7710
J. Phys. Chem. 1996, 100, 7710-7712
Dipole Polarizability, Cauchy Moments, and Related Properties of Hg Dirk Goebel and Uwe Hohm* Institut fu¨ r Physikalische und Theoretische Chemie der Technischen UniVersita¨ t Braunschweig, Hans-Sommer-Strasse 10, D-38106 Braunschweig, F.R.G. ReceiVed: January 24, 1996X
We report interferometric measurements of the frequency dependence of the refractive index n and the dipole polarizability R of Hg vapor. From the measured dispersion of the dipole polarizability the Cauchy moments S(-2) ) 33.91(34), S(-4) ) 279.8(8.4), and S(-6) ) 14 100(2810) have been obtained. The static limit S(-2) is in good accordance with ab initio calculations and results obtained from dielectric constant measurements. The Cauchy moments are used to predict approximations of the dispersion interaction energy constant C6 ) 255 au and the second hyperpolarizability γ(0) ) 2.2 × 104 au.
1. Introduction The refractive index n(λ,F,T) and the dipole polarizability R(λ) are fundamental electrooptical properties of matter. However, very few precise experimental data of n(λ,F,T) and R(λ) of metal atoms are available.1 The only exception is mercury, for which the frequency dependence of the refractive index is known in the wavelength range between 700 and 189 nm.2 But Wolfsohn2 has only reported the temperatures at which his measurements were performed. The corresponding density F of the mercury vapor must be calculated via vapor pressure data of Hg. This makes the calculation of the dipole polarizability from the refractive index and density data a little bit uncertain, which calls for a remeasurement of n and R of mercury. At present there is a rapidly increasing interest in the polarizability of heavy atoms3-5 and molecular compounds containing these elements.6 For quantum-chemical calculations the dipole polarizability of these elements provide a severe test case for the implementation of accurate relativistic corrections. Moreover, knowledge of R(λ) of mercury is extremely important in the evaluation of depolarized7 and polarized8 interaction induced light scattering spectra of Hg. On account of this renewed interest in the refractive index and polarizability of mercury, we have investigated the dependence of the refractivity [n(λ,F) - 1] on the wavelength λ and the density F of evaporized Hg. The measurements of [n(λ,F) - 1] were performed at three different wavelengths between 632.99 and 325.13 nm using a traditional interferometric technique. Additionally, we have recorded quasi-continuous refractive index spectra of Hg vapor via dispersive Fourier transform spectroscopy (DFTS)9,10 in the visible wavenumber range. 2. Experimental Section All interferometric measurements were performed with an evacuated Michelson interferometer.11 Absolute measurements of the refractive index were performed at vacuum wavelengths of λ1 ) 632.99 nm, λ2 ) 543.516 nm, and λ3 ) 325.13 nm, using HeNe and HeCd lasers as radiation sources. Mercury of mass m was filled into a cylindrical quartz cell of length l/2 ≈ 0.5 m and volume V ) 26.5 cm3. The cell was evacuated, fused off, and placed into one arm of the interferometer. The sample was evaporized by heating the cell with a pipe furnace to the
final temperature T. The temperature T was chosen high enough to ensure complete evaporation of the Hg sample. During evaporation, the resulting interference fringe shift ∆N was counted and the refractivity was obtained via n - 1 ) ∆Nλ/l, giving an uncertainty of ∆(n - 1)/(n - 1) ≈ 0.01. All results are summarized in Table 1. After complete evaporation, a 100 W halogen lamp was used as a white-light radiation source and dispersive Fourier spectra were recorded by moving one end mirror of the interferometer. Fourier transformation of the resulting interferogram yielded a quasi-continuous refractive index spectrum in the wavenumber range between σ ) 1/λ ) 10 000 cm-1 and σ ) 20 000 cm-1. This relative spectrum is fixed to the refractivities measured at λ1 and λ2. More details of the experimental procedure can be found in ref 9, where we have examined the complex refractive index of NO2/N2O4 and in refs 10 and 12, where the dipole polarizability of Cd has been obtained by interferometric high-temperature measurements. 3. Results and Discussion The dipole polarizability R(λ) for the three discrete wavelengths λ1, λ2, and λ3 is calculated via the Lorentz-Lorenz relation
n(λ,F)2 - 1
4 ) πNARF n(λ,F) + 2 3 2
(1)
where NA is Avogadro’s constant. We have used a constraint fit to ensure that n ) 1 at F ) 0. From this, the dipole polarizabilities R(λ1) ) 5.297(46), R(λ2) ) 5.428(47), and R(λ3) ) 6.615(49) have been obtained (all polarizabilities in 10-30 m3). Our DFTS measurements yield a quasi-continuous polarizability curve, with a resolution of about 5 cm-1. The arithmetic mean of 14 DFTS measurements is shown in Figure 1 together with our polarizability value obtained at λ3. The open squares indicate the polarizabilities obtained from Wolfsohn’s refractive index data.2 Overall, very good agreement can be found between our new values and the results of Wolfsohn.2 While Wolfsohn’s data extend far into the UVfrequency range, our measurements are closer to the static dipole polarizability of Hg. In order to obtain this static limit, we have fitted our data to a three-term Cauchy formula 2
* Author for correspondence. X Abstract published in AdVance ACS Abstracts, April 1, 1996.
