# Correlation Between NMR Coupling Constants and σ-Donating Properties of N-Heterocyclic Carbenes and Their Derivatives

Radhika Gupta, Gilles Frison

► To cite this version:

Radhika Gupta, Gilles Frison. Correlation Between NMR Coupling Constants and σ-Donating Properties of N-Heterocyclic Carbenes and Their Derivatives. Chemistry - A European Journal, 2025, 31 (4), pp.e202403403. 10.1002/chem.202403403. hal-05003420

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# Correlation Between NMR Coupling Constants and σ-Donating Properties of N-Heterocyclic Carbenes and Their Derivatives

Radhika Guptaª and Gilles Frison*a,b

a. LCM, Ecole polytechnique, CNRS, IP Paris, 91128 Palaiseau, France.

b. Sorbonne Université, CNRS, Laboratoire de Chimie Théorique, 75005 Paris, France. E-mail: gilles.frison@cnrs.fr

## Abstract

Is the ¹J<sub>C-H</sub> coupling constant for protonated carbene a relevant measure of its σ-donation ability? This paper addresses this question by comparing calculated ¹J<sub>C-H</sub> values with various experimental and theoretical approaches across a broad spectrum of carbene compounds. We examined Arduengo-type NHCs based on the 2-imidazolylidene scaffold and many other derivatives with modified frameworks, including carbenes with extended, saturated, or conjugated rings, reduced heteroatom stabilization, alternative heteroatoms, permanently charged carbenes, acyclic carbenes, amidocarbenes, and cyclic amino(alkyl/aryl) carbenes, carbodicarbenes and carbodiphosphoranes. Our findings reveal a nuanced relationship between different parameters associated with σ-interaction, such as ¹J<sub>C-H</sub>, Huynh electronic parameter (HEP), σ-donation from ETS-NOCV, and lone pair energy. Notably, the best correlation was observed between ¹J<sub>C-H</sub> and the ETS-NOCV method, particularly for mono- and diaminocarbenes, highlighting the utility of ¹J<sub>C-H</sub> in comparing σ-donation among structurally similar carbene types. However, the use of ¹J<sub>C-H</sub> as a universal measure across all carbene classes appears limited, especially when considering carbenes with significantly different structural frameworks. While HEP is less effective for carbenes with diverse structural backbones, our study suggests that ¹J<sub>C-H</sub> has potential across a broader range of systems. Additionally, the analysis demonstrates that lone pair energy reflects basicity rather than σ-donor ability.

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CC-BY 4.0

For the purpose of Open Access, a CC-BY public copyright licence has been applied by the authors to the present document and will be applied to all subsequent versions up to the Author Accepted Manuscript arising from this submission.
Introduction

The coordination properties of ligands have a major impact on the chemical
characteristics of the complexes they form. Their significance is illustrated by their
influence on numerous factors, ranging from bond lengths[1] and bond dissociation
energies[2] to radical stabilization capability.[3] These coordination properties play a
particularly significant role in the catalytic activity of their complexes with transition
metals. Therefore, to predict the efficiency of catalysts, ligand descriptors have
undergone significant development in recent years.[4] N-heterocyclic carbenes (NHCs)
and related C-donor ligands, known to cover a large range of electronic properties,[5]
are widely used in homogeneous catalysis.[6] Many experimental descriptors of their
bonding capability, such as the Tolman (TEP)[7], Lever (LEP)[8] and Huynh (HEP)[9]
electronic parameters or other indicators based on NMR approach, have been
proposed.[10,11] TEP and LEP (also referred to as ligand electrochemical parameter) both
measure the overall donation capability of a ligand L, using respectively the CO
stretching frequencies in the IR spectrum of [Ni(CO)₃L] or other tetrahedral complexes,
and the redox potential of [MCl(COD)L] complexes (M = Rh, Ir).[12] For its part, HEP is
based on the ¹³C NMR chemical shift of *trans*-[PdBr₂(Pr₂-bimy)L] complexes and
primarily measures the σ-donor strength of ligand L. The π-acidity of carbenes can be
estimated experimentally by measuring the ³¹P or ⁷⁷Se chemical shift of carbene-
phosphinidene[13] or carbene-selenium[14] adducts, respectively. The theoretical study
of these descriptors has proven to be very useful in establishing the electronic
properties of carbenes or confirming the relevance of these descriptors,[15] as
illustrated by the work of Gusev[16] and of Nolan and Cavallo.[17]

