Calculation of second virial coefficients using ab initio intermolecular pair potentials for F2-F2 and H2-F2 dimers

Chemical Physics 485–486 (2017) 67–80

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Calculation of second virial coefficients using ab initio intermolecular pair potentials for F2-F2 and H2-F2 dimers Tat Pham Van a,⇑, Ulrich K. Deiters b a b

Faculty of Science and Technology, Hoa Sen University, Viet Nam Institute of Physical Chemistry, University of Cologne, Luxemburger Str. 116, D-50939 Köln, Germany

a r t i c l e

i n f o

Article history: Received 4 May 2016 In final form 6 January 2017 Available online 17 January 2017 Keywords: Ab initio 5-site potentials Second virial coefficients Ab initio interaction energy

a b s t r a c t The ab initio intermolecular pair potentials of dimers F2-F2 and H2-F2 were calculated from all constructed orientations, using the level of theory CCSD(T) and basis sets aug-cc-pVmZ (m = 2, 3, 23). The complete basis set limit aug-cc-pV23Z was extrapolated by ab initio interaction energies at the level of theory CCSD(T) with two basis sets aug-cc-pVmZ (m = 2, 3). Then the quantum mechanical results were used for constructing two new 5-site potential functions by fitting ab initio energies of dimers F2-F2 and H2F2. The correlation between ab initio and the fitted ab initio energies of 5-site pair potentials for dimers F2-F2 and H2-F2 is appeared by fitness values R2 in range 0.99749–0.99997. The fitted potentials were used in standard thermodynamic relations to obtain the second virial coefficients and the results were compared to experimental data. Ó 2017 Elsevier B.V. All rights reserved.

1. Introduction Hydrogen, Fluorine and the mixture hydrogen-fluorine are used in several industrial areas. Hydrogen in its liquid form has been used as a fuel in space vehicles for years [1]. It could become the most important energy carrier of tomorrow [2]. Liquids hydrogen and fluorine can be used such as the usual liquid fuels for rocket engines [3]. The National Aeronautics and Space Administration (NASA) is the largest user of liquid hydrogen in the world [3,4]. The knowledge of thermodynamic properties of the pure substances hydrogen, fluorine and the mixture and hydrogen-fluorine are important for practical applications. It is also necessary for their safe use [4,5]. Computer simulations have expanded in number, complexity, and importance over the last many years [6]. Computers permit the study of systems for which analytical solutions are not available or require approximation techniques. Molecular simulations have been used for various studies [7]. The properties of the studied systems are determined solely by the intermolecular forces and energies [6,7]. Therefore, simulations have become a necessary tool for studying fluids and fluid mixtures. They can generate structural and thermodynamic as well as transport properties consistently without the need to introduce artificial simplifications as required by integral equation techniques, and statistical thermodynamic ⇑ Corresponding author. E-mail address: [email protected] (T.P. Van). http://dx.doi.org/10.1016/j.chemphys.2017.01.003 0301-0104/Ó 2017 Elsevier B.V. All rights reserved.

perturbation theory [6,7]. Computer simulation techniques such as Monte Carlo and molecular dynamics require a knowledge of intermolecular potentials. Monte Carlo simulation is used to calculate widely phase equilibria of both polymeric and low molecular organic substances [8]. The field of phase equilibria simulation is now highly developed and very broad in techniques and applications. Some recently published works, are de Pablo et al. 1998 and 2000 [9,10], Delhommelle et al. 2000 [11], Martin and Siepmann 1998 and 1999 [12,13] and Spyriouni et al. 1998 [14]. Gibbs ensemble simulation had been developed by Panagiotopoulos 1987 [8,15,16]. The basic idea in the Gibbs ensemble method is to simulate phase coexistence properties by following the evolution in phase space of a system composed of two distinct regions. The two regions in the simulation system represent the two coexistence phases, e.g. a vapor in equilibrium with a liquid at saturation. Vapor-liquid equilibria can also be estimated with several equations of state and with Monte Carlo simulation using analytic potential functions, in which the usual procedure is to assume a simple model potential, e.g., the Lennard-Jones pair potential 1924 [17] and the Morse potential 1929 [18], fit its parameters to suitable experimental data, and then to perform the simulation. Such a simulation is no longer predictive, because it requires an experimental input of the same kind that it produces. This can sometimes be a severe limitation, namely if experimental data are scarce.

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T.P. Van, U.K. Deiters / Chemical Physics 485–486 (2017) 67–80

Recently an alternative approach has become feasible, for which the name ‘‘global simulation” has been coined by H. Popkie et al. 1973 [19]. It consists of a calculation of intermolecular potentials by quantum mechanical methods, followed by computer simulations and eventually calculations with equations of state fitted to the simulation results, in order to obtain properties that are not accessible to simulations. Such global simulations have been reported for the noble gases, where it is now possible to predict the vapor-liquid phase equilibria without recourse to experimental data with an accuracy comparable to the experimental uncertainty. One of the first attempts that achieved near-experimental accuracy was that of Deiters, Hloucha and Leonhard 1999 [20] for neon. Further global simulation attempts for noble gases were published by the groups of Eggenberger and Huber [21–24], Sandler [25], and Malijevsky´ [26]. Using a functional form for the dispersion potentials of argon and krypton proposed by Korona et al. [27], Nasrabad and Deiters 2003, 2004 even predicted phase high-pressure vaporliquid phase equilibria of noble-gas mixtures [28–32]. Other mixed-dimer pair potentials for noble gases were published by López Cacheiro et al. 2004 [33], but they were not used for phase equilibria predictions, yet. The development of ab initio pair potentials for molecules is much more complicated because of the angular degrees of freedom of molecular motion, but for some simple molecules such potentials have already been constructed: Leonhard and Deiters 2002 used a 5-site Morse potential to represent the pair potential of nitrogen [34] and were able to predict vapor pressures and orthobaric densities. In recent attempts Tat and Deiters also proposed the 5-site ab initio pair potentials for hydrogen and mixture hydrogen-oxygen [35]. Bock et al. 2000 used a 5-site pair potential for carbon dioxide [36]. In recent years there are some articles for constructing ab initio potential surfaces of fluorine proposed by Karimi-Jafari [37], and Hassan Sabzyan proposed the intermolecular potential energy surface for the F2-F2 system using MP2/6-31G⁄ calculations [38]. The ab initio intermolecular potentials of F2-F2 was also improved by CCSD(T) and QCISD(T) calculations using the aug-cc-pVDZ and aug-cc-pVTZ basis sets proposed by Mansoor Namazian et al. [39]. But those were not fitted for the Morse-style potential functions to calculate the virial coefficients of fluorine. In this work we reported the construction of the angular orientations of the dimers F2-F2 and H2-F2; the calculation of the ab initio intermolecular energies; correction of the ab initio energy results for the basis set superposition error (BSSE) with the counterpoise method; extrapolation of the interaction energies to the complete basis set limit aug-cc-pV23Z; construction of new 5-site ab initio intermolecular potentials of dimers F2-F2 and H2-F2 along the proposed potentials of dimers hydrogen, hydrogen-oxygen [35] and hydrogen chloride [41]; calculation of the second virial coefficients of dimer F2-F2 and the cross second virial coefficients of dimer H2F2 by 4D numerical integrals eventually including corrections for quantum effects; the accurate comparison of the obtained virial coefficients of this work with experimental data and with those from the correlation equations and equations of state.

