J. Phys. Chem. 1082, 86, 2794-2796
2794
van der Waab Coefflclents for Interactions Involving Sulfur Hexafluorlde Russell T Packt Theoretical Chemlsby Group (T- 1 4 , Los Alemos National bbOr6tOry, Los Alemos, New Mexbo 87545 (Received: January 19, 1982; I n Final Form: March 22, 1982)
Experimental frequency-dependentpolarizabilities and Pade approximant methods are used to calculate values and error bounds for the oscillator strength sums of SF6and the van der Waals c6 dispersion coefficients for the interaction of SF6 with SF6, coz, NH3, H20, NzO, CO, NO, H2, N2,02,He, Ne, Ar, Kr, Xe, H, Li, Na, K, Rb, Cs, N, and 0.
Introduction In other work in which we are determining intermolecular potentials for the interaction of SF6with noble gas atoms, it has been found that estimates of the van der Waals c6 coefficients for the interactions of SF6with other atoms and molecules are needed but not available. In this paper we use experimental frequency-dependent polarizability data and the Pade approximant methods of Langhoff and Karplus’ to determine those coefficients. It is well-known2that the c6 coefficients for the interactions of 2-state diatomic or linear polyatomic molecules with S-state atoms have a spherical part and an anisotropic part whose angle dependence is given by the Legendre polynomial P2. However, the c6 for the interaction of the octahedral SF6 molecule in its ground (Al,) electronic and vibrational state with S-state atoms is spherically symmetric. This is easily demonstrated by direct generalization of the usual deri~ations.~BHowever, it is even more easily seen by noting that the interaction between an SF6 molecule and any species A must be invariant if any of the symmetry operations of the O h group are performed on the SF,. The lowest-rank nonspherical quantity that transforms like AI, under O h is a spherical tensor T4of rank 4. This anisotropy can first occur in the van der Waals series in the coefficient Cg,and c6 is thus spherical under rotations of the SFe. Of course, in the interactions of SF6with other molecules, the c6 can depend on the angular coordinates of the other molecule, but, in the present paper, we give only the spherical part of c6 for such cases. Because the lowest nonzero permanent multipole moment of SF, is its hexadecapole (1 = 4)moment, the first induction term in the interaction of SF6with an S-state atom does not occur until C12. In the interaction of SF6with dipolar molecules, there is an induction contribution to C6due to the dipole moment of the other molecule; however, that contribution can easily be ~ a l c u l a t e dand ~ , ~added by the reader, and we here present only the dispersion contribution. We follow essentially the same procedure here as prev i ~ u s l y . ~The - ~ spherical, dispersive part of the van der Waals coefficient for the interaction of X = SF6 with species A is given (in atomic units) by
where the f,, are oscillator strengths and the on = En - E o are excitation energies or frequencies. The sum includes integration over the continuum and the prime implies omission of the n = 0 and m = 0 terms. Equation 1 can be written in the form6 Also Adjunct Professor of Chemistry, Brigham Young University. 0022-3654/82/2086-2794$01.25/0
where a&y) is the polarizability of species C at imaginary frequency w = z = iy (3)
Following Langhoff, Gordon, and Karplus,lJ we make the Cauchy expansion of eq 3 a&)
= & S c ( - 2 j - 2)(02)’
(4)
J
where the S&)
are oscillator strength sums S C ( ~=) C r f n ( c ) [ o n ( C ) I k n
(5)
and determine the Sc(-2j - 2 ) by fitting eq 4 to experimental frequency-dependent polarizability data subject to Stieltjes constraints.’ The resulting Sc(-2j - 2 ) are used to construct Pade approximants which are upper and lower bounds to a&) and which can be rearranged into the form of eq 3 with a finite number of effective oscillator strengths and frequencies, so that the integral ( 2 ) can be performed analytically to yield simple but rigorous bounds to c6 in the form of eq 1 but with a finite number of terms. For molecules such as SF6which have infrared-active vibrational modes, there is a contribution to eq 3 and 1 from excited vibrational states of the ground electronic state as well as excited electronic states.* In what follows we show that this infrared contribution to c6 is negligible for all the interactions considered here, but the contribution to asFs(0) is substantial. If one were calculating a true ab initio rigid-rotor potential for an SF6interaction, so that the positions of the nuclei within the SF6were fixed, one should use only the electronic (ultraviolet) part of a and C6in such a potential. However, if one wishes to obtain (1)P. W.Langhoff and M. Karplus, J. Chem. Phys., 53, 233 (1970); J. Opt. SOC.Am., 59,863(1969);P. W.Langhoff, J. Chem. Phys., 57,2604 (1972). (2) See, for example, R. T Pack, J. Chem. Phys., 64,1659(1976),and references therein. (3)See, for example, G. C. Nielson, G. A. Parker, and R. T Pack, J. Chem. Phys., 64,2055 (1976). (4)See, for example, G. A. Parker and R. T Pack, J.Chem. Phys., 64, 2010 (1976). (5)R. T Pack, J. Chem. Phys., 61,2091 (1974). (6)H.B. G. Casimir and D. Polder, Phys. Rev., 73, 360 (1948). J. Chem. Phys., (7)P. W.Lanehoff, R. G. Gordon, and M. KWD~US, 55, 2126 (1971). (8)E. A. Gislason, F. E. Budenholzer, and A. D. Jorgensen, Chem. Phys. Lett., 47,434 (1977).