Recently, the magnitude of the ¹J<sub>C-H</sub> coupling constants in azolium-type NHC precursors
(as well as the ¹J<sub>C-Se</sub> coupling constants in NHC=Se complexes) has been proposed by
Ganter as a promising measure of the σ-donation of NHCs.[18] This descriptor is
assumed to be strongly related to the s-character of the carbon involved in the C-H
bond,[19-22] with larger ¹J<sub>C-H</sub> value indicating weaker σ-donor ability. It is particularly
attractive for estimating the σ-donation of divalent carbon ligand, as it only requires
the ¹³C or ¹H NMR spectrum of the readily available NHC precursor azolium salt.[23] On
the other hand, the ¹J<sub>C-Se</sub> coupling constant has been shown to provide valuable
information on NHC ligands; however, its measurement is hindered by the low natural
abundance of ¹³C and limited Se sensitivity. In this context, the evaluation of ¹Jc-H coupling constants has emerged as a more practical alternative for characterizing the σ-donating ability of NHC ligands. Despite recent works showing minimal effect (Δ¹Jc-H < 0.3 Hz) for concentration and counteranion and modest impact for the solvent nature (Δ¹Jc-H < 1.5 Hz),[23,24] there is no systematic study assessing the influence of these parameters on the ¹Jc-H value. It is also noted that the ¹Jc-H descriptor still faces several issues.[10] Indeed, salts derived from different carbenes, for which chemical evidence indicates different donation capabilities, have been measured with different counterions but still have the same ¹Jc-H value.[10,18,25] Moreover, the number of available experimental data for this recent descriptor is somewhat limited and does not cover the full diversity of carbene ligands and their derivatives.[23,25-37] Finally, no comprehensive comparison with other experimental methods has provided a clear conclusion on its suitability, with only one recent study comparing ¹Jc-H and HEP.[37] This study, limited to pyrazole- and indazole-derived NHC, concluded that ¹Jc-H and HEP parameters estimate the σ-donor strength similarly, although HEP is more accurate in evaluating subtle differences between molecules. Another commonly used indicator in the literature to assess the σ-donating ability of a carbene is the energy of its highest occupied molecular orbital (HOMO) associated with the lone pair on the carbene carbon. For example, Ganter et al.[18] and Landais et al.[38] reported good linear correlations (R² between 0.91 and 0.95) between the HOMO energy and the ¹Jc-Se or ¹Jc-H coupling constant of selected carbenes, supporting the idea that these descriptors all reflect the σ-donation ability of the carbene. However, it should be noted that these satisfactory correlations required the exclusion of certain compounds, likely indicating a partial disagreement between these descriptors, the reason for which remains unknown.

Interested in the ability of computational methods to accurately describe the bonding of NHCs and related divalent carbon compounds[39,40] in transition metal[41,42] and main group complexes,[43-44] we report in this context a computational assessment of the ¹Jc-H coupling constant as a descriptor of the σ-donation capability. This study aims to enhance our understanding of its relevance and generality. To achieve this, we conducted a theoretical study on a substantial and diverse set of NHC and related divalent carbon compounds (Fig. 1). For these compounds, various descriptors related
to the σ-donation capability of the carbene were compared to identify agreements and discrepancies among them.

## Computational methods

All structures were fully optimized without any symmetry constraints at the DFT B3LYP/TZVP level with the Gaussian09 package. When multiple conformers were identified for a compound, the one with the lowest electronic energy was selected. Calculations of the ¹H and ¹³C chemical shifts as well as the ¹J<sub>C-H</sub> coupling constants were performed using Gaussian09 at the B3LYP/(aug)-cc-pVTZ//B3LYP/TZVP level, either in the gas phase or in solution, where solvent effects were incorporated using the IEFPCM polarizable continuum model. The %age of s character of the carbene's lone pair (%S<sub>LP</sub>) and of the C-H bond (%S<sub>σ(C-H)</sub>) were obtained via Natural Bond Orbital (NBO) analysis using NBO6 software, based on the B3LYP/TZVP electronic structure. The ETS-NOCV method was employed to decompose the X-H bond energy and estimate the carbene's σ-donation term, ΔE<sub>ETS-NOCV</sub>. These calculations were performed using the ADF2021.105 software at the B3LYP/TZ2P//B3LYP/TZVP level. Further details on the computational methods used are provided in the Supporting Information.

## Results and discussion

In the first step, we evaluated the ability of DFT modeling to reproduce the available experimental data from the literature. We identified 64 azolium cations derived from carbenes for which at least one experimental ¹J<sub>C-H</sub> value was available,[18,23-37] with values ranging from 183 to 233 Hz. Their list is detailed in Table S2.

