Ab initio calculation of the intermolecular potential and prediction of second virial coefficients for dimer H2-H2

656 Journal of Chemistry, Vol. 45 (5), P. 656 - 660, 2007 Ab initio calculation of the intermolecular potential and prediction of second virial coefficients for dimer H2-H2 Received 20 January 2007 Pham Van Tat1, U. K. Deiters2 1Department of Chemistry, University of Dalat 2University of Cologne, Germany summary The intermolecular pair potentials of the dimer H2-H2 were constructed from quantum mechanics using method CCSD(T) with Dunning's correlation-consistent basis sets aug-cc-pVmZ (m = 2, 3, 4); ab initio energies were extrapolated to the complete basis set limit. The interaction energies were corrected for the basis set superposition error (BSSE) with the counterpoise scheme. The second virial coefficients of hydrogen resulting from ab initio potential functions were obtained by integration; corrections for quantum effects were included too. The results agree well with experimental data. Keywords: ab initio potential, dimer H2-H2, second virial coefficients, BSSE. I - Introduction Hydrogen and the mixture hydrogen-oxygen are used in several industrial applications. It could become the most important energy carrier of tomorrow [3]. Liquid hydrogen, oxygen are the usual liquid fuels for rocket engines [2]. The National Aeronautics and Space Administration (NASA) is the largest user of liquid hydrogen in the world [1, 4]. Computer simulations have become indispensable tools for studying fluids and fluid mixtures. One of the first attempts Nasrabad and Deiters predicted phase high-pressure vapour- liquid phase equilibria of noble-gas mixtures [5, 6] from the global simulations using the intermolecular potentials. Other mixed-dimer pair potentials for noble gases were published by López Cacheiro et al. [7], but not used for phase equilibria predictions, yet. Leonhard and Deiters used a 5-site Morse potential to represent the pair potential of nitrogen [8] and were able to predict vapour pressures and orthobaric densities. Bock et al. also used a 5-site pair potential for carbon dioxide [9]; Naicker et al. developed the 3-site pair potentials for hydrogen chloride [11]; they predicted successfully the vapour-liquid phase equilibria of hydrogen chloride with GEMC (Gibbs Ensemble Monte Carlo simulations [12]. Recently Diep and Johnson carried out the ab initio calculations with post-SCF methods MPn (n = 2, 3, 4) and CCSD(T) with the basis sets aug-cc-pVmZ (m = 2, 3, 4) and the complete basis set limit [13, 14]. In this work we report quantum mechanical calculations at a sufficiently high level of approximation to obtain pair potential data of the dimer H2-H2. These data are then represented by analytical pair potential functions. These in turn are used to calculate 657 2nd virial coefficients. The 2nd virial coefficents can then be compared with experimental data as far as such data are available. They can furthermore be used to determine the parameters of a suitable equation of state. II - Computational details 1. Molecular Orientation Hydrogen molecule is represented as 5-site model with two sites placed on the atoms H, one site in the center of gravity M, and two sites halfways between the atoms and the center N; the interatomic distance is set to 0.74130 Å for hydrogen [15]. The intermolecular pair potential is a function of distance r (between the centers of gravity) and three angular coordinates,
, , 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
, , and  were varied from 0 to 180o with increment 45o. Figure 1: 5-site model and special orientations for quantum chemical approach 2. Quantum chemical calculations There are several post-SCF methods that can capture at least a part of the electron correlation effects. Experience shows that especially the CCSD(T) method appears to account for the most significant electron correlation effects. The Dunning's correlation-consistent basis sets [16] aug-cc-pVDZ (for hydrogen: 5s2p/3s2p), aug- cc-pVTZ (for hydrogen: 6s3p2d/4s3p2d) and aug-cc-pVQZ (for hydrogen: 6s3p2d1f/4s3p2d1f) were used in this work. The ab initio energies were corrected for BSSE with the counterpoise correction method proposed by Boys and Bernardi [17]: Eint = EAB - (EAb + EaB) (1) 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 complete basis set limit [18]: E(m) = E()+ cm-3 (2) with m = 2 for aug-cc-pVDZ or 3 for aug-cc- pVTZ. Ab initio calculations were carried out with the Gaussian03 program package [19]. 