S0022-3654(96)00231-6 CCC: $12.00
R(ω) ) ∑S(-2k - 2)ω2k k)0
© 1996 American Chemical Society
(2)
Properties of Hg Vapor
J. Phys. Chem., Vol. 100, No. 18, 1996 7711
TABLE 1: Experimental Conditions and Measured Refractivities n - 1 of Hg Vapor at Different Wavelengths λa 105(n - 1)
a
T/K
106m/kg
F/mol m-3
l/m
λ ) 632.99 nm
λ ) 543.516 nm
578 573 604 576 569 563 560 561 576 577 578 656 664 658 658 657 656
8.72(4) 8.72(4) 8.72(4) 8.72(4) 20.06(2) 20.06(2) 20.06(2) 20.06(2) 29.70(2) 29.70(2) 29.70(2) 128.45(2) 128.45(2) 128.45(2) 128.45(2) 128.45(2) 128.45(2)
1.640(20) 1.640(20) 1.640(20) 1.640(20) 3.774(32) 3.774(32) 3.774(32) 3.774(32) 5.587(46) 5.587(46) 5.587(46) 24.16(19) 24.16(19) 24.16(19) 24.16(19) 24.16(19) 24.16(19)
0.998442 0.998439 0.998457 0.998441 0.998989 0.998986 0.998984 0.998984 1.000469 1.000470 1.000470 1.003878 1.003883 1.003879 1.003879 1.003879 1.003878
3.3474 3.3493 3.4615
3.4023 3.5384 3.5165
8.3322 8.4590 8.5224
8.5146 8.6099 8.7323
12.8753 12.9259 13.0018 48.2090 48.1118 48.0867 48.0424 48.0804
13.2013 13.2692 13.3371 49.5024 49.2855 49.2694 49.1743 49.2640
λ ) 325.13 nm
4.0298
10.0811
60.3999
In all cases, the volume of the sample cell is V ) 26.5(2) cm , and the uncertainties in T and l are 1 K and 4 × 10
-5
3
Figure 1. Dispersion of the dipole polarizability of mercury: (b) this work, absolute determination at 325.13 nm; (s) this work, determined by dispersive Fourier transform spectroscopy with a resolution of 5 cm-1; (]) calculated from refractive index data of Wolfsohn.2
where R(ω) is the polarizability in atomic units (1 au ) 1.481 847 4 × 10-31 m3), ω is the frequency in atomic units (ω/au ) 4.556 335 × 10-6(σ/cm-1)), and S(k) are known as Cauchy moments. We have observed that a one-term Kramers-Heisenberg dispersion formula
ω02 R(ω) ) R(0) 2 ω0 - ω2
(3)
where ω0 is an effective transition frequency in the order of the ionization potential, is not as accurate as eq 2 for representing our measured polarizabilities. The Cauchy moments S(k) can be used to get approximations of the dispersion interaction energy constant C6 ≈ 3S2(-2)ω0/4, the second hyperpolarizability14 γ(0) ≈ 27.36S(-3)S(-4)/S(-2) ≈ 27.36S(-4)[S(4)/S(-2)](1/2), and the r-6 term of the mean incremental pair polarizability and its anisotropy.10,13 In particular, we obtain the Cauchy moments S(-2) ) 33.919(7) au, S(-4) ) 279.8(2.4) au, and S(-6) ) 14064(203) au. The values in parentheses are the rms errors of the fit, whereas the systematic errors in S(-2) and S(-4) are 1 and 3%, respectively. The uncertainty of S(-6) is in the order of 20%. The static polarizability S(-2) is in very good agreement with the result obtained from the dielectric constant of Hg reported by Wu¨sthoff15 of S(-2)
m, respectively.
