# Do bonds need a name?: New classification of Non-Covalent Interactions

Katarzyna Kate Zator, Eline Desmedt, Mercedes Alonso, Julia Contreras-García

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## To cite this version:

Katarzyna Kate Zator, Eline Desmedt, Mercedes Alonso, Julia Contreras-García. Do bonds need a name?: New classification of Non-Covalent Interactions. 2026. [hal-05505609](https://hal.science/hal-05505609v1)

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# Do bonds need a name?: New classification of
## Non-Covalent Interactions

Katarzyna J. Zator,*,† Eline Desmedt,‡ Mercedes Alonso,‡ and Julia

Contreras-García*,†

†Laboratoire de Chimie Théorique, Sorbonne Université and CNRS, 4 Pl Jussieu, F-75005,
Paris, France

‡Vrije Universiteit Brussel, Research Group for General Chemistry - Algemene Chemie
(ALGC), Pleinlaan 2, Brussels B-1050, Belgium

E-mail: katarzyna.zator@sorbonne-universite.fr; julia.contreras.garcia@sorbonne-universite.fr

### Abstract

Non-covalent interactions (NCIs) span a wide range of electronic behaviours, yet traditional names such as hydrogen bond or halogen bond group are used for interactions with very different energetic origins. We propose that a more robust classification should rely on electronic characteristics in analogy with the long-standing covalent/ionic distinction for strong interactions. In the case of non-covalent interactions, the extent of directionality versus non-directionality determines the structural organisation and dynamic behaviour of interactions, being the source of information used for selectivity or drug design. From an energetic viewpoint, it can be understood as the ratio between electrostatic and dispersive contributions. However, this characterisation is cumbersome from a computational viewpoint. We show that these characteristics can be captured by a simple density-based descriptor, $q_{NCI} / V_{vdW}$, which distinguishes directional, electrostatically driven interactions from more diffuse dispersion-dominated contacts.
Application to model systems and benchmark datasets (S66, X40, NCIAtlas-2421 complexes) shows that this descriptor reveals substantial variation hidden by atom-centred nomenclature and provides a unified, transferable framework for analysing weak interactions in big systems such as macrocyclic host-guest structures or ligand-protein complexes.

Introduction

Non-covalent interactions (NCIs) are fundamental forces in chemistry, biology, and materials science,¹⁻³ governing everything from protein folding and molecular recognition to crystal packing and supramolecular assembly.⁴⁻⁷ Historically, NCIs have been classified into types—such as hydrogen bonds, halogen bonds, or dispersion forces—based on the identity of the atoms involved and certain geometrical or energetic features.⁸

However, advances in spectroscopy, quantum chemistry, and energy decomposition analyses have revealed that these interactions form a continuum of behaviours that transcend atom-based nomenclature. Hydrogen bonding, once considered a uniquely directional and atom-specific interaction,⁹ is now recognised in a variety of weak and unconventional forms.¹⁰ At the same time, newer bonding types—halogen, ¹¹⁻¹³ chalcogen, ¹⁴,¹⁵ pnictogen, ¹⁶,¹⁷ and even carbon bonding¹⁸—have emerged as analogues with similar directional character, often driven by electropositive regions known as σ or π-holes.¹⁹ These developments blur the boundaries between classical categories and raise fundamental questions about how NCIs should be defined and classified.

Localised interactions are widely exploited in molecular recognition, asymmetric catalysis, and crystal engineering, where controlled orientation and high chemical specificity are required.²⁰⁻²² In contrast, van der Waals interactions arise predominantly from dispersion forces and exhibit delocalised electron density distributions that lack strong directionality. Although individually weak, these contributions are essential for hydrophobic complementarity in protein–ligand binding, for stabilising soft condensed phases, and for determining
molecular packing in crystals and materials.²³ Distinguishing between localised and delocalised interaction character is thus essential for understanding reactivity and design principles across chemical systems, yet traditional atom-centred classifications often obscure this electronic continuum.²⁴⁻²⁶

In this context, rather than focusing on the elements involved, we argue that a more robust and transferable framework can be built focusing on these localised versus delocalised characteristics. We propose a density-based approach for classifying NCIs that bridges visual, topological, and energetic perspectives. We analyse a broad range of systems — from weak hydrogen bonds to halogen interactions — across standard datasets (S66, X40, NCIAtlas),²⁷⁻³⁰ demonstrating how their defining characteristics can be unified through electron density features. Specifically, we introduce the ratio of NCI charge to van der Waals volume ($\frac{qNCI}{V_{vdW}}$) as a semi-quantitative descriptor that distinguishes localised (electrostatic) from delocalised (dispersion-dominated) interactions. This descriptor offers a scalable and physically grounded alternative to atom-based taxonomies, aligning better with the electronic organisation that governs non-covalent interactions. It invites a re-evaluation of traditional labels across the periodic table—suggesting that only genuinely distinct energetic or density characteristics merit new names.

