Daniel Hrivňák a, Karel Oleksy a, René Kalus a a Department of Physics, University of Ostrava, Ostrava, 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). Stable Structures of the Small and Medium- Size Singly Ionized Helium Clusters Daniel Hrivňák a, Karel Oleksy a, René Kalus a a Department of Physics, University of Ostrava, Ostrava, 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 INPUT POTENTIALS RESULTS – STABLE STRUCTURES OF He N + TRIATOMICS-IN-MOLECULES METHOD (TRIM) is energy of the adiabatic (stationary) state. 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- (R. A. Aziz,, A. R. Jansen, M. R. Moldover, PLR 74 (1995) 1586, HFD – B3 – FCI1) and three-body (N. Doltsinis, Mol. Phys. 97 (1999) ) potentials for helium. E J (ABC) … energy of an ionic (ABC) fragment in the electronic ground (J = 1) and the first two excited (J = 2,3) states, taken from ab initio calculations (I. Paidarová, R. Polák, 2006) on He 3 + : method CASSCF(5,10) / icMRCI (5 active electrons in 10 active orbitals) [1] basis set d-aug-cc-pVTZ program package MOLPRO Comparison with literature method E min R e D e [hartree][bohr][eV] QICSD(T), aug-cc-pVTZ [2] QICSD(T), aug-cc-pVQZ [2] MRD-CI, cc-pVTZ [3] this work [1] H.-J. Werner and P. J. Knowles, J. Chem. Phys. 89, 5803 (1988); P. J. Knowles and H.-J. Werner, Chem. Phys. Letters 145, 514 (1988) [2] M. F. Satterwhite and G. I. Gellene, J. Phys. Chem. 99, 1339 (1995) [3] E. Buonomo et al., Chem. Phys. Letters 259, 641 (1996) TRIM Hamiltonian Hamilton Matrix where General theory: R. Kalus, Universitas Ostraviensis, Acta Facultatis Rerum Naturalium, Physica-Chemia 8/199/2001. GENETIC ALGORITHM DESCRIPTION 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. N Front viewSide View Energy [1] [eV] Core Charges [%] N Front viewSide View Energy [1] [eV] Core Charges [%] TECHNICAL DETAILS 1.Random generation of the initial population. 2.For each population: 2.1. Copy two best individuals to next generation (elitism) Select two individuals A, B by the roulette wheel Crossover of individuals A, B (one-point cut of all coordinates) Two point crossover of A, B (exchange of two nuclei locations) IF (random < rotation_probability) THEN invert each nuclei along the centre of mass for individuals A, B IF (random < mutation_probability) THEN mutate A and B (inversion of random bits in one randomly selected nucleus) Repeat 2.2. – 2.6. until next generation is completed Move randomly one nucleus for 30% of individuals (in the case of stagnation 80%, the best individual is unchanged). 3.Repeat 2. until STOP condition is fulfilled (number of generations greater then limit AND changes of the best individual fitness less then limit AND number of epochs greater then limit). Four parallel populations were simultaneously evolved. If stagnation in population 1, 2 or 3 occurred, the best individual of it was copied to population 4 and new population was created – new epoch began. Symposium on Size Selected Clusters, 2007, Brand, Austria Main parameters:  number of parallel populations = 4  number of individuals in each population = 24  probability of mutation = 0.1  probability of rotation = 0.1  number of bits per coordinate = 16  number of generations = tens of thousands V. Kvasnička, J. Pospíchal, P. Tiňo, Evolučné algoritmy, Slovenská technická universita, Bratislava H. M. Cartwright, An Introduction to Evolutionary Computation and Evolutionary Algorithms, in R. L. Johnston, Application of Evolutionary Computation in Chemistry, Springer 2004 D. M. Deaven, K. M. Ho, Physical Review Letters 75 (1995) 288 J. J. Collins and Malachy Eaton. Genocodes for genetic algorithms. In Osmera [158], pages ga97n J. J. Collins. S. Baluja and R. Caruana, "Removing the genetics from the standard genetic algorithm," Proceedings of ML-95, Twelfth International Conference on Machine Learning, A.Prieditis and S. Russell (Eds.), 1995, Morgan Kaufmann, pp B.Hinterding, R., H. Gielewski, and T.C. Peachey The nature of mutation in genetic algorithms. In Proceedings of the Sixth International Conference on Genetic Algorithms, L.J. Eshelman, ed San Francisco: Morgan Kaufmann. [1] Zero energy level is set as energy of isolated atoms.