Figure 2 shows a remarkable agreement between experimental and theoretical ¹J<sub>C-H</sub> coupling constants for these 64 molecules, with a good coefficient of determination (R² = 0.98). The calculated coupling constant ¹J<sub>C-H</sub><sup>DFT</sup> can therefore be used to predict the expected experimental value ¹J<sub>C-H</sub><sup>pred</sup> using formula (1).

$$ ^{1}J_{C-H}^{pred} = (0.847 \pm 0.017) * ^{1}J_{C-H}^{DFT} + 26.3 \pm 3.7 \quad (1) $$
An excellent agreement is obtained between the predicted ¹J<sub>C-H</sub><sup>pred</sup> and experimental ¹J<sub>C-H</sub><sup>exp</sup> values, as demonstrated by the low root mean square deviation (RMSD = 2.3 Hz) and maximum absolute deviation (MAD = 8.7 Hz). The low uncertainty values for the slope and intercept confirm the quality of the correlation. The fact that the scaling factor is measurably different from 1, along with a notable offset, is likely attributable to the computational level used in these DFT-based NMR calculations, although further verification of this point would be beneficial. Excluding the outlier **52H** enhances the agreement between experiment and calculations (Fig. S2; R<sup>2</sup> = 0.98, RMSD = 2.1 Hz, MAD = 4.1 Hz). It should also be noted that the discrepancy between the experimental and theoretical values is not significantly greater than the experimental error. In fact, the determination of ¹J<sub>C-H</sub> values experimentally may involve an uncertainty that can be challenging to estimate, typically around 0.5 to 1.0 Hz,<sup>[45]</sup> partly due to line broadening. This confirms the strong ability of DFT to estimate this parameter. It is noteworthy that the largest deviation is observed for compound **52H**, for which the experimental value has been debated (vide supra),<sup>[10,18,25]</sup> highlighting the need to treat some of these data with caution.
<figure><img src="image_3.png" /><figcaption>**Figure 1.** Azolium cations XH (X = 1-124) computed in this study, classified in order of increasing $^{1}J_{C-H}$ obtained in the gas phase at the B3LYP/(aug)-cc-pVTZ//B3LYP/TZVP level (Hz). The hydrogen atom shown in red is the atom removed to lead to the corresponding NHC ligand X (X = 1-124). Basis sets cc-pVTZ and aug-cc-pVTZ are used respectively for atoms in blue and black (including H atom in red) for the NMR calculations. (continued)</figcaption></figure>
<figure><img src="image_4.png" /><figcaption>Figure 1. continued</figcaption></figure>
<figure><img src="image_5.png" /><figcaption>**Figure 2.** Correlation of computed $^1J_{C-H}^{DFT}$ (IEFPCM(solvent)-B3LYP/(aug)-cc-pVTZ//B3LYP/TZVP) versus experimental $^1J_{C-H}^{exp}$ coupling constants for 64 azolium cations.</figcaption></figure>

The Huynh Electronic Parameter (HEP) is known to measure the σ-donation ability of classical Werner-type ligands, such as NHCs and related C-donor ligands.[9] It is therefore expected to be similar to the $^1J_{C-H}$ coupling constant. However, the comparison between experimental HEP and $^1J_{C-H}$ measurements is limited by the lack of data. Furthermore, the solvent and counterion used to measure $^1J_{C-H}$ are not always the same, while HEP is consistently measured in CDCl₃, which can slightly distort the comparison. To eliminate any potential bias, we have calculated the $^1J_{C-H}$ coupling constant in CDCl₃ for 42 NHCs with known HEP values.[9a,25,35-37,46-53]

Figure 3 highlights significant discrepancies between HEP and $^1J_{C-H}$ values, with no significant relationship found between the two ($R^2$=0.25). Moreover, the data points tend to cluster according to carbene classes (Figure 4), highlighting distinct trends and variations within specific carbene families. This clustering indicates that $^1J_{C-H}$ values are less influenced by substituents compared to HEP. For instance, the range of $^1J_{C-H}$ values for imidazol-2-ylidenes (A; 226.7-233.2 Hz) covers only 8% of the overall range of values (181.3-259.4 Hz) obtained for these 42 ligands, while the range of HEP values for these 7 "classical" NHCs (177.2-181.2 ppm) covers more than 40% of the overall HEP value
range (176.1-185.1 ppm). This suggests that ¹J<sub>C-H</sub> is primarily sensitive to the nature of
the ligand backbone, which will need confirmation through further studies involving a
larger number of molecules, at least for some families.

