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J. Phys. Chem. 1992,96,9194-9197
Olson, K. F. J . Am. Chem. Soc. 1982, 104, 3740. (e) Gassman, P. G.; Yamaguchi, R. Tetrahedron 1982.38, 1113. (f) Roth, H. D.; Schilling, M. L. M.; Gassman, P. G.; Smith, J. L. J. Am. Chem. Soc. 1984, 106, 2711. (2) (a) HOGS.;Livneh, M.; Cohen, D. J. Am. Chem. Soc. 1987,109,5149. (b) Hoz, S.; Basch, H.; Cohen, D. J . Am. Chem. Soc. 1987, 109, 6891. (3) Gerson, F.; Qin, X.-Z.; Ess, C.; Kloster-Jensen, E. J. Am. Chem. Soc. 1989, I 11,6456. (4) (a) Masamune, S.;Kabe, Y.; Collins, S.; Williams, D. J.; Jones, R. J . Am. Chem. Soc. 1985,107,5552. (b) Jones, R.; Williams, D. J.; Kabe, Y.; Masamune, S.Angew. Chem., Int. Ed. Engl. 1986,25, 173. (c) Kawase, T.; Batcheller, S.A.; Masamune, S . Chem. Lett. 1987, 227. (5) Schkyer, P. v. R.; Sax, A. F.; Kalcber, J.; Janoschek, R. Angew, Chem., b r . Ed. Engl. 1987, 26, 364. (6) (a) Dabisch, T.; Schoeller, W. W. J. Chem. Soc., Chem. Commun. 1986,896. (b) Schoeller, W. W.; Dabisch, T.; Busch, T. Inorg. Chem. 1987, 26, 4383. (7) Nagase, S.; Kudo, T. J . Chem. Soc., Chem. Commun. 1988, 54. (8) Boatz, J. A.; Gordon, M. S. J . Phys. Chem. 1989, 93, 2888. (9) Nagasc, S.;Nahno, M. J. Chem. Soc., Chem. Commun. 1988,1077. (IO) Francl, M.M.; Pietro, W. J.; Hehre, W. J.; Binkley, J. S.M.;Gordon, M. S.; DeFrees, D. J.; Pople, J. A. J. Chem. Phys. 1982, 77, 3654. (11) Wadt, W. R.; Hay, P. J. J . Chem. Phys. 1985,82, 284. (12) Huzinaga, S.;Andzelm, J.; Klobukowski, M.; Radzio-Andselm, E.; Sakai, Y .; Tatewaki, T. Gaussian Basis Sets for Molecular Calculations; Elsevier: New York, 1984. (13) Gordon, M. S.; Binkley, J. S.;Pople, J. A.; Pietro, W. J.; Hehre, W. J. J . Am. Chem. Soc. 1982, 104, 2797. (14) Bobrowicz, F. W.; Goddard, G. W. Modern Theoretical Chemistry, Vol. 3, Methods ofElectronic Structure Theory; Schaefer 111, H. F., Ed.; Plenum Press: New York, 1977; p 79. (15) (a) Binkley, J. S.;Pople, J. A. lnr. J . Quantum Chem. Symp. 1975, 9, 229. (b) Pople, J. A.; Binkley, J. S.;Seeger, R. Int. J . Quantum Chem.
