7301
J. Phys. Chem. 1992,96,7301-7307
potentials to be included in intermolecular interactions studies, the ultimate goal toward which this model is directed. The usefulness of this approximate p(r) and their derived p e tentials in local density type mol& calculations will be analyzed in the near future. In this context, the inclusion of the model into efficient techniques for the decomposition of intermolecular interaction energies is one of the immediate problems to be treated. Related to this application, the estimation of intra- and intermolecular correlation effects by means of any suitable local density existing approximation with our analytic model p(r) will be also explored. This could permit a fast and easy computation of approximate correlation energies without basis set dependency problems.
Acknowledgment. Financial support from the Spanish Direccion General de Investigacion Cientfica y Tecnica (DGICYT), under Grant No. PS89-0025, is gratefully acknowledged.
References and Notes (1) Weinstein, H.; Politzer, P.; Srebenik, S.Theor. Chim. Acta 1975,38, 159.
Politzer, P.; Parr, R. G. J. Chem. Phys. 1976, 64, 4634. (a) Schmider, H.; Sagar, R. P.; Smith, V. H. J . Chem. Phys. 1991,94, (b) Kohout, M.; Savin, A.; Preuss, H. J. Chem. Phys. 1991,95,1928. Pacios, L. F. J . Phys. Chem. 1991, 95, 10653. (a) Politzer, P., Truhlar, D. G., Eds.; Chemical Applications of Atomic and Molecular Electrostatic Potentials; Plenum: New York, 1981, (b) Buckingham, A. D.; Fowler, P. W.; Huston, J. M. Chem. Rev. 1988,88,963. (6) Sokalski,W. A.; Sneddon, S. F. J. Mol. Graphics 1991, 9, 74. (7) (a) Talman, J. D.; Shadwick, W. F. Phys. Rev. A 1976, 14, 36. (b) Talman, J. D.Compur. Phys. Commun. 1989, 54, 85. (8) Politzer, P. J. Chem. Phys. 1980, 72, 3027. (9) Chandler, J. P. QCPE 1976, 13, 307. This routine is based on the SIMPLEX optimization procedure: Nelder, J. A.; Mead, R. Compur. J. 1965, (2) (3) 8627. (4) (5)
7, 308. (10) Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; Vetterling, W. T. Numerical Recipes. The Art of Scientific Computing Cambridge University Press: Cambridge, UK, 1989; Chapter 3. (11) Boyd, R. J. J. Chem. Phys. 1977, 66, 356. (12) Steiner, E. J . Chem. Phys. 1963, 39, 2365. (13) Sham, L. J.; Kohn, W. Phys. Reu. 1966, 145, 561. (14) Slater, J. C. The Self-Consistent Field for Molecules and Solids; McGraw Hill: New York, 1974; Vol. 4. (15) Nagy, A. Inr. J. Quantum Chem. 1987.31, 269. (16) Press, W. H.; et al. in ref 10, Chapter 4. (17) Wolfram, S.J. Mathemcrtica. A System for Doing Mathematics by Computer, 2nd ed.; Addison-Wesley: Redwood City, CA, 1991.
Anisotropic Molecular Polarizabilities, Dipole Moments, and Quadrupole Moments of (CH2)2X, (CH9)2X, and C4H4X(X = 0, S, Se). Comparison of Experimental Results and ab Initio Calculations Michael H. Coonan, Ian E. Craven, Mark R. Hesling, Geoffrey L. D. Ritchie,*,' and Mark A. Spackman Department of Chemistry, University of New England, Armidale, New South Wales 2351, Australia (Received: March 11, 1992)
Measurements of the temperature dependence of the Cotton-Mouton effects, together with the Rayleigh depolarization ratios and mean polarizabilities,of species in the series (CH2)2X,(CH3)2X, and C4H4X (X = 0, S) are reported and analyzed to yield optical-frequency (632.8 nm) principal polarizabilities of the relevant molecules. In addition, the previously developed 6-31G (+sd+sp) basis set is used to obtain SCF and MP2-level mean and anisotropic polarizabilities and also dipole and quadrupole moments for all nine molecules in the series (CH2)2X,(CHJ2X, and C4HJ (X = 0,S,Se). The results elucidate the electronic charge distributions in these molecules.
Introduction The determination, by experimental and theoretical methods, of accurate molecular polarizabilities and multipole moments continues to be of interest, not only because of the intrinsic value of these properties but also because their ab initio calculation is an important test of the methods of computational quantum chemistry. However, relatively few studies, especially interactive experimental and theoretical studies, have been reported of extended series of molecules, from which useful structural generalizations can be drawn. In this investigation, measurements of the temperature dependence of the Cotton-Mouton effect (magnetic-field-induced birefringen~e)~-~ were combined with Rayleigh depolarization ratios and mean polarizabilities to derive the optical-frequency prinoipal polarizabilities of molecules in the series (CH2)2X,(CH3)2X,and C4H4Xwith X = 0 and S. The results were complemented and extended by SCF and MP2-level calculations, with the previously developed 6-3 l G (+sd+sp) basis set? of the mean and anisotropicoptical-frequency polarizabilities, and also the electric dipole and quadrupole moments, of all nine species in the series (CH2)2X,(CH3)zX,and C4H4Xwith X = 0, S,and Se (Le., oxirane, thiirane selenirane; dimethyl ether, sulfide, and selenide; furan, thiophene, and selenophene). Analysis of the data elucidates the electronic charge distributions in these molecules. 0022-365419212096-7301$03.00/0
Theory
The low-density molar Cotton-Mouton constant, ,C, of a diamagnetic species is, in SI units6 ,C = 2nV,&3(n2
+ 2)2]-1[(nll- nl)K2]g=o
(1)
= ( N A ~ 0 2 / 2 7 0 ~ o )+l A(W-' ~ [ 4 x X X- X) + olyy(xJy-
x) + ~ z z ~ x -XI11 z z
(2)
where eq 1 is a measure of the refractive index difference, nilnL, induced in the gas by the magnetic induction B, and eq 2 is the theoretical relationship3between the observed birefringence and fundamental molecular properties. In eq 2, Aq (=qafl,afl 1 /3qee,flfl) is the anisotropy in the magnetic hyperpolarizability; am,aw,a,,and xu, xyy,xzare, for the molecules of interest here, the principal elements of the optical-frequency polarizability and the magnetizability, respectively; and x (=1/3xea)is the mean magnetizability. Obviously, eq 2 has the form ,C = P + Q T l (3) so that the Cotton-Mouton constant of an anisotropic molecule should exhibit a linear dependence on the reciprocal of the temperature. If the effect is measured over a range of temperature such that a plot of ,C against T'can be reliably extrapolated to T I = 0, the intercept P, given by 0 1992 American Chemical Society
Coonan et al.