2. Computational details 2.1. Molecular orientation In this work hydrogen and fluorine molecules are represented as 5-site models, with two sites placed on the atoms (H or F), one site in the center of gravity (M), and two sites halfways between the atoms and the center (N). The molecules are treated as rigid; the interatomic distances are set to 0.74140 Å for hydrogen and 1.41190 Å for fluorine [42]. As hydrogen and fluorine are linear molecules, the intermolecular pair potential is a function

of distance r (distance between the centers of gravitiy) and three angular coordinates, a, b, and /, which are explained in Fig. 1. Interaction energies were calculated for all values of r from 2.6 to 15 Å with increment 0.2 Å; the angles a, b, and /, were varied from 0 to 180° with increment 45°. Care was taken to recognize identical configurations in order to reduce the computational workload. 2.2. Quantum chemical calculations To calculate ab initio interaction energies the Hartree-Fock method provides approximate solutions to the Schrodinger equation by replacing the real electron-electron interaction with an average interaction. The electron correlation (EC) energy is the energy difference between the Hartree-Fock and the lowest possible energy in each basis set. For systems and states where correlation effects are important, the Hartree-Fock results will not be satisfactory [43,44]. QM methods have been developed to include some effects of electron correlation [45,46,37]. The coupled-cluster correlation correction method presents one of the most successful approaches to account for the many-electron molecular systems. It might applied to relatively large systems and is capable of recovering a large part of the correlation energy [43,45,46]. However the Coupled-Cluster wave function provides an accurate correlation to the Hartree-Fock description [43,46]. Because of the diffuse, wide-range nature of dispersion force fields it is necessary to adopt appropriate basis sets. Here we use the correlation-consistent basis sets of Dunning et al. [47]: augcc-pVDZ (for hydrogen: 5s2p/3s2p; for fluorine: 10s5p2d/4s3p2d) and aug-cc-pVTZ (for hydrogen: 6s3p2d/4s3p2d; for fluorine: 11s6p3d2f/5s4p3d2f). The ab initio energy results were corrected for the basis set superposition error with the counterpoise correction method proposed by Boys and Bernardi [43,48]

DE ¼ EAB  EAb  EaB

ð1Þ

where EAB denotes the total electronic energy of a dimer AB, EAb the energy of a dimer consisting of an A atom and a B ghost atom (an atom without nucleus and electrons, but having its orbitals), and EaB vice versa. The electronic energies are then extrapolated to the basis set limit [43,46,49–51]:

DEðmÞ ¼ DEð1Þ þ cm3

ð2Þ

with m = 2 (for the aug-cc-pVDZ basis set) or 3 (for aug-cc-pVTZ). If results for two basis sets are available, it is possible to calculate the energy value for an infinite basis set from Eq. (2); this result is referred to as aug-cc-pV23Z below. In order to verify that the calculation method and the different basis sets can yield an adequacy to describe for dimers H2-F2 and F2-F2, we approached to perform calculations without restrictions of the intramolecular bond length of molecules H2 and F2 in optimal T-configuration: The results, listed in Table 1, show that the CCSD(T) calculation with the extrapolated basis set gives results within 1% uncertainty of the experimental values. This also proves that the molecules are not significantly deformed upon making contact with each other, or that vibrational excitation is not to be expected with low-speed collisions. All quantum chemical calculations were carried out with the Gaussian03TM program package [45,52]. The influence of the choice of the theory levels with various basis sets is shown in Figs. 2 and 3 for four special molecular configurations for F2-F2 and H2-F2 dimers, as given in Fig. 1. In all cases the extrapolation to the basis set limit has a significant effect on the calculated energies. We can use the basis set limit aug-cc-pV23Z for all calculations in this work, because of

T.P. Van, U.K. Deiters / Chemical Physics 485–486 (2017) 67–80

69

Fig. 1. 5-site molecular model and some selected orientations used for quantum chemical calculations.

Table 1 Comparison of interatomic lengths/Å of molecules H2 and F2 in dimer H2-F2 for T orientation (a = 90, b = 0, / = 0) at the theoretical levels CCSD(T)/aug-cc-pVmZ (m = 2, 3, 4, 23, 34), respectively. Basis set

rF-F/Å

rH-H/Å

aug-cc-pVDZ aug-cc-pVTZ aug-cc-pV23Z aug-cc-pVQZ aug-cc-pV34Z Exp [42]

1.45016 1.41823 1.40479 1.41309 1.41092 1.41190

0.76176 0.74296 0.73504 0.74199 0.74158 0.74140

aug-cc-pV34Z the calculation time for each configuration can be taken a lot of hours. So computational cost will be more expensive than the use of basis sets limit aug-cc-pV23Z. In Table 1 the difference errors between two basis sets aug-cc-pV23Z and aug-cc-pVQZ are insignificant for ab initio calculation in this work. The locations and depths of the potential minima for four different dimer configurations are shown in Table 2. The potential energies of the dimers F2-F2 and H2-F2 for all distances and orientations, calculated with the CCSD(T) method and the aug-cc-pVmZ basis sets, are given in Table 3. 3. The pair potential function

the extrapolated results of interatomic lengths derived from two basis sets limit aug-cc-pVmZ (m = 23, 34) are not different from experimental value. But in case of the use of basis set limit

For the calculation of virial coefficients, especially for the use within computer simulations, it is necessary to represent the ab ini-

Fig. 2. Comparison between intermolecular potentials of dimer F2-F2 calculated with method CCSD(T) for various basis sets: j: aug-cc-pVDZ; s: aug-cc-pVTZ; d: aug-ccpVQZ; –: aug-cc-pV23Z.

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Fig. 3. Comparison between intermolecular potentials of dimer H2-F2 are calculated with method CCSD(T) for various basis sets; for an explanation of the symbols see Fig. 2.

Table 2 Comparison of minimum interaction energies Emin/lEH and equilibrium distance rmin/Å of dimers F2-F2 and H2-F2 for selected orientations in Fig. 1 were calculated at the level of theory CCSD(T)/aug-cc-pVmZ (m = 2, 3, 23), respectively. Angle/degree

Dimer F2-F2 m=2

m=3

m = 23

a

b

/

rmin/Å

106Emin/EH

rmin/Å

106Emin/EH

rmin/Å

106Emin/EH

0 90 90 90

0 0 90 90

0 0 0 90

4.8 3.8 3.4 3.2

7.730 398.390 250.900 313.590

4.4 3.6 3.2 3.0

132.120 547.080 410.000 513.590

4.4 3.6 3.2 3.0

202.000 616.926 501.941 637.511

Angle/degree

Dimer H2-F2 m=2

m=3

m = 23

a

b

/

rmin/Å

106Emin/EH

rmin/Å

106Emin/EH

rmin/Å

106Emin/EH

0 90 90 90

0 0 90 90

0 0 0 90

4.2 3.6 3.4 3.4

48.3917 277.2440 122.5280 119.0350

4.0 3.6 3.2 3.2

87.0103 412.3610 206.9500 195.9250

4.0 3.4 3.2 3.2

105.4125 487.5303 252.7370 240.9529

tio pair potential data by an analytic function. As the molecules considered here are not spherical, the construction of an accurate, yet not too computationally expensive function is not a trivial task. Modelling the molecular anisotropy by spherical harmonics is possible in principle, but not attempted here. Instead, a multi-center potential is proposed. Previous works on nitrogen [34], hydrogen and hydrogen-oxygen [35] and carbon dioxide [36] have shown that 5-site intermolecular models represented the ab initio data well, and that 5-site models were sufficient. In principle such a 5-site model leads to a pair potential consisting of 25 spherical site-site interactions, but because of molecular symmetry only six different site-site interaction potentials have to be fitted for dimer F2-F2, and 8 for dimer H2-F2. Our two new 5-site functional forms were explained in this work. The first one is based on a 5-site pair potential developed