Symp. 1976, 10, 1. (c) Krishnan, R.; Frisch, M. J.; Pople, J. A. J. Chem. Phvs. 1980. 72. 4244. i 1 6 ) Schlegel, H. B. J . Phys. Chem. 1988, 92, 3075. (17) W a r , H. 0.;Dupuis, M. Chem. Phys. Lett. 1987, 142, 59. (18) (a) Reed, A. E.; Weinstock, R. B.; Weinhold, F. J. Chem. Phys. 1985, 83,735. (b) Reed,A. E.;Curtiss, L. A.; Weinhold, F. Chem. Reu. 1988,88, 899. (19) Bally, T. J . Mol. Struct. (Theochem) 1991, 227, 249. (20) Collins, S.; Dutler, R.; Rauk, A. J. Am. Chem. Soc. 1987,109,2564. (21) Nagase, S. Angew. Chem. Int. Ed. Engl. 1989, 28, 329. (22) Nagase, S.Polyhedron 1991.10, 1299. (23) Nagase, S: Kudo, T.1.Chem. Soc., Chem. Commun. 1990, 630. (24) Matsumoto, H.; Miyamoto, H.; Kojima, N.; Nagai, Y.; Goto, M. Chem. Lett. 1988, 629. (25) Sita, L. R.; Bickerstaff, R. D. J . Am. Chem. Soc. 1989, 111, 3769. (26) Gordon, M. S.; Nguyen, K. A.; Carroll, M. T. Polyhedron 1991,10, 1247. (27) Seeger, R.; Pople, J. A. J . Chem. Phys. 1977, 66, 3045. (28) Pyykko, P. Chem. Rev. 1988,88, 563. (29) Kudo, T.; Nagase, S.,to be published. (30) Frisch, M. J.; Head-Gordon, M.; Schlegel, H. B.; Raghavachari, K Binkley, J. S.;Gonzalez, C.; DeFrees, D. J.; Fox, D. J.; Whiteside, R. A.; Sager, R.; Melius, C. F.; Baker, J.; Martin, R.; Kahn, L. R.; Stewart, J. J. P.; Fluder, E. M.; Topiol, S.; Pople, J. GAUSSIAN 88, Gaussian, Inc., Pittsburgh, PA, 1988. (31) Frisch, M. J.; Head-Gordon, M.; Trucks, G. W.; Foresman, J. B.; Schlegel, H. B.; Raghavachari, K.; Robb, M.A.; Binkley, J. S.; Gonzalez, C.; DeFrees, D. J.; Fox, D. J.; Whiteside, R. A.; Seeger, R.; Melius, C. F.; Baker, J.; Martin, R. L.; Kahn, L. R.; Stewart, J. J. P.; Topiol, S.;Pople, J. A. GAUSSIAN 90, Gaussian, Inc., Pittsburgh, PA, 1990. (32) Dupuis, M.; Watts, J. D.; Villar, H. 0.; Hurst, G. J. B. HONDO 7, QCPE, Indiana University.
Expllclt Perlodlc Trend of van der Waals Radll Remi Cbauvint Laboratoire de Synthbe Asymztrique, Batiment 420, Universitt de Paris-Sud, 91405 Orsay, France (Received: January 21, 1992)
van der Waals radii of covalently bonded nonmetal atoms are suggested to vary in a row of the periodic table as cN/ In (cN), where N is the number of valence electrons, and in a column as ab-'/("'), where n'is the Born exponent. The reliability of the analytically homogeneous set of van der Waals radii given by the empirical formula 3 is shoyn by comparison with reference values issued from Pauling's and Bondi's recommendations. The latter are interpolated or extrapolated by means of an extended correlation with covalent or atomic radii. The values obtained are transferable to atoms in current molecular the discrepncy packings resulting from an averagedispersion effect: this standard compression of the atomic spheres rationbetween the van der Waals radii serving to estimate the distance of closest approach between nonbonded atoms (rw)and the equilibrium values (r*) occurring in phenomenological van der Waals potentials of molecular mechanics force fields. A quantitative correlation between rw and r* is outlined.
Introduction One distinguiihes between three degrees of molecular modeling, in which atomic features and a molecular descriptor give a molecular representation: (1) ab initio or semiempirical modelsbasis sets of atomic orbitals and a Hamiltonian serve to construct molecular orbitals; (2) molecular mechanics (MM) modelsatomic parameters and a force field (FF)serve to calculate the conformation of minimumsteric energy;' and (3) cap and spheres and CPK models-atomic parameters and a molecular graph serve to construct molecular skeletons. The distance between two atoms is the sum of their covalent or ionic radii if they are held together by a covalent or ionic bond (cap cut off), and it is greater than the sum of their van der Waals radii if the atoms are nonbonded and nonbonded to a same atom. London attractive forces and van der Waals repulsive forces are represented in phenomenological van der Waals potentials of M M FFs, but the former are not explicitly taken into account 'Present address: Centre de Recherche R o w 1 Uclaf, 102 route de Noisy, 93230 Romainville, France.
