Introduction

Molecular recognition and self-assembly of biomolecules and nanomaterials1,2,3,4 are governed mostly by non-covalent interactions5,6,7 including hydrogen bonding8,9,10, π-interactions11,12,13,14,15 and halogen bonding16,17,18,19,20,21,22. Quite often, molecular conformations are determined by electrostatic interactions which are generally described by isotropic atomic charges. However, atomic charges in molecules are not isotropic. As a simple example, the σg molecular orbital (MO) of an H2 molecule shows overlap between 1s atomic orbitals (AOs) of the two atoms. This overlap increases the electron density (ρe) between the two nuclei (i.e., over the cylindrical surface surrounding the bond), which results in negative electrostatic potential (ESP). Then, the decreased ρe outside the two nuclei (around bond-ends) results in positive ESP. Thus, the quadrupole moment (Qzz) for the case where the two atoms are along the z-axis is positive (0.45 debye·Å). Such phenomena could be expected for all the homonuclear diatomic molecules regardless of the σ- or π-type overlap, as can be seen from ESP maps of H2 (σg MO) and C2 (πu MO) shown in Figure 1a. The anisotropic charge distributions result in highly direction-specific interactions best exemplified by halogen bonding, the origin of which is generally explained by the concept of σ-hole23,24,25,26,27. This anisotropy in charge distribution is visualized in F2 and Cl2 ESP maps (Figure 1a). It can be understood as a “hole” residing in the antibonding σ*-orbital. Since this σ-hole, in principle, forms at both ends of every σ-bond, one might expect that this concept would be applicable to all the elements.

Figure 1
figure 1

ESP maps and σ-bonding valence MO.

(a). Ground/first excited states of A2 (A: N, O, F, P, S, Cl) at the MP2/aVTZ level of theory (density isovalue: 0.001 au.). H2, B2, C2, CO (isoelectronic to N2) are added for comparison. Superscript “t” means triplet state. The Be2 and tF2 are not drawn because of dissociation at the CCSD(T)/aVTZ level. The bond-length (d in Å), quadrupole moment (Qzz in debye · Å) and dipole moment (μ in debye) are given at the optimized CCSD(T)/aVTZ geometry. (b). σ-bonding orbitals of the ground-state structures.

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In this regard, we have calculated the ESP maps for homonuclear diatomic molecules of second and third row elements. We note that, for N2 and O2, the covering of σ-hole by the nonbonding electrons inverts the typical anisotropy (Figure 1a). N2 shows negative ESP at bond-ends and positive ESP around the cylindrical-bond-surface. A similar result was also reported very recently by Hobza and coworkers28. A simple AO-overlap concept cannot explain the N2 case. The positive ESP around the cylindrical-bond-surface between two N nuclei of the N≡N bond is even more difficult to understand, since compounds with C≡C bonds usually undergo facile reactions with electrophiles29.

To understand this puzzling anomaly, we investigated the difference in MOs and natural bonding orbitals (NBOs) for the ground states of N2, tO2, F2, P2, tS2 and Cl2 and their first excited states (tN2, O2, tF2. tP2, S2 and tCl2), where the superscript “t” denotes triplet-state. The differences in orbital hybridization, bond-length and nuclear-charge turn out to be important factors in anisotropy. To investigate the anisotropy effect on the measurable quantities, we studied the interaction of homonuclear diatomic molecules with themselves, positively/negatively charged ions (Na+, Cl−), a water molecule and a benzene molecule, using Møller-Plesset second-order perturbation theory (MP2) and coupled-cluster theory of singles, doubles and perturbative triples excitations (CCSD(T)) at the complete basis set (CBS) limit. Here we discuss the anomaly arising from the charge anisotropy of N2 and tO2 in particular.

Results

Ground states of N2, tO2 and F2

ESP contour maps of representative homonuclear diatomic molecules at their optimized geometries are shown in Figure 1a along with their bond distance (d) and Qzz. Inspection of the Qzz values of second-row homonuclear diatomic molecules (tB2, C2, N2, tO2, F2) reveals that Qzz decreases abruptly to a negative value at N2 and again increases to a positive value at F2. This sign change in Qzz indicates the inversion of anisotropy at N2. Consistent to this observation, the ESPs of N2 and tO2 are negative at bond-ends and positive around the cylindrical-bond-surface, unlike other ground state species. The issue of Qzz and its relation to ESP was also addressed in literature recently28,30.

