J. Phys. Chem. 1985,89, 301 1-3016
3011
Polarizability Density of Inert Gas Atom Pairs. 1 William H. Orttung Department of Chemistry, University of California, Riverside, California 92521 (Received: November 2, 1984; In Final Form: January 23, 1985)
The polarizabilities of He2, Ne2, and Ar2 were investigated in the context of the continuum dielectric model. For He2 in a parallel field, the polarization and bound charge density functions were compared with analogous ab initio quantum results after integrating over planes perpendicular to the internuclear axis. Good agreement with the quantum functions was obtained for an additive combination of slightly distorted atomic polarizability density functions. The same polarizability density also correctly predicted the perpendicular polarizability of He,. To improve both integration and plotting of widely spaced points, a “continuous curvature” algorithm was developed as an alternative to the commonly used four-point cubic polynomial interpolation between functional points.
Introduction The idea of using the continuum dielectric formalism to evaluate polarizability interactions within a molecule is not new, but progress toward quantitative results has been slow. Mathematical difficulties arising from spatially varying susceptibility and irregular shape have required the development of new methods, and there has been a lack of agreement about the manner in which the formalism can be applied within molecules. These topics have recently been discussed.’s2 The goal of the present work is to discover the simplest atom pair polarizability density function that will reproduce the detailed quantum results via the standard continuum dielectric formalism. Since the dielectric calculations are much easier than quantum calculations, it is hoped that the model may eventually be useful for evaluating interactions in more complex systems. For the present, detailed comparison of the model with quantum results for simple systems continues to provide interesting insights. Oxtoby and Gelbart3 explored distributed polarizability functions for inert gas atoms and considered the polarizabilities of atom pairs in collisions. Oxtoby4 later refined their ideas. Our work on simple atomic systems was stimulated by the qualitative success of their pioneering effort. Kress and Kozak5 did quantum calculations of the He2 and Ne, polarizability tensors for separations between 2.5 and 9.0 au using a large Gaussian basis set. Dacre6 carried out extensive quantum calculations of the polarizability components of a wide range of inert gas atom pairs as a function of internuclear distance. H e obtained self-consistent field Hartree-Fock results and also investigated electron correlation effects. Frommhold’ reviewed both the experimental and interpretive aspects of diatom polarizabilities. In a related area, Bader* discussed the subtleties of quantum charge distributions in diatomic and other molecules. Huntg recently introduced a “nonlocal polarizability density” using the earlier formulation of Maaskant and Oosterhoff.l0 (This (1) Orttung, W. H.; Vosooghi, D. J . Phys. Chem. 1983,87, 1432. (2) Orttung, W. H.; Julien, D. St. J . Phys. Chem. 1983, 87, 1438. (3) (a) Oxtoby, D. W.; Gelbart, W. M. Mol. Phys. 1975, 29, 1569. (b) Mol. Phys. 1975,30,535. The idea of distributed polarizability was suggested by: (c) Frisch, H. L.; McKenna, J. L. Phys. Rev. A . 1965, 139, 68. (d) Thiemer, 0.; Paul, R. J. Chem. Phys. 1%5,42,2508. The idea of representing an atom by a uniform dielectric (or conducting) sphere is very old. See: (e) Bottcher, C. J. F. “The Theory of Electric Polarization”, 2nd ed.; Elsevier: Amsterdam, 1973; Vol. I . (4) (a) Oxtoby, W. J . Chem. Phys. 1978, 69, 1184. (b) J . Chem. Phys. 1980, 72, 5 17 1 . (5) Kress, J. W.; Kozak, J. J. J . Chem. Phys. 1977, 66, 4516. (6) (a) Dacre, P. D.; Mol. Phys. 1978, 36, 541. (b) Can. J. Phys. 1981, 59, 1439. ( c ) Can. J . Phys. 1982, 60, 963. (d) Mol. Phys. 1982, 45, 1, 17. (e) Mol. Phys. 1982, 45, 17. ( f ) Mol. Phys. 1982, 47, 193. (7) Frommhold, L. Adu. Chem. Phys. 1981, 46, 1 . (8) Bader, R. F. W. In “The Force Concept in Chemistry”; Deb, B. M., Van Nostrand Reinhold: New York, 1981; Chapter 2. (9) (a) Hunt, K. L. C. J . Chem. Phys. 1983, 78,6149. (b) J . Chem. Phys. 1984, 80, 393. (10) Maaskant, W. J. A,; Oosterhoff, L. J. Mol. Phys. 1964, 8, 319.
