Toward a Rigorous Definition of a Strength of Any Interaction Between

May 18, 2017 - Strength of interaction between Bader's atomic basins, enclosed by zero-flux surfaces of electron distribution, was proposed to be a me...
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Toward a Rigorous Definition of a Strength of Any Interaction Between Bader’s Atomic Basins Ivan V. Ananyev,*,†,‡ Valentina A. Karnoukhova,†,‡ Artem O. Dmitrienko,† and Konstantin A. Lyssenko† †

X-ray Structural Laboratory, A. N. Nesmeyanov Institute of Organoelement Compounds RAS, 119991 Moscow, Russian Federation N. D. Zelinsky Institute of Organic Chemistry RAS, 119991 Moscow, Russian Federation

‡

S Supporting Information *

ABSTRACT: Strength of interaction between Bader’s atomic basins, enclosed by zero-flux surfaces of electron distribution, was proposed to be a measure of elastic deformation of an interaction. The set containing 53 atomic aggregate and covering all range of interaction strength (from van der Waals interactions to triple covalent bonds) was calculated by DFT and perturbation theory methods. Further analysis was performed to seek correlations between various local quantities based on electron density and effective force constants of stretching diatomic vibrations. The linear trend between effective force constants and the potential energy density at the (3, −1) critical point of electron distribution was found. This correlation was improved by the integration of the potential energy density over an interbasin zero-flux surface of electron density. Simple mechanical explanation of established trends is presented. The correlations can be further used to at least semiquantitatively compare any pair of interactions between Bader’s atomic basins.

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Here an important role of the (3, −1) saddle critical point of ρ(r) should be emphasized. This point corresponds to the edge of the connectivity graph and lies on the zero-flux surface between only two atomic basins (interbasin zero-flux surface, e.g., Ω).1,14 In the literature, the (3, −1) point is commonly called as the bond critical point (BCP) that reflects its role to be both necessary and sufficient conditions (indication) of the presence of exchange interaction channel (bonding interaction) between two atomic basins.22 It should be remembered that BCP is merely an indicator of interbasin interaction, given that it is only the most preferable channel of electronic exchange between basins but is not the whole interaction itself. However, this saddle point, rather than the Ω surface, is broadly considered as an indicator and even descriptor of chemistry’s Holy Graila chemical bond. Such simplification is easy to understand since the analysis of several isolated points is a very convenient and not demanding way to study features of interatomic bonding. Various local quantities, including ρ(r), its second derivatives, energy densities and other ρ(r)-based functions, are commonly computed at BCPs to provide information on interactions between atoms. Usually, the main task of further analysis is to describe bonding using relations between these quantities and conventional chemical concepts. Many correlations have been explored and successfully used so

INTRODUCTION Nowadays, Bader’s theory of “Atoms in Molecules” (AIM)1 is one of the most powerful methods to study various features of electronic structure within atomic aggregates, from small molecules (e.g., refs 2−4) to (bio)macromolecules (e.g., refs 5−8) and crystals.9,10 The core of the AIM theory is the topographical (usually called as topological) analysis of the realspace scalar field−electron density function ρ(r)11 which plays a key role in any property of an atomic aggregate12 and can be both computed using quantum chemistry methods and reconstructed from experimental data.9,13 The AIM topographical analysis allows one both to divide a structure into the so-called atomic basins containing a nucleus and enclosed by zero-flux surfaces of ρ(r) and to elucidate topological features (apexes, edges, faces, cavities) of a structure composed by these basins by a search of ρ(r) critical points.11,14 Namely, if the ρ(r) function is far from a catastrophic situation, the type of a critical point defined by the signature of the ρ(r) Hessian matrix corresponds to the number and mutual location of atomic basins which share this point. In other words, each critical point of a particular type corresponds to a specific topologic object which is believed to describe a chemically recognizable fragment of a structure: apexes−atomic nuclei, edges−chemical bonds, faces−rings, and cavities−cages. Aside from several special cases (see, e.g., refs 15−21) the whole set of critical points unambiguously defines a graph of atomic connectivity using concept of atomic basins sharing the same zero-fluxsurface. © 2017 American Chemical Society