7302 The Journal of Physical Chemistry, Vol. 96, No. 18, 1992 (4)
determines AT, and the slope Q,given by
Q=
( N ~ ~ 2 / 2 7 0 c o k ) [ ~ , ( ~ x x+- ~a )y y ( ~ y y - ~+) d
no. of x z z
- x)I (5)
provides one equation in the three principal polarizabilities. It remains to recall that the Rayleigb depolarization ratio, po = G/c, and the isotropic refractive index, n, are expressible in terms of the polarizabilities as' Spo(3
- 4po)-'
= K2 = [(a, - ayy)2 + (ayy
-
+ ( a r z - axx)*1/18a2 (6)
and
(2VIndN*)(n -1) = a =Xb
x x
+ ayy + azz)
(7)
so that simultaneous solution of eqs 5-7, in conjunction with the known magnetizabilities, yields all three components of the anisotropic polarizability. ExperimentalSection
Samples used in the experimental work were as follows: oxirane (Matheson, >99.7%) and dimethyl ether (Matheson, -99.9%), used without further treatment; thiirane (Aldrich, 98% stabilized with 0.5% n-butyl sulfide) and dimethyl sulfide (Merck, 199%), purified by fractional distillation under reduced pressure; furan (Merck, -98%), fractionally distilled from phosphorus pentoxide; and thiophene (Merck, =99%), refluxed with and fractionally distilled from sodium. The purities of oxirane, dimethyl ether, and dimethyl sulfide were confmed by gas chromatographicand mass spectrometric analyses; and, as a general procedure, the compounds that are liquids (thiirane, dimethyl sulfide, furan, and thiophene) were subjected to freeze-pumpthaw-distill cycles in the vapor-handling system h e d i a t e l y prior to the measurements. Unfortunately, the three selenium compounds were not examined: as noted below, selenirane has not yet been isolated, and dimethyl selenide and selenophene are less than attractive subjects for experimental work. Apparatus and procedures for observations of the magneticfield-induced birefXngences (Cotton-Mouton effects) of the vapors were as previously described? For five of the six oxygen and sulfur compounds measurements over ranges of pressure and temperatun were possible, and the temperature dependences were adequately determined. However, thiirane decomposed or polymerized when heated only slightly above ambient temperature, and the investigation of this species had to be abandoned. Recorded density second virial coefficients9 were used to calculate the number densities from the measured pressures; Cotton-Mouton second virial coefficients were not detectable under the conditions of thesc experiments. The results are summarized in Table I, where the uncertainities shown are based on the standard deviations derived from the least-squares straight lines. Previous studies of these molecules, and their selenium analogues, are limited to measurements of the solution-phase Cotton-Mouton constants of furan, thiophene and selenophene dissolved in cyclohexane.1° Equipment developed in these laboratories for measurements, at temperatures up to -90 OC, of the vapor-phase Rayleigh depolarization ratios, po, of relatively involatile and perhaps weekly anisotropic species has also been The high-temperature capability made it possible to maintain much higher vapor pressures in the cell and, even in the presence of an optical filter to remove extraneous vibrational Raman contributions, adequately measurable intensities were achieved in the depolarizedcomponent of the scattered light. The observations, summarized in Table 11, were confined to those compounds, other than the selenium compounds, for which depolarization ratios at 632.8 nm have not previously been reported. For three of the four species examined, ambient temperatures (-20-25 "C) provided sufficient vapor pressures; for thiophene (bp 84 "C), an oven temperature of 77 OC and pressures of -15-20 kPa (=100-150 mmHg) were used.
T/K
press.
447.6 420.6 404.6 379.2 363.6 351.9 327.0 326.8 325.2 307.3 306.2 302.3 302.1 293.6 292.7
10
13 10 13 17 9 8 14 8 9 9 15 18 14 7
maX dkPI Oxirane 101 110
100 117 108 99 116 117 109 120 109 136 129 118 100
1@1,C/m5A-2
mol-' 0.390 f 0.023 0.495 f 0.049 0.457 f 0.061 0.615 f 0.033 0.685 f 0.037 0.566 f 0.087 0.602 f 0.050 0.616 f 0.047 0.722f 0.041 0.830 f 0.048 0.715 f 0.043 0.796 f 0.066 0.776f 0.047 0.901 f 0.071 0.755 f 0.041
414.5 395.3 381.5 362.9 340.2 328.5 316.5 303.2 298.5 294.9
Dimethyl Ether 300 299 300 299 11 399 12 399 14 488 14 446 12 426 14 445
-0.553 0.021 -0.567 f 0.034 -0.633 f 0.020 -0.740 f 0.028 -0.724 f 0.014 -0.785 f 0.027 -0.768 f 0.012 -0.808 f 0.023 -0,819f 0.026 -0.829 f 0.010
443.9 424.1 414.2 404.4 387.9 383.6 363.1 341 .O 333.6 3 18.2 317.2 301.2 293.2
16 13 19 9 15 11 15 10 7 7 7 12 7
Dimethyl Sulfide 114 117 116 106 118 98 113 101 98 40 93 43 34
-0,795f 0.034 -0.857 f 0.034 -0.892 0.035 -0.994 f 0.042 -0.977 f 0.049 -0.973 f 0.037 -1.099 i 0.039 -1.164 f 0.031 -1.179 f 0.048 -1.234 f 0.083 -1.217 f 0.042 -1.350f 0.079 -1.396 f 0.126
445.0 444.7 419.4 384.7 351.5 326.9 307.1 294.6
6 4 5 5 5 6 5 6
446.0 408.5 384.3 373.1 350.6 325.8 315.3 294.1
6 6 7 6 5 7 6 7
9 8 9 8
Furan 55 55 42 55 55 55
53 55 Thiophene 23.1 24.3 25.8 20.2 22.3 20.4 7.7 6.8
*
10.09 f 0.07 10.00 f 0.02 10.55 f 0.14 11.68 0.07 12.52 f 0.04 13.55 f 0.06 14.44 f 0.12 14.89 f 0.04
*
18.9 f 0.2 20.6 f 0.2 22.3 f 0.3 23.0 i0.2 23.6 f 0.2 26.0 f 0.2 26.4 0.5 28.1 f 0.4
TABLE 11: Rsyieigb D@arizatiom btkm rad M c u pduluwitkr of TLiiruc, DhactLyI SulfUe, haq rad TLl0pb.C at 632.8 am (CHzM (CHM C4H4O c4h45 T/'C 25 24 21 774 0.51 f 0.02" 0.506 f O.OISb 1.31 f 0.02' 1.58 0.02' 1@PO 8.10 f 0.08' 10.7 0.1' 7.69 f 0.0EC 8.40 f 0.08' IO%/ C m2 V-1
*
a Vibrational Raman contribution excluded. *Vibrational Raman contribution included. CVapor-phaszvalue, determined by the method described in ref 14 (- text). 'Liquid-phase value, calculated from refractive index and density of pure liquid.