0 1982 American Chemical Society
van der Waals Coefficients Involving SF,
The Journal of Physical Chembtty, Vol. 86, No. 14, 1982 2705
TABLE I: Oscillator Strength Sums of SF, a
k
Suv(k)
TABLE 11: Effective Oscillator Strengths and Frequencies of SF, Used in the Calculations of C, a
SIR(k)
70.000 (2.40 t 0.02) x 10-4 -2 30.38 t 0.02 13.83 t 0.07 -4 32.6 i 4.9 -6 6 8 0 i 310 The two columns are the ultraviolet and infrared contributions. The S I R ( k ) for large negative k are not useful and not reproduced. Hartree atomic units,
0
a potential that can be used in comparisons with room temperature experimental data, one should use the total a and c6 because, at low collision energies, the vibrational motions are rapid compared to the relative motion, and both the electronic and vibrational motions adjust adiabatically during the interaction and thus contribute to a and c6. We report herein the infrared and ultraviolet contributions to a separately, so that the reader can, for example, easily calculate the induction contribution to c6 for an SF6-dipolar molecule interaction appropriate to either fixed nuclei or adiabatic vibrational motion.
Calculations and Results The frequency-dependent polarizability is obtained from the dielectric constant E and refractive index q via the Clausius-Mosotti and Lorenz-Lorentz equations9
('-') - =4:n( q2-1) E +2 +2
0)= 47rn
q2
(6)
where n is the number density of SF6molecules. For the polarizability at almost zero frequency, we used the microwave data of Tipton, Deam, and Boggs,'O which give a = 44.21 f 0.05a03,in good agreement with the dielectric constant measurements of Hostika and Bose" which yield a = 44.27 f 0 . 0 7 ~ 2 .Earlier measurements12appear to be less accurate. At higher frequencies the only data used were those of Watson and R a m a s ~ a m y , who ' ~ measured the refractive index of SF6at five frequencies in the visible provide no range. The only other optical data a~ailable'~ additional information. To fit the available data for SF6, we first write eq 3 in the form a ( w ) = ff'R(W)
+ aW(w) =
6
An n=1[1 - ( w / o , ) ~ ]
+ CSW(-2j - 2)(w2)' J
(7)
where A , = fn/w,2. The infrared contribution is not expanded because the small w, from the two infrared-active vibrational modes make the IR sum rules diverge rapidly. Also, with no available measurements of a ( w ) at IR frequencies there is no need to include the finite line widths and the actual experimental vibraused in earlier tional frequencies are used for the w, in (7). To estimate the f, and A , for n = 1 and 2 we used the absorption data (9)See, for example, J. 0. Hirschfelder, C. F. Curtiss, and R. B. Bird, "Molecular Theory of Gases and Liquids",Wiley, New York, 1954,pp 861 and 883. (10)A. B. Tipton, A. P. Deam, and J. E. Boggs,J. Chem. Phys., 40, 1144 (1964). (11)C. Hostika and T. K. Bose, J. Chem. Phys., 60, 1318 (1974). (12)R. M.Fuoss, J. Am. Chem. SOC.,60,1633 (1938);R. Linke, 2. Phys. Chem., B48, 193 (1948). (13)H. E.Watson and K. L. Ramaswamy, h o c . R. SOC.London, Ser. A , 156,144 (1936). (14)W. Klemm and P. Henkel, 2.Anorg. Allg. Chem., 213,115(1933).
