Interaction second virial coefficients of dimer co-Co from new ab initio potential energy surface

506 Journal of Chemistry, Vol. 47 (4), P. 506 - 510, 2009 INTERACTION SECOND VIRIAL COEFFICIENTS OF DIMER CO-CO FROM NEW AB INITIO POTENTIAL ENERGY SURFACE Received 2 May 2008 PHAM VAN TAT Department of Chemistry, University of Dalat abstract The new 5-site ab initio intermolecular interaction potentials of dimer CO-CO were constructed from quantum mechanics using method CCSD(T) with Dunning's correlation- consistent basis sets aug-cc-pVmZ (m = 2, 3) [7]; ab initio energies were extrapolated to the complete basis set limit aug-cc-pV23Z. The ab initio intermolecular energies were corrected for the basis set superposition error (BSSE) with the counterpoise scheme [8]. The interaction second virial coefficients of dimer CO-CO resulting from the 5-site ab initio analytical potential functions obtained by integration; first-order corrections for quantum effects were included too. The results agree well with experimental data. Keywords: Second virial coefficients, 5-site ab initio potentials. I - Introduction The knowledge of thermodynamic properties of the pure substance CO-CO is important for practical applications. It is also necessary for its safe use. Computer simulations have become indispensable tools for studying pure fluids and fluid mixtures and understand macroscopic phenomena. One of the first attempts Nasrabad and Deiters predicted phase high-pressure vapour - liquid phase equilibria of noble-gas mixtures [1, 2, 4] from the global simulations using the intermolecular potentials. Other mixed-dimer pair potentials for noble gases were published by Lãpez Cacheiro et al. [3], but not used for phase equilibria prediction, yet. This work presents quantum mechanical calculations at a sufficiently high level of approximation to obtain pair potential data of carbon monoxide using the high level of theory CCSD(T) with Dunning's correlation-consistent basis sets aug-cc-pVmZ (m = 2, 3) [7]; the complete basis set limit aug-cc-pV23Z is obtained by ab initio intermolecular energies [8]. These ab initio energy results are corrected for the basis set superposition error (BSSE) with the counterpoise method. Two new 5-site ab initio potentials are developed for the dimer CO-CO; the interaction second virial coefficients of dimer carbon monoxide are compared with the experimental data and with those from the Deiters equation of state [15]. II - Computational Details 1. Molecular orientation Carbon monoxide asymmetric molecule is represented as 5-site model with two sites placed on the atoms C and O, one site in the center of gravity M, and two sites halfways between the atoms and the center N and A; the interatomic distance is set to 1.128206 Å for molecule CO [6]. The intermolecular potential is a function of distance r (between the centers of gravity) from 2.8 to 15 Å with increment 0.2 Å and three angular coordinates, α, β, and φ from 507 0 to 180o with increment 45o, which are explained in Fig. 1. Figure 1: 5-site model of dimer CO-CO and special molecular orientations 2. Quantum chemical calculation The method CCSD(T) and the correlation- consistent basis sets of Dunning et al. [7]: aug- cc-pVDZ (for oxygen: 10s5p2d/4s3p2d, for carbon: 9s4p1d/3s2p1d), aug-cc-pVTZ (for oxygen: 12s6p3d2f/5s4p3d2f, for carbon: 15s6p3d1f/9s5p3d1f) were used. 