3. Potential function Two new 5-site pair potential functions were developed by incorporating the repulsive and the dispersive contribution from the terms of the site-site pair potentials in publications [11], [9] and [8]. The damping functions were chosen from the potentials in [8, 9, 20].    = = =   ++= 5 1 5 1 10,8,6 0 21 4 )()( i i n ij ji ijn ij ij n ija rij e r qq rf r CrfeDu ijij 
(3) 658 with 15 )2(2 1 )1()( += ijijrija erf  and ijij r ij erf =1)(2    = = =   ++= 5 1 5 1 12,10,8,6 0 21 4 )()( i i n ij ji ijn ij ij n ijb rij e r qq rf r CrfeDu ijij 
(4) with  = = 10 0 1 ! )( 1)( k k ijijr ijb k r erf ijij  and ijij r ij erf =1)(2 Here the rij site-site distances, the qi electric charges of sites, and the n ijC dispersion coefficients; the leading dispersion term is always proportional to r. The two models differ mostly in the choice of the damping functions f1a(rij) and f1b(rij). 4. Fitting potential function The adjustable parameters of the ab initio pair potential functions can be estimated by nonlinear least-square fitting to the ab initio interaction energy values. Table 1: The statistical results for fitting the intermolecular potentials Eq. 3 and Eq. 4. The values are in µEH Residual Potential R2 rms min max Eq. 3 0.9999 0.2329 -6.311 6.284 Eq. 4 0.9999 0.4258 -7.449 7.911 The fit process has to be carried out by two steps. The global minima are coarsely located by means of the Genetic algorithm, and the parameters resulting from the Genetic algorithm are used as initial values for the Marquardt- Levenberg algorithm. The values of root mean-square deviations (rms), multiple correlation coefficients (R2), and residuals of the fitted analytical potential functions are given in table 1. The statistical estimates here are important for assessing the fitting quality. The residual area for the potential Eq. 3 resulting from the least-square fit is narrower than the residual area of the potential Eq. 4. So the fitting quality for Eq. 3 is better, but this difference is insignificant for 930 interaction energy points. This turned out to be very satisfactory. III - Second virial coefficients 1. Classical virial coefficients The classical virial coefficients 0clB of hydrogen resulting from the formula Eq. 6 using the ab initio 5-site pair potentials Eq. 3 and Eq. 4 are depicted in Figs 2a and 2b, respectively. Furthermore, Diep et al. [13,14] computed the second virial coefficients at level of theory. CCSD(T) with complete basis set limit using the path integral and semiclassical method, respectively. In recent publication the virial coefficients of hydrogen were calculated by Wang [21] using the Lennard-Jones 6-12 potential as described in Fig 2. The classical 2nd virial coefficients 0clB of hydrogen resulting from the new potentials Eq. 3 and Eq. 4 at the level of theory CCSD(T) with complete basis set limit aug-cc-pV23Z turn out to be in good agreement with experimental data and those in publications [13, 14, 21, 25]. 659 50 100 150 200 250 300 350 400 -20 -10 0 10 20 T/K B 2( T) /c m 3 m ol -1 a) 50 100 150 200 250 300 350 400 -20 -10 0 10 20 T/Kb) B 2(T )/c m 3 m ol -1 Figure 2: Second virial coefficients 0clB of hydrogen resulting from the pair potential functions: a): Eq. 3 and b): Eq. 4 at level of theory CCSD(T); ----: aug-cc-pVDZ; ---: aug-cc-pVTZ; : aug-cc- pVQZ; : aug-cc-pV23Z; : experimental data [23, 24]; : Lennard-Jones potential by Wang [21]; : spherical harmonic potential by Etters and Diep [25, 13, 14]. 2. Quantum corrections In the case of hydrogen the matter is more complicated because of quantum effects. The first-order quantum corrections to the second virial coefficient of linear molecules by Pack [10] and Wang [21] can be written as:              += 212102 21 2 )(12 11)/exp(1 2   dddrdruH Tk Tku ddu NB B B A (5) Here NA is Avogradro’s constant, kB Boltzmann’s constant, T the temperature, and u(r;
