J. Phys. Chem. C 2007, 111, 15565-15568
15565
Third Virial Coefficients of Argon from First Principles† Alexandr Malijevsky´ Institute of Theoretical Physics, Faculty of Mathematics and Physics, Charles UniVersity Prague, V HolesˇoVicˇ ka´ ch 2, 180 00 Praha 8, Czech Republic and E. Ha´ la Laboratory of Thermodynamics, Institute of Chemical Process Fundamentals, Academy of Sciences of the Czech Republic, 165 02 Prague 6, Suchdol, Czech Republic
Frantisˇek Karlicky´ and Rene´ Kalus Department of Physics, UniVersity of OstraVa, 30. dubna 22, 701 03 OstraVa 1, Czech Republic
Anatol Malijevsky´ * Department of Physical Chemistry, Institute of Chemical Technology, 166 28 Prague 6, Czech Republic and Center of Biomolecules and Complex Molecular Systems, Institute of Chemical Technology, 166 28 Prague 6, Czech Republic ReceiVed: March 10, 2007; In Final Form: May 28, 2007
Third virial coefficients of argon have been calculated using a recent ab initio state-of-the-art pair potential and a new ab initio three-body potential. The theoretical results have been compared with experimental data in the range of temperatures from 130 to 673 K. The comparison reveals an excellent agreement between the theoretical and the newest experimental results: For all considered temperatures, differences between the theoretical and the experimental values are within uncertainties of the experimental data. The theoretical third virial coefficients reported in this work are also compared with those obtained using alternative theoretical approaches.
I. Introduction The goal of the molecular theory is an accurate (quantitative) prediction and interpretation of the properties of macroscopic systems without any adjustable parameters, that is, using only universal constants such as the charge of electron, velocity of light, and the Planck constant. The simplest real systems to reach the goal are rare gases for their sphericity and nonpolarity. Among them, argon is the most suitable for several reasons. First, it is the most experimentally explored rare gas; second, quantum effects at moderate temperatures and pressures are not too important, and third, the number of electrons is a reasonable compromise between helium and xenon, which is important in quantum chemical calculations. The simplest bulk quantity directly connected with two-body (pair) interactions is the second virial coefficient. Early calculations were based on the Lennard-Jones 12-6 and related potentials. It is well-known that these simple potentials with parameters such as and σ fit to the second viral coefficient or low-density viscosity data are not sufficiently accurate for a large range of state variables (temperature, density); see, for example, ref 1. Later, more sophisticated empirical and semiempirical pair potentials with a number of free parameters (their values were obtained by multi-property fits to second virial coefficients, low-density transport properties, spectral data, scattering data, etc.) combined with known dispersion coefficients (C6, C8, C10, etc.) were proposed.2 As a next step, attempts to calculate pair intermolecular potentials ab initio, that is, without any parameter fitted to experimental data, were made; see, for example, ref 3 and references therein. †
Part of the “Keith E. Gubbins Festschrift”. * Corresponding author.
Pair intermolecular potential is the only input for theoretical calculations of low-density bulk quantities. For a dense gas, liquid, and solid, three-body interactions must be taken into account (see below). It is believed that four-body and higher interactions are not important. Anyway, there are no tools to extract them with a reasonable accuracy using either theoretical methods or experimental data, at present. The simplest bulk property directly connected with threebody intermolecular interactions is the third virial coefficient, C. It can be written as a sum of three contributions
C ) Cadd + Cnadd + Cquant,
(1)
where Cadd is a part of the third viral coefficient calculated using pair intermolecular interactions only, Cnadd is a contribution of three-body interactions, and Cquant is a quantum correction to the classical third virial coefficient. It holds
Cadd ) - 16π2N2rm6
∫0∞∫r∞∫sr+s f(r) f(s) f(t) t dt s ds r dr
(2)
Cnadd ) 16π2N2rm6
∫0∞∫r∞∫sr+s e(r) e(s) e(t) [exp(-βu3) 1] t dt s ds r dr, (3)
Cquant ) N2h2rm4β2/(m)
∫0∞∫r∞∫sr+s f(s) f(t) e(r)
(
d2u2 2
dr
+
)
2 du2 t dt s ds r dr. (4) r dr
Here, f(r) ) e(r) - 1 ) exp[-βu2(r)] - 1 is the Mayer function, u2(r) is the pair potential, u3 ≡ u3(r, s, t) is the three-body
10.1021/jp071939a CCC: $37.00 © 2007 American Chemical Society Published on Web 07/21/2007
