Inorganic Materials, Vol. 37, No. 9, 2001, pp. 871–885. Translated from Neorganicheskie Materialy, Vol. 37, No. 9, 2001, pp. 1031–1046. Original Russian Text Copyright © 2001 by Batsanov. Van der Waals Radii of Elements S. S. Batsanov Center for High Dynamic Pressures, Mendeleevo, Solnechnogorskii raion, Moscow oblast, 141570 Russia Received February 14, 2001 Abstract—The available data on the van der Waals radii of atoms in molecules and crystals are summarized. The nature of the continuous variation in interatomic distances from van der Waals to covalent values and the mechanisms of transformations between these types of chemical bonding are discussed. INTRODUCTION der Waals radius with the quantum-mechanical require- ment that the electron density vary continuously at the The notion that an interatomic distance can be periphery of atoms. thought of as the sum of atomic radii was among the most important generalizations in structural chemistry, In this review, the van der Waals radii of atoms eval- treating crystals and molecules as systems of interact- uated from XRD data, molar volumes, physical proper- ing atoms (Bragg, 1920). The next step forward in this ties, and crystal-chemical considerations are used to area was taken by Mack [1] and Magat [2], who intro- develop a universal system of van der Waals radii. duced the concept of nonvalent radius (R) for an atom situated at the periphery of a molecule and called it the atomic domain radius [1] or Wirkungsradius [2], ISOTROPIC CRYSTALLOGRAPHIC implying that this radius determines intermolecular dis- VAN DER WAALS RADII tances. Later, Pauling [3] proposed to call it the van der Kitaigorodskii [4, 5] was the first to formulate the Waals radius, because it characterizes van der Waals principle of close packing of molecules in crystalline interactions between atoms. He also showed that the phases. He assumed that the van der Waals areas of van der Waals radii of nonmetals coincide with their peripheral atoms in neighboring molecules are in con- ionic radii and exceed their covalent radii (r), typically tact but do not overlap (rigid-atom model), because the by 0.8 Å. repulsive forces between closed electron shells rise Initially, only x-ray diffraction (XRD) data, molar sharply with decreasing intermolecular distance. He volume measurements, and crystal-chemical consider- made up a system of van der Waals radii as consistent ations were used to determine R. Later studies extended as possible with the intermolecular distances in organic the range of experimental approaches and culminated compounds. His radii differed little from Pauling’s in a complete system of the van der Waals radii of free (Table 1). and bound atoms. Comparison of the results obtained The system of van der Waals radii was further by various physical methods made it possible to assess refined by Bondi [6, 7]. His detailed tables were very the accuracy and locate the applicability limits of the popular among chemists, even though the values of R van der Waals radii and to reconcile the concept of van were criticized in a number of works [8]. Bondi not Table 1. Crystallographic van der Waals radii of nonmetals R, Å Author, year HFClBrIOSNC Pauling, 1939 1.2 1.35 1.80 1.95 2.15 1.40 1.85 1.5 1.70 Bondi, 1964 1.20 1.47 1.75 1.85 1.98 1.52 1.80 1.55 1.70 Zefirov, 1974 1.16 1.40 1.90 1.97 2.14 1.29 1.84 1.50 1.71 Gavezzotti, 1983–1999 1.17 1.35 1.80 1.95 2.10 1.40 1.85 1.50 1.70 Batsanov, 1995 1.80 1.90 2.10 1.51 1.80 1.68 Wieberg, 1995 1.5 1.8 1.9 2.1 1.5 1.8 1.6 1.7 Rowland, 1996 1.10 1.46 1.76 1.87 2.03 1.58 1.81 1.64 1.77 0020-1685/01/3709-0871 $25.00 © 2001 MAIK “Nauka/Interperiodica” 872 BATSANOV R RM RX AA M X r rX dMX Fig. 1. Schematic showing the formation of an A2 molecule. Fig. 2. Schematic showing the formation of an M–X bond. only determined R from structural data but also calcu- molecular compounds [20, 21]. Let us outline these lated R by adding 0.76 Å to covalent radii and evaluated approaches and the obtained results. R from thermodynamic and physical properties. The R Since the valence state of a metal remains values recommended by Bondi are also listed in unchanged upon a polymorphic transformation, while Table 1. the bond number increases, the electron density redis- Zefirov and Zorkii [9, 10] corrected some of the van tributes between ligands, der Waals radii and made a number of new suggestions, ∆V ==2∆V 3∆V …, (1) in particular, that R should be calculated from the short- 1 2 3 est intermolecular distances (D), ensuring a three- where ∆V is the volume common to the two atomic dimensional system of contacts, since other peripheral spheres (Fig. 1), and the subscripts