
==== Front
J Chem Theory Comput
J Chem Theory Comput
ct
jctcce
Journal of Chemical Theory and Computation
1549-9618
1549-9626
American Chemical Society

39208255
10.1021/acs.jctc.4c00784
Article
van der Waals Radii of Free and Bonded Atoms from Hydrogen (Z = 1) to Oganesson (Z = 118)
https://orcid.org/0000-0003-3069-2522
Charry Jorge
https://orcid.org/0000-0002-1012-4854
Tkatchenko Alexandre *
Department of Physics and Materials Science, University of Luxembourg, L-1511 Luxembourg City, Luxembourg
* Email: alexandre.tkatchenko@uni.lu.
29 08 2024
10 09 2024
20 17 74697478
18 06 2024
20 08 2024
31 07 2024
© 2024 The Authors. Published by American Chemical Society
2024
The Authors
https://creativecommons.org/licenses/by-nc-nd/4.0/ Permits non-commercial access and re-use, provided that author attribution and integrity are maintained; but does not permit creation of adaptations or other derivative works (https://creativecommons.org/licenses/by-nc-nd/4.0/).

Reliable numerical values of van der Waals (vdW) radii are required for constructing empirical force fields, vdW-inclusive density functional, and quantum-chemical methods, as well as for implicit solvent models. However, multiple definitions exist for vdW radii, involving either equilibrium or the closest contact distances between free or bonded atoms within molecules or crystals. For the paradigmatic case of the hydrogen atom, its reported vdW radius fluctuates between 2.15 and 3.70 Bohr depending on the definition, leading to a high uncertainty in calculations and different conceptual interpretations of noncovalent interactions. In this work, we systematically review different definitions and methodologies to establish the free and bonded vdW radii for hydrogen, based on equilibrium vdW distances in noncovalently bonded molecules, enveloping electron density cutoffs, noncovalent positron bonds in hydrogen anion dimer, vacuum virtual photon cloud caused by the hydrogen atom, and atomic dipole polarizability. By doing so, we show that the vdW radius of the free hydrogen atom is 3.16 ± 0.06 Bohr. By employing the most general and elegant definition of atomic vdW radius as a function of the atomic polarizability, we tabulate consistent values of vdW radii for all atoms in the periodic table up to Z = 118.

European Research Council 10.13039/501100000781 NA Fonds National de la Recherche Luxembourg 10.13039/501100001866 13590856 document-id-old-9ct4c00784
document-id-new-14ct4c00784
ccc-price
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pmcIntroduction

Effective noncovalent radii for free and bonded atoms are widely used to describe a broad set of physicochemical phenomena, especially those where chemical bonds interplay with noncovalent interactions. In particular, the concept of van der Waals (vdW) radius goes back to 1932, starting with the idea of Mack1 and Magat2 to introduce a specific radius, which describes a distance an atom maintains from other atoms in the case of noncovalent interactions. There are two distinct categories of vdW radii: crystallographic and equilibrium.3−5 The first one corresponds to the closest approach of atoms in a crystal, as originally defined by Pauling6 and Bondi.7 In a more recent and deeper study, Alvarez analyzed millions of atomic distance distributions to a probe atom (usually oxygen), finding a clear separation of bonded and intermolecular nonbonded vdW contacts in crystal structures.8 This approach has an important limitation: when two molecules approach, their equilibrium distance is reached when the sum of all (many-body) repulsive interactions balances the attractive ones. Therefore, atoms can come closer than the sum of their respective vdW radii.4,9 On the other hand, equilibrium vdW radii are based on the distance in the vdW potential minimum between two isolated atoms.3 Taking this definition, Batsanov tabulated reference vdW radii values for many atoms by performing an extensive compilation of molecular systems in which these atoms are in a nearly isolated state or forming weak bonds with the environment, in addition to further data extrapolating to effectively neutral atoms.4

However, vdW radii defined by the aforementioned approaches are not unique since experimental structures yield a distribution of noncovalent interatomic distances. Hence, those vdW radii are statistical in nature. Furthermore, Batsanov does not report a unique vdW radius for the hydrogen atom.4