originally by Bock et al. [36] for carbon dioxide and later used by Leonhard and Deiters [34] for nitrogen and by Tat and Deiters [35] for hydrogen:

uðr ij ; a; b; /Þ ¼

5 X 5 h X

Dije



1  eaij ðrij bij Þ

2

i¼1 i¼1

þf 1 ðr ij Þ

X C ij qi qj n þ n r 4 p e0 rij ij n¼6;8;10

1



!# ð3Þ

 15 with f 1 ðrij Þ ¼ 1 þ e2ðdij rij 2Þ . The second functional from was originally proposed by Naicker et al. [41] as a 3-site model and then utilized by the authors [35] as a 5-site model for H2-H2 and H2-O2 dimers:

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T.P. Van, U.K. Deiters / Chemical Physics 485–486 (2017) 67–80 Table 3 Pair interaction energies calculated with the CCSD(T) method and the basis sets aug-cc-pVmZ, m = 2, 3, 23*. r/Å

a

b

/

106u/EH F2-F2

r/Å

106u/EH H2-F2

m=2

m=3

m = 23

3.8 4 4.2 4.4 4.6 4.8 5.2 5.4 5.6 15

0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0

1507.75 499.98 148.6 34.08 0.01 7.73 4.55 1.9 0.19 0.02

1057.7 187.76 79.44 132.12 118.41 91.01 47.23 33.84 24.33 0.03

m=2

m=3

m = 23

868.47 56.49 175.32 202 168.2 126.03 65.18 47.27 34.64 0.05

3.4 3.6 3.8 4 4.2 4.4 4.6 5 6.5 15

505.38 136.96 1.18 43.31 48.39 41.16 31.47 16.05 0.43 0.08

335.88 28.5 70.33 87.01 75.99 58.85 43.1 21.53 0.97 0.08

264.52 17.17 99.44 105.41 87.61 66.3 48 23.84 1.2 0.08

3.6 3.8 4 4.2 4.4 4.6 5 8 15

45 45 45 45 45 45 45 45 45

0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0

536.43 182.98 349.4 334.09 273.87 212.72 122.17 4.14 0.17

289.26 365.5 490.75 439.45 347.76 262.29 144.62 5.02 0.18

185.34 442.24 550.18 483.75 378.83 283.13 154.06 5.39 0.18

3.2 3.4 3.6 3.8 4 4.2 4.8 8 15

681.25 116.64 89.38 144.7 141.62 119.69 55.62 1.64 0.01

412.55 67.83 212.51 224.31 192.02 151.53 64.95 1.82 0.01

299.41 145.5 264.36 257.82 213.24 164.93 68.88 1.9 0.01

3.2 3.4 3.6 3.8 4 4.2 4.4 4.6 5.4 8 15

90 90 90 90 90 90 90 90 90 90 90

0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0

776.13 132.78 380.96 398.39 346.68 282.41 223.71 175.26 67.88 6.93 0.34

472.78 356.72 547.08 513.15 420.02 326.86 250.28 191.46 71.08 6.79 0.3

345.24 450.88 616.93 561.4 450.86 345.55 261.45 198.27 72.43 6.73 0.28

3 3.2 3.4 3.6 3.8 4 4.2 4.4 5 9 15

965.21 115.51 194.63 277.24 269.03 229.62 184.64 144.24 66.65 1.88 0.09

591.55 173.79 400.75 412.36 352.55 280.14 215.8 164.44 73.91 1.99 0.1

434.22 295.6 487.53 469.25 387.72 301.4 228.92 172.94 76.97 2.03 0.11

3.5 3.7 3.8 4 4.2 4.4 4.6 5 6 15

135 135 135 135 135 135 135 135 135 135

45 45 45 45 45 45 45 45 45 45

0 0 0 0 0 0 0 0 0 0

633.37 49.58 84.33 198.4 213.62 192.62 161.68 105.09 35.89 0.32

382.63 128.62 230.99 292.81 271.31 227.04 182.38 113.48 37.09 0.28

277.21 203.55 292.65 332.51 295.57 241.51 191.08 117.01 37.59 0.26

3 3.2 3.4 3.6 3.8 4 4.2 4.6 6 15

1175.25 343.85 5.84 132.56 161.05 150.16 126.84 81.36 16.99 0.08

879.42 137.76 143 220.86 217.03 185.87 150.14 92.09 18.35 0.08

754.86 50.99 200.75 258.04 240.6 200.91 159.95 96.61 18.92 0.08

3 3.2 3.4 3.6 3.8 4 4.2 4.4 6 15

90 90 90 90 90 90 90 90 90 90

45 45 45 45 45 45 45 45 45 45

0 0 0 0 0 0 0 0 0 0

1343.97 202.2 182.1 275.86 265.86 224.67 179.48 139.68 20.42 0.22

926.04 110.38 406.71 427.25 363.21 286.41 219.45 166.62 23.38 0.2

750.32 241.81 501.15 490.9 404.14 312.37 236.26 177.95 24.62 0.19

3.2 3.4 3.6 3.8 4 4.2 4.4 4.6 6 15

511.89 134.73 20.27 72.91 81.59 73.79 61.54 49.6 10.19 0.03

289.83 5.71 106.1 125.41 114.64 95.45 76.26 59.84 11.65 0.04

196.34 64.84 142.25 147.51 128.55 104.57 82.46 64.16 12.26 0.05

3.6 3.8 4 4.2 4.4 4.6 4.8 5 6 15

45 45 45 45 45 45 45 45 45 45

45 45 45 45 45 45 45 45 45 45

0 0 0 0 0 0 0 0 0 0

1040.07 302.81 31.59 57.02 76.43 71.2 58.88 46.07 11.63 0.07

716.93 70.83 132.59 169.79 152.49 122.86 94.96 72.35 19.21 0.1

581.06 26.71 201.62 217.2 184.47 144.58 110.13 83.4 22.4 0.11

3.2 3.4 3.6 3.8 4 4.2 4.4 4.6 6 15

643.66 193.19 9.81 54.34 67.93 62.28 51.07 39.84 5.71 0.04

452.7 64.38 75.03 109.17 103.19 85.14 66.18 50.05 6.88 0.03

372.3 10.15 110.75 132.25 118.04 94.76 72.55 54.35 7.38 0.02

2.8 3 3.2 3.4 3.6 3.8 4 4.2 5 9

90 90 90 90 90 90 90 90 90 90

90 90 90 90 90 90 90 90 90 90

0 0 0 0 0 0 0 0 0 0

885.84 79.38 191.33 250.9 232.91 192.13 150.36 114.97 38.78 0.64

394.61 260.74 410 386.33 317.69 247.82 189.15 143.25 48.68 1.03

188.07 403.75 501.94 443.27 353.34 271.24 205.46 155.14 52.84 1.19

2.6 2.8 3 3.2 3.4 3.6 3.8 4 5 9

1063.97 327.46 14.57 98.21 122.53 112.55 92.74 72.83 18.67 0.22

636.37 45.28 162.09 206.95 190.64 156.73 122.35 93.17 22.85 0.32

456.32 73.53 236.47 252.74 219.31 175.33 134.82 101.74 24.61 0.36

(continued on next page)