The Journal of Physical Chemistry, Vol. 96, No. 18, 1992 7303
Anisotropic Molecular Polarizabilities
TABLE IIk Cotton-Mouton Effects,Mean PoLriubiKties, Raykigh DepoluimtioaRatios, md Derived Optid-Frapency (X Anisobopic Pohriubilities of (cH2)2X (CHl)& md cj4x (X = 0, s)o PropcrtY (CH2)20 (CH3)zO (CHM C4H40 -0.33 f 0.10 0.05 f 0.07 0.13 f 0.10 102’P/m5 A-2 mol-’b 0.38 f 0.12 10Z4Q/m5A-2 mol-’ Kb 0.336 f 0.035 -0.261 f 0.022 -0.434 f 0.038 4.28 f 0.05 -0.8 & 0.3 0.1 i 0.2 0.3 f 0.3 l@Ag/C m2 V-I T2 1.0 f 0.3 l@9(x,x - x)/J T2 1@9(xyy x)/J T2 10m(xzz x)/J T2 1 P a / C m2 V-I
-
1O2P0
10Qa,/C m2 V-’ l P a Y y / Cm2 V-I IO”a,,/C m2 V-l’
-0.03 f 0.06’ 10.22 f 0.04 -10.19 f 0.07 4.91 i 0.09 0.295 f 0.0101 4.71 i 0.28 5.58 f 0.09 4.44 f 0.28
0.66 f 0.33d -5.76 f 0.39 5.09 f 0.51 5.81 f 0.OG 0.35 f 0.018 5.19 f 0.40 6.60 f 0.22 5.64 f 0.60
-1.39 f 0.16e -2.51 f 0.19 3.91 i 0.25 8.40 f 0.08 0.506 f 0.015 8.56 i 0.18 9.72 f 0.22 7.04 f 0.12
23.8 f 0.11 19.0 f 0.1 -42.9 f 0.2 8.10 f 0.08h 1.31 f 0.02 8.79 f 0.37 9.76 f 0.42 5.75 f 0.13
= 632.8 am) C4HS 0.7 f 1.0 8.16 f 0.37 1.8 f 2.5 27.4 f 0.d 28.0 f 0.7 -55.5 f 0.9 10.7 f 0.1* 1.58 f 0.02 13.1 f 0.8 11.7 f 1.0 7.28 i 0.46
‘Locations of molecular axes: x and y directions in plane of heavy atoms, x coincident with direction (- to +) of molecular dipole moment; z direction perpendicular to plane. bSee text; ,C = P + QTl (eq 3). CRcference15a. dReference 15b. CRefercnce15c. /Reference 15d. XReference 12. *Reference 10. ‘Preferred solution to eqs 5-7, deduced by comparison with theoretical values (Table VI). 1.0
Depolarization ratios have previously been reported for dimethyl sulfide in the vapor phase but at different wavelengths;12 and for furan and thiophene in the liquid and dilutesolution states.13 In addition to Rayleigh depolarization ratios, mean opticalfrequency polarizabilities, a,can, if necessary, also be obtained from the present experiment. As noted by Alms et al.,14 it is easily established that where a and aArare the mean polarizabilities of the species of interest and argon, respectively, and and & are the corresponding intensities, determined at the same gas density, of the vertically polarized components of the 90° scattered light. Obviously, this method is a valuable alternative to measurements of either vapor- or liquid-state refractive indices and densities; but in this investigation only thiirane was so examined.
Experimental Results As noted above, the molar Cotton-Mouton constant is expected to show a linear dependence on the reciprocal of the temperature. Figure I displays plots of ,C against T’, and Table I11 contains the intercepts and slopes of the least-squaresstraight lines, together with values of the magnetic hyperpolarizability anisotropy Aq (derived from eq 4) and the molecular polarizabilities azx,ayy, azz(derived from eqs 5-7 and the known molecular magnetizabilities15). Note that the molecular axes are defined such that the x and y axes are in the plane of the heavy atoms, with x coincident with the direction (- to +) of the molecular dipole moment; the z axis is perpendicular to the plane of the heavy atoms. Once again, for reasons that have been emphasized in previous the magnetic hyperpolarizability anisotropies of these medium-sized (and, in some cases, relatively involatile) species are rather imprecisely determined by experiments of this kind. Nevertheless, it is of interest that the proportion of the CottonMouton constant at 298 K that originates in the temperatureindependent contribution appears to be unusually large for (CH2)*0(-4 f 15%) and certainly larger than for (CH&O, (CH3)*S,C4H40, and C4H4S( 6 f 9%, -10 f 8%,3 f 1% and 3 f 4%). Studies of the temperature dependence of the Cotton-Mouton effect have now yielded values of Aq for a large number of anisotropic molecules, perhaps 30 in total; but, unfortunately, trends or anomalies in the data have tended to be obscured by the experimental errors. The usefulness of the Cotton-Mouton effect, in conjunction with the Rayleigh depolarizationratio and the mean polarizability, as a route to molecular polarizabilities originates in the availability, particularly from studies of the microwave Zeeman effect, of freemolecule magnetizabilitiesfor a wide range of dipolar species. Of course, simultaneous solution of eqs 5-7, one of which is quadratic, would be expected to result in two sets of principal polarizabilities that are consistent with the experimental data. In the past, it has often been necessary to invoke qualitative arguments, for example, bond-additivity models, to resolve the am-
1
I
I 3.0
2.0
.r
-0.4
3.5
2.5
3
-.