The vdW radius is a broadly useful concept when calculating or analyzing noncovalent interactions. For example, the nature of a given bond can be inferred by inspecting interatomic distances and comparing them to the sum of respective vdW radii.10−12 The vdW radii are also widely used in calculations of noncovalent interactions,13−21 for example, by rescaling the free (isolated) vdW radius to an effective radius that resembles an atomic size within a molecule. In practice, effective radii are employed in polarizable force fields,3,20,22−24 in the construction of the atomic cavity in solvation models,17,25−27 and as distance parameters for range separation damping functions in theoretical models for dispersion corrections, such as the Tkatchenko–Scheffler (TS) method,28 the many-body dispersion (MBD) formalism,29−31 as well as certain implementations of Grimme’s DFT-D methods32−34 and the exchange–dipole moment (XDM) model.35−37

Since the atomic vdW radius is an effective quantity, several definitions exist for it. For noble-gas atoms, the vdW radius is conveniently defined as half of the equilibrium distance in the respective homonuclear dimers. This definition is rational, since noble-gas atoms are noncovalently bonded upon interacting with each other. Other chemical elements need to be considered in molecular environments where they become effectively closed shell due to their covalent bonding within the corresponding molecule and can interact noncovalently with other atoms or molecules. However, such an approach can yield a large range of values for atomic vdW radii since it depends on the chemical environment of each atom.3,4,17,21,38,39

The effect of the environment is attributed to two factors. First, the electronic density changes depending on the surrounding atomic environment in a molecule, caused by the formation of covalent bonds and intramolecular interactions. Second, further changes are induced by the intermolecular attractive and repulsive vdW interaction between molecules.9 The latter effect can be controlled by selecting molecular systems with low contact areas as well as molecular fragments with low permanent dipoles and low atomic polarizabilities. The effect of the electronic density changes can be determined by taking into account the changes in atomic partial charges40,41 or atomic volumes, as we will further discuss later in the text. Hydrogen atoms are the paradigmatic case for the so far ambiguous definition of the atomic vdW radius. For hydrogen, one can find different vdW radii lying between 2.1 and 3.7 Bohr.4,6,7,9,42−46 Arguably, hydrogen is the “fruit fly” of chemistry, for which a consistent definition of the vdW radius would be critical to further our understanding of vdW interactions and solvation in chemical and biological systems. However, this seems to be a perplexing and unresolved puzzle.3,4,7,12,40,42−48

In this work, we explore and discuss several approaches toward a consistent determination of the vdW radius, focusing on the hydrogen atom. To estimate the effective vdW radii, we employed the equilibrium distance of neutrally charged hydrogen atom systems, whose properties are largely determined through vdW interactions. Those effective vdW radii are rescaled to free vdW radii by employing the ratio between free and effective Hirshfeld lengths of an atomic density in a molecule. In addition, our findings are compared with previously reported equilibrium values, positron-induced bonding between hydrogen anions,49,50 and system-independent methodologies such as atomic density cutoffs delineating an atomic size,51 vacuum virtual photon cloud caused by the hydrogen atom,52 as well as estimations from the atomic dipole polarizability.53 The latter approach is then employed to calculate the recommended set of equilibrium vdW radii of atomic elements from Hydrogen to Oganesson (1 ≤ Z ≤ 118).

