Communication pubs.acs.org/jchemeduc
Facets of van der Waals Radii That Are Not Commonly Included in Undergraduate Textbooks Philip A. W. Dean* Department of Chemistry, University of Western Ontario, London, Ontario N6A 5B7, Canada ABSTRACT: Several important aspects of van der Waals radii are not commonly included in undergraduate textbooks: anisotropy; physical phase-dependence; dependence on other atoms in a molecule, especially directly bonded atoms; and influence over detailed molecular structure and molecular stability, through intramolecular interactions. A brief discussion of these aspects may be appropriate for lecture material in an upper-level course. KEYWORDS: Second-Year Undergraduate, Upper-Division Undergraduate, Graduate Education/Research, History/Philosophy, Interdisciplinary/Multidisciplinary, Misconceptions/Discrepant Events, Descriptive Chemistry, Enrichment/Review Materials, Molecular Properties/Structure “packing radii” and vdW radii as synonyms, whereas Kitaigorodskii5,6a used the term “intermolecular radii” for what are now more commonly called vdW radii. Today, the vdW radii most commonly quoted and used by chemists appear to be based on those of Bondi.9 These singlevalued radii were obtained by critically evaluating radii obtained from solids, liquids, and gases against a variety of criteria and are independent of the direction of nonbonded approach, as represented in Figure 1a for an atom A, which is covalently
T
he radii called van der Waals (vdW) radii are named after Johannes D. van der Waals, 1837−1923, the Nobel prizewinning Dutch thermodynamicist.1 The doctoral thesis of van der Waals in 1873 contained the derivation of the “equation of state” that ties together the gaseous and liquid states and that to this day bears his name. The vdW equation may be written as (P + n2a/V 2)(V − nb) = nRT
in which n, P, V, R, and T have their usual meanings and a and b are empirical constants (now known as vdW constants). For gases, this equation is an amended form of the ideal gas law, PV = nRT. The two additional terms allow for the nonideal behavior of real gases. van der Waals recognized the a term as allowing for weak attractive interactions (of some type unknown to him) between molecules; these interactions are now known as vdW interactions or forces. The b term was recognized as arising from the part of total volume denied to the gas by the finite volumes of the molecules (or atoms, for monatomic gases) themselves. The van der Waals b values provide one method of finding what are now called vdW radii. If we consider 1 mol of a chemically inert monatomic gas, for simplicity, the closest that any atom may approach another is the distance 2rAtom, the diameter of one atom. The distance 2rAtom may be obtained from the equation b = 2πN0(2rAtom)3/3, in which N0 is Avogadro’s number (e.g., ref 2). The radius rAtom is an example of a vdW radius (of the gaseous atom, here): the effective radial extent of the atom, the distance at which a second atom can just make nonbonded contact. For gaseous molecules, rMolecule, the effective radius of a molecule, replaces rAtom in the vdW equation. Overall, relatively little use has been made of rMolecule and much more emphasis has been given to the vdW radii of the individual peripheral atoms in molecules. Emsley3 refers to the vdW radius of a peripheral atom in a molecule as “the face that the atom presents to the world beyond the molecule”. Values of the vdW radii of the outer atoms have been obtained in several different ways. Those of Pauling4a and Kitaigorodskii,5,6a for instance, were based on interatomic distances obtained by crystallography.7 Pauling and Hayward8 used © XXXX American Chemical Society and Division of Chemical Education, Inc.
Figure 1. Diagrammatic representations of van der Waals radius, or radii, for an atom A covalently bound to atom B: (a) the van der Waals radius has single value, r, and (b) the van der Waals radii differ along and perpendicular to the bond direction, ra and rp, respectively, with rp ≥ ra.