0022-3654/85/2089-3011$01.50/0
quantity differs from the “local” polarizability density of the continuum dielectric formalism. See the Discussion section below.) Finally, the closely related electron gas model (with additive atomic overlap) has been used for both pair potential” and diatom polarizability calculations.12 The finite element method (an extension of the variation method) has been developed to provide accurate solutions of the Poisson partial differential equation of the continuum dielectric f0rma1ism.l~ In the present context, the electrostatic potential can be evaluated for a system with spatially varying susceptibility in an applied field. (The atomic polarizability density is formally identical with the dielectric susceptibility in this model.) If a susceptibility function is assumed, then the electrostatic potential can be calculated. All other quantities of interest can be calculated from the electrostatic potential. These include electric field, polarization (induced dipole density), bound charge density, and integral properties such as overall polarizability. The above approach allowed fitting of the quantum results for isolated He, Ne, and Ar atoms2 by the following procedure: a modified Slater orbital functional form was assumed for the atomic polarizability densities, and the bound charge density (from the dielectric calculation) was fitted to the quantum charge increment (in an applied field) by variation of the parameters in the polarizability density function. The resulting atomic polarizability density functions provide the starting point of the present research.
Methods Dielectric Formalism. A more detailed discussion of the dielectric formalism and its application to atomic interiors has been given.’S2 The Poisson partial differential equation may be solved for the electrostatic potential 6 if the dielectric tensor e and free charge density p are specified. ( p is taken as zero in the present context.) The electric field and polarization are given by
E = -V$
(la)
1 P = -(e
- l).E = X-E 47r where X is the dielectric susceptibility (or polarizability density) tensor. The polarization P is the induced dipole density. The bound charge density (induced by the applied field) is pb = -v’P
(2)
It is formally analogous to the quantum charge increment in a field. Note that pb is a scalar and P is a vector. The overall polarizability of the diatom in a uniform applied field Eiis (1 1) (a) Gordon, R. G.; Kim, Y. S. J . Chem. Phys. 1972, 56, 3122. (b) Kim, Y. S.; Gordon, R. G. J. Chem. Phys. 1974, 60, 1842. (12) (a) Harris, R. A.; Heller, D. F.; Gelbart, W. M. J. Chem. Phys. 1974, 61, 3854. (b) Heller, D. F.; Harris, R. A.; Gelbart, W. M. J . Chem. Phys. 1975,62, 1947. See also ( c ) Cina, J. A.; Harris, R. A. J. Chem. Phys. 1984, 80, 329 and other papers cited therein. (13) Orttung, W. H. Ann. N.Y. Acad. Sei. 1977, 303, 22.
0 1985 American Chemical Society
Orttung
3012 The Journal of Physical Chemistry, Vol. 89, No. 14, 1985 ai=
'sPi Ei
v
du, i = p or s
(3)
where p and s refer to parallel and perpendicular applied fields, and the volume Vis large enough to include all contributions from P,. The important quantities in the following calculations are the global polarizability components, a p and as,and the density functions, P and pb. Finite Element Method. The basic techniques for solving the Poisson equation have been d e ~ c r i b e d . ~The , ~ diatomic calculations for parallel field were done in cylindrical coordinates (z,r&), and were effectively two-dimensional because of the cylindrical symmetry. Perpendicular field calculations required three-dimensional meshes.13 From eq 1 and 2, it can be seen that the polarization and bound charge density require first and second derivatives of the potential. In the earlier atom calculations,2 the bound charge density was not given with adequate precision by the direct approach (differentiating the finite element interpolating functions), and the alternate "potential fit method" was developed and found to give more precise results. An extension of this method to linear molecules is described below. Potential Fit Method. The form chosen for the potential function should be consistent with the molecular symmetry. For a diatomic or linear molecule, the symmetry is C,, for parallel applied field. For perpendicular applied field, the symmetry is C2, if the molecule has a center of symmetry, and C, (or Clh) otherwise. The following form is appropriate in each of these cases:
most of the contribution to the induced dipole arises. The r line results were reasonable but not particularly informative. For these reasons, the "cylinder integral" extension (described in the next section) was adopted and gave appropriate and comprehensible results. The potential fit method was not attempted for perpendicular applied fields because of the greater difficultly of the three-dimensional calculations. Only the overall polarizability was evaluated for perpendicular fields. Cylinder Integral Method. For unit parallel applied fields, integration of the density functions, Pz(z,r) and pb(z,r), over rand 4 yields one-dimensional densities, defined as
