ value of 1.72 D.⁴⁵ Like in formaldehyde, one can find a series of Stark measurements relying on various spectroscopic techniques for the lowest $^1A_2$ ES. Quite a range of magnitudes have been reported for $\mu^{\text{ES}}$: 0.79±0.04 D (absorption spectroscopy),¹⁸⁰ 0.838±0.008 D (laser-induced fluorescence excitation),¹⁸¹ 0.850±0.002 D (microwave-optical double resonance),¹⁶⁸ and 0.815±0.020 D (intermodulated fluorescence).¹⁷¹ Obviously, the error bars of these measurements are not overlapping, but the latter work warns that values between 0.77 and 0.93 D can be obtained.¹⁷¹ Our TBEs of 0.88±0.01 D (GS geometry) and 0.89±0.01 D (ES geometry) are therefore obviously compatible with the experimental measures. The only previous wave function-based TBE we are aware of is the CC2/aug-cc-pVQZ value of 0.96 D estimated by Hellweg (ES geometry),⁴⁵ that appears approximately 0.10 D too large. For the second ES (of Rydberg nature), we know only one measurement of the Stark effect (on the absorption spectrum) that led to a $\mu^{\text{ES}}$ value of -2.2 ± 0.3 D.¹⁶⁹ Theory clearly confirms the flip of the dipole as compared to the GS, but our TBEs are significantly larger than this experimental value, irrespective of the considered geometry: -3.27 (GS geometry) and -3.05 D (ES geometry). Given the significant basis set dependence of $\mu^{\text{ES}}$ of this state, one clearly needs to be cautious but it is nevertheless

likely that the experimental value of -2.2 D is too low. To the best of our knowledge, there is no previous published value of $\mu^{\text{ES}}$ for the $A_1$ ES. Concerning the oscillator strengths, the previous TBEs are likely our CC3/aug-cc-pVTZ values: 0.012 ($B_1$) and 0.178 ($A_1$),⁵ which are consistent with the new values listed in Table 9. On the “experimental side”, an estimate of 0.38 was proposed for the valence transition,¹⁸² but it is based on an empirical ratio of 10 compared to an earlier estimate of the oscillator strength for the corresponding Rydberg ES of formaldehyde.¹⁷⁵ We trust that our current TBEs are significantly more accurate.

## 3.8 Nitroxyl and fluorocarbene

Table 10 provides the dipole moments and transition energies of the lowest ES of HNO. Although this transition is not strictly forbidden by symmetry, all methods return very low $f$ values (< 0.001). Thus, we have not bothered reporting the values of the oscillator strength.

<table><caption>Table 10: Ground-state dipole moment $\mu^{\text{GS}}$, vertical transition energies $\Delta E_{\text{vert}}$, and excited state dipole moments, $\mu_{\text{vert}}^{\text{ES}}$ and $\mu_{\text{adia}}^{\text{ES}}$, determined of nitroxyl. We report the norm of the dipoles. See caption of Table 1 for details.</caption><thead><tr><th rowspan="2">Basis</th><th rowspan="2">Method</th><th colspan="2"><sup>1</sup>A'</th><th colspan="2"><sup>1</sup>A'' (Val, n → π*)</th></tr><tr><th>μ<sup>GS</sup></th><th>ΔE<sub>vert</sub></th><th>μ<sup>ES</sup><sub>vert</sub></th><th>μ<sup>ES</sup><sub>adia</sub></th></tr></thead><tbody><tr><td rowspan="4">6-31+G(d)</td><td>CCSD</td><td>1.902</td><td>1.802</td><td>1.982</td><td>2.111</td></tr><tr><td>CCSDT</td><td>1.876</td><td>1.797</td><td>1.948</td><td>2.076</td></tr><tr><td>CCSDTQ</td><td>1.869</td><td>1.799</td><td>1.938</td><td>2.063</td></tr><tr><td>CCSDTQP</td><td>1.868</td><td>1.800</td><td>1.927</td><td>2.062</td></tr><tr><td rowspan="4">aug-cc-pVDZ</td><td>CCSD</td><td>1.701</td><td>1.779</td><td>1.719</td><td>1.840</td></tr><tr><td>CCSDT</td><td>1.667</td><td>1.767</td><td>1.681</td><td>1.799</td></tr><tr><td>CCSDTQ</td><td>1.658</td><td>1.770</td><td>1.670</td><td>1.785</td></tr><tr><td>CCSD</td><td>1.722</td><td>1.756</td><td>1.727</td><td>1.850</td></tr><tr><td rowspan="2">aug-cc-pVTZ</td><td>CCSDT</td><td>1.683</td><td>1.737</td><td>1.688</td><td>1.807</td></tr><tr><td>CCSD</td><td>1.735</td><td>1.753</td><td>1.735</td><td>1.859</td></tr><tr><td rowspan="2">aug-cc-pVDZ</td><td>CCSD</td><td>1.695</td><td>1.778</td><td>1.709</td><td>1.831</td></tr><tr><td>CCSDT</td><td>1.661</td><td>1.766</td><td>1.671</td><td>1.790</td></tr><tr><td rowspan="2">aug-cc-pVTZ</td><td>CCSD</td><td>1.720</td><td>1.755</td><td>1.724</td><td>1.847</td></tr><tr><td>CCSDT</td><td>1.681</td><td>1.737</td><td>1.685</td><td>1.804</td></tr><tr><td rowspan="2">aug-cc-pVQZ</td><td>CCSD</td><td>1.735</td><td>1.753</td><td>1.679</td><td>1.795</td></tr><tr><td>CCSD</td><td>1.674</td><td>1.740</td><td>1.676</td><td>1.791</td></tr><tr><td rowspan="2">aug-cc-pVTZ</td><td>TBE<sup>a</sup></td><td>1.69±0.01</td><td>1.73±0.01</td><td>1.69±0.01</td><td>1.80±0.01</td></tr><tr><td>TBE<sup>b</sup></td><td></td><td></td><td></td><td></td></tr><tr><td rowspan="2">Lit.</td><td>Th.</td><td>1.654<sup>c</sup></td><td>1.74<sup>d</sup></td><td></td><td></td></tr><tr><td>Exp.</td><td>1.67±0.03<sup>e</sup><br>1.62±0.02<sup>h</sup></td><td>1.63<sup>f</sup></td><td>1.69±0.01<sup>g</sup></td><td></td></tr></tbody></table>

a,b See corresponding footnotes in Table 7; cCCSD(T)/CBS value from Ref. 47; dUnpublished exFCI/aug-cc-pVTZ value (from our groups); eFrom microwave spectroscopy (Ref. 183); f 0-0 energy from Ref. 184 and references therein. gMicrowave optical double resonance value from Ref. 185 hFrom Stark effects measurements (Ref. 186).

As can be seen in Table 10, the convergences with respect to the CC excitation order and basis size are rather fast: quadruples tune the dipole par ~-0.01 D only, and basis set extension beyond triple-$\zeta$ is unnecessary. In other words CCSDT/aug-cc-pVTZ provides very accurate estimates and the CBS extrapolations come with small error bars. One notes that the geometrical relaxation of the ES increases the predicted dipole by ca. +0.11 D for all methods. Our TBE/CBS for $\mu^{\text{GS}}$, 1.69±0.01 D, is slightly above the Hai and Head-Gordon value (1.654 D), but the two available experiments