Tables for
Volume F
Crystallography of biological macromolecules
Edited by E. Arnold, D. M. Himmel and M. G. Rossmann

International Tables for Crystallography (2012). Vol. F, ch. 22.1, pp. 708-709   | 1 | 2 |

Section van der Waals radii

M. Gersteina* and F. M. Richardsa

aDepartment of Molecular Biophysics & Biochemistry, 266 Whitney Avenue, Yale University, PO Box 208114, New Haven, CT 06520, USA
Correspondence e-mail: van der Waals radii

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For all the calculations outlined above, the hard-sphere approximation is used for the atoms. (One must remember that in reality atoms are neither hard nor spherical, but this approximation has a long history of demonstrated utility.) There are many lists of the radii of such spheres prepared by different laboratories, both for single atoms and for unified atoms, where the radii are adjusted to approximate the joint size of the heavy atom and its bonded hydrogen atoms (clearly not an actual spherical unit).

Some of these lists are reproduced in Table[link]. They are derived from a variety of approaches, e.g. looking for the distances of closest approach between atoms (the Bondi set) and energy calculations (the CHARMM set). The differences between the sets often come down to how one decides to truncate the Lennard–Jones potential function. Further differences arise from the parameterization of water and other hydrogen-bonding molecules, as these substances really should be represented with two radii, one for their hydrogen-bonding interactions and one for their VDW interactions.

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Standard atomic radii (Å)

For `*' see following notes on specific sets of values. Bondi: Values assigned on the basis of observed packing in condensed phases (Bondi, 1968[link]). Lee & Richards: Values adapted from Bondi (1964)[link] and used in Lee & Richards (1971)[link]. Shrake & Rupley: Values taken from Pauling (1960)[link] and used in Shrake & Rupley (1973)[link]. >C= value can be either 1.5 or 1.85. Richards: Minor modification of the original Bondi set in Richards (1974)[link]. (Rationale not given.) See original paper for discussion of aromatic carbon value. Chothia: From packing in amino-acid crystal structures. Used in Chothia (1975)[link]. Richmond & Richards: No rationale given for values used in Richmond & Richards (1978)[link]. Gelin & Karplus: Origin of values not specified. Used in Gelin & Karplus (1979)[link]. Dunfield et al.: Detailed description of deconvolution of molecular crystal energies. Values represent one-half of the heavy-atom separation at the minimum of the Lennard–Jones 6–12 potential functions for symmetrical interactions. Used in Nemethy et al. (1983)[link] and Dunfield et al. (1979)[link]. ENCAD: A set of radii, derived in Gerstein et al. (1995)[link], based solely on the ENCAD molecular dynamics potential function in Levitt et al. (1995)[link]. To determine these radii, the separation at which the 6–12 Lennard–Jones interaction energy between equivalent atoms was 0.25 [k_{B}T] was determined (0.15 kcal mol−1; 1 kcal = 4.184 kJ). CHARMM: Determined in the same way as the ENCAD set, but for the CHARMM potential (Brooks et al., 1983[link]) (parameter set 19). Tsai et al.: Values derived from a new analysis (Tsai et al., 1999[link]) of the most common distances of approach of atoms in the Cambridge Structural Database.

Atom type and symbolBondi (1968)[link]Lee & Richards (1971)[link]Shrake & Rupley (1973)[link]Richards (1974)[link]Chothia (1975)[link]Richmond & Richards (1978)[link]Gelin & Karplus (1979)[link]Dunfield et al. (1979)[link]ENCAD derived (1995)CHARMM derived (1995)Tsai et al. (1999)[link]
[-\hbox{CH}_{3}] Aliphatic, methyl 2.00 1.80 2.00 2.00 1.87 1.90 1.95 2.13 1.82 1.88 1.88
[-\hbox{CH}_{2}-] Aliphatic, methyl 2.00 1.80 2.00 2.00 1.87 1.90 1.90 2.23 1.82 1.88 1.88
[\gt\!\hbox{CH}-] Aliphatic, CH 1.70 2.00 2.00 1.87 1.90 1.85 2.38 1.82 1.88 1.88
[{\geq}\hbox{CH}\!=] Aromatic, CH 1.80 1.85 * 1.76 1.70 1.90 2.10 1.74 1.80 1.76
[\gt\!\hbox{C}\!=] Trigonal, aromatic 1.74 1.80 * 1.70 1.76 1.70 1.80 1.85 1.74 1.80 1.61
[-\hbox{NH}_{3}^{+}] Amino, protonated 1.80 1.50 2.00 1.50 0.70 1.75 1.68 1.40 1.64
[-\hbox{NH}_{2}] Amino or amide 1.75 1.80 1.50 1.65 1.70 1.70 1.68 1.40 1.64
[\gt\!\hbox{NH}] Peptide, NH or N 1.65 1.52 1.40 1.70 1.65 1.70 1.65 1.75 1.68 1.40 1.64
[=\!\hbox{O}] Carbonyl oxygen 1.50 1.80 1.40 1.40 1.40 1.40 1.60 1.56 1.34 1.38 1.42
[-\hbox{OH}] Alcoholic hydroxyl 1.80 1.40 1.60 1.40 1.40 1.70 1.54 1.53 1.46
[-\hbox{OM}] Carboxyl oxygen 1.80 1.89 1.50 1.40 1.40 1.60 1.62 1.34 1.41 1.42
[-\hbox{SH}] Sulfhydryl 1.80 1.85 1.85 1.80 1.90 1.82 1.56 1.77
[-\hbox{S}-] Thioether or –S–S– 1.80 1.80 1.85 1.80 1.90 2.08 1.82 1.56 1.77

Perhaps because of the complexities in defining VDW parameters, there are some great differences in Table[link]. For instance, the radius for an aliphatic CH (>CH=) ranges from 1.7 to 2.38 Å, and the radius for carboxyl oxygen ranges from 1.34 to 1.89 Å. Both of these represent at least a 40% variation. Moreover, such differences are practically quite significant, since many geometrical and energetic calculations are very sensitive to the choice of VDW parameters, particularly the relative values within a single list. (Repulsive core interactions, in fact, vary almost exponentially.) Consequently, proper volume and surface com­parisons can only be based on numbers derived through use of the same list of radii.

In the last column of the table we give a recent set of VDW radii that has been carefully optimized for use in volume and packing calculations. It is derived from analysis of the most common distances between atoms in small-molecule crystal structures in the Cambridge Structural Database (Rowland & Taylor, 1996[link]; Tsai et al., 1999[link]).


Rowland, R. S. & Taylor, R. (1996). Intermolecular nonbonded contact distances in organic crystal structures: comparison with distances expected from van der Waals radii. J. Phys. Chem. 100, 7384–7391.
Tsai, J., Taylor, R., Chothia, C. & Gerstein, M. (1999). The packing density in proteins: standard radii and volumes. J. Mol. Biol. 290, 253–266.

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