Stable
S
tructures of the
S
mall and
M
edium

S
ize
S
ingly
I
onized
H
elium
C
lusters
Daniel Hrivňák
a
,
Karel Oleksy
a
,
Ren
é Kalus
a
a
Department of Physics, U
niversity of Ostrava, Ostrava, Czech Republic
F
inancial support
:
the Grant Agency of the Czech Republic (
g
rant
s
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
x
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) 847

852
)
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. Paidarov
á, 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 2000.1
C
omparison with literature
method
E
min
R
e
D
e
[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. Ph
ys. 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
B
asis
N
multielectron wave functions of the form
where
N
is
number of
He
atoms
,
n
=2
N

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 view
Side View
Energy
[1]
[eV]
Core Charges [%]
4

2.566
51

46
5

2.59
7
51

47
6

2.6
30
51

47
7

2.66
5
51

47
8

2.70
1
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

2.91
1
49

49
1
7

2
.
9
20
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
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 2000.
H. M. Cartwright, An Introduction to Evolutionary Computation and Evolutionary Algorithms, in R. L. Johnston, Application of
Evo
lutionary
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 23

30. ga97n
J
.
J
.
Collins. S. Baluja and R. Caruana, "Removing the genetics from the standard genetic algorithm," Proceedings of ML

95, Twelfth In
ternational 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 Six
th
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.
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