1 1.327 X 1.693 0.002 799 9 2 2.264 X 12.139 0.004 319 0 3u 0.017 0.591 0.172 02 31 28.273 30.378 0.964 73 4u 69.982 29.787 1.532 78 0.0 41 0.0 u and 1 label the parameters obtained from the upper and lower bounding Pade approximants. Hartree atomic units. TABLE 111: Spherical van der Waals C, Dispersion Coefficients for the Interaction of SF, with Various Partnersa partner c,UV Cetot uncertainty 848 854 k 200 SF, t 50 395 398 CO, 268 269 f 50 NH, * 40 195 196 HZO 393 394 k70 k 48 267 268 NO t 50 247 248 93.4 93.8 t8.0 HZ 246 247 k 35 NZ +45 234 0 2 233 k4.4 He 34.8 35.0 i:13 Ne 76.4 76.6 k 40 Ar 238 239 t60 Kr 335 337 5 120 Xe 592 594 k5.0 H 65.0 65.4 ?: 25 Li 508 521 k 65 Na 621 635 * 100 K 926 949 t 120 Rb 107 1 1096 * 70 cs 876 905 f 22 N 140 14 1 k 22 0 114 114 a Hartree atomic units (e2ao5).
a:o
and dipole matrix elements of Fox and Person'5 and Person and Overend16with the relation fn
= 2unl1~
(8)
However, the low-frequency polarizability data are more accurate than the absorption data, and so an iterative procedure was adopted in which the spectral data were then, estimates of the UV sum used to first estimate dR; rules were obtained to fit the optical data and estimate aw, and then the microwave frequency data were used to refine the f,, and A, for n = 1 and 2, adjusting their magnitudes but letting the spectral data determine the ratio A2/A1. The adjustment required is small and the procedure converges rapidly. Because a ( w ) is known at so few optical frequencies, only the Sw(k) for lz = -2, -4, and -6 can be determined when they and higher coefficients are required to satisfy the Stieltjes constraints.' The results of this procedure are in Table I, where one should note that S(0) = 70 is the number of electrons in SF6, and S(-2) = a(0) so that the total static polarizability is aIR(0)+ (~""(0) = 4 4 . 2 1 ~ ~The ~ . uncertainties shown in Table I are the variations that allow satisfaction of the Stieltjes constraints. Because of the limited frequency range of the data, the actual uncertainties in S(-4) and S(-6) may be even larger than shown, as observed for other systems by Zeiss and Meath." The effective oscillator strengths and (15)K.Fox and W. B. Person, J. Chem. Phys., 64,5218 (1976). (16)W. B. Person and J. Overend, J. Chem. Phys., 66,1442 (1977).
2798
J. Phys. Chem. 1082, 86, 2796-2799
frequencies obtained for the IR and for the upper bounding [2,1] and lower bounding [1,0] Pade approximants are listed in Table 11. The small number of SUV(k)determinable from the data do not allow calculation of the lower [2,1] approximant. For consistency we could have used the [l,O] approximants to determine both the upper and lower bounds; however, the resulting C6 values differ negligibly from those obtained with the parameters of Table 11. The c6 coefficients for the interaction of SF6 with a number of atoms and molecules were calculated with the data in Tables I and I1 and similar data for the other species obtained from Langhoff and Karplus,' our earlier w ~ r k and, , ~ , ~for those systems for which they were given, the more accurate data of Zeiss and Meath.15 The results are in Table 111. The values shown are the means of the Pade bounds obtained by using the parameters of Table 11. However, the bounds have been extended to include the additional uncertainty generated by the uncertainties in the S ( k ) shown in Table I. The bounds are still not rigorous because, as noted above, the full uncertainites of S(-4) and S(-6) are not known. (17) G. D. Zeiss and W. J. Meath, Mol. Phys., 33, 1155 (1977).
