A further focus on penetration indices of misfit van der Waals crusts

A further focus on penetration indices of misfit van der Waals crusts Jorge Echeverr´ ıa a and Santiago Alvarez b The recent proposal of the van der Waals crust as the spherical section comprised between the atomic radius and the van der Waals radius of an element generated a commentary by Grochala pointing out some supposed weaknesses of the approach (https://doi.org/10.1039/D3SC90191B). Our inital study is complemented herein by an analysis of the effects of the different size of the two interacting crusts, which shows that the penetration index defined earlier can adequately handle the misfit crust interactions. In a recent paper published in this journal, 1 we analysed in depth the potential application of a penetration index between the van der Waals crusts of two atoms A and B, p AB , 2 as an alternative to the use of raw interatomic distances for the analysis of a variety of bonding situations, from weak van der Waals interactions all the way to strong multiple covalent bonds. The only pieces of information needed are the covalent and van der Waals radii of each of the two elements, r A, r B ,v A and v B , as well as the interatomic distance d AB (Fig. 1a). The interpenetration of the two crusts is dened as i AB =v A +v B −d AB (1) Dividing the interpenetration by the sum of the van der Waals crust widths (w A and w B , Fig. 1a and eqn (2)), and multiplying by 100, we obtain the penetration index as a percentage of the sum of the two crust widths (eqn (3)). w A =v A −r A ;w B =v B −r B (2) pAB ¼100 iAB wAþwB (3) Simultaneously to the appearance of that paper, W. Grochala published a commentary warning about the problems that might arise when using such a penetration index for pairs of atoms with dissimilar sizes and different van der Waals crust widths. 3 Although we had pinpointed such a possible problem in the text and given a simplied preliminary analysis in an Appendix, we think that the concern expressed by Grochala deserves a more detailed analysis that we present here. For an AB atom pair, we can differentiate the penetration of atom A into the van der Waals crust of atom B (p A , eqn (4)) from the penetration of atom B in the crust of atom A (p B , eqn (5)). Herewith we consider atom A to have a wider crust than atom B, regardless of the values of their covalent and van der Waals radii, i.e.,w A $w B . pA¼100 iAB 2wB (4) pB¼100 iAB 2wA (5) From eqn (4) and (5) it follows that p A $p B in all cases. Notice also that the two degrees of penetration depend on the crust widths and on the distance, since i AB depends linearly on d AB , not on the atomic radii. Finally, it must be noted that p A and p B are calculated relative to the width of one crust each, whereas p AB is calculated with respect to the sum of the two widths, and for that reason we introduce a factor 2 in the denominator of eqn (4) and (5). To explore the relationships between the three penetration parameters we compare in Fig. 2 their values at some special interatomic distances for the case of two crusts of the same width (le Fig. 1 (a) Definition of the covalent (r) and van der Waals (v) radii, the width of the van der Waals crust (w) and the interatomic distance (d) for two nearby atoms A and B. (b) Parameters needed to define the penetration of atom A into the van der Waals crust of atom B and vice versa (eqn (4) and (5)). a Instituto de S´ ıntesis Qu´ ımica y Catalisis Homog´ enea (ISQCH) and Departmento de Qu´ ımica Inorg´ anica, Facultad de Ciencias, Universidad de Zaragoza, Pedro Cerbuna 12, 50009 Zaragoza, Spain. E-mail: Jorge. [email protected] b Department de Qu´ ımica Inorg` anica i Org` anica, Secci´ o de Qu´ ımica Inorg` anica, and Institut de Qu´ ımica Te` orica i Computacional, Universitat de Barcelona, Mart´ ıi Franqu` es 1-11, 08028 Barcelona, Spain. E-mail: [email protected].es Cite this: Chem. Sci.,2024,15,12166 All publication charges for this article have been paid for by the Royal Society of Chemistry DOI: 10.1039/d4sc03827d rsc.li/chemical-science 12166 |Chem. Sci.,2024,15,12166–12168 © 2024 The Author(s). Published by the Royal Society of Chemistry Chemical Science COMMENTARY Open Access Article. Published on 15 July 2024. Downloaded on 11/28/2024 1:01:31 PM. This article is licensed under a Creative Commons Attribution-NonCommercial 3.0 Unported Licence. View Article Online View Journal | View Issue column) and for two mist van der Waals crusts (right column), deduced by introducing the chosen distances into eqn (3)–(5). Notice that the expressions in the right hand column can incorporate the w A =w B cases (lehand side) by simply replacing in all the expressions the “<” and “>”symbols by “#”and “$”, respectively. First, we consider the case in which two van der Waals spheres are barely in contact (Fig. 2a). By denition, the three penetration parameters are 0% both for crusts with the same width (Fig. 2a, le) or for mist crusts (Fig. 2a, right). In a second case (Fig. 2b), the vdW crust of atom A reaches the valence sphere of atom B. The three penetration parameters adopt a value of 50% if the two crust widths are the same (Fig. 2b, le), but for mist crusts only p A has that value, while p B is less than 50% (Fig. 2b, right). For two