Dimethyl Ether
-0.8
-1.2
I
(v
0
I
I
I
i
2.0
2.5
3.0
3.5
-’
l o 3 T I K-’ Flgure 1. Temperature dependence of the vapor-state Cotton-Mouton effects of (CH2)20, (CH3)2O, (CH3)2S, C4H40, and C4H& biguity. In the present study, however, high-level ab initio calculations, described in the next section, have sewed to discriminate between the alternatives, and no ambiguity remains. For three of the five molecules in Table 111(oxirane, furan, and thiophene), eqs 5-7 were immediately solvable; but for the other two (dimethyl ether and sulfide) they were not, and for these it was necessary to vary the least precise parameters (the slope, Q, and the magnetizabilities, x, etc., in eq 5 ) through the ranges defmed by the quoted experimental uncertainties in order to locate the real solutions. A similar procedure was used, for all five molecules, to estimate the likely errors in the derived polarizabilities.
xu-
7304 The Journal of Physical Chemistry, Vol. 96, No. 18, 1992
C-X C-C C-H CXC HCH
e“
1.436 1.462 1.082 61.2 115.6 22.2
1.431 f 0.002 1.466 f 0.002 1.085 f 0.003 61.6 116.6 f 0.3 22.0 f 0.3
1.811 1.479 1.080 48.2 114.9 27.7
1.815 f 0.003 1.484 f 0.003 1.083 f 0.002 48.3 f 0.3 115.8 f 0.2 28.2 f 0.2
1.981 1.470 1.080 43.5 115.6 27.3
“Bond lengths in angstroms (1 A = lo-’’’ m), angles in degrees. *Hirose, C. Bull. Chem. SOC.Jpn. 1974, 47, 1311-1318. ‘Okiye, K.; Hirose, C.; Lister, D. E.; Sheridan, J. Chem. Phys. Le??.1974, 24, 1 1 1-113. dAngle between bisector of HCH angle and line defined by C-C bond.
Calculations Geometries. Geometries for molecules in the (CHJ2X and (CHJ2X series were obtained at the MP2 level of theory with the 6-31G* basis sett6~l7 for H, C, 0, and S and the HUZ-SV* basis set for Se. The latter was derived from the minimal (4333/433/4) basis of Huzinaga and co-workers;’*the valence s and p functions were split to give (43321/4321/4), and the basis was augmented by a single d-type polarization function with Q(Se) = 0.338, as re”mended.l8 It is, therefore, related to but more flexible than the 3-21G* basis set of Dobbs and Hehre,Ig and it is identical with that used by Dukezoin recent work on gallium compounds. The geometrical parameters that emerge are given in Tables IV and V, which also include the relevant data from the experimental substitution (r,) structures. It is immediately obvious that the calculated bond lengths and angles are in excellent agreement with the experimental data; but, as previously noted by Simandiras and co-workers,21such agreement may be fortuitous, because experimental r, bond lengths are generally longer than re values and, as well, MP2-level calculated re values, as reported here, are typically larger than experimental re values. The calculated structure given here for selenirane is apparently the first for this molecule at any level of theory; no experimental structures have been reported, and it seems that the compound has never been isolated, although substituted seleniranes have been implicated in recent synthetic work.22 From the results in Table IV, selenirane would be expected to be highly strained but nevertheless very similar to its oxygen and sulfur analogues. The C-Se bond is longer in selenirane than in dimethyl selenide, and similar lengthenings are predicted and have been observed for the C-S and C-O bonds in the relevant species. A study of the thermodynamic stability of selenirane was not undertaken in the present work, although it was verified that the results in Table IV are for a stationary point on the potential energy surface; all eigenvalues of the MP2 second-derivative matrix are positive and the lowest vibrational frequency is 523 cm-I. Because the results for the (CH2)2Xand (CH3)2Xseries (Tables IV and V) indicated little difference between calculated parameters and experimental r, parameters, MP2 geometries were not determined for the larger molecules of the C4H4Xseries, and reported substitution s t r u c t u r e ~ ~ were ~ - ~used ~ for calculations of
Coonan et al. electric properties. The experimental geometries were also used by De Broucksre and in calculations of one-electron properties of furan and thiophene. Moreover, Simandiras and co-workers21have already reported MP2/DZP geometries for furan and thiophene, and it is unlikely that the basis set used here would yield significantly different geometries. Other relevant structural studies have included SCF, MP2, and CISD/DZP structures for (CH2)20;27SCF, CISD, and TCSCF structures, with DZP and TZ2P basis sets, for (CH2)2S;28 SCF/3-21G* geometries for C4H4X (X = 0, S, Se);29 and SCF/3-21G and SCF/3-21G* geometries for (CH&X and C4H4X(X = 0, S,Se).30 Ekctric Properties. Computations of electric dipole polarizabilities and dipole and quadrupole moments followed the procedure of the earlier in~estigation.~~ For H, C, 0, and S the 6-31G (+sd+sp) basis was utilized; this is simply the standard 6-31G basis16 with an additional diffuse s-type function on all atoms and additional diffuse p and d-type functions on H and on C, 0, and S,respectively. Details of the choice of exponents and the application of this basii set to a large number of polyatomic molecules have already been given.5a For Se, the HUZ-SV basis described above, augmented by a single diffuse s-type function with a,(Se) = 0.03 11 and a set of diffuse d-functions with ad(Se) = 0,090, was used. The d-function exponent for Se was optimized in calculations of the mean polarizability of H2Se, in the manner previously described.5a An account of the development and applicability of this basis set, denoted HUZ-SV (+sd) will appear el~ewhere.~~ In this study, SCF and MP2 values of static and optical-frequency polarizabilities and dipole and quadrupole moments were calculated; but because the main concern of this paper is the comparison of results from experiment and theory, only the optical-frequency (632.8 nm) polarizabilities are explicitly reported here. MP2 corrections to the static polarizabilities and electric moments are defined as the derivatives of the second-order corrections to the energy.32 Optical-frequency polarizabilities were obtained at the SCF level by a time-dependent coupled Hartree-Fock procedure” and at the MP2 level by addition of the SCF-level dispersion correction to the MP2-level static polarizability: Alternatively, this procedure can be viewed as a correlation correction to the SCF-level optical-frequency polarizability:
+
ru%”(632.8) = ~tzg~(632.8) [a%p2(-) -
(10)
and can, therefore, be expected to be reliable provided the correction is virtually independent of frequency. The same approach has been used by others in studies of sulfur dioxide34and cyclohexane,5band it has, more recently, been shown to be an excellent approximation in the cases of formaldehyde and amm~nia.’~ All calculations were performed with the Cambridge Analytic Derivatives Package ( C A D P A C ) ~on ~ the Gould NP1 in the Computer Centre of the University of New England. The theoretical results, together with the experimental results from the present and previous investigations, are summarized in Tables VI and VI1
TABLE V Calculated aud Experimental Structures of (CH3),X (X = 0, S,Sep
(CHd20 calc
c-x
C-H, C-H,
cxc
1.412 1.094 1.085 111.0
exptb 1.410f 0.003 1.110f 0.005 1.091 f 0.007 111.7 f 0.4 110.8 f 0.4
(CH3)zS
(CHASe
calc
expte
calc
exptd
1.803 1.087 1.086 98.4
1.802 f 0.002 1.091 f 0.002 1.091 f 0.002 98.9f 0.2 110.8 106.6 109.5 f 0.3 109.6 f 0.3
1.951 1.085 1.085 95.8 110.1 107.1 110.5 109.4
1.945 f 0.001 1.096 f 0.004 1.085 f 0.013 96.3 f 0.2 110.3 f 0.3 105.0f 0.7 109.9 f 0.5 110.6 f 0.5
111.3 XCH, 1 1 1.5 XCH, 107.1 107.2 f 0.6 107.6 109.4 Ha-4 108.3 108.7 f 0.5 108.6 HaCH, 109.2 109.6 f 0.6 “Bond lengths in angstroms (1 A = m), angles in degrees; the H atoms of both CH3 group have a staggered conformation with respect to rotation about the C-O bonds. bBlukis, U.; Kasai, P. H.; Myers, R. J. J . Chem. Phys. 1963, 38, 2753-2760. cPierce,L.; Hayashi, M. J . Chem. Phys. 1961, 35, 479-485. dPandey, G.K.; Dreizler, H. 2.Nuturforsch. A 1977, 32, 482-484.