bound to atom B and has a vdW radius r. A bright student might wonder why vdW radii should be single-valued in molecules, and not be dependent on the angle that any nonbonded contact at atom A makes with the reference direction provided by the covalent A−B linkage. In fact, Bondi considered it “well known that many [atoms in molecules] are more nearly pear-shaped” than spherical (and, indeed, data included in ref 9a show some vdW radii not to be independent of the angle of nonbonded approach), but chose to work with atoms that were segments of spheres for the sake of expediency. Despite their popularity with chemists, and as Rowland and Taylor10 have emphasized, Bondi’s vdW radii were originally intended for calculation of molecular volumes and, according to
A
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Bondi himself,9a “may not always be suitable for the calculation of contact distances in crystals”, which nevertheless seems to be one of their major current uses!11 Bondi’s initial paper9a also notes the tacit assumption of “the invariance of the vdW radius of an atom under the most drastic environmental changes, i.e., irrespective of its chemical combination and its nearest nonbonded neighbors as well as the phase state in which it is found.” This particular caveat is examined below. From nonbonded distances in the crystal structures of I2(s) and X2(s) (X = Cl, Br, or I), Kitaigorodskii6b and Nyburg,12 respectively, showed that in the crystalline dihalogens, the vdW radii of the halogen atoms had cylindrical symmetry, with flattening of the vdW radius along the direction of the halogen−halogen bond (“polar flattening”12). If we use the labels ra and rp for the vdW radii along and perpendicular to the covalent bond direction, respectively, as represented in Figure 1b for atom A, Kitaigorodskii noted that in I2(s) ra was 0.6 Å smaller than a single-valued vdW radius (r) of I, whereas Nyburg showed that ra and rp were 1.67 and 1.90, 1.64 and 2.01, and 1.76 and 2.16 Å for X = Cl, Br, and I, respectively. Theoretical calculations confirm that ra < rp for the atoms in a variety of isolated X2 molecules.13 Thus, there is clear evidence that for at least some common terminal atoms a single value of the vdW radius may not always be appropriate. vdW radii in the gas phase differ from those in the crystalline state. This difference has been attributed to the absence of extensive intermolecular interactions in the gas, compared with the solid.14,15 As an example, experimental data give values of 1.32−1.36 and 1.94 Å for ra and rp of Cl in gaseous Cl2,13 which can be compared with the values above found for Cl2(s); the larger anisotropy, rp − ra, of the radii observed for the gas phase versus the condensed phase is not uncommon.13 Evidently vdW radii are not, in general, independent of “phase state”. Nyburg and co-workers12,16 scrutinized appropriate subsets of crystal structure data available to them for a variety of terminal atomsN, O, F, S, Cl, Se, Br, and Hattached to C. They interpreted the data in terms of the terminal atoms having, generally, vdW surfaces that are spheroids of revolution, characterized by two perpendicular vdW radii, as for A in Figure 1b. However, when Rowland and Taylor10 carried out a systematic study of organic crystal structures reported up to 1996, again a selected subset of all available data, they found that intermolecular distances involving halogens and S agree with Bondi’s (single-valued) vdW radii remarkably well, those involving C, N, and O less so (but still to within about 0.05 Å) and that Bondi’s value for the vdW radius of H may be too high by about 0.1 Å in these organic crystal structures.17 Recently, Kertesz and Eramian18 have confirmed that in crystal structures the vdW surfaces about terminal S, Se, Br, and I atoms do show polar flattening. Reversibly formed chemical species held together by vdW forces (or, more generally, by forces that do not include covalent bond formation or interaction between charged entities19) are called vdW complexes or molecules. In gaseous vdW complexes of the type B···Cl−X (B = base, X = F or Cl), B···Cl distances depend on both B and X.20 Variation of B···Cl distance with X is always in the order B···Cl−F < B···Cl−Cl; for example, the C···Cl distances are 2.77 or 3.13 Å in linear OC··· Cl−X for X = F or Cl. From data for a range of B···Cl−X complexes, the effective axial vdW radius of the Cl in Cl−X was found to be 1.28 ± 0.11 or 1.55 ± 0.07 Å for X = F or Cl, respectively, compared with Bondi’s value of 1.75 Å for the isotropic vdW radius of Cl. Similarly, when B is the rare gas Ar,
