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10th CONFERENCE
on

DYNAMICAL SYSTEMS THEORY AND APPLICATIONS
Decem ber 7-10, 2009. L Сd , Pol a nd

ABOUT EFFECTIVE MODES METHOD FOR DESCRIPTION OF INTERNAL DYNAMICS OF WEAKLY BOUND CLUSTERS
Elena Belega, Evgeny Cheremukhin 1 , Dmitry Trubnikov
Abstract: New approach to describe the internal dynamics of weakly bound clusters is presented. It is based on the method of main components of nonlinear dynamical systems which was adapted to study collective modes behavior in molecular systems. The method was applied to extract the collective modes in small water clusters. It is found that the number of active collective modes depends on the initial cluster excitation and that the internal energy is partitioned nonuniformly among the modes. Calculated collective modes were used to estimate the internal temperature of small water clusters and energy intervals for their quasi-phases. 1. Introduction Study of weakly bound clusters plays an important role in various areas of science, in particular, in physical chemistry, molecular biology, in the theory of phase transitions in finite-size systems, in the theory of nonlinear dynamic systems. Investigation of movement of particles in the clusters conducts to understanding the mechanisms of isomerization and fragmentation and to revealing their dynamic features. The study is faced with some principal difficulties: all types of particle motion are mixed. As a result the dynamics is sophisticated and can not be described by well known method of normal modes because of interaction of different types of vibrations in weakly bound clusters. This interaction influences on energy redistribution between internal degrees of freedom and, consequently, on the rate constant of isomerization and monomolecular fragmentation. Describing the internal dynamics in the weakly bound clusters it is more reasonable to talk not about the degrees of freedom but about collective motions of the particle in the clusters. The highdimensional nature of the dynamics is often a substantial obstacle to clarifying the exact mechanisms

1

Work partially supported by grant RFBR 09-07-00505a.


of collective motions and new approaches to identify a small number of collective variables that play a predominant role in the structural dynamics of clusters are very important. The authors of given paper suggest to use the main components method, which has been adapted to describe the internal dynamics of weakly bound atomic clusters [1] to study the collective motions in small water clusters. 2. Method of effective modes in molecular dynamics Let us describe the internal motion of dynamical system in momentum subspace dimension N . Multidimensional vector of momentum p(t ) of system with

E

p N

with

n

particles and masses

mi ( i = 1, n ) can be presented as superposition of N collective modes ( N = 3n ) at each moment
of time

t [0, t0 )
p( t ) =

(where t0 ­ time of observation over the system): ( p ( t ), e k )e k .


k =1

N

Specially ordered orthonormal basis vector set

{ek }k

=1, N

E

p N

determines the main directions in

the phase space with the maximum kinetic energies components. Effective collective mode (i.e. main component of movement) is the projection of kinetic

^ momentum p k (t ) = (p (t ), e k )e k at every direction ek . Such basis can be constructed and ordered as a
result of solution of eigen values task:

p ( t ) p* ( t ) e k = k2 e k ,
p* ( t ) p ( t ) g k = k2 g k , k = 1, N ,
where g
k

- singular vectors conjugated to ek . Values

1 2 ... n 0, are sorted in the

nonascending order. The first effective mode has the maximum value of the total kinetic energy of dynamical system. First

m ( m N ) time-averaged effective modes have more kinetic energy than others m modes
The value

that are calculated with any another orthonormal basis set.

k =

2 k


k =1

N

2 k

gives the contribution of the k-th mode to common dynamics in new coordinate system.


Contribution of k-th mode in simultaneous kinetic energy of the system can be estimated by formula:
2 Ek ( t ) = k2 g k (t )


i =1

n

(eik )2 . 2mi

Time-averaged kinetic energy

E
2

k

in k-th mode can be calculated as:

E

k

=

k2
2T


i =1

n

(e )
i k

mi

= E k ,

where E ­ mean kinetic energy of system during the time of observation

t

0

.
^ E

^ As a result K-dimensional ( K N ) approximation of p(t ) with the mean kinetic energy
be calculated as:

can

^ p(t ) =


k =1

K

^ ( p ( t ), e k ) e k , E =


k =1

K

E

k

Time-averaged square error of such approximation can be predicted by formula:
K

^ E=

k = K +1



N

Ek .