Received: February 15, 2017 Revised: May 16, 2017 Published: May 18, 2017 4517

DOI: 10.1021/acs.jpca.7b01495 J. Phys. Chem. A 2017, 121, 4517−4522

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(binding energy which is equal to BDE in diatomics) suffers from electronic relaxation effects which depends on the system’s nature and the model of calculation (homolysis, heterolysis). In multiatomic species, the process of breaking of a chemical bond is additionally followed by relaxation of nuclei which also has its own contribution into the BDE value.36 In other words, the BDE values can be considered as a strength of interbasin interaction only in systems where these relaxation effects are negligible that is in line with the types of bonding which can be successfully analyzed by the EML correlation. Clearly, relaxation effects cannot affect any measure of elastic deformation of an interaction. It was shown for many chemical bonds (including weak noncovalent interactions) that their strength can be adequately estimated and compared by analyzing force constants of elastic atomic (nuclei) vibrations.37−41 This method implies the partitioning of measured or calculated normal vibrational modes (which always have complex multiatomic coordinates in any case but diatomics) into diatomic contributions.35,37 This partitioning procedure, providing effective force constants (eFC) of stretching vibrations of each atomic pair within an adiabatic potential, allows to range all chemical bonds by their strength.42 The elastic properties of a chemical bond are also believed to be better estimators of bonding strength.43 The current work presents an improvement of the EML correlation which was done by a reasonable choosing of another, more elastic property describing bonding strength, rather than BDE valueseffective force constants.35 We transposed the eFC concept to the framework of interbasin interactions, simply supposing that the more elastic is the bonding the more pronounced is electron “sharing” between atomic basins. The set of molecules, ions and molecular complexes was calculated to analyze electron density features and eFC as a measure of elastic deformation. The set of systems was designed using model chemical considerations such that it covers the full range of strength of bonding between atoms (from the weakest in Ar2 to the strongest NN in HNNH2+) and contains H-bonded associates, simple polyatomic molecules and diatomic species with formally different bond order. Since there are other local quantities, rather than v(r)BCP, providing good estimations of bonding strength (see ref 30, for example), all local quantities commonly analyzed at BCP were compared to eFC values (see also the Methods and Tables S1−S11 of the Supporting Information containing all calculated data). For each system from the set two types of calculations were performed with the aug-cc-PVQZ basis sets:44,45 using the DFT theory (the B98 functional46,47) and the perturbation theory (MP448). The B98 functional was chosen as providing a very good approximation of the electron density of high-level ab initio calculations.49 The fourth order perturbation theory was chosen merely as an example of the post-HF wave function methods. Concerning this work, both types of calculations gave very similar results with a small discrepancy between them (see Table S12 in the Supporting Information and the Methods). Below, the data from the B98 calculations are discussed.

