First-principles prediction of the density correction to the Clausius–Mossotti function of strongly polar gases: the unresolved ammonia λ discrepancy from Van Vleck's translational fluctuation calculation

Van Vleck's quantum-mechanical revision (1937) of the translational fluctuation effect — the pair-collision contribution to the density dependence of the Clausius–Mossotti function of a gas — predicts $\lambda\approx 0.7$ for ammonia, substantially larger than the experimental value $0.3$ (Keyes and Kirkwood, 1930) and worse than the preceding classical estimate $\approx 0.4$; the note leaves the discrepancy unexplained. The same coefficient is the second dielectric virial coefficient $B_\epsilon=\lambda P_0^2$ of the modern dielectric virial expansion. Later work rebuilt the theory on intermolecular-potential grounds (Harris and Alder, 1953; Buckingham and Pople, 1955) and extended it with quantum corrections, induction, and shape terms (MacRury and Steele, 1974), but for strongly polar gases the calculated $B_\epsilon$ still overestimates the sparse experimental data (for NH$_3$: 990 versus 440 $\mathrm{cm^6\,mol^{-2}}$ at 100 °C in 1955; polarizable models still overpredict in 1974), no ab initio (first-principles, parameter-free) prediction validated against experiment exists for any strongly polar molecule (modern ab initio $B_\epsilon$ covers only He, Ne, Ar), and the experimental basis for NH$_3$ has not been renewed since 1948. The problem — predicting $\lambda$ (equivalently $B_\epsilon$) for polar molecular gases such as NH$_3$ from molecular theory in agreement with experiment, and identifying the physics responsible for the persistent overestimate — remains open.

Background

The relative permittivity (dielectric constant) $\epsilon$ of a gas increases with density. In the Debye–Langevin picture, treating each molecule as a polarizable point dipole and using the Lorentz local field (the electric field acting on a molecule sitting in a small spherical cavity carved out of the polarized medium), the Clausius–Mossotti function $(\epsilon-1)/(\epsilon+2)$ times the molar volume $V$ equals the molar polarization $P_0=(4\pi/3)N_A(\alpha+\mu^2/3kT)$, where $\alpha$ is the molecular polarizability, $\mu$ the permanent dipole moment, $N_A$ Avogadro's number, $k$ Boltzmann's constant, and $T$ the absolute temperature. $P_0$ is itself density-independent, so any observed density dependence of $(\epsilon-1)/(\epsilon+2)\,V$ reflects intermolecular interactions. That dependence is conventionally expanded as

$\frac{\epsilon-1}{\epsilon+2}\,V \;=\; P_0\left[1+\frac{\lambda P_0}{V}+\omega\left(\frac{P_0}{V}\right)^2+\cdots\right],$

with $V$ the molar volume. In the modern literature the same physics is written as the dielectric virial expansion $(\epsilon-1)/(\epsilon+2)V_m=A_\epsilon+B_\epsilon/V_m+C_\epsilon/V_m^2+\cdots$, in which $A_\epsilon=P_0$ and $B_\epsilon=\lambda P_0^2$ is the second dielectric virial coefficient.

Kirkwood (1936) showed that for nonpolar gases the leading deviation from the Clausius–Mossotti coefficient, caused by density fluctuations associated with the transient clustering of molecules during collisions (the "translational fluctuation" effect), is a small correction $\gamma\approx 0.1$ for most substances. Van Vleck (July 1937) extended the calculation to polar molecules classically and found adequate agreement with the measurements of the dielectric constant of ammonia as a function of temperature and density by Keyes and Kirkwood (1930). Van Vleck (December 1937) revised the calculation quantum-mechanically: the induced polarization of two colliding molecules was modeled as harmonic oscillators coupled by the dipole–dipole interaction — the same model London used for dispersion forces, in which quantized zero-point motion of the coupled oscillators produces the attractive van der Waals interaction — with the molecular rotational coordinates treated classically as slowly varying parameters. The resulting formula reduces, for nonpolar molecules, to Kirkwood's expression (resolving an earlier disagreement between the two authors about the role of the van der Waals term), but for NH$_3$ it yields $\lambda\approx 0.7$, appreciably larger than the experimental value $\lambda=0.3$ and worse than the earlier classical value $\approx 0.4$. The note leaves the source of this discrepancy unexplained and does not identify which physical ingredient is missing; predicting $\lambda$ for polar molecules such as NH$_3$ therefore remained an open problem.

