Monte Carlo simulations of thermodynamic properties of argon, krypton and xenon in liquid and gas state using new ab initio pair potentials

Size: px

Start display at page:

Download "Monte Carlo simulations of thermodynamic properties of argon, krypton and xenon in liquid and gas state using new ab initio pair potentials"

Walter Phillips
6 years ago
Views:

1 MOLECULAR PHYSICS, 2NOVEMBER 23, VOL. 11, NO. 22, Monte Carlo simulations of thermodynamic properties of argon, krypton and xenon in liquid and gas state using new ab initio pair potentials ALEXANDR MALIJEVSKY * and ANATOL MALIJEVSKY Department of Physical Chemistry, Institute of Chemical Technology, Prague 6, Czech Republic (Received 27 June 23; revised version accepted 21 October 23) The internal energies and compressibility factors of argon, krypton and xenon have been simulated using recent state-of-the-art ab initio pair intermolecular potentials and the best semi-empirical pair potentials, and the Axilrod Teller Muto three-body term. The results are compared with experimental data for both sub-critical and super-critical temperatures and for densities ranging up to a 2.5 multiple of the critical density. Both the ab initio and semi-empirical results for argon are in very good agreement with the experimental ones. For krypton and xenon, the ab initio results are worse than the semi-empirical results but they are still acceptable. 1. Introduction One of the dreams of theoretical chemistry and physics is to determine quantitatively bulk thermodynamic properties of gases and liquids from first principles, i.e. without parameters fitted to experimental data. The only input that the statistical thermodynamics of fluids needs to reach the aim is the total potential energy U N of a system of N particles as a function of their mutual positions and orientations. The simplest real systems on which theoretical tools can be tested are rare gases. For them the internal energy can be written in the form U N ¼ X i;j u 2 ðr ij Þþ X i;j;k u 3 ðr ij ; r ik ; r jk Þþ; ð1þ where r ij are the inter-particle distances. The leading term in this expansion, a sum of the pair potentials, u 2, is the most important. It is well known that simple pair potentials, such as the Lennard-Jones potential, cannot describe different bulk properties that depend only on the pair interactions over a wide range of state variables and with accuracy comparable to the experimental one. Even for the second virial coefficient no pair of parameters and is able to give results within experimental errors over the whole range of temperatures for which experimental data are available, see e.g. [1]. The best present pair potentials are based on the semiempirical approach: the functional form of u 2 ¼ u 2 ðrþ is *Author for correspondence. Alexandr.malijevsky@ vscht.cz theoretically grounded, but the free parameters it contains are fitted to microscopic and bulk data depending only on the pair-wise interactions (rotational vibrational spectra of rare gas dimers, collision crosssections, second virial coefficients, low density transport properties), see [2] for earlier results. The most accurate pair potentials for argon, krypton and xenon are given in [3 5], respectively. The above-mentioned semi-empirical pair potentials, although highly accurate, do not meet the final goal prediction of bulk properties without adjustable parameters. An alternative to them are ab initio pair potentials which use no input constants but electron charge, Planck s constant, etc. Such potentials were proposed for helium [6], neon [7, 8], argon [8, 9], krypton [1] and xenon [11, 12]. Due to an exponential progress in computer technologies and a similar progress in numerical methods, the ab initio approach is becoming a more and more prospective and attractive route. Let us return to equation (1). The next term in the expansion is the three-body potential, u 3. The leading contribution to this term is the Axilrod Teller Muto potential [13] u 3 ¼ 1 þ 3 cos 1 cos 2 cos 3 ðr 12 r 13 r 23 Þ 3 taking into account triple-dipole (DDD) interactions. There is empirical evidence that the influence of higherorder dispersion terms (DDQ, DQQ,...) and short-range contributions to bulk fluid properties fortuitously cancels (see e.g. [14 18] and references therein). Thus, the DDD ð2þ Molecular Physics ISSN print/issn online # 23 Taylor & Francis Ltd DOI: 1.18/