To find a qualitative reasoning for the prominent difference between N2 and F2, we take both ρe and nuclear charge into account. Nitrogen has a smaller effective nuclear-charge (Zeff(N2p) = 3.83) than fluorine (Zeff(F2p) = 5.10)31. Therefore, at some distances from the bond-ends the ρe(N) becomes higher than ρe(F), as can be seen from the σ-bonding MOs shown in Figure 1b. Furthermore, since the bond length of N2 (1.10 Å) is much shorter than that of F2 (1.42 Å), the large electron-population required by the N≡N triple bond cannot be accommodated within such a small space between the two N nuclei. A nodal plane bounded by a positively charged region near each nucleus and a negatively charged region somewhat away from the N nucleus is formed outside each N nucleus. This is because a large fraction of electron-population in a large space outside the N nucleus (though not dense) screens out the small effective positive charge of the N nucleus beyond a certain distance from the nucleus. As a result, a small portion of the total electron-population in the N≡N bond and the summed nuclear-charge of two closely adjacent N nuclei make the cylindrical-bond-surface electrostatically positive. In contrast, a small electron-population of the F-F single bond can be easily accommodated in a reasonably large space between the two F nuclei. In F2, a large fraction of the total electron-population stays around the bond-ends, whereas only a small portion of the total electron-population stays outside the two F nuclei. Therefore, the regions outside the F-F bond-ends are positively charged due to the large nuclear-charge and small electron-population, whereas the cylindrical-bond-surface between the two F nuclei is negatively charged due to the large fraction of electron-population. In the case of the ground triplet state tO2, its molecular size and electronic properties are between those of N2 and F2, as visualized from their σ-bonding MOs (Figure 1b). The bond-ends are nearly neutrally charged or very weakly positively charged. Even though the effective charge of O (Zeff(O2p) = 4.45) is almost in between those of N and F, the tO2 double-bond-length (1.21 Å) is still short, closer to the bond-length of N2 than F2 and so the cylindrical-bond-surface of tO2 is still positively charged and its bond-ends are nearly neutral but weakly electrostatically negative. The most electrostatically positive site is the flat potential region −60° < θ < 60° around the O nucleus.

To explain in a more quantitative manner, we computed the NBOs of N2, tO2, F2 and their third-period analogues. From the s-p hybridization characters of σ-lone pairs lying on the bond axis, we find an important difference in p-character among the lone pairs on N2 (37%), O2 (18%) and F2 (5%) (Supplementary Table S1), which is due to the bond-length, the energy-gap between s and p orbitals and nuclear-charge. For this reason, the 2s-electrons of N atoms spill out of the bonding region upon the formation of N2, making bond-ends negatively charged. On the other hand, the 2s-electrons (95%) of F atoms stay localized upon the formation of F2 (not compensating for the σ-holes at bond-ends), making bond-ends positively charged. The charge anisotropy of tO2 lies in between those of N2 and F2, featuring a near-flat ESP on the density isosurface. As for a large nuclear-charge, the s-orbital is favored in order to screen the nuclear-charge but not sufficient enough, giving positive ESP around the bond-ends. For a small nuclear-charge, the somewhat p-like electron-population can be widely dispersed, resulting in negative ESP due to the still significant ρe in the bond-ends. This can be seen from the ESP map of CO (isoelectronic to N2), where the electron-population around the C atom is dispersed, while that around the O atom is highly contracted, as noted from the HOMO of CO in Figure 1b.

