Ab initio calculation of intermolecular potentials and second virial coefficients for monte carlo simulation of dimer n2–n2 - Nguyen Thanh Duoc

AB INITIO CALCULATION OF INTERMOLECULAR POTENTIALS AND SECOND VIRIAL COEFFICIENTS FOR MONTE CARLO SIMULATION OF DIMER N2–N2 Nguyen Thanh Duoc(1) (1) Thu dau Mot University Received 23 June 2018, Accepted 10 July 2018 Email: ntduocag@gmail.com Abstract The results presented in this paper are the ab initio intermolecular potentials and the second virial coefficient, B2 (T) of the dimer N2-N2. These ab initio potentials were proposed by the quantum chemical calculations at high level of theory CCSD(T) with basis sets of Dunning’s valence correlation-consistent aug-cc-pVmZ (m = 2, 3); these results were extrapolated to complete basis set limit aug-cc-pV23Z. The ab initio energies of complete basis set limit aug-cc-pV23Z resulted from the exponential extrapolation were used to construct the 5-site pair potential functions. The second virial coefficients for this dimer were predicted from those with four-dimensional integration. The virial coefficients of them described accurately by the ab initio Lennard-Jones and Morse-style potentials. These potentials were also used for prediction of thermodynamics properties of vapor-liquid equilibria by Gibbs Ensemble Monte Carlo simulation (GEMC). The simulation results were also compared to those from equation of state. The obtained results turned out to be good agreement with experimental data. Keywords: Ab initio intermolecular potentials, virial coefficients, GEMC simulation 1. INTRODUCTION Computer simulations have become indispensable tools for studying fluids and fluid mixtures [[1],[2]]. They can generate structural, thermodynamic as well as transport properties consistently without the need to introduce artificial simplifications as required by, e.g., integral equation techniques, statistical thermodynamic perturbation theory. Computer simulation techniques, Monte Carlo as well as Molecular Dynamics, cannot work without some input, however: It is necessary to know the interaction potentials of the systems under study. The usual procedure is to assume a simple model potential, e.g., the Lennard-Jones pair potential, fit its parameters to suitable experimental data, and then to perform the simulation. Such a simulation is no longer predictive, because it requires experimental input of the same kind that is produces. This can sometimes be a severe limitation, namely if experimental data are scarce. In this work we report the development of 5-site intermolecular pair potentials for nitrogen from ab initio calculations at theoretical level CCSD(T)/aug-cc-pVmZ (m = 2, 3, 23). The second virial coefficients of the dimers N2 is calculated by using two ab initio potentials to test the accuracy of these two potentials. The thermodynamic data of vapor-liquid equilibria for liquid systems N2 gas enable to be predicted by using these potentials for the Gibbs ensemble Monte Carlo simulation (GEMC). The simulation results obtained by the Lennard-Jones and Morse-style pair potentials ab initio are compared with experimental data and those from Peng-Robinson equation [12]. 2. COMPUTATIONAL METHOD 2.1. Ab initio calculation The ab initio potential energy surfaces of dimers N2-N2 was constructed by selecting over 1000 molecular configurations. Ab initio energy calculations were carried out on a uniform grid of angular orientations constructed by a permutation a of 0 to 180o, b of 0 to 180o, f = 0 – 180o with increments 45o, and center-of-gravity distances r of 2.8 to 15 Å with increment 0.2 Å. The 5-site intermolecular models for these dimers were illustrated in Figure 1. Figure 1. 