15566 J. Phys. Chem. C, Vol. 111, No. 43, 2007
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potential, β ) 1/kBT, rm is a position of the minimum of the pair potential, is its depth, m is the mass of the molecule (atom), and N is the Avogadro constant. Variables r, s, t are interparticle distances4 reduced by rm. In eq 4, only quantum corrections to Cquant caused by pair interactions are taken into account; three-body corrections have been omitted as they are small. Otherwise, eqs 2-4 are exact in the range of temperatures where the third virial coefficients are measured. The only inputs to eqs 2-4 are the pair and three-body potentials, u2(r) and u3(r,s,t), respectively. The most accurate pair potential of argon is a semiempirical potential of Aziz.5 However, present state-of-the-art computational science allows for the calculation of the pair potential for argon with almost the same accuracy from first principles.3 The three-body potential was often ignored in the past; see, for example, ref 1. However, its omission causes serious errors at low and mediate temperatures; see ref 6 and references therein. For example, at T ) 150 K, the three-body contribution to the third virial coefficient of argon is 30% of the total value.7 There are no sufficiently accurate experimental data (virial coefficients, spectral data, transport properties, scattering measurements) which allow for the construction of empirical threebody potentials in the same manner as it has been done for pair potentials.2,5 Thus, there remain two routes. The first is based on a truncated dispersion series8
u3(r, s, t) ) u3(DDD) + u3(DDQ) + u3(DQQ) + u3(DDO) + u3(QQQ) + ... (5) where DDD denotes dipole-dipole-dipole interaction, DDQ is dipole-dipole-quadrupole, DDO is dipole-dipole-octupole, and so forth. The triple dipole as represented by AxilrodTeller-Muto (DDD) term9
u3(DDD) )
ν(1 + 3 cos θ1 cos θ2 cos θ3) (rst)3
(6)
is often used to model u3 as it is believed that the higher order terms in the dispersion expansion, DDQ, DQQ, and so forth, fortunately cancel. A disadvantage of the dispersion series is that it diverges at short interparticle distances. It was partially corrected to charge-overlap effects by Varandas.10 The second route is based on ab initio calculations of u3 pioneered for neon by Ermakova et al.11 and for argon by Mas et al.12 by employing their SAPT approach (see ref 13 and references therein) and extended recently to supramolecular calculations by Karlicky´ and Kalus.14 A problem with the third virial coefficient is that it is difficult to extract its experimental values from volumetric measurements15 with sufficient precision. Thus, earlier attempts to judge various calculations of the third virial coefficient were inconclusive. Only recently, Tegeler et al.16 precisely recalculated experimental data of Gilgen et al.17 These new data help theory to judge more reliably the accuracy of different theoretical approaches. The aim of this paper is to calculate third virial coefficients of argon using the ab initio pair potential of ref 3 and the new ab initio three-body potential and to compare them thoroughly with available experimental data as well as with earlier theoretical estimates. II. Theory The ab initio pair potential of Slavı´cˇek et al.3 was used in this work. It was obtained using CCSD(T) all electron calcula-
tions with an extended basis set (aug-cc-pV6Z) combined with bond functions (spdfg). The calculated pair potential was tested on experimental data for the second virial coefficients, spectral characteristics, and scattering data. An excellent agreement between the theoretical and the experimental values was found, similar to that of the semiempirical pair potential of Aziz.5 The potential was tested (together with the three-body DDD term) on dense fluid thermodynamic properties,18 again in accordance with experimental data. The three-body potential was calculated using the supramolecular approach, the CCSD(T) correlation method, and the d-aug-cc-pVQZ basis set as implemented in the MOLPRO package.19 The points included in those calculations cover a broad range of Ar3 configurations that can be expected in condensed argon. For short interatomic distances, the reliability of the three-body potential is guaranteed up to Ar3 energies of ∼104 cm-1. For long interatomic distances, the calculations were performed so that the potential could be smoothly linked to the asymptotic expansion of eq 5 with the interaction coefficients taken from independent, presently the most accurate ab initio calculations by Thakkar et al.20 Detailed tests we performed for several D3h geometries of Ar3 indicate that the calculations were converged within