far; examples being estimations of the covalent/ionic contribution into bonding from the ∇2ρ(r)BCP values and energy densities,23,24 calculations of the bond order using the ρ(r)BCP values25,26 and/or eigenvalues of the ρ(r) Hessian at BCP,27 estimations of the directionality of bonding from the ρ(r) Hessian at BCP,27,28 estimations of the bonding strength (as bond dissociation energy) from electronic potential energy density (virial field),29 and kinetic energy density30 at BCP (v(r)BCP and g(r)BCP, respectively). Nevertheless, the bridge between physically grounded local quantities and model concepts of theoretical chemistry (properties describing chemical bonds) cannot be ab initio. Accordingly, none of mentioned correlations is both quantitative and universal, i.e., can be used for quantitative comparison of any pair of bonding situations. Moreover, the soft condition of the “chemical bond” itself (see, for instance, the classic book by Pauling31) allowed some authors to undermine the podium occupied by BCPs and Ω surfaces as indicators of chemical bonds, conclusively demonstrating the presence of chemical bonding without BCPs (for instance, in the metals’ π-complexes17 and metal clusters16) and the absence of chemical bonding between atomic basins sharing a Ω surface and a BCP (the well-known problem of H···H interactions in the biphenyl molecule19−21). By all this means, neither conditions nor properties of an interbasin interaction defined within the AIM theory can be simply related to the conditions and properties of a chemical bond. The current work presents an attempt to get further insight into the interbasin interaction: while its indicators are given within the AIM theory, intrinsic properties conclusively describing this interaction are not so unambiguous. Since any sought property of the interbasin interaction has to be of a quantitative and universal character, one of the mentioned above correlations sounds the most promising though was initially established as a measure of a chemical bond property. Namely, the correlation between the bond dissociation energy (BDE) and the v(r)BCP value29 can be chosen as a starting point to find a universal correlation between local quantities analyzed within the AIM framework and a strength of interbasin interaction. Initially designed by Espinosa, Molins and Lecomte for hydrogen bonds in model associates,29 the semiempirical correlation between BDE and the virial values at BCP (hereafter, EML) was further extended on other weak bonding interactions10 and more strong bonds such as some covalent coordinate bonds between d-metals and ligands.32,33 For all these bonding’s types the EML correlation provides a rather accurate estimations of measurable BDE values and consequently allows of at least semiquantitative comparison of interactions. At the same time, while the choice of virial values at BCP within the initial EML concept, though being a quite arbitrary one, clearly accounts for the tendency of electrons to be shared between two atomic nuclei, the reasonable explanation of this correlation has not been done yet. Attempts to extend the EML correlation on stronger covalent bonds were unsuccessful;34 that is, in our opinion, due to the limited character of the model chemical concept used in this correlation. Indeed, the soft condition of a chemical bond makes the energy of its breaking to be an unsatisfying property of an interbasin interaction (see above). Moreover, the BDE can hardly be a universal property of interaction between two atomic basins due to the following known reasons.35 A theoretical estimation of plastic deformations of bonding

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METHODS Computational Details. Theoretical calculations of a set of isolated atomic aggregates were performed using the Gaussian09 program.50 The set contains Ar2, BeH+, BeH·, BeO, BN, BO, Cl2, ClF, ClO, CN−, CO, F2, H2, HBr, HCl, He2+, HF, LiCl, LiF, LiH, LiO, N2, NaCl, NaF, NaH, Ne2+, NO+, O2, O2−, O2+, OF, OH−, OH·, SH−, SH·,HCCF, HCCLi, 4518

