5120
J . Phys. Chem. 1989, 93, 5120-5123
Nucleophilic Addition to Activated Double Bonds: Predictions of Reactivity from the Laplacian of the Charge Density Marshall T. Carroll, James R. Cheeseman, Department of Chemistry, McMaster University, Hamilton, Ontario, Canada L8S 4M1
Roman Osman,* and Hare1 Weinstein Departments of Physiology and Biophysics and of Pharmacology, Mount Sinai School of Medicine of the City Unioersity of New York, New York. New York 10029 (Received: November 10, 1988; I n Final Form: February 21, 1989)
The reactivities of a series of molecules in a Michael addition reaction are analyzed on the basis of properties expressed in the Laplacian of the charge density distribution. The charge densities of structurally optimized acrylic acid (AA), methacrylic acid (MAA), acrylonitrile, and acrolein have been calculated by ab initio quantum mechanical methods with various basis sets. The relative reactivities of the activated double bond in these molecules toward a nucleophilic attack, predicted by the values of 0 2 p at the corresponding critical point, are in accord with experiment. The changes in the atomic properties of two reactants, AA and MAA, upon the nucleophilic addition of F are examined by using the theory of atoms in molecules. The changes in these properties provide a quantitative description of A A F and M A A F carbanion formation which is related to the size of the region of charge depletion on the attacked atom.
Introduction
The reactivities of activated double bonds toward nucleophilic attack are directly correlated with the electron-withdrawing tendency of the group linked to the double bond.’ The commonly accepted explanation for this correlation is based on the ability of the electron-withdrawing group to stabilize the negative charge formed on the carbon adjacent to the site of nucleophilic attack. In one type of nucleophilic attack at an activated double bond, the Michael addition reaction, the formation of such an intermediate carbanion is followed by a protonation that leads to stable adducts. It has been proposed that congeners of acrylic acid, which contain a double bond activated by an electron-withdrawing group, may add to biopolymers by the Michael reaction2 For example, acrylonitrile (ACN) and acrylamide (ACA) have been demonstrated to add to DNA in ~ i t r o ,possibly ~ , ~ following the mechanism of a Michael addition reaction. The addition of a nucleophile (fluoride anion) to the activated double bonds of acrylic acid (AA) and methacrylic acid (MAA) was studied recently as a model for such a reaction with DNA.S Analysis of the potential energy surfaces for these reactions and of the energy terms calculated for the various stages indicated the source of the difference in the reactivities of AA and MAA; it showed that this difference depends mainly on the properties of the isolated species. The reactivity characteristics evidenced in the molecular electrostatic potential (MEP) maps and in the Mulliken atomic charges of the isolated species suggested a basis for the difference in the reactivities of the two molecules, in agreement with the calculated potential energy surfaces. The favorable agreement between these results from simulations of the nucleophilic attack and the experimental findings showing the lower biological activity of MAA derivatives compared to AA derivatives supports the hypothesis that Michael addition in biological systems is a likely mechanism for the toxicity of derivatives of acrylate^.^*^ It also underscores the power of molecular properties as predictors of reactivity. In the present work the reactivities of a series of such activated double bonds, including those found in AA, MAA, ACN, and ( I ) Rappoport, Z . Acc. Chem. Res. 1981, 14, 7. Patai, S.; Rappoport, Z . In The Chemistry of Alkenes: Patai, S . , Ed.; Wiley: New York, 1964;
Chapter 8. (2) Eder, E.; Henschler, D.; “decker, T. Xenobiotica 1982, 12, 831. (3) Solomon, J. J.; Cote, I . L.; Wortman, M.; Decker, K.; Segal, A. Chem.-Biol. Interact. 1984, 5 1 , 167. (4) Solomon, J. J.; Fedyk, J.; Mukai, F.; Segal, A. Cancer Res. 1985, 45, 3465. ( 5 ) Osman, R.; Namboodiri, K.; Weinstein, H.; Rabinowitz. J. R. J . Am. Chem. SOC.1988, 110, 1701. (6) Moore, M. M.;Amtower. A.; Doerr, C.; Brock, K. H.; Dearfield, K. L. Enuiron. Mutagen. 1988. 11. 49.