The values



k

give the kinetic energy distribution among collective degrees of freedom called

collective modes. The effective dimension Keff

(
N

Keff N of the momentum subspace E

)

p N

in which

the most significant movement of dynamical system is concentrated is calculated as:

K

eff

= exp -


k =1

k

lnk .

(1)

The effective dimension reflects the number of internal degrees of freedom filled by kinetic energy and can be used to calculate the internal temperature of the cluster. The temperature is the parameter which is borrowed from statistical physics for describing an internal state of the cluster. Standard estimation of cluster temperature calculated as:

T=
where k is Boltzmann constant and

2E kB K
,
df

(2)

B

K

df

is the number of internal degrees of freedom. For non-

rotating cluster placed in mass center K

df

= 3n - Q - 6 , where Q ­ number of rigid bonds in the

cluster. Definition (2) works well for the clusters consisting of identical atoms and is not correct for


complex multicomponent atomic and molecular systems which are clusters of water. For the last an assumption of equipartition of energy among internal degrees of freedom is not valid. Using effective phase dimension Keff instead Kdf in (2) gives the temperature

Tef

f

that can estimate temperature of

micro canonical ensemble of molecular systems more accurately. 2. Analyses of small water clusters dynamics For application of the effective modes method the internal dynamics of small water clusters has been considered. Water clusters are important weakly bound complexes due to the crucial role which they play in natural processes. They belong to a class of nonrigid systems with very complicated internal dynamics. Study of the dynamics can clarify on microscopic scale a wide range of chemical problems from hydrogen bond rearrangement rules to thermodynamic properties of water. Water pentamer and tetramer (nmol = 4, 5) were chosen for the study. Coordinates and momenta of the atoms in the cluster were calculated by classical molecular dynamics. Classical Hamiltonian of cluster can be defined as follows:

H ( p1 (t ), ..., p n (t ), r1 (t ), ..., rn (t )) =


i =1

n

pi2 (t ) + U (r1 (t ), ..., rn (t )) , 2mi

where U (r1 (t ), ..., rn (t )) - potential energy of interaction between particles. Generation of different initial states of non-rotated bound cluster with initial energy H 0 = U 0 + E0 was performed by MonteCarlo method, where energy. The value

U 0 - is potential energy of global minima of cluster', E0 - is initial kinetic

U 0 and initial coordinates of atoms ri (0) can be found once by well known

thermal annealing method for example. Setting up the initial excitation value

E0

it can possible to

generate randomly initial atoms momenta p i (0 ) that must satisfy system of equations:


where


n

n

i =1

p i (0 ) = 0 , ri (0 ) , p i ( 0 ) ] = 0 ,

[
i =1 n



i =1

p i2 (0 ) = E0, 2mi

[,]

is the vector product. For the study rigid TIP5P [2] potential was used to describe

interaction between molecules in the cluster. The initial geometries of clusters obtained in this paper are placed in Fig.1.


Fig.1 Water tetramer (left) and pentamer (right) initial geometry (global miminum of potential TIP5P energies: U0=-1.233 eV for tetramer and U0=-1.653 eV for pentamer). For Hamilton equations numerical integration the next parameters of trajectories were chosen: the integration step h was equal to 1 fs, observed time interval t0 = 2 - 5 ns. The Verlet algorithm provided with the accuracy of the integration in the third sing in the value of the total energy. The values of all parameters presented in this work were obtained by averaging on 10 trajectories for each initial excitation energy E0 of the cluster. Quasi-states of cluster of two types (A, B) of atoms usually can be determined using Lindemann parameter that calculated as:

2 ( AB ) = N AB ( N AB - 1)
Where


i< j

N

AB

rij2 - rij rij

2

,

(3)

N

AB - number of pairs of atoms of types A and B,

r

ij

- distance between them. Solid-

liquide states boundary of cluster usually determined by criteria ( both clusters.

> 0.1

) and placed in Fig. 2 for

Fig.2 Lindemann' indexes for water tetramer (left) and for water pentamer (right) in dependence of excitation energy

E0

.