2 3336 A. Malijevsky and A. Malijevsky term is generally considered to be a reasonably accurate effective three-body potential in the fluid region. Very little (practically nothing) is known about the influence of u 4 and higher-order terms in equation (1) on bulk fluid properties (see e.g. [19]). This is for two reasons: (i) ab initio calculations of higher-order potentials are too computer-time demanding and (ii) there are no experimental values of microscopic and bulk properties to compare the theoretical results with (for example, no sufficiently accurate experimental fourth virial coefficients are available). Anyway, it is generally believed that the influence of higher-order terms in equation (1) on bulk fluid properties is negligible. Most calculations of thermodynamic properties of fluids are based on the semi-classical approximation. In very precise calculations, quantum corrections should be considered. This is usually done using the first term in the Wigner Kirkwood expansion of the Helmholtz free energy, F in powers of h 2 (see e.g. [2]) F Q =ðnk b TÞ ¼ h 2 2 =ð24pmþ Z 1 gðrþ d 2 u 2 dr 2 þ 2 r du 2 dr r 2 dr þoðh 4 Þ; ð3þ where the symbols have their usual meaning. There are a number of works concerning calculations/ simulations of dense fluid thermodynamic quantities using semi-empirical and/or ab initio pair potentials, and three-body potentials. These works had a long tradition (see e.g. [21, 22]). More recently Ermakova et al. [23] simulated structural, thermodynamic and transport properties of argon; Kirchner et al. [24] simulated the properties of neon. Vogt et al. [25] used Gibbs ensemble simulations for vapour liquid equilibrium curves for neon and argon. Marcelli and Sadus [16, 26] reported similar simulations for argon, krypton and xenon. Leonhard and Deiters performed simulations for neon and argon [8] and for nitrogen [18], and Nasrabad and Deiters [1] for krypton. Recently, Slavı ček et al. [27] published new ab initio pair potentials for heavier rare gases calculated at the CCSD(T) level. For argon the aug-cc-pv6z basis set by Dunning [28] was used, augmented by the spdfg set of bond functions and with the core corrections. For krypton and xenon, the relativistic effective core potentials constructed for use with the valence aug-ccpvqz basis augmented with the spdf bond functions set were used (see [27] for details). The calculated values of the pair potentials were fitted to the HFD-B [29] analytical form where ( FðrÞ¼ exp ½ðDr1 1Þ 2 Š; r < D; 1; r D; and where r is the interatomic separation distance. In this paper, we present new simulated values of thermodynamic properties of argon, krypton and xenon in the liquid and dense gas region obtained using the above-mentioned ab initio pair intermolecular potentials [27] and the DDD three-body potential. We compare the simulated results with those simulated using the best semi-empirical pair potentials [3 5] and the same three-body contribution, and with experimental data [3]. 2. Computer simulations We have simulated the internal energy and pressure of argon, krypton and xenon using the inter-particle interactions described by equations (2) and (3). We employed the conventional canonical ensemble Monte Carlo method in periodic boundary conditions, with the total number N ¼ 5 particles in a cubic box. We started from a random initial configuration. Following the initial equilibration stage, each run was divided into 1 basic simulation blocks, in which 5 measurements were performed to collect internal energy, pressure and pair distribution function data. 1 trial moves per particle were made between each measurement. The acceptance ratio was adjusted to 3 35%. Approximately 1 8 equilibrium configurations were generated in each run. The internal energy was computed in the standard way [31, 32]. It was corrected for the finite size effects as follows. For the two-body interactions the nearest image boundary condition method was used, see Appendix for details. For the three-body interactions, the cut-off distance r cut ¼ 2:5r min was chosen where r min is the position of the pair potential minimum; only triangles with all sides less then the cut-off distance were used. At several state points we used a larger cut-off distance r cut ¼ 3:r min. We found that the thermodynamic results were almost identical. Therefore, no corrections for larger distances were done. The pressure, p, was computed using the virtual change of volume method, see e.g. [32]. As a test of the consistency of the developed computer codes, the pressure was also computed via the virial theorem ð5þ u 2 ðrþ¼a exp ðr þ r 2 Þ FðrÞ C 6 r 6 þ C 8 r 8 þ C 1 r 1 ; ð4þ pv ¼ Nk b T 2 3 W; ð6þ