0022-3654/92/2096-9194S03.00/0
by models where atoms are represented by hard spheres. The radius of these spheres is the concern of this study.
The Hard Spbere Model The notion of shape, volume, and free surface for molecules is based on the obsemation of a lower limit far the distance between nonbonded atoms. The spherical representation of atoms inside molecules is admitted not only in CPK models but also in M M FFs (isotropic van der Waals potentials') and in ab initio models (sphere fits to total electron density"). Discussing molecular ~ .steric ~ effects in chemical packing in liquid or solid ~ t a t e ,or reactions? the selection of a set of van der Waals radii appears somewhat arbitrary, due to the variability of "contact distances" among X-ray diffraction data: atomic spheres are more or less compressed by their neighbors and warp to a limited extent.6 In this context, van der Waals radii are not quantified with a better precision than a few hundredths of an angstr6m. In addition, even if the hard sphere model is supposed to absorb some dispersion effect into the radius values, it does not account for the vibrational states of van der Waals contacts.' Consequently, the radius of 0 1992 American Chemical Society
The Journal of Physical Chemistry, Vol. 96, No. 23, 1992 9195
Explicit Periodic Trend of van der Waals Radii
mulation of radii of ions with a closed nth valence shell is written:Eb,lO*
TABLE I: Row Coartmts C, Calculated from Interatomic Dk?.tances 2r,(X) (in A) M d from Differential Scattering by van der wads Compkxes X-X"-'* n n' X r d x ) (4 C" 1 5 He 1.485 0.3822 Ne 1.551 0.5510 2 7 3 4 5 6
9 10 12 14
1.880 2.004 2.181
Ar
Kr Xe Rn
ri = C$(Zed
(2)
with
0.6679 0.7120 0.7749
AZefd =
cn 1
n'is the Born exponent of the valence shell, CN, I,and Z, denote the coordination number, the charge, and the effective nuclear charge of the ion, respeatively; z'is the charge of the counterion(s) in a binary compound,Z1and c, is a coefficient depending upon the rare gas electronic configuration of the ion. 2, varies as the number N of valence electrons in the neutral atom, and, by analogy, we assume that the variation of van der Waals radii keeps the form
an atom in current chemical environments is not defined from the shortest nonbonded distance but from an average of short nonbonded distances. The recommendations of Pading& and Bondi9 are the most used values. Correlations of these radii with electron density values2 and with other atomic parameters such as the de Broglie wave length AB of outermost valence electrons (r, AB/2), the ionic radius, or the covalent radius (seenext section) have been attempted? Whereas ionic,8bJocovalent,"." and metallic12radii can be systematically estimated by empirical formula, standard van der Waals radii are less unanimously accepted: in order to avoid cheating in the search for a periodic equation for van der Waals radii,I3*reference values have to be assigned by a unique correlation.
