An approach to the resolution of dispute on collective atomic interactions

arising from [S.Sowlati-Hashjin] et al. Nature Communications [10.1038/s41467-022-29504-0] (2021)

The previous active discussion in the reply¹ (henceforth, P1 and its authors be aP1) on the article² (P2 and its authors be aP2) concerns the application of two different energy partitioning approaches to examples such as LiCX₃ (X = F, Ph). These systems were initially involved in the original work³, where aP1 introduced a new idea of bonding called collective interactions. The energy partitioning method used in article³ was the theory of interacting quantum atoms (IQA)⁴. This approach deals with the 3D-space partitioning into atomic regions that can be easily imagined. All its constituent energy terms directly correspond to the classical non-relativistic Hamiltonian kinetic and potential contributions⁴. Despite its clarity, it has an inherent flaw that is vaguely mentioned when discussing IQA results: the dramatic dependence of the results on the chosen technique of atomic partition of a molecule.

The second method, an energy decomposition analysis (EDA), which was involved in the discussion², operates with orbitals in functional Hilbert space⁵. Its results are not so easy to interpret and understand directly. This approach also relies on some questionable assumption, namely the use of unoptimized (and often highly multiconfigurational) abstract radical moieties to study interactions. Moreover, an actively used quantity, such as Pauli repulsion, can be defined in several different ways and does not have a unique definition⁶. But the main drawback in the context of the current discussion, as aP1 rightly emphasized, is that the EDA approach, in principle, cannot assess the strength of a given interatomic interaction for a particular A ⋯  B pair. For such purposes, aP2 turns to the curious pictures of overlapping Singly-Occupied molecular orbitals (SOMO) to gain a possible insight into the contribution of specific interactions.

As can be seen from the active discussion^1,2, the two approaches lead to contradictory results, completely different in their interpretation (nature). aP1 claim that the Li ⋯  C interaction is destabilizing and the LiCF₃ molecule exists due to Li ⋯  3F bonding electrostatic interactions^1,3. Namely, the Li ⋯  C contact has a magnitude of 94 kcal/mol, which is determined mainly by the very repulsive Coulomb term together with the much smaller stabilizing V_xc part (−21 kcal/mol)³. The expected Li ⋯  CF₃ bonding is achieved through three strong electrostatically stabilizing Li ⋯  F contacts (3 × −80 = −240 kcal/mol)³. However, aP2 came to completely opposite results and argue that the Li ⋯  CF₃ interaction is essentially covalent in nature (orbital interaction energy term, ΔE_oi = −95 kcal/mol) due to the Li − C bond and the electrostatic contribution from the three Li ⋯  F contacts (ΔE_el = −30 kcal/mol)².

The main goal of the present article is to shed light in the right direction in resolving controversies using the energy partitioning scheme specially developed by the author, combining two approaches into one powerful tool. This allows all terms of the EDA analysis to be assigned to specific atomic and interatomic fragments of the molecule, similar to IQA. Thus, it helps to gain a deeper insight into conventional EDA data without resorting to uninformative SOMO isosurface images actively used in ref. ².

IQA results for p-LiCF₃ by different atomic partitions

Before going on to the main discussion, I will start with an introductory example of the two opposite results that we can get for p-LiCF₃ system using the IQA approach with different atomic partitions. One of them relies on the regular zfs-atoms⁷ (according to Bader), which are always used by adherents of the IQA methodology⁴. The second method uses polyhedra⁸.

To perform a polyhedral partitioning, we must specify the appropriate atomic radii to reasonably adjust the relative sizes of the atoms involved. The single-bond covalent radii of carbon (75 pm) and fluorine (64 pm)⁹ are used for the constituents of CF₃ moiety. However, for lithium the choice is not so obvious. Indeed, the average covalent radius proposed in ref. ⁹ for lithium (128 pm; the same value was also found in ref. ¹⁰) is so large that results in a significant penetration of Li into the space of fluorines in i-LiCF₃. Thus, Li acquires an excess of electrons, thereby creating an unphysical negative charge and a noticeable positive charge on the fluorines. This results in an unrealistically high energy of Li − CF₃ interaction  > 600 kcal/mol (see Table SII in SI), arising from the doubtful extremely covalent nature of this bonding.