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]. 3. Potential functions In this work two our new 5-site pair potentials were developed from [4] for dimer CO-CO: 5 5 ( ) 2 1 1 1 1 6,8,10 0 ((1 ) 1) ( ) 4 ij ij ij ij r i jij n e ij n i i n ij ij q qCu D e f r r r α β πε − − = = = ⎡ ⎤⎛ ⎞= − − + +⎢ ⎥⎜ ⎟⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦ ∑∑ ∑ (3) 5 5 ( ) 2 2 2 1 1 6,8,10,12 0 ((1 ) 1) ( ) 4 ij ij ij ij r i jij n e ij n i i n ij ij q qCu D e f r r r α β πε − − = = = ⎡ ⎤⎛ ⎞= − − + +⎢ ⎥⎜ ⎟⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦ ∑∑ ∑ (4) With 2( 2) 15 1( ) (1 )ij ij r ijf r e δ− − −= + and 10 2 0 ( ) ( ) 1 ! ij ij k r ij ij ij k r f r e k δ δ− = = − ∑ Here the rij site-site distances, the qi, qj 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 f1(rij) from [5] and f2(rij) from [10]. Iii - Results and Discussion 1. Fitting potential function The optimal adjustable parameters of the potential functions Eq.3 and Eq.4 were estimated by nonlinear least-square fitting to the ab initio intermolecular energy values. 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 C α N A A N O C O M M φ β L: α = 0, β = 0, φ = T: α = 90, β = 0, φ = 0 H: α = 90, β = 90, φ = 0 X: α = 90, β = 90, φ = 90 508 initial values for the Marquardt-Levenberg algorithm. a) b) Figure 2: Comparison of ab initio and calculated energies: a) Eq. 3; b) Eq. 4 The multiple correlation coefficients (R2) of the fitted analytical potential functions Eq. 3 and Eq. 4 are given in Fig. 2a and Fig. 2b. The difference between them is insignificant for 780 ab initio interaction energy points. 2. Classical virial coefficient The classical virial coefficients 0clB of dimer CO-CO resulting from the formula Eq. 7 using the ab initio 5-site pair potentials Eq. 3 and Eq. 4 are depicted in Figs. 3a and 3b, respectively. a) b) Figure 3: Second virial coefficients 0clB of carbon monoxide resulting from the ab initio 5-site pair potentials: a): Eq. 3 and b): Eq. 4 at theoretical level CCSD(T); ⋅⋅⋅⋅ : aug-cc-pVDZ; ----: aug-cc- pVTZ; ⎯: aug-cc-pV23Z; •: experimental data [13,14]; {: Deiters equation of state, EOS-D1 [15]. 3. Quantum corrections In this case the matter is more complicated because of quantum effects. The first-order quantum corrections to the second virial coefficients of linear molecules by Pack [11] and Wang [12] can be written as 200 300 400 500 600 -40 -30 -20 -10 0 10 20 B 2(T )/c m 3 m ol -1 T/K 200 300 400 500 600 -40 -30 -20 -10 0 10 20 B 2(T )/c m 3 m ol -1 T/K 0 2000 4000 6000 -1000 0 1000 2000 3000 4000 5000 6000 7000 ab initio energies /μEH fit te d en er gi es /μE H R2 = 0.99785 0 15000 30000 45000 0 10000 20000 30000 40000 50000 fit te d en er gi es /μE H ab initio energies/μEH R2 = 0.99749 509 2 0 1 2 1 22 1 2 11 exp( ) 1 12( )2 A B B N uB H u dr dr d d k T k Tu d d ⎧ ⎫⎡ ⎤⎪ ⎪= − − + Ω Ω⎨ ⎬⎢ ⎥Ω Ω ⎪ ⎪⎣ ⎦⎩ ⎭∫ ∫ ∫ ∫∫∫ (5) Here NA is Avogradro’s constant, kB Boltzmann’s constant, T the temperature, and u(r; α, β, φ) the pair potential; H0 is the translation-rotation Hamiltonian for a pair of molecules. This expression can be broken down into a classical term and first-order quantum corrections (radial part, angular part proportional to I-1 (moment of inertia), angular