3 Simulations of thermodynamics of heavier rare gases 3337 where the virial W is W ¼ 1 2 * + X N i¼1 r i and h...i denotes the average value. Neglecting higher-order terms in equation (1), the virial becomes a sum of two- and three-body contributions where and W ¼ W ð2þ þ W ð3þ ; * W ð2þ ¼ 1 X 2 ðr ij Þ r ij ij i;j * W ð3þ ¼ 1 3 ðr ij ; r ik ; r jk 3 ðr ij ; r ik ; r jk Þ r ij þ r ik i;j;k ik 3 ðr ij ; r ik ; r jk Þ þ r jk : jk It can be shown that in the case of the DDD threebody potential, W ð3þ is in a simple relation to the total three-body energy i;j;k ð7þ ð8þ ð9þ X W ð3þ DDD ¼9 u 3;DDD : ð11þ 2 This relation was used as an additional test for the reliability of our simulations. The pressure was calculated using equations (6), (8), (9) and (11). The simulation results obtained using the virial theorem were tested by comparing with those obtained using the virtual volume change method. The differences were negligible. We have not calculated the quantum corrections, see equation (3), to the simulation results for the following reason. For the correction to the internal energy, E Q, and compressibility factor, z Q, it holds, respectively E Q ¼ F Q Q V and z Q ¼ Q Nk b T : ð13þ In the numerical implementations the partial derivatives are approximated by ratios of differences. To do this one should simulate the radial distribution, gðrþ, in equation (3) at one more density and one more temperature at least, at each state point. This would increase computer simulation time three times. In order to roughly estimate magnitudes of quantum corrections we used results given in [22]. They were calculated using the reference-hypernetted chain approximation (RHNC) [33] and older semi-empirical pair potentials than those considered in this work. 3. Results and discussion We have simulated the internal energy and pressure (compressibility factor) for both ab initio and semiempirical pair potentials, with the DDD three-body interaction term included. Table 1 compares experimental and theoretical excess internal energy of argon. In column 3, experimental data taken from Vargaftik s monograph [3] are given. Column 4 shows pair potential contributions to the internal energy. Strictly speaking the contributions are not just pair. They have been calculated as averages over configurations where the DDD three-body interactions have been taken into account. The next column shows the three-body contributions. In column 6, deviations of the ab initio results from the experimental results are shown, and in column 7, deviations from the experimental data are shown where the semi-empirical pair potential [3] has been used. In general, the deviations from the experimental data are small. Surprisingly, the ab initio results are even better than the semi-empirical ones. Table 2 compares the compressibility factors of argon. It is known that pressure (or compressibility factor) in the high-density liquid region is very sensitive to minor errors in the input quantities. The deviations from the Table 1. Internal energies, ðe E id Þ=ðNk b TÞ, of argon. Expt denotes experimental values [3]. Pair denotes pair contribution to the internal energy (see main text for explanation). Three denotes the three-body contribution. D ab in is the deviation of results of this work from the experimental results and D emp is the deviation of results calculated using the semi-empirical pair potential [3] and equation (2)

4 3338 A. Malijevsky and A. Malijevsky Table 2. Compressibility factors, z ¼ pv=ðnk b TÞ, of argon. Notation as in table Table 5. Internal energies, ðe E id Þ=ðNk b TÞ, of xenon. Notation as in table 1, but the semi-empirical pair potential was taken from [5] Table 3. Internal energies, ðe E id Þ=ðNk b TÞ, of krypton. Notation as in table 1, but the semi-empirical pair potential was taken from [4] Table 6. Compressibility factors, z ¼ pv=ðnk b TÞ, of xenon. Notation as in table 1, but the semi-empirical pair potential was taken from [5] Table 4. Compressibility factors, z ¼ pv=ðnk b TÞ, of krypton. Notation as in table 1, but the semi-empirical pair potential was taken from [4] experimental data in the liquid region are therefore higher than in the case of the internal energy, but they are still acceptable. In tables 3 and 4, the same quantities are compared for krypton, and in tables 5 and 6 for xenon. In all cases, the semi-empirical results are better than the ab initio results. This is not surprising; the ab initio pair potentials for krypton and xenon cannot be obtained at the same level of accuracy as for argon due to the larger number of electrons and relativistic effects. We shall now consider possible sources of deviations of the theoretical results from the experimental data. There are four sources of errors: (i) computer simulation errors, (ii) pair potential errors, (iii) neglecting the quantum corrections and (iv) three-body potential errors. ii(i) Statistical errors of the present simulations are small. For example, for argon at T=T c ¼ :9 and = c ¼ 2, the uncertainty in the simulated average value (estimated from the values of 1 sub-runs) is of the order of 1 5 for the internal energy and of the order of 1 4 for the compressibility factor. These errors are negligible in comparison with the differences between the simulated and experimental data (see tables 1 and 2). Finite size errors (5 particles in the basic box) are small but not negligible (of the order of.1 in reduced units). In the case of the pair contributions, they were corrected to the infinite size of the system (see Appendix). The cutoff correction to the three-body potential is negligible. i(ii) Another source of errors is the uncertainty in the pair potential. For argon (tables 1 and 2), the results obtained by two mutually independent pair potentials are similar; thus this source of errors is also small. For krypton and xenon (tables 3 to 6) the semi-empirical results are better than the ab initio ones.