PI(])is the induced dipole density and pb(l) is the bound charge density, each per unit length along z. Qb, the total bound charge and the integral of pb(l) over the whole molecule, is zero in general. a p may be calculated by either of the following expressions:
Although the basic finite element calculation gives a better estimate of ap(as the integral of P, over the mesh) than eq 10, these expressions allowed checking of the internal consistency of the calculations. A more useful result is obtained hy considering the analogue of eq 2 for the one-dimensional densities: pb(') =
m
4(z7,4) = C4,(z,r)P,(cos 4)
(4)
n=O
where P,(cos 4) is a Legendre polynomial. Only the n = 0 term (Po = 1) occurs for parallel applied field, and only the n = 1 term is likely to be important for perpendicular applied field (at least for the lighter atoms2). If the polarizability density tensor X(z,r) is diagonal in a local z,r,4 coordinate system, then we may speak of the principal values, X,, X,, and X,. For cylindrical coordinates E, = -a4/az, E, = -a+/ar, E , = -(l/r) ad/a4 (5) P, = XzE,, P, = X,E,,
(6)
P, = X,E,
A straightforward derivation substituting eq 4 and 5 into eq 1 and 2 yields a general result for P and P b in an applied field. For parallel applied field Pb = ax,/az a+,/az
+ x, aZ+,/az2 +
(X,/r
+ axr/ar)
a40/ar + x,az40/ar2 (7)
where the third and fourth terms reduce to 2X, d2+o/arz along the z axis (r = 0). For perpendicular applied field, the n = 1 (cos 4) term gives Pb = [ax,/az a41/az
ax,/ar) a + , / a r
+ xza241/azz + (X,/r + + X, a2&/ar2 - (X,/rz)
COS
4 (8)
where the third and fourth terms reduce to 2Xr a241/dr2when r = 0. As may be seen from eq 5-8, the functions b0(z,r) and 41(z,r) are needed to evaluate Pb and components of P. Since the algebraic form of and d1is undoubtedly complex, no attempt was made to obtain the global functional forms. Instead, values of 4 along z or r at the three nodes nearest to the point of interest were used to estimate the first and second derivatives from a quadratic fit. In the following, the potential fit method is applied only to the case of a parallel applied field. The mesh must be rectangular along the z or r line for which values are to be calculated. The mesh must also be fine-grained because of the strong inhomogeneity of the atomic X functions. Exploratory calculations with suitable meshes were carried out to obtain p b along the axis ( r = 0) and also along r lines for particular z values. The axis results were not particularly useful because of the strong variations encountered near the nuclei and because the axis is not representative of the outer regions, where
-dP,(')/dz
(11)
Equation 11 integrates directly (in contrast to eq 2) to yield
where PZ(l)is zero at large z. Equation 12 is important in the present considerations because it allows the quantum charge increment to be converted to a polarization (or charge shift), which is more easily compared with the dielectric model calculations. In summary, the cylinder integral method gives smoothly varying one-dimensional densities incorporating the contributions of all distances from the axis and allows comparison of either polarization or bound charge density with the quantum calculations. Integration Methods. The functions to be integrated were usually in the form of lists of unevenly spaced values from a finite element calculation (e.g., nodal values along a z or r direction). The independent variable (z or r) was often relatively widely spaced in terms of the functional variability. Initially, four consecutive points were fitted to a cubic equation, and the integral between the middle two points was evaluated a1gebrai~ally.l~The same algorithm was also used in our curve plotting programs. Although this algorithm is widely used, and usually works well if the points are close relative to function variations, considerably difficulty was experienced with oscillation of the fit. The errors in the integrals were in the 10% range. One solution to the problem would have been to use more refined finite element meshes, but this would not have been cost effective (particularly for future calculations on larger molecules). The problem was solved by the introduction of a simpler and more effective interpolation algorithm. For four consecutive points, the second derivatives at the second and third points can be estimated from points 1-3 and 2-4 by using second differences (or quadratic fits). If the function between points 2 and 3 is still to be represented by a cubic equation of the form y =
c, + c1, + c2x2+ c3x3
then JJ"
= 2C2
+ 6C3,
so that Cz and C, can be evaluated from the values of y" at points (14) Bevington, P. R. "Data Reduction and Error Analysis for the Physical Sciences"; McGraw-Hill: New York, 1969; p 271.