On the other hand, the use of different ionic radii leads to very comparable data (see Table SII in Supplementary Information (SI)) and does not fundamentally change the conclusions obtained in the work. However, since the minimum values of these radii in ref. ¹¹ were obtained for lithium halides and chalcogenides, where it exists precisely in the Li⁺ variant, then for our systems, in which lithium cannot be considered as a pure monocation, it was more reasonable to select a balanced average value of r_Li (see Section I in SI for the details). It is the value of 85 pm, close to the average value over ionic radius range of 59–106 pm presented in ref. ¹¹, that gives the charge of polyhedral Li atom, which agrees well with aP2 Voronoi deformation density charges data². Therefore, evaluated energy data are expected to be consistent with aP2 EDA results as well.

We first discuss the atomic charges in p-LiCF₃ calculated using the two models. For Bader atoms, carbon has an unusually high positive charge (1.2e). To a synthetic chemist, this value may seem incredible or even impossible, since such carbon center in an organometallic compound cannot be so positively charged. Because then it will behave very unpredictably in chemical reactions and will be very reactive. Such tendency of Bader’s approach to overestimate the magnitude of the atomic charge in some cases is well known^12,13. Of course, the Li atom also has a high positive charge of 0.92e. However, it is not even bonded with the halogen atom, but formally with carbon. Such charges are the main source of the large Li − C electrostatic repulsion (94 kcal/mol) and much smaller covalent binding (exchange-correlation part if the interaction energy, V_xc = −20.4 kcal/mol) according to our calculated IQA results (Table SIII in SI). In contrast, each fluorine atom exhibits a strong ionic attraction with lithium (−79 kcal/mol) along with negligible covalent contribution (V_xc = −1 kcal/mol). All these obtained results are completely consistent and very close to the corresponding IQA data presented in ref. ³.

If we now move on to the ‘polyhedral’ atoms, the situation changes completely. We found the following charges: q(C) = 0.32e, q(Li) = 0.51e (very close to the result 0.49e in ref. ²), and q(F) = −0.27 (cf. the MKS charge q(F) = −0.21e in CHF₃ in ref. ¹⁴). These are quite reasonable values (charges obtained by other approaches are discussed in the Section I of SI), corresponding to the not very negative fluorines and not very positive carbon in organolithium compound (even taking into account that it has three electron-withdrawing F atoms). Before interaction with Li, the frozen ⋅CF₃ radical had q(C) = 0.49e and q(F) = −0.16e, the carbon charge being consistent with the MKS charge q(C) = 0.56e in CHF₃¹⁴. As we see, 0.5e is transferred in the direction Li → CF₃.

The IQA data on the polyhedral partition of p-LiCF₃ are summarized in Table 1 (all corresponding results obtained on zfs-atoms are also evaluated and presented in Table SIII of SI). From here we observe the Li − C interaction as a covalent bond with a moderate strength E_i = −67.2 kcal/mol, arising from a strong V_xc term (−81 kcal/mol) with a much smaller electrostatic destabilization of 13.6 kcal/mol. Such behavior is very common for covalent bonds. Each F atom attracts Li with E_i = −15.2 kcal/mol (it has a small V_xc part of  −3.8 kcal/mol). The latter result is quite reasonable, since the already covalently bonded lithium is a closed shell atom that do not need to interact strongly with F. It should be added that the C − F pair is a strong covalent bond E_i = −183 kcal/mol (it consists of classic E_cl = 20 and solely quantum V_xc = −203 kcal/mol), formed by a very high degree of covalency, which is in good agreement with high homolytic dissociation energy¹⁴. These results contradict the calculated zfs-IQA data for C − F bond: E_i = −437 kcal/mol, that includes E_cl = −310 and V_xc = −127 kcal/mol, which show that C − F bond is by on 70% ionic.

Table 1.

EDA (orbital interaction E_oi, electrostatic V_el and Pauli E_pauli) and IQA (exchange-correlation V_xc and classic E_cl) terms (in kcal/mol) for p-LiCF₃ with polyhedral partition approach

	EDA			IQA		suma
	Eoi	Vel	Epauli	Vxc	Ecl	Ei/def
Li − C	−51.7	18.9	−34.4	−80.9	13.6	−67.2
Li ⋯  F	−9.4	−1.4	−4.5	−3.8	−11.5	−15.2
Li − CF3	−80.0	14.7	−47.8	−92.2	−20.8	−113.0
Li-	43.7	−17.2	4.8	38.7	−7.5	31.2
-CF3	−56.2	−26.8	88.8	−95.8	101.2	5.4
Mol.	−92.5	− 29.4	45.8	−149.3	72.9	−76.3
Ref. 2	−95.5	−29.7	46.6			−78.6

^aThe sum correponds to the interaction E_i (deformation E_def) energies for pairs (fragments).