T h e Journal of Physical C h e m i s t r y , Vol. 96, No. 18, 1992 7305
Anisotropic Molecular Polarizabilities
TABLE Vk
Cdculated rad Experimental Optierl-Frequency (A = 632.8 nm) Polarizabilities of (CHI)]X, (CHJ2X, 8 d C4H4X (X = 0, S,
SCF ax,
a,, a,,
a lo2& u,,
a,, aZz
a 102po
4.08 5.18 3.96 4.41 0.460 4.77 5.95 4.67 5.13 0.382
calc MP2
scaled expt (CH2)ZO 4.41 4.64 4.71 f 0.28 5.34 5.63 5.58 f 0.09 4.23 4.46 4.44 f 0.28 4.66 4.9Ic 0.327 0.295 f 0.01W (CH3120 5.47 5.19 f 0.40 5.18 6.60 f 0.22 6.32 6.68 5.28 5.64 f 0.60 4.99 5.50 5.81e 0.342 0.35 f O.OlC C4H4O
SCF
calc MP2
8.47 6.78 5.84 7.03 0.714
8.66 7.03 6.21 7.30 0.578
7.62 9.08 6.37 7.69 0.615
8.04 9.42 6.78 8.08 0.527
scaled (CH2)2SC 9.12 7.41 6.54
expt
SCF
7.69 f 0.08 0.51 f 0.02 (CH3)2S 8.56 f 0.18 8.36 9.79 9.72 f 0.22 7.05 7.04 f 0.12 8.4W 0.506 f 0.015 C4H4S
10.70 7.76 7.10 8.52 0.998 9.05 10.59 7.48 9.04 0.585
calc MP2 scaled (CH2)2Sea 10.73 10.78 8.01 8.04 7.60 7.64 8.78 0.746 (CH3)2Se 9.46 9.77 10.85 11.2 8.04 8.31 9.45 0.437 C4H4Se
expt
(8.829
(9.7V)
8.59 8.68 8.91 8.79 f 0.37 12.37 12.64 12.9 13.1 f 0.8 14.21 14.57 15.4 (14.2h) 9.16 9.47 9.72 9.76 f 0.42 11.29 11.75 12.0 11.7f1.0 12.14 12.66 13.3 (13.4h) a,, 5.38 5.52 5.67 5.75 f 0.13 6.83 7.08 7.22 7.28 f 0.46 7.70 8.06 8.49 (9,45h) a 7.71 7.89 8.10 f 0.08' 10.16 10.49 10.7 f 0.1' 11.35 11.76 12.4 f 0.1' 102po 1.37 1.38 1.31 f 0.02 1.63 1.59 1.58 f 0.02 1.68 1.59 'Polarizabilities expressed here as 1040a,,/C m2 V-I, etc. The relationship for interconversion from SI units (C m2 V-I) to cgs electrostatic units (csu) is (a/esu) = 106(4mo)-1(a/C m2 V-I) = 0.8988 X 10l6( a / C m2 VI).bLocations of molecular axes: x and y directions in plane of heavy atoms, x coincident with direction (- to +) of molecular dipole moment; z direction perpendicular to plane. 'Thiirane was observed to decompose and/or polymerize when heated, so that the temperature dependence of the Cotton-Mouton effect could not be determined. dSelenirane has not yet k e n reported as an isolable compound. eReference 12. /Estimated from bond refractions tabulated in: Le Fbvre, R. J. W. Molecular Refractivity and Polarizability. In Aduances in Physical Organic Chemistry; Gold, V., Ed.; Academic Press: London, 1965; Vol. 3, pp 1-90. #Estimated from the liquid-skate refractive index and density of diethyl selenide. Reference 10. These are apparent polarizabilities derived from measurements of the dilute-solution Kerr and Cotton-Mouton effects of selenophene in cyclohexane. Such results are not rigorously comparable with free-molecule values; note, however, that the relationship a,, > a,, >> a,, was correctly established. 'Calculated from the liquid-state refractive index and density; see ref IO for details. a,,
ayy
TABLE VI1 Calculated and Experimental Dipole and Quadrupole Moments of (CH2)2X, (CH3)]X, and C4H4X ( X = 0, S, calc SCF p,
8.27 -15.07 10.18 4.89
p,
5.64 -9.09 11.66 -2.57
e,, eyy e,, e,, e,, e,,
calc MP2 expt (CH2)20 6.91 6.27 f 0.03' -13.66 -12.3 0.2' 9.32 8.7 f 0.21 3.7 f 0.4 4.33 (CHI)@ 4.91 -9.04 11.59 -2.56 C4H40
4.37 f 0.03d -6.7 i 1.7' 11.0 f 2.0' -4.3 3.3'
SCF 7.91 -4.24 6.50 -2.26 6.42 -7.67 15.14 -7.47
calc MP2 (CH2)2S 6.37 -3.18 5.76 -2.58 (CH3)2S 5.64 -6.96 14.38 -7.42 C4H4S
expt
SCF
6.14 f 0.07e -1.7 f 2.3' 4.0 f 2.0' -2.3 f 3.3'
8.27 6.40 2.34 -8.74
5.00 f 0.03e -5.0 f 1.7' 10.7 f 1.7' -5.7 f 2.7'
6.19 -2.64 14.33 -11.68
'
MP2 (CH2)2Se 6.19 6.08 2.38 -8.47
expt
5.28 4.70 f 0.07f -1.91 13.59 -11.68 C4H4Se