The Journal of Physical Chemistry, Vol. 89, No. 14, 1985 3013
Inert Gas Atom Pairs TABLE I: Parallel and Perpendicular Polarizabilities for Inert Gas Diatoms Assuming Additive Overlap of Unperturbed Atomic Polarization Densities in the Continuum Dielectric Formalism”
Ob
0.9 1.8 2.4 3.0 3.6 4.243 5.1 6.0 7.5 .Dc
2.234 2.484 2.753 2.785 2.167 2.728 2.695 2.663 2.643 2.627 2.6 12
2.302 2.415 2.520 2.571 2.587 2.596 2.612
4.093 4.470 5.139 5.377 5.402 5.336 5.233 5.124 5.054 5.000 4.940
4.193 4.403 4.651 4.797 4.851 4.882 4.940
16.08 17.66 21.27 23.83 25.44 25.99 25.64 24.62 23.65 22.61 21.46
.I
R
16.45 17.29 18.54 19.60 20.13 20.53 20.94 21.46
“All values are in atomic units (0.529 17 A/au). bCoalesced atom pair. CDoublethe isolated atom values, which were given by meshes as similar as possible to those used for the diatom calculations. Atom values obtained from refined meshes were 1.303, 2.463, and 10.70 au, respectively, for He, Ne, and Ar.Z
2 and 3. Equation 13 can then be used with the function values a t points 2 and 3 to evaluate C, and C , . For the first and last intervals of the overall range, the y” of the middle interval is extrapolated. The physical meaning of this procedure is that the second derivative of the fit varies linearly from one point to the next and is continuous at each point. The slope is not required to be continuous at the points, but this freedom has not caused problems. The calculations are simpler than for the older algorithm because two pairs of equations are easier to solve than one set of four. The algorithm is probably more effective because it weights the middle two points more heavily and seems to suppress oscillations. We are not aware of any discussion of this “continuous curvature” algorithm in the 1 i t e r a t ~ r e . I ~ Simple tests were carried out with the function, xe-x, which is similar to the functions of interest. The errors were reduced to the 1% range for the spacing of points dictated by our meshes. In addition, all of the curves in the figures of this paper were computer drawn by using this algorithm for interpolation. The results are generally very close to what would be drawn by hand. Quantum Programs. Ab initio Hartree-Fock self-consistent field calculations were carried out on a VAX 111750 computer using the HONDO program system.I6 The programs were modified to allow applied electric fields (by recognizing that the field perturbation terms of the Hamiltonian combine with the nuclear attraction terms and give one-electron integrals that are only slightly different from the overlap integrals). The charge densities were evaluated from the HONDO orbital coefficients by a Fortran430 microcomputer program. The cylinder integrals and the overall integral of the moment of the density were routinely evaluated. In addition, the cylinder integrals were integrated by eq 12 to give a quantum estimate of the distribution of polarization in the parallel field.
Calculations and Results Global Polarizabilities for Additive Overlap of Unperturbed Atoms. The atomic polarizability density functions2 were used additively without modification in a preliminary calculation of the overall diatomic polarizability components, apand a,. The Poisson equation was solved by the finite element method, using (15) Bowyer, A.; Woodwark, J. “A Programmer’s Geometry”; Butterworths: London, 1983. (16) (a) Dupuis, M.; Rys, J.; King. H. F. J. Chem. Phys. 1976,65, 11 1. (b) Dupuis, M.; Wendoloski, J. J.; Spangler, D. Nor/.Resour. Comput. Chem. SofZwure Cur. 1980, 1, QGO1. (c) HONDO 0 Version 1.02, Rev 1, 10/80 QGOl.2, from Wendoloski, J. J. Du Pont Central Research Laboratory, Wilmington, Delaware. (17) Diercksen, G. H. F.; Sadlej, A. J. Chem. Phys. 1983, 77, 429. (18) Beck, D. R.;Nicolaides, C. A. Chem. Phys. Lert. 1977, 49, 357. (19) Dalgarno, A.; Kingston, A. E. Proc. R. SOC.London, Ser. A 1960, 259, 424.