Lindemann' parameter usually works well in systems like rare-gases clusters when isotropic types of interactions between all particles are present only. In particles with weak anisotropic bonds like water clusters that parameter doesn't fully describe liquid-solid phase transition of cluster. Analysis of parameter (3) in application for water clusters (parameters (OO), (OH ), ( HH ) for different types of atoms) can bring typically four energy regions of cluster' states: Solid-like state of cluster ( < 0.1 ); E0<0.15 eV for tetramer and E0<0.2 eV for pentamer. Only small vibrations of all atoms are presented. `Pre-melting' state of cluster (

· ·

> 0.1 ); 0.15
for pentamer; There is only hydrogen atoms' activity are observed but oxygen atoms (water molecules as whole) construct stationary (with lifetime (structural isomers).

>1

ps) spatial structures

·

Liquid-like state (characterized by strong growth of ); 0.5


1ps) are viewed.

·

Monomolecular fragmentation (characterized by stabilization of tetramer and E0>0.65 eV for pentamer.



); E0>0.6 eV for

Fig.3. Effective dimensions of momentum subspace of phase spaces of water tetramer and pentamer) in dependence of excitation energy

E0

.


Lindemann' parameter estimates particles activity in a general way and doesn't divide types of motions. Effective dimension of momentum phase space Keff of water clusters is placed in Fig.3. Dependence Keff from E0 also can help understand dynamics activity in every quasi-phase. On the example of pentamer Keff has values 13-14 in solid-like phase, stable (~15) in pre-melting state and going up to 17-18 in fragmentation region. It is interesting that in region E0=0.3-0.4 eV value Kef
f

starts to growth due to Lindemann' index in that region have a plateau. Such behavior can be explained by faster cluster structure isomerization and motion of cluster became non-planar. For tetramer that region gives E0=0.25-0.35eV. At energies lower that threshold there is only hydrogen atoms' activity are observed but water molecules as whole construct stationary planar structural isomers with seldom transitions (



1

ps). Collective motion of oxygen atoms performs mostly in

plane. At higher energies nonplanar structures of cluster can be created sometimes so the effective dimension Keff of phase space starts to growth again. Other application of parameter Keff is correction of micro canonical temperature of cluster. As it is seen in Fig.4 temperature Teff calculated by formula (2) using Keff as parameter has values above ~100K higher than standard one T in all ranges of energies. Someone can conclude from Fig.4 that mean kinetic energy by effective directions in the phase space (like number of freedom) has not equipartition distribution for small water clusters.

Fig.4. Temperatures of water tetramer and pentamer in dependence of excitation energy standard approach (2) with number of freedom Kdf momentum phase space Keff (Teff-curves) as parameter.

E

0

using

(T-curves) and effective dimension of


4. Conclusions Method of main components for collective motion analysis of weakly bonded molecular systems is presented in paper. Collective modes for water tetramer and pentamer have shown that mean kinetic energy has not equipartition distribution by different types of motion. Method of main components can help to determine number of active modes at any energy of cluster. Growing up excitation energy gradual modes activation is performed and redistribution of kinetic energy by modes is viewed. Using effective dimension of the momentum phase space together with Lindemann' parameter brings more rich information about phase quasi-states of cluster and detects isomerization and fragmentation thresholds. Method of main components can be easily adapted to other molecular systems. References 1. Belega E. D., Rybakov A. A.,. Trubnikov D. N, Chulichkov. Effective Dimension of a Trajectory in the Visualization of Dynamical Systems. // A. I. Comp. Math. and Math. V.42. 2002. P.1817-1823. 2. M.W. Mahoney, W.L. Jorgensen. A five-site model for liquid water and the reproduction density anomaly by rigid, nonpolarizable potential functions. // J. Chem. Phys. V.112. P.8910-8922. Phase Phys. of the 2000.

Belega Elena (edbelega@gmail.com) Eugene Cheremukhin (eacherem@gmail.com) Dmitry Trubnikov (tdn@phys.chem.msu.ru) Chemical Department, M.V.Lomonosov Moscow State University, Vorobyevy Gory, b.1, 119991, Moscow, Russia