part proportional to μ-1 (reduced mass)): 0 1 1 1 cl a a( ) ( ) ( ) ( ) ( )r IB T B T B T B T B Tm= + + + (6) The classical virial coefficient is given by: 2 0 2 cl 0 0 0 0 sin sin exp 1 4 A B N uB d d d r dr k T π π π ϕ β β α α ∞ ⎛ ⎞⎛ ⎞= − − −⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠∫ ∫ ∫ ∫ (7) The first-order correction terms can write as: 222 1 2 3 0 0 0 0 ( ) sin sin exp 96 ( ) A r B B N u uB T r drd d d k T k T r π π π β α α β φμ ∞ ⎛ ⎞ ∂⎛ ⎞= −⎜ ⎟⎜ ⎟∂⎝ ⎠⎝ ⎠∫ ∫ ∫ ∫ h (8) 1 2 1 2 1 2 22 1 a 2 0 0 0 0 21 1 2 2 1 2 ( ) sin sin exp ( ) ( , , ) 48( ) ( 1) ( 1) 2 2 A I l l l l l l l l lB B N uB T u r A k T k T l l l l r drd d d I I π π π β α α β φ α β φ ∞ ⎛ ⎞= − −⎜ ⎟⎝ ⎠ ⎛ ⎞+ +× +⎜ ⎟⎝ ⎠ ∑∫ ∫ ∫ ∫h (9) 1 2 1 2 1 2 22 1 a 2 0 0 0 0 2 2 sin sin exp ( ) ( , , ) 48( ) ( 1) 2 A l l l l l l l l lB B N uB u r A k T k T l l r drd d d r π π π μ β α α β φ α β φμ ∞ ⎛ ⎞= − −⎜ ⎟⎝ ⎠ +× ∑∫ ∫ ∫ ∫h (10) The terms 1 2 1 2 ( ) ( , , )l l l l l lu r A α β φ represent a spherical harmonics expansion of the interaction potential. All these integrals were evaluated numerically with a 4D Gauss-Legendre quadrature method [16]. The resulting virial coefficients of dimer CO-CO included the first-order quantum corrections, as shown in Fig. 4, due to the effects of relative translational motions, and the molecular rotations. The second virial coefficients of dimer CO-CO obtained from the two new ab initio pair potential functions Eq. 3 and Eq. 4 are very close to experimental data, as described in Fig. 4. The discrepancies between them are insignificant. The interaction 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 dimer CO-CO even at high temperatures. Of these corrections, only the radial term is important; the angular terms are usually much smaller in size. 510 Figure 4: Second virial coefficients 2 ( )B T of carbon monoxide included first-order quantum corrections resulting from the ab initio 5-site pair potentials: ⎯: Eq. 3 and ----: Eq. 4 at theoretical level CCSD(T)/aug-cc-pV23Z(this work); others: see explanation in Fig. 3 IV - Conclusion We conclude that two our new ab initio 5- site pair potentials developed for the dimer CO- CO are reliable for predicting the thermodynamic properties. In coming work we will report the use of these ab initio 5-site pair petentials Eq. 3 and Eq. 4 for the Gibbs ensemble Monte Carlo (GEMC) simulation of vapor-liquid phase equilibria for pure liquid carbon monoxide. The thermodynamic behaviour of this system will be predicted by GEMC simulation. 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. 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. I wish to thank the members of the steering and the executive Committee for this overseas training project. References 1. A. E. Nasrabad and U. K. Deiters, J. Chem. Phys., 119, 947 - 952 (2003). 2. A. E. Nasrabad, R. Laghaei, and U. K. Deiters, J. Chem. Phys., 121, 6423 - 6434 (2004). 3. J. Lãpez Cacheiro, B. Fernandez, D. Marchesan, S. Coriani, C. Hattig, and A. Rizzo. J. Mol. Phys., 102, 101 - 110 (2004). 4. K. Leonhard and U. K. Deiters. Mol. Phys., 100, 2571 - 2585 (2002). 5. S. Bock, E. Bich, and E. Vogel, Chem. 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Elsevier, New York (1970). 200 300 400 500 600 -40 -30 -20 -10 0 10 20 B 2(T )/c m 3 m ol -1 T/K