-
I
I
I
I
2.00
0.00
.,
l
I
I
I
e. 00
6. Od
4.00 R (AU)
Figure 1. Parallel and perpendicular polarizability increments relative to infinite separation, AaPand Ass, for Hez. (The upper curve of each
pair is the parallel component.) -, dielectric calculatiod with additive overlap of unperturbed atomic polarizability densities (0, 2-dimensional calculation;0,3-dimensional calculation. - - -,quantum calculation: A, Kress and K o ~ a k V, ; ~ Dacre;6cX, present calculation. ---, point polarizability interaction model. I
I
I
0.00
I
,
I/
I
2. 00
\
I
I
I
I
4. 00 R (AU)
I
I
6. 00
I
I
I
I
8. 00
Figure 2. As in Figure 1, but for the neon atom pair.
meshes of the type developed earlier.13 apand a, were obtained as the integral of the polarization vector over all contributing elements near the atoms for unit applied field. The apvalues were obtained with two- and three-dimensional meshes, and the asvalues were obtained from three-dimensional meshes. The two-dimensional calculations were done on IBM 3701 155, VAX 1 11750, and Northstar Horizon computers, with essentially perfect agreement. All of the three-dimensional calculations were done on the VAX 111750. Although the meshed3 were relatively crude compared to those used to obtain the atomic polarizability density functions,2 the errors in the overall polarizability components appear to be only a few tenths of a percent (because local inadequacies tend to average out in integral results). Results for He,, Ne,, and Arz are presented in Table I and plotted in Figures 1-3. The figures show the increment relative to a pair of isolated atoms. Zero separation corresponds to coalesced atoms: Hez becomes Be, Ne, becomes Ca, and Ar,
3014
Orttung
The Journal of Physical Chemistry, Vol. 89, No. 14, 1985 TABLE I11 Helium Atom Gaussian Basis Set“ shell or contraction type exponents
S
1-7
192.4388, 28.95149, 6.633653, 1.879204, 0.589851, 0.193849, 0.131 Pb 0.585, 0.206;0.082 DC 0.27
8-10 11
Adapted from HuzinagaZ2and ponent. ‘Six functions per exponent.
*Three functions per ex-
TABLE I V Results of HONDO Self-consistent Field Quantum Calculations on the Helium Atom and Helium Atom PaiP
basis
0
uiI
I
I
2. 00
0.00
I
1
I
4.00
8.00
6.00
R (AU)
Figure 3. As in Figure 1, but for the argon atom pair. The krypton coalesced atom value is also shown ( 0 ) at R = 0.
6s (GTO) 7s,8p,5d (STO) 5s (STO) 5s (STO-6G), lp,ld (STO-4G) 7s (GTO) 7s,3p,ld (GTO) 7s,3p,ld (GTO)
fieldb total energy Helium Atom n 2.861 116391 P n 2.8616199 n 2.86103578 n n p
polarizability component
1.3252
c
c 1.313
quantity
7s,3p,ld (GTO)
He? Ne, Polarizability Increments CAP 43 (Bey 166 (Ca)d CSC, Aa,/Aa/ 3.67/-0.26 6.94/-0.49 DID at Rcsc, A a p / A a , 0.341-0.11 0.32/-0.11
-5.43 (Kr)‘ 30.14/-2.11 0.34/-0.11
Internuclear Distances RDID(a,=m) = (2a0)’/3 1.4 1.7 Rcsc = 2 ( 4 ” 3 2.2 2.7 lJg 4.8 5.3 r0g 5.4 6.0
2.8 4.4 6.3 7.1
CSR = a0’/’ CSR of CAP =
CY,,^'/^
Radii 1.1 (0.58) 3.6 (1.91)
1.35 (0.71) 5.5 (2.91)
C
Helium Atom Pair ( R = 3 au) 5s (STO-6G), n 5.708527 13 lp,ld (STO-4G)
6a 2.4678 2.6299
S
TABLE 11: Miscellaneous Values of Interest in Connection with Figures 1-3”gb
22 23 24 6a
1.322
2.861 117819 2.861 123 331 2.861 123987
Ar,
ref
P n s p
5.708 684054 5.708685278 5.708685356
C
2.447 2.603
A field strength of 0.001 au was a All values are in atomic units. used; n, s, and p stand for no field, perpendicular field, and parallel field. cPresent work.
