Size Singly Ionized Helium Clusters

libyantawdryAI and Robotics

Oct 23, 2013 (3 years and 5 months ago)

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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.