0022-365418912093-5 120$01.50/0
acrolein (ACR), toward a nucleophile in a Michael addition reaction are predicted on the basis of properties of the above isolated molecules expressed in the Laplacian of the calculated charge distribution. The Laplacian, defined as vZp= a Z p / a X 2 a Z p / a y 2+ a2p/az2 (1)
+
determines regions of local charge concentration and depletion’ and correspondingly determines sites susceptible to electrophilic or nucleophilic attack. The Laplacian determines the relative susceptibility to attack in a generalized acid-base reaction and also predicts the relative geometry of approach of the acid and base molecule^.^^^ Motivated by these features, the theory of atoms in molecule^,^ and in particular the results of the V 2 p analysis, are used to predict the order of reactivities of the four molecules in the Michael addition reaction and also the approach angles of the nucleophile. To relate such results to the frame of the previous analy~is,~ the present work also examines the changes in atomic properties of the reactants AA and MAA upon interaction with F in the simulated Michael addition. Methods A . Molecular Structures. Restricted Hartree-Fock calcula-
tions of the isolated molecular species were performed with the 6-3 IG//6-31G, 6-3 lG**//6-31G, and 6-31++G//6-31++G basis sets (denoted schemes a, b, and c, respectively) using GAUSSIAN The four molecules ACR, AA, ACN, and MAA are very nearly planar and for simplicity will be assumed planar. Complexes of AA and MAA with F were calculated by using scheme c. The p and V 2 p analyses and calculations of atomic properties were carried out using the PROAIM series of programs.” B. The Laplacian of the Charge Density. The calculation and extensive analysis of molecular properties based on the Laplacian of the charge density are well-d~cumented.’-~We review briefly some of the properties of the analysis used in the present work. The topology of the charge density p provides a faithful mapping of the models of atoms, bonds, and molecular structure9 but does (7) Bader, R. F. W.; MacDougall, P. J. J . A m . Chem. SOC.1985, 107, 6788. Bader, R. F. W.; MacDougall, P. J.; Lau, C. D. H. J . A m . Chem. Soc. 1984, 106, 1594. Bader, R. F. W.; Essen, H. J . Chem. Phys. 1984,80, 1943. (8) Carroll, M. T.; Chang, C.; Bader, R. F. W. Mol. Phys. 1988, 63, 387. Bader, R. F. W.; Chang, C. J . Phys. Chem., in press. (9) Bader, R. F. W. Arc. Chem. Res. 1985, 18, 9. Bader, R. F. W.; Nguyen-Dang, T. T. Adu. Quantum Chem. 1981, 14, 6 3 . Bader, R. F. W.; Nguyen-Dang, T. T.; Tal, Y . Rep. Prog. Phys. 1981, 44, 893. (10) Binkley, J. S.; Frisch, M. J.; DeFrees, D. J.; Raghavachari, K.; Whiteside, R. A.; Schlegel, H. B.; Fluder, E. M.;Pople, J . A. GAUSSIAN 82; Carnegie-Mellon University: Pittsburgh, 1983. (11) Biegler-Konig, F. W.: Bader, R. F. W.; Tang, T. H. J . Comput. Chem. 1982, 3, 3 1 7 .
0 1989 American Chemical Society
The Journal of Physical Chemistry, Vol. 93, No. 13, 1989 5121
Nucleophilic Addition to Activated Double Bonds not offer any indication of the presence of shell structure in an atom or of maxima in a molecular charge distribution corresponding to bonded or nonbonded pairs of electron^.^ The latter characteristics, anticipated on the basis of the Lewis model or the model of localized electron pairs, are, however, recovered in the properties of V2p. The form of V2p for an isolated atom reflects the shell structure by exhibiting a corresponding number of pairs of spherical shells of local charge concentration and charge depletion. Upon chemical combination, local centers of diarge concentration and charge depletion are created within the v4lence shell (Figures 1 and 2). The number, locations, and magnitudes of the charge concentrations are found to be in general agreement with the properties that are ascribed to the bonded and nonbonded electron pairs in models of electronic s t r ~ c t u r e . ~ Sites of electrophilic and nucleophilic attack in a molecule correlate respectively with the sites of maximum charge c,oncentration and charge depletion. The Laplacian of p at a point r provides a measure of the difference between the average value of p at points neighboring r and its value at r. When V2p(r)< 0, p(r) is greater than its average value at neighboring points; when V2p(t) > 0, the opposite is true. This difference between the local and average value of p is maximal at the extrema in V2p. The locations of maxima, minima, or saddle points in a scalar function such as the Laplacian are determined by the positions of critical points in this function, Le., where V(V2p) = 0. Whether the function is a maximum, minimum, or a saddle at a given critical point is determined by the curvatures of the function at that point. Each critical point is classified by the number of nonzero eigenvalues (its rank) and the algebraic sum of the