## Theory

Along with our characterisation of non-covalent interactions, we will resort to the Non-Covalent Interaction (NCI) index and Energy decomposition analyses (EDAs). Both are described in the coming subsections.

### NCI

The Non-Covalent Interaction (NCI) index³¹,³² enables the detection and characterisation of weak interactions of varying strengths through the analysis of the reduced density gradient,
$s(\mathbf{r}):$

$$s(\mathbf{r}) = \frac{|\nabla \rho(\mathbf{r})|}{2(3\pi^2)^{1/3} \rho(\mathbf{r})^{4/3}} \quad (1)$$

The NCI approach reveals both stabilising (e.g., hydrogen bonds, van der Waals forces) and destabilising (e.g., steric clashes) interactions directly in real space, offering a chemically intuitive perspective. Such interactions are typically characterised by low values of both $s$ and $\rho$. As depicted in Figure 1, hydrogen bonds manifest as disc-like isosurfaces, reflecting localised two-body interactions, whereas more delocalised interactions appear as extended surfaces.

<figure><img src="image_3.png" /><figcaption>**Figure 1:** Acetic acid - aniline complex with NCI surfaces ($s=0.5$, colour range: -0.03 ≤ sign($\lambda_2$)$\rho$ ≤ 0.03 a.u.). The NCI calculation was performed with DFT (PBE0/def2-SVP) electron densities.</figcaption></figure>

Bonding patterns may be further classified through examination of the electron density and the eigenvalues ($\lambda_1$, $\lambda_2$, $\lambda_3$) of its Hessian matrix. This allows interactions to be categorised as: (i) strongly attractive (high $\rho$, $\lambda_2 < 0$, e.g., hydrogen bonds); (ii) very weak (low $\rho$, e.g., van der Waals interactions); and (iii) repulsive steric clashes (high $\rho$, $\lambda_2 > 0$).

Quantitative measures such as NCI region charges and volumes can be obtained by integrat-
ing the relevant quantities over the interaction region,³³ Ω<sub>NCI</sub>:

$$V_{NCI} = \int_{\Omega_{NCI}} d(\mathbf{r}) \qquad (2)$$

$$q_{NCI} = \int_{\Omega_{NCI}} \rho(\mathbf{r}) d(\mathbf{r}) \qquad (3)$$

For example, for the aniline-acetic acid complex in Figure 1 these values are $q_{NCI} = 0.613$ and $V_{vdW} = 63.360$ (a.u.). The region can be further constrained to a particular range of interaction, e.g., within the sign($\lambda_2$)ρ range of -0.02 and 0.02 a.u. $V_{vdW}$ we obtain the NCI volume of only the van der Waals interactions.³¹,³⁴

## Energetic contributions

Energy decomposition analyses (EDAs) are a widely used method for investigating contributions to binding. The Morokuma-type energy decomposition method is the most common scheme, which is implemented in the ADF software.³⁵,³⁶ This EDA scheme characterises three physical terms:³⁷

$$\Delta E_{int} = \Delta V_{elst} + \Delta E_{Pauli} + \Delta E_{oi} = \Delta E_{steric} + \Delta E_{oi} \qquad (4)$$

In this framework, the energy is decomposed into electrostatic ($\Delta V_{elst}$), Pauli repulsion ($\Delta E_{Pauli}$), and orbital term ($\Delta E_{oi}$).

This partition therefore follows a division into a classical regime of electrostatic attraction between the unperturbed charge distributions of fragments A and B, $\rho_A$ and $\rho_B$, resulting in the combined density $\rho_A + \rho_B$. For example, for the aniline-acetic acid complex in Figure 1, $\Delta V_{elst}=-6.97$ kcal/mol.

The modulation of the electrostatic term by effects such as charge transfer, polarisation, and electron-pair bonding is captured by the orbital interaction energy. In the aniline-acetic acid
complex, this accounts for $\Delta E_{oi}$=-3.08 kcal/mol of the interaction.

The quantum detail is supplemented by the antisymmetrisation and normalisation of the
corresponding wavefunction that generates the Pauli repulsion term. This also accounts
for destabilising interactions between occupied orbitals and reflects the steric repulsion be-
tween molecular fragments. Therefore, by combining $\Delta V_{elst}$ and $\Delta E_{Pauli}$, a steric interaction
term for neutral fragments, $\Delta E_{steric}$,³⁸ can be obtained. In the aniline-acetic acid complex,
$\Delta E_{Pauli}$ is 6.56 kcal/mol, leading to $\Delta E_{steric}$=-0.41 kcal/mol.

To obtain a semi-quantitative estimate of the dispersion contribution, a D3(BJ) (Becke-
Johnson-corrected Grimme's empirical dispersion) correction was applied to DFT energies.³⁹
This will be noted as $E_{disp}$. For the aniline-acetic acid complex we have that $E_{disp}$=-3.28
kcal/mol.