there is general agreement that in linear Ar···Cl−X, the calculated21−23 Ar···Cl distance for X = Cl exceeds the 3.330 Å measured24 in molecular beam experiments for X = F. It must be concluded that in the in Ar···Cl interaction, ra of Cl, or Ar, or both, must vary with X; the primary effect should be in the radius of Cl. These examples show that the vdW radius or radii of an atom may not be independent of the nature of (an)other atom(s) directly attached to it, that is, its “chemical combination”.25 vdW interactions may occur not only intermolecularly but also intramolecularly. A general definition of such interactions is “The attractive or repulsive forces between molecular entities (or between groups within the same molecular entity [my italics]) other than those due to bond formation or to the electrostatic interaction of ions or of ionic groups with one another or with neutral molecules”.19 Bartell27 showed that bond angles X−C− Y and C-to-C distances in various substituted hydrocarbons could be accounted for very well in terms of intramolecular repulsions between nonbonded geminal atoms (or groups of atoms) and relaxation of such repulsions; he modeled the vdW surfaces of the nonbonded atoms as “hard spheres”. This model has been extended by Glidewell28 to a wider range of substances in which other main group atoms occupy the central position.29 For species containing terminal F, Cl, O, and OH groups attached to central elements from main groups, there is evidence that the hard-sphere “ligand intramolecular contact radii” depend on the specific central atom to which the groups are attached;30 a great deal of structural chemistry can then be rationalized in terms of ligand close packing about the central atom. NMR chemical shifts have been found to be useful indicators of intramolecular (as well as intermolecular) vdW interactions (for examples, see ref 31). Further, intramolecular vdW interactions are important in determining the relative stabilities of molecular species.14,32 For example, intramolecular vdW interaction energy must be included in calculations to obtain satisfactory relative energies of straight-chain alkanes versus their branched-chain isomers using the popular density functional approach, especially for large molecules.32 In summary, vdW (nonbonded) radii, and corresponding vdW interactions between nonbonded atoms, have a venerable history and continue to be an active area of interest. From their advent, it has been recognized that tabulated vdW radii are average values that may not yield correct minimum nonbonded distances in all situations; discrepancies of up to about 10% have been noted for organic structures. Most undergraduates will only ever see single-valued vdW radii, most commonly those based on the work of Bondi, which were introduced with important, though widely ignored, caveats. These single-valued radii are widely used in crystallography, but appear to have been tested systematically against only selected subgroups of crystallographic data, those pertaining to organic structures. A significant body of research shows that in general, vdW radii may not be independent of the direction of a nonbonded interaction(s). The radii are also affected by the physical phase of the molecule(s) of interest; radii appropriate for the solid state are not generally appropriate for the gas phase or for calculations of properties for isolated molecules. Further, the vdW radius of an atom is affected by the nature of other atoms bound to it. Therefore, great caution must be exercised in choosing appropriate vdW radii for any particular application; for some applications, the appropriate value(s) may simply not be available in the literature. It would seem that at the very least B
