1430
J. Phys. Chem. 1903, 87, 1438-1444
Polarizablllty Densities within Atoms. 2. Hellum, Neon, and Argon Wllllam H. Orttung’ and Dell St. Jullen’ Department of Chemistry, University of Calitornie, Rivers&, Calltomie 9252 1 (Received: January 7, 1982; I n Final Form: June 28, 1982)
Quantum-mechanicalcharge density incrementa, Ap(r,O), in uniform applied fields were used to evaluate formal polarizability densities, X ( r ) ,for rare gas atoms. The Poisson equation of continuum electrostatics was solved for the electrostatic potential by the finite-elementmethod with an assumed X ( r ) ,and the bound-chargedensity function, pb(r,O), was calculated. X ( r ) was then varied by successive approximations to achieve a fit of pb to Ap. A generally satisfactory functional form for the many-electron X ( r ) was suggested by the analytic result for the hydrogen atom, multiplied by a power of the radius. The results are discussed in terms of the field distribution and shielding within the atom. As a byproduct of the calculations, the orbital contributions to Ap were evaluated as a function of radius.
Introduction Continuum dielectric models of molecular systems are at least as old as the subject of physical chemistry. However, the ability to consider realistic shapes and spatial variation of properties has been achieved only recently. The finite-element method, a powerful general technique for solving partial differential equations such as the Poisson equation; is now being developed for electrostatic problems in molecular The continuum dielectric formalism needs little justification for application to proteins and larger molecules. However, in 1975, it was shown that it could also provide interesting results for isolated atoms and atom pairs in applied electric field^.^^^ For an exploratory application of our methods at the atomic level, it was assumed that the Polarizability density of argon was a linearly decreasing “tent” function of the radiusa3This assumption gave reasonable results for the polarizability tensor of a pair of argon atoms in contacts3 The linear polarizability function was clearly a crude approximation, and the present paper describes our efforts to develop more satisfactory distributed polarizability functions for three of the rare gases. The subtle problems of applying the continuum dielectric model to atomic interiors have been discussed in the first paper of this series,’ and reference to other recent literature is given there. The present paper is a straightforward application of the dielectric formalism to closed-shell atoms. The deduced polarizability densities are consistent with the principles of the continuum dielectric theory and should be useful for comparison with the results of more sophisticated future theories. We hope that the results may also be suggestive for a variety of other investigations in which better results might be obtained by the use of more realistic polarizability functions. Beyond the most immediate application to the polarizability tensor of an atom pair, diatomic and simple polyatomic molecules are of interest. In simple polar molecules, the interaction of polarizability and distributed “embedded charge” is sensitive to the distribution of both (1) Present address: Department of Chemistry, Cornel1 University, Ithaca, NY 14850. (2) G. B. Kolata, Science, 184, 887 (1974). (3) W. H. Orttung, Ann. N . Y . Acad. Sci., 303, 22 (1977). (4) W. H. Orttung, J. Am. Chem. SOC.,100, 4369 (1978). (5) D. W. Oxtoby and W. M. Gelbart, Mol. Phys., 29, 1569 (1975). (6) D. W. Oxtoby and W. M. Gelbart, Mol. Phys., 30, 535 (1975). (7) W. H. Orttung and D. Vosooghi, J . Phys. Chem., preceding paper in this issue.
charge and p~larizability.~ In the Kirkwood-Westheimer theory; the use of a variable atomic polarizability density at the important solute-solvent interface might lead to improved results.