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The Journal of Physical Chemistry A H2O, HBrO, HClO, HCN, HCNH+, HNC, HNNH2+, H2O··· HOH, HF···HF, F2HN···HF, F3N···HF, FH2N···HF, H3N···HF, HOH···NH3, MeO(H)...HOH, and MeOH···OH2. Calculations of two types were performed: (1) within the density functional theory using the B98 functional, 46,47 (2) within the perturbation theory at the MP4(SDQ) level.48 The basis set was always Dunning’s aug-cc-PVQZ.44,45 The geometry of each aggregate was fully optimized to the equilibrium state and the type of saddle point was confirmed by normal vibrational mode calculations. Tight optimization criteria and ultrafine grids were used in each calculation. Force constants for diatomic species, being equal to eFCs, were derived by the treatment of computed normal-mode frequencies with reduced masses μ according to the μ = (m1·m2)/(m1 + m2) formula where m1 and m2 are atomic masses of corresponding atoms. Available experimental frequencies for diatomic species51 confirm the reliability of normal mode calculations. The eFCs values for multiatomic species and H-bonded associates were estimated by calculations of second derivatives of second order polynoms approximating changes of energy along coordinates corresponding to stretching of atoms in interest (for H-bonded associates only formally noncovalent interactions were considered). For this purpose, five single point calculations were performed in each case with the step of 0.003 Å for the corresponding internuclear separation. Any change of the step value, used for these calculations, affected the eFC values only negligible. In particular, for the systems with strong, formally triple bonds, where the electronic relaxation effects can be pronounced enough, the decrease of the step up to 0.0001 Å led to an increase of eFC values on not more than 0.2 mdyn· cm−1. For H-bonded associates the BDE and binding energy values were also calculated using the zero-point corrected energies of optimized and nonoptimized fragments, respectively. Topography Analysis and Integration. Topography analysis of the ρ(r) function calculated for equilibrium nuclei coordinates was performed using the MultiWFN52 and AIMAll53 programs. Only expected critical points of ρ(r) were found with their total set satisfying the Poincare−Hopf relation.1 At BCP corresponding to an interaction in interest, the ρ(r), ∇2ρ(r) values and ρ(r) Hessian components λi together with energy densities were calculated for the whole calculated set of systems. The topography analysis of the v(r) function was carried out using MultiWFN, the values of v(r) at its (3, +1) saddle point corresponding to an interaction in interest were calculated for the whole calculated set. For the whole calculated set, the integration of v(r) over the Ω surface was done using AIMAll. For the purpose of comparison of four correlations (see further discussion and Figure S1), only diatomic species were treated with the MultiWFN and a selfwritten simple code which provides a measure of the surface integrals as follows. From a BCP or a (3, −1) saddle point of −v(r), one gradient path within an interbasin surface (from ρ(r) or -v(r), respectively) was calculated with 10000 steps of 0.001 au (0.00052918 Å). At each point the v(r) value is calculated to be further used to estimate volumes of embedded cylinders defined in the (x′, y′, v(r)) space, where x′ and y′ are coordinates of a plane in which the area of the largest cylinder base would be equal to the area of interbasin surface defined in the real (x, y, z) space. A difference between cylinder volume was calculated as dif = (ri+1 − ri)2·π·v(r), where ri+1 and ri are vectors defining external and internal points, respectively, of the

calculated gradient path. The sum of dif over i was assigned as an estimation of the v(r) integral over the interbasin surface. Correlations Treatment. Note that the experimental BDE values, which are available for several diatomic species,51 do not correlate reasonably with the eFC values. For the whole calculated set all basic BCP descriptors (ρ(r), v(r), kinetic and full energy densities, ∇2ρ(r), Hessian components) together with the v(r) value at its (3, +1) point were investigated and the v(r)BCP values were found to be the best predictors for eFC for both B98 and MP4 data. Better model performance was explored for both B98 and MP4 only for kH values (see Results and Dicussion). The correlations can be probably improved by increasing accuracy of v(r) calculations: while the usage of available experimental frequencies51 as eFCs in diatomics does not improve any of presented correlations, some deterioration of correlations if using v(r) from MP4 (see Table S12 in Supporting Information) is clearly caused by the absence of exact method to calculate accurately the v(r) distribution from post-HF methods. Presented correlations are obviously not exact and may somehow vary if only another method of calculation is used; however, the general trend is expected to be always the same and the interaction’s data calculated at the same level of theory can be adequately compared.

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RESULTS AND DISCUSSION In the case of one-variable fitting for the whole set of species, only the v(r)BCP value correlates with the eFC values well enough (Figure 1). This correlation (hereafter, eFC/v(r)BCP) is

Figure 1. Linear correlation between eFC (mdyn·Å−1) and v(r)BCP (a.u.e. × a0−3) for the whole set of calculated species: from Ar2 to HNNH2+.

approximately linear within the whole range of interactions (R2 = 0.9559) and satisfies the boundary condition at v(r)BCP → 0 quite well. Notably, there are several systems with strong, formally triple bonding (HCCF, NO+, HNNH2+, HCCLi) which deviate from the eFC/v(r)BCP regression curve quite significantly (30−40%). Large electron relaxation effects can affect potential energy surface (and, hence, eFC values) in strong-bonded systems even in the vicinity of equilibrium, that may serve as a possible reason for the mentioned deviations. However, a decrease of the step value, while scanning potential energy surface, changes the eFC values only insignificantly (see the Methods). At the same time, an unsatisfactory accuracy of the calculated virial field may also be the reason for the mentioned deviations. Namely, additional calculations using 4519