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(13) Batsanov, S. S. Anisotropy of Atomic van der Waals Radii in the Gas-Phase and Condensed Molecules. Struct. Chem. 2000, 11, 177− 183 and references contained within. (14) Allinger, N. L. Calculation of Molecular Structure and Energy by Force-Field Methods. Adv. Phys. Org. Chem. 1976, 13, 1−82. (15) Batsanov., S. S. Calculation of van der Waals Radii of Atoms from Bond Distances. J. Mol. Struct.: THEOCHEM 1999, 468, 151− 159. (16) Nyburg, S. C.; Faerman, C. H.; Prasad, L. A Revision of van der Waals Atomic Radii for Molecular Crystals. II. Hydrogen Bonded to Carbon. Acta Crystallogr., Sect. B 1987, B43, 106−110 and references contained within. (17) Earlier, Pauling (ref 4) considered his vdW radii to be reliable only to 0.05 or 0.10 A, whereas Kitaigorodskii noted that intermolecular radii are not strictly constant and that experimental values for intermolecular distances may differ from the appropriate sum of intermolecular radii by ± 5% (ref 6b) and “not infrequently... about 10%” ( ref 5b). (18) Kertesz, M.; Eramian, H. Anisotropy of van der Waals Atomic Radii of Selected Main Group Elements. Abstr. 244th ACS Natl. Meeting & Exposition, Philadelphia, PA, Aug 2012; Abstr. COMP-179. (19) IUPAC. Compendium of Chemical Terminology, 2nd ed. (the “Gold Book”). Compiled by McNaught, A. D.; Wilkinson, A.; Blackwell Scientific Publications: Oxford, 1997. XML on-line corrected version: http://goldbook.iupac.org (2006-) created by M. Nic, J. Jirat, B. Kosata (accessed Oct 2013); updates compiled by A. Jenkins, http://dx.doi.org/10.1351/goldbook (accessed Dec 19, 2012). (20) Karan, N. K.; Arunan, E. Chlorine Bond Distances in ClF and Cl2 Complexes. J. Mol. Struct. 2004, 688, 203−205 and references contained within. (21) Tao, F.-M.; Klemperer., W. The van der Waals Potential Energy Surfaces and the Structures of ArClF and ArCl2. J. Chem. Phys. 1992, 97, 440−451. (22) Rohrbacher, A.; Williams, J.; Janda., K. C. Rare Gas-Dihalogen Potential Energy Surfaces. Phys. Chem. Chem. Phys. 1999, 1, 5263− 5276. (23) Experiment shows that the ground-state molecule for X = Cl is T-shaped (ref 22 and references contained within). (24) Harris, S. J.; Novick, S. E.; Klemperer, W.; Falconer, W. E. Intermolecular Potential Between an Atom and a Diatomic Molecule: The Structure of ArClF. J. Chem. Phys. 1974, 61, 193−197. (25) A more extreme comparison has been given by Huheey (ref 26): the vdW radii of Xe in Xe(s) and XeF4(s) are 218 and ca. 170 pm (2.18 and ca. 1.70 Å), respectively, the difference being attributed to the high electronegativity of F causing a contraction of the Xe atom in XeF4. An analogous explanation might be used to account for the way in which the vdW radius of Cl in B···Cl−X varies with X (see text), the electronegativities of the two possible terminal X atoms being F > Cl. (26) Huheey, J. E. Inorganic Chemistry, 3rd ed.; Harper & Row: New York, 1983; pp 256−257. (27) Bartell, J. S. Molecular Geometry. J. Chem. Educ. 1968, 45, 754− 767 and references contained within. (28) Glidewell, C. Intramolecular Non-Bonded Atomic Radii: New Data and Revised Radii for p Elements. Inorg. Chim. Acta 1979, 36, 135−138 and references contained therein. (29) Pauling (ref 4b) had earlier drawn attention to the fact that X− E−Y angles for a range of “quadricovalent” atoms in a variety molecules appeared to reflect the different vdW radii of X and Y to a large extent. (30) Gillespie, R. J.; Robinson, E. A. Molecular Geometry of “Ionic” Molecules: A Ligand Close-Packing Model. Adv. Mol. Struct. Res. 1998, 4, 1−41 and references contained therein. (31) (a) Evans, D. F. Solvent Shifts of Fluorine Nuclear Magnetic Resonance Spectra. J. Chem. Soc. 1960, 877−880. (b) Boden, N.; Emsley, J. W.; Feeney, J.; Sutcliffe, L. H. The Effects of Π-Distribution and Intramolecular Fields on the 19F Nuclear Magnetic Resonance Shielding in Substitute Perfluorobenzenes. Mol. Phys. 1964, 8, 133− 149. (c) Li, S.; Allinger, N. L. Intramolecular van der Waals’ Interactions and 1H Chemical Shifts: Steric Effects in Some Cyclic
undergraduates should be made aware that single-valued vdW radii they see may be very approximate; if a few students then ask why, this communication will have been worthwhile. The interested reader is directed to references 33−36 for further discussion of the multiple facets of vdW radii.
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AUTHOR INFORMATION
Corresponding Author
*E-mail:
[email protected]. Notes
The authors declare no competing financial interest.
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ACKNOWLEDGMENTS I am grateful to R. J. Gillespie, Department of Chemistry, McMaster University, Hamilton, Ontario, for helpful discussions and to the Faculty of Science, University of Western Ontario, for a (postretirement) appointment as Adjunct Research Professor. I thank the reviewers and the editorial staff of the Journal for their helpful constructive criticisms.