Meanwhile, the quality of the atomic charge is a very debatable thing, since this quantity does not have a strict definition (see the discussion of charges in the SI). However, it is not difficult to estimate their magnitude based on the chemical behavior of the substance or comparison with alternative partitioning schemes. As it was demonstrated, the described IQA results for the LiCF₃ system lead to completely contradictory conclusions if for the same energy partitioning scheme we choose a different atom in the molecule formalism. Thus, when using IQA, it is important to critically compare results based on classical Bader atoms with other partitioning schemes.

Polyhedral atomic partitions of the EDA components

Lets us now discuss the results obtained using the EDA approach with the complete atomic and interatomic contributions of all its constituent terms, in the following simple manner, relevant for discussion:

$$\mathrm{\Delta }F=F\left(\mathrm{Li}\text{-}\right)+F\left({\text{-CF}}_3\right)+F_i\left({\text{Li-CF}}_3\right),$$

where F = E_oi, V_el, E_pauli are standard components of the usual EDA⁵.

We discuss here only the results for the polyhedral partition. All similar EDA data, but based on the Bader’s partition, are also calculated, but are not discussed here for the reasons stated above (due to their dubiousness) and are presented in SI (Tables SIII and SIV) for the curious reader. Only the homolytic mechanism of formation of LiCF₃ is considered, because all our results based on the polyhedral approach indicate a predominantly covalent nature of the Li–CF₃ interaction. In the case of p-LiCF₃, the energy of ionic dissociation is about 2.5 times higher (150 vs 64 kcal/mol²) than homolyic one and does not correspond to the probable dissociation channel in vacuum.

p-LiCF₃ case

According to aP2², the most important stabilizing contribution to the overall Li ⋯  CF₃ interaction is ΔE_oi component. Moreover, based on this result, it is argued² that Li − C bond behaves like a typical electron-pair bond between Li⋅ and  ⋅CF₃ fragments and is therefore a polar covalent bond. The electrostatic contribution to the Li − C bonding is about a third of the covalency (−30 vs  −95 kcal/mol). The atomic partitioning of these energies opens up the possibility to clarify many hidden details and to look at some conclusions in a completely different way.

If we now turn to our data (Table 1), we find that  −52 kcal/mol of the total −93 kcal/mol corresponds to direct Li − C binding and only  −10 kcal/mol additionally to each Li ⋯  F pair, giving thus in a total 87% (−80 kcal/mol) of the molecular orbital interaction contribution ΔE_oi = −93 kcal/mol. Another 13% comes from the intragroup terms of Li (destabilization, +44 kcal/mol) and CF₃ (stabilization,  −56 kcal/mol). The electrostatic part, which has a predominantly stabilizing character (−29 kcal/mol), does not make a stabilizing contribution to the Li − CF₃ interaction. It noticeably (by 19 kcal/mol) destabilizes the Li − C bond and, to a lesser extent, the overall Li − CF₃ interaction (15 kcal/mol). This fact contradicts the aP2 assumption² about the presence of a stabilizing ionic component in the Li − C bond, which was made on the basis of the total value ΔV_el = −30 kcal/mol. The direct electrostatic stabilization occurs in the intragroup Li (−17 kcal/mol) and CF₃ (−27 kcal/mol) moieties. Despite the overall Pauli repulsion of 46 kcal/mol, this quantity indicates a strong stabilization of the Li − C bond (−34 kcal/mol) and an overall intergroup bonding up to  −48 kcal/mol for the Li − CF₃ interaction. Thus, it contributes additionally to the covalency of Li − C along with the E_oi energy. The repulsive parts of the Pauli term E_pauli are indeed observed, but within the Li and CF₃ fragments (Table 1) mainly due to the large kinetic contribution (ΔT = 21 (Li) and 132 (CF₃) kcal/mol).