Ground states of P2, tS2 and Cl2

The anisotropy of ESP of the third-period equivalents is less prominent than their second-period equivalents. Owing to the increased bond-length, the ESP is negative near the mid-region of the cylindrical-bond-surface because the effect of two nuclear-charges is sub-additive. The lengthening of bonds also leads to the decrease in 3s-3p orbital mixing, resulting in the localization of σ-lone pairs in 3s-orbitals. For example, the bond length of Cl2 (1.99 Å) is longer than that of F2. Its bond-ends are more positively charged than that of F2, but the overall pattern of ESP is alike. Analogously, one could expect that the ESP of the triplet ground state tS2 is also similar to that of tO2. However, since the S = S bond-length (1.90 Å) is much longer than that of tO2 (1.21 Å), the ESP contribution from the two nuclei is rather weak at the mid region of the cylindrical-bond-surface of tS2. Therefore, the mid-region of tS2 is electrostatically negative unlike that of O2, which in turn results in electron deficiency in the tS2 bond-ends. Overall, tS2 behaves rather similarly to F2/Cl2. P2 forms a triple-bond with one σ-bonding and two π-bonding MOs. However, the large internuclear separation (1.92 Å) makes the mid-bond highly electrostatically negative, thereby resulting in electrostatically positive bond-ends. Thus, P2 behaves oppositely to N2, but rather similarly to F2/Cl2 despite the fact that N and P belong to the same group V.

Excited States

The anisotropic ESPs of the excited states of homonuclear diatomic molecules are also shown in Figure 1a. ESP patterns of the excited states are in many cases opposite to the ground state. Such trends appear for the excited state of all other homonuclear diatomic molecules. Since the excited states show ESP patterns different from or opposite to the corresponding ground state, one can imagine laser-controlled on-off motion which can lead to the design of molecular flippers or nanomechanical devices including molecular switches32.

We consider the cases where N2, tO2 and F2 are excited to triplet, singlet and triplet states, respectively. The excited triplet tN2 has one σg bonding, one πux bonding, one half-occupied πuy bonding and one half-occupied πuy* antibonding. Since one of the π bonds is lost upon excitation, the bond distance is lengthened to 1.22 Å. As such, the electron-population between two N nuclei in tN2 no longer spills over outside the N nuclei, in contrast to the overcrowded electron-population between the two closely bound N nuclei in the ground singlet N2. The large electron-population between two N nuclei in the πuy MO cancels the depleted electron-population in the πuy* MO. Additionally, the σg bonding makes the bond-ends electrostatically positive along the z-axis by the overlap between the two pz orbitals. On the other hand, the πux MO induces negative ESP due to highly increased ρe on the top and bottom of the cylindrical-bond-surface (top view of tN2 in Figure 1a), while it introduces positive ESP due to the depleted ρe on the front and back of the cylindrical surface (bottom front view of tN2 in Figure 1a). The effective MOs for tN2 are a half σg bond, one πux bond and a half πuy bond. In the case of the singlet O2, one σg bond, two πu bonds and one πg* bond (the effective MOs: one σg bond, one πux bond) behave similarly as in the N2 case. Also, in the case of the singlet F2, one σg bond, two πu bonds, one and a half πg* bond and a half σu* bond behave similarly as in the N2 case. This is explained by cancellation between bonding and antibonding such that the resulting effective MOs are a half σg bond and a half πu bond, which is similar to the singlet O2 and the triplet tN2. The excited states tP2, S2 and tCl2 show similar trends, as discussed for tN2 and O2. In the tP2 case, the contrast between the maximum and minimum ESP is slightly weaker than in S2 and tCl2.

Even when an electron is fully excited to a cationic state, the anisotropic charge distribution can still be seen though the polarization effect is diminished by the charge effect. The discussion along with the ESPs of ionized homonuclear diatomic molecules and the issues of MO energy level diagrams31,33 for the charged states is in Supplemenatry note 1.

van der Waals atomic radii in homodiatomic molecules

In molecular interactions the electrostatic interactions (Eel) often govern molecular structures. The van der Waals interactions composed of the dispersion energies (Edisp) and exchange repulsion energies (Eexch) determine molecular size. The van der Waals radii of atoms are generally treated isotropically because their features are considered to be hardly susceptible to the environmental effects. Nevertheless, we note a significant anisotropy in van der Waals radius (rv) of each atom in N2, tO2 and F2, namely, a significant difference among the θ = 0° direction (atoms end), the θ = 90° direction and the central direction from a nucleus to the plane bisecting two nuclei (Figure 2). By excluding the hard wall radius (rw) of Ar (1.685 Å), the rw values (in Å) for the cases of θ = 0°/θ = 90°/central are 1.60/1.58/1.66 for N2, 1.35/1.42/1.50 for tO2 and 1.10/1.33/1.40 for F2 and their van der Waals radii (rv) are considered to be 1.122( = 21/6) times that of the rw. The rw decreases moving right in the periodic table from N2 to F2. The rw changes depending on orientation angles, showing anisotropic behavior. The rw(central) is the largest, while the rw(θ = 0°) tends to have a smaller value. It is because overall the cylindrical-bond-surface has much denser electron-population due to two adjacent positively charged nuclei, while the bond-ends have less dense and diffuse electron-population. In the case of N2 the rw(θ = 0°) is similar to but slightly larger than rw(θ = 90°) because of significantly large electron-population along the bond-ends. F2 shows much more anisotropic behavior in the van der Waals radius in that the rw(θ = 0°) is much smaller than the rw(central), which reflects the σ-hole effect. The angular dependence of the size of the atom was experimentally noted and theoretically described within halogen bonding in terms of polar flattening34,35,36.