Results and Discussion

Let us start by reviewing a few relevant values of hydrogen vdW radii compiled from the literature, collected in Figure 1. Overall, different methodologies provide a wide range of values with some notable variations even within each category. Regarding the crystallographic vdW radius, the original value by Bondi7 is the most commonly accepted, as has been confirmed by recent revisions.8,48,54,56 For molecular systems, Batsanov reported three possible averaged values for the equilibrium vdW radii for many atoms: Re, as a fitting parameter in a vdW potential molecular mechanics model,9RG determined experimentally from the structure of molecules containing noble-gas atoms (Rg-M or Rg-M2), and R0 is a radius defined as the extrapolated zero charge limit of the effective radii of M for MXn systems as a function of the partial charge on M for a given set of X halogens. Despite presenting three estimated vdW radii for hydrogen, Batsanov did not recommend a particular value, probably due to the large discrepancy among them. This is in contrast to the rest of the atomic elements, where the recommended value is an average value from the three definitions. The atomic probes also display a wide range of values depending on the target system, the probe, and the nature of their bonding.8,14,45 Finally, atomic sizes from a fixed electron density contour value are intrinsically dependent on the choice of the density cutoff.46,51,55,57,58 We revisit the last three definitions in what follows.

Figure 1 Literature values for van der Waals radii of hydrogen are grouped into four categories. Crystallographic: from distances of closest approach between atoms in crystal structures. Equilibrium: based on the minimum of vdW interaction potential between two nearly isolated atoms. Atomic probes: derived from the closest approach distance in the interaction between an atom and a probe. Density topology: based on defining a density cutoff to delineate a region containing a percentage of the electron density.

Such variations in the vdW radius could lead to drastic effects. For example, in Figure 2, we show how a variation of the hydrogen vdW radius could affect the hydrogen–hydrogen vdW interaction energy EvdW as estimated from a Lennard-Jones potential23,24,59−61 for a local minima structure of a water cluster (H2O)30621

2

3

In that model, for two atoms I and J separated by a distance Rij, the vdW energy depends only on a well-depth parameter ϵ and on their effective vdW radii RvdWeff; both are combined using the widely used Lorentz–Berthelot rules to describe the interaction of nonbonded atoms.23,24,59−61 Additionally, the vdW radii are also employed as a cutoff value to calculate EvdW if the internuclear distance is larger than their respective sum of vdW radii. Therefore, increments of vdW radii also reduce the number of vdW interacting atoms, nvdW, counteracting the reduction in the interaction energy. This effect is evidenced by the exchange between the flat and descending regions of both quantities in Figure 2. The variations of EvdW and nvdW are directly linked to the geometrical distribution of the water cluster. Evidently, even small changes in the vdW radii can yield large differences in vdW interactions, emphasizing the importance of revealing physically motivated and reliable values for atomic vdW radii.

Figure 2 Variation of the hydrogen–hydrogen van der Waals energy EvdWH–H and the number of vdW interacting hydrogen atoms nvdWH–H with respect to the hydrogen vdW radius, assuming a Lennard-Jones potential for a cluster of 30 water molecules. Well-depth potential ϵ of 0.0007 kcal/mol (hydrogen bonded to oxygen).23 The effective vdW radii RvdWeff were computed from eq 4 by using PBE0/aug-cc-pVTZ atomic and molecular densities. The gray region indicates the most common range of radii employed for hydrogen in the prior literature.

The values of vdW radii depend on the chemical environment of atoms in the molecules. In model vdW potentials, the effective vdW radii are usually estimated from reference vdW radius for free atom by rescaling it with the ratio between the free atomic volume VA, as16,18,20,28,31,634

with the atomic volume defined as5

for each atom A centered at the position RA. The atomic densities in a molecule, nAH, can be computed using the Hirshfeld partitioning of the density,18,646

where nAfree(r) is the electronic density for the free atom A and n(r) is the electronic density of the full molecular system at a point r in the space.

Figure 3 shows the distribution of the Hirsfield ratios Veff/Vfree of H, C, N, O, S, and Cl atoms for a set of optimized organic molecules in the QM7-X data set containing 42k molecules.65 From this plot, it is evident that hydrogen atoms exhibit the largest and most variable change in volume when they are in a molecule. This reinforces previous observations that the hydrogen vdW radius is strongly affected by the chemical environment.16−18,63,66−68 Therefore, care is needed when attempting to define the free-atom vdW radius based on molecular systems.