Methods General. The properties of the continuum dielectric formalism may be deduced from the Poisson partial differential equation V*(e.Vrj) = -4rp
(1)
where q5 is the electrostatic potential, p is the free-charge density, and t is the dielectric constant tensor. Other electrostatic quantities may be evaluated from 4. The electric field and polarization are given by
E = -Vd P = ( 1 / 4 ~ ) ( t- l)*E= X*E
(2a) (2b)
where X is the dielectric susceptibility tensor (formally identical with a polarizability density tensor). The polarizability a of an atom in an applied field E, is given by
(3) and the bound-charge density induced by an applied field is Pb
= -v*P
(4)
Numerical. Solution of the Poisson equation for general problems of molecular interest is now possible by the finite-element extension of the variation m e t h ~ d .In ~ this technique, eq 1 is converted to the equivalent variational integral, and the space of the problem is divided into “finite elements”. The trial functions (also called shape or interpolating functions) are not global, but are restricted to single elements, and are taken as simple polynomials controlled by coefficients whose values correspond to the unknown potentials, rjn at nodal points of the elements. Linear, quadratic, and cubic polynomials were used as needed in the present calculations. It may be noticed that a of eq 3 requires a first derivative of 4, while pb of eq 4 requires a second derivative. Both properties may be evaluated from the finite-element interpolating functions, but the strong inhomogeneity of atomic susceptibility functions makes the pb calculation more difficult. We have calculated Pb by two methods. In the first (within the finite-element method), we used
0022-3654/83/2087-1438$01 .50/0 0 1983 American Chemical Society
The Journal of Physical Chemistry, Vol. 87, No. 8, 1983
Polarizability Densities within Atoms
smaller elements and averaged Pb estimates for a given node over calculations in adjacent elements. The second approach involved fitting 4(r,e) to a Legendre polynomial series, as described in the following paragraph. If the potential of the atom in a field is expressed as
X(r)=
(7)
(8)
The expression from quantum mechanics has the form’
where Pn(cos 0) is a Legendre polynomial, then a straightforward derivation, substituting eq 5 into eq 2 and 4, yields Pb
+ 2Zr2)e-2zr
P’ = (dP/dE)E=O
(5)
n=O
f 6r
for a uniform field.7is The charge increment per unit applied field is defined as
m
4 = C Rn(r) P n ( C 0 S 6)
$(
1439
2 [XnRn”+ (p X r + Xr’)Bn‘+ l)Rn
1
Pn(cos 0) (6)
where single and double primes indicate first and second derivatives with respect to the argument r. X,(r) and X&r) are the radial and angular components of the susceptibility tensor. It is interesting to note that, if only one Rn(r) is nonzero, then Pb has the same angular dependence as the potential 13 according to the dielectric theory. To apply eq 5 and 6, we solved the Poisson equation by the finiteelement method, using a mesh whose elements were defined by circular arc8 and rays from the origin. The nodal values of the potential at each radius were fitted to Po, PI, P2,and P3 by least squares. The coefficients of the fit are the Rn(r). This method gave smoother and more reliable results for P b than the first method, described in the preceding paragraph. The input and output of all calculations were in atomic units. Each atom was scaled so that the van der Waals radius was close to unit radius on the mesh. The results were independent of scale factor variations of at least f20%. The calculations were done in cylindrical coordinates, with the field along the symmetry axis. The mesh was a two-dimensional slice through the axis and could be restricted to the upper half of the atom because of symmetry considerations. The outer boundary was cylindrical, with a radius and height of 10 mesh units each. Nodes on the boundary were assigned potential values appropriate for a uniform applied field, plus a small correction for the polarized atom. Boundary perturbations of the results were investigated and were found to have no influence on the results presented here. A series of increasingly refined finite-element meshes was used in the early part of the work. The final version had 12 circular arcs between r = 0 and 1,and 3 between 1and 1.5. Radial edges (rays) were placed every 11.25’ out to 3.5 mesh units. Elements with cubic interpolating functions extended to r = 2.75 and elements with quadratic functions extended to r = 6.5. The mesh had 168 elements, 743 variable nodes, and 72 boundary nodes and was computer generated and indexed? The maximum bandwidth of the equations to be solved was 45. A typical calculation took about 1min on an IBM 3701155.