-
rw
= CXN)
C,, is supposed to be constant in a row of the periodic table as soon as the octet rule is satisfied. Since an increase of the nuclear charge outweighs the coulombic repulsion within the electron cloud, a regular decrease of van der Waals radii from the left to the right of the periodic table could be expected2 as for ionic radii. However, the accompanying increase of the number of electrons would cause an opposite effect (contrary to ionic bonds, the shared electrons in covalent bonds are not completely located around one of the nucleiz2). In the reference set, a minimum occurs in each row between the VIA atom (0, S, Se, Te) and VIIIA atom (rare gas), i.e., between N = 6 and N = 8.19 In order to represent the variation, we attempt to use the function
Reference Set of van der Waals Radii Pauling's van der Waals radii r, are grossly related to covalent radii r, by r, = r, 0.80 A for VA, VIA, and VIIA elements.8a Bondi proposed r, = r, + 0.76 A and recommended 1.70 A for carbon, 2.10 A for silicon, 2.19 A for germanium, and 2.27 A for tinas We found that the latter values also fit the above relationships, provided that the covalent radius r, is replaced by the atomic radius r,, (for coordination number 12). Equation 1 affords mean values of Bondi's and Pauling's recommendations: r,(ref) = r 0.78 A (1)
+
CN
+
f(N) = wherein c depends on the maximum number of electrons in the valence shell (two or eight). The OcCuTrence of a minimum could reflect the variation of the natural bonding symmetry in a row of the periodic table: it is completely spherical for rare gases, tetrahedral for the group IVA, and "much less spherical" for the intermediate groups. Although Pauling and Bondi decided to recommend slightly smaller van der Waals radii for the halogens than for the group VIA elements (parallel to ionic and covalent radii), their respective average values are almost identical. Therefore, it is not a priori incorrect to assume that the minimum occurs when the valence shell is occupied by as much nonbonding pairs as covalent bonding pairs, Le., for the VIA elements: c is defined by requiring that f(6) is a minimum, namely c = e / 6 = 0.453. This analysis does not work for the first row (H, He). Assuming a proportionality with the nonmetal section of other rows,23the minimum should formally occur for N = 2/3, and the constant c for hydrogen and helium is set at c = e / ( 2 / 3 ) = 4.077. Paulhg calculated the C,, values from distances measured between
wherein r = r, in the groups VA, VIA, VIIA, and r = r, in the group IVA. The reference set is completed by van der Waals radii of rare gases. Values reported by Bondis are almost identical to more recent determinations:l4 He-He,l5 Ne-Ne,I6 Ar-Ar,17 Kr-Kr,'* and Xe--Xe'8 distances were accurately measured by differential scattering, and calculationspredict the same values.'9 Due to this precision, van der Waals radii of rare gas need no correlation of type 1.20 Within the well accepted framework of pairwise additivity of van der Waals interactions, the values measured in rare gas dimers are transferable to atoms in a crystal lattice: the chemical homogeneity of the reference set will be discussed in the last section.
Periodic Equation for van der Waals Radii Pauling observed that ionic radii (ri)are linked to van der Waals radii by r, = ri ro where ro = 0 for VIA and VIIA elements, and ro = 0.2 A for VA elements (Table 11), and Pauling's for-
+
TABLE II: Comparison of Calculated Values of rr by Means of Formula 3 with Commonly Accepted Valuesc
H He
Ge As
Se Br
Kr
1.11 1.49
2.17 1.97 1.94 1.96 2.00
1.10 1.49'
(2.15) 1.98 1.94 1.92 2.000
1.2
2.0 2.00 1.95
1.0-1.2 1.40
2.19 1.85 1.87 1.85 2.02
C N
1.q 1.53
0
1.50
1.51
1.40
F
1.51 1.55 2.36 2.15 2.11 2.13 2.18
1.50 1.55" (2.40) 2.18 2.14 2.11 2.18'
1.35
Ne Sn
Sb Te I Xe
1.69 1.53
1.5
2.2 2.20 2.15
1.70 1.55-1.60 1.50 1.50 1.54 2.27 2.06 1.96 2.16
Si P S C1 Ar
Pb Bi Po At Rn
2.04 1.85 1.82 1.84 1.88 2.52 2.29 2.25 2.27 2.33
2.10 1.84 1.80 1.77 1.88' (2.53) (2.24) (2.24) (2.23)
1.9 1.85 1.80
2.10 1.80 1.85 1.75 1.88 2.02
'Values collected in ref 14. bMean values of Bondi's and Pauling's correlations with covalent radius (eq 1). cValues recommended by Pauling .~ in brackets are merely obtained by extrapolation from from ionic radii with a precision less than 10.05 A.*. dValues recommended by B ~ n d i #Data cq 1.