Figure 2
figure 2

Anisotropic hard wall distance (Å) of an atom N/O/F in the molecule N2/O2/F2 for three different orientations (0°, 90°, central): N (3.29, 3.26, 3.34), O (3.03, 3.11, 3.18), F (2.78, 3.02, 3.08) at the CCSD(T)/CBS level (see text for van der Waals radius).

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Interactions of homonuclear diatomic molecules with ionic species Na+ and Cl−

Physical manifestation of the ESP maps can be understood by considering the interaction energy between homonuclear diatomic molecules with a positively or negatively charged ion (Na+ or Cl−) (Figure 3). The bond-ends of N2 favor cationic species, whereas those of F2/Cl2/tS2 favor anionic species. tO2 favors cationic species at bond-ends, but behaves somewhat isotropically towards anionic species. P2 behaves nearly isotropically towards cationic species, but strongly favors the θ = ±60° direction towards anionic species.

Figure 3
figure 3

Interactions of homodiatomic molecules with Na+ (upper panel) and Cl− (lower panel) calculated at MP2/aVTZ (contours at −80 to 10 kJ/mol), distance d in Å at CCSD(T)/aVTZ and interaction energy E (Ee) in kJ/mol at CCSD(T)/CBS.

dn is the shortest distance from a nucleus to an ion in the linear structure; dm is the distance from the mid-point of a molecule to an ion on a perpendicular position; dm' in Cl2 is the distance from a nucleus to an ion along the θ = 60° direction. Superscript “t” means triplet state. The lowest MP2/aVTZ Ee in the contour map obtained with the fixed nuclei geometry of each free dimer is similar to the CCSD(T)/CBS Ee’s at the CCSD(T)/aVTZ optimized geometry. An exception is F2 for which the lowest MP2/aVTZ Ee in the contour map (for the fixed F-F bond-length 1.40 Å and the F…Cl distance 2.55 Å) is only −23 kJ/mol, but the fully optimized CCSD(T)/CBS Ee is −90 kJ/mol because the F-F distance is highly increased to1.84 Å and the F…Cl distance is drastically decreased to 1.95 Å like F−…F-Cl).

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N2 strongly interacts with Na+ around the bond-ends (interaction energy Ee = −30 kJ/mol at the distance from the ion to the nearest nucleus (dn) of 2.53 Å according to the CCSD(T)/CBS energy and CCSD(T)/aVTZ optimized geometry. In contrast, F2 strongly interacts with Na+ around the cylindrical mid-surfaces of the bond (Ee = −14 kJ/mol) at the distance from the ion to midpoint of the two nuclei (dm) of 2.54 Å. tO2 behaves between N2 and F2, but slightly more closely to N2, because Na+ favors the bond-ends of tO2 (Ee = −16 kJ/mol, dn = 2.57 Å). P2 shows almost isotropic potential (Ee = −37 kJ/mol at dm = 3.05 Å; Ee = −34 kJ/mol at dn = 2.98 Å). tS2 strongly favors Na+ around the cylindrical mid-surfaces of the bond (Ee = −34 kJ/mol at dm = 2.91 Å; Ee = −18 kJ/mol at dn = 2.92 Å). Cl2 shows Ee = −34 kJ/mol on the cylindrical mid surface (dm = 2.76 Å), but no binding along the z axis.