signs of the eigenvalues (its signature). A minimum in -V2p, identifying a local depletion of charge, exhibits three positive curvatures and is a (3,+3) critical point. A maximum in -V2p, Le., a local concentration of charge, is a (3,-3) critical point. Saddle points may be either (3,+1) or (3,-I) critical points. The valence shell of charge concentration (VSCC) of a free atom possesses a spherical surface over which the charge is maximally concentrated. Thus, the curvature of -V2p normal to this surface, the radial curvature, is negative. The two remaining curvatures, those tangential to the surface, are equal to zero. In general, this surface persists when the atom is in chemical combination (the derivative of -V2p normal to the surface is still zero and the corresponding curvature is negative), but the surface is no longer one of uniform concentration as the tangential curvatures assume either positive or negative values. If a local maximum is formed on the surface, then the two tangential curvatures of -V2p are negative while in the formation of a local minimum these same two curvatures assume positive values. Reference to relief and contour maps of V2p for ACR (Figures 1 and 2 ) indicates that the radial curvature of -V2p is negative for the VSCC of C1 of ACR and that this atom still possesses a surface in which -V2p is a maximum in all radial directions from the carbon nucleus. The two regions of charge depletion on C1 above and below the molecular plane, however, exhibit (3,+ 1) critical points in -V2p. Thus, the remaining two curvatures are positive, -V2p is a minimum in the surface of charge concentration, and charge is locally depleted in the vicinity of these two critical points. The Laplacian of p has ben shown7to be directly related to the local contributions to the electronic energy of a system through ( h 2 / ( 4 m ) ) V 2 p ( r )= V(r) + 2G(r) where V(r) is the potential energy density (its integral over all space yields the total electronic potential energy V) and G ( r ) is the kinetic energy density (its integral over all space yields the total kinetic energy of the electrons, 7'). Since V(r) < 0 and G(r) > 0 for all r, eq 2 states that the lowering of the potential energy dominates the total energy in regions where electronic charge density is concentrated, that is, where V2p < 0. The integral of the Laplacian of p vanishes for an isolated atom l V 2 p ( r ) d r = $dS
Vp(r)=n(r)= 0
(3)
because of the vanishing of V p on the surface of the atom at
Figure 1. Relief maps of -V2p for the acrolein molecule in the plane containing the nuclei (top) and in the perpendicular plane along the C1C2 axis (bottom). The largest valence-shell charge concentrations are found in the nonbonded region of the oxygen atom. While C1 exhibits a shell of charge concentration (both planes show a "lip" around the inner-shell region of charge depletion), V2p is actually positive over much of this shell in the perpendicular plane; i.e., only the curvature of -V2p along a radial line out from the C1 nucleus is negative, and the two curvatures tangent to the surface are positive. The core, or first quantum shell, of each atom exhibits a spikelike charge concentration a t the nucleus surrounded by a deep region of charge depletion. The arrow in the bottom diagram denotes the site of nucleophilic attack at C1. The charge density is generated by using scheme b.
infinity. Thus, the concentration of charge at certain distances from the nucleus (where V2p< 0) results in an equivalent depletion of charge at other distances (where V2p> 0) as demanded by the vanishing of the integral in eq 3. The corresponding alternation in the preponderance of either the potential or kinetic energy contributions in each shell, eq 2, also averages out on integration over an atom to yield the average values demanded by the virial theorem V+2T=O
(4)
From the local point of view afforded by eq 2, the virial theorem is obtained as a consequence of V2p vanishing over an isolated atom or over an entire molecule. An atom in a molecule, as defined by the topological properties of p , is also bounded by a surface through which the flux in the gradient of p is zero as in eq 3. Thus, the integral of the Laplacian of p over a bound atom L? also vanishes and the virial theorem is again obtained. ( h 2 / ( 4 m ) ) f i 2 p ( r )d r = V(L?) + 2T(L?) = 0
(5)
When an isolated atom enters into chemical combination, the surface of zero flux in Vp and its attendant physical consequences are preserved. The creation of local concentrations of bonded and nonbonded charge within the valence shell of an atom upon
5122
The Journal of Physical Chemistry, Vol. 93, No. 13, 1989
Carroll et al. TABLE I: Properties of the Laplacian of the Charge Density for the Isolated Reactants" LMClC2 ACR 113.7 AA 114.2 ACN 115.1 MAA 115.8
1031V2~lb 95.2 92.7 82.0 75.5
E.