9196 The Journal of Physical Chemistry, Vol. 96, No. 23, 1992
monovalent anions and cations having the same rare gas configuration: N a + F (Ne), K+Cl- (Ar),Rb+Br- (Kr), and Cs+I- (Xe).8b Following this process, we calculate the row constants C,, for van der Waals radii from interatomic distances in van der Waals dimers of rare gases (N = 8, Table I):I4 In (8e/6) e,,= 8e/6 r,(X) O.35528rw(X),X = Ne, Ar, Kr, Xe
Testing a formal analogy with Pauling’s eq 2 for ionic radii, we found that C,, depends on the Born exponent n’(Tab1e I) according to
c,, e &)-1/W-I) The fit is surprisingly accurate for (I = 1.166 and 6 = 87.313, and the correlation coefficient equals 0.99991. The nature of the repulsive interaction between nonbonded neutral atoms is not electrostatic, and the Occurrence of the exponent -l/(n’-1) is heuristic.24 Nonetheless, van der Waals radii are periodically estimated by means of eq 3 CN r, = 1.166(87.3)-1/(d-1) (3) In (cn? with c = 4.077 and n’ = 5 for n = 1 and c = 0.453 and n’ = 7, 9,10, 12, 14 for n = 2-6. Results are listed in Table 11. A salient agreement is obtained with eq 1, which itself reprcduces an average of Pauling’s and Bondi’s recommendations. For the sixth row of the periodic table, no experimental data are available, but the values inferred by eq 3 are still very satisfactory with respect to those of eq Equation 3 is based on the assumption c = e/6. Other assumptions (e/8 < c < e/6) can be defended, but van der Waals radii of rare gases are subjected to take the same well established values: for each c, new row constants C,, are calculated. If the is still assumed, then the value 87.3 is still form C,,= a(b)-l/(d-l) assigned to 6, and 3.282 In (8c)/(8c) = a. A short calculus provides alternative sets of van der Waals radii by
-
rw = 0.410(87.3)-’/(“-’)N In (8c)/ In (Nc) (for the second to the fifth rows). Nonbonded distances depend on many factors: temperature, pressure size, and electronegativity of substituents, hybridization state etc., ... In addition, van der Waals radii slightly depend on the relative orientation of the atoms in contact, and isotropic models have a limited scope. Allinger corrected the spherical model for anisotropic effects: oxygen and nitrogen lone pairs were assigned to substituents having their own and the distance to a hydrogen atom in a head on approach was defined with an offset of the potential center 7.7% into the C-H bonds.27 If needed, the basic values of eq 3 can be corrected for some of the aforementioned effects.’3b In the same spirit, univalent ionic radii for coordination number six were calculated by Pauling (C,, = 1 in eq 2) and were corrected for lower valence states and other coordination numbers.10a*8b van der Waals radii of some ‘naked”,28bonded,43and charged” metal atoms have been reported. Covalently bonded metal atoms surrounded by donor ligands f&g their outermat electron shells (rule of 18 electrons for transition metals) are hidden from nonbonded contacts: their van der Waals radii have to be evaluated from independent studies. Adjustment of the van der Waals radius of boron for MM2 calculations lead to 1.98 A,31 while a direct application of eq 3 ( N = 3) gives 2.45 A. However, the vacant orbital of boron atoms is more or less filled by donor ligands (e.g., solvent): if 50% of one of the two electrons forming the B-:donor bond is incorporated into the atomic valence shell, then N = 3.5 and eq 3 leads to r,(B) = 1.90 A. This provides a hint to apply eq 3 to the rest of the periodic table, and the precise relationship between r, and the values suitable for MMFF‘s is discussed below.
