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Cluster
Studies Group
Theoretical
Studies of Metal Clusters
and Cluster-Ligand Systems:
Unraveling the Complexity
J. Jellinek
Novel dynamical/thermodynamical analysis schemes
for finite systems
An important part of our program is development of principally
new concepts and theoretical methodologies and analysis tools
especially tailored for the small and mesoscale size ranges.
These tools are indispensable for unraveling, understanding,
and describing the challenging added complexities of finite
systems. A recent development in this area is formulation of a
new scheme in which thermodynamical/statistical notions and
concepts, such as temperature, equipartition, degrees of
freedom, finite size effects, etc. are extracted directly from the
dynamical (time-) evolution of systems. The central difference
between our new scheme and the traditional statistical analyses
is that no a priori assumption on equal probability of all the
(relevant, e.g., vibrational) degrees of freedom (or equivalently,
on rapid redistribution of the energy between all the degrees of
freedom) is involved. The new thermodynamical/statistical
notions and concepts capture the essence and reflect the
specificity of the actual dynamics of the systems, irrespective
of whether they are ordered or chaotic (or, using the traditional
terminology, "statistical" or "nonstatistical"). A new equipartition
postulate, which assures that the dynamical temperature
(computed from the time-evolution of the systems) preserves
the intensivity property of the thermodynamical temperature,
is formulated. This postulate leads to introduction of the
so-called active or dynamical degrees of freedom ddf, which
in contrast to the traditional (kinematical) degrees of freedom
are not necessarily integers, and which allow for an exact
quantification of the effects of distribution and redistribution
of energy between the subsystems (degrees of freedom) of a
system. The ddf can be computed for individual atoms or
groups of atoms (in general, arbitrary subsystems), and those
with the same values of the ddf are combined into "classes of
dynamical similarity". The number of classes of dynamical
similarity and their content (i.e., the number and nature of the
subsystems in them) depend on the system energy (or
temperature). These classes are analysis tools that allow one
to understand and characterize the most intricate behavior of
systems. They are expected to be especially powerful in
applications to complex systems, such as those of low or
no symmetry and/or comprised of more than one type of
atoms. The evolution of these classes with energy (or
temperature) gives a comprehensive and detailed
understanding and description of energy- (or temperature-)
driven phenomena. The finite size effects are incorporated
through an explicit function (or functions). In cases when the
dynamics involve all the (relevant) degrees of freedom with
equal probability our scheme coincides with the traditional
statistical analyses. Fig. 1 shows
the ddf per atom as a
function of energy for four subsystems of an initially
icosahedral Al55. These represent the central atom and the
three shells, respectively. The evolution of the ddf with energy
reflects the stagewise transition from a solidlike to a liquidlike
state. The stage of surface melting is signified by merging of
the ddf values for the shells 2 and 3. Because of its generality
our new analysis is expected to find many applications in
diverse areas of physics and chemistry.
Density functional studies of silver clusters and
silver cluster-ligand systems
We carried out gradient-corrected density functional studies
of structural and electronic properties of bare Agn, n=2-8,
clusters and (AgX)m, X=Cl, Br, m=1-4, complexes. Exhaustive
search for isomeric forms of the neutral, anionic, and cationic
species was performed without imposition of symmetry constraints.
Normal mode frequencies were computed to separate structures
corresponding to minima of the corresponding potential energy
surfaces from those representing transition state configurations.
For each isomer the binding energy, ionization potentials (vertical
and adiabatic), electron affinities, and vertical detachment energies
were computed. The nature of bonding in the (AgX)m complexes
has been characterized. The studies have also been extended to
Agn(C2H4)m complexes, which
were investigated experimentally
by M. Knickelbein in our group. The interest in these systems
stems from their relevance to the technologically important reaction
of epoxidation of ethylene to form ethylene oxide, a common
feedstock in the production of detergents, synthetic fibers, and
other products. A key step in the reaction is the adsorption of
ethylene on the surfaces of silver particles and formation of
Agn(C2H4)m complexes. The
measurements revealed that the
ionization potentials (IPs) of these complexes are lower than those
of the corresponding bare Agn clusters. Our density functional
studies of Ag3C2H4 identified two isomeric
forms with binding
energies of 0.82 eV and 0.47 eV, respectively (cf. Fig.
2).
One of the intriguing results of the computations is that whereas
the IP of the more stable isomer I is by 0.37 eV lower than that
of Ag3, which is in quantitative agreement with the measured
data, the IP of the higher energy isomer II is by 0.16 eV higher
than that of the bare silver trimer. The IP, therefore, emerges
here as a label that allows one to identify isomer I as the
dominant, if not exclusive, structure of the Ag3C2H4
complexes
produced in the experiment. A complementary corroboration
of this structure comes from the comparison of the computed
fundamental vibrational frequencies with the measured infrared
spectra. In addition to the neutral Ag3C2H4
we have
characterized also the cationic forms of the complex. The transition
states along the different isomerization pathways have also been
identified. Computations are in progress on the neutral and
cationic Ag5C2H4 species.