<figure><img src="image_6.png" /><figcaption>**Figure 3.** Comparison of computed ¹J<sub>C-H</sub><sup>DFT</sup> (IEFPCM(CHCl<sub>3</sub>)-B3LYP/(aug)-cc-pVTZ//B3LYP/TZVP) and experimental HEP values for 42 ligands. See Figure 4 for the NHC classes **A-K** and the definition of **R**.</figcaption></figure>

A careful examination of Figure 3 suggests that ¹J<sub>C-H</sub> is particularly sensitive to two parameters: the nature of the atoms bonded to the carbenic carbon atom, and the Y-C<sub>carbene</sub>-Y angle. Increasing the electronegativity of atoms in the α- and, to a lesser extent, β-position relative to the C<sub>carbene</sub> atom results in a higher ¹J<sub>C-H</sub> value, in line with the expected decrease in σ-donor character. For example, considering NHC based on a conjugated five-membered ring, an increase in ¹J<sub>C-H</sub> values follows the increase in electronegativity of the ring atoms: H (α-C,N + β-N) < G (α-C,N + β-N,N) < A (α-N,N) < B (α-N,N + β-N) < J (α-N,O). It is noteworthy that this trend is also generally observed for HEP: the higher the electronegativity of neighboring atoms, the lower the HEP value, indicating a decrease in σ-donation ability.

For carbenes with the same neighboring atoms (α-N,N), increasing the Y-C<sub>carbene</sub>-Y angle along
the series **A** < **C** < **D** < **E** < **F** (Table 1) leads to a decrease in ¹J<sub>C-H</sub>, suggesting an increase in σ-
donation ability. This effect aligns with chemical intuition, as increasing the Y-C<sub>carbene</sub>-Y angle
should increase the p-character of the lone pair on the carbene carbon (or its percentage of p-character in the C-H bond), thereby enhancing its σ-donor character (vide infra). It is important to note that, unlike ¹J<sub>C-H</sub>, HEP values appear relatively independent of variations in the Y-C<sub>carbene</sub>-Y angle.

<figure><img src="image_7.png" /><figcaption>**Figure 4.** Classes of NHC and related C-donor ligands</figcaption></figure>

<table><caption>**Table 1.** Y-C<sub>carbene</sub>-Y bond angle ranges by classes (in °) for NHC considered in Figure 3, computed at the B3LYP/TZVP level.</caption><thead><tr><th>Class</th><th>Y-C-Y (XH)<sup>a</sup></th><th>Y-C-Y (X)<sup>b</sup></th></tr></thead><tbody><tr><td>A</td><td>108.8 – 109.3</td><td>101.8 – 102.5</td></tr><tr><td>C</td><td>110.8 – 111.2</td><td>103.7 – 104.2</td></tr><tr><td>D</td><td>114.3 – 115.3</td><td>106.1 – 107.4</td></tr><tr><td>E</td><td>125.2 – 131.4</td><td>116.0 – 122.3</td></tr><tr><td>F</td><td>133.1</td><td>123.8</td></tr></tbody></table>

<footnote><sup>a</sup> range of Y-C<sub>carbene</sub>-Y bond angles (in °) for XH; <sup>b</sup> range of Y-C<sub>carbene</sub>-Y bond angle (in °) for X.</footnote>
The σ-donor strength of NHCs for a given substituent is compared in Figure 3. 7 ligands (18, 21, 25, 47, 81, 96, and 114 – indicated by pink arrows in Figure 3) possess two benzyl substituents (R = Bn), and the comparison of HEP and ¹J<sub>C-H</sub> values for these compounds shows good agreement, with the σ-donor character increasing in the order B < A / C < D < E. The only inconsistency is between classes A and C, with the HEP scale indicating that imidazoline-2-ylidene (A) is a better σ-donor than benzimidazolylidene (C), while the ¹J<sub>C-H</sub> scale indicates the opposite. This disagreement is also observed for the Me and <sup>i</sup>Pr substituents. The aromatic character of the β-carbon atoms in C, intermediate between the sp<sup>2</sup>-hybridized carbons of A and the sp<sup>3</sup>-hybridized carbons of D, suggests that the σ-donor strength of class C falls between those of classes A and D, in agreement with the result given by the ¹J<sub>C-H</sub> values.

As seen previously, the substituents play only a minor influence on the ¹J<sub>C-H</sub> value,
contrary to what is observed for HEP. Nevertheless, for a specific class of NHC, as
observed for pyrazolin-3-ylidene (H),<sup>[37]</sup> a correlation is roughly obtained between ¹J<sub>C-H</sub>
and HEP values, illustrating that these two parameters measure a comparable effect
for the substituents. However, caution is required when extending this correlation to
different classes of NHC. Figure 3 shows a general trend, mainly disturbed by the
compounds of classes K, E, and to a lesser extent D. Interestingly, the reliability of the
HEP values of the most outliers (8,<sup>[36]</sup> 18, 24, 31<sup>[25]</sup>) has been questioned due to
anisotropic effect. The correlation between HEP and ¹J<sub>C-H</sub> remains very modest
(R<sup>2</sup>=0.58) when these four ligands were excluded as outliers and improves noticeably
(R<sup>2</sup>=0.84) when all compounds of classes K, E, and D were excluded (Figure S3).