5 Simulations of thermodynamics of heavier rare gases 3339 (iii) The influence of quantum effects on the thermodynamic quantities increases with decreasing molecular mass, decreasing temperature and increasing density. To estimate it we used results shown in tables 6 to 8 (columns labelled by Q) in [22]. For argon at T=T c ¼ :9 and = c ¼ 2:5 (the lower temperature and the highest density considered in this work) the quantum correction to the internal energy and compressibility factor is about.5. At the higher temperature (T=T c ¼ 1:5) the correction does not exceed.1. For krypton and xenon the corrections are negligible. (iv) Errors caused by using the DDD term only for modelling the three-body interactions can be judged only indirectly. 71 out of 84 values of D in the tables are positive. These systematic deviations cannot be caused by the errors discussed in (i) and (ii). It seems that rather large three-body contributions to the thermodynamic quantities (see columns 5 in the tables) are not accurate enough. This hypothesis can be supported by the fact that the semi-empirical results are worse for krypton and xenon where the three-body contributions are larger than in the case of argon. 4. Conclusions We have simulated the internal energies and the compressibility factors of argon, krypton and xenon using both new ab initio pair potentials and the best literature semi-empirical pair potentials, and the Axilrod Teller Muto three-body potential. We have shown that the theoretical results (using both ab initio and semi-empirical pair potentials) for argon in the dense gas and liquid regime agree with experiment within the chemical accuracy. For krypton and xenon, the theoretical results are worse than the semiempirical but they are still in reasonable agreement with experiment. Thermodynamic properties of rare gases are experimentally well explored. Is there any sense in performing painstaking calculations such as those presented in this paper and in the works cited in section 1? We believe that it is. It is much more economical and less time consuming to calculate bulk properties at extreme temperatures and pressures than to measure them. Rare gases (the simplest real systems) are shopwindows for the theory of fluids. They open a window for calculations of thermo-physical properties of substances of more practical interest. We acknowledge support of the Ministry of Education, Youth and Sports of the Czech Republic under the grant No. LNA32 (Center for complex molecular systems and biomolecules) Appendix When the nearest image boundary condition approximation is used in a basic cubic box it holds Z 1 Z 1 Z 1 u 2;corr ¼ 8 u 2 dx dy dz L=2 L=2 L=2 Z 2=L Z 1 Z 1 L=2 L=2 Z 2=L Z 2=L Z 1 L=2 u 2 dx dy dz u 2 dx dy dz; where L is the edge of the cubic box. The symmetry numbers (8, 24, 24) can be obtained from a general expression n2 d, where n is a number of geometrical objects of the cube, 8 vertices, 12 edges and 6 faces, and d is their dimension,, 1, 2, respectively. We used the first two long-range terms in equation (3), C 6 r 6 and C 8 r 8 (FðrÞ ¼1), and neglected all higher-order terms. We rewrote the equation into a form more convenient for numerical evaluation u 2;corr ¼ 8 Z 2=L Z 2=L Z 2=L C 8 C 6 ½ð1=x 2 Þþð1=y 2 Þþð1=z 2 ÞŠ x 2 y 2 z 2 dxdydz ½ð1=x 2 Þþð1=y 2 Þþð1=z 2 ÞŠ 4 Z L=2 Z 2=L Z 2=L C 6 ½ð1=x 2 Þþð1=y 2 Þþz 2 Š 3 C ½ð1=x 2 Þþð1=y 2 Þþz 2 Š 4 x 2 y 2 dxdydz Z L=2 Z L=2 Z 2=L C 6 ½ð1=x 2 Þþy 2 þ z 2 Š 3 C 8 1 ½ð1=x 2 Þþy 2 þ z 2 Š 4 x 2 dxdydz: The integrals were calculated numerically using the trapezoidal rule. References [1] MALIJEVSKÝ, A., MAJER, V., VONDRA K, P., and TEKA č, V., 1986, Fluid Phase Equilib., 28, 283. [2] AZIZ, R. A., 1984, Inert Gases, edited by M. L. Klein (Berlin: Springer-Verlag).