First principles-based Monte Carlo studies of
lithium clusters
The approach is based on a hybrid Hartree-Fock(HF)/density
functional (LYP) electronic structure treatment combined with a
J-walking Monte Carlo sampling of the nuclear configuration
space. Simulated cooling from many different high-energy
configurations was used to identify the equilibrium structures.
Four isomeric forms of Li8 and three isomeric forms of Li9+
were found. To explore the thermal behavior of the clusters,
J-walking MC simulations were performed at 20 temperatures
covering the range of 10-700 K. 3·105 to 4·105 nuclear
configurations were sampled at each temperature, and the
energy of each configuration was computed at the HF/LYP
level. The quantitative analysis was performed in terms of
distributions of the potential energy and short-walk averaged
potential energy at different fixed temperatures, caloric curve
(long-walk averaged potential energy as a function of
temperature), and configurational heat capacity as a function
of temperature. The graph of the latter
for Li8 is shown in
Fig. 3. The peak in the graph is a signature of a solid-to-
liquid-like transition. The broadness of the peak indicates
the temperature range over which the transition takes place.
This study is among the most extensive first principles-based
explorations of thermal properties of clusters to date.
Structural and thermal properties of bimetallic clusters
In addition to size, bimetallic, or more generally two-component,
clusters offer two additional "knobs" for tuning their properties.
These knobs are the choices of the component elements and
their relative concentrations. Using a semiempirical many-body
potential, we performed large-scale numerical simulation studies
of mixed Ni/Al clusters of different sizes. For each size all the
possible compositions were considered. For each composition
a number of different geometric structures (isomers) of the clusters
was explored, and for each isomer the equilibrium structures and
energies of its homotopic forms (which correspond to different
distributions of the component elements between the sites of an
isomer) were mapped out. We have introduced an exact energy-
based quantitative measure of the degree of mixing (mixing
coefficient) applicable to any potential and have shown that the
large manifold of the possible structural forms can be subdivided
into classes within which the energy ordering of the homotops is
defined by the mixing coefficient. General trends in the energy
spectra of the structural forms as a function of the composition
were uncovered. The dynamical/thermal properties were studied
over a broad range of energy/temperature sufficient to observe a
solid-to-liquid-like transition. We gave an analysis of the size-
and composition-specific mechanisms of this transition and have
shown that these mechanisms can be understood in terms of the
size- and composition-specific features of the energy spectra
of the corresponding structural forms. Our theoretical explorations
are complemented by recent experimental studies of mixed
Ni/Al clusters in the group (see Experimental
Cluster Chemistry
Studies ).
Computation of electron binding energies within
the density functional theory
It is well known that the Kohn-Sham eigenenergies of the
traditional (time-independent) density functional theory (DFT)
correspond to "quasiparticles" rather than to real electrons.
Consequently, these eigenenergies have to be corrected if one
wants to compute electron binding energies within DFT. Recently
we formulated a general phenomenological scheme, which allows
one to compute the correction term for any electron of the system.
The merits of the scheme are that it uses only ground state
properties rigorously defined within DFT and it is applicable to
any version of the DFT. Extensive tests on atoms and molecules
prove its accuracy. DFT studies of structural and electronic
properties of magnesium clusters We carried out gradient-
corrected DFT studies of neutral and charged magnesium
clusters. Equilibrium structures, binding energies, and electronic
properties such as ionization potentials, electron affinities,
vertical detachment energies, and nature of bonding were
characterized. We have applied our new correction scheme
to compute electron binding energies relevant to the
photoelectron spectroscopy measurements on anionic
magnesium clusters performed recently by K. Bowen's
group (Johns Hopkins University). The goal of these
measurements was to test the insulator-to-metal transition
in magnesium clusters as a function of cluster size. The results
of our computations are not only in excellent agreement with
the measured data, but they also help to understand and
interpret them. Other areas of our theoretical studies include
detailed investigations of cluster-molecule collision systems;
dynamical and statistical analyses of cluster fragmentation
(this work includes development of novel statistical analyses
schemes); development of new, more adequate many-body
potentials (this includes formulation of new fitting procedures);
structural and electronic properties of constrained clusters, etc.
Selected Recent Publications
ALLOY CLUSTERS: STRUCTURAL CLASSES, MIXING,
AND PHASE
CHANGES, J. Jellinek and E. B. Krissinel, In Theory of Atomic and
Molecular Clusters with a Glimpse at Experiments, J. Jellinek, Ed.,
Springer-Verlag, Heidelberg, 1999, p. 277
ON THE PROBLEM OF FITTING MANY-BODY POTENTIALS:
1.