Figure 3 Whisker plot of Hirshfeld volume ratios of H, C, N, O, S, and Cl atoms for the optimized structures within the QM7-X data set of organic molecules with up to 7 heavy (non-hydrogen) atoms.65

For this work, in order to reduce the influence of the chemical environment, for all calculated effective vdW radii, we estimate the vdW radius for the free hydrogen atom by rescaling them with the ratio between the free atomic length L, and the effective Hirshfeld lengths, employing new proposed scaling laws,69,70 as7

This procedure is then the inverse of eq 4, commonly used to obtain effective vdW radii from free reference values.16,18,20,28,31,63 The atomic length LA in the above equation is calculated with the Cartesian component of the position operator8

In eq 7, the exponent 4/7 is determined using two recently proposed scaling laws involving those quantities and the dipole polarizability. First, the dipole polarizability with vdW radius follows as α1 ∝ RvdW7.53,70 Second, in ref (69), a four-dimensional scaling law α1 ∝ CL4 has been proposed between the dipole polarizability and a characteristic length L of a quantum system defined via the Euclidean norm of the position operator r, and C is a dimensionless excitation-energy ratio, which in this case will be assumed to be same between the free atom and the effective atom in a molecule. This new proportional relationship of eq 7 provides a smaller rescaling factor when the effective size of the atom is smaller than the free atom, compared to the standard classical assumption63 given by RvdW,A = (VA)1/3.

Let us now examine the definition of equilibrium vdW radii for noncovalently interacting hydrogen systems. One of the earliest definitions of a vdW radius was established by Bondi in terms of the distance at which the Pauli repulsion balances the attraction forces between the two atoms.7 Nevertheless, the above definition must be adopted carefully since there is no unique approach to quantify noncovalent attractive and repulsive forces between two atoms in a molecule. Among the many existing quantum chemistry approaches, symmetry-adapted perturbation theory (SAPT)71−74 allows computing and decomposing the total interaction energy between two noncovalently bonded subsystems into physically intuitive terms such as electrostatics, induction, exchange repulsion, and dispersion Eint = Eelst + Eexch + Eind + Edisp, while keeping the accuracy of interaction energies close to the CCSD(T) “gold standard”. For this reason, we employed the SAPT methodology to explore the aforementioned balance of attractive and repulsive forces in the interaction of vdW systems. The Supporting Information provides a more detailed description of the calculations. The SAPT forces were computed by numerical differentiation of each energy component with respect to the internuclear separation . All SAPT and CCSD(T) calculations presented in this work were obtained with the PSI4 code.75

To provide the first estimate of hydrogen vdW radius, we examine a simple system containing neutral hydrogen atoms—the hydrogen molecule in its triplet state.76 This system exhibits a very small binding energy of 4.3 cm–1 at an internuclear separation of 7.85 Bohr.77 Here, the SAPT decomposition indicates that at the equilibrium distance of the triplet H2, the main contribution to the attractive forces comes from the dispersion terms by 83%, while electrostatics is 11%, induction is 5%, and the repulsive force is only due to exchange, suggesting that vdW interactions play a substantial role in the binding mechanism of triplet H2. Therefore, we pencil in 3.92 Bohr as a first estimation of the effective hydrogen vdW radius [using half of the equilibrium distance in triplet H2 at the CCSD(T)/CBS level of theory], and the same value is obtained for the corrected free vdW radii in this case due to the small variation in size at such large distances.

The next simplest vdW system exhibiting bonding between neutral hydrogen atoms is the hydrogen molecule dimer. Here, the full-dimensional potential-energy surface is rather complex.26,78−81 However, several specific configurations exist where the intramolecular structure of two hydrogen molecules is conserved, and the binding of the two monomers is dominated by vdW forces.79,82 For this study, we have selected the collinear, parallel, perpendicular, and shifted parallel configurations (see Figure 4), given that previous SAPT analyses indicate that their equilibrium geometry is caused by a balance between exchange and dispersion interactions.81,82

Figure 4 Schematic representation of the selected hydrogen molecule dimers. Solid lines represent the fixed distances, and dotted lines represent variable intermolecular distances.