, , ) the pair potential; H0 is the translation-rotation Hamiltonian for a pair of molecules. The classical virial coefficient is given by: drr Tk udddNB B A cl 2 00 0 2 0 0 1expsinsin 4    ! " ## $ %  ! " ## $ % =  

 (6) All these integrals were evaluated numerically with a 4D Gauss-Legendre quadrature method [22]. The 2nd virial coefficients B2(T) including quantum corrections at the level of theory CCSD(T) with the complete basis set limit aug-cc- pV23Z are presented in Fig 3. IV - Conclusion The second virial coefficients of hydrogen obtained from the two potential functions Eq. 3 and Eq. 4 are very close to experimental data, as described in Fig 3. The discrepancies between them are insignificant. The second virial coefficients are generated almost within the uncertainties of the experimental measurements. The first-order quantum corrections contribute significantly to the second virial coefficients of hydrogen even at high temperatures. Of these corrections, only the radial term is important; the angular terms are usually much smaller in size. This turned out that new ab initio pair potentials of the dimer hydrogen are reliable for predicting the thermodynamic properties. 660 Figure 3: Second virial coefficient B2(T) of hydrogen are calculated using the pair potentials (this work): —: the pair potential Eq. 3; ....: the pair potential Eq. 4; : experimental data [23,24]; : path integral [13] and : semi- classical method [14]. Acknowledgments: The Regional Computer Center of Cologne (RRZK) contributed to this project by a generous allowance of computer time as well as by efficient software support; we wish to thank Dr. L. Packschies for technical help with the Gaussian03 software. We would like to thank Prof. Dr. J. Karl Johnson (University of Pittsburgh, USA) for providing the virial coefficient results. References 1. V. P. Dawson. A History of the Rocket Engine Test Facility at the NASA Glenn Research Center, NASA, Glen Research Center, Cleveland (2004). 2. L. J. Krzycki. Build and test small liquid- fuel rocket engines, Rocketlab, Oregon (1967). 3. E. Larry. Homebrew fuel cell and Hydrogen Electrolyser, Home Power, 85, 52 - 59 (2001). 4. T. M. Haarmann and W. Koschel. Proc. Appl. Math. Mech., 2, 360 - 361 (2003). 5. A. E. Nasrabad and U. K. Deiters. J. Chem. Phys., 119, 947 - 952 (2003). 6. A. E. Nasrabad, R. Laghaei, and U. K. Deiters. J. Chem. Phys., 121, 6423 - 6434 (2004). 7. J. López Cacheiro, B. Fernandez, D. Marchesan, S. Coriani, C. Hattig, and A. Rizzo. J. Mol. Phys., 102, 101 - 110 (2004). 8. K. Leonhard and U. K. Deiters. Mol. Phys., 100, 2571 - 2585 (2002). 9. S. Bock, E. Bich, and E. Vogel. Chem. Phys., 257, 147 - 156 (2002). 10. R. T. Pack. J. Chem. Phys., 78, 7217 - 7222 (1983). 11. P. K. Naicker, A. K. Sum, and S. I. Sandler. J. Chem. Phys., 118, 4086 - 4093 (2003). 12. A. Z. Panagiotopoulos. Mol. Phys., 61, 813 - 826 (1987). 13. P. Diep and J. K. Johnson. J. Chem. Phys., 112, 4465 - 4473 (2000). 14. P. Diep and J. K. Johnson. J. Chem. Phys., 113, 3480 - 3481 (2000). 15. L. E. Sutton. Table of Interatomic Distances and Configurations in Molecules and Ions. Chemical Society, London, 18 (1965). 16. R. A. Kendall, T. H. Dunning, Jr., and R. J. Harrison. J. Chem. Phys., 96, 6796 - 6806 (1992). 17. S. F. Boys and F. Bernardi. Mol. Phys., 19, 553 - 566 (1970). 18. P. L. Fast, M. L. Sanchez, and D. G. Truhlar. J. Chem. Phys., 111, 2921 - 2926 (1999). 19. Gaussian03. Revision B.02. Gaussian Inc., Wallingford, CT, USA (2003). 20. K. T. Tang and J. P. Toennies. J. Chem. Phys., 80, 3726 - 3741 (1984). 21. W. F. Wang, J. Quant. Spectrosc. Radiat. Transfer, 76, 23 - 30 (2003). 22. W. Squire. Integration for Engineers and Scientists. Elsevier, New York (1970). 23. J. H. Dymond and E. B. Smith. The Virial Coefficients of Pure Gases and Mixtures. Clarendon Press, Oxford (1980). 24. D. R. Lide. Handbook of Chemistry and Physics. CRC Press, Raton, 85th edition (2000). 25. R. D. Etters, R. Danilowicz, and W. England. Phys. Reviews A, 12, 2199 - 2212 (1975). 50 100 150 200 250 300 350 400 -20 -10 0 10 20 T/K B 2(T )/c m 3 m ol -1 661 26.