-
ID 0 I
I
I
I
I
I
HE2, 3 AU.
Ep
J
2.2 (1.16) 2.6 (1.38)
“See the text for an explanation of the quantities shown. All values are in atomic units except that those in parentheses are in A. bThe following abbreviations are used in this table: DID, point polarizability interaction model; CAP, coalesced atom pair; CSC, conducting spheres in contact; CSR, conducting sphere radius. cQuantum calculation: ab = 45.7 au.” dQuantum calculation: aCa= 169 au.18 eExperimental.19 fconducting spheres in contact.,O g r o and u are the distances for the minimum in the van der Waals interaction and for zero potential energy.2’
becomes Kr. Polarizability values for the coalesced atoms are shown in Table 11. The predictions of the point polarizability interaction model shown in Figures 1-3 are given by cyp
- 2ao = 2 a o ( 2 a o / R 3 ) /1( - ~ L Y ~ / R ~(15) )
LY,-
2a0 = 2aO(-aO/R3) /( 1
+ a0/R3)
(16)
3 Equation 15 becomes infinite when R is reduced to ( 2 a 0 ) 1 /and eq 16 becomes -2a0 at R = 0. If the atoms are thought of as conducting spheres of radius r (a = s), then the spheres overlap if R (=2r) is less than 2 0 1 ~ ’ / (See ~ . Table I1 for numerical values of these quantities.) The Poisson equation was solved analytically for conducting spheres down to the contact distance.20 The results of Hartree-Fock self-consistent field quantum calculation^^^^ in Figures 1-3 show that additive overlap of the (20) Levine, H. B.; McQuarrie, D. A. J . Chem. Phys. 1968, 49, 4181. (21) Hirschfelder, J. 0.;Curtiss, C. F.; Bird, R. B. ‘Molecular Theory of Gases and Liquids”; Wiley: New York, 1954.
8 8 I. 0.00
2. 00
4.00
6. 00
2 (AU)
Figure 4. 2-moment of the r,&cylinder integral of the He, charge increment. The atom separation is 3 au in a parallel applied field of 1 au. (The center of the pair is at the origin, and only the +z half is shown.) 0,-, dielectric calculation with additive overlap of unperturbed atomic polarizability densities; - -,quantum calculation.
-
unperturbed polarizability density functions (in the dielectric model) is a poor assumption for separations less than u (the distance of zero potential energy), especially for the parallel field. Density Functionsfor Additive Overlap of Unperturbed Atoms. To reveal the local origins of the above discrepancy with the overall quantum results, the Gaussian basis set of Table 111 was used in calculations on He and He, with an without an applied field. The 6 GTO (Gaussian type orbital) basis of Huzinaga22was supple~
~
~~~
(22) Huzinaga, S. J . Chem. Phys. 1965, 42, 1293
Inert Gas Atom Pairs
The Journal of Physical Chemistry, Vol. 89, No. 14, 1985 3015
0
m
I
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I
I
TABLE V Distortion Coefficients of the He, Atomic Polarizability Density Functions Giving Best Agreement of the Dielectric Calculation with Quantum Results within the Framework of Eq 2 0 " ~ ~
I
eq 20
dielectric
quantum
R
c1
c2
c3
ap
as
ffp
ffs
2.5 3.0 3.5 4.0
1.01 0.99 1.00 1.00
-0.26 -0.16 -0.10 -0.05
-0.07 -0.05 -0.03 -0.01
2.636 2.605 2.646 2.673
2.404 2.444 2.499 2.551
2.622 2.603 2.637 2.663
2.340 2.447 2.517 2.557
"All values are in atomic units. 'Using the Gaussian basis set of Table 111.