The equilibrium distances for hydrogen molecule dimers are summarized in Table 1, as well as the relative contributions of each SAPT2 + 3(CCD) force component. The intramolecular H–H distance was fixed at 0.74 Å. Regarding the SAPT force decomposition, it can be concluded that the collinear and parallel configurations are the arrangements where two hydrogen molecules interact mostly due to the balance of dispersion and exchange forces. Contrary to perpendicular and shifted parallel configurations, where the electrostatic component is not negligible, this indicates that even for relatively simple molecules in different geometrical configurations, the equilibrium distance can vary significantly depending on the nature of the underlying interaction. Additionally, in a parallel configuration, all four atoms are directly involved in the intermolecular interaction, increasing the exchange repulsion at short distances, which can be compensated only at longer distances with the slower decaying dispersion terms. Interestingly, in a colinear configuration, there is a cancellation of forces between electrostatic and induction terms, favoring the dispersion and exchange balance and leading to an effective vdW radius of 2.89 Bohr, and an estimated free vdW radius of 3.13 Bohr.

Table 1 Equilibrium Distances (in Bohrs) for the Hydrogen Molecule Dimers and Atomic Probes Obtained with CCSD(T)/CBS and SAPT2+3(CCD)/aug-cc-pVQZa

system	Req CCSD(T)	Req SAPT						
H2–H2 (CL)	5.77	5.59	0.86	–0.11	0.89	0.14	-0.97	
H2–H2 (PA)	6.71	6.75	0.91	0.03	1.00	0.06	–0.91	
H2–H2 (PE)	5.63	5.58	0.61	0.28	1.00	0.11	–0.61	
H2–H2 (SP)	6.43	6.46	0.81	0.13	1.00	0.06	–0.81	
H2–He	5.72	5.65	0.74	0.17	1.00	0.09	-0.74	
H2–Ne	5.49	5.65	0.69	0.26	1.00	0.05	–0.69	
H–He	6.66	6.96	0.74	0.17	1.00	0.09	-0.74	
H–Ne	6.88	6.70	0.64	0.32	1.00	0.04	–0.64	
H2(13Σu+)	7.85	7.85	0.83	0.11	1.00	0.05	–0.83	
a Also shown is the ratio of each SAPT force with respect to the total attractive F– or repulsive forces F+. The largest force ratio for each group is highlighted in bold.

Another common approach to infer a vdW radius is the use of noble-gas atoms as atomic probes, since the latter form noncovalent interactions with other atoms and molecules.8,42,83−85 Here, the vdW radius simply corresponds to the difference between the equilibrium distance of the complex and the vdW radius of the atomic probe. For this work, we focused on systems with neutral hydrogen atoms, and for the probe, we selected helium and neon atoms to estimate the vdW radius in the complexes Rg-H2 (with a fixed H–H length of 0.74 Å) and Rg-H, see Table 1. By analyzing the contribution of each SAPT force with respect to the total attractive forces, the dispersion term is, in fact, the major contributor. However, in the case of neon probes, the electrostatic force is considerably high, as previously observed by Mantina.45 Therefore, the helium probe seems to be a more suitable atom than neon for effective vdW radii, with values of 2.92 and 3.86 Bohr for closed-shell and open-shell hydrogen atom, respectively. This corresponds to estimated free vdW radii of 3.15 and 3.86 Bohr, respectively.