3.50 2.52 2.205 f 0.02W 3.03 1.51 1.83 f 0.03h 2.89 1.07 1.328 f 0.033' -1.01 -1.39 0.7 f 1.3"' 1.91 2.08 5.7 f 5.3"' 6.72 5.87 7.0 f 5.7" 21.85 19.61 19.7 f 1.0"' 22.72 19.51 22.0 f 5.0"' 21.82 18.89 21.7 f 7.3" -20.84 -18.22 -20.3 1.3"' -24.63 -21.60 -27.7 f 7.3"' -28.54 -24.76 -28.7 i 9.3n Dipok moments expressed here as 10'px/C m and quadrupole moments as lo"e,,/C m2, etc. The relationships for interconversion from SI units (C m, C m2) to cgs electrostatic units ( a u ) are (w/esu) = 10-18(p/D) = 106(10/4~c0)1/2(p/~ m) = 0.2998 X 1012(p/Cm);and (e/csu) = los. ( 1 0 / 4 ~ e ~ ) ~ / ~ (m2) e / C= 0.2998 X lOI4(8/C m2). bLocations of molecular axes: x and y directions in plane of heavy atoms, x coincident with direction (- to +) of molecular dipole moment; I direction perpendicular to plane. 'Cunningham, G.L.; Boyd, A. W.; Myers, R. J.; Gwinn, W. D.; Le Van, W. I. J. Chem. Phys. 1951, 19,676-685. dBlukis, U.; Kasai, P. H.; Myers, R. J. J. Chem. Phys. 1963,38,2753-2760. 'Pierce, L.; Hayashi, M. J. Chem. Phys. 1961, 35,479-485. 'Beecher, J. F. J. Mol. Spectrosc. 1966, 21, 414-424. tSirvetz, M. J. Chem. Phys. 1951, 19, 1609-1610. "gats, T.; Kozima, K. J. Mol. Spectrosc. 1972, 42, 38-46. 'Brown, R. D.; Burden, F. R.; Godfrey, P. D. J. Mol. Spectrosc. 1968, 25, 415419. 'Reference 15a. kSutter, D. H.; Flygare, W. H. Mol. Phys. 1969, 16, 153-164. 'Reference 15b. "'Reference 15d. "Czieslik, W.;Sutter, D.; Dreizler, H.; Norris, C. L.; Rock, S.L.; Flygare, W. H. Z . Naturforsch. A 1972, 27, 1691-1694. p,
e,, e,, e,,
and compared in the discussion that follows.
Discussion Pduipbilities. It is, of course, appropriate f i t to seek to test the reliability of the procedure that has been exploited here as a route to experimental values of anisotropic molecular polarizabilities. Fortunately, one of the nine molecules in Table VI, dimethyl ether, has previously been examined by a well-established and independent method that involves analysis of measurements of the temperature dependence of the Kerr effect, instead of the Cotton-Mouton effect, in conjunction with the Rayleigh depolarization ratio and mean polarizability of the species.37 The polarizabilities (expressed here as 10aa,,/C mz V-I, etc.) that
were so derived (5.28 f 0.13, 6.69 f 0.17, 5.46 f 0.14) are in excellent agreement with those in Table VI (5.19 f 0.40, 6.60 f 0.22,5.64 f 0.60); indeed, the precision appears to be slightly higher in the earlier results than in the present results. In this respect, it should be borne in mind that, in favourable circumstances?* the Kerr effect method provides a direct determination of the polarizability, ax,, in the direction of the dipole moment; and also that the Cotton-Mouton effect method propagates the relatively large uncertainties (see Table 111) associated with the Zeeman effect magnetizabilities. On the other hand, the temperature dependence of the Kerr effect is more complicated than that of the Cotton-Mouton effect (of a diamagnetic molecule), and additional information (such as an estimate of the second Kerr
7306 The Journal of Physical Chemistry, Vol. 96, No. 18, 1992
Coonan et al. here were obtained by the microwave spectroscopic method; and, because the quadrupole moment emerges as the small difference between large and oppositely signed terms that involve the g value and the magnetic anisotropy, the results are in some cases rather uncertain. From Table VI1 it can be seen that in the (CH2)zX and (CH3)zXseries the experimental results are better predicted by the MP2 values than by the SCF values; and in both series the computed moments are larger than those observed. For the C4HJ series, the comparison is complicated by the relatively large experimental errors. In relation to the much clearer trends through the X = 0, S, Se progressions, the two factors (variations in heteroatom electronegativity and molecular geometry) mentioned above are again relevant. However, the predominant influence arises from the fact that for dipolar molecules the numerical values of all three components of the quadrupole moment arc dependent on the choice of origin,for obvious reasons of conveniencehere uniformly defined as the center of mass. Because of the symmetry of these molecules, the origin dependence of the quadrupole moment can be expressed simply as
eid = e, - 2/Lxf
Figure 2. van der Waals surfaces for (CH2)2X,(CHJ2X, and CIHIX (X = 0, s, SC).