Thus, as a final result, we can state that Li^+1/2 (a half cation) strongly interacts with ${\text{CF}}_3^{-1/2}$ fragment with energy of  −113 = −128(E_oi + E_pauli) + 14.7(V_el) kcal/mol. Moreover, about 60% (−67 kcal/mol) of this energy corresponds to the Li − C covalent bond and the remaining 40% belongs to the three Li ⋯  F contact (−15 × 3 kcal/mol). This result is significantly different in nature from the IQA data discussed above and also in refs. ^1,3, which describe all these interactions as predominantly ionic with the main bonding contribution from Li ⋯  F contacts. In the context of our results, the strong Coulomb attraction (E_i = −80 kcal/mol) between Li and F and repulsive E_i = +73 kcal/mol for Li − C evaluated in the present work and by aP1 using the zfs-atoms³, should be interpreted with caution. It thus can be concluded that our data are in good agreement with reported P2 results², with the exception of some findings about specific pairwise interactions and the stabilizing contribution to the interaction energy from the Pauli “repulsion” term.

i-LiCF₃ case

The results obtained for i-LiCF₃ are even more intriguing. Several interesting contradictions were found, which aP1 carefully pointed out¹ and which aP2 did not explain in ref. ². First, there is a smaller SOMO’s overlap for i-LiCF₃ (0.22) compared to p-LiCF₃ (0.32) with a significantly larger ΔE_oi (−155 vs  −96 kcal/mol, respectively). The second issue concerns the implicit assignment by aP2 of all EDA terms, primarily to the Li − C interaction². The aP1 made fairly right comment¹ that such an assignment requires an atomic partitioning approach, whereas conventional EDA is not the case. Therefore, aP2 could not confidently draw such a conclusion based on their data.

Lets try to resolve such contradictions between two distinctly different approaches. Our partitioning analysis data (Table 2) shows that only about (−46 kcal/mol) of the total ΔE_oi = −152 kcal/mol corresponds to the direct Li − CF₃ bonding. And almost all of this is observed for Li ⋯  3F contacts (−18 × 3 = −54 kcal/mol). It should be noted that, in contrast to the aP2 observations², we find that the Li − C stabilizing interaction is not observed at all (E_oi = +8 kcal/mol). This is the main reason explaining the reduction in SOMO overlap for i-LiCF₃ mentioned above. Another part of ΔE_oi is accounted for by the stabilizing CF₃ (−130 kcal/mol) and destabilizing Li (+24 kcal/mol) fragments. The electrostatic term are almost negligible for Li − CF₃: out of the total ΔV_el = −88, only a small (+5.4 kcal/mol) destabilizing contribution. Moreover, a strongly stabilizing V_el is observed within the Li (−39 kcal/mol) and CF₃ (−54 kcal/mol) parts.

Table 2.

EDA (orbital interaction E_oi, electrostatic V_el and Pauli E_pauli) and IQA (exchange-correlation V_xc and classic E_cl) terms (in kcal/mol) for i-LiCF₃ with polyhedral partition approach

	EDA			IQA		suma
	Eoi	Vel	Epauli	Vxc	Ecl	Ei/def
Li − C	7.9	5.1	−10.2	−10.5	13.4	2.8
Li ⋯  F	−17.9	0.1	−44.1	−53.5	−8.4	−61.9
Li − CF3	−45.7	5.4	−142.4	−170.9	−11.8	−182.7
Li-	24.1	−39.3	−38.0	31.5	−84.7	−53.2
-CF3	−130.2	−54.0	304.9	−55.6	176.3	120.7
Mol.	−151.9	−87.9	124.5	−195.1	79.8	−115.3
Ref. 2	−155.4	−91.6	128.7			−118.3

^aThe sum correponds to the interaction E_i (deformation E_def) energies for pairs (fragments).

Finally, the Pauli term stabilizes very strongly the Li − CF₃ interaction (−142 = 3 × −44 (Li ⋯  3F) −10 (Li − C)), namely three times more than for the pyramidal case. This interesting observation was not even supposed by aP2 in ref. ². On the contrary, they usually assumed that the Pauli term reduces the strength of the overall Li − CF₃ interaction. Thus, we can combine the Pauli contribution and the orbital interaction term to assign this sum the stabilizing part of the covalent bonding. Unlike p-LiCF₃, the huge destabilizing Pauli repulsion is concentrated only within CF₃ moiety (+305 kcal/mol).

Lets summarize results as follows. We can state that Li (q = 0.4e) in i-LiCF₃ interacts strongly with the CF₃ fragments (q(C) = 0.18e and q(F) = − 0.19e) with a very high total energy  −183 = −46(E_oi) + 5.4(V_el)−142(stabilizing Pauli term). Moreover, almost all of this energy corresponds to “collective” Li ≺ 3F interactions. The Li ⋯  C contact has a weak destabilizing character of 3 kcal/mol. This lies in complete contradiction with the assumptions of aP2² and is completely agreed with the conclusion of aP1 (even for zfs-partition)^1,3.

