OSTRAVA

1. OSTRAVA Stable Structures of the Small and Medium- Size Singly Ionized Helium ClustersDaniel Hrivňáka, Karel Oleksya, René KalusaaDepartment of Physics, University of Ostrava, Ostrava, Czech RepublicFinancial support: the Grant Agency of the Czech Republic (grants No. 203/02/1204 and 203/04/2146), Ministry ofEducation of the Czech Republic (grant No. 1N04125). TRIATOMICS-IN-MOLECULES METHOD (TRIM) General theory: R. Kalus, Universitas Ostraviensis, Acta Facultatis Rerum Naturalium, Physica-Chemia 8/199/2001. TRIM Hamiltonian N multielectron wave functions of the form Basis where Nis number of He atoms, n=2N-1 is number of electrons, ai 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. Hamilton Matrix where is energy of the adiabatic (stationary) state. Coefficients xKJ are calculated using the DIM method; in case the three-body correction to the He3+ interaction energy is a small perturbation, the resulting Hamiltonian matrix is expected to be correct up to 1st order of perturbation theory. GENETIC ALGORITHM DESCRIPTION INPUT POTENTIALS • Random generation of the initial population. • For each population: • 2.1. Copy two best individuals to next generation (elitism). • 2.2. Select two individuals A, B by the roulette wheel. • 2.3. Crossover of individuals A, B (one-point cut of all coordinates). • 2.4. Two point crossover of A, B (exchange of two nuclei locations). • 2.5. IF (random < rotation_probability) THEN invert each nuclei along the centre of mass for individuals A, B. • 2.6. IF (random < mutation_probability) THEN mutate A and B (inversion of random bits in one randomly selected nucleus). • 2.7. Repeat 2.2. – 2.6. until next generation is completed. • 2.8. Move randomly one nucleus for 30% of individuals (in the case of stagnation 80%, the best individual is unchanged). • 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. Eneut(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) 847-852) potentials for helium. EJ(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 He3+: method CASSCF(5,10) / icMRCI (5 active electrons in 10 active orbitals) [1] basis set d-aug-cc-pVTZ program package MOLPRO 2000.1 Comparison with literature method EminReDe [hartree] [bohr] [eV] QICSD(T), aug-cc-pVTZ [2] -7.896672 2.340 2.598 QICSD(T), aug-cc-pVQZ [2] -7.902103 2.336 2.640 MRD-CI, cc-pVTZ [3] -7.8954 2.34 2.59 this work -7.897021 2.339 2.639 [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) • V. Kvasnička, J. Pospíchal, P. Tiňo, Evolučné algoritmy, Slovenská technická universita, Bratislava 2000. • H. M. Cartwright, An Introduction to Evolutionary Computation and Evolutionary Algorithms, in R. L. Johnston, Application of EvolutionaryComputation 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 23--30. 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, • Prieditis and S. Russell (Eds.), 1995, Morgan Kaufmann, pp. 38--46. • Hinterding, R., H. Gielewski, and T.C. Peachey. 1995. The nature of mutation in genetic algorithms. In Proceedings of the Sixth International Conference on Genetic Algorithms, L.J. Eshelman, ed. 65--72. San Francisco: Morgan Kaufmann. RESULTS – STABLE STRUCTURES OF HeN+ TECHNICAL DETAILS • 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 [1] Zero energy level is set as energy of isolated atoms. Symposium on Size Selected Clusters, 2007, Brand, Austria