Three-body Interactions in Rare Gases. René Kalus, František Karlický, Michal F. Jadavan Department of Physics, University of Ostrava, Ostrava, Czech Republic Financial support: the Grant Agency of the Czech Republic (grant. no. 203/02/1204), the Ministry of Education, Youth, and Sports of the Czech republic (grant no.FRVŠ 1122/2003, grant no. IZEP ) Abstract Analytical formula Configurations Methods and basis sets Convergence Calculations Results Least-square fits Outlooks Praha Ostrava Three-body contributions to the interaction energy of Ar 3 have been calculated using the HF – CCSD(T) method and multiply augmen- ted basis sets due to Dunning and co-workers [1]. The long-range, asymptotic part of the three-body energy has been represented by semi-empirical formulae [2] with the adjustable parameters taken from a previous ab initio calculation [3]. The calculations have been performed using the MOLPRO 2002 suite of quantum-chemistry programs. The basis set superposition error has been treated through the counterpoise correction by Boys and Bernardi [4], the convergence to the complete basis set has been analyzed, and both the triple- excitation and core-electron contributions to the three-body energy of Ar 3 have been assessed. The ab initio data have been represented by an appropriate analyti- cal formula [5]. [1] see, e.g., T. H. Dunning, J. Phys. Chem. A 104 (2000) [2] see, e. g., M. B. Doran and I. J. Zucker, Sol. St. Phys. 4 (1971) 307. [3] A. J. Thakkar et al., J. Chem. Phys. 97 (1992) [4] S. F. Boys and F. Bernardi, Mol. Phys. 19 (1970) 553. [5] inspired by V. F. Lotrich and K. Szalewicz, J. Chem. Phys. 106 (1997) This work is the first step in a series of calculations of the three- body non-additivities we plan for heavier rare gases, argon – xenon. The aim of these calculations consists in obtaining reliable three-body potentials for heavier rare gases to be employed in subsequent molecular simulations. Firstly, a thorough analysis of methods and computations is proposed for the argon trimer. In addition to the all-electron calculations presented here, which use Dunning’s multiply augmented basis sets of atomic orbitals, further calculations will be performed for argon including mid-bond functions, effective-core potentials, and core-polarization potentials. Secondly, properly chosen mid-bond functions, optimized for dimers, will be combined with effective-core potentials and with core-polarization potentials to calculate three-body energies for krypton and xenon. Our previous work on krypton and xenon two- body energies [1] indicates that this way may be rather promising. [1] P. Slavicek et al., J. Chem. Phys. 119 (2003) 2102 The geometry of Ar 3 is described by 1) the perimeter of the Ar 3 tri- angle, p, and by 2) the two smaller angles in this triangle,  and . Angles  and  used in our ab initio calculations have been cho- sen so that they spread over all the relevant three-body geometries in both an fcc and an hcp argon crystal.  correlation method – CCSD(T)  only valence electrons have been correlated  basis sets – x-aug-cc-pVNZ with N = D,T,Q and x = s,d,t  MOLPRO 2002 suite of ab initio programs  computer – PIV 2.3 GHz, 2 GB RAM, and about 10 GB HDD requested In addition to what is stated above, some additional calculations have been performed at the CCSD level to assess the importance of triple excitations, and some test calculations have also been carried out with all electrons correlated to assess how much the excita- tions of inner-shell electrons are important in the calculations of the three-body inter- actions. Three-body energies for D 3h geometry and p erimeter = 3 x A. Differences between t-aug-cc-pVQZ and d-aug-cc-pVQZ. Contributions due to triple-excitations and core correlations. For each geometry of Ar 3 given in the Configurations panel, three- body energies have been calculated at the CCSD(T) + d-aug-cc- pVQZ level for a sufficiently broad range of perimeters. The calcula- tions seem to be converged within about several tenths of K at this level. To save computer time, the following procedure has been adopted: 1.A small set of energies has been calculated for a small number of perimeters, at both the aug-cc-pVTZ and the d-aug-cc-pVQZ level. This has usually taken 7 – 10 days of computer time. 2.Differences between the aug-cc-pVTZ and the d-aug-cc-pVQZ energies have been fitted to an analytical formula. 3.A much larger set of energies has been calculated (20 – 40) at the aug-cc-pVTZ level (about one hour of computation). 4.The aug-cc-pVTZ energies of (3) have finally been corrected using the formula of (2). Differences between the aug-cc- pVTZ and d-aug-cc-pVQZ energies for a selected C 2v geometry, and the corresponding least-square fit. The ab initio points calculated for a given geometry of the Configu- rations panel (defined by a pair of angles  and  ) have been fitted to an analytical formula where A k,  k and  k are adjustable parameters, parameters C k have been calculated from the asymptotic expressions and F(p) is the damping function introduced in panel Analytical formula. Examples of least-square fits The three-body energy of an argon trimer,  E 3, depends on a li- near parameter defining its size and on two angular parameters corresponding to its shape. In this work, perimeter p of Ar 3 is used as the linear parameter and two smaller angles in Ar 3,  and  are used as the shape-related parameters. The following analytical representation is used for this dependence: where  is the permutation operator, The f kl functions are obtained from least-square fits of  E 3 calcula- ted for selected trimer configurations. In the equation for  E 3 (asymp), functions W k are taken from perturba- tion theory, K = DDD 3, DDQ 3, DQQ 3, DDO 3 and QQQ 3, and cons- tant parameters Z k from independent ab initio calculations.