Computational Methods and Data Sourcing

EDA calculations were performed with the ADF 2022.105 software at the PBE0/TZP level of theory.³⁶,⁴⁰ Dispersion energies were computed for each functional with D3(BJ) correction with the Psi4 software.³⁹,⁴¹,⁴² QTAIM analysis was performed with AIMAll software.⁴³ NCIPLOT-4.0³² was used for NCI analysis: electron densities, charges, volumes and isosurfaces were computed using DFT (PBE0/def2-SVP) densities (with ρ=0.2, s=1.0, γref=0.75); or promolecular densities for the supramolecular complex (with ρ=0.2, s=1.0, γref=0.85) according to Ref 34, and the NCI isosurfaces were subsequently visualised with VMD version 1.9.2.⁴⁴

The non-conventional hydrogen bond complex geometries were obtained from Ref. 45 and
were verified as PES minima. The S66 geometries were obtained from the original publica-
tion,²⁹ and their EDA energies were shared by the authors of Ref.⁴⁶ The NCIA Atlas geometries
were obtained from the dedicated database.²⁷,²⁸,⁴⁷,⁴⁸
# Results and Discussion

## Classifying localised vs delocalised interactions

To establish a reference framework for our analysis, we first contrast the two limiting regimes of non-covalent interactions. Localised NCIs, such as hydrogen and halogen bonds, exhibit anisotropic electron density distributions that produce directional geometries and are primarily stabilised by electrostatic and induction effects.⁴⁹⁻⁵¹ These interactions underpin molecular recognition, asymmetric catalysis, and crystal engineering, where orientation and specificity are crucial.⁵²⁻⁵⁴ In contrast, van der Waals interactions are dominated by dispersion forces and display delocalised, non-directional electron behaviour.⁵⁵⁻⁵⁷ Although individually weak, their cumulative effect is essential for hydrophobic complementarity in protein–ligand binding,⁵⁸,⁵⁹ for stabilising soft condensed phases, and for dictating packing in molecular and supramolecular solids.

<figure><img src="image_4.png" /><figcaption>**Figure 2:** 3D plots for weak hydrogen bond complexes: a) HCCH···OH₂, b) HCCH···HLi, c) FH···HLi, d) HCCH···(π)HCCH, e) HOH···(π)HCCH, f) FH···(π)HCCH. NCI isosurfaces correspond to s=0.5 and a colour scale of -0.04 < sign(λ₂)ρ < +0.04 a.u. q is NCI charge and *V*<sub>vdW</sub> is volume of vdW-region NCI. The NCI calculation was performed with promolecular electron densities.</figcaption></figure>

We showed in a previous publication⁴⁵ that the integrals over the NCI region can provide
information on the directionality of the interaction. A small showcase non-conventional
hydrogen bond example is collected in Fig. 2. Visually, we see that **hb3** shows a compact
disc shape with a strong blue colour, which corresponds to a strongly directional interaction.
Instead, **hb1** - **hb2** show greener surfaces, though still round in shape. These correspond
to weaker localised interactions. Correspondingly, their volume is bigger, and their charge is
smaller. Finally, **hb4** - **hb6** have larger and less rounded surfaces. These are good examples
of $\pi$ interactions, with much bigger volumes which typically lead to either intermediate or
delocalised interactions.

All in all, electrostatic interactions lead to localised NCI regions, where electron density and
charge are higher, whereas dispersion interactions lead to more spread out volumes but lower
charge densities. This can be reflected by the $\frac{qNCI}{V_{vdW}}$ ratio, which showed to be higher for more
localised and electrostatic bonds and lower for dispersion-driven interactions. We note that
there have been analogous examples in the literature where the $V_{elst}$ and $E_{disp}$ values have
been used as input molecular parameters for ML models to predict molecular properties
or binding energy,⁶⁰,⁶¹ where the authors suggested the interaction of two terms could be
key to determining the bonding nature. Further, Towne, et. al. used the electrostatic-to-
dispersion energy (E/D) ratio from SAPT to classify interactions.⁶² As far as big systems
are concerned, these two components have also been used to represent secondary structures
in contact maps.⁶³

Hence, we decided to investigate these behaviours of NCI charges and volumes on bigger
test sets and their potential relationship with the energetic terms. The S66 dataset²⁹ con-
tains small neutral molecular complexes showing common types of NCIs in biomolecules
(electrostatic, dispersion, and a combination of both). The X40 dataset,³⁰ comprising 40
non-covalent complexes of organic halides, halo-hydrides, and halogen molecules, was also
included to introduce a variability of directional interactions. The complexes from the pre-
vious publication⁴⁵ have also been included under the name “xB”. Figure 3 shows the
semi-quantitative correlation between the directional/non-directional features as recovered
by the NCI integrals and the electrostatic/dispersion terms from EDA and Grimme's dispersion correction<sup>46</sup> (note that the DFT analogue is available in S.I.).