alC1)
3.7 6.2 7.4
0.082 0.092 0.145 0.086
d(CH,)l) 0.093 0.101
0.153 0.078
Results are from scheme b calculations. Angles are in degrees. All other values are in a u except the experimental activation energy of amine addition, E,, which is in kcal bThese are the absolute values of V2p evaluated at the extrema of the local regions of charge depletion, M, in the VSCC of C1. To the number of significant figures reported here, the two nearly equivalent M in the VSCC of C1 in each reactant have identical values for 1031V2pland the M C l C 2 angle. As found previously,',* the region of largest nonbonded charge concentration in the nucleophile (e.g., the maxima in -V2p in the VSCC of the fluoride anion or the VSCC of nitrogen in NH,) aligns with the region of largest charge depletion in the electrophile (Cl). The greater the charge depletion, the more susceptible C l is to nucleophilic attack. The extent of charge depletion is measured by the value of the minimum in -V2p in the VSCC of C1. For the molecules calculated here, these values (Table I) show that the order of decreasing reactivity to nucleophilic attack (and Michael addition) is ACR > AA > ACN > MAA. This prediction is in agreement with experimentally determined activation energies for nucleophilic attack by aminesI2and also accounts for the greater biological activity of AA derivatives compared to MAA
\
derivative^.^
-
kigure 2. Contour plots for two planes of acrolein, one being the plane containing the nuclei (top) and the other a plane perpendicular to this
and along the C1C2 axis (bottom). The dashed (solid) lines denote regions of charge concentration (depletion). The Laplacian is also negative within the region bounded by the innermost solid contour enclosing each C and 0 nucleus. The positions of the bonded and nonbonded charge concentrations are denoted by solid squares. The lower diagram shows the two (3,+1) critical points (denoted by solid triangles) which determine the sites of nucleophilic attack at C1. Starting at a zero contour, contour values change in steps of f 2 X IO", f4 X lo", and f8 X 10" with n beginning at -3 and increasing in steps of unity chemical combination must therefore result in the formation of equivalent regions of charge depletion. The reaction of a nucleophile with an electrophile, or of a base with an acid, is the combination of a center of charge concentration with one of charge depletion within the valence shells of the respective atoms. Their reaction corresponds to the combination of a region of excess potential energy with one of excess kinetic energy (eq 2) to yield a linked pair of atoms for which the virial theorem is satisfied for each atom separately as well as for the combined pair (eq 5).
Results A . Analysis of V 2 p in the Isolated Reactants. The (3,+1) critical points of V 2 pdisplayed in Figures 1 and 2 correlate with centers of nucleophilic attack. These (3,+1) critical points on C1 in all four molecules form angles with the CI=C2 bond axis (denoted MClC2 in Table I) of approximately 1 1 5 O . Thus, a nucleophile is predicted to approach C1 from above or below the molecular plane along a path forming an angle of 115' with the Cl=C2 bond axis. This prediction agrees favorably with the ab initio calculations of the potential energy surface given by Osman et aL5 where LFCICZ is 114.4O and 114.5' for the stable carbanions of AA and MAA with F, respectively. Notably, the Laplacian predicts an off-angle approach to C1 by a simple determination of the position of a particular critical point in the Laplacian, with no recourse to the calculation of the potential energy surface.