Discussion From Table I11 it is clear that the small amount of data available for sF6has led to rather large uncertainties. One also sees that we have reported more significant figures than are justified. This is necessary to show the size of the infrared contribution to c6 which is always much smaller than the uncertainty. It should be noted that, for N20, CO, and NO, c6totcontains the partners NH,, H20, the IR contributions due to SF6 but not for the partner, as the input data on these molecules only included the UV contributions. However, even for the partners SF6 and COz, the IR contribution from both molecules accounts for less than 1% of c6 and is completely negligible compared to the uncertainty. The only cases in which C6IRis more than 1% of the total are when the partners are alkali atoms, and then Cem becomes as much as 3% of Cgbt but is still considerably less than the uncertainty. In conclusion, we have here determined the first realistic c 6 coefficients for SF6interactions. The uncertainties are larger than we would like but cannot be decreased without additional information. Acknowledgment. This research was supported by the U S . Department of Energy.
Vibrational Energy Transfer and Pyrolysis of Nitromethane by the Varlable Encounter Method W. Yuan,+ B. S. RablnovHch,' and R. Tosa Lbpartmnt of Chemistry BG10, Universtty of Washington, Seef#e, Washington 98195 (Received: January 13, 1982; In Final Form: February 8, 1982)
The pyrolysis of nitromethane has been studied by the variable encounter method (VEM) at temperatures from 816 to 1092 K with two reactors of differing geometry with fused silica surfaces. The probability of reaction per collision with the reador surface was measured. The down energy transition jump size, (AI?'), was determined. It decreased with increasing wall temperature. A comparison is made of (AI?') with previous results reported to date for other substrate molecules. Nitromethane is one of the more efficient energy transfer agents. However, in addition to the size (vibrational eigenstate density) and the polarity of the molecules, the nature of the hot surface seems also to play a role.
Introduction Nitromethane is a widely investigated species because of its importance in a number of phenomena including the mechanism of gas-phase nitration of hydrocarbons, the mechanism of formation of photochemical smog, and the chemistry of propellants. A large amount of work1+ has been devoted to the study of the kinetics and mechanism of the thermal decomposition of nitromethane. However, virtually no attention has been paid to the study of vibrational energy transfer involving this species. Of course, collisional activation and deactivation is a fundamental physical process of great ubiquity. Since nitromethane is a highly polar compound, it is also of interest to study the behavior of this molecule by the VEM method which has been applied recently to a variety of molecules, principally nonpolar hydrocarbon^.^^^ By this technique, vibrational Visiting scholar; permanent address: Department of Chemistry and Chemical Engineering, Quinghua University, Beijing, China. 0022-3654/82/2086-2796~01.2510
relaxation in the transient region may be studied, in principle, on a collision-by-collision basis. In this method, a molecule equilibrated at some low temperature in a reservoir flask is allowed to enter a reactor of variable dimensions, say cylindrical, heated to reaction ~~
~
(1) Dubikhin, V. V.; Nazin, G. M.; Manelis, G. B. Akad. Nauk SSSR, Chemical Sci. Diu.Bull. 1971, 1247. ( 2 ) Crawforth, C. G.;Waddington, D. J. J. Phys. Chem. 1970, 74,2793; Trans. Faraday SOC.1969,65, 1334. (3) Nazin, G. M.; Manelis, G . B.; Dubovitskii,F. I. Russ. Chem. Reu. 1963, 37, 603. (4) Zaalonko, I. S.; Kogarko, S. M.; Mozzhukhin, E. B.; Petrov, Yu. P. Kinet. Catal. 1972,13, 1001. (5) Glanzer, K.; Troe, J. Helo. Chim. Acta 1972,55, 2884; Troe, J. J. Chem. Phys. 1977,66,4758. (6) Bemon, S. W.; O'Neal, H. E. Natl. Stand. Ref. Data Ser., Natl. Bur. Stand. 1970, No. 21, 473. (7) Kelley, D. F.; Zalotai, L.; Rabinovitch, B. S. Chem. Phys. 1980,46, 379. ( 8 ) Yuan, W.; Tosa, R.; Chao, K.-J.; Rabinovitch, B. S. Chem. Phys.
Lett. In press.
0 1982 American Chemical Society