<figure><img src="image_5.png" /><figcaption>**Figure 3:** Scatter plot for the $\frac{q_{NCI}}{V_{vdW}}$ ratio and the ratio of $\Delta V_{elst}$ from EDA and $\Delta E_{disp}$ from Grimme's dispersion correction for the investigated complexes with xB (weak hydrogen, halogen and pnictogen bonds from Ref. 45), S66 and X40 set. Best-fit line is a quadratic function: $y = 2.478 \times 10^{-5}x^2 + 1.152 \times 10^{-3}x + 7.033 \times 10^{-3}$. Pearson's R<sup>2</sup> = 0.92. The two main outliers are: AcOH···uracil (above the line) and HF···methylamine (below the line). S66 energetic data provided by the authors of Ref. 39, and we thank C. Fonseca for the S66 energetic partition data used in this graph.</figcaption></figure>

Given the correspondence between directional/delocalised energetic and promolecular density features, we further analysed the $q_{NCI}$ and $V_{vdW}$ characteristics of these compounds and the agreement with their original classification. Indeed, the datasets were originally divided into three families:<fnref n="1" />

* Directional (localised): S66 electrostatic interactions and X40 H-bonded systems,

<footnote n="1">¹the full list of compounds and their families as provided in the original publications<sup>29,30</sup> are given in S.I.</footnote>
* Non-directional (delocalised): S66 dispersion and X40 dispersion, induction, dipole-dipole, $\pi$-$\pi$ stacking,

* Mixed/not-preclassified: S66 mixed interactions and X40 sigma-hole interactions. This group was created to separate interactions that were not clear in terms of energetics.

<figure><img src="image_6.png" /><figcaption>**Figure 4:** NCI charge-vdW volume scatter plot for combined S66 and X40 complexes divided into polar, non-polar, and mixed/not-preclassified complexes dependent on the leading contribution to binding. The cross points (xB), complexes discussed in the ref.⁴⁵ The darker-coloured navy and purple dots have their NCI surfaces depicted on the right.</figcaption></figure>

Figure 4 shows the NCI charge and volume characteristics of these groups. It can be seen that this classification is to an excellent extent recovered by the $\frac{q_{NCI}}{V_{vdW}}$ ratio. Directional complexes (in blue) achieve higher $\frac{q_{NCI}}{V_{vdW}}$ values than delocalised ones (in green), whereas mixed/not-preclassified interactions (in purple) occupy the space in between the two trends. The best-fit lines show a ratio of $\frac{3}{125}$ and $\frac{1}{125}$ a.u., respectively. Moreover, based on the typical $\frac{q_{NCI}}{V_{vdW}}$ values of the presented groups, it is possible to classify the interactions in terms of their density properties: interactions would be localised with a ratio greater than $\frac{1}{50}$ a.u., delocalised with a ratio lower than $\frac{1}{100}$ a.u., and mixed interactions if their $\frac{q_{NCI}}{V_{vdW}}$ value is in between those two values. The S.I. contains a parallel analysis done with DFT
(PBE0/def2-SVP) electron densities, which, despite showing more scatter, had similar $\frac{q_{NCI}}{V_{vdW}}$ ratio values.

Note that the $\frac{q_{NCI}}{V_{vdW}}$ values typical of hydrogen bonds and dispersion interactions also allowed us to classify the non-standard NCIs (halogen bonds, non-conventional hydrogen bonds, etc.) using the ratio as a uniform metric, prompting an overall more coherent description of these bonds with reference to other types.

Finally, it is also worth mentioning the outliers, i.e., some molecules move away from their initial classification. In other words, the $\frac{q_{NCI}}{V_{vdW}}$ criterion allows for rechecking the initial classification of complexes into directional or delocalised, now based on density criteria. The most notable reclassified molecules are highlighted in Figure 4, with their corresponding points plotted in darker shades of the original colours (dark purple for the outliers of mixed interactions and navy for the outliers of localised interactions). In total, there are three mixed outliers (pentane···AcOH, pentane···AcNH₂, and benzene···trifluoromethyl-iodide) and three localised-ouliers (MeNH₂···peptide, MeNH₂···MeOH, and peptide···peptide). In both cases, the initial classification attributes a more localised nature of the interaction than the density analysis.

In all cases, the EDA analysis agreed with a significant $E_{disp}$ contribution to overall binding. For the mixed outliers, it was even the case that dispersion dominated the binding. For the localised layers, the interaction was overall driven by electrostatics, but dispersion represented 20-25% of the total energy, which was a much more significant fraction than for other polar complexes.