THE MINIMAL MAXIMUM ERROR SCHEME AND THE PARADIGM
OF METAL SYSTEMS, M. J. Lopez and J. Jellinek, J. Chem. Phys. 110,
8899 (1999)
AB INITIO MONTE CARLO STUDIES OF SMALL LITHIUM
CLUSTERS,
S. Srinivas and J. Jellinek, Phys. Stat. Sol. B 217, 311 (2000)
CHARGE TRANSFER AND FRAGMENTATION IN CLUSTER-ATOM
COLLISIONS, O. Knospe, J. Jellinek, U. Saalman, and R. Schmidt,
Phys. Rev. A 61, 022715-1 (2000)
THEORETICAL INVESTIGATIONS OF SILVER CLUSTERS
AND
SILVER-LIGAND SYSTEMS, S. Srinivas, U. A. Salian, and J. Jellinek,
In Metal-Ligand Interactions in Biology, Chemistry and Physics,
N. Russo and D. R. Salahub, Eds., Kluwer Academic Publishers, Dordrecht,
2000, p. 295
REACTIONS OF SMALL CLUSTERS WITH DIATOMIC
MOLECULES:
MD SIMULATIONS OF D2+Nin (n=7-10) SYSTEMS, P. Durmus,
M. Boyukata,
S. Ozcelik, Z. B. Guvenc, and J. Jellinek, Surf. Sci. 454, 310
(2000)
ON THE TEMPERATURE, EQUIPARTITION, DEGREES
OF FREEDOM,
AND FINITE SIZE EFFECTS - APPLICATION TO ALUMINUM CLUSTERS,
J. Jellinek and A. Goldberg, J. Chem. Phys. 113, 2570 (2000)
STRUCTURE AND DYNAMICS OF Pd13 CLUSTERS, M.
Boyukata,
M. Karabacak, S. Ozcelik, Z. B. Guvenc, and J. Jellinek, Bulg. J. Phys.
27, 110-114 (2000)
MOLECULAR DYNAMICS STUDY ON STRUCTURES AND
MELTING OF Pd9 CLUSTERS, M. Karabacak, M. Boyukata, S. Ozcelik,
Z. B. Guvenc, and J. Jellinek, Bulg. J. Phys. 27, 123-126 (2000)
STRUCTURE AND MELTING OF Cu13 CLUSTERS, S.
Ozcelik,
Z. B. Guvenc, and J. Jellinek, Bulg. J. Phys. 27, 127-130 (2000)
THE TEMPERATURE EFFECT ON CLUSTER-MOLECULE
INTERACTIONS D2 + Ni14, P. Durmus, S. Ozcelik, Z.
B. Guvenc,
and J. Jellinek, Bulg. J. Phys. 27, xxx-xxx (2000)
DYNAMICS OF THE D2 + Ni(100) COLLISION SYSTEM:
ANALYSIS
OF THE REACTIVE AND INELASTIC CHANNELS, M. Boyukata,
Z. B. Guvenc, B. Jackson, and J. Jellinek, Int. J. Quantum Chem. 84,
48 (2001)
STRUCTURE AND REACTIVITY OF Nin
(n=7-14, 19) CLUSTERS,
M. Boyukata, Z. B. Guvenc, S. Ozcelik, P. Durmus, and J. Jellinek, Int.
J.
Quantum Chem. 84, 208 (2001)
THEORETICAL INVESTIGATIONS OF THE INTERACTION
OF SILVER
TRIMER WITH ETHYLENE MOLECULE, U. Salian, S. Srinivas, and
J. Jellinek, Chem. Phys. Lett. 345, 312-318 (2001)
THEORETICAL AND EXPERIMENTAL STUDIES OF THE
STRUCTURES
OF 12-, 13- AND 14-ATOM BIMETALLIC NICKEL/ALUMINUM CLUSTERS,
E. F. Rexer, J. Jellinek, E. B. Krissinel, E. K. Parks, and S. J. Riley,
J. Chem.
Phys. 117 (1), 82-94 (2002)
MAGNESIUM CLUSTERS: STRUCTURAL AND ELECTRONIC
PROPERTIES AND THE SIZE-INDUCED NONMETAL-TO-METAL
TRANSITION, J. Jellinek and P. H. Acioli, J. Phys. Chem. A 106,
10919-10925 (2002)
ELECTRON BINDING ENERGIES OF ANIONIC MAGNESIUM
CLUSTERS AND THE NONMETAL-TO-METAL TRANSITION,
P. H. Acioli and J. Jellinek, Phys. Rev. Lett. 89 (21), 213402/1-213402/4
(2002)
CONVERTING KOHN-SHAM EIGENENERGIES INTO ELECTRON
BINDING ENERGIES, J. Jellinek and P. H. Acioli, J. Chem. Phys.
118 (17), 7783-7796 (2003)
Books edited
THEORY OF ATOMIC AND MOLECULAR CLUSTERS
WITH A GLIMPSE
AT EXPERIMENTS, J. Jellinek (Ed.), Springer-Verlag, Heidelberg, Germany,
1999
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