Evolutive Ensemble Thermodynamics of Nanoclusters
.
J. Mulia,
A. Tame
z,
C.R
.
Sandoval
.
Facultad de Ciencias, Universidad Autónoma del Estado de México, Instituto
Literario
#
100 Toluca
,
Estado de México,
México. C.P. 50000
.
atm@uaemex.mx
Abstract
Thermodynamic properties of nanoclusters are derived through a statistical determination of their
potential energy surface using a genetic algorithm combined
with a conjugate gradient local
minimization. Stationary points, their interconnections, and a mea
sure of their contribution to the
configurational density of states are determined. This information, combined with knowledge of the
kinetic degrees of freedom, is sufficient to yield the total density of states, allowing the calculation of
thermodynamic p
roperties. There is no calculation of the forces, Boltzmann
weights, nor are
Newton’s equations of motion solved. Problems arrising from potential energy barriers are non

existent and ergodicity is greatly improved over
the mo
lecular dynamics or Monte Carl
o
approaches. Since evaporation is a natural part of this evolutive ensemble, the thermodynamics
can be extended into the liquid

to

gas transition region. Application is
made to the Lennard

Jones
clusters of size 7, 13 and 38 atoms.
A great diversity of
unusual behavior has been observed
for nanoclusters; solid

solid phase
transitions
[1], pre

melting, a continuous “melting” transition
region in which solid and liquid

like
phases coexist
[2], ebullition, negative heat capacities in the microcanonical
ensemble [3, 4
] as well
as a nontrivial
dependence of most properties on cluster size
[5
]. This flavor in the dynamics is a
result of the
unique energy distributions of the lowest lying minima
and their saddle points for each
particular type
and size of cl
uster. This, in turn, is a direct consequence
of the interplay between
inter

atomic interaction
range and the small finite size of nanoclusters.
A common error in many of
the analyses has been
to confuse the initial condition dependent dynamic
behavior wit
h true
thermodynamic behavior of nanoclusters. Nanoclusters are small systems which
may have little
total energy compared to the often
large energy barriers separating different regions of
the
energetically available phase space. Nanoclusters,
either in ex
periment or in simulation, may,
therefore,
easily become trapped and thus prevented from visiting
all of the energetically allowed
phase space. Convergence
in simulation or experiment does not imply
thermodynamic equilibrium
.
On raising the energy
sufficie
ntly, suddenly new regions of phase space
become accessible. In
we
have shown how these
problems can lead to a particular dynamical behavior
that, if confused with
thermodynamics, can lead to
the erroneous determination of negative heat capacity
in
nanoclu
sters.
Therefore, unlike in the case of the bulk, where
the initial state morphological
symmetry can be controlled
and where extremely high energy barriers pre

vent the observation of
more than one symmetry in
the solid, accurate modeling of the thermodyna
mic
behavior of
nanoclusters requires sampling the contribution
of many different symmetry configurational
structures relevant to the energetically allowed phase
space. Much of this phase space may not be
connected
isoenergetically but may be isolated by h
igh
energy barriers, frustrating the approach to
ergodicity
in the traditional molecular dynamics or Monte
Carlo approac
hes. The purpose of this
is
to
describe an alternative approach to determining the
thermodynamics of nanoclusters based on a
statistical
determination of the potential energy surface using
a genetic algorithm combined with a
conjugate
gradient local optimization which is immune to the
problems associated with trapping.
Both the molecular dynamic and Monte Carlo approaches to the thermodyn
amics of nanoclusters
sufferfrom problems of ergodicity and quasi

ergodicity. Ensuring ergodicity requires large duration
trajectories, making large systems not amenable, because of CPU time limitations, to the molecular
dynamic or Monte Carlo approaches.
A further problem is that large duration trajectories increase
the probability of evaporation, rendering the trajectory useless. Avoiding quasi

ergodicity due to
potential energy barriers and evaporation in the molecular dynamics and Monte Carlo approaches
,
requires special considerations which complicate the simulation and further reduce efficiencies and
reliability. Since the evolutive statistical approach described here concentrates on mapping the
potential energy surface and not simulating a single traj
ectory over the surface, problems of
ergodicity are reduced and quasi

ergodicity is non

existent. The natural inclusion of evaporation in
the evolutive ensemble allows the determination of thermodynamic properties into the liquid to gas
transition region.
No forces nor Boltzmann weights are calculated, neither are Newton’s equations
of motion solved. The resulting efficiency of the approach facilitates larger systems and more
realistic modeling (for example ab initio calculations).
References:
[1]
J. P.
Neirotti, F. Calvo, D. L. Freeman, J. D.Doll, J. Chem. Phys. 112 (2000) 10340.
[
2]
D. J. Wales, Mol. Phys. 78 (1993) 151.
J.P.K. Doye and D.J.Wales, J. Chem. Phys. 102
(1995) 9659.
[3]
K. Michaelian, A. Taméz, I. L. Garzón, Chem.
Phys. Lett. 370
(2003) 654.
[4]
E. M. Pearson, T. Halicioglu, and W. A. Tiller,
Phys. Rev. A 32 (1985) 3030.
[
5]
I. L. Garzón, K. Michaelian, M. R. Beltrán, A.
Posada

Amarillas, P. Ordejón, E. Artacho, D.
Sánchez

Portal, and J. M. Soler, Phys.
Rev. Lett.
81
(1998) 1600.
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