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T.P. Van, U.K. Deiters / Chemical Physics 485–486 (2017) 67–80

Table 3 (continued) r/Å

a

b

/

106u/EH F2-F2 m=2

m=3

m = 23

m=2

m=3

m = 23

15

90

90

0

0.12

0.12

0.12

15

0.02

0.02

0.02

3 3.2 3.4 3.6 3.8 4 4.2 4.4 5 9 15

90 90 90 90 90 90 90 90 90 90 90

45 45 45 45 45 45 45 45 45 45 45

45 45 45 45 45 45 45 45 45 45 45

1415.68 259.31 147.62 258.25 258.35 222.58 180.06 141.42 66.47 2.03 0.23

995.73 54.81 370.27 405.65 351.56 280.9 217.39 166.49 76.02 2.18 0.2

819.16 186.88 463.88 467.63 390.75 305.42 233.09 177.03 80.04 2.24 0.19

3 3.2 3.4 3.6 3.8 4 4.2 4.4 5 9 15

1260.06 452.07 94.91 47.68 92.26 95.52 84.02 69.19 34.53 1 0.04

926.92 228.21 46.71 134.33 145.39 129.11 106.14 84.27 40.19 1.09 0.04

786.65 133.95 106.35 170.82 167.76 143.25 115.46 90.62 42.57 1.13 0.04

3.6 3.8 4 4.2 4.4 4.6 4.8 6 9 15

45 45 45 45 45 45 45 45 45 45

45 45 45 45 45 45 45 45 45 45

45 45 45 45 45 45 45 45 45 45

703.39 113.4 85.14 134.25 130.08 109.84 87.44 18.36 0.89 0.12

424.76 89.12 227.93 231.07 194.39 152.88 117.16 24.45 1.45 0.13

307.61 174.27 287.97 271.78 221.43 170.98 129.66 27.01 1.69 0.13

3 3.2 3.4 3.6 3.8 4 4.2 5 6 15

1536.77 560.19 138.73 27.95 81.66 88.27 77.74 28.93 7.99 0.02

1251.85 365.3 7.3 114.36 137.45 124.17 101.09 34.19 9.24 0.01

1131.88 283.24 48.04 150.75 160.93 139.29 110.93 36.41 9.76 0

2.6 2.8 3 3.2 3.4 3.6 3.8 4 5 9 15

90 90 90 90 90 90 90 90 90 90 90

90 90 90 90 90 90 90 90 90 90 90

45 45 45 45 45 45 45 45 45 45 45

2058.68 492.05 85.33 261.58 281.3 246.39 198.61 154.02 40.43 0.89 0.14

1433.27 29.64 400.63 462.23 405.51 324.64 250.41 190.26 49.54 1.21 0.14

1170.31 164.78 533.2 546.59 457.73 357.54 272.19 205.5 53.37 1.34 0.14

2.6 2.8 3 3.2 3.4 3.6 3.8 4 5 9 15

1096.04 345.28 23.76 93.89 120.93 112.53 93.59 74.12 20.04 0.37 0

667.26 64.31 151.43 201.77 188.8 156.84 123.48 94.77 24.25 0.46 0

486.72 53.99 225.19 247.19 217.38 175.49 136.06 103.46 26.02 0.5 0

3 3.2 3.4 3.6 3.8 4 4.2 4.4 6 15

90 90 90 90 90 90 90 90 90 90

45 45 45 45 45 45 45 45 45 45

90 90 90 90 90 90 90 90 90 90

1516.76 333.99 102.53 234.12 246.79 217.93 179 142.16 22.98 0.25

1091.11 16.44 324.38 378.27 336.34 273.09 213.96 165.43 25.1 0.23

912.14 117.08 417.66 438.88 373.99 296.28 228.66 175.21 25.99 0.22

3 3.2 3.4 3.6 3.8 4 4.2 4.4 6 15

1167.15 392.89 55.46 74.88 111.47 109.39 94.2 76.81 12.57 0.05

832.18 167.47 87.24 162.3 165.22 143.49 116.76 92.26 14.13 0.05

691.14 72.56 147.33 199.11 187.85 157.85 126.26 98.76 14.79 0.05

3.4 3.6 3.8 4 4.2 4.4 5 6 15

45 45 45 45 45 45 45 45 45

45 45 45 45 45 45 45 45 45

90 90 90 90 90 90 90 90 90

1535.92 349.4 80.62 207.1 219.01 192.67 97.49 29.46 0.22

1230.87 124.03 243.68 318.4 291.14 238.3 111.03 32.79 0.21

1102.61 29.27 312.24 365.2 321.47 257.49 116.72 34.19 0.21

3 3.2 3.4 3.6 3.8 4 5 6 15

1302.96 418.89 45.5 93.77 130.22 125.11 40.6 12.63 0.03

1011.88 217.25 90.18 182.38 187.12 161.67 46.09 13.95 0.03

889.32 132.34 147.31 219.69 211.07 177.06 48.4 14.5 0.03

2.6 2.8 3 3.2 3.4 3.6 3.8 4 5 6 15

90 90 90 90 90 90 90 90 90 90 90

90 90 90 90 90 90 90 90 90 90 90

90 90 90 90 90 90 90 90 90 90 90

1213.39 155.26 218.86 313.59 300.68 253 200.62 154.72 41.59 13.33 0.17

622.63 282.02 513.59 499.45 415.54 325.76 249.13 188.82 49.98 16.18 0.16

374.24 465.88 637.51 577.6 463.83 356.35 269.53 203.16 53.51 17.38 0.16

2.6 2.8 3 3.2 3.4 3.6 3.8 4 5 6 15

1131.78 365.19 34.06 88.98 119.04 112.35 94.36 75.39 21.29 6.82 0.01

701.21 85.18 139.66 195.93 186.59 156.7 124.45 96.26 25.64 8.09 0.01

519.91 32.72 212.8 240.95 215.03 175.38 137.12 105.04 27.47 8.63 0.01

3.4 3.6 3.8 4 4.2 4.4 4.6 6

45 45 45 45 45 45 45 45

135 135 135 135 135 135 135 135

45 45 45 45 45 45 45 45

1278.5 280.21 95.39 210.68 222.89 198.65 165.19 34.89

988.09 67.43 244.21 307.94 283.13 235.03 187.34 36.53

865.99 22.03 306.78 348.83 308.46 250.33 196.65 37.22

3 3.2 3.4 3.6 3.8 4 4.2 6

1190.62 352.03 0.14 127.35 156.35 145.94 123.1 15.97

895.85 146.78 136.98 216.06 212.83 182.07 146.69 17.32

771.73 60.36 194.71 253.42 236.61 197.28 156.62 17.89

r/Å

106u/EH H2-F2

73

T.P. Van, U.K. Deiters / Chemical Physics 485–486 (2017) 67–80 Table 3 (continued)

*

r/Å

a

b

/

106u/EH F2-F2

r/Å

106u/EH H2-F2

m=2

m=3

m = 23

15

45

135

45

0.3

0.3

m=2

m=3

m = 23

0.3

15

0.07

0.07

3.6 3.8 4 4.2 4.4 4.6 4.8 6 15

180 180 180 180 180 180 180 180 180

135 135 135 135 135 135 135 135 135

45 45 45 45 45 45 45 45 45

536.43 182.98 349.4 334.09 273.87 212.72 161.79 32.12 0.17

289.26 365.5 490.75 439.45 347.76 262.29 194.75 37.77 0.18

0.07

185.34 442.24 550.18 483.75 378.83 283.13 208.61 40.15 0.18

3.2 3.4 3.6 3.8 4 4.2 4.6 6 15

453.84 32.52 118.49 153.93 144.13 120.2 73.49 12.56 0.01

282.23 86.41 198.81 206.93 178.95 143.29 84.23 13.73 0.01

209.98 136.49 232.63 229.25 193.61 153.01 88.75 14.22 0.01

:‘‘2300 denotes the extrapolation from sets 2 and 3.