On the other hand, for Cl−, N2 strongly interacts with it around the mid-point of the cylindrical-bond-surface (Ee = −9 kJ/mol at dm = 3.55 Å); in contrast, F2 strongly interacts along the bond-ends along the z axis (Ee = −90 kJ/mol, dn = 1.95 Å). tO2 behaves almost in between N2 and F2, but again slightly more closely to N2, because the cylindrical-bond-surface is more favored (Ee = −6 kJ/mol at dm = 3.57 Å; Ee = −5 kJ/mol at dn = 3.40 Å). P2 shows the strongest interaction (Ee = −62 kJ/mol) at the distance from the mid-point of a molecule to an ion around θ = 60° from each nucleus (dm') (3.07 Å). tS2 gives Ee = −40 kJ/mol at the bond-ends (dn = 2.97 Å) and Cl2 shows Ee = −102 kJ/mol at the bond-ends (dn = 2.33 Å). As such, we confirmed that the interaction of homonuclear diatomic species with closed-shell ions is mainly determined by the ESP of the diatomic species.

Interactions of the homodiatomic molecules with H2O

Water moisture is present in air, which is composed predominantly of N2 and O2. In clouds, on wet surfaces and on the surface of water in rivers, lakes and sea, the water molecules and clusters interact with N2 and O2 in the atmosphere. Even though individual interaction is small in magnitude, their abundance in the huge atmospheric space on earth is enormous. For this reason, understanding their accurate interactions is highly important. Figure 4a shows the interactions of H2O with N2, tO2, O2 and F2. N2 interacts strongly with the H of H2O on the bond-ends (interaction energy Ee = −5.17 kJ/mol) and weakly with the O of H2O on the cylindrical-bond-surface (Ee = −2.96 kJ/mol). However, in the case of F2, the F-F bond-ends interact strongly with the O of H2O (Ee = −5.71 kJ/mol), while the cylindrical-bond-surface interacts with the O of H2O (Ee = −4.70 kJ/mol). In the cases of tO2 and O2, the O at an edge of the cylindrical-bond-surface (making an angle of ~60° with respect to the z axis) interacts with H of H2O (Ee = −2.33 kJ/mol and Ee = −6.37 kJ/mol, respectively). At the interface between water and the atmosphere, the H atoms in H2O tend to interact strongly with the bond-ends of N2, while the interaction with tO2 is somewhat weaker. Given that the O atoms in water are better stabilized by coordinating H atoms of other water molecules than the H atoms stabilized by O atoms of other water molecules, as noted in water clusters37 and water surfaces38,39, such interaction would help H atoms in the water surface interact with the bond-ends of N2 molecules.

Figure 4
figure 4

Interactions of homonuclear dimers X2 (or Y2) with water, X2/Y2 and benzene.

(a). Interaction of a single water molecule with N2/tO2/O2/F2. The binding energies Ee are given for both the most stable structure interacting with H of a water molecule and that with O of a water molecule. For the less stable structure between the H and O interaction cases, the water molecule is given with a half-tone color. (b). Interactions of N2 with N2 and F2. (c). Interactions for benzene-X2 (X = N, tO, O, F, Cl). All binding energies Ee are given in kJ/mol at the CCSD(T)/CBS level. Each distance marked in a dotted line is given in Å (in parentheses) for the CCSD(T)/aVTZ optimized geometry (In the case of benzene-X2, only the interacting distance was optimized at the CCSD(T)/aVTZ level using the BSSE-corrected MP2/aVTZ geometry).

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Interactions in homo-dimers and hetero-dimers of the homonuclear diatomic molecules

The structures of homo-dimers and hetero-dimers for the homonuclear diatomic molecules are shown in Figure 4b. Binding energies of these structures are governed by the electrostatic interaction and van der Waals interaction. Using symmetry adapted perturbation theory (SAPT)40,41, we performed energy decomposition with the asymptotically corrected (AC) PBE0 functional and the aVTZ basis set on the MP2/aVTZ optimized geometry. We analyzed the SAPT interaction energy components [electrostatic energy (Ees), effective induction energy (Eind* = Eind + Eexch-ind), effective dispersion energy (Edisp* = Edisp + Eexch-disp), effective exchange repulsion (Eexch*: sum of the first order perturbation terms)42, remaining higher order correction term (EHF) and total SAPT interaction energy Etot] (Supplementary Table S2). For most of the structures studied here, |Edisp*| is much larger than |Ees|. In this case, Edisp* tends to be partly cancelled by Eexch* at the equilibrium structure except some special cases where |Ees| is almost equivalent to or larger than |Edisp*|. Since Eexch*, Edisp* and Ees are proportional to r−12, r−6 and r−1, respectively, (where r is the interatomic distance), Eexch* and Edisp* which have sharp slopes with respect to r tend to be cancelled out to give an energy minimum point, while Ees showing a much weaker slope tends to change the minimum point slightly. This is the reason why Etot is close to Ees in most cases shown in Supplementary Table S2 and also in many other cases13,14,42,43.