The discrepancy between HEP and ¹J<sub>C-H</sub> raises the question of which of these two σ-
donation scales should be used to compare ligands of different classes. To address this
question, we used the ETS-NOCV method, which allows decomposition of a ligand
binding energy within a complex into its different contributions, including σ-donation
(noted ΔE<sub>σ ETS-NOCV</sub>). To avoid large computations, we calculated the σ-donation
component of the C-H bond in the azolium cation where the carbene is coordinated to
the proton as the Lewis acid. The ETS-NOCV calculations were performed for the 42
molecules with available HEP data. The comparison between ΔE<sub>σ ETS-NOCV</sub> and ¹J<sub>C-H</sub> shows
reasonable agreement (Figure 5A; R<sup>2</sup> = 0.86). However, compounds 8H and 123H,
belonging to classes K and J, respectively, are far from the correlation line. Excluding
these outliers leads to a coefficient of determination of 0.91 (Figure S4). This indicates
that the ETS-NOCV and ¹J<sub>C-H</sub> approaches provide comparable estimates of the σ-donor
strength for carbenes stabilized by two flanking nitrogen centers (classes A-F) and for
other NHC including one α-N and one α- sp<sup>2</sup>-hybridized C centers (classes G-I).

<figure><img src="image_8.png" /><figcaption>**Figure 5.** Correlation of the ETS-NOCV energy associated with the carbene-H⁺ σ-donation interaction (ΔE<sub>σ ETS-NOCV</sub>) for 42 **XH** complexes versus A) computed ¹J<sub>C-H</sub>DFT (IEFPCM(CHCl₃)-B3LYP/(aug)-cc-pVTZ//B3LYP/TZVP) and B) experimental HEP values. The main outliers in Figure 3 (orange square) are included in the correlation.</figcaption></figure>
Excluding the 4 main outliers **8, 18, 24**, and **31** from Figure 3, or all compounds of classes **K, E, and D** has a minimal effect on the correlation, with a slight increase of $R^2$ to 0.91 and 0.92, respectively (Figure S5). On the other hand, comparing the σ-donation from ETS-NOCV calculations and HEP values for the 42 compounds shows a lack of correlation (Figure 5B; $R^2$ = 0.08). Exclusion of the previous outliers (**8, 18, 24**, and **31**) or all compounds of classes **K, E, and D**, significantly improves the correlation (Figure S6; $R^2$ = 0.47 and 0.84, respectively), which however remains lower than that obtained for $^1J_{C-H}$. The deceptive result for HEP may be related to several factors. It is known that the paramagnetic contribution to the NMR shift depends on various factors, such as the nature of the magnetically coupled orbitals, their spatial arrangement, and the relevant carbene-centered orbital energies. Thus, steric effects or anisotropy effects can have a substantial influence, leading to HEP values that are problematic. The presence of compounds **8, 18, 24**, and **31** significantly weakens the correlations with HEP, yet does not notably affect the correlation between $^1J_{C-H}$ and $\Delta E_\sigma^{ETS-NOCV}$. This suggests that the HEP values for these four compounds may be unreliable. The choice of protonated complexes for the ETS-NOCV calculations could also contribute to the disappointing results in the comparison with HEP. The organometallic fragment [PdBr₂(Pr₂-bimy)] used for HEP measurement is significantly larger than a proton, and carbenes with flexible substituents, such as benzyl groups, may adopt different conformations in the measurements of HEP and $\Delta E_\sigma^{ETS-NOCV}$. However, studies on different conformers (see SI) do not improve the agreement between HEP measurements and $^1J_{C-H}$ or $\Delta E_\sigma^{ETS-NOCV}$. Finally, it is also possible that $\Delta E_\sigma^{ETS-NOCV}$ may follow different trends when the carbene is bound to a proton or a metal fragment, although this aspect is beyond the scope of this study. Overall, these comparisons between $^1J_{C-H}$, HEP, and $\Delta E_\sigma^{ETS-NOCV}$ suggest that $^1J_{C-H}$ is a more reliable indicator of σ-donating ability than HEP when comparing carbenes of different classes. However, for specific carbenes, such as acyclic alkylamino- (**K**) and aminooxy-carbene (**J**), even $^1J_{C-H}$ may not provide an accurate estimation

To further test the validity of $^1J_{C-H}$ compared to ETS-NOCV, we expanded our study to include more carbenes, adding 48 structures of classes **A-I**, which include positively charged carbenes, to the original 40 compounds. The results confirm the strong correlation between ETS-NOCV and $^1J_{C-H}$ for these classes, except for those with Cl
substitutions (Figure 6; R² = 0.90 excluding the three chlorinated compounds **105, 113** and **117**).