Yet another definition for the vdW radius was proposed by Bader, based on the average distance from the nucleus, where the electron density falls to 0.001 electrons per Bohr3. This particular value was chosen to correlate well with crystallographic vdW radii for neutral46,51,57,68 and ionic atoms.58 An alternative approach consists of analyzing the free-atom electron densities compared to the electron density of corresponding noble-gas atom at a distance of their vdW radii from the nuclei.28 This approach has been employed in the Tkatchenko–Scheffler method28 to obtain the reference atomic vdW radii. In the Supporting Information, we illustrate the above procedure, and we compared several quantum chemistry methods and basis sets for the helium atom, showing no remarkable difference, where the highest level is FCI/CBS with a density contour of 0.0008 au at a distance from the nuclei equal to its RvdW of 2.65 Bohr.4 Boyd46,86 showed that below an electron density of 0.001 au, the relative radii of atoms are nearly invariant to small changes in the density cutoff, and still satisfy the periodic trends in a given period and group of the periodic table. Therefore, it is a suitable option to choose the distance at the RvdW of helium originally defined by Bondi,7 which has been widely employed for estimating the vdW radii of other elements using atomic probes42,45,83−85 Additionally, Rahm et al.46 showed how using the density cutoff metric provides vdW radii that correlate linearly with the equilibrium dimer distance for all noble gases. Under the above considerations, we obtained a value of 3.10 Bohr for the vdW radii of the free hydrogen atom.

A slightly more exotic approach for estimating interacting (noncovalent) atomic radii is based on the equilibrium distance of diatomic anions bound by positrons.49,50,87,88 These are systems in which a positron plays a major role in neutralizing and bonding two otherwise repulsive anionic atoms, forming a stable compound at a specific internuclear distance. One common feature in the studied positron-bonded systems49,87 is the small change in the electronic density distribution and the lack of electronic covalent bonds between atoms.89 For example, two or one positron can bind together two hydrogen anions in 2 e+[H–H–] or e+[H–H–], with an equilibrium distance of 6.00 and 6.36 Bohr, respectively. Because of the resemblance between the vdW radii of neutral atoms and their anionic radii as first found by Pauling,6 we can use the equilibrium distance of the positron-bonded systems to propose an estimation of the hydrogen anionic radius and, by extension, the hydrogen atom vdW radius. In such systems, the positron orbital creates an attractive force between two repulsive closed-shell anions until reaching the interpenetrability limit of the electronic clouds. This is not unlike vdW systems where the Pauli repulsion balances the attraction forces between the noncovalently bonded closed-shell atoms.7 Hence, we suggest an estimate of 3.00 to 3.18 Bohr for the vdW radius of free hydrogen atom from the hydrogen anion dimer bonded by one or two positrons.

The atomic polarizability (with units of volume) is evidently connected to the vdW radius since both properties measure effective atomic size. Indeed, a direct connection between the atomic dipole polarizability and vdW radius was established53 by studying the balance between Pauli exchange repulsion and London dispersion attraction forces for two interacting quantum Drude oscillators, obtaining the following final relation:709

where αfsc is the fine-structure constant, ϵ0 is the permittivity of free space, and a0 is the Bohr radius. The numerical value of the prefactor in the above equation was initially found by fitting the scaling law using reference dipole polarizabilities and the equilibrium vdW radii of Batsanov.4 Therefore, using the well-known exact polarizability of the hydrogen atom of 9/2 au (see, for example, ref (69)) and the inverse of eq 9, we obtain a free hydrogen vdW radius of 3.16 Bohr.

Yet another approach to define the vdW radius of free atoms is to employ first-principles of quantum electrodynamics (QED)90−93 for an atom interacting with the everpresent electromagnetic field. An atom spontaneously emits and absorbs radiation, with the equilibrium state of the atom–field system only achieved for large enough time scales. Hence, at any given instant, the atom is surrounded by a cloud of virtual photons. The properties of this photon cloud can be calculated from QED. Such calculations are rather involved, but they have been carried out for hydrogen by Passante and collaborators.52 In particular, the radius of the photon cloud (in the electromagnetic field reference frame) for the dominant 1s-2p transition of hydrogen atom is Bohr, where c is the speed of light and E1s–2p is the 1s-2p transition energy. To transform this radius to the atomic reference frame, one needs to multiply Rph by αfsc4/3, as explained in ref (70). This yields a radius of 3.25 Bohr. Since the virtual photon cloud provides an alternative representation of polarization from the point of view of the electromagnetic field, the connection between scaled Rph and the atomic vdW radius is not surprising. We remark that higher-energy transition contributions to Rph would slightly decrease its value,52 getting closer to the vdW radius calculated from the atomic polarizability and highlighting an intimate connection between QED and eq 9.