mist atoms, a further approach is required to get the VdW sphere of atom B in contact with the valence sphere of A (Fig. 2b0), giving p A > 50% and p B =50%, while p AB is still less than 100%. Finally, when the two valence spheres establish contact, i.e., when the interatomic distance is identical to the covalent radii sum (Fig. 2c), the interpenetration index p AB is 100% regardless of the widths of the two van der Waals crusts. The individual penetration p A reaches values higher than Fig. 2 Values adopted by the individual and diatomic penetration indices at special interatomic distances. The left column shows the case of two atoms of identical curst widths but different atom sizes and the right column a case of two atoms with misfit crust widths. © 2024 The Author(s). Published by the Royal Society of Chemistry Chem. Sci.,2024,15,12166–12168 | 12167 Commentary Chemical Science Open Access Article. Published on 15 July 2024. Downloaded on 11/28/2024 1:01:31 PM. This article is licensed under a Creative Commons Attribution-NonCommercial 3.0 Unported Licence. View Article Online 100%, but those of p B are smaller than 100%. All in all, it is seen that the evolution of the three penetration parameters as the interatomic distance decreases present each a clearly predictable behaviour. Still more interesting, the individual penetrations p A and p B can be obtained from the interpenetration index p AB according to eqn (6) and (7). Conversely, p AB can be obtained from the individual penetrations by means of eqn (8). pA¼pAB 21þwA wB(6) pB¼pAB 21þwB wA(7) pAB ¼2pApB pAþpB (8) Those relationships are illustrated in Fig. 3, where we plot the individual penetration parameters p A and p B as a function of the interpenetration index p AB , for two atom pairs: one with practically identical crust widths, Ta–O(w Ta = 0.83, w O =0.84 Å), and one with very different widths, Zn–Te (w Zn =1.17 Å, w Te =0.61 Å). In the rst case (Fig. 2, le column), the individual penetrations p A and p B are equal to the interpenetration index p AB at any interatomic distance, whereas for atom pairs with different crust widths (e.g., Zn and Te) p A increases at a higher rate than p B as the interatomic distance decreases (i.e.,asp AB increases). Conclusion Even for mist van der Waals crusts the interpenetration index p AB carries information on the different le-right (p A ) and right-le(p B ) penetrations. Those differences depend only on the relative widths of the two interpenetrating crusts (w A and w B ), but not on the atomic sizes (r A ,r B ,v A and v B ), as seen in eqn (6)–(8). pA pB ¼wA wB (9) The deviation of the p A and p B values from p AB for an AB atom pair with mist van der Waals crusts can be expressed by their quotient, which is found to be identical to the quotient of the crust widths (eqn (9)). For those reasons, we think that it is advisable to use preferentially the interpenetration index p AB , which is conceptually clearer, implies no loss of information on the individual penetration indices p A and p B , and is simpler to use. Data availability No primary research results, soware or code have been included and no new data were generated or analysed as part of this comment. Author contributions Both authors contributed to the discussion, writing and revision of the original dra. Conflicts of interest There are no conicts to declare. Acknowledgements This work has been supported by the Spanish Ministry of Economy and Competitiveness (PGC2018-093863-B-C21 and PID2022-140244NB-I00), the Spanish Structures of Excellence Mar´ ıa de Maeztu program (grant MDM-2017-0767) and the Generalitat de Catalunya –AGAUR (grant 2017-SGR-1289). The allocation of computer time at CSUC is also acknowledged. J. E. is indebted to the Spanish MICCIN for a Ram´ on y Cajal research contract (RYC-2017-22853), and to the Gobierno de Arag´ on-ESF program (Research Group E07_23R) for nancial support. References 1 J. Echeverr´ ıa and S. Alvarez, The Borderless World of Chemical Bonding Across the Van der Waals Crust, Chem. Sci., 2023, 14, 11647– 11688. 2 D. M. Gil, J. Echeverr´ ıa and S. Alvarez, Tetramethylammonium Cation: Directionality and Covalency in Its Interactions with Halide Ions, Inorg. Chem., 2022, 61, 9082–9095. 3 W. Grochala, A focus on penetration index –a new descriptor of chemical bonding, Chem. Sci., 2023, 14, 11597– 11600. 4 B. Cordero, V. G´ omez, A. E. PlateroPrats, M. Rev´ es, J. Echeverr´ ıa, E. Cremades, F. Barrag´ an and S. Alvarez, Covalent Radii Revisited, Dalton Trans., 2008, 2832–2838. 5 S. Alvarez, A Cartography of the Van der Waals Territory, Dalton Trans., 2013, 42, 8617–8636. Fig. 3 Dependence of the individual penetration parameters p A and p B on the penetration index p AB for the Zn–Te and O–Ta atom pairs, with crust width ratios w A /w B =1.92 and 1.01, respectively. The points a, b, b0and c correspond to the special distances specified in Fig. 2. All values calculated from standard covalent 4 and van der Waals 5 atomic radii at varying A–B distances. 12168 |Chem. Sci.,2024,15,12166–12168 © 2024 The Author(s). Published by the Royal Society of Chemistry Chemical Science Commentary Open Access Article. Published on 15 July 2024. Downloaded on 11/28/2024 1:01:31 PM. 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