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. ., . ., . ., . .

(, )
. . , . . [1]. . . -- . () - , [2, 3]. , . . -- ,
424

. . . -- -- 2006, . 2, . 424-436 Rybakov A. A. at al -- MCE -- 2006, v. 2, p. 424-436

, , . , , [4]. : [59]. , , ' ' ' '. , [10]. . -- [11-14]. . n- , . : m (m=1,2,...,n) . , . p(t ) n t [ 0, T ] :

p(t)=

(
k =1

n

e k p(t ) ) e k , (< pp* >)e k = k2 e k .

(1)

425

7. , Part 7. Mathematical Models In Biology, Ecology And Chemistry

ek ( e k *e j = k j ) (1). ek ( ), . ( (p(t ) e k )(p(t ) e j )* = k j k2 ). k-

( Ekin ) k / E

kin

= k2 /

k =1

n

k2 .
k

, ( Ekin ) :

( Ekin ) k = [ ( p(t ) e k )] / 2.
2

(2)

, m , m , [15]. , . , , ek. , . -

426

. . . -- -- 2006, . 2, . 424-436 Rybakov A. A. at al -- MCE -- 2006, v. 2, p. 424-436

, [12]. . , . . , , , , - , . mn m n [15]:

mn (ti ) = arctan ( -(p(t ) e n ) /(p(t ) em )).
; , . , (2), . , N , i - . [16] , :

Erot (t ) = 1 / 2 ( M (t ) ) , (t ) = I -1(t ) M,
M -- , -- , I (t ) -- :

I (t ) =

i
i

mi ri 2 (t ) - i 2 (t ) ,
mi i (t ) i (t ); , ( ) = x, y, z.

I (t ) =-

427

7. , Part 7. Mathematical Models In Biology, Ecology And Chemistry

p e k 3N- , N . (1) , : 3N 3 M = ( p(t ) e k ) ri (t ) Ч ( e k )i . k =1 i =1 , : 3N 1 Erot = ( Erot )k , ( Erot )k = (p(t ) e k 2 k =1 ( E
rot k

-

)

i =1

3

ri (t ) Ч ( e

ki

)

(t ),

(3)

)

k- -

. , k- , : ( Evib ) k = ( Ekin ) k - ( Erot ) k . , . neff

neff = 10

-

k

ak log ak

,

ak ( , ) k- . neff ( ) ( ). [17].

428

. . . -- -- 2006, . 2, . 424-436 Rybakov A. A. at al -- MCE -- 2006, v. 2, p. 424-436

. . 2 12 6 3p 3 i H = 2m + U( r ij ), U( rij ) = 4 U 0 - , (4) r i =1 i< j rij ij

r ij = q i - q j , q i pi -- i -
, m - . , , -- -- - (4), U 0 -- - , . . , m , U 0 . , , m = 39.945 а. е. м . U 0 = 99.55 см -1 = 3.757 / 21/ 6 A .
2 = / mU 0 2

:

-

= 2 . 93 10 - 2 , [6]. E =-3 , = 6 2 . . M max

429

7. , Part 7. Mathematical Models In Biology, Ecology And Chemistry

, , . . [6]. 2500 . , , . . ( ) , [18].
neff
5 4 3 2 1
0,2
rot

6

0,4

0,6

0,8

M 1,0

rot . 1. n eff M ( '' ) ( '' ) . = -1.5

.
430

. . . -- -- 2006, . 2, . 424-436 Rybakov A. A. at al -- MCE -- 2006, v. 2, p. 424-436

. . 1 ro ( n efft ) . ro , n efft , . , , . ro 0.2 0.7 n efft , .
1,5 1,0 0,5 0,0 -0,5

1,5 1,0 0,5 0,0 -0,5

-1,0 -1,5 0
1,5

-1,0
20 40
1,5

60

80

10 0

-1,5
0,10

t

0

20

40
0,10

60

80

t

100

1,0

1,0

0,05
0,5 0,5

0,05

0,0 0,00 0,25

0,0 0,50 0,00

0,00
0,25

t

-1

0,50

0,00

0,25

0,00 0,50 0,00

0,25

t

-1

0,50

. 2. ( , - ) () () t -- -- . =-1.5, M = 0.61 Mmax

431

7. , Part 7. Mathematical Models In Biology, Ecology And Chemistry

. 2 . . . . 3 12 . .
1000

12
750 2 1 500 250 0 0 250 500 750

t

1000

.3. 12 ( 1) ( 2) t

. 4 ( n vib ) . eff n vib eff ,

432

. . . -- -- 2006, . 2, . 424-436 Rybakov A. A. at al -- MCE -- 2006, v. 2, p. 424-436