Table 4 Optimized parameters of 5-site potential Eq. (3). For all interactions dij = 2.0 Å1 were assumed. partial charges for hydrogen: qN/e = 0.078329, qH = 0; for fluorine: qN/e = -0.781897, qF = 0; qM = 2qN; EH = 4.359782.1018 J (Hartree energy unit).

a/Å1

b/ Å1

C6/EHÅ6

C8/EHÅ8

C10/EHÅ10

Site-site interaction of dimer F2-F2 F-F 4.597  100 F-N 9.075  102 F-M 6.699  101 M-M 1.587  102 M-N 7.659  101 N-N 3.479  102

5.279 5.019 5.020 3.483 5.252 3.432

1.618 0.653 1.246 0.434 0.689 0.558

6.953  101 1.264  102 1.452  102 1.523  101 5.025  101 1.105  102

7.817  102 1.123  103 1.365  103 1.584  103 1.074  103 1.618  103

3.836  103 6.146  103 8.344  103 3.639  103 2.171  103 5.309  103

Site-site interaction of dimer H2-F2 H-F 1.300  100 H-N 5.649  100 H-M 2.780  100 N-F 2.075  101 N-N 3.192  101 N-M 1.620  101 M-F 1.563  101 M-M 4.245  101

1.360 1.286 1.070 1.401 1.951 1.979 1.437 1.096

0.081 5.149 2.074 1.052 0.689 0.342 0.490 1.479

2.576  100 7.132  101 1.090  102 5.370  101 1.743  102 1.222  102 1.098  102 1.278  102

1.426  101 3.761  102 6.449  102 2.594  102 9.437  102 7.723  102 5.266  102 4.217  102

3.503  101 6.514  102 1.198  103 5.393  102 1.556  103 1.049  103 1.079  103 1.685  103

Interaction

De/EH

uðr ij ; a; b; /Þ ¼

5 X 5 h X

Dije

  2 1  eaij ðrij bij Þ  1

i¼1 i¼1

þf 2 ðr ij Þ with f 2 ðr ij Þ ¼ 1  edij rij

X

qi qj C ijn þ n r 4 p e0 rij n¼6;8;10;12 ij

!# ð4Þ

P10

ðdij rij Þk . k¼0 k!

Here Dije is the depth parameter of potential, e0 is the dielectric constant, rij denote site-site distances, qi and qj are electric charges of sites, and C ijn are dispersion coefficients; the leading dispersion term is always proportional to rnij . The two models differ mostly in the choice of the damping function f1(rij) proposed by Bock et al. [36], and Tang and Toennies damping function f2(rij) [40]. The site charges qi and qj are evaluated by fitting to the electrostatic potential of the molecule. In the 5-site model (Fig. 1) the auxiliary sites N, placed on the molecular axis half-ways between the outer sites (H or F) and the center M, bear each a charge of +q, and the central site M a charge of 2q. The outer sites have no electric charge. The optimized parameters of the ab initio intermolecular pair potential functions can be estimated by nonlinear least-square fitting to the ab initio interaction energy values resulting from the ab initio calculations. However, this fit proved to be very difficult, because of the object potential functions of the fitting problem have many local minima. Consequently the fit process has to be carried out by two steps. For the first step the global minima are coarsely located by means of the genetic algorithm. Then these initial parameters are optimized with the Marquardt-Levenberg

algorithm. The results – the parameters of the interaction potentials – are given in Tables 4 and 5. The fitting quality of ab initio energies of dimers F2-F2 and H2-F2 for the 5-site intermolecular potentials Eqs. (3) and (4) is also shown in Figs. 4 and 5, respectively. 4. Second virial coefficients Virial coefficients are related to intermolecular potentials by rigorous statistical thermodynamic theory; the second virial coefficient depends on the pair potential only. On the other hand, at least second virial coefficients have been determined experimentally for many gases. The calculation of the second virial coefficients from ab initio potential functions is a stringent and necessary test for the usefulness of such ab initio potentials. In this section, the numerical results for the second virial coefficients of the F2-F2 and H2-F2 dimers are presented. The case of these dimers is more complicated because of quantum effects, in which the interaction consists of atoms or molecules with small masses or small moments of inertia. These can in principle be obtained from a perturbation expansion of Planck’s constant as given in Eq. (5) [53]. The first order quantum corrections to the second virial coefficients of these linear molecules can be calculated using Eqs. (3) and (4). which have been worked out by Pack [54] and Wang [55]. Following the latter, the virial coefficient up to first order can be written as BðTÞ ¼

#!  " ! ! NA u 1 1  exp H u d r 1 d r 2 dX1 dX2 1þ 0 2 kB T 2VdX1 X2 12ðkB TÞ

ð5Þ

74

T.P. Van, U.K. Deiters / Chemical Physics 485–486 (2017) 67–80

Table 5 Optimized parameters of 5-site potential Eq. (4). For all interactions dij = 5.0 Å1 were assumed. See Table 1. for the partial charges.

a/Å1

b/Å1

C6/EHÅ6

C8/EHÅ8

C10/EHÅ10

C12/EHÅ12

site-site interaction of dimer F2-F2 F-F 8.817  102 F-N 1.183  102 F-M 3.168  102 M-M 6.776  101 M-N 6.875  102 N-N 2.934  102

3.244 3.159 3.120 2.580 3.241 3.121

0.078 0.922 0.671 0.261 0.331 0.803

6.216  101 1.340  102 1.982  102 5.875  102 2.896  102 1.160  103

1.232  103 3.337  103 4.944  103 6.408  103 1.992  103 1.577  104

5.451  103 1.151  104 1.157  104 5.099  104 4.292  104 6.314  104

1.270  103 6.940  103 3.852  103 1.070  105 7.173  104 1.621  105

site-site interaction of dimer H2-F2 H-F 3.046  101 H-N 2.781  101 H-M 6.524  101 N-F 4.202  101 N-N 5.116  101 N-M 4.467  101 M-F 2.489  101 M-M 2.554  101

2.520 2.269 2.938 2.054 2.141 2.852 2.104 1.981

0.225 0.043 0.280 0.488 0.255 0.128 0.313 0.034

5.047  101 1.703  102 1.448  102 5.855  101 3.163  102 1.909  102 8.171  100 6.141  101

2.794  101 3.353  102 1.240  102 3.163  102 2.657  102 6.620  102 4.329  102 2.261  103

3.642  102 1.500  103 2.068  103 2.159  102 2.259  102 5.222  103 5.434  102 1.624  104

2.116  102 1.537  103 2.185  103 2.525  102 1.404  103 8.645  103 4.225  102 2.434  104

Interaction

De/EH

Fig. 4. Correlation between ab initio and fitted ab initio energies for dimer F2-F2: a) Eq. (3); b) Eq. (4).