5-site model of dimers N2-N2 and special molecular orientations The theoretical level CCSD(T) and the correlation-consistent basis sets proposed by Dunning et al. The ab initio energies were corrected for the basis set superposition error (BSSE) [8]: ΔEint = EAB - (EAb + EaB) (1) ΔE(m) = ΔE(∞) + cm-3 (2) 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. With m = 2 (for the aug-cc-pVDZ basis set) or 3 (for the aug-cc-pVTZ basis set), the complete basis set limit aug-cc-pV23Z was calculated by ab initio intermolecular energies ΔE(m). Ab initio calculations were carried out with the Gaussian03 program package [9]. The potential energy surfaces of special orientations were constructed by ab initio intermolecular energies, as shown in Figure 2. The optimal geometry parameters of monomer nitrogen obtained by quantum calculations ab initio at the level of theory CCSD(T) with basis sets aug-cc-pVmZ (m = 2, 3, 23), as given in Table 1. Figure 2: Ab initio potential surface of special configurations for dimers N2-N2: I(0-0-0), T(90-0-0), H(90-90-0) and X(90-90-90) Table 1. Optimization geometry of monomer nitrogen Nitrogen Method bond N-N Ref. pVDZ 1.121 Ǻ This work pVTZ 1.104 Ǻ This work pV23Z 1.097 Ǻ This work Exp. 1.100 Ǻ [13] All the optimal adjustable parameters of the potentials Eq.3 and Eq.4 were drawn out by nonlinear least-square fitting to the ab initio energy points. This fit process has to be carried out by two steps, as used in [6]. The global minima are coarsely located by means of the differential evolution algorithm (DE), and the parameters resulting from this algorithm are used as initial values for the Marquardt-Levenberg algorithm. 2.2. Ab initio pair potentials In this work two new 5-site pair intermolecular potentials were developed from Lennard-Jones potential and Morse potential for dimer N2-N2 as [3,4,5]. The ab initio Morse-style pair potential proposed by Naiker [4] is developed to use for dimer N2: (3) With, and In this work the ab initio Lennard-Jones-style intermolecular pair potential is also constructed to apply for dimer N2: (4) With Wherethe well-depth parameter; aij the potential well width parameter; dij,the position parameters of the potential energy well, for all the interactions between site i on molecule a and site j on molecule b; rij site-site distances; the qi, qj electric charges of sites, and the dispersion coefficients; the leading dispersion term is always proportional to r; fn(dijrij) and f1(rij) the Tang-Toennies damping function [7]. 2.3. Second virial coefficients The second virial coefficient is an important property for assessing two-body interactions, so it is a good way for testing any pair potential. The classical virial coefficient relates to the pair potential function: (5) Where u(a,b,f, r) the pair potential function. A Monte Carlo integration was used for evaluating this 4D integral over the molecular orientation vectors r, a, b and f. Figure 3. Comparison of second virial coefficients for dimer nitrogen using ab initio pair potentials Eq. 3 and Eq. 4 with first-order quantum correction (this work) with those from Peng-Robinson Equation of State Eq.6 and experimental data [11]. In this work the Peng-Robinson equation also was used to calculate the virial coefficients of dimer N2-N2: (6) Where P pressure, Vm relative volume; T temperature; Tc critical temperature ; Pc critical pressure ; w acentric factor ; a, b Peng-Robinson constant and R molar gas constant. 2.3. Simulation runs The Ab initio pair potentials Eq. 3 and Eq. 4 were used in Gibbs ensemble Monte Carlo (GEMC) simulations to predict the vapor-liquid equilibria of pure fluid nitrogen. The GEMC simulations were carried out in the NVT ensemble (GEMC-NVT) with 512 molecules [10,14]. The temperatures used for all the simulation runs were less than the critial points of pure fluids nitrogen. The simulation equilibration between two phases required 2.0 x 105 cycles. All movements were performed randomly with defined probabilities. The simulation data were exported using block averages with 1000 cycles per block. The simulations were started with equal densities in two phases. The simulation systems were equilibrated for about 1.0 x 105 cycles. 