0 I
0
0.00
I
2.00
Z
(AU)
rri
I
I
mented by a seventh s-type function, three p-type functions, and one d-type function. The supplementary functions were suggested by Dacre's calculation.6a A field strength of 0,001 au was used in all calculations. Overall numerical results are shown in Table IV. The parallel and perpendicular polarizabilities were within 1% of Dacre's S C F results," and the density funct'ion increments in the field were found to be insensitive to additional changes in the basis for all present purposes. A comparison of the quantum and dielectric model results is shown in Figures 4 and 5 for 3 au separation of the atoms. It is clear that the polarizability density is too high between the atoms, but only slightly low on the outer side of the atom. (This conclusion is more easily reached from Figure 5 than from Figure 4.) Analogous plots for an isolated atom on the same scale as Figure 5 do not shqw noticeable differences between the quantum and dielectric results. Correction for Pair Perturbations. It is useful to consider the simplest quantum description of interacting atoms to see what kind of modification of the polarizability density might be effective. The simplest bonding molecular orbital for a pair of electrons, each originally in an orbital of the bonding atoms, is
# = $a + f i b
1
I
Figure 5. As in Figure 4, but for the cylinder integral of the polarization vector rather than the charge increment.
HE2.
I
I
I
I
3 AU. PART A
I
HEz.
I
3 AU. PART B
t
(17)
if normalization factors are neglected. The electron density will have the form
By analogy, this result suggests the form
x = cl[xa + x b + cZ(x&b)l/z]
(19)
for the polarizability density of an electron pair bond such as occurs in Hz.(A similar result is obtained if the valence bond orbital is considered.) On the other hand, if IC/, and fib eachcontain two electrons (as in Hez), the two electron pairs are assigned to the plus and minus (bonding and antibonding) combinations in eq 17, and the cross term of eq 18 cancels in the density of four electrons in the simplest approximation. Thus eq 19 would not be appropriate for a nonbonded atom pair such as He2. Consideration of Figure 5 suggested that a simple distortion of the atomic functions within the additive approximation should reduce the difference from the quantum results. 'The following function was then explored: L
X(z,r) = C X i [ C l i= 1
+ (-1)'C2zi + C,z?]
where atom 1 is a t z = +R/2 and atom 2 is at z = -R/2 and X i is the polarizability density of the independent atom. zi is measured from the center of atom i to the z value of the function point. The fit for R = 3 au was obtained by successive approximation, Le., by adjustment of C1, C,,and C3of eq 20 in dielectric model (23) Sitter, Jr., R. E. Thesis, State University of New York at Buffalo, Buffalo, New York, 1969; University Microfilms, Ann Arbor, MI, 70-17366. (24) Clementi, E.; Roetti, C. At. Data Nucl. Data Tables 1974, 14, 177.
0.00
2.00
4.
00
Z (AU)
Figure 6. Polarizability density function for He2 at R = 3 au. (a) The additive combination of unperturbed atomic functions: -, r = 0., - - - , r = 0.5; and ---, r = 1.0 au. (b) Difference of the additive combination of distorted atomic functions (eq 20) from the curves of Figure 6a, the line definitions are the same as in part a. The center of the molecule is at z = 0.
calculations. For the numerical values of the C,shown in Table V (for a range of R), the deviation of the dielectric calculation from the quantum was no longer visible on the scale of Figure 5 . The overall polarizability component values are also shown in Table V for the dielectric and quantum calculations. The changes in the polarizability density function indicated in Table V are a rather small fraction of the total function. The additive unperturbed function is shown in Figure 6a for R = 3 au, and the difference of the additive distorted pair function from it is shown in Figure 6b. Although the largest values are along the axis, it should be remembered that the smaller off-axis values make a greater contribution to the overall property. (The cylinder integral is not appropriate here because X alone does not make an additive contribution to any overall molecular property.) Figure 7 shows the difference of the quantum charge density of He2 and two isolated helium atoms. The charge density along z (r = 0) is compared to the cylinder integral (over r, 4 at each z ) . The difference again suggests the importance of the off-axis
Orttung I
I
I
0 O
I
I
HE^, an. PART A
G-
0 f
I:
0 I
I
0. 00
I
I
2. 00
1
4.00
I
I 6. 00
Z (AU)
Figure 7. Quantum charge density of He2 at R = 3 au. -, cylinder integral (over r, 6 at each z); ---,along z ( r = 0). (a) The function for the basis of Table 11; and (b) the difference of Hez from 2He (isolated atoms). The center of the molecule is at z = 1.5 au.
contribution. It is also of interest to compare Figures 6 and 7 (after shifting origins). The change of the polarizability density function (in going from isolated atoms to the atom pair is clearly quite different from the change in the charge distribution, although both changes are small.