The quantity $\lambda P_0^2=B_\epsilon$ is precisely the modern second dielectric virial coefficient, and the problem was subsequently attacked on that footing. Harris and Alder (1953) computed the leading pair term from intermolecular potentials for ammonia, methyl fluoride, and hydrogen fluoride, finding agreement "as good as can be expected" given the unsatisfactory precision of the experimental data. Buckingham and Pople (1955) derived the general statistical-mechanical series for $B_\epsilon$ of polar gases in terms of pair correlation functions for a Stockmayer potential (a Lennard-Jones 12:6 core plus a point-dipole–point-dipole interaction), with systematic corrections for molecular polarizability (dipole–induced–dipole effects), quadrupole moments, and anisotropic short-range repulsion (molecular shape); for NH$_3$ at $100^\circ$C their calculation still overestimates experiment by roughly a factor of two ($B_\epsilon$ calculated 990 versus measured 440 $\mathrm{cm^6\,mol^{-2}}$). High-precision measurement programs of the early 1970s (Sutter and Cole; Barnes and Sutton; Kong's related shape-factor theory) targeted halogenated methanes and other polar gases and continued to find model failures, for example fitted shape parameters of the wrong sign for fluoroform. MacRury and Steele (1974) recomputed $B_\epsilon$ for water and ammonia including translational and rotational quantum corrections, dipole–induced–dipole terms, and polarizability: polarizable models still overpredict the sparse ammonia data, and sharply different potential-parameter sets fit the data equally well, so the available measurements cannot discriminate among models. Later work largely moved elsewhere — to rigorous results for idealized hard-sphere and dumb-bell models (Buckingham and Joslin, 1980), to liquid-state theories of polarizable polar fluids (Høye and Stell, 1980; Thompson, Schweizer and Chandler, 1982, whose mean-spherical-approximation treatment — an exactly solvable liquid-state theory — of fluids whose molecular polarizability is represented by quantum Drude oscillators, that is, by harmonically bound charges, cites that note), and to collision-induced spectroscopy (spectral signals produced by the interaction within transient molecular pairs; Bose, 1985) — while ab initio predictions of $B_\epsilon$ validated against experiment exist only for atomic gases (helium; helium, neon, argon), where they underpin dielectric-constant gas thermometry, the metrological use of gas permittivity to determine temperature and pressure.

Problem Statement

Predict, from molecular theory, the coefficient $\lambda(T)$ — equivalently the second dielectric virial coefficient $B_\epsilon(T)=\lambda P_0^2$ — that governs the leading density dependence of the Clausius–Mossotti function of polar molecular gases, with ammonia (NH$_3$) as the required benchmark and a class-wide treatment of other strongly polar gases welcome but not required. For ammonia, determine the function $\lambda(T)$ (equivalently $B_\epsilon(T)$) that agrees with the measured coefficient at the $100^\circ$C benchmark ($\lambda\approx 0.3$, equivalently $B_\epsilon\approx 440\ \mathrm{cm^6\,mol^{-2}}$, a margin far below the historical quantum-mechanical overestimate of $0.7$) and that follows the observed temperature dependence over the range of the available measurements (the right sign and order of magnitude). That result must be obtained from the molecular model rather than by fitting adjustable parameters to the target datum. Taking the measured ammonia coefficients as the reference, the problem poses a second question: why does the molecular model overestimate them? The persistent theoretical overestimate (Van Vleck 1937: $\lambda\approx 0.7$ from the quantum-mechanical translational-fluctuation formula, $\approx 0.4$ from the classical one, versus the experimental value $0.3$ from the Keyes–Kirkwood measurements; Buckingham and Pople 1955: $B_\epsilon$ calculated 990 versus 440 $\mathrm{cm^6\,mol^{-2}}$ at $100^\circ$C; MacRury and Steele 1974: polarizable models still overpredict the sparse ammonia data) must be resolved into its physical contributions, with the dominant contribution and its approximate magnitude identified. What the 1937 note left open is why theory persistently overestimates the coefficient for strongly polar gases such as NH$_3$, and which calculation of $\lambda(T)$/$B_\epsilon(T)$ agrees with reliable experimental data.