Fig. 5. Correlation between ab initio and fitted ab initio energies for dimer H2-F2: a) Eq. (3); b) Eq. (4).

Here NA is Avogradro’s constant, kB Boltzmann’s constant, T the temperature, and u(r, a, b, u) the pair potential; its parameters, the center-center distance and the relative orientation angles must !

be calculate from the center vectors r i and the absolute orientations Oi. H0 is the translation-rotation Hamiltonian for a pair of molecules. In this work Eq. (5) is used for calculating the quantum corrections to the virial coefficients of the dimers F2-F2 and H2-F2. It is broken down into a zeroth order (classical) term and first-order quantum corrections (radial part, angular part proportional to I1

(moment of inertia), angular part proportional to l1 (reduced mass)): ð0Þ

ð1Þ

ð1Þ B2 ðTÞ ¼ Bcl ðTÞ þ Bð1Þ r ðTÞ þ Ba;I ðTÞ þ Ba;l ðTÞ

ð6Þ

In Eq. (6) the virial coefficients B2(T) can be computed by the two ð0Þ

following steps: the classical second virial coefficients Bcl

are

calculated from Eq. (7). And the first-order correction terms, Bð1Þ r ; ð1Þ

ð1Þ Ba;I and Ba; l are calculated using the Eqs. (8)–(10). The total

75

T.P. Van, U.K. Deiters / Chemical Physics 485–486 (2017) 67–80

Table 6 Second virial coefficients B2(T) of fluorine as a function of temperature (given in cm3/mol). Methods Eqs. (3) and (4) are the two ab initio pair potentials (this work). D1-EOS: ð0Þ Deiters equation of state [57,58] and virial equation of state Eq. (14) [59,63,64]. exp.: experimental data [60–62]; virial values Bcl : classical results obtained from pair potentials; ð1Þ ð1Þ values Bð1Þ , B and B : quantum corrections; B (T): total virial coefficient. 2 r a;l a;I T(K)

Method

Bcl

Bð1Þ r

Ba;I

Bð1Þ a;l

B2 ðTÞ

Ref.

90

Eq. (3) Eq. (4) D1-EOS Eq. (14) exp. exp.

191.180 173.065

0.18516 0.14564

0.02538 0.05156

0.0024 0.0033

190.967 172.864 191.5 199.164 208.7 191.0

This work This work [57,58] [59,63,64] [60] [61]

100

Eq. (3) Eq. (4) D1-EOS Eq. (14) exp. exp.

156.094 142.027

0.16025 0.15576

0.03518 0.02341

0.0025 0.0106

155.896 141.837 154.13 162.293 156 156.10

This work This work [57,58] [59,63,64] [60] [61]

110

Eq. (3) Eq. (4) D1-EOS Eq. (14) exp. exp. exp.

129.935 119.066

0.14767 0.13545

0.01764 0.0268

0.002 0.0075

129.768 118.896 126.880 134.763 134.3 130.1 171.0

This work This work [57,58] [59,63,64] [60] [61] [62]

125

Eq. (3) Eq. (4) D1-EOS Eq. (14) exp. exp.

101.589 94.2294

0.54453 0.26773

0.05442 0.03244

0.003 0.0021

100.987 93.9271 97.684 104.662 101.8 101.7

This work This work [57,58] [59,63,64] [60] [61]

140

Eq. (3) Eq. (4) D1-EOS Eq. (14) exp. exp. exp.

81.4776 76.5069

0.79348 0.19292

0.08608 0.0403

0.0012 0.0024

80.5968 76.2713 77.164 83.152 81.5 81.5 113.0

This work This work [57,58] [59,63,64] [60] [61] [62]

145

Eq. (3) Eq. (4) D1-EOS Eq. (14) exp. exp.

76.0106 71.6567

0.69341 0.47222

0.05998 0.04268

0.0033 0.0025

75.2539 71.1393 71.63 77.310 75.9 75.9

This work This work [57,58] [59,63,64] [60] [61]

150

Eq. (3) Eq. (4) D1-EOS Eq. (14) exp. exp.

71.0214 67.2143

0.60294 0.67348

0.06077 0.04438

0.003 0.0055

70.3547 66.4909 66.603 71.992 70.9 70.9

This work This work [57,58] [59,63,64] [60] [61]

155

Eq. (3) Eq. (4) D1-EOS Eq. (14) exp. exp.

66.4473 63.1268

0.69652 0.86576

0.06157 0.04602

0.0035 0.0026

65.6858 62.2124 62.02 67.135 66.3 66.3

This work This work [57,58] [59,63,64] [60] [61]

165

Eq. (3) Eq. (4) D1-EOS Eq. (14) exp. exp.

58.3449 55.8495

0.84495 0.18875

0.06283 0.04876

0.0093 0.0028

57.4279 55.6092 53.971 58.592 55.7 58.2

This work This work [57,58] [59,63,64] [60] [61]

200

Eq. (3) Eq. (4) D1-EOS Eq. (14) exp. exp. exp.

37.6058 37.0192

0.81083 0.59788

0.02856 0.01826

0.0018 0.0014

36.7646 36.4017 33.878 37.297 35.9 37.6 47.0

This work This work [57,58] [59,63,64] [60] [61] [62]

230

Eq. (3) Eq. (4) D1-EOS Eq. (14)

25.7783 26.2051

0.17594 0.37255

0.04369 0.03552

0.0025 0.0022

25.5561 25.7949 22.694 25.576

This work This work [57,58] [59,63,64]

ð0Þ

ð1Þ

(continued on next page)

76

T.P. Van, U.K. Deiters / Chemical Physics 485–486 (2017) 67–80

Table 6 (continued) T(K)

ð0Þ

Method

Bð1Þ r

Bcl

ð1Þ

Bð1Þ a;l

Ba;I

exp. exp. exp.

B2 ðTÞ

Ref.

25.1 25.9 32.0

[60] [61] [62]

250

Eq. (3) Eq. (4) D1-EOS Eq. (14) exp. exp.

19.8561 20.7978

0.13516 0.31533

0.0647 0.05762

0.0026 0.0023

19.6536 20.4226 17.068 19.762 19.7 20.0

This work This work [57,58] [59,63,64] [60] [61]

260

Eq. (3) Eq. (4) D1-EOS Eq. (14) exp. exp. exp.

17.3484 18.5141

0.54829 0.76975

0.03907 0.03246

0.0023 0.002

16.7587 17.7099 14.651 17.289 17.3 17.5 25.0

This work This work [57,58] [59,63,64] [60] [61] [62]

300

Eq. (3) Eq. (4) D1-EOS Eq. (14) exp. exp.