The most stable structures for the N2 dimer (N2 − N2) are the displaced-Parallel (Pd) shape (Ee = −1.24 kJ/mol) and the L-shape (Ee = −1.22 kJ/mol). In the Pd structure the negatively charged bond-end of one molecule is on top of the positively charged bond-surface of the other molecule and vice versa. In the L-shape the negatively charged bond-end of one molecule is directly pointing to the positively charged bond-surface of the other molecule. In both Pd and L shapes, the electrostatic energy (Ees = −0.74 and −0.76 kJ/mol, respectively) is important. In the case of the hetero-molecular interaction between N2 (which has electrostatically negative bond-ends) and F2 (which has electrostatically positive bond-ends), a linear structure is the most stable in the potential energy surface (Ee = −2.24 kJ/mol) where the key energy contribution is the electrostatic energy (Ees = −2.58 kJ/mol). In the homo-dimer systems, while Ees is important, the dispersion term Edisp* related to the van der Waals radius and van der Waals interaction can also be important in determining their structures (see Supplementary Note 2 for the details).

Interactions of the homonuclear diatomic molecules with benzene

The interactions of homonuclear diatomic molecules with benzene (Bz) are shown in Figure 4c. Hobza and coworkers carried out a similar study44. Bz has negatively charged electron clouds both above and below the Bz-plane, while being surrounded by positively charged H atoms around the edge. The parallel structure is the most stable (Ee = −6.3/−5.5 kJ/mol) for N2/tO2, since the electrostatically negative surface of the Bz-plane favorably interacts with the electrostatically positive cylindrical-bond-surface of N2/tO2. For the interaction between Bz and F2/Cl2 the T-shape-on-bond structure is the most stable (Ee = −6.0/−12.7 kJ/mol), as the most electrostatically negative CC aromatic bond of the Bz-plane favorably interacts with an electrostatically positive bond-end of F2. Based on this information, we can understand the interactions of these diatomic molecules with Bz and further with graphene. In general, for the Bz-A2 complexes, even though the dispersion energy is dominant in magnitude, the anisotropic charge distribution in A2 plays an important role in determining the most stable structures (Supplementary Note 3).

We performed SAPT/DFT calculations to decompose the interaction energy into physically meaningful components (Supplementary Table S2). For Bz-N2, the most stable parallel structure (P) shows strong electrostatic energy (Ees = −4.55 kJ/mol), while the effective dispersion and exchange energies nearly cancel each other (Edisp* + Eexch* = −0.25 kJ/mol); thus, this structure is electrostatically driven. For Bz-O2 (singlet), the most stable parallel (P) structure shows strong electrostatic energy (Ees: −13.01 kJ/mol), while Edisp* + Eexch* + EHF (2.55 kJ/mol) is slightly positive; thus, it forms an electrostatically driven structure. For Bz-F2, the most stable T-shape-on-bond structure (Tb) exhibits a large electrostatic energy (Ees = −5.87 kJ/mol), while the Edisp* + Eexch* (4.38 kJ/mol) is positive; as such it forms an electrostatically driven structure. A similar behavior is noted in Bz-Cl2, but with stronger interaction energy. The most stable Tb structure shows strong electrostatic energy (Ees: −13.10 kJ/mol), while the Edisp* + Eexch* (8.17 kJ/mol) is positive, forming an electrostatically driven structure. In general, for the Bz-A2 complexes, even though the magnitude of the effective dispersion is very large, the anisotropic charge distribution in A2 plays a very important role in determining the structures.