hyperpolarizability, Y ~ from , ESHG measurements) is usually required.as From such considerations,and a comparison of the two sets of experimental results with the scaled (see below) theoretical predictions, it is reasonable to conclude, once again,3g that the Cotton-Mouton effect method can yield molecular polarizabilities of high accuracy and acceptable precision. It has previously been shown5that the 6-31G (+sd+sp) basis set, which was developed specifically for such calculations, yields Mp2-level mean and anisotropic optical frequency polarizabilities that are only about 5% smaller than experimental values. The results in Table VI further demonstrate the validity of this conclusion; men for (CHJ2Se and (CHJ2Se, for which there are no direct experimental data, the calculated mean polarizabilities are close to (but slightly less than) the best indirect estimates. If the calculated principal polarizabilities for each molecule are scaled by the ratio (typically =1.02-1.06) of the experimental to the calculated mean polarizability, scaled theoretical polarizabilities that are in excellent agreement with the experimental results are obtained. It follows, therefore, that the scaled theoretical polarizabilities for (CH&S, (CH2)+3e, (CH3)2Se,and C4H4Se,for which experimental values of the principal polarizabilities are not available, can reasonably be considered to be of high accuracy. Finally, it is of interest to consider the relative magnitudes of the two in-plane (Le., the plane of the heavy atoms) components of the polarizability. In the (CH3)2Xseries the relationship is uniformly a, < a,,; but in the closed-ring compounds of the (CH2)2Xand C4H4Xseries it is a, < a,, for X = 0 but a, > ap for X = S and Se. Figure 2, which displays the van der Waals surfacg of these molecules, provides a clear pictorial explanation of these trends. Dipok Mownts. In relation to the dipole moments of these molecules, the overall outcome that is apparent from the data in Table VI1 is that, with the diffusely polarized 6-31G (+sd+sp) basis set, the experimental results are approached much more closely by the MP2 values than by the SCF values, as expected. In the (CH2)2Xand (CH3)2Xseries, the MP2 values are higher than those ohparved by 4-10% and =12%, respectively. However, in the C4H4Xseries, where the dipole moments are very much smaller because of conjugative rearrangements of the electronic charge distribution$,the MP2 values overestimate the experimental value for X = 0 but underestimate those for X = S and Se. The weak trends that are evident on replacement of 0 by S and, in turn, Se are qualitatively explicable in terms of two main factors: the decrease in electronegativity (0> S = Se), which tends to decrease the dipole moment, and the decrease in the CXC angle (Tables IV and V; C4H40 106.6O, C4H4S 92.2O, C4H4Se 87.7O 23-25), which tends to increase the dipole moment. Qdrupole Momeets. It must first be noted that all of the experimental quadrupole moments for the molecules of interest
(1 1)
where the primed subscripts indicate a new origin and the unprimed subscripts indicate an old origin, both on the x axis, /L is the molecular dipole moment, and x'is the distance in the direction of the dipole moment (- to +) through which the origin is shifted. As 0 is replaced by S and, in turn,by Se,theorigin (Le., the center of mass) moves closer to the heteroatom (i.e., x'is negative), so that €3, would be expected to increase and both e ,, and 8, to decrease in magnitude. Obviously, these are the trends that are observed in most cases,but any more detailed analysis would involve at least all three of the factors mentioned here. Some useful insights into the nature of the charge distributions in these molecules can be drawn from simple qualitative interpretations of the quadrupole moments. For this purpose, it is necessary to recall that the definition of the quadrupole moment isM
and analogous expressions follow for the other components. In the case of oxirane, for example, 8, and e,,are found to be negative and positive in sign, respectively, from which it can be inferred that ( x 2 ) is larger and (y2) and (z2) are smaller than would be required to give zero quadrupole moments, Le., the electronic charge distribution is concentrated and extended along the symmetry axis and somewhat depleted in the vicinity of the H atoms. For furan, as for other aromatic molecules, 8, is strongly and characteristically negative, a result of redistribution of electronic charge from the H atoms and the H-C bonds toward and inside the periphery of the ring; in consequence, ( x 2 ) and especially are smaller and (9) is larger than if this annponent were zero. Although rationalizations of this kind may seem obvious, it must be emphasized, once again, that the traceless quadrupole moment is rather a subtle property that originates in a slight (and frequently unpredictable as to sign) imbalance between the electronic and nuclear charge distributions within the molecule.
v)
s-ry In this study, measurements of the temperature dependence of the vapor-statc magnetooptical Cotton-Mouton effect have bem combined with Rayleigh depolarization ratios and mean polarizabilities at 632.8 nm to derive reliable Optical-frequencyprincipal
Anisotropic Molecular Polarizabilities polarizabilities for five of the six molecules in the series (CH2)*X, (CH3)2X,and C4H4X(X = 0,S). At the same time, the 6-31G (+sd+sp) basis set, developed specifically for calculations of polarizabilities, has been used to obtain SCF and MP2-level mean and anisotropic optical-frequency polarizabilities, and also dipole and quadrupole moments, for all nine species in the series (CH2)2X,(CH3)2X,and C4H4X(X = 0,S,Se). A comparison of the experimental and theoretical results confirmed the earlier conclusion that this basis set yields polarizabilities that are only about 5% smaller than observed values. In addition, the calculated dipole and quadrupole moments were critically assessed against literature data, and trends in the electronic charge distributions in these groups of molecules were considered.
Acknowledgment. Australian Postgraduate Research Awards (to M.H.C. and M.R.H.), financial support from the Australian Research Council (to G.L.D.R. and M.A.S.), and technical assistance from Dr. D. R. Laver and Mr. R. Stankey are gratefully acknowledged. Registry No. Oxirane, 75-21-8; thiirane, 420-12-2; selenirane, 66121-2; dimethyl ether, 115-10-6; dimethyl sulfide, 75-18-3; dimethyl selenide, 593-79-3; furan, 110-00-9; thiophene, 110-02-1; sclenophene, 288-05-1.
References and Notes (1) The Cotton-Mouton effect measurements reported here were performed in the School of Chemistry, University of Sydney, New South Wales 2006, Australia. (2) Cotton, A.; Mouton, H. C. R. Hebd. Seances Acad. Sci. 1905, 141, 317-319.349-351. (3) Buckingham, A. D.; Pople, J. A. Proc. Phys. Soc., London, Secr. B 1956,69, 1133-1 138. (4) Buckingham, A. D.; Prichard, W. H.; Whiffen, D. H. Trans. Faraday S&. 1%7,63; 1057-1064. (5) (a) Spackman, M. A. J. Phys. Chem. 1989, 93, 7594-7603. (b) Spackman, M. A. Chem. Phys. Lerr. 1989, 161,285-290. (6) For a summary of the algebraic and numerical factors required to convert the relevant electric and magnetic properties from SI to cgs units we:
Lukins, P. B.; Laver, D. R.; Buckingham, A. D.; Ritchie, G. L. D. J . Phys. Chem. 1985.89, 1309-1312. (7) Bridge, N. J.; Buckingham, A. D. Proc. R. Soc. London, Ser. A 1966,
295, 334-349. (8) (a) Lukihs, P.B.; Buckingham, A. D.; Ritchie, G. L. D. J. Phys. Chem. 1984,88,2414-2418. (b) Craven, I. E.; Hesling, M. R.; Laver, D. R.; Lukins, P. B.; Ritchie, G. L. D.;Vrbancich, J. J. Phys. Chem. 1989,93,627-631. (c) Coonan, M. H.; Ritchie, G. L. D. J. Phys. Chem. 1991, 95, 1220-1223. (9) Dymond, J. H.; Smith, E. B. The Vinal Cafficients of Pure Cares and Mixrures; Clarmdon Rtss: Oxford, U.K., 1980. (10) Dennis, G. R.; Gentle, I. R.; Ritchie, G. L. D.; Andrieu, C. G. J . Chem. Soc., Faraday Trans. 2 1983, 79, 539-545. (11) Craven, I. E.; Hcsling, M. R.; Ritchie, G. L. D. Chem. Phys. Lett. 1991, 185, 371-374. (12) Bogaard, M. P.; Buckingham, A. D.; Pierens, R. K.; White, A. H. J. Chem. Soc., Faraday Trans. 1 1978, 74. 3008-3015. (13) (a) LeFi?vre, C. G.; Le FZvre, R. J. W.; Purnachandra Rao, B.; Smith, M. R. J. Chem. Soc. 1959, 1188-1192. (b) Canselier, J.-P.; Cllment, C. J . Chim. Phys. PhysXhim. Bioi. 1978, 75, 880-888.