To understand the reason for this mismatch, their NCI surfaces are also depicted in Figure 4. In the case of the localised-outliers (navy) complexes, there is a localised NH···H interaction, but accompanied by an important delocalised interaction associated with CH₃ units. For that reason, their $\frac{q_{NCI}}{V_{vdW}}$ was found to be lower, aligning more closely with the characteristics of the mixed category. Similarly, the mixed outliers (dark purple) complexes had even more diffuse surfaces, associated with a less polar nature of interaction, as has been found for
the complexes in the delocalised category. This again appears to be due to the geometry
of the complex, whereby the surface area of contact extends beyond the mere interaction of
functional groups to contact between larger portions of the molecular van der Waals surface.
Overall, the presence of these outliers highlights the fact that initial classifications require
rethinking in terms of the energy/density characteristics of the full molecule and not just
the main/more localised interaction.

Reclassification of big datasets

The fact that several compounds are found outside their original classification, and that
this can be verified at a cheap (promolecular density) level, pushed us to verify big datasets
present in the literature. Noticeably we have looked at the following NCIAtlas datasets:27,28,47,48

* HB375: hydrogen-bonded complexes, of which we selected 223 complexes with only the more rigid hydrogen bond definition of $X - H \cdots Y$, where X, Y = N/O only,

* D1200: dispersion complexes with multiple elements, spanning hydrocarbons, noble gases, halides, (per)oxides, ketones, aldehydes, sulfides, borates, nitriles and azo complexes, as well as aromatic and heteroaromatic species,

* HB300SPX: hydrogen-bonded complexes including interactions of S, P, and halogens

* SH250: dataset of halogen bonds extending to those including in S, Se, P, and As.

Figure 5 shows the $q_{NCI}-V_{vdW}$ scatter plot as in Fig. 4 for the different datasets alongside
the best-fit lines for the edge polar/local and non-polar/delocalised cases from Figure 4.
The datasets show a recognisable distribution in families. Nevertheless, a notable scatter
is present (even overlapping), leading again to the conclusion that i) main families can be
defined, but ii) no interaction class has a strictly identifiable characteristic from the energetics
or from the electron density viewpoint. Accordingly, one can also anticipate considerable
variation in both electrostatic and dispersion interactions, and hence variable macroscopic
behaviour (solubility, reactivity).

<figure><img src="image_7.png" /><figcaption>**Figure 5:** NCI charge-vdW volume scatter plot for NCIAtlas datasets representing common NCI types. The solid lines correspond to best-fit lines from Figure 4 for hydrogen bonds (blue) and dispersion (green). The different datasets can be distinguished by colour and marker. The NCI calculations were performed with promolecular electron densities.</figcaption></figure>

Considering the variability within each class, the results support the idea of avoiding NCI
definitions based solely on atomic composition. Instead, we propose that the $\frac{E_{elec}}{E_{disp}}$ is a sim-
ple measure of the bonding characteristics that spans the NCI chemical space, and that the
$\frac{q_{NCI}}{V_{vdW}}$ ratio can provide a fast and easy estimate of it. The ratios relate to the characteris-
tics (localised/delocalised) and energetics (polar/non-polar), thus providing an informative
characterisation.

Moreover, once again, when the datasets are examined using the ratio as a distinguishing
factor, the division varies from the one created by element-dependent definitions. Every
dataset contains at least (de)localised and mixed complexes, and HB375, HB300SPX, and
SH250 contain all three types (see Tables S4-6 in S.I. for the complete re-classification of the
NCIAtlas database). Some examples have been highlighted in Figure 6, where we showcase
the initial classification with symbols, and the energetic/density one in framed categories.
As an example, our initial aniline-acetic acid complex (Figure 1) is classified as delocalised,
despite having an NH-OC hydrogen bond. Like many other complexes, it additionally con-
tains a big non-polar contact surface between the two molecules, which, when considered
in the overall interaction, changes its overall classification to more delocalised than initially
expected: the $\frac{q_{NCI}}{V_{vdW}}$ ratio was 0.010, and the $\frac{\Delta V_{elst}}{\Delta E_{disp}}$ ratio was 2.13, making the complex in
line with the delocalised dataset. Once again, we find that the classification of the overall
molecule is not in agreement with its total interaction due to a big dispersion coming from
other parts of the molecule. This issue will be dealt with in the coming section.

Scale-up: application to complex systems

It should be noted that the calculation of the $\frac{q_{NCI}}{V_{vdW}}$ ratio from promolecular densities (i.e.
atomic densities that do not require an SCF calculation) gives a very similar pattern to
that shown in Fig. 5 (see S.I.'s Figure S4), which points at the possibility of using this
classification of non-covalent interactions in bigger systems.

However, for interactions in big systems, a single $\frac{q_{NCI}}{V_{vdW}}$ index would further average the nature
of the interaction over the full system as happened for the smaller datasets. To remedy this,
it is necessary to be able to provide $\frac{q_{NCI}}{V_{vdW}}$ ratios for different parts of the molecule. To
that end, we have created a clustering algorithm that separates the surface into distinct
interactions using NCI critical points as clustering centres and attributing volumes following
an ‘euclidean’ metric (scikit-learn’s NearestNeighbors algorithm$^{64}$).

