nanomaterials. design and simulation, 2007, p.329

⑦
18
THEORETICAL AND COMPUTATIONAL CHEMISTRY
Nanomaterials:Design and Simulation
THEORETICAL AND COMPUTATIONAL CHEMISTRY
SERIES EDITORS
Professor P. Politzer
Department of Chemistry
University of New Orleans
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VOLUME 1
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VOLUME 2
Modern Density Functional Theory: ATool
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Molecular Electrostatic Potentials: Concepts
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Recent Developments and Applications of Modern
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Pauling’s Legacy: Modern Modelling of the
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VOLUME 9
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VOLUME 10
Valence BondTheory
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VOLUME 11
Relativistic