Calculations and Results Hydrogen Atom. Although a single point electron in rapid motion about a point nuclear charge might seem to be an unlikely choice (among atomic and molecular systems) for modeling by the continuum dielectric formalism, a preliminary study of this system was found to be rewarding in insights and methods of value for more complex atoms. The formal polarizability density obtained from quantum mechanics has the form
(9)
where
p’(r) = ( 1 / ~ ) ( 2 Z+r
= n-1
1 -X&n r2
p’(r,O) = p’(r) cos I3
(10)
In the following, p’ will denote the radial factor, p’(r),unless stated otherwise. Since the contribution of a spherical shell of thickness dr to the total induced moment of the atom is dp’ = (4n/3)r3p’ dr
(11)
after angular averages are carried out, we conclude that r3p’ is the proper function of p’ to plot for attempts at fitting p’ by the classical theory. By similar reasoning, r2 is the proper weighting factor for the charge density and (if the variation of the field over the atom is small) for the polarizability density. Since solutions of the Poisson equation automatically contain interaction effects of distant charge elements that do not actually occur in a one-electron system, we used a different procedure for our first calculation. The potential was taken to be that of a uniform applied field, I#J = -Ez, and the bound-charge density was calculated from eq 4, 2b, and 2a by using X ( r ) from eq 7 and the finite-element programs. Good agreement was obtained with eq 9 and 10 for unit field. We then solved the Poisson equation for a uniform applied field and the same X ( r ) function. The total polarizability from eq 3 was found to be 15.5% low. The correct total value could easily be obtained by a 21.5% increase of eq 7 (at each value of r ) . However, this simple adjustment gave a calculated P b that was not in good agreement with p’ from eq 9 and 10. In Figure 1, Pb from the solution of the Poisson equation for the unadjusted X ( r ) of eq 7 is compared with p’ of eq 10. The source of the discrepancy is the nonphysical self-interaction of distant charge elements introduced by the use of the Poisson equation. The field acting on the electron is falsely reduced in the inner regions by shielding from polarized shells further out, and falsely increased in the outer regions by dipole fields from polarized shells closer in. Gradients in both E and X contribute to Pb, and each must be considered in an analysis of the result for Pb
It is natural to ask at this point whether a generalization of eq 7 such as
X ( r ) = (1/4a)(ao + alr
+ a2r2)e*g
(12)
could be adjusted so that the correct p’ could be obtained from the Poisson equation. The question seems to have a negative answer in the sense that an unreasonably large value of a. would be required to counter the nonphysical shielding introduced by the Poisson equation at small r. We did not pursue this point since the Poisson equation is more appropriate for many-electron systems, where interelectron interactions occur and self-interaction is relatively less important. (8) D.W. Oxtoby, J. Chem. Phys., 69, 1184 (1978).
The Journal of Physical Chemistry, Vol. 87, No. 8, 1983
1440
Orttung and St. Julien
,
I
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Quantum Charge Densities of Rare Gases. Wave functions for isolated rare gas atoms in uniform applied fields are available from Sitter's LCAO-STO Hartree-Fock calculation^.^ Similar calculations have also been carried out by Clementi and RoettilO for zero applied field. Detailed comparison of r2p(r)plots for the Clementi-Roetti double-{ and Hartree-Fock values with Sitter's results at 90° to a small applied field showed that the three sets of results were almost identical in all cases. A further check on the Sitter orbital coefficients was carried out by verifying orbital orthonormality for each atom (He, Ne, Ar) and field strength to five decima1s.l' Since Sitter did not provide results for zero field, the difference of the p values for the two lowest fields was divided by the field difference to approximate the low-field derivative
=A~/AE
\
\
/
-
= (ap/amE=,,
\
I
Flgwe 1. Hydrogen atom functions. Top: Polarizability density from quantum mechanics. Middle: Charge increment in small applied field from quantum mechanics (- -) and the Poisson equation (-). The values along the z axis (parallel to the applied field) are shown. Bottom: Electric field within the atom, according to the Poisson equation solution.
pi
\
I
I
I
I
\,TOTAL
0
\
I
I
I
I
I
I
2
3
4
5
r
Flgure 2. Radial induced dipole density contributions of the neon orbitals. evaluated from Sitter's coefficients.'
'1
4.8
I
I
I
I
Ar
' ,I-\
1 I I
4.0
i
/
\,TOTAL \
\
/
\ \
\
- 1.6u
(13)
pr was
then fitted to a Legendre polynomial expansion at each radius: 3
p'(r,e) = C
n=O
~ ~ 'P,(COS ( 4 e)
(14)
(Note that the n subscript of eq 14 differs from the lz subscript used in paper l.7) The p i coefficients were generally very small relative to the pl', but the pol and p i were not always negligible, especially in the case of Ar. For (9) R. E. Sitter, Jr., Thesis, State University of New York at Buffalo, Buffalo, NY, 1969, University Microfilms, Ann Arbor, MI, 70-17366. (10) E. Clementi and C. Roetti, A t . Data N u l . Data Tables, 14, 177 (1974). (11) This procedure led to the discovery of a printout error on p 74 of Sitter's thesisQin the 15th coefficient of the 30 orbital of Ar at E = 0.006 au. The exponent -1 was printed as -2.