9.63409 11.5282

0.28788 0.63636

0.07426 0.02146

0.0025 0.0012

9.26948 10.8692 10.9314 9.494 9.7 9.5

This work This work [57,58] [59,63,64] [60] [61]

quantum-corrected second virial coefficient B2(T) is defined as the sum of the contributions from the expressions Eqs. (7)–(10). The classical part is given by

B0cl ðTÞ ¼  

NA 4 Z p 0

Z 2p Z p 0

Z0

sin a

sin b    u 1  exp  r 2 drdadbd/ kB T

1

0

ð7Þ

z ¼ 1 þ cc0

The first-order correction terms can be written as:

Brð1Þ ðTÞ ¼

ð1Þ

Z p

Z 2p Z p

2

NA h

sin b sin a 3 0 0 96lðkB TÞ 0   2 Z 1 u @u  exp  r2 drdadbd/ k @r T B 0

Ft ¼

Z 2p Z p

2

Z p

sin b

Z

ð8Þ

1

sin a

1 2

ð1Þ Ba; l ðTÞ ¼ 

Z p

Z 2p Z p

2

NA h

2

sin b

Z sin a

1

4n  2n2 ð1  nÞ3



 abq  2 ~ c wð TÞ þ F With h 0 iþ1 c2 T

3 X 6 X 10 X

pijk ðc  1Þk T~ j ðbqÞ t i

ð11Þ

k¼0 j¼0 i¼0

  u exp  2 kB T 0 0 0 48ðkB TÞ 0   X l1 ðl1 þ 1Þ l2 ðl2 þ 1Þ 2  ul1 l2 l ðrÞAl1 l2 l ða; b; /Þ  þ r drdadbd/ ð9Þ 2I1 2I2 l l l N A h

Ba;I ðTÞ ¼ 

which yields correct critical temperatures, pressures, and densities of pure components. Deiters also proposed an extension to binary mixtures. Because of the equation D1-EOS originally proposed by Deiters could not be integrated analytically, so Deiters exchanged a part by a polynomial series. In this work we use the Deiters equation of state to calculate the second virial coefficients for fluid fluorine [57,58]:

  u exp  kB T

0 0 0 48ðkB TÞ 0 X lðl þ 1Þ 2 ul1 l2 l ðrÞAl1 l2 l ða; b; /Þ  r drdadbd/  2lr 2 l l l

where

pijk

is

constants

~ ¼ Tðexpð ~ wðTÞ T~ 1 Þ  1Þ;

obtained

from

Deiters;

c ¼ 1  0:697816ðc  1Þ2 ; pffiffi p 2 h0 = 7.0794046, c0 = 0.6887; T~ ¼ ckT e ; n ¼ 6 bq. and

For pure-component EOS parameters: ke ¼ a. The parameters of Deiters’s EOS can be determined from the critical data of the pure component. In this work we applied Deiters’s program ThermoC [58] to determine the mixture parameters set by taking into account the van der Waals 1-fluid theory [58]. 6. Virial equation of state for mixture

ð10Þ

1 2

The terms ul1 l2 l ðrÞAl1 l2 l(a, b, u) represent a spherical harmonics expansion of the interaction potential. All these integrals were evaluated numerically with a 4D Gauss–Legendre quadrature method [56]. The results for the second virial coefficients are presented in Tables 6 and 7. Figs. 6 and 7 similarly demonstrate the influence of the basis set. It must be pointed out that the basis sets most commonly used for calculations of chemical bonds, turn out to be totally insufficient for the purpose of calculating intermolecular potentials. 5. Deiters equations of state (D1-EOS) Deiters constructed an equation of state D1-EOS based on perturbed hard chain theory with the aim to obtain this equation

To the best of the author’s knowledge, there is no experimental data for the second virial coefficient of the H2-F2 mixture in the literature. In this work, as an alternative method, the virial equation of state is used to calculate the virial coefficients using the critical parameters and the acentric factor of the respective pure fluids. The virial equation can be written in a power series with respect to density for fluids [59,64]:

Z ¼ 1 þ Bmix q þ C mix q2 þ . . .

ð12Þ

where B, C are the second and third virial coefficients. The attractive interactions are characterized by B11, B22, and B12, the mixing rule for Bmix is given by

Bmix ¼ x2i Bii þ 2xi xj Bij þ x2j Bjj where B is a function of T. Bij is cross virial coefficient.

ð13Þ

77

T.P. Van, U.K. Deiters / Chemical Physics 485–486 (2017) 67–80 Table 7 Cross second virial coefficients B2(T) of mixture hydrogen-fluorine. For an explanation of the other properties see Table 6. T(K)

Method

Bcl

Brð1Þ

Ba;I

ð1Þ Ba; l

B2 ðTÞ

Ref.

50

Eq. (3) Eq. (4) D1-EOS Eq. (14).

127.161 130.274

0.25665 0.25984

0.04059 0.02854

0.0094 0.0012

126.854 129.984 127.746 129.687

This work This work [57,58] [59,63,64]

60

Eq. (3) Eq. (4) D1-EOS Eq. (14)

94.3719 95.5777

0.94286 0.65438

0.0358 0.03717

0.0063 0.0089

93.387 94.8772 95.002 96.1375

This work This work [57,58] [59,63,64]

70

Eq. (3) Eq. (4) D1-EOS Eq. (14)

72.5241 73.4276

0.73608 0.7549

0.07981 0.06559

0.0045 0.0026

71.7037 72.6045 72.314 73.0398

This work This work [57,58] [59,63,64]

80

Eq. (3) Eq. (4) D1-EOS Eq. (14)

57.3616 58.2263

0.84995 0.5929

0.05766 0.07121

0.0072 0.0051

56.4468 57.557 56.106 56.5935

This work This work [57,58] [59,63,64]

90

Eq. (3) Eq. (4) D1-EOS Eq. (14)

46.3721 47.3029

0.8418 0.85496

0.09613 0.07564

0.0092 0.0072

45.425 46.3652 44.152 44.4864

This work This work [57,58] [59,63,64]

100

Eq. (3) Eq. (4) D1-EOS Eq. (14)

38.0614 39.0935

0.92346 0.81835

0.09564 0.09038

0.0073 0.0065

37.035 38.1782 35.080 35.3093

This work This work [57,58] [59,63,64]

110

Eq. (3) Eq. (4) D1-EOS Eq. (14)

31.5313 32.6713

2.17599 2.13952

0.04835 0.09574

0.0029 0.0062

29.3041 30.4298 28.025 28.1774

This work This work [57,58] [59,63,64]

120

Eq. (3) Eq. (4) D1-EOS Eq. (14)

26.2336 27.4782

2.29254 2.09754

0.07961 0.09299

0.0027 0.0015

23.8588 25.2862 22.421 22.5159

This work This work [57,58] [59,63,64]

160

Eq. (3) Eq. (4) D1-EOS Eq. (14)

12.1012 13.7132

2.73251 2.89917

0.08481 0.09443

0.0037 0.0084

9.28021 10.7112 8.457 8.41899

This work This work [57,58] [59,63,64]

200

Eq. (3) Eq. (4) D1-EOS Eq. (14)

4.02889 5.95276

2.29647 2.83244

0.08819 0.07572

0.0019 0.0035

1.64233 3.0411 1.228 1.1252

This work This work [57,58] [59,63,64]

250

Eq. (3) Eq. (4) D1-EOS Eq. (14)

1.72024 0.52143

1.58046 2.6632

0.04442 0.07062

0.0019 0.0037

3.34704 2.21611 3.849 3.99524

This work This work [57,58] [59,63,64]

300

Eq. (3) Eq. (4) D1-EOS Eq. (14)