Table 1 showcases the results for the aniline-acetic acid complex. Since it has four NCI
intermolecular minima, we obtain an effective division of the total interaction region into 4
distinct interactions. The delocalised parts now lead to ratios around 0.01 (delocalized) for
the alkyl chain, whereas it is in the mixed ratio for the NH-O interaction.

This approach is thus applicable to big systems, where EDA or even DFT would be out of
<figure><img src="image_8.png" /><figcaption>**Figure 6:** Example complexes from NCIAtlas database that have been classified contrary to chemical intuition, where hydrogen bonds are localised, sigma holes and SPX hydrogen bonds are mixed, and dispersion interactions are delocalised. NCI isosurfaces correspond to s=0.5 and a colour scale of -0.04 < sign($\lambda_2$)$\rho$ < +0.04 a.u. The $\frac{q_{NCI}}{V_{vdW}}$ ratios are given in a.u.</figcaption></figure>
<table><caption>**Table 1:** Bond critical point-divided surface $\frac{q_{NCI}}{V_{vdW}}$ ratios for the aniline-acetic acid complex.</caption><thead><tr><th>Contact</th><th>distance / Å</th><th>q<sub>NCI</sub>/V<sub>vdW</sub></th></tr></thead><tbody><tr><td>NH-OC</td><td>2.25</td><td>0.012</td></tr><tr><td>CO<sub>π</sub>-CN<sub>π</sub></td><td>3.65</td><td>0.010</td></tr><tr><td>CO<sub>π</sub>-CC<sub>π</sub></td><td>3.32</td><td>0.009</td></tr><tr><td>CH-CC<sub>π</sub></td><td>2.96</td><td>0.009</td></tr></tbody></table>

<figure><img src="image_9.png" /><figcaption>**Figure 7:** Macrocycle host - guest complex with the NCI surface clustered with selected $\frac{q_{NCI}}{V_{vdW}}$ ratios represented. NCI isosurfaces correspond to s=0.5 and a colour scale of -0.04 < sign($\lambda_2$)ρ < +0.04 a.u. The NCI calculations were performed with promolecular electron densities. The $\frac{q_{NCI}}{V_{vdW}}$ are given in a.u. and colour-coded with blue: localised, orange: intermediate, green: delocalised.</figcaption></figure>
<table><caption>**Table 2:** Close (≤ 2.9 Å) contacts in the macrocycle-ligand complex (more multiatomic contacts are present at longer distance). The atom numbering follows the order from the X-ray file in Ref.⁶⁵ (cif: RACKAH).</caption><thead><tr><th>Atoms interacting</th><th>Macrocycle Atom</th><th>Ligand Atom</th><th>distance / Å</th><th>q<sub>NCI</sub>/V<sub>vdW</sub></th></tr></thead><tbody><tr><td colspan="5" style="text-align: center; font-weight: bold;">Localised</td></tr><tr><td>CO-NH</td><td>H104</td><td>O7</td><td>2.03</td><td>0.040</td></tr><tr><td>CO-NH</td><td>H74</td><td>O7</td><td>2.03</td><td>0.040</td></tr><tr><td>CO-NH</td><td>H103</td><td>O8</td><td>2.05</td><td>0.039</td></tr><tr><td>CO-NH</td><td>H173</td><td>O8</td><td>2.05</td><td>0.039</td></tr><tr><td>NH-πCC</td><td>C53</td><td>H9</td><td>2.27</td><td>0.031</td></tr><tr><td>NH-πCC</td><td>C29</td><td>H10</td><td>2.31</td><td>0.029</td></tr><tr><td colspan="5" style="text-align: center; font-weight: bold;">Intermediate</td></tr><tr><td>CH-πCC</td><td>C17</td><td>H14</td><td>2.70</td><td>0.016</td></tr><tr><td>CH-πCC</td><td>C66</td><td>H12</td><td>2.80</td><td>0.014</td></tr><tr><td>CH-πCC</td><td>C64</td><td>H11</td><td>2.89</td><td>0.013</td></tr><tr><td>CH-πCC</td><td>C13</td><td>H13</td><td>2.88</td><td>0.013</td></tr><tr><td colspan="5" style="text-align: center; font-weight: bold;">Delocalised</td></tr><tr><td>CO-HC</td><td>H81</td><td>O7</td><td>2.55</td><td>0.010</td></tr><tr><td>CO-HC</td><td>H109</td><td>O7</td><td>2.61</td><td>0.009</td></tr><tr><td>CO-HC</td><td>H115</td><td>O8</td><td>2.55</td><td>0.010</td></tr><tr><td>CO-HC</td><td>H85</td><td>O8</td><td>2.67</td><td>0.009</td></tr></tbody></table>