-1.6 0
I
I
I
I
2
3
r
I
I
4
5
Figure 3. Radial induced dipole density contributions of the argon orbitals, evaluated from Sitter's coefficients.'
Ar, however, we found p i = 2pd, so that pr was always zero in a plane perpendicular to the applied field and passing through the atomic nucleus. (Recall that Po + 2P2= 3 cos2 0.) In the present paper, only the p l r coefficients are used for fitting by the dielectric formalism. In the Hartree-Fock approximation, each orbital contributes independently to the charge density, and also to
The Journal of Physical Chemistry, Vol. 87, No. 8, 1983 1441
Polarizabliity Denstties within Atoms
.20
,
for 2 = 2 is also shown. It is much too small, indicating tighter binding of the electrons than actually occurs. It might be remarked that 2 = 1.6 is close to the value of 2 = 1.7 that minimizes the total energy in the analogous elementary variation treatment in quantum mechanics. To fmd a X ( r )that reproduces the Sitter p i values more closely, we solved the Poisson equation repetitively, with successive improvements in the functional form of X(r). The lack of a simple analytic relation between X and p’ ruled out straightforward use of the least-squares procedure. However, rational variation of X ( r ) was simplified by giving it an analytic form. After some exploration, a generalized form of the hydrogen atom function (eq 7) was chosen:
I
He
/‘-y
.ia
= I .6 I 5
i .I 6
.I4
.I2 - :m
.IO
m
X=
.oa
ill
fi(l/4*)[aoi + alir + a2ir2]Pre-ae1r (16)
Self-interaction of the electrons is automatically excluded by this choice. If we also set 2 = 2, then so is the interaction of the electrons with each other. By reducing 2, we introduce partial interaction of the electrons as a nuclear screening effect. If we take 2 = 1.615, the total polarizability agrees with Sitter’s value of 1.322 au. (2 = 1.597 gives the experimental value of 1.384 au.12) In Figure 4, we compare the functional form of pl’ from Sitter’s results with p’ (=2p’H of eq 10) according to the above approximation. The agreement is surprisingly good considering the crudity of the approximation. The result
where the f i and ai are adjustable parameters. a, was taken as 2(n - l),where n denotes the shell, in analogy to a Slater orbital. Terms were added one by one, moving from large to small radii. Existing terms usually had to be adjusted when new terms were added. Before actually trying to fit the quantum charge increment function by solving the Poisson equation, we considered the possibility of correcting for electron self-interaction. If an atom contains N polarizable electrons, the average susceptibility at a given point will be that of N electrons. However, the field acting on a particular electron at a given point will have contributions from only N - 1 electrons. If each electron is assumed to have an average formal susceptibility X , ( r ) ,then the total susceptibility to be used in solving the Poisson equation may be taken as ( N - l)X,, while the susceptibility for calculating P and pb may be taken as N X , . This simple method of correction is similar to that of Fermi and Amaldi in the electron gas model of the atom.13 The main difficulty in applying this method is that it is only convenient if all electrons can be assigned the same X, function. This is reasonable for isolated He, Ne, and Ar atoms, where only one shell makes any appreciable contribution to the polarizability. However, for atom pairs (where the present results may be applied), different X , functions must be defined about the two centers. For this reason, we only explore this correction in depth for He, where (N - 1)/N is l / p For Ne and Ar, (N - 1)/N = 7/8 for the contributing shell. In all three cases, if the correction is not made, then the X ( r )that gives the best fit of pl’ will be larger for small r and slightly smaller for large r. Helium was first fitted by using the electron self-interaction correction described above. The outer part of pl’(r)was fitted by a single term of eq 16. The best zi value was quickly found to be 1.45, in close (but perhaps fortuitous) agreement with the smaller of the double-l values.l0 The residual from the these fit, shown in Figure 5, was then fitted by a second term of eq 16. The residual of the two-term fit is also shown in Figure 5. The two-term fit gave a = 1.317,0.4%below Sitter’s value (1.322). The small residual near the origin could have been reduced by adding a third term, but this step was not felt to be justified. The parameters of the polarizability density function are shown in Table I, and the function is plotted in Figure 6. The effects of varying the individual parameters were complex, and it would be difficult to attach much significance to the magnitude of any one. Only the overall fit is meaningful, and eq 16 provided an efficient
(12) A. Dalgarno and A. E. Kingston, h o c . R. SOC.London, Ser. A , 259,424 (1960).