3. RESULTS AND DISCUSSION The critical temperature Tc/ K and density rc/ g.cm-3 of the pure fluid nitrogen was derived from least-squares fits to the orthobaric densities using the relations Eq.7 of the rectilinear diameter law [10]. The orthobaric diagrams of them at various temperatures resulting from the intermolecular potentials Eq.3 and Eq.4 were pointed out in Figure 4. Figure 4. Vapor-liquid coexistence diagram of nitrogen; experimental data and critical temperature [13]; PR-EOS [12]. The experimental-critical densities and temperatures of pure fluid nitrogen was also shown in there, respectively. (7) Where rl and rv are the coexistence liquid density and vapor density, b is the critical exponent (b = 0.325). A and B are adjustable constants. The critical pressure Pc/ MPa was calculated with the Antoine equation: (8) Where A, B and C are Antoine constants. The relation between vapor pressure, heat of vaporization DHvap and temperature at the standard state P0 = 0.101 MPa is given by the Clausius-Clapeyron equation (9) Here the slope and the intercept of lnP are proportional to DHvap and DSvap. The critical properties of the pure fluid nitrogen could not be calculated directly from the simulation, but they could be obtained from the orthobaric densities of vapor-liquid equilibria by the least-square fit to the relations Eq.8. The critical pressures of nitrogen were calculated from the Antoine equation Eq. 8 as shown in Table 2. The results agreed reasonable well with experimental data. The thermodynamic properties of these fluids are also shown in Table 2. Table 2. Critical properties of nitrogen resulting from the GEMC-NVT simulation results using equations Eq.3 and Eq.4; EOS-PR: Peng-Robinson equation of state [12]; Exp.: experimental values Nitrogen Method Tc/ K rc/ g.cm-3 Ref. Eq.3 124.432 0.3125 This work Eq.4 132.876 0.3284 This work EOS-PR 126.143 0.3233 [12] Exp. 126.200 0.3140 [13] The discrepancies between predicted results and experimental data are insignificant. 4. CONCLUSION The vapour-liquid coexisting phase and thermodynamic properties of the pure fluids nitrogen gas was calculated successfully with the developed computer simulation program GEMC-NVT using 5-site intermolecular potential functions Eq. 3 and Eq. 4 resulting from ab initio energy calculations. The simulation results agree well with critical experimental data and those from Peng-Robinson equation of state (EOS-PR). REFERENCES A. Z. Panagiotopoulos (1987). Direct determination of phase coexistence properties of fluids by Monte Carlo simulation in a new ensemble. Mol. Phys., 61, 813-826. D. S. Wilson and L. L. Lee (2005). Chemical potentials and phase equilibria of Lennard-Jones mixtures: A self-consistent integral equation approach. J. Chem. Phys., 123, 044512. K. Leonhard, U. K. Deiters (2002). Monte Carlo simulations of nitrogen using an ab initio potential. Mol. Phys., 100, 2571–2585. P. K. Naicker, A. K. Sum, S. I. Sandler (2003). Ab initio pair potential and phase equilibria predictions for hydrogen chloride. J. Chem. Phys., 118, 4086-4093. A. Z. Panagiotopoulos (1987). Direct determination of phase coexistence properties of fluids by Monte Carlo simulation in a new ensemble. Mol. Phys., 61, 813-826. Pham V. T., U. K. Deiters (2007). Ab initio calculation of the intermolecular potential and prediction of second virial coefficients for dimer H2-H2. Journal of Chemistry, vol. 5, No. 5, 656-660. K. T. Tang, J. P. Toennies (1984). An improved simple model for the Van der Waals potential based on universal damping functions for the dispersion coefficients. J. Chem. Phys., 80, 3726-3741. S. Rybak, B. Jeziorski, K. Szalewicz (1991). Symmetry-Adapted Perturbation Theory of Intermolecular Interactions. H2O and HF Dimers. J. Chem. Phys., 95, 6576-6601. C. G. Gray, K. E. Gubbins (1984). Theory of Molecular Fluids. Vol. 1: Fundamentals. Oxford University Press, Oxford, 230-350. M. P. Allen, D. J. Tildesley (1991). Computer Simulation of Liquids. Clarendon Press, Oxford, 210-300. J. H. Dymond, E. B. Smith (1980). The Virial Coefficients of Pure Gases and Mixtures. Clarendon Press, Oxford, 100-300. U. K. Deiters. ThermoC project, homepage: Database of National Institute of Standards and Technology NIST, Acknowledgment: We would like to thank Assoc. Prof. Dr. Pham Van Tat for providing and making available his computer programs.