Chauvin
Connection with Models Involving Explicit Dispersion Terms van der Waals interactions are mostly described by LennardJones or Buckingham potentials,but many other fomulations (e.g., the Hill equation32)are used33 VL.-,(d) = c
[(d*/d)” - (n/6)(d*/d)6] (Lennard-Jones) (n/6) - 1 sE [exp(-nd/d*) - (ne-”/6) X = (n/6) - 1 (d+/d6] (Buckingham) where d is the actual interatomic distance, n characterizes the steepness of the repulsive term (n = 14?4 9?5.36or 12). and c is the well depth reached at the distance d*. The equilibrium distance is often approximated as a sum d* = r* + r n of atomic ”van der Waals radii”. On average, van der Waals radii in Table I1 (r,) are 0.2 A smaller than equilibrium radii r*, and even larger differences (r* - r, = 2 A) were assumed in a force field for amides.” Many sets of r* values are found in the literat~re.W~9~’ For instance, putative r* values for halogens were simply assigned to the van der Waals radii of the * raregasaugmented by 0.05 A.38 For carbon and hy&=ua were determimd by fits to thermochemical data, but von R. Schleyer reported that, depending on the method used, r* ranges from 1.2 to 2.17 A for H and from 1.11 to 1.89 A for C.33By beginning with rare gases, AUinger extrapolated “equilibrium radii” (r*) parallel to the trcnd evidenced by Pauling and Bondi for “radii of closest approach” ( P , ) . ~ ~ Whereas d* is entirely defined as the position of the potential minimum, the distance of closest approach, exemplifed by d, = r, r,’, lacks a precise definition. The latter is now discussed by muming Allinger’s illustrationz Two ran gas atoms X are not pressed to each other by other f o m than their spbcific van der Waals interactions, and they lie at the position of the corresponding energy minimum, Le., 2r*(X) A apart. Suppose that instead of two atoms, two a h e molecules lie side by ~ide.3~ If two facing hydrogens on the two chains lie at the equilibrium distance 2r*(H), there are still many pairs of atoms (e.g., pairs of internal carbons) which are far from their ideal equilibrium distance: as they are attracted to each other, they draw the fonner facing hydrogens nearer than 2r+(H). The observationsof Pauling and Bondi show that this packing effect is significantly standardized for each atom: in “current chemical environments”, the mini” distance is close to 2rw(H)< W(H) for hydrogen atoms and is close to 2rw(X) = 2r*(X) for rare gas atoms X. Nonbonded distances which are smaller than the distance of closest approach d, are observed when additional effects to the standard van der Waals packing Occur, e.g., hydrogen bonding or polar interactions. Although intermolecular and intramolecular van der Waals interactions are mostly described by identical potentials, the conformation of strained molecules is dictated by compressing effects which are much more important than the van der Waals packing invoked above. Some nonbonded H-nH and C-C distances have been reported to be sisnificantly shorter than twice the van der Waals radii: exo,exo-tetracyclo6.2.1.1.13~6.02~7]dodecane displays dH-H= 1.75 < 2r,(H) = 2.22 and d,.,, = 3.11 < 2r,(C) = 3.66 A. These short contact distances are well predicted by MM, provided that the excess van der Waals contribution to the steric energy is balanced by lower bond stretching, angle bending, or torsional contributions.@ The clarified discrepancy between r* and r, can be formulated by r* a kNrwrwhere kN is a constant for each column of the periodic table. By definition of the standard attractive effect absorbed in r,, kV1IlA = 1.00 (rare gas). For other columns, we found that kvnA = 1.06, VIA = 1.10, kVA= 1.11, klvA = 1.03 and are suitable to reproduce the r* values recommended by Allinger .22
+
6
Conclusion More and more attention is being paid to van der Waals b~nding.l~-~I Although hard sphere models alone are not suitable to describe this bonding, a standard attractive component has been
Explicit Periodic Trend of van der Waals Radii physically and quantitatively incorporated into the radius values for sphem and caps models. A parametrizationof van der Waals potentials by a consistent "softening process" of both the repulsive and attractive potentials between "adhesive"hard spheres of radii r, and r,' will be shortly proposed.