Extending the comparison to the entire set of 124 structures shown in Figure 1 significantly degrades the correlation (R² = 0.64). However, it is noteworthy that the structures most divergent from imidazol-2-ylidenes, such as diaminocyclopropenylidene (**122**, class S),[54] carbodicarbenes (**2-4**, class U),[55] or carbodiphosphoranes (**1** and **6**, class T),[56] exhibit the largest deviations from the correlation line obtained for classes **A-I**. Limiting the comparison to structures containing at least one nitrogen atom directly bonded to the carbenic carbon (i.e., mono- and diaminocarbenes classes **A-Q**), whether cyclic or acyclic, results in an imperfect but acceptable correlation (R² = 0.84). It should be noted, however, that this agreement is only achieved by excluding certain outliers, namely the three class **A** Cl-substituted carbenes and the three class **K** acyclic (aryl)amino carbenes **7, 8** and **20** (Figure 4, R¹ = aryl group).[36,57] Conversely, cyclic(aryl)amino carbenes (**36, 38** and **41**, class I),[58] cyclic(alkyl)amino carbenes (**12, 15, 16** and **22**, class Q)[59] and their class **K** (R¹ = alkyl group) acyclic counterpart **5**,[60] and the monosubstituted carbene **17** (Figure class **K**, R¹ = H),[61] among others, fit well within this correlation.

<figure><img src="image_9.png" /><figcaption>**Figure 6.** Correlation of ΔE<sub>σ</sub><sup>ETS-NOCV</sup> vs computed <sup>1</sup>J<sub>C-H</sub><sup>DFT</sup> (B3LYP/(aug)-cc-pVTZ//B3LYP/TZVP) for 85 (blue, **A-I** classes but 3 chlorinated compounds noted **A<sup>Cl</sup>** excluded as outliers), 108 (green, **A-Q** classes but 3 acyclic amino(aryl)carbenes, noted **K<sup>Ar</sup>**, and **A<sup>Cl</sup>** excluded) and 124 (red, all classes) **XH** structures.</figcaption></figure>
The HOMO energy of a carbene is frequently used in the literature to assess its σ-donation ability. For the 124 structures studied, it is important to note that while the HOMO typically corresponds to the σ-type lone pair on the carbene carbon, this is not true for 18 structures. In these cases, this lone pair is found in the HOMO-1 or HOMO-2 orbitals. For example, in carbodiphosphoranes (1 and 6) and carbodicarbenes (2-4), the HOMO is a π-type lone pair on the carbene carbon. Similarly, in cationic carbenes 117 and 120, the HOMO is a π orbital of an aryl substituent, and in carbenes 29, 83, 94, 101, 115, and 116, an extended (hetero)π-system results in a π-type HOMO (Figure 7).

Therefore, the energy of the carbene lone pair (E<sub>LP</sub>), corresponding to the HOMO energy in most cases but to the HOMO-1 or HOMO-2 energy in others, was compared to ¹J<sub>C-H</sub> and ΔE<sub>σ</sub><sup>ETS-NOCV</sup> values. Contrary to what is suggested in the literature, there is a weak correlation between E<sub>LP</sub> and ¹J<sub>C-H</sub> (Figure 8). While charged molecules are clear outliers, their exclusion only modestly improves the correlation (R<sup>2</sup> = 0.70). Furthermore, restricting the analysis to specific classes of compounds does not significantly enhance the correlation. A similar result is observed when comparing E<sub>LP</sub> to ΔE<sub>σ</sub><sup>ETS-NOCV</sup> (Figure S8), confirming that σ-donation is poorly estimated by E<sub>LP</sub>.

<figure><img src="image_10.png" /><figcaption>**Figure 7.** Plot of the MO corresponding to the carbene σ-lone pair and the highest MO for ligands **4**, **15**, **29**, **98** and **120**. Isosurface value: 0.07 a.u.</figcaption></figure>
<figure><img src="image_11.png" /></figure>

$$y = -26.731x + 63.721$$
$$R^2 = 0.697$$

<figure><img src="image_12.png" /></figure>

$$y = -86.685x - 1641.8$$
$$R^2 = 0.910$$

**Figure 8.** Correlation of the energy of the carbene's lone pair (E<sub>LP</sub>) (B3LYP/TZVP) and (A) ¹J<sub>C-H</sub> (B3LYP/(aug)-cc-pVTZ//B3LYP/TZVP) and (B) proton affinity (PA, B3LYP/TZVP) for ligands **1-124**. Charged molecules X (X = **72, 106, 109, 117, 119** and **120**, red square) are excluded from the correlations in (A).
The very low E<sub>LP</sub> values for positively charged carbenes illustrate that E<sub>LP</sub> depends not only on the overlap and energy of the atomic orbitals used to construct the molecular orbitals but also on the total intramolecular electronic interactions. In other words, E<sub>LP</sub> might not merely estimate σ-donation ability, which pertains solely to the orbital component of a σ bond, but rather the carbene's ability to form a σ interaction, including both the orbital and electrostatic contributions to bonding. Thus, E<sub>LP</sub> could be considered an estimate of the nucleophilicity of the carbene, or more precisely, its basicity, since E<sub>LP</sub> does not account for steric effects on the interaction energy with an electrophilic site.