. , [9, 11]. , n vib eff , . , . 4 . 5, , [9] , , , . , .
vib

neff

6 5 4 3 2 1
0,0 0,2 0,4 0,6 0,8 M 1 ,0

. 4. n vib ( eff '' ) ( '' ) . = - 1.5

. ,
433

7. , Part 7. Mathematical Models In Biology, Ecology And Chemistry

. , , . , , .
/ E kin, % E 60 40 20 0
0,0 0 ,2 0,4 0 ,6 0,8 M 1 ,0
vib

100 80

. 5. ( '' ) ( '' ) . = -1.5

1. .., .., .., .. // .: , 1991. 2. Leitner D.M., Berry S.R., and Whitnell R.M. Quantum chaos of Ar3: Statistics of eigenvalues // J. Chem. Phys. --1989. -- Vol. 91. -- 6. -- P. 3470-3476. 3. Amitrano C., Berry R.S. Probability distributions of local Liapunov exponents for small clusters // Phys. Rev. Lett. -- 1992. -- Vol. 68. -- 6. -- P. 729-732. 4. Berry R.S. Introductory lecture. Clusters, melting, freezing and phase transitions // Chem. Soc. Faraday Trans. -- 1990. -- Vol. 86. -- 13. -- P. 2343-2349.

434

. . . -- -- 2006, . 2, . 424-436 Rybakov A. A. at al -- MCE -- 2006, v. 2, p. 424-436

5. Jellinek J.J. and Jasien P.J. Dynamical effects in the phase change behavior of small cklusters // in 'The Structure of Small Molecules and Ions', edited by R. Naaman and Z. Vager (Plenum, New York, 1988), P. 39-41. 6. .., .., .., .. // . 1997. -- . 355 -- 6. -- C. 750-753. 7. Yurtsever E. and Elmaci . Chaotic behavior of triatomic clusters // Phys. Rev. A. -- 1997. -- 1. -- Vol. 55. -- P. 538-544. 8. Yurtsever E. Chaos in rotating triatomic clusters // Europhys. Lett. -- 1997. -- Vol. 37 (2). -- P. 91-96. 9. Yurtsever E. Angular-momentum-driven chaos in small clusters // Phys. Rev. A. -- 1998. -- Vol. 58. -- 1. -- P. 377-382. 10. Belega E.D., Trubnikov D.N., Lohr. L.L. Effect of rotation on internal dynamics and phase-space structure of rare-gas trimers // Phys. Rev. A --. 2001. -- Vol. 63. -- 043203. 11. .., .., .., .. // . -- 2002. -- .42. -- 12. -- . 1891-1898. 12. .., .., .., .. // . . '. . ', .10., 2. - .-: ' ', 2003. -- . 330-344. 13. .., .., .., .. // . , 2004. -- 5. -- . 15-21. 14. .., .., .., .. // . . . '. . '. . 12. .-: ' ', 2005. -- . 721-732. 15. Palacios A., Gunaratne G. H., Gorman M., and Robbins K. A. KarhunenLoиve analysis of spatiotemporal flame patterns // Phys. Rev. E. -- 1998. -- 5. -- Vol. 57. -- P. 5958-5971. 16. Li D.H. and Jellinek J. Separation of the Energy of Overall Rotation in Any N-Body System // Phys. Rev. Lett. -- 1989. -- Vol. 62. -- 3. -- P. 241-244. 17. . . // . -- 1988. -- . 42. -- 7. -- C. 397-442. 18. Benettin G., Galgani L., Strelcyn J.M. Kolmogorov entropy and numerical experiments // Phys. Rev. A. 1976. Vol. 14. 6. P. 2338-2345.

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7. , Part 7. Mathematical Models In Biology, Ecology And Chemistry

ORDER AND CHAOS IN INTERNAL DYNAMICS OF NONRIGID SYSTEMS Rybakov A. A., Belega E. D., Trubnikov D. N., Chulichkov A. I.
(Russia, Moscow)

The influence of non-rigid rotation on internal dynamics of triatomic argon clusters has been studied. The method of partitioning of the kinetic energy into rotational and vibrational components was suggested. It has been found that the most significant factor for the chaotic behavior of the cluster is the partitioning of the vibrational and rotational energies among the modes.

436