6 334 A. Malijevsky and A. Malijevsky [3] AZIZ, R. A., 1993, J. chem. Phys., 99, [4] DHAM, A. K., ALLNATT, A. R., MEATH, W. J., and AZIZ, R. A., 1989, Molec. Phys., 67, [5] DHAM, A. K., MEATH, W. J., ALLNATT, A. R., AZIZ, R. A., and SLAMAN, M. J., 199, Chem. Phys., 142, 173. [6] AZIZ, R. A., JANZEN, A. R., and MOLDOVER, M. L., 1995, Phys. Rev. Lett., 74, 1586; JANZEN, A. R., and AZIZ, R. A., 1997, J. chem. Phys., 17, 914. [7] EGGENBERGER, R., GERBER, S., HUBER, H., and SEARLES, D., 1991, Chem. Phys., 156, 395; EGGENBERGER, R., GERBER, S., HUBER, H., and WELKER, M., 1994, Molec. Phys., 82, 689. [8] LEONHARD, K., and DEITERS, U. K., 2, Molec. Phys., 98, 163. [9] WOON, D. E., 1993, Chem. Phys. Lett., 24, 29. [1] NASRABAD, A. E., and DEITERS, U. K., 23, J. chem. Phys., 11, 947. [11] RUNEBERG, N., and PYYKKO, P., 1998, J. quantum Chem., 66, 131. [12] FAAS, S., VAN LENTHE, J. H., and SNIJDERS, J. G., 2, Molec. Phys., 98, [13] AXILROD, B. M., and TELLER, E., 1943, J. chem. Phys., 11, 299; MUTO, Y., 1943, Proc. Phys. math. Soc. Jpn, 17, 629. [14] ANTA, J., LOMBA, E., and LOMBARDERO, M., 1997, Phys. Rev. E, 55, 277. [15] LOTRICH, V. F., and SZALEWICZ, K., 1997, J. chem. Phys., 16, [16] MARCELLI, G., and SADUS, R. J., 1999, J. chem. Phys., 111, [17] BUKOWSKI, R., and SZALEWICZ, K., 21, J. chem. Phys., 114, [18] LEONHARD, K., and DEITERS, U. K., 22, Molec. Phys., 1, [19] VAN DER HOEF, M. A., and MADDEN, P. A., 1999, J. chem. Phys., 111, 152. [2] HANSEN, J. P., and WEIS, J. J., 1969, Phys. Rev., 188, 314. [21] BARKER, J. A., and HENDERSON, D., 1976, Rev. mod. Phys., 48, 587. [22] MALIJEVSKY, A., POSPI SˇL, R., and LABI K, S., 1987, Molec. Phys., 61, 533. [23] ERMAKOVA, E., SOLCA, J., HUBER, H., and WELKER, M., 1994, J. chem. Phys., 12, [24] KIRCHNER, B., ERMAKOVA, E., SOLCA, J., and HUBER, H., 1998, Chem. Eur. J., 4, 383. [25] VOGT, P. S., LIAPINE, R., KIRCHNER, B., DYSON, A. J., HUBER, H., MARCELLI, G., and SADUS, R. J., 21, Phys. Chem. chem. Phys., 3, [26] MARCELLI, G., and SADUS, R. J., 2, J. chem. Phys., 112, [27] SLAVI CˇEK, P., KALUS, R., PASˇKA, P., ODVARKOVA, I., HOBZA, P., and MALIJEVSKÝ, A., 23, J. chem. Phys., 119, 212. [28] DUNNING, JR, T. H., 2, J. phys. Chem. A, 14, 962. [29] AZIZ, R. A., and CHEN, H. H., 1977, J. chem. Phys., 67, [3] VARGAFTIK, N. B., 1975, Tables of Thermophysical Properties of Liquids and Gases (New York: Wiley). [31] ALLEN, M. P., and TILDESLEY, D. J., 22, Computer Simulations of Liquids (Oxford: Clarendon Press). [32] FRENKEL, D., and SMIT, B., 22, Understanding Molecular Simulations (New York: Academic Press). [33] LADO, F., FOILES, S. M., and ASHCROFT, N. W., 1983, Phys. Rev. A, 28, 2374.

Solution of the Electronic Schrödinger Equation. Using Basis Sets to Solve the Electronic Schrödinger Equation with Electron Correlation

Solution of the Electronic Schrödinger Equation Using Basis Sets to Solve the Electronic Schrödinger Equation with Electron Correlation Errors in HF Predictions: Binding Energies D e (kcal/mol) HF Expt