To confirm this hypothesis, the proton affinity (PA) of the carbenes and their interaction energy with H⁺ (E<sub>int</sub>) were calculated for all compounds. E<sub>int</sub> includes not only ΔE<sub>o ETS-NOCV</sub> but also the electrostatic contribution and other orbital interactions such as protonation-induced π-polarization of the ligand.[43] The nearly perfect correlation between E<sub>int</sub> and PA (Figure S7, R<sup>2</sup> = 0.996) indicates that protonation-induced geometric deformation of the compounds marginally affects the coordination properties of these ligands. The satisfactory agreement between PA and E<sub>LP</sub> (Figure 8, R<sup>2</sup> = 0.91),[62] in line with previous results,[41,63] is significantly better than that between E<sub>LP</sub> and <sup>1</sup>J<sub>C-H</sub>. This confirms that E<sub>LP</sub> provides a rough estimate of the carbene's basicity. Indeed, the strength of the σ<sub>C-H</sub> bond formed upon protonation is based not only on the carbene's σ-donation ability but also on the electrostatic component of the interaction.[64] In the case of the protonated structures XH (X = 1-124), the electrostatic term associated with the C-H bond, calculated by the ETS-NOCV method (ΔE<sub>elstat</sub>), ranges between -511 and 97 kJ/mol (see Table S6). It is positive or only slightly negative for cationic carbenes. For neutral carbenes, it varies from -511 to -154 kJ/mol, showing a variability that exceeds that of σ-donation, which lies between -846 and -656 kJ/mol. The poor agreement between E<sub>LP</sub> and <sup>1</sup>J<sub>C-H</sub> arises largely from this high variability in ΔE<sub>elstat</sub>.

Next, the <sup>1</sup>J<sub>C-H</sub> coupling constant was compared to the hybridization of C<sub>carbene</sub>. This hybridization can be measured for the carbene, with the percentage of s character of the σ-lone pair (%S<sub>LP</sub>), or for the corresponding azolium with the percentage of s character in the C-H bond (%S<sub>σ(C-H)</sub>). Surprisingly, these two parameters only moderately correlate. In most cases, %S<sub>LP</sub> is larger than %S<sub>σ(C-H)</sub>, in agreement with the widening of the Y-C<sub>carbene</sub>-Y bond angle due to protonation (Figures S11-S12). The greater spatial extension of the carbene lone pair compared to the corresponding σ<sub>CH</sub> orbital, as well as the shortening of the Y-C bond lengths upon protonation due to lower
π-delocalization at Ccarbene, explain this larger Y-Ccarbene-Y bond angle.[39] Carbodiphosphoranes T and to a lesser extent carbodicarbenes U do not fit into this picture because their protonation induces less π-delocalization from the occupied pπ orbital of the divalent carbon. The resulting longer C-Y bonds induce lower steric repulsion and allow a narrower Y-C-Y bond angle. %SLP is a relatively poor estimator of σ-donation, as its comparison with ¹Jc-H or ΔEσETS-NOCV reveals a weak correlation (R² = 0.77-0.81, Figure S13). Conversely, the comparison between %Sσ(C-H) and ¹Jc-H across compounds 1-124 is more satisfying, although still not excellent (R² = 0.86, Figure 9). The linear relationship between the calculated ¹Jc-H and %Sσ(C-H) is expressed by equation (2).

$$ ^{1}J_{C-H}^{\text{DFT}} = (7.379 \pm 0.264) \, ^{1}\text{S}_{\sigma(C-H)} - 34.9 \pm 8.9 \qquad (2) $$

Combined with equation (1), which related calculated and experimental coupling constants, we obtain ¹Jc-H Exp ≈ 6.25 %Sσ(C-H) – 3.3. This relationship aligns approximately with the empirical formula established by Günther,[65] based on the three molecules CH₄ (sp³ carbon), CH₂=CH₂ (sp² C), and HC≡CH (sp C): ¹Jc-H ≈ 5.00 %Sσ(C-H).[66] This confirms the close link between the coupling constant ¹Jc-H and the carbon hybridization in the C-H bond.