With the aim to integrate all our efforts to define consistent free and bonded vdW radii for hydrogen, in Table 2, we summarize all calculations presented in this work. The methodologies discussed can be categorized into three groups: free-atom-based (dipole polarizability in eq 9, density cutoff, and QED photon cloud), interaction of H2 with closed-shell systems (H2 dimers and noble-gas probes with H2), and open-shell systems (H2 triplet state and noble-gas probe with H atom). In the latter group, for vdW systems where the hydrogen atom is not covalently bonded, the largest vdW radii are obtained. This result is in line with the value R0 reported by Batsanov,4 which corresponds to a hydrogen atom approaching a halide dimer.40 Therefore, it clearly belongs to a different category of vdW radii. For the second group, we have vdW systems at equilibrium distances bonded mainly by intermolecular interactions involving neutrally charged closed-shell hydrogen atoms or in a nearly isolated neutral state. Effective vdW radii in this category are in agreement with the averaged Allinger’s equilibrium radii Re, Batsanov’s radii RG, and positron-bonded hydrogen anions, having values of 2.98 ± 0.10 Bohr. Since bonded atoms are usually more compact than free atoms, effective radii are lower than free radii. The small variation between all effective radii, estimated free-atom radii, and system-independent definitions indicates a high degree of consistency. Therefore, despite lacking a unique definition for the vdW radius, our comprehensive set of estimations results in a value of 3.16 ± 0.06 Bohr for the free hydrogen atom, excluding the radius from the open-shell hydrogen system. This reduced range provides a significant improvement compared to the scattered results available in the literature as summarized in Figure 1.

Table 2 Summary of Effective RvdWeff and Free RvdWfree vdW Radius Obtained from Different Methodologies (in Bohr)a

method	RvdWeff	RvdWfree	
α1/7	–	3.16	
ρ cutoff	–	3.10	
QED photon cloud	–	3.25	
H2–H2 (CL)	2.89	3.13	
H2–He	2.92	3.15	
e+[H–H–]88	3.18	–	
2 e+[H–H–]50	3.0	–	
Allinger, Re(3)	2.95	–	
Batsanov, RG4	2.95	–	
average	2.98 ± 0.10	3.16 ± 0.06	
H–He	3.86	3.86	
H2 (13Σu+)	3.92	3.92	
Batsanov, R04	3.70	–	
a Average and standard deviation are provided between all of the closed-shell and system-independent definitions.

In Figure 5, we further explore the differences between open-shell and closed-shell systems by comparing the SAPT force components along the intermolecular distance. Here, we observe that for interactions involving open-shell hydrogen atoms, such as H–H 13Σu+ and H–He, all of the force components are always higher, in absolute value, than those involving interactions between hydrogen in molecules, such as H2–H2 CL and H2–He. Therefore, in the former cases, the force balance between the faster-decaying exchange repulsion and attractive forces occurs at larger distances, while in the latter, when hydrogen is covalently bonded to another atomic fragment, all of the intermolecular force components are weaker resulting in shorter vdW equilibrium distances. Based on the above reasons, we could interpret the vdW radii of 3.16 as a radius for describing vdW interactions with covalently bonded hydrogen, and 3.8 bohr for a noncovalently bonded case.

Figure 5 Comparison of SAPT forces components split into exchange Fexch (solid lines) and attractive forces −Fdisp+elst+ind (dashed lines in opposite signs) as a function of the intermolecular distance. (a) Comparison of the He atomic probes approaching hydrogen in H–He against H2–He. (b) Comparison of hydrogen–hydrogen vdW contact in H–H triplet state against H2–H2 in collinear configuration. Vertical lines indicate the equilibrium distance, where the net forces are zero.