References

1. J. H. Van Vleck, 'Revised Calculation of the Translational Fluctuation Effect in Gaseous Dielectrics', The Journal of Chemical Physics 5, 991 (1937). DOI: 10.1063/1.1749980, https://doi.org/10.1063/1.1749980
2. J. H. Van Vleck, 'On the Role of Dipole-Dipole Coupling in Dielectric Media', The Journal of Chemical Physics 5, 556–568 (1937). DOI: 10.1063/1.1750074, https://doi.org/10.1063/1.1750074
3. J. G. Kirkwood, 'On the Theory of Dielectric Polarization', The Journal of Chemical Physics 4, 592–601 (1936). DOI: 10.1063/1.1749911, https://doi.org/10.1063/1.1749911
4. F. G. Keyes and J. G. Kirkwood, 'The Dielectric Constant of Ammonia as a Function of Temperature and Density', Physical Review 36, 1570–1575 (1930). DOI: 10.1103/PhysRev.36.1570, https://doi.org/10.1103/PhysRev.36.1570
5. L. Onsager, 'Electric Moments of Molecules in Liquids', Journal of the American Chemical Society 58, 1486–1493 (1936). DOI: 10.1021/ja01299a050, https://doi.org/10.1021/ja01299a050
6. F. London, 'The general theory of molecular forces', Transactions of the Faraday Society 33, 8 (1937). DOI: 10.1039/tf937330008b, https://doi.org/10.1039/tf937330008b
7. A. van Itterbeek and K. de Clippeleir, 'Measurements on the dielectric constant of gaseous ammonia, carbon oxide and hydrogen as a function of pressure and temperature', Physica 14, 349–356 (1948). DOI: 10.1016/0031-8914(48)90052-4, https://doi.org/10.1016/0031-8914(48)90052-4
8. F. E. Harris and B. J. Alder, 'Intermolecular Potentials from Dielectric Polarization in Polar Gases', The Journal of Chemical Physics 21, 1351–1357 (1953). DOI: 10.1063/1.1699219, https://doi.org/10.1063/1.1699219
9. M. Katayama, 'On the Temperature Dependence of the Molecular Polarization of Ammonia', The Journal of Chemical Physics 22, 1473–1476 (1954). DOI: 10.1063/1.1740441, https://doi.org/10.1063/1.1740441
10. A. D. Buckingham and J. A. Pople, 'The dielectric constant of an imperfect non-polar gas', Transactions of the Faraday Society 51, 1029–1035 (1955). DOI: 10.1039/tf9555101029, https://doi.org/10.1039/tf9555101029
11. A. D. Buckingham and J. A. Pople, 'The statistical mechanics of imperfect polar gases. Part 2.—Dielectric polarization', Transactions of the Faraday Society 51, 1179–1183 (1955). DOI: 10.1039/tf9555101179, https://doi.org/10.1039/tf9555101179
12. C. L. Kong, 'Dielectric Second Virial Coefficients of Simple Polar Gases', The Journal of Chemical Physics 53, 1522–1524 (1970). DOI: 10.1063/1.1674205, https://doi.org/10.1063/1.1674205
13. H. Sutter and R. H. Cole, 'Dielectric and Pressure Virial Coefficients of Imperfect Gases. I. Polar Halogenated Methanes', The Journal of Chemical Physics 52, 132–139 (1970). DOI: 10.1063/1.1672657, https://doi.org/10.1063/1.1672657
14. A. N. M. Barnes and L. E. Sutton, 'Equilibrium properties of polar gases. Part 2.—The second dielectric virial coefficients of some polar gases', Transactions of the Faraday Society 67, 2915–2925 (1971). DOI: 10.1039/tf9716702915, https://doi.org/10.1039/tf9716702915
15. T. B. MacRury and W. A. Steele, 'Quantum effects on the dielectric virial coefficients of polar gases', The Journal of Chemical Physics 61, 3352–3365 (1974). DOI: 10.1063/1.1682498, https://doi.org/10.1063/1.1682498
16. T. B. MacRury and W. A. Steele, 'Calculations of the second virial coefficients for water and ammonia', The Journal of Chemical Physics 61, 3366–3375 (1974). DOI: 10.1063/1.1682499, https://doi.org/10.1063/1.1682499
17. A. D. Buckingham and C. G. Joslin, 'The second dielectric virial coefficient of dipolar hard-sphere fluids', Molecular Physics 40, 1513–1516 (1980). DOI: 10.1080/00268978000102461, https://doi.org/10.1080/00268978000102461
18. J. S. Høye and G. Stell, 'Dielectric theory for polar molecules with fluctuating polarizability', The Journal of Chemical Physics 73, 461–468 (1980). DOI: 10.1063/1.439897, https://doi.org/10.1063/1.439897
19. M. J. Thompson, K. S. Schweizer and D. Chandler, 'Quantum theory of polarization in liquids: Exact solution of the mean spherical and related approximations', The Journal of Chemical Physics 76, 1128–1135 (1982). DOI: 10.1063/1.443081, https://doi.org/10.1063/1.443081
20. T. K. Bose, 'A Comparative Study of the Dielectric, Refractive and Kerr Virial Coefficients', in Phenomena Induced by Intermolecular Interactions, edited by G. Birnbaum, pp. 49–66, Plenum Press (1985). DOI: 10.1007/978-1-4613-2511-6_3, https://doi.org/10.1007/978-1-4613-2511-6_3
21. R. Moszynski, T. G. A. Heijmen and A. van der Avoird, 'Second dielectric virial coefficient of helium gas: quantum-statistical calculations from an ab initio interaction-induced polarizability', Chemical Physics Letters 247, 440–446 (1995). DOI: 10.1016/0009-2614(95)01271-0, https://doi.org/10.1016/0009-2614(95)01271-0
22. B. Song and Q.-Y. Luo, 'Accurate second dielectric virial coefficient of helium, neon, and argon from ab initio potentials and polarizabilities', Metrologia 57, 025007 (2020). DOI: 10.1088/1681-7575/ab62c3, https://doi.org/10.1088/1681-7575/ab62c3