reach. As a first example, Figure 7 shows a macrocycle host-guest⁶⁶ complex with important areas of hydrogen bonds, van der Waals and also some weak hydrogen interactions. The localised interactions would have been drawn in the general $q_{NCI}/V_{vdW}$ ratio, but the separate description of the interactions allows us to find the four CO-N anchoring hydrogen bonds (most directional ones) with respectively $q_{NCI}/V_{vdW}$=0.039-0.04 a.u (see Table 2). It is also possible to identify two non-conventional NH - benzene hydrogen bond ($q_{NCI}/V_{vdW}$=0.031-0.029 a.u.) and several localised to intermediately localised interactions involving benzene rings ($q_{NCI}/V_{vdW}$=0.013-0.016 a.u.). The good complementarity in the inclusion complex is finally ensured by several delocalised interactions ($q_{NCI}/V_{vdW}$=0.009-0.1 a.u.).

In order to show the potential use for biomolecules, Figure 8 shows a protein-ligand⁶⁷ pocket, and Table 3 shows the close interactions listed. It is easily seen that two main regions of interaction appear in the pocket, close to the sulphur terminal region (see Table 3): a classical NH-O hydrogen bond where the oxygen atom comes from residue Asp289 ($q_{NCI}/V_{vdW}$=0.024 a.u.)
and a non-conventional but well localised CH-O hydrogen bond with the oxygen of Asp93 ($\frac{q_{NCI}}{V_{vdW}}=0.022$ a.u.). Several intermediate interactions also appear, which correspond to non-conventional hydrogen bonds (carbon donors and acceptors, $\pi$ acceptors). It is interesting to note how easily this approach enables identifying these non-conventional interactions, and even to tell apart the one that is actually stronger and more localised. Table 3 also includes the distances, which allows us to see that the density characteristics are not only dictated by the distance but atomic composition is also very relevant: the CH-O (Asp93-OD2) interaction at 2.1Å is more localised than the CH-C (Ile171-HD11) interaction at 1.93Å. All other contributions are mainly related to complementarity through van der Waals interactions.

<figure><img src="image_10.png" /><figcaption>**Figure 8:** BACE-1 Protein-ligand complex (PDB ID: 3KMX) with the NCI surface clustered with selected $\frac{q_{NCI}}{V_{vdW}}$ ratios represented. NCI isosurfaces correspond to $s=0.5$ and a colour scale of $-0.04 < \text{sign}(\lambda_2)\rho < +0.04$ a.u. The NCI calculations were performed with promolecular electron densities. The $\frac{q_{NCI}}{V_{vdW}}$ are given in a.u. and colour-coded with blue: localised, orange: intermediate, green: delocalised.</figcaption></figure>
<table><caption>**Table 3:** Close (≤ 2.9 Å) contacts in BACE-1 protein-ligand complex. The atom numbering follows the order from the X-ray file.</caption><thead><tr><th>Atoms interacting</th><th>Protein Atom</th><th>Ligand Atom</th><th>distance / Å</th><th>q<sub>NCL</sub>/V<sub>vdW</sub></th></tr></thead><tbody><tr><td colspan="5" style="text-align: center; font-weight: bold;">LocaliSed</td></tr><tr><td>CO-HNH</td><td>Asp289-OD2</td><td>HN22</td><td>1.90</td><td>0.024</td></tr><tr><td>CO-HC</td><td>Asp93-OD2</td><td>H112</td><td>2.10</td><td>0.022</td></tr><tr><td colspan="5" style="text-align: center; font-weight: bold;">Intermediate</td></tr><tr><td>CH-HC</td><td>Ile171-HD11</td><td>H82</td><td>1.93</td><td>0.015</td></tr><tr><td>CH-HN</td><td>Gly291-HA2</td><td>H1</td><td>2.06</td><td>0.013</td></tr><tr><td>CH-HC</td><td>Trp176-HZ2</td><td>H72</td><td>2.21</td><td>0.012</td></tr><tr><td>CH-HC</td><td>Leu92-HD2</td><td>H2</td><td>2.25</td><td>0.011</td></tr><tr><td>COπ-HC</td><td>Asp93-OD2</td><td>H3</td><td>2.49</td><td>0.014</td></tr><tr><td>CCπ-HC</td><td>Tyr132-CD2</td><td>H5</td><td>2.62</td><td>0.012</td></tr><tr><td colspan="5" style="text-align: center; font-weight: bold;">Delocalised</td></tr><tr><td>OH-HC</td><td>Tyr132-OH</td><td>H113</td><td>2.60</td><td>0.010</td></tr><tr><td>CH-HC</td><td>Leu92-HD21</td><td>H73</td><td>2.62</td><td>0.008</td></tr><tr><td>CH-HC</td><td>Leu92-HD23</td><td>H73</td><td>2.64</td><td>0.008</td></tr><tr><td>CH-OC</td><td>Thr293-HG1</td><td>H102</td><td>2.64</td><td>0.007</td></tr><tr><td>CH-HC</td><td>Ile171-HD11</td><td>H72</td><td>2.69</td><td>0.009</td></tr><tr><td>CH-OC</td><td>Gly291-O</td><td>H103</td><td>2.70</td><td>0.005</td></tr><tr><td>CH-HC</td><td>Trp176-HH2</td><td>H72</td><td>2.76</td><td>0.007</td></tr><tr><td>C=Oπ-HC</td><td>Gln73-O</td><td>H92</td><td>2.77</td><td>0.009</td></tr><tr><td>CC-HC</td><td>Tyr132-CZ</td><td>H113</td><td>2.79</td><td>0.010</td></tr><tr><td>CH-HC</td><td>Leu92-HD2</td><td>H93</td><td>2.80</td><td>0.003</td></tr><tr><td>CH-HC</td><td>Gly74-HA2</td><td>H101</td><td>2.83</td><td>0.005</td></tr><tr><td>CH-Clσ</td><td>Gly135-HA3</td><td>Cl1</td><td>2.86</td><td>0.008</td></tr><tr><td>CO-HC</td><td>Gly291-O</td><td>H2</td><td>2.88</td><td>0.010</td></tr></tbody></table>
This approach thus enables us to accurately classify localised/delocalised interactions in
systems of arbitrary size.