(13) P. Gombas, “Die statistische Theorie des Atoms and Ihre Anwendungen”, Springer-Verlag, Vienna, 1949.
.o E .O4 .O2 0
Flgure 4. Results for the helium atom polarizability density function approximated by the sum of two hydrogen-like contributions. 2 = 1.615 gives the correct total polarizability for Sitter’s calculation.
the moments of the charge density. The radial contribution of the ith orbital to the induced dipole moment, (api/ar)/E,was calculated from the Sitter coefficients, and the results for the orbitals of the dominant shells of Ne and Ar are shown in Figures 2 and 3. These values include hyperpolarizability contributions which are negligibly s m d for present purposes. Sitter’s resultse for the total polarizability were somewhat lower than experimental values. Thus, the ratio, calculated to experimental, is 1.32211.384, 2.37412.663, and 10.774/11.080 for He, Ne, and Ar, respectively, so that the calculations are 4.5,10.8, and 2.8% below e ~ p e r i m e n t .Since ~ our results are based on the calculated p{(r),they contain a similar error in comparison with experimental results. Helium Atom. Before attempting a Poisson equation fit of the Hartree-Fock results, we tried a simple approximation of the polarizability density of helium. Each electron was assumed to be in a hydrogen-like orbital, so that XHe= 2xH, where X H is from eq 7. The total polarizability of helium is then7 “He
= 2(9/2)Z4
(15)
Orttung and St. Julien
The Journal of Physical Chemistty, Vol. 87, No. 8, 1983
1442
.
’
*
~
J
0 I
I
I
I
-
3
2
I
0
4
5
6
r
Figwe 5. Fit of the helium atom quantum charge Increment functlon by a twc-term polarlzaMvty de~nsllyfunction In the PoissOn equation, correcting for self-hrteractkn of the electrons. Resldual from the outer term (- -1 and residual from both terms (- -). (-)
-
-
TABLE I: Polarizability Density Function Parameters of Eq 16
atom i
Ha Heb HeC NeC
Arc
1 1 2 1 2 3 1 2 3 1 2 3
fi
aai
3 2.07 1.00 1.00 3.00 2.07 1.90 2.30 3.00 1.00 22.0 2.91 1.80 1.00 38.9 360 -0.80 0.80 1.90 76.0 1.10 15000 0.90 1
aii
azi
ar
aei
6 0.80 9.00 0.80 9.00 6.00 1.00 6.00 6.00 1.00 6.00 6.00
2 2.90 7.65 2.90 7.65 8.00 2.30 5.82 9.20 3.10 5.30 9.20
0 0 0 0 0 0 2 2 2 4 4 4
2 2.90 5.10 2.90 5.10 8.00 3.42 5.82 9.20 3.14 5.30 9.20
Theoretical, 2 = 1. With electron self-interaction correaction. The values shown were used to solve the Poisson equation. Doubled values were used to calculate c y , pb, and the field at the nucleus. No correction for selfinteraction. The same values were used to solve the Poisson equation and to calculate the other quantities mentioned in footnote b. means of achieving it. Equation 16 is effective because it seems to reflect the physical reality of the problem, in addition to providing considerable flexibility. The average electric field within the polarized atom is also of interest. In the context of the dielectric model, there are actually two fields of interest (neglecting fields that already exist before the applied field is turned on). The field experienced by an electron consists of the applied field and the field of the other electron, while the field experienced by a small test charge (or by the nucleus) consists of the applied field and the fields of the two electrons. (We will refer to these fields as partial and total, respectively.) Since 4, the solution of the Poisson equation with the self-interaction correction, corresponds to one
0
0
I
2
3
r
4
5
6
Flgure 8. Comparative plots of the polarlrabiilty denslty functions used in the Polsson equatlon solutions. The lower curve for helium (- -) contained the self-interaction correction.