A careful examination of the data in Figure 9 shows that the correlation is much stronger when the analysis is restricted to similar compounds. Specifically, the analysis of 47 neutral 5-membered ring diaminocarbenes (types A-D and compounds 67 and 76 of type P; in dark blue in Fig. 9) demonstrates an excellent agreement between %Sσ(C-H) and ¹Jc-H (R² = 0.95). A similar correlation is observed for 21 neutral extended-ring and acyclic diaminocarbenes (types E, F and O and compounds 40, 50, 51 and 55 of type P; in gray), albeit with a shift of approximately +7 Hz for the same value of %Sσ(C-H). This suggests that the constraint of the 5-membered ring alters the relationship between these two parameters to some extent.
<figure><img src="image_13.png" /><figcaption>**Figure 9.** All compounds (black) or type-specific (colors) compounds correlations of %S<sub>σ(C-H)</sub> vs computed <sup>1</sup>J<sub>C-H</sub><sup>DFT</sup> (B3LYP/(aug)-cc-pVTZ//B3LYP/TZVP). Charged (unfilled squared) and iodine-substituted (unfilled triangle) compounds are excluded as outliers in type-specific compounds correlations.</figcaption></figure>

Similarly, shifting from diaminocarbenes to monoaminocarbenes results in a shift of approximately –8 Hz, while maintaining an excellent correlation (R² = 0.98 for the 30 compounds in classes G-I, L, N, and Q, with compound 57, substituted by an iodine atom, excluded from the correlation; in yellow). The comparison of correlation slopes, including uncertainties, for the different curves in Figure 9 (7.379 ± 0.264 for the full sample; 8.016 ± 0.375 for the dark blue set; 7.576 ± 0.392 for the gray set; and 7.472 ± 0.222 for the yellow set), suggest that the data for the different groups studied are only slightly different statistically. Other changes in the electronegativity of Y atoms bonded to C<sub>carbene</sub> or the presence of a permanent positive charge also modify <sup>1</sup>J<sub>C-H</sub> for the same value of %S<sub>σ(C-H)</sub>. Thus, while the linear relationship between <sup>1</sup>J<sub>C-H</sub> and %S<sub>σ(C-H)</sub> is relevant at the first order, it must be modulated at the second order by various structural factors such as the Y-C<sub>carbene</sub>-Y angle, the nature of the Y atoms, the presence of a permanent charge, or certain atoms.

Finally, we investigated whether carbene properties could be determined using proton or carbon chemical shifts instead of the <sup>1</sup>J<sub>C-H</sub> coupling constant. It is indeed tempting,
for example, to assess the coordination properties of carbene ligands using the isotropic chemical shielding of the acidic proton. The calculated data, which do not suffer from counterion effects, can serve as a pertinent test. Unfortunately, the comparison is clear: all comparisons between ¹H or ¹³C chemical shifts and the other parameters studied previously show a lack of correlation (Figure S14).

## Conclusion

In this computational study, the ¹J<sub>C-H</sub> coupling constants of a large series of azolium cations were calculated to evaluate whether this easily accessible experimental property can be used to gauge the σ-donation ability of the corresponding carbene ligand. The calculated ¹J<sub>C-H</sub> coupling constants were compared with other experimental data as well as other calculated electronic parameters related to σ-donation of the carbenes. The main conclusion is that ¹J<sub>C-H</sub> of azolium compound is a more reliable experimental probe, compared to HEP or ¹H or ¹³C NMR shifts, to measure the σ-donation capability of the corresponding carbene. However, the comparison with the calculated σ-donation contribution from the ETS-NOCV method highlights certain issues with ¹J<sub>C-H</sub>, especially when dealing with very different compounds. Its effectiveness as a descriptor is reasonable for most mono- and diaminocarbenes but diminishes when comparing them with acyclic (aryl)amino carbenes or C-ligands with different backbones. In the same vein, while both HEP and ¹J<sub>C-H</sub> correlate well within specific carbene classes regarding the effects of substituents, caution is needed when extending this correlation across different carbene classes. Another significant conclusion is that the energy of the carbene's lone pair, often used to describe σ-donation ability, turns out to be a poor indicator of this property. Conversely, E<sub>LP</sub> is useful for assessing the carbene's basicity, which is not provided by ¹J<sub>C-H</sub>. Finally, these findings confirm that the hybridization of the carbon in the C-H bond of the azolium, rather than that of the carbene's lone pair, approximately correlates with ¹J<sub>C-H</sub>. Nevertheless, other factors such as the Y-C<sub>carbene</sub>-Y bond angle and the nature of the atoms bonded to the carbene carbon affect this correlation.

## Conflicts of interest

There are no conflicts to declare.
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[67] The uncertainties in the intercept further support this similarity: -34.9 ± 8.9 for the full sample; -55.7 ± 9.7 for the dark blue set; -34.1 ± 12.0 for the gray set; and -44.9 ± 7.5 for the yellow set.
**Table of Content entry**

The comparison of σ-donation indicators, including the $^1J_{C-H}$ coupling constant, calculated for a wide range of diverse carbenes, provides detailed insights into their strengths and limitations.

<figure><img src="image_14.png" /></figure>