-
It is found that schemes a and c, which exclude polarization functions, do not produce Laplacian fields exhibiting the (3,+1) critical points in the VSCC of C1. However, regardless of the scheme used, the magnitudes of V 2 p at the saddle points in the valence shell of charge depletion of C1 predict the same order of reactivity as above. B. Atomic Properties. The atomic average of the charge of an atom in a molecule q ( R ) is given in Tables I and 11. The integral of p ( r ) over the basin of an atom, i.e., over the subspace of real three-dimensional space occupied by the atom,9 gives the electron population of that atom, N(R), and q(Q) = z n - N Q )
(6)
where Zn is the nuclear charge. The values of q(C1) and q((CH2)1),where the latter symbol denotes the sum of the charges on C1 and the two hydrogens bonded to it, do not parallel the reactivities (Table I). Thus, the net charge on the electrophilic group of the Michael addition is not a good predictor of the reactivity of C1 to nucleophilic attack: as q(C1) and q((CH2)l) become more positive, the molecule does not become more susceptible to nucleophilic attack. In fact, it has previously been shown*that q(Q) cannot be used to predict correctly the reactivities of sites. The average kinetic energy of an atom in a molecule, T(R), is readily obtained by integration of a corresponding density over the basin of the atom. The atomic virial theorem9
E(R) = -T(R)
(7)
is used to obtain the atomic energy E(R). It is the changes in E(R) upon the addition of a nucleophile that are most important, and these values are used in Table I1 to describe the addition of the fluoride anion to the activated double bond in AA and MAA. As previously disc~ssed,~ the course of the addition of F to AA and MAA is very similar. A stable hydrogen-bonded complex is formed (Scheme I of ref 5 ) as an intermediate. Further approach of F to C1 raises the energy of this complex until a transition state (TS) is reached. From this TS the system moves down the potential energy surface to produce a stable carbanion (SC). The only significant difference between the potential energy ( 1 2 ) Morton, M.; Landfield, H. J . A m . Chem. SOC.1952, 74, 3 5 2 3 . (13) Wiberg, K . B.; Bader, R. F. W.; Lau, C. D. H. J . A m . Chem. SOC. 1987, 109, 1001.
The Journal of Physical Chemistry, Vol. 93, No. 13, 1989 5123
Nucleophilic Addition to Activated Double Bonds
TABLE 11: Atomic Properties and Their Changes for Complexes of AA and MAA with Fa complexes isolated reactants Drouertv 4(Q)
Rb
F c1 c2 (Ha+Hb) 1
(HI2 (CH$ COOH
F c1 c2 (Ha+Hb) 1
(H)2 (CH3)2
COOH total
AAF
MAAF
sc
TS
sc
0.305 0.419 -0.099 -0.120 -0.084 N/A -0.421
0.160 0.277 -0.028 -0.016 N/A -0.126 -0.267
0.309 0.368 -0.045 -0.131 N/A -0.163 -0.338
-1.000 0.044 0.002 0.101 0.054 N/A' -0.201
-1.000 0.024 -0.018 0.085 N/A 0.120 -0.21 1
TS 0.141 0.250 -0.062 -0.017 -0.062 N/A -0.250
-99.41 74 -37.8037 -37.8728 -1.2082 -0.6031 N/A -1 88.0544
-99.4174 -37.8070 -37.8783 -1.2149 N/A -39.6342 -188.0314
-23.86 85.22 6.28 -24.41 -12.93 N/A -40.53
-99.35 160.08 19.45 -53.96 -15.75 N/A -24.23
-16.71 90.61 4.90 -25.41 N/A -19.43 -40.51
-88.43 160.39 12.55 -62.29 N/A -19.70 -1 2.70
-265.5422
-304.5658
-10.23
-13.76
-6.55
-10.18
AA
MAA
'For the isolated reactants AA and MAA, q(R) and E(R) values are listed in au. For the complexes the changes in these values are listed (complex minus reactant), and the energy changes are in kcal mol-'. The EsCF(in au) and -V/Tvalues for AA, MAA, F, A A F (TS and SC), and M A A F (TS and SC) are -265.54244 and 1.999971 15, -304.565 91 and 1.9997455 2, -99.417 38 and 2.00062525, -364.97598 and 2.00025903, -364.981 92 and 2.000 201 89, -403.994 45 and 2.000 070 34, and -403.999 5 1 and 2.000 019 08, respectively. The virial ratio is needed in the derivation of the energy of an atom from its average kinetic energy via the atomic virial theorem.I3 The quantity I X N ( 0 ) - N(molecu1e)l gives a measure of the integration error and never exceeds 0.001 electron. The quantity I X E ( 0 ) - EsCFl never exceeds 0.50 kcal mol-'. The values of AESCF (= EscF(complex) - EscF(reactants)) in kcal mol-' for A A F (TS and SC) and M A A F (TS and SC) are -10.14, -13.86, -7.00, and -10.18, respectively. bThe symbols (Ha+Hb)l, (H)2, (CH3)2, and COOH refer to the sum of the atomic properties for the two hydrogens, H a and Hb, bonded to C1, the atomic properties of the hydrogen bonded to C2 in the acrylic acid systems, the sum of the atomic properties for the three hydrogens and one carbon comprising the methyl group in the methacrylic acid systems, and the sum of the atomic properties of the atoms comprising the carboxylic acid group in all systems, respectively. 'Not available.