.
electron in unit applied field (E,,, = l),then the partial field is -V4 = El, + Eapp (17) where El, is the field at a point due to the shift of one electron. Similarly, the total field is 2E1,+ Eapp,or E , = -2V4 - Ea,, (18) The value of Ebt along the z axis is plotted in Figure 7 for Ea,, = 1 au. The total field must be zero at the origin if the nucleus is to remain stationary. The small deviation of the calculated result from zero (+0.02au) is probably due to the imperfect fit of the quantum charge increment by our X function. Sittere obtained a nuclear shielding factor, /3 (=-E,t/Eapp)of 1.000544. A three-term fit without the self-interaction correction was then carried out, and the parameters are given in Table I. The total polarizability, CY = 1.303 au, was 1.4% below Sitter’s value. The residual was slightly larger than that of the two-term fit (with the self-interaction correction) shown in Figure 5. The corresponding total field deviated somewhat more strongly near the origin, as may be seen in Figure 7. A t this point, we remark that the hydrogen atom field shown in Figure 1may also be thought of as the total field in the present context; it does not go to zero at the origin because the X function was not adjusted to fit Pb to p’. Neon Atom. The Sitter $ p i values calculated according to eq 13 are shown in Figure 8. The most obvious qualitative difference from helium is the negative dip at r < 1 au. From Figure 2, this effect arises from the 2p, orbital. A three-term fit without the self-interaction correction was carried out by using the same procedures as in the case of helium. The parameters are listed in
Polarizability Densities within Atoms
The Journal of Physical Chemistry, Vol. 87, No. 1.4 I
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-
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w
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Figure 0. Fit of the argon atom quantum charge increment function (-) by a three-term polarizability density function in the Poisson equation (- - -). No correction is made for self-interaction. I
1.2
0
O
I
I
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1
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Ar
I
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6
r Flgure 7. Electric field variation within the rare gas atoms according to the Poisson equation solutions. For helium, the results with selfinteraction mectlOn (-) and without (- - -) are shown. For neon and argon, self-interaction was not included, and the dashed lines simply indicate physically expected behavior near the origin.
I
!
Ne .30-
.2 5 -
I
0
:
0-
I
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I
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5
6
.20I
4-
2
Ar
.15-
.4c I
O
I
2
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3 r
4
5
6
Figure 10. Fit of the argon atom quantum charge increment function by the linearly decreasing polarizability density function in the Poisson equation (- -).
(-)
Figure 8. Fit of the neon atom quantum charge increment function by a three-term polarizability density function in the Poisson equation (- -). No correction is made for self-interaction.
(-)
-
Table I and the function is plotted in Figure 6. The Pb obtained is compared to pl' in Figure 8. The total polarizability of 2.463 au was 3.8% above Sitter's value (2.374). It was not possible to fit the negative contribution by a
-
solution of the Poisson equation, since a large negative dielectric constant would have been required, but could not be used in the Poisson equation. The total field is plotted in Figure 7. The deviation from expected behavior near the nucleus is due to the lower quality of the fit in this region in comparison with helium. The fit does not include the negative polarizability density or the 1s electron contribution. The latter is a negligible part of the total polarizability density but has a larger effect on the field at the nucleus. Argon Atom. The fitting procedure for argon was similar to that for neon, except that the n = 3 shell electrons
1444
J. Phys. Chem. 1083, 8 7 , 1444-1440
dominate the polarizability density. The Sitter pl' values from eq 13 are compared with the three-term fit without the self-interadion correction in Figure 9. The parameters of the polarizability density function are listed in Table I and the function is plotted in Figure 6. The total polarizability of 10.702 au was 0.7% below the Sitter value (10.774). The total field is plotted in Figure 7. In an earlier calculation for the argon atom,3 the polarizabilitydensity was represented by a linearly decreasing function, as illustrated in Figure 10. The Poisson equation prediction of pl' for this function is also shown in Figure 10. The large discrepancy illustrates the sensitivity of pl' to the form of X ( r ) . The linearly decreasing X ( r ) was also used to evaluate the pair polarizability tensor of argont3 with reasonable results. We conclude that the pair polarizability is probably not as sensitive to the form of X ( r ) as the charge increment. (New calculations of the pair polarizability are now being carried out.)