) 33.7(1.3) au. Our result is also in fair agreement with recent relativistic MP4/QCISD(T) calculations performed by Pyykko¨ and co-workers3 giving S(-2) ) 33.44 au. Our static polarizability is, however, slightly larger than calculated by Kello¨ and Sadlej,4 who obtained 31.8 au with a quasi-relativistic CCSD(T) technique. Our measured result may help to decide between the two different techniques of calculation of Pyykko¨ and co-workers and of Kello¨ and Sadlej. To proceed with our analysis, an effective transition frequency of ω0 ) 0.296 au is obtained from eq 3. Hence, from our approximations given above we obtain C6 ≈ 255 au and γ(0) ≈ 2.2 × 104 au. The value of C6 obtained in this way is known to be too small by 10-15%.16 Nevertheless, it is in fair agreement with the experimental result of Stwalley and Kramer17 of 240 au and calculations of Maeder and Kutzelnigg18 of 222 au. It does, however, show a large deviation from the value of 399 au calculated in ref 3 and from an estimation given by Hensel et al.7 of 418 au. At present, the variety of these results calls for further examinations of C6 of mercury. On the other hand, our approximation of the second hyperpolarizability is in the order of the scaled result given by Hensel et al.7 of γ(0) ) 1.6 × 104 au. The r-6 terms of the incremental mean pair polarizability and its anisotropy are also in fair agreement with what is used in the interpretation of collision-induced light scattering experiments of mercury vapor. For the r-6 part of the mean incremental pair polarizability we obtain13 4S(-2)[S2(-2) + 2.85S(-4)] ) 2.64 × 105 au and for the anisotropy13 6S(-2)[S2(-2) + 1.14S(-4)] ) 2.99 × 105 au. These results are in fair agreement with the values used by Hensel et al.7,8 of 3.23 × 105 and 4.00 × 105 au, respectively. The difference between ours and Hensel’s results in these two invariants of the incremental pair polarizability tensor of mercury are mainly due to the difference in the C6 values. This again reflects the necessity of further determinations of the dispersion interaction energy constant C6 of mercury. Acknowledgment. Generous grants from the Deutsche Forschungsgemeinschaft and Fonds der Chemischen Industrie as well as technical assistance by B. Gerke-Kothe and A. Kru¨ger are gratefully acknowledged. References and Notes (1) Lide, D. R., Ed. CRC Handbook of Chemistry and Physics, 73rd ed.; CRC Press: Boca Raton, FL, 1992. (2) Wolfsohn, G. Z. Phys. 1933, 83, 234.
7712 J. Phys. Chem., Vol. 100, No. 18, 1996 (3) Schwerdtfeger, P.; Li, J.; Pyykko¨, P. Theor. Chim. Acta 1994, 87, 313. (4) Kello¨, V.; Sadlej, A. J. Theor Chim. Acta 1995, 91, 353. (5) Kadar-Kallen, M. A.; Bonin, K. D. Phys. ReV. Lett. 1994, 72, 828. (6) Kirpekar, S.; Oddershede, J.; Jensen, H. J. A. J. Chem. Phys. 1995, 103, 2983. (7) Barocchi, F.; Hensel, F.; Sampoli, M. Chem. Phys. Lett. 1995, 232, 445. (8) Sampoli, M.; Hensel, F.; Barocchi, F., private communication. (9) Goebel, D.; Hohm, U.; Kerl, K.; Tru¨mper, U.; Maroulis, G. J. Phys. Chem. 1994, 98, 13123.
Goebel and Hohm (10) (11) (12) (13) (14) (15) (16) (17) (18)
Goebel, D.; Hohm, U. Phys. ReV. 1995, A52, 3691. Hohm, U.; Kerl, K. Meas. Sci. Technol. 1990, 1, 329. Goebel, D.; Hohm, U. Microchim. Acta, in press. Hohm, U. Chem. Phys. Lett. 1993, 211, 498. Hohm, U. Chem. Phys. Lett. 1991, 183, 304. Wu¨sthoff, P. Ann. Phys. 1936, 27, 312. Hohm, U. Chem. Phys. 1994, 179, 533. Stwalley, W. C.; Kramer, J. J. Chem. Phys. 1968, 49, 5555. Maeder, F.; Kutzelnigg, W. Chem. Phys. 1979, 42, 95.
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