Among the many possible definitions of the vdW radius, the one derived from the dipole polarizability in eq 9 stands out for several reasons. First, it is system-independent, opposite to statistically derived radii from molecular systems with intrinsic variations due to different chemical environments. Second, the numerical value of the vdW radius obtained from the polarizability also coincides with the average value obtained from all of the free-atom definitions in Table 2. Third, this definition uniquely and unambiguously connects the vdW radius—a geometrical property—with an observable electronic property—the dipole polarizability—for all chemical elements.

Having established the vdW radius for hydrogen, in Figure 6, we reexamined the behavior of the formula in eq 9 for the periodic table, employing recommended atomic dipole polarizabilities95 and recommended equilibrium vdW radii from Batsanov4 when available. For comparison purposes, we added the correlation with the lowest and highest RvdW value previously reported for hydrogen, see Figure 1. It is evident now that for the hydrogen atom, our recommended vdW radius of 3.16 Bohr adjusts perfectly to the linear tendency with respect to all of the other atoms. Furthermore, the excellent correlation between both α1/7 and RvdW in eq 9 (R2 = 0.9877) allows us to define the free atomic vdW radii solely in terms of the atomic polarizability in a consistent manner for all atomic elements. Hence, it becomes possible to compute the vdW radius for elements where no previous data has been reported as summarized in the periodic table of Figure 7 for all elements from Hydrogen (Z = 1) to Oganesson (Z = 118).

Figure 6 Correlation between RvdW and RvdW(α) obtained from the atomic dipole polarizability using the inverse of eq 9 for all atoms. vdW radii from ref (7) for noble gases ref (94), for Rn, and ref (4) for all remaining atoms except Z = 58–71, 84–85, 87–88, 92–118. Dipole polarizability values are taken from ref (95) for all atoms except Z = 116.

Figure 7 Recommended values for free atomic van der Waals radius (in Bohr) for elements Z = 1 to Z = 118, obtained from the recommended dipole polarizabilities compiled in ref (95) by using eq 9. Estimated uncertainties are given in parentheses.

Conclusions

In conclusion, we have carried out a comprehensive analysis of different methodologies to calculate the free-atom vdW radii for hydrogen. Our examination of multiple definitions derived from system-independent approaches, as well as estimations from molecular systems where a hydrogen atom is found in a nearly closed-shell isolated state, shows that the RvdWfree for the hydrogen atom is 3.16 ± 0.06 Bohr. Remarkably, a previously revealed connection between the vdW radius and atomic dipole polarizability53,70 allowed us to define the free vdW radii for all atoms in a consistent manner solely based on their electronic polarizability. These new values can be used to consistently compute the effective vdW radius of an atom in a molecule, enabling accurate calculations of noncovalent interactions. Furthermore, the effect of the molecular environment of vdW radii can be taken into account by rescaling them using projected electron densities. The RvdWeff can be used in force field parametrization as another criterion to train or to classify different types of bonded hydrogen atoms in a molecule, instead of using chemical intuition to define atomic type according to the chemical environment.23,24 It is reassuring that widely different definitions for the vdW radius converge to essentially the same value when straightforward physicochemical arguments are used for extrapolating to the free-atom limit. In addition, the excellent agreement between the polarizability formula and explicit QED calculations hints at a deeper definition of the vdW radius based on an atom interacting with the quantized electromagnetic field. More work is needed to extend this connection to many-electron atoms.

Supporting Information Available

The Supporting Information is available free of charge at https://pubs.acs.org/doi/10.1021/acs.jctc.4c00784.Recommended vdW radii; correction to free atomic vdW radii; SAPT interaction energies and force curves decomposition; and basis set convergence on density contour cutoff values (PDF)

Supplementary Material

ct4c00784_si_001.pdf

The authors declare no competing financial interest.

Acknowledgments

The authors acknowledge financial support from the Luxembourg National Research Fund: AFR PhD Grant “POMO(AFR PhD/19/MS, GrNum:13590856) and from the European Research Council (ERC-AdG FITMOL). The authors thank Dmitry V. Fedorov, Matteo Barborini, Matej Ditte, and Péter Szabó for many fruitful discussions.
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