specify the bond atoms of the molecule may be not in contact, and the number. It is easily seen that determination of R by averaging all intermolecular dis- 2 2 tances may be unjustified. Although they highlight the ∆V = ---π()Rr– ()2Rr+ . (2) statistical nature of van der Waals radii and their vari- 3 ability within a few tenths of an ångström, depending Given that, in polymorphic transformations, a decrease on various structural factors, the radii in their system in the intermolecular contact area is accompanied by an are given with an accuracy of a few thousandths of an increase in covalent bond length [22], to each coordina- ångström (Table 1). tion number Nc there correspond particular R and r. A classical approach to the determination of the van Therefore, Eq. (1) takes the form der Waals radii of organogens was proposed in [11–14] ()2()()2()… (Table 1). The system of van der Waals radii of nonmet- R1 – r1 2R1 + r1 ==2 R2 – r2 R2 + r2 , (3) als elaborated by Rowland and Taylor [15] for struc- valid for univalent atoms. In the case of multivalent tural organic chemistry was based on a wealth of statis- metals, the left-hand side of Eq. (3) should be multi- tical data. plied by valence v if single-bond radii are used as r [3]. The van der Waals radii of the halogens and carbon As a result, we obtain in inorganic compounds were first determined by Paul- ()2()()2() ing [3], who used data on layered compounds such as R1 – r1 R1 + r1 = nRn – rn Rn + rn , (4) CdCl2 and graphite. Later, the van der Waals radii of where n = Nc/v. Taking into account that the maximal halogens and chalcogens were evaluated from a large N in metals is 12, one can use Eq. (4) to calculate R body of experimental data [16]: OH 1.51, Cl 1.80, c Br 1.90, I 2.10, S 1.80, Se 1.85, and Te 2.02 Å. corresponding to the largest change in Nc upon the polymorphic transformation. Such calculations were The van der Waals radii of metals are difficult to carried out [17] under the assumption that, upon an determine directly, because there is only a small num- increase in Nc, the decrease in R is proportional to the ber of structures in which metal atoms can be in contact increase in r (Table 2). with another molecule; with the development of struc- In another approach to determining R, the M–X tural chemistry, the number of such structures bond is also thought of as the region common to the increases. Table 2 lists the crystallographic van der atomic spheres M and X, with an internuclear distance Waals radii determined to date [16–19]. dMX (Fig. 2). It is easily seen that Recently, the van der Waals radii of metals were cal- ()2 2 1/2 culated from structural data for metals [17] and their RM = RX + dMX – 2dMXrM . (5) INORGANIC MATERIALS Vol. 37 No. 9 2001 VAN DER WAALS RADII OF ELEMENTS 873 Table 2. Crystallographic van der Waals radii of metals R, Å Atom [17] [20] [21] empirical [25] (9) Li 2.24 2.25 2.24 2.05 2.1 Na 2.57 2.40 2.39 2.3 K 3.00 2.67 2.67 2.7 Rb 3.12 2.78 2.76 2.8 Cs 3.31 2.90 2.93 2.9 3.1 2.9 Cu 2.00 2.16 2.14 2.09 1.9 1.9 Ag 2.13 2.25 2.29 1.9 2.0 Au 2.13 2.18 2.0 Be 1.86 2.03 1.99 1.86 1.8 Mg 2.27 2.22 2.25 2.10 2.2 Ca 2.61 2.43 2.27 2.17 2.6 Sr 2.78 2.54 2.40 2.36 2.7 Ba 2.85 2.67 2.55 2.52 2.8 Zn 2.02 2.09 2.16 2.12 2.1 2.0 Cd 2.17 2.18 2.29 2.1 Hg 2.17 2.15 2.28 1.75 [18] 1.95 2.1 Sc 2.28 2.37 2.18 2.3 2.25–2.41 Y 2.45 2.47 2.38 1.8 2.4 2.33–2.46 La 2.51 2.58 2.59 2.6 2.30–2.44 B 1.74 1.87 1.72 1.7 1.83 1.75 1.91–2.00 Al 2.11 2.19 2.17 1.97 2.02 2.0 2.29–2.38 Ga 2.08 2.17 2.16 2.14 2.05 2.0 1.95–2.02 In 2.24 2.28 2.29 2.32 2.2 2.09–2.15 Tl 2.25 2.29 2.42 2.05 1.92 2.3 2.01–2.07 Ti 2.14 2.30 2.22 2.0 2.1 2.11–2.19 Zr 2.25 2.38 2.31 2.3 2.2 2.20–2.29 Hf 2.24 2.34 2.29 2.3 2.04–2.12 Si 2.06 2.06 2.19 2.15 1.95 2.10–2.22 Ge 2.13 2.10 2.22 2.0 2.0 1.91–2.02 Sn 2.29 2.21 2.29 2.40 2.2 2.2 2.02–2.10 Pb 2.36 2.24 2.46 2.3 2.3 2.00–2.08 V 2.03 2.14 2.0 2.0 2.05–2.19 Nb 2.13 2.34 2.15 2.1 1.96–2.10 Ta 2.13 2.26 2.22 2.2 1.93–2.05 As 2.16 2.05 2.14 1.96 2.0 1.92 Sb 2.33 2.20 2.32 2.2 2.06–2.10 Bi 2.42 2.28 2.40 2.22 2.40 [19] 2.4 2.04–2.08 Cr 1.97 2.27 2.05 2.0 2.0 2.06–2.20 Mo 2.06 2.29 2.16 2.2 2.1 1.92–2.00 W 2.07 2.23 2.14 2.2 2.1 1.85–1.93 Mn 1.96 2.15 2.23 [24] 2.00 2.0 2.00–2.14 Tc 2.04 2.11 2.1 1.93–2.02 Re 2.05 2.21 2.11 2.1 1.78–1.86 Fe 1.96 2.19 1.98 2.1 2.0 2.05–2.25 Co 1.95 2.16 1.97 2.0 1.94–2.14 Ni 1.94 2.14 1.97 2.0 1.88–2.11 Ru 2.02 2.05 2.16 2.1 1.89–1.97 Rh 2.02 2.04 2.1 1.83–1.90 Pd 2.05 2.14 2.1 1.87–1.96 Os 2.03 2.02 2.1 1.76–1.82 Ir 2.03 2.01 2.1 1.76–1.82 Pt 2.06 2.15 2.1 2.02 2.1 1.77–1.85 Th 2.43 2.54 2.50 2.25 U 2.17 2.51 2.45 1.94 * Minimal radius from M···H van der Waals distances for R(H) = 1.2 Å [32].