Heteronuclear diatomic molecules

The ESP maps for heteronuclear diatomic molecules of nonmetallic elements are computed at MP2/aVTZ level. The charge analysis according to ESP (somewhat different from the NBO charge analysis, for example, as in CO for which the NBO charges give the wrong dipole direction) of heteronuclear diatomic molecules shows: Cδ+ Nδ−, Cδ− Oδ+, Cδ− Fδ+, Nδ+ Oδ−, Nδ− Fδ+, Oδ+ Fδ−, Nδ− Pδ+, Oδ− Sδ+, Fδ− Clδ+, Cδ− Pδ+, Cδ− Sδ+, Cδ− Clδ+, Nδ− Sδ+, Nδ− Clδ+ and Oδ− Clδ+ (Figure 5). This result is counterintuitive, since the more electronegative element is positively charged except for a few cases. Therefore, we speculate that the electrons tend to be populated to reduce the electrostatic imbalance, i.e., to neutralize the electrostatical positivess around the nuclei, but not sufficiently. Therefore, the regions around the nuclei with the larger nuclear-charge (i.e., higher electronegativity) tend to be electrostatically positive except for the group I–III elements which tend to be strongly electrostatically positive. Indeed, such a trend holds for almost all the cases of the above hetero-diatomic molecules. As for the three exceptional cases Cδ+ Nδ−, Nδ+ Oδ− and Oδ+ Fδ− (three left-top ESP maps in Figure 5), the two nuclear-charges are similar (the nuclear-charge difference is only one). Therefore, their positiveness/negativeness could depend delicately on MOs. CN is isoelectronic to N2+, the ionized state of N2. An electron can be easily detached from the less positively charged nucleus C in CN− isoelectronic to N2, which provides the electrostatically positive site for the C atom. The electronic behaviors of NO (isoelectronic to O2+) and OF (isoelectronic to F2+) can be explained similarly. However, we observe a tendency for diatomic molecules of nonmetals to have opposite atomic charges against those predicted by Pauling's electronegativity when the two elements are at least two groups apart.

Figure 5
figure 5

ESP of heteronuclear diatomic molecules (MP2/aVTZ).

q: NBO charge (au), μA: dipole moment (debye), where the subscript A designates the atom to which the dipole direction is pointing from the molecular center. The dipole vectors are along the right direction except the left direction of the three cases of CN, NO and OF (for which the nuclear charge difference between two atoms is only 1).

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Discussion

We analyzed the anisotropic charge distribution and anisotropic van der Waals radii of atoms in diverse diatomic molecules to understand intriguing novel molecular interactions. We scrutinized molecular interactions of various diatomic molecules of Group (IV–VII) elements (which disfavor anionic species) with themselves, a cation(Na+)/anion(Cl−), H2O and benzene. Though there have been some discussions on such group elements interacting with anionic sites or themselves, the clear understanding was lacking. For accurate description of their subtle interactions, we note that the anisotropy in charge distribution around the atoms which arise from a number of factors including MO, nuclear charge and bond length should be considered. The fundamental understanding of the origin and characteristic features of anisotropic noncovalent interactions could be utilized in novel molecular recognition, assembling, engineering and dynamical control.

Methods

The ESP at each point in space is defined as in Equation (1), where ZI and RI are the nuclear charge and position, respectively.

The quadrupole moment Qzz is defined as in Equation (2), which becomes more negative/positive as the electron-population gets contracted/expanded toward the z-axis (i.e., as the electron-population is more oblated/prolated in the diatomic molecule).

The CCSD(T)/CBS limit (ECBS) of an interaction energy (E) is evaluated based on the extrapolation method45,46 exploiting that the basis set error in the electron correlation energy is proportional to N−3 for the aug-cc-pVNZ (aVNZ) basis set (ECBS = [EN*N3 − EN−1*(N − 1)3]/[N3 − (N − 1)3]). Here, CCSD(T)/aVTZ and CCSD(T)/aVQZ energies at the CCSD(T)/aVTZ optimized geometries were used for the extrapolation to the CBS limit. Ab initio calculations were carried out using Gaussian [Frisch, M. J. et al. Gaussian 09, revision A.02 (Gaussian, Inc., 2009)] and Molpro [Werner, H.-J. et al. Molpro quantum chemistry package, version 2012.1, http://www.molpro.net./ (2012) (date of access: 01/06/2012)] packages.