Conclusion

We propose an approach to the classification of non-covalent interactions based on electron
density. The shape and integrals of the NCI regions enable a coarse-grained classification of
non-covalent interactions in terms of their electrostatic and/or dispersive nature as a sign
of their directionality. More specifically, larger NCI volumes and charges within the attrac-
tive region denote electrostatic non-covalent interactions and are often associated with more
localised interactions with high $\frac{q_{NCI}}{V_{vdW}}$. Instead, dispersive NCIs are characterised by signif-
icantly larger $V_{vdW}$ but not necessarily higher $q_{vdW}$ than polar ones. This ratio correlates
well with independent energetic decompositions, offering a simple proxy for the electrostatic-
to-dispersion character of a given interaction. Moreover, the underlying NCI calculations are
furthermore scaling much better than similar measures, e.g., SAPT0, taking merely seconds
for SAPT's minutes for systems of up to 50 atoms. Hence, this new approach allows the
efficient ab-initio classification of non-covalent interactions in systems of all sizes.

After verifying 2421 compounds with main group elements,<fnref n="2" /> trends in $\frac{q_{NCI}}{V_{vdW}}$ support a contin-
uous but meaningful categorisation scheme: localised interactions above $\frac{1}{50}$ a.u., delocalised
below $\frac{1}{100}$ a.u., and mixed cases in between. It should be noted that these ratios are valid
both for DFT (where the functional made little difference to NCI indices³⁴) and promolec-
ular calculations. Via a simple partition of the interacting volumes, this classification has
also been made available to big systems, showcased by a macrocycle and a ligand-protein
interaction.

Two main conclusions can be drawn from our analysis of thousands. On the one hand, the use
of mere chemical intuition for the classification of compounds leads to an underestimation of
the role of dispersion in their stabilization. This should be taken into account when making

<footnote n="2">²Applications to transition metals will be the aim of a future publication.</footnote>
datasets of small molecules, where the division of interacting parts might also show to be
necessary.

On the other hand, we see that the division between different families in terms of atomic
composition (hydrogen bonds, halogen bonds, etc) lacks generality in terms of energetic
grounds. Indeed, these classifications were typically introduced to identify strong localized
interactions that allow chemical design. Although they apply for most compounds in their
families, numerous exceptions are also present. Hence, the use of atom-based nomenclatures
can hide weak delocalized non covalent interactions that do not fulfill the original naming
purpose. We propose that the localized/delocalized classification allows to overcome this
caveat, providing a fast characterization of the interactions in terms of their physical and
chemical characteristics (electrostatic/dispersion, localised/delocalised), which allow a better
grasp of their expected chemical behaviour, in the same spirit as covalent-polar-ionic for
stronger bonds.

Acknowledgments

We thank Tatiana Woller and Roberto Boto for preliminary data in Ref. 45 that led to this ar-
ticle. We thank ANR TcPredictor S22JRAR036, ANR Fisciency S23JRAR060, Emergence-
SU H2Ox S23JR31014, and Sorbonne University Alliance iDream for funding, and GENCI
A0040710377 for computational resources. M.A. thanks the VUB for the Strategic Research
Program awarded to the ALGC research group. E.D. acknowledges the financial support
from the Fund for Scientific Research - Flanders (FWO-11E0321N). We thank C. Fonseca
for the S66 energetic partition data.

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