Stable Structures of the Small and Medium- Size Singly Ionized Helium ClustersDaniel HrivňákDaniel Hrivňákaa, , Karel OleksyKarel Oleksyaa, , RenRené Kalusé Kalusaa

aa Department of Physics, UDepartment of Physics, University of Ostrava, Ostrava, Czech Republicniversity of Ostrava, Ostrava, Czech Republic

FFinancial supportinancial support:: the Grant Agency of the Czech Republic ( the Grant Agency of the Czech Republic (ggrantrantss No. 203/02/1204 No. 203/02/1204 and 203/04/2146 and 203/04/2146), Ministry of), Ministry of Education of the Czech Republic (grant No. 1N04125)Education of the Czech Republic (grant No. 1N04125)..

OSTRAVA

INPUT POTENTIALS

RESULTS – STABLE STRUCTURES OF HeN+

TRIATOMICS-IN-MOLECULES METHOD (TRIM)

-2 -1( ) ( )

TRIM1 1 1 1

1 3ˆ ˆ ˆH H H2 2

N N N NABC A

A B A C B A

N

N

(123) (123) (123)11 12 13(123) (123) (123)

(123) 12 22 234 (123) (123) (123)

13 23 33(123)neut

0

0H (He )

0

0 0 0

U U U

U U U

U U U

E

is energy of the adiabatic (stationary) state. Coefficients KJ 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.

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-pVTZprogram package MOLPRO 2000.1

Comparison with literature

method Emin Re De

[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)

TRIM Hamiltonian

Hamilton Matrix

3( ) ( ) ( ) ( )

1

,ABC ABC ABC ABCKL J KJ LJ

J

U E

3

( ) ( )

1

ABC ABCJ KJ K

K

E

where

General theory: R. Kalus, Universitas Ostraviensis, Acta Facultatis Rerum Naturalium, Physica-Chemia 8/199/2001.

GENETIC ALGORITHM DESCRIPTION

Basis1 1 2 2

1(1) (2) (3) (4)... (2 1)... ( 1) ( ) , 1,2,..., ,

!K K N Na a a a a K a n a n K N

n

N multielectron wave functions of the form

where N is 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.

N Front view Side View

Energy[1] [eV]Core Charges [%]

4 -2.566

51-46

5 -2.597

51-47

6 -2.630

51-47

7 -2.665

51-47

8 -2.701

51-48

9 -2.739

24-52-24

10 -2.778

51-46

N Front view Side View

Energy[1] [eV]Core Charges [%]

11 -2.814

51-47

12 -2.843

51-47

13 -2.864

51-47

14 -2.894

51-48

15 -2.902

51-48

16 -2.911 49 - 49

17 -2.920 49 - 49

TECHNICAL DETAILS

1. Random generation of the initial population.2. 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).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

Length of the calculations for 50000 generations

0200400600800

1000120014001600

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Size of cluster N

Tim

e [m

in]

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

1exp

2fitness energy

2, energy of the best individual

( )energy absolute value of potential energy

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 Evolutionary Computation in Chemistry,

Springer 2004D. M. Deaven, K. M. Ho, Physical Review Letters 75 (1995) 288J. J. Collins and Malachy Eaton. Genocodes for genetic algorithms. In Osmera [158], pages 23--30. ga97nJ. 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. 38--46. B. 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.

[1] Zero energy level is set as energy of isolated atoms.