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REFERENCES
(1) (a) Oesper, R. E. Johannes Diderik van der Waals. J. Chem. Educ. 1954, 31, 599. (b) Johannes Diderik van der Waals − Biographical. http://www.nobelprize.org/nobel_prizes/physics/laureates/1910/ waals-bio.html (accessed Oct 2013). (c) Johannes Diderik van der Waals, http://en.wikipedia.org/wiki/Johannes_Diderik_van_der_ Waals (accessed Oct 2013). (2) Moelwyn-Hughes, E. A. Physical Chemistry, 2nd corrected ed.; The Macmillan Co.: Oxford, 1964; p 594 et seq. (3) Emsley, J. The Elements, 3rd ed., 2nd reprint; Oxford University Press: Oxford, U.K., 2000; p 3. (4) Pauling, L. The Nature of the Chemical Bond, 3rd ed,; Cornell University Press: Ithaca, NY, 1960; (a) p 257 et seq; (b) p 114−116. (5) (a) Kitaigorodskii, A. I. Molecular Crystals and Molecules. Phys. Chem. 1973, 29, 10−18 and references contained within. (b) Kitaigorodskii., A. I. Non-Bonded Interactions of Atoms in Organic Crystals and Molecules. Chem. Soc. Rev. 1978, 7, 133−163 and references contained within. (6) Kitaigorodskii, A. I. Organic Chemical Crystallography; Consultants Bureau: New York, 1961; (a) p 5−8, and references contained within; (b) pp 141, 167. (7) Pauling ( ref 4a) noted that for several nonmetals, the vdW radius approximately equals the radius of the anion. He also noted that, for the elements he examined, vdW radius is generally 0.75−0.83 Å greater than the single-bond covalent radius, with the sum of covalent radius and plus 0.80 giving vdW radii “to within their limit of reliability”. (8) Pauling, L.; Hayward, R. The Architecture of Molecules; W.H. Freeman and Co.: San Francisco, CA, 1964; p 58. (9) (a) Bondi, A. van der Waals Volumes and Radii. J. Phys. Chem. 1964, 68, 441−450. (b) Bondi, A. van der Waals Volumes and Radii of Metals in Covalent Compounds. J. Phys. Chem. 1966, 70, 3006−3007. (10) Rowland, R. S.; Taylor, R. Intermolecular Nonbonded Contact Distances in Organic Crystal Structures: Comparison with Distances Expected from van der Waals Radii. J. Phys. Chem. 1996, 100, 7384− 7391. (11) The vdW radii quoted on the Web site of the Cambridge Crystallographic Data Centre (CCDC) are based in part on those of Bondi: Non-Bonded Contact Definition; van der Waals Radii, http:// isostar.ccdc.cam.ac.uk/help/IsoStar/isostar.3.44.html (accessed Oct 2013) and CSD Elemental Radii, http://www.ccdc.cam.ac.uk/ SupportandResources/Resources/pages/Resources.aspx?mc=-1&sc=1&rt=9&p=15 (accessed Oct 2013). (12) Nyburg, S. C. ‘Polar Flattening’: Non-Spherical Effective Shapes of Atoms in Crystals. Acta Crystallogr., Sect. A 1979, A35, 641−645 and references contained within. C
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Systems. Tetrahedron 1988, 44, 1339−1350. (d) Chestnut, D. B.; Wright, D. W.; Krizek., B. A. NMR Chemical Shifts and Intramolecular van der Waals Interactions: Carbonyl and Ether Systems. J. Mol. Struct. 1988, 190, 99−111 and references contained within. (e) Jaime, C. Empirical Computation of Carbon-13 NMR Chemical Shifts. Magn. Reson. Chem. 1990, 28, 42−46. (f) Boykin, D. W.; Hertzler, R. L.; Delphon, J. K.; Eisenbraun, E. J. Oxygen-17 NMR Studies on Alkylindanones: Steric Effects. J. Org. Chem. 1989, 54, 1418−23 and references contained within. (32) Schwabe, T.; Grimme., S. Theoretical Thermodynamics for Large Molecules: Walking the Thin Line between Accuracy and Computational Cost. Acc. Chem. Res. 2008, 41, 569−579. (33) Batsanov, S. S. van der Waals Radii of Elements. Inorg. Mater. 2001, 37, 871−885. (34) Zefirov, Y. V. van der Waals Radii and Current Problems of Their Application. Russ. J. Inorg. Chem. 2001, 46, 646−650. (35) Schiemenz, G. P. The Sum of van der Waals RadiiA Pitfall in the Search for Bonding. Z. Naturforsch. 2007, 62b, 235−243. (36) Hu, S.-Z.; Zhou, Z.-H.; Robertson, B. E. Consistent Approaches to van der Waals Radii for Metallic Elements. Z. Kristallogr. 2009, 224, 375−383.
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