1. Daniel Hrivňák a, František Karlický a, Ivan Janeček a, Ivana Paidarová b, and René Kalus a a Department of Physics, University of Ostrava, Ostrava, Czech Republic b J. Heyrovsky Institut of Physical Chemistry, Prague, Czech republic Financial support: the Grant Agency of the Czech Republic (grants No. 203/02/1204 and 203/04/2146), Ministry of Education of the Czech Republic (grant No. 1N04125). Semiempirical Modelling of He N + Clusters Daniel Hrivňák a, František Karlický a, Ivan Janeček a, Ivana Paidarová b, and René Kalus a a Department of Physics, University of Ostrava, Ostrava, Czech Republic b J. Heyrovsky Institut of Physical Chemistry, Prague, Czech republic Financial support: the Grant Agency of the Czech Republic (grants No. 203/02/1204 and 203/04/2146), Ministry of Education of the Czech Republic (grant No. 1N04125). OSTRAVA RESULTS – COMPARISON WITH AB INITIO DATA 2 THEORY I – DIATOMICS-IN-MOLECULES METHOD (DIM) General theory: F. O. Ellison, J. Am. Chem. Soc. 85 (1963), 3540. Application to He N + : Knowles, P. J., Murrel, J. N., and Hodge, E. J., Mol. Phys. 85 (1995), 243. Ovchinnikov et al., J. Chem. Phys. 108/22 (1998), 9350. Diatomic inputs DIM Basis N multielectron wave functions of the form where N is number of He atoms, n=2N-1 is number of electrons, a i is helium 1s-spinorbital with centre on i-th atom (dash over a label denotes opposite spin orientation), || represents Slater determinant (antisymetrizator). K-th wavefunction of the base represents electronic state with the electron hole on K-th helium atom. DIM Hamiltonian P-th Q-th P-th Q-th P-th Hamilton Matrices a) Overlap neglected b) Overlap included P-th Q-th P-th Q-th where and Overlap matrix for He 3 + where and is overlap integral of atomic orbitals localized on the J-th and K-th atom respectively. For hydrogen-like orbitals it has the form as Thee potential energy curves for He 2 and He 2 + : E neut (r) – ground-state for He 2 [R. A. Aziz, A. R. Janzen, and M. R. Moldover, Phys. Rev. Letters 74 (1995) 1586 ]. E u + (r), E g + (r) – ground state ( 2  u + ) and first excited state ( 2  g + ) of He 2 + [F.X. Gadéa, I. Paidarová, Chem. Phys. 209 (1996) 281. J. Xie, B. Poirier, and G. I. Gellene, J. Chem. Phys. 122 (2005) Art. No. 184310. ] where Symmetric (true) configuration results from ab-initio model and DIM with overlap model with Z ef <2. Asymmetric (false) configuration results from standard DIM model and from DIM with overlap model with Z ef >= 2. CONCLUSIONS Median of deviations: DIM – 137 meV Overlap (FIT) – 122 meV Overlap (Z = 1.9) – 65 meV Median of deviations: DIM – 168 meV Overlap (FIT) – 143 meV Overlap (Z = 1.9) – 67 meV Median of deviations: DIM – 117 meV Overlap (FIT) – 96 meV Overlap (Z = 1.9) – 52 meV 1 Fitted generalized overlap formula: 2 See this poster session, Paidarová a kol., Ab initio calculations on He 3 + of interest for semiempirical modelling of He n +. 1 1 1  Standard DIM method gives no satisfactory results for He n + clusters. Stable configuration of He 3 + trimer in DIM approach is a linear asymmetrical instead of the linear symmetrical, for example 3.  Inclusion of the overlap to the DIM method gives relevant changes of results. Parameters of the overlap formula can be set properly to minimize deviation between ab initio data and resulted data from DIM model. The most important parameter is effective atomic number Z ef.  Correct (i. e. symmetrical) stable configuration of He 3 + results from model DIM + overlap with Z ef < 2. The best agreement with ab initio data has been achieved for values of Z ef between 1.6 and 1.9, but resulting typical deviation about 60 meV in potential energy is not quite satisfactory.  Next possibility to enhance accuracy of the DIM + overlap method is to fit some parameters in the overlap formula. How indicate our first results, this way is not very hopeful.  We want to attain really better results by using the so called triatomics-in-molecules method (TRIM). In opposite of the DIM method, the TRIM method organically involves three-body corrections to the diatomic energies.  As an input to the TRIM method serve three-atomic potential energy hypersurfaces for three lowest energy levels. Construction of these accurate hypersurfaces is our topical goal.  The semiempirical methods mentioned above are based on the semi-classical Born-Oppenheimer approach, whose application to the lightweight helium atoms is quite limited. The main advantage of these methods is their computational inexpensivity.  It will be necessary to use some fully quantum method for more exact results (Path Integral Monte Carlo, Diffusion Monte Carlo etc.). THEORY II – TRIATOMICS-IN-MOLECULES METHOD (TRIM) Coefficients  KJ are calculated using the DIM method; in case the three-body correction to the He 3 + interaction energy is a small perturbation, the resulting Hamiltonian matrix is expected to be correct up to 1 st order of perturbation theory. E neut (ABC) … energy of a neutral (ABC) fragment in the electronic ground-state, calculated using semiempirical two- and three-body potentials for helium, E J (ABC) … energy of an ionic (ABC) fragment in the electronic ground (i = 1) and the first two excited (J = 2,3) states, taken from ab initio calculations on He 3 + (see also this poster session: I. Paidarová et al., Ab initio calculations on He 3 + of interest for semiempirical modelling.) TRIM Hamiltonian Triatomic inputs Hamilton Matrix, where where is energy of the adiabatic (stationary) state General theory: R. Kalus, Universitas Ostraviensis, Acta Facultatis Rerum Naturalium, Physica-Chemia 8/199/2001. 3 Very good known results, see Knowles, P. J., Murrell, J. N., Mol. Phys. 87 (1996), 827, for example.