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have tried to provide an explanation of the correlation by considering an interbasin interaction more accurately as Ω surfaces and invoking basic concepts of classical mechanics’. One can consider a space between atomic nuclei as a model “bar” (or “coil”) which determines the vibrational forces acting on nuclei. Since this “bar” is composed by electrons, its density varies in space. In terms of classical physics this “bar” can be characterized by a Hooke’s elastic modulus kH (J·m−2) while Young’s elastic modulus kY (J·m−3) is related to features of the “bar’s” material at its any point and describes a small volume with a constant electron density. Since energy of an elastic deformation of the “bar” is a potential one by its sense, it might be simply proposed that the v(r) at a point is the kY modulus at this point. Obviously, such “bar” would be stretched in regions with smaller density to a larger extent and one cannot limit such stretching process to the only one point (BCP). If the forms of atomic basins are not influenced by small nuclei vibrations and the motion of these basins coincides with the nuclei motion, then it is again evident that the Ω surface is more feasible to describe strength of interbasin interaction35,54elastic forces governing vibrational motion of nuclei and corresponding atomic basins in the vicinity of equilibrium. Consequently, the v(r)BCP value is merely an estimation of the distribution of potential energy density over the Ω surface. Summing up the kY modulus over the Ω surface (“bar’s width”) gives the whole elastic force between atomic basins (J·m−1) while dividing this sum by the internuclear separation (“bar’s length”) provides a measure of the kH modulus of the “bar”. Indeed, such simple treatment gives a linear correlation between kH and eFC values (Figure 3) that is statistically

four DFT functionals (B98, PBE0, B3LYP, and B972) were performed for the NO+ system which violates the eFC/v(r)BCP trend to the largest extent. The v(r)BCP value was found to be rather sensitive to the type of DFT functional: it varies within a range of at least 0.1 au Still, for the whole set of species correlations with any other local quantity at BCPs (even kinetic energy density that was also shown30 to provide accurate estimations of BDE for some interactions) were found to be inadequate (see table S12 in Supporting Information). Even for interbasin interactions of “H-bonding” type the eFC/v(r)BCP correlation was found to be much better than the traditional EML scheme (R2 = 0.9726 vs 0.8978, see Figure 2). The established eFC/v(r)BCP correlation

Figure 2. A comparative plot showing the linear correlations between eFC or BDE (mdyn·Å−1/a.u.e.) and v(r)BCP (a.u.e. × a0−3) values for H-bonds.

implies that the virial value at BCP is more likely an estimation of the eFC of diatomic stretching vibration around equilibrium state rather than a measure of energy of interaction breaking. Thereby the eFC/v(r)BCP correlation provides a bridge between electronic potential energy distribution in equilibrium state and characteristics of nuclei motion around it. This is in line with the Hohenberg−Kohn theorem12 which is at the core of modern theoretical chemistry and, particularly, of the AIM theory. The universal character of the eFC/v(r)BCP correlation to some extent argues for the reasonability of the EML-based BDE estimations for weak interactions, since the BDE and eFC values should be respectively intercorrelated as measure of plastic and elastic deformation of interactions in cases where formation/breaking of an interaction perturbs a nuclear− electronic system insignificantly and relaxation contributions are negligible. Though presented eFC/v(r)BCP correlation can be considered as being of a universal and at least semiquantitative character, its physical grounding is still questionable. For instance, the eFC/v(r)BCP correlation suffers from the dimensionality problem: the force constants can be expressed as J·m−2 while the v(r)BCP value at a point is only J·m−3. We

Figure 3. Linear correlation of eFC and kH (both in J·m−2) for the whole set of calculated species: from Ar2 to HNNH2+.