The Journal of Physical Chemistry, Vol. 96, No. 18, 1992 1307 (14) Alms,G. R.; Bumham, A. K.; Flygare, W. H. J . Chem. Phys. 1975, 63, 3321-3326. (15) (a) Hamer, E.; Sutter, D. H. Z.Narurforsch. A 1976,31,265-271. (b) Benson, R. C.; Flygare, W. H. J. Chem. Phys. 1970,52,5291-5298. (c) Hamer, E.; Sutter, D. H.; Dreizler, H. Z . Naturforsch. A 1972, 27, 1159-1 164. (d) Sutter, D. H.; Flygare, W. H. J . Am. Chem. Soc. 1969, 91, 4063-4068. (16) (a) Hehre, W. J.; Ditchfield. R.; Pople, J. A. J . Chem. Phys. 1972, 56, 2257-2261. (b) Dill, J. D.; Poplc, J. A. J. Chem. Phys. 1975, 62, 2921-2923. (c) Francl, M. M.; Pictro, W. J.; Hehre, W. J.; Binkley, J. S.; Gordon, M. S.;De Frees, D. J.; Pople, J. A. J . Chem. Phys. 1982, 77, 3654-3665. (17) Hariharan, P. C.; Pople, J. A. Theor. Chim. Acta 1973,28,213-222. (18) Huzinaga, S.;Andzelm, J.; Klobukowski, M.; Radzio-Andzelm, E.;
Sakai, Y.; Takewaki, H. Gaussian Basis Sets for Molecular Calculations; Elsevier: New York, 1984. (19) Dobbs, K. D.; Hehre, W. J. J . Comput. Chem. 1986, 7, 359-378. (20) Duke, B. J. J . Mol. Srrucr. 1990, 208, 197-204. (21) Simandiras, E. D.; Handy, N. C.; Amos, R. D. J . Phys. Chem. 1988, 92, 1739-1742. (22) See, as examples: (a) Van Ende, D.; Krief, A. Tetrahedron Left. 1975,2709-2712. (b) Ando, W.; Kumamoto, Y.; Tokitoh, N. Tetrahedron Lett. 1987, 28, 2867-2870. (23) (a) Bak, B.; Christensen, D.; Dixon, W. B.; Hansen-Nygaard, L.; RastrupAndersen, J.; Schottlaender, M. J. Mol. Specrrosc. 1%2,9, 124-129. (b) Mata, F.; Martin, M. C.; Soerensen, G. 0. J. Mol. Srrucr. 1978, 48, 157-163. (24) Bak, B.; Christensen, D.; Hanscn-Nygaard, L.; RastrupAndersen, J. J. Mol. Specrrosc. 1961, 7, 58-63. (25) Pozdeev, N. M.; Akulinin, 0. B.; Shapkin, A. A,; Magdesieva, N. N. Zh. Srrukr. Khim. 1970. 11. 869-874. (26) De Brouckh. d.;Nieuwpoort, W. C.; Broer, R.; Berthier, G. Mol. Phys. 1982,45,649-661. (27) Simandiras, E. D.; Amos, R. D.; Handy, N. C.; Lee, T. J.; Rice, J. E.; Remington, R. B.; Schaefer, H. F. J . Am. Chem. Soc. 1988, 110, 1388-1 393. (28) (a) Allen, W. D.; Bertie, J. E.; Falk, M. V.; Hess, B. A.; Mast, G. B.; Othen, D. 0.;Schaad, L. J.; Schaefer, H. F. J. Chem. Phys. 1986, 84, 4211-4227. (b) Fowler, J. E.; Alberts, I. L.; Schaefer, H. F. J . Am. Chem. Sot. 1991, 113,4768-4776. (29) Baldridge, K. K.; Gordon, M. S. J. Am. Chem. Soc. 1988, 110, 4204-4208. (30) Yadav, V. K.; Yadav, A.; Poirier, R. A. J . Mol. Srrucr. 1989, 186, 101-1 16. (31) Dougherty, J.; Spackman, M. A., manuscript in preparation. (32) Amos. R. D. Chem. Phvs. Lett. 1982.88. 89-94. (33) Amos; R. D.; Handy, N: C.; Knowles,'P. J.; Rice, J. E.; Stone, A. J. J . Phys. Chem. 1985,89, 2186-2192. (34) Bacskay, G. B.; Rendell, A. P. L.; Hush, N. S.J. Chem. Phys. 1988, 89, 5721-5730.(35) Rice, J. E.; Handy, N. C. J . Chem. Phys. 1991, 94, 4959-4971. (36) Amos, R. D.; Rice, J. E. CADPAC: The Cambridge Analyric Derivatives Package, Issue 4 . e Cambridge, 1987. (37) Bogaard, M. P.; Buckingham, A. D.; Ritchie, G. L. D. J . Chem. Soc., Faraday Trans. 2 1981, 77, 1547-1551. (38) Gentle, I. R.; Laver, D. R.; Ritchie, G. L. D. J . Phys. Chem. 1990, 94, 3434-3437. (39) (a) Lukins, P. B.; Ritchie, G. L. D. J . Phys. Chem. 1985, 89, 1312-1314. (b) Lukins, P. B.; Ritchie, G. L. D. J. Phys. Chem. 1985,89, 3409-341 1. (40) (a) Buckingham, A. D. Electric Moments of Molecules. In Physical
Chemistry, An Advanced Treatise; Eyring, H., Henderson, D., Yost, W., Eds.; Academic Press: New York, 1970; Vol. 4, pp 349-386.