Discussion The data presented in Figures 1-3 and Table I1 reveal a number of interesting properties of inert gas atom pairs. Within the dielectric model, additive overlap of unperturbed atom polarizability densities predicts too large an anisotropy as the atoms first come within the van der Waals minimum, and too small an anisotropy as the overlap of the electron clouds becomes more important. The anisotropy for additive overlap goes through a maximum at about the separation corresponding to the equivalent conducting spheres in contact. Perhaps the most striking aspect of Figures 1-3 is the steep rise of the quantum Aa, as the separation becomes smaller than that of conducting spheres in contact. For He, and Nez, the polarizabilities of the corresponding united atoms (Be and Ca) are very large (Table 11). For Ar,, the polarizability of Kr is less than that of two Ar atoms, but Figure 3 shows that Acu, must be very large while the atoms are approaching the united state. (The agreement of the experimental krypton point with our dielectric calculation for 2XA, (at R = 0) is interesting, but possibly fortuitous.) A qualitative reason for the very large polarizabilities at short separations may be seen from a consideration of the united-toseparat‘ed atom correlation diagram. For the simplest case of He,,
the a,ls and u,*ls orbitals are occupied in the separated atoms, and the lsa, and 2su, orbitals are occupied in the united atom (Be). In the absence of a field, the a,*ls and a,2s orbitals cross without mixing as the atoms converge, but the mixing in a field when the orbitals are of similar energy could explain a large polarizability on the way to convergence. After convergence, the looser binding and larger radius of the 2su, electrons are undoubtedly responsible for the large polarizability of Be. Qualitatively similar phenomena undoubtedly occur with the conversion of Ne, to Ca and Ar, to Kr. The point polarizability interaction model gives cup values that are too large at longer distances and too small values at the distance of conducting spheres in contact. (See Table 11, lines 2 and 3.) Since a conducting sphere (having infinite susceptibility) is the most polarizable of spherical systems, its radius (for a given polarizability) must represent the minimum sise of the system. Thus, it seems that the point polarizability interaction model lacks a physical justification for distances closer than donducting spheres in contact. This aspect of the point polarizability interaction model is of greater concern for bonded atoms than for the nonbonded atom pairs considered here. It is interesting that the dielectric model can be made to agree with the quantum results by a slight distortion of the atomic densities and without the need for nonadditivity or local anisotropy of the density function. (This result for nonbonded atoms may not carry over to bonded systems, which are currently being investigated.) It should be mentioned that expanded difference plots of the results presented here do show systematic deviations of the dielectric model from the quantum results. It is possible that better fits might be obtained by a small scaling of the atom functions. (This question is still under investigation.) The distortion coefficients (eq 20, Table V) seem to vary smoothly with separation. It would be of theoretical interest to explore the quantum calculations between R = 0 and 3 au. None of the models discussed here seem able to easily explain the large increases in the quantum cup at very short separations. It is of interest to compare Hunt’s “nonlocal” polarizability density, u(r,r’): with the “local” polarizability density X(r) of the continuum dielectric theory. Following Maaskant and Oosterhoff,’O Hunt’s function is defined by their eq 2.27:
P(r) =
s
u(r,r’)-e(r’) dr’
mol
(21)
where e(r’) is the field within the molecule arising from sources external to the molecule.. In the standard form of the continuum dielectric theory, the above equation takes the form
P(r) = X(r).e(r)
+ X ( r ) - sVT ( r - r’)-X(r’)-[e(r’) + e’(r’)] dr’
where T(r) is the dipole propagator tensor, (3rr - r21)/r*, and e’(r’) is the field at r’ arising from sources in other parts of the same molecule. Equation 22 is consistent with solutions of the Poisson equation. Huntgbhas discussed the relationship between eq 21 and 22. It should be noted that the present work deals only with the linear response to the applied field and that consideration of hyperpolarizability effects awaits further development of the theory. Acknowledgment. I a m indebted to Dr. Brian M. Pierce (Hughes Aircraft Co., Long Beach, CA) for providing access to the HONDO programs.’6c The research was supported by grants from the Committee on Research and from the Academic Computing Center of the University of California, Riverside. Registry No. He,, 12184-98-4; Ne,, 12185-05-6; Ar,, 12595-59-4.
