1. Praha Ostrava Abstract Ab initio calculations Potential energy surface fit Outlooks Financial support: the Ministry of Education, Youth, and Sports of the Czech Republic (grant no. IN04125 – Centre for numerically demanding calculations of the University of Ostrava), the Grant Agency of the Faculty of Science, University of Ostrava (grant. no. 1052/2005) Three-body contributions to the interaction energy of Ar 3 have been calculated using the HF – CCSD(T) method and multiply augmented basis sets due to Dunning and co-workers [1]. 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 [2], 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. A complete nonaditive potential has been fitted to 330 computed points and compared with other points. Choice of points Methods and basis sets  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 Analysis of convergence Analytical formula Calculations Jacobi coordinates (r, R,  ): r represents the “shortest” Ar-Ar separation, R denotes distance between the “remaining” Ar atom and the center-of-mass of the previous Ar-Ar fragment, and  is the angle between r and R vectors. Accurate ab initio three-body potential for argon František Karlický and René Kalus František Karlický and René Kalus Department of Physics, University of Ostrava, Ostrava, Czech Republic Hypersurface Tests Three-body energies for D 3h geometry and p erimeter = 3 x 3.757 A. Differences between t-aug-cc-pVQZ and d-aug-cc-pVQZ. References [ 1] see, e.g., T. H. Dunning, J. Phys. Chem. A 104 (2000) 9062. [2] S. F. Boys and F. Bernardi, Mol. Phys. 19 (1970) 553. [3] inspired by V. Špirko and W. P. Kraemer, J. Mol. Spec. 172 (1995) 265. [4] see, e. g., M. B. Doran and I. J. Zucker, Sol. St. Phys. 4 (1971) 307. [5] A. J. Thakkar et al., J. Chem. Phys. 97 (1992) 3252. [6] P. Slavicek et al., J. Chem. Phys. 119 (2003) 2102.  The geometry of Ar 3 has been described by the perimeter of the Ar 3 triangle, p, and by the two smaller angles in this triangle,  and .  Angles  and  used in our ab initio calculations have been chosen so that they spread over all the relevant three-body geometries in both an fcc and an hcp argon crystal. Contributions due to triple-excitations and core correlations. For each choice of Ar 3 geometry, three-body energies have been calculated at the CCSD(T) + d-aug-cc-pVQZ level for a sufficiently broad range of perimeters. The calculations seem to be converged within about several tenths of K at this level. To save computer time, the following procedure has been adopted:  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.  Differences between the aug-cc-pVTZ and the d-aug-cc-pVQZ energies have been fitted to an analytical formula.  A much larger set of energies has been calculated (20 – 40) at the aug-cc-pVTZ level (about one hour of computation).  Finally, the aug-cc-pVTZ energies have been corrected. Differences between the aug-cc-pVTZ and d-aug- cc-pVQZ energies for a selected C 2v geometry, and the corresponding least-square fit. Examples of results Our three-body potential (2-D cut) for argon compared with Axilrod-Teller-Muto u DDD term The least-square fit compared with additionally calculated points - 1D cut along a line in the  -  plane (see section “choice of points”), perimeter of Ar 3 triangle is fixed. The Axilrod- Teller (u DDD ) term is also added. 1D cut for fixed r,  Jacobi coordinates r R  Three-body potential has been represented by a sum of short-range [3] and long-range terms expressed in Jacobi coordinates, suitably damped by function F. Geometrical factors W lmn were derived from third-order perturbation theory [4], force constants Z lmn are taken from previous independent ab initio calculations [5]. Method  least-square fit using Newton-Raphson method  achieved standard deviation 0,4 K  maximum fit deviation 1 K from ab initio points  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 [6] indicates that this way may be rather promising.  Our potential will be tested by computing experimentally measurable results: vibrational spectra of trimers, third virial coefficient, crystal structures and their binding energies. V 3 (r = 3.757 Å,R,  ) sections of three-body potential for argon l, m, n = D-dipole, Q-quadrupole, O-octupole V 3 (r,R,  =  /2) sections of three-body potential for argon comparison our potential Axilrod-Teller-Muto u DDD term