Interaction collectivity index

It is interesting to discuss the interaction collectivity indices (ICI) proposed in P1³. Although in the mentioned work the ICI were based on the V_xc and Coulomb energies, here we additionally propose to use the orbital interaction energies (ICI_oi) and the sum of the E_oi and Pauli terms (ICI_op) for specific atomic pairs to estimate the ICI(Li) in two LiCF₃ examples.

For the pyramidal molecule we observe two different results (Table 3). Using the indices ICI_oi and ICI_op calculated for polyhedral atoms, we find no significant collectivity in the Li–CF₃ interaction, but a predominantly contribution from Li − C bond. The same result is observed when we use ICI_xc for both molecular partitions. This agrees well with aP1 conclusions³ about the insensitivity of ICI_xc to the chosen atomic domain. However, when using zfs-atoms, the calculated ICI_oi and ICI_op become 2–3 times smaller (Table 3), which indicates a significant degree of collectivity. This observation once again confirms the dubious adequacy of using zfs-partitioning to study the examples under discussion.

Table 3.

Various interaction collectivity index (${\mathbf{ICI}}_F\left(\mathrm{Li}\right)=F\left(\mathrm{Li}\text{-C}\right)/F\left(\mathrm{Li}\text{-}{\mathrm{CF}}_3\right)$)^a data for two partitions

	p-LiCF3			i-LiCF3
	ICIoi	ICIop	ICIxc	ICIoi	ICIop	ICIxc
polyhedra	0.646	0.674	0.877	−0.174	0.012	0.062
zfs-atoms	0.205	0.399	0.893	0.019	0.078	0.030
zfs Ref. 3			0.910			0.022

^aSee the text discussion for the collectivity indexes definition.

In the case of i-LiCF₃, the IQA on zfs-atoms gives ICI_xc = 0.03 (0.02 in ref. ³). According to our data (from E_oi) we found ICI_oi = −0.17 (a small negative value). If we add the Pauli term, then ICI_op will be close to ICI_xc obtained using the V_xc energy. All these findings are consistent with aP1 conclusions^1,3. The i-LiCF₃ molecule does indeed have a multicenter collective character induced by three Li − F interactions. In this example, the type of atomic basins does not qualitatively affect the obtained ICI results.

Conclusions

After careful evaluation of the calculated data, we can conclude that:

• When using an IQA scheme, great care must be taken when checking results based on Bader atoms. Namely, the magnitude of atomic charges should be analyzed by comparison with other alternative methods. Otherwise, the results may be very controversial from a chemical point of view, e.g., a very large electrostatic contribution to truly covalent bonds.

• In the case of the conventional EDA technique, it is highly desirable to provide additional analysis of how the EDA terms are distributed over one- and two-center components. This is exactly the approach developed and used in this article. It proves to be a very useful tool that helps to analyze and interpret EDA data in more depth.

• It is believed that fulfillment of the mentioned conditions will prevent future disagreements such as those discussed concerning the two articles in question.

As a result of such a comprehensive approach, it can be clearly stated that, in contrast to the pyramidal form p-LiCF₃ (existing predominantly due to Li − C bond), the more stable inverted structure i-LiCF₃ is organized as a result of collective bonding between Li and three fluorine atoms through a moderate covalent 2e-multicenter interaction with an energy of  −62 kcal/mol per one Li − F pair.

Methods

For consistent comparison of the discussed results, the geometry of p-LiCF₃ was taken from ref. ³. Such xyz-data for the inverted i-LiCF₃ system was missed there, so it was optimized by the M06-2X functional using def2-TZVPP basis set, as proposed in ref. ³. It should be noted that close geometries were obtained in ref. ² using another basis set TZ2P. All IQA and EDA energy decomposition analyzes were performed using the TWOe code⁸ (ver. Z2), which includes now a developed Model DFT approach for correct atomic and interatomic partitioning of the M06-2X xc-terms. Additional data on charges, zfs-based IQA and EDA results, and geometries used are available in Supporting Information free of charge.

Author contributions

All work was performed by the single author.

Peer review

Peer review information

Nature Communications thanks William Tiznado and the other, anonymous, reviewer(s) for their contribution to the peer review of this work.

Data availability

All data generated or analyzed during this study are included in this published article and its supplementary information file.

Competing interests

The author declares no competing interests.

Supplementary information

The online version contains supplementary material available at 10.1038/s41467-024-54552-z.