Discussion The preceding calculations illustrate the deduction of atomic polarizability density functions from quantum charge density increments using the Poisson equation form of the continuum dielectric theory. By successive approximations, reasonably close fits of Pb to pl' were obtained. The accuracy of the results is limited by several factors. First, the quantum calculationsgare at the Hartree-Fock level and do not include all effects of electron
correlation. The uncertainty of the quantum results seems to be about 1070, and the deviation of our fit from the quantum results is in the 5-10% range. Second, the classical dielectric formalism does not recognize the nonlocal effects of electronic m ~ m e n t aand , ~ the use of our results with nonuniform applied fields will lead to additional uncertainties. It is interesting to note that the negative polarizability density region of neon could not be fitted by the continuum dielectric model. It also seems impossible to obtain an angular dependence other than Pl(cos 0) from the dielectric theory if the polarizability density is taken as an isotropic function of r. Thus, the Po and P2(cos8) contributions in the quantum results for argon were beyond the scope of the present calculations. The possible effects of anisotropic polarizability density7remain to be explored in this connection. Our results for the polarizability density functions (Figure 6) are more compressed than some of the earlier estimates, suggesting that the polarizability anisotropy of atom pairs will follow the point polarizability interaction model more closely near the potential minimum. At shorter distances typical of bonded atoms, the point polarizability interaction model will remain inadequate, however. Registry No. He, 7440-59-7; Ne, 7440-01-9; Ar,7440-37-1; H, 12385-13-6.
Pressure Dependences of Rates of Nucleophilic Attack by Water In Aqueous Solution. Hydrolyses of Methyl p -Nitrobenzenesulfonate and Ethyl Trlchioroacetate Joseph L. Kurz" and John Y.-W. Lu Department of Chemlstry, Washlngton Unhws@, St. Louls. Mlssourl 63 130 (Received: September 9, 1982)
The pressure dependences of the rates of two formally uncatalyzed reactions of water in aqueous solution have been measured in the range 0-2 kbar at 25 "C. A plot of in k vs. P for the hydrolysis of ethyl trichloroacetate (water-catalyzedwater addition, followed by ethanol elimination) is smoothly and gently curved with AV* = -33 f 3 cm3 mol-l near P = 0. The corresponding plot for the hydrolysis of methyl p-nitrobenzenesulfonate (direct methyl transfer from ArSOC to H20) shows an abrupt change in slope near 0.1 kbar; the slope between 0 and 0.07 kbar corresponds to AV* = -24 f 5 cm3mol-', while that between 0.14 and 2.07 kbar corresponds to AV* = -6.3 f 0.1 cm3mol-'. Comparison to the data of Baliga and Whalley for CH3Brhydrolysis at higher temperatures suggests that the pressure dependences of rates of methyl transfer to water may commonly show such changes in slope and that increasing temperature shifts the slope change to higher pressure and decreases its sharpness.
Introduction
Measurement of the pressure dependence of a reaction rate allows evaluation of the volume of activation (AV') and its pressure dependence. As is evident from a recent review,' few precise measurements of this type exist for reactions of solvent water in dilute aqueous solutions. In fact, for reactions in water and particularly in water-organic cosolvent mixtures, the observed pressure dependences are sufficiently diverse and complex to have prompted an admonition to avoid the use of such solvents? (1) Asano, T.; le Noble, W. J. Chem. Rev. 1978, 78, 407-89. (2) Reference 1, p 441. 0022-3654/03/2087-1444$01 .SO10
However, evidence from kinetic isotope effects and transition-state acidities implies that methyl transfer to water in aqueous solution may not follow the traditional SN2 mechanism, and suggests that a change in solvent configuration may be the principal contributor to the acThus,since different tivation process for such structural configurationsof solvent water are likely to have (3) Kurz, J. L. Acc. Chem. Res. 1972,5, 1-9. (4) Kurz, J. L.; Lee, Y.-N. J. Am. Chem. SOC. 1976, 97, 3841-2. (5) Kurz,J. L.; Lee, J. J. Am. Chem. SOC.1980, 102, 5427-9. (6) Kurz, J. L.; Lee, J.; Rhodes, S. J. Am. Chem. SOC.1981, 103, 7651-3. (7) Treindl, L.; Robertson, R. E.; Sugamori, S. E. Can. J.Chem. 1969, 47. 3397-404.
0 1903 American Chemical Society