4.81646 2.34253

1.74384 3.06065

0.07652 0.09833

0.0084 0.0073

6.64525 5.50883 6.848 7.01982

This work This work [57,58] [59,63,64]

ð0Þ

ð1Þ

Long Meng and et al. [63] and Estela-Uribe and Jaramillo [64] published empirical correlation equations for second virial coefficients which are based on the corresponding-states approach of Lee and Kesler [65]. In their work, binary interactions are characterized by so-called pseudocritical parameters, which are interpolations of the pure-fluid critical temperatures and densities. The mixture virial coefficients [59,64] can be calculated by

Bij ðTÞ ¼

! RT c;ij ½B0 ðT r;ij Þ þ xij B1 ðT r;ij Þ with T r;ij ¼ T=T c;ij pc;ij

ð14Þ

B0 ðT r;ij Þ ¼ 0:083 

B1 ðT r;ij Þ ¼ 0:139 

T 1:6 r;ij 0:172 T 4:2 r;ij

aij qc;ij ð1  kij ÞðT c;i T c;j Þ1=2 ; and kij ¼ 1  1 þ ð21:8=M ij TÞ ðqc;i qc;j Þ1=2 



q1=3 ¼ 0:5ð1 þ dij Þ q1=3 þ q1=3 and Z c;ij ¼ 0:5ðZ c;i þ Z c;j Þ c;ij c;i c;j pc;ij ¼

ð17Þ

ð18Þ

 Z c;ij qc;ij Rð1  kij ÞðT c;i T c;j Þ1=2 1  1 M i þ M 1 with M 1 ij ¼ j 2 1 þ ð44:2=Mij TÞ ð19Þ

And

0:422

T c;ij ¼

ð15Þ

ð16Þ

Here Mi and Mj denote molar masses of molecule fluorine and hydrogen, Mij an ‘‘interaction molar mass” [64,65]; R = 8.314471 J mol1 K1 is the universal gas constant. The binary interaction parameter kij for system hydrogen–fluorine is taken from Refs. [63,64]. The binary interaction parameters aij and dij are set to zero. The results, presented in Table 6, show a remarkably good agreement with the predictions from quantum mechanics.

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ð0Þ

Fig. 6. Comparison of classical virial coefficients Bcl for dimer F2-F2 are calculated with method CCSD(T) for various basis sets: - - -:aug-cc-pVDZ; ---: aug-cc-pVTZ; —: augcc-pV23Z; s: D1-EOS; j: Eq. (14); 4: Exp. [60]; *: Exp. [61]; h: Exp. [62]. Using potentials: a) Eq. (3) and b) Eq. (4).

ð0Þ

Fig. 7. Comparison of classical virial coefficients Bcl for dimer H2-F2 are calculated with method CCSD(T) for various basis sets; for an explanation of the symbols see Fig. 6.

Fig. 8. Second virial coefficients B2(T) are calculated with method CCSD(T) with basis set limit aug-cc-pV23Z; Symbols: - - -: potential Eq. (3); —: potential Eq. (4); j: Eq. (14); s: D1-EOS; 4: Exp. [60]; *: Exp. [61]. a) for dimer F2-F2 and b) for dimer H2-F2.

7. Discussion The predicted values for the second virial coefficients of fluorine shown in Table 6 are in excellent agreement with the experimental data. It appears that values obtained with Eq. (3) are marginally better than those obtained with Eq. (4), although the latter function has more adjustable parameters. For the dimer F2-F2 the second virial coefficients B2(T) were also calculated in the temperature range 90–K using the level of theory

CCSD(T) with the basis set limit aug-cc-pV23Z. The results without quantum effects are shown in Table 6, and Fig. 8a. It turned out that the results derived from Eqs. (3) and (4), were close to the experimental data. The second virial coefficients B2(T) resulting from such pair potential functions are also close to those obtained with Deiters equation of state (D1-EOS), as can be seen in Fig. 8a. However, there are still some small differences. Consequently, quantum effects are also considered here for the dimers F2-F2 and H2-F2. Especially the cross second virial coefficients of the

T.P. Van, U.K. Deiters / Chemical Physics 485–486 (2017) 67–80

hydrogen-fluorine system, in which it consists of a quantum component, is not found in the literature. Those are calculated here in the temperature range 50–300 K including the quantum effects. The accurate prediction of cross second virial coefficients from an ab initio pair potential function without recourse to experimental data is important, and the method CCSD(T), applied to basis set limit aug-cc-pV23Z, is able to calculate the second virial coefficients of fluorine almost within uncertainties of the experiments as is seen in Fig. 8. The resulting virial coefficients of fluorine also included the first-order quantum corrections, due to the small effects of relative translational motions, and the molecular rotations. The values of second virial coefficients for fluorine with the quantum effects shown in Table 6 too. It turns out that quantum corrections have a small contribution to the second virial coefficients of fluorine over a wide temperature range. Table 6 shows that the contribution of translational motions Bð1Þ in the correcr tions is more important; the molecular rotations (i.e. the angular ð1Þ

terms Ba;l and Bð1Þ a;l ) are usually much smaller. The calculated virial coefficients of fluorine resulting from the ab initio pair potentials Eqs. (3) and (4) were compared with those predicted with Deiters equation of state (D1 EOS) [57,58] and with the experimental data [60–62]. It appeared that the differences are very small. The results derived from Deiters equation of state [57,58], are in excellent agreement with the experimental data. This is a suitable way for testing the accuracy of the results resulting from the ab initio pair potentials. The empirical correlation equations of Elliott and Lira [59] and Estela-Uribe and Jaramillo [64] were used for mixture hydrogenfluorine too. The results, presented in Table 7, show a remarkably good agreement with the predictions from quantum mechanics. More important is the fact that an accurate prediction of second virial coefficients from an ab initio pair potential without recourse to experimental data is possible, and that the CCSD(T) method, applied to basis sets aug-cc-pVDZ and aug-cc-pVTZ, and followed by an extrapolation to the basis set limit aug-cc-pV23Z, is evidently able to generate virial coefficients almost within the uncertainties of the experiments (see Fig. 8). It is worth noting that quantum corrections contribute significantly to the cross second virial coefficients of mixture hydrogen-fluorine. Of these corrections, only the radial term is important; the angular terms are usually smaller. In particular these corrections for dimer fluorine are insignificant. Experimental values for the cross second virial coefficients of the hydrogen–fluorine interaction are difficult to find in the literature. We conclude that two our developed ab initio intermolecular pair potentials for dimers F2-F2 and H2-F2, derived from accurate potential energy surfaces at the high level of theory CCSD(T) with basis set limit aug-cc-pV23Z, are reliable and usable for the prediction of virial coefficients. So the ab initio pair potentials for the hydrogen–fluorine interaction become to be important for calculation of thermodynamic properties from Monte Carlo simulation results can be useful, if experimental data are scarce. Acknowledgments The Regional Computer Center of Cologne (RRZK) contributed to this project by a generous al-lowance of computer time as well as by efficient software support; we especially wish to thank Dr. L. Packschies (RRZK) for technical help with the Gaussian03 software. We would like to thank Dr. P. K. Naicker and Dr. A. K. Sum (University of Delaware) for making available their computer program for virial coefficents. Furthermore I would like to thank the Government and the Ministry of Education and Training of Vietnam for the financial support over three years within the Vietnamese overseas scholarship program.

79

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