References and Notes (1) For a general introduction to molecular mechanics, sa:Boyd, D. B.; Lipkowitz, K. B. J . Chem. Educ. 1982, 69, 269, and references therein. (2) (a) Franc], M. M.; Hout, R. F., Jr.; Hehre, W. J. J. Am. Chem. SOC. 1984, 106. 563. (b) Deb, B. M.; Singh, R.; Sukumar, N. J. Mol. Srrucr. (Theochem) 1992,259, 121. (3) Gaveuotti, A. Nouo. J. Chim. 1982, 6,443. Gavezzotti, A. J. Am. Chem. Soc. 1983,105,5220. Gavezzotti, A. J. Am. Chem. Soc. 1985,107, 962. Marplo, J.; Pires de Matos, A. Polyhedron 1989,8, 2431. (4) Mingos, D. M. P.; Rohl, A. L. J. Chem. Soc., Dalton Trans. 1991, 3419. (5) Sa,for example: Charton, M. J . Am. Chem. Soc. 1969, 91, 615. Tolman, C. A. Chem. RN. 1977,77,313. Charton, M. J. Org. Chem. 1978, 43, 3995. Chauvin, R.; Kagan, H. B. Chirality 1991, 3, 242. (6) Kitaygorodsky, A. I. Tetrahedron 1%1,14,230. (7) Ewing, G. E. Angew. Chem., Inr. Ed. Engl. 1972,II, 486. (8) (a) Pauling, L. The Nature Of The Chemical Bond, 3rd ed.; Cornell University Press: Ithaca, NY, 1960; Chapter 7, p. 257. (b) The Nature Of The Chemical Bond; Chapter 13, p 505. (c) The Narure Of The Chemical Bond; Chapter 7 , p 221. (9) Bondi, A. J. Phys. Chem. 1964,68,441. (10) (a) Pauling, L. J. Am. Chem.Soe. 1927,49,765. (b) For more recent dismssiona, see,for example: J e n l d ~H. , D. B.; Thakur,K. P. J. Chem. Educ. 1979,56,576. Raymond, K. N.; Eigcnbrot, Jr., C. W. Acc. Chem. Res. 1980, 13,276. Sol&-Corrta, H.; Gdmez-Lara, J. J. Chem. Educ. 1987,64,942. Olszewski, S.; Kwiatkowski, T. J. Mol. Structure (Theochem) 1991,235,391. (ll)OKetffe, M.; Brese, N. E. J. Am. Chem. Soc. 1991, 113, 3226. (12) Ping, M.; Xiubin, L.; Yuankai, W. J . Chem. Educ. 1990, 67, 218. (13) (a) At most thra variables occur in a 'periodic equation": the column and the row of the atom in the periodic table and eventually a t h i coordinate related to electronegativity: for a discussion, see: Allen, L. C. J. Am. Chem. Soc. 1992, 114, 1510. (b) A slight variation of c with electronegativities in eq 4 could be envisioned. (14) Hobza, P.; Zahradnik, R. Top. Curr. Chem. 1980, 93, 53. (15) Burgmans, A. L. J.; Farrar, J. M.; Lee, Y . T. J . Chem. Phys. 1976, 64, 1345. (16) Farrar, J. M.; Lee, Y. T.; Goldman, V. V.;Klein, M. L. Chem. Phys. Lett. 1973, 19, 359. (17) Parson, J. M.; Siska, P. E.; Lee, Y. T. J . Chem. Phys. 1972,56,1511. (18) Barker, J. A,; Watts, R. 0.;Lee, J. K.; Schafer, T. P.; Lee, Y. T. J . Chem. Phys. 1974.61, 3081. (19) It could be argued that the radii adopted for rare gas are not of the same nature as the other ones: they correspond to an equilibrium and not to a lower limit of nonbonded distances. It must be stressed that the referencc set is established with respect to standard environments which are defined for each atom and which cannot be compared anyway. The distances serving to define all the r,(ref)'s are equilibrium distances for the environment in which