curves for AA and MAA is in the relative energies of the TS and S C systems from the separated reactants. The TS of the F addition to MAA is -7.00 kcal mol-' relative to the separated reactants compared to an energy of -10.15 kcal mol-' for AA; the SC is -10.18 kcal mol-' for MAA compared to -13.87 kcal mol-' for AA. Thus, the presence of a methyl group on C2 in MAA causes a reduction in the stabilization of TS and SC by about 3 kcal mol-] compared to the separated molecules. This result is consistent with both the lower stability of carbanions formed from MAA and a slower rate of nucleophilic addition to methacrylates compared to acrylates.I2 In general, the results in Table I1 show that the changes incurred in the atoms upon forming the TS continue in the same direction as the SC is formed. We can therefore restrict our discussion hereafter to the differences in q(0) and E ( 0 ) between SC and the isolated reactants. For both stable carbanions, the following observations are made: Both F and C1 lose electrons, and the lAq(0)l values are largest for these atoms. The remaining atoms all gain charge. Most of the charge is gained by the electronwithdrawing group COOH. The magnitudes of Aq are similar for the two carbanions except in the activated double bond. The gain of electrons in C2 of A A F is small but more than twice that in C2 of M A A F . The change in charge is large on C1, which loses 0.51e more upon A A F formation than upon M A A F formation. The energy changes in the atoms of the complexes, aE(0),are fairly similar for A A F and M A A F . The fluoride anion, the hydrogens, and the COOH and CH3 groups are stabilized while the carbons of the activated double bond are destabilized upon carbanion formation. The fluoride anion is stabilized the most (Table 11). and the difference in M ( 0 )values between A A F and M A A F is greatest for F and COOH (compare columns 5 and 7 of Table 11). The C1 is destabilized the most, more than 8 and 13 times than the destabilization of C2 in A A F and M A A F , respectively. A factor contributing to this large destabilization is the larger magnitude of Aq(Cl), the largest change in atomic charge of all atoms in the carbanions.
its largest region of charge concentration with the largest region of charge depletion in the valence shell of the electrophile (Cl). The nucleophile is predicted to approach C1 from above or below the molecular plane along a path forming an angle of 1 15O with the Cl=C2 bond axis. This prediction, based on the properties of V 2 p , agrees favorably with ab initio r e s ~ l t s .The ~ degree of reactivity of C1 to nucleophilic attack is predictable from the relative size of the regions of charge depletion on C1, expressed in the values of V 2 p at the corresponding critical point. From these criteria, the predicted order of reactivities (from most to least reactive) is ACR > AA > ACN > MAA. These results are in accord with experiment.12 The calculated atomic properties provide a quantitative description of A A F and M A A F carbanion formation. Because C1 has a larger region of charge depletion in AA than MAA, one might expect F to donate more electronic charge to AA than to MAA. This is not the case. The Aq(F) values are nearly identical for both reactions. However, F is more stabilized by AA than by MAA (by 10.9 kcal mol-') consonant with the larger positive environment in AA generated by the larger charge depletion on C1. Thus, it appears that the stabilization of the incoming nucleophile is related to the size of the region of charge depletion on the attacked atom.
Conclusion An incoming nucleophile, such as NH, or F, initially aligns
Registry No. AA, 79-10-7; MAA, 79-41-4; ACN, 107-13-1; ACR, 107-02-8; A A F , 120771-44-0; M A A F , 120771-45-1.
-
Acknowledgment. The work was supported in part by the U S . Environmental Protection Agency under cooperative research agreement CR 8142292 (R.O. and H.W.). H.W. is a recipient of a Research Scientist Development Award (DA-00060) from the National Institute on Drug Abuse. A generous grant of computer time from the University Computing Center of the City University of New York is gratefully acknowledged. The donation of an FPS-264 by Floating Point Systems to McMaster University is gratefully acknowledged. The research described has been reviewed by the Health Effects Research Laboratory, U S . Environmental Protection Agency, and approved for publication. Approval does not signify that the contents necessarily reflect the views and policies of the Agency, nor does mention of trade names or commercial products constitute endorsement or recommendation for use.