better than the eFC/v(r)BCP trend presented above (R2 = 0.9686 vs 0.9558); this fact shows the reliability of these classical model considerations. Notably, the combination of simple mechanical models and electronic potential energy density was already used to calculate the metal’s mechanical properties by means of potential energy formula for the uniform electron gas.55 Our estimations of diatomic force constants from potential energy density using classical mechanics’ concepts are in line with these investigations. The role of the v(r) function in the presented correlations puts a question as to whether the eFC values should be estimated at v(r) critical points rather than at ρ(r) ones. Indeed, considering that the molecular graphs based on the −v(r) and ρ(r) distributions are regarded as homeomorphic,56 it can be supposed that the least force required to elastically 4520

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deform internuclear electronic “bar” should be imposed at the zero-flux surface between two basins of the −v(r) function (or at the corresponding (3, +1) critical point of v(r)). In this area electronic potential energy density is always not less than at the Ω surface (or, respectively, at BCP). The corresponding correlations between the eFC and v(r)int or the eFC and v(r)pot values (where v(r)pot is the v(r) value at its (3, +1) saddle point, and v(r)int is integral of −v(r) over its interbasin surface, divided by the internuclear separation) are statistically worse (see Figure S1 in the Supporting Information), that once again points out a specific physical meaning of Ω surfaces, and BCPs as their estimators. At the same time, all established correlations between eFC and density of electronic potential energy are very similar that argues for the key role of electronic virial field as the measure of forces acting on atoms.

AUTHOR INFORMATION

Corresponding Author

*(I.V.A.) [email protected]. ORCID

Ivan V. Ananyev: 0000-0001-6867-7534 Author Contributions

The manuscript was written through contributions of all authors. All authors have given approval to the final version of the manuscript. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS I.V.A. is grateful to for Grant No. 16-33-60133 of the Russian Foundation for Basic Research for financial support. K.A.L., V.A.K. and A.O.D. were supported by the Russian Science Foundation, Project No. 14-13-00884. We thank Prof. K. Yu. Suponitsky and M. G. Medvedev for discussion.

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CONCLUSIONS The presented trends between eFC values and v(r) distribution still cannot be considered as ultimate and physically grounded, despite the mechanical model proposed to explain them. However, the following conclusions can be accurately made. Any interbasin interaction defined within the AIM theory is suggested to be described by its strength which is a universal and quantitative property of this interaction. This property can be defined as the amount of electronic potential energy between two osculating atomic basins (on their mutual boundary) which corresponds to elastic forces (eFC values) governing a stretching diatomic vibration. These forces are probably the first example of the dynamic properties of atomic aggregates to be estimated via the AIM theory. In concordance with the DFT theory one can expect the existence of a more general relation between electronic potential energy distribution and parameters of nuclei vibrations. From the practical point of view, our results provide an opportunity to easily estimate the eFC values using energy density from theoretically calculated one-particle density matrix (or experimental X-ray diffraction data) without performing resource-demanding calculations of normal vibrational modes. We believe that such eFC estimations can be further used to parametrize force fields describing the whole range of possible interatomic interactions including the weak ones which are of a special interest in supramolecular chemistry and structural biology. Since the EML correlation between energies of chemical bond dissociation and v(r)BCP values can be considered as a special case of trends established in this work, all chemical bonds may now be at least semiquantitatively ranged by their strength using eFC/v(r)BCP and eFC/kH correlations if a chemical bond is thought to correspond to an interbasin interaction. It implies that the presented correlations may also be of a great significance upon the analysis of stability and properties of any atomic aggregate (molecular or supramolecular) and estimations of particular interactions’ contributions.

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REFERENCES

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ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.jpca.7b01495. Additional correlation plots, results of calculations, and data on established correlations (PDF) 4521

DOI: 10.1021/acs.jpca.7b01495 J. Phys. Chem. A 2017, 121, 4517−4522

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DOI: 10.1021/acs.jpca.7b01495 J. Phys. Chem. A 2017, 121, 4517−4522