The Journal of Physical Chemistry, Vol. 96, No. 23, 1992 9197 they are measured (see discussion in the last section). (20) This is fortunate because the covalent radius of a rare gas X is d e f d only in the charged species [X-XI' wherein the approximate spherical geometry of the electron cloud around X differs to the natural sphericity of the electron cloud around X in the monoatomic molecule [XI that we are referring to. (21) Remy, H. Treatise on Inorganic Chemistry; Kleinberg, J., Ed.; Elsevier Publishing Co.: New York, 1956; Vol. l , p 214. (22) Allinger, N. L. Adu. Phys. Org. Chem. 1976, 13, 1. (23) Let us consider that hydrogen belongs to the group VIIA, and that 6 = (2Nvlla + Nlv,,)/3 = (2 X 7 + 4)/3. Replacing N~V,by No = 0 and Nvlla by 1, the minimum should occur for N = (2 X 1 0)/3 = 2/3. (24) A formal electrostatic model also describes metallic bonding: Rioux, F. J. Chem. Educ. 1985,62. 383. (25) It must be noted that the remarkable fit of the eq 1 by the eq 3 cannot be used as a periodic equation of covalent radii which are determined with a higher precision: eq 3 does not reproduce the well established decay of the covalent radius between the groups VIA and VIIA. (26) Allinger, N. L.; Chung, D. Y. J. Am. Chem. SOC.1976,98, 6798. Rofeta, S.,Jr.; Allinger, N. L. J. Am. Chem. Soc. 1985,107,1907. Spherical oxygen and nitrogen atoms are again considered in MM3 force fields: Allinger, N. L.; Rahman, M.; Lii, J.-H. J. Am. Chem. SOC.1990,112, 8293. Schmitz, L. R.; Allinger, N. L. J. Am. Chem. SOC.1990, 112, 8307. (27) Lii, J.-H.; Allinga, N. L. J. Am. Chem. Soc. 1989,111,8576. Allinger, N. L. J . Am. Chem. Soc. 1977, 99, 8127. (28) Balfour, W. J.; Douglas, A. E. Can. J. Phys. 1970,48,901. Balfour, W. J.; Whitlock, R. F. Can. J . Phys. 1975, 53, 472. (29) Bondi, A. J. Phys. Chem. 1966, 70, 3006. (30) Wipff, G.; Weiner, P.; Kollman, P. J . Am. Chem. Soc. 1982, 101, 3249. Kollman, P. A,; Wipff, G.; Singh, U.C. J. Am. Chem. Soc. 1985,107, 2212. Bartolotti, L. J.; Pedersen, L. G.; Charifson, P. S . J. Compur. Chem. 1991, 12, 1125. (31) Bcmardi, A.; Capelli, A. M.; Gennari, C.; Goodman, J. M.; Paterson, I. J. Org. Chem. 1990, 55, 3576. Goodman, J. M.; Khan, S.D.; Paterson, I. J. Org. Chem. 1990, 55, 3295. (32) Hill, T. L. J. Chem. Phvs. 1948, 16. 399. (33) Williams, J. E.; Stang, P . J.; von R. Schleyer, P. Ann. Reo. Phys. Chem. 1968. 19. 531. (34) Williams, D. E. J . Chem. Phys. 1966,45, 3770. (35) Ermer, 0.; Lifson, S . J. Am. Chem. SOC.1973,95,4121. (36) Hagla, A. T.; Huler, E.; Lifson, S . J. Am. Chem.Soc. 1974, %, 5319. (37) Weiner. S.J.; Kollman, P. A.; Case, D. A.; Singh, U. C.; Ghio, C.; Alagona, G.; Rofeta, S.,Jr.; Weiner, P. J . Am. Chem. Soc. 1984, 106,765. (38) Meyer, A. Y.; Allinger, N. L. Tetrahedron 1978, 31, 1971. (39) For van der Waals potentials between hydrocarbons, see ref 40 and the following: Jorgensen, W. L.; Madura, J. D.; Swenson, C. J. J. Am. Chem. Soc. 1984,106,6638. Szczesniak, M. M.;Chalasinski, G.; Cybulski, S.M.; Scheiner, S . J: Chem. Phys. 1990, 93,4243. (40)Allinger, N. L.; Yuh, Y. H.; Lii, J.-H. J . Am. Chem. Soc. 1989,111, 8551 and references therein. (41) Saykally, R. J. Acc. Chem. Res. 1989, 22, 295. Canceill, J.; Cesario, M.; Collet, A.; Guilhem, J.; Lacombe, L.; Lozach, B.; Pascard, C. Angew. Chem., Int. Ed. Engl. 1989, 28, 1246. Bieler, C. R.; Janda, K. C. J . Am. Chem. Soc. 1990.112,2033. Reynolds, C. H. J. Am. Chem. Soc. 1990, I12, 7903. Wales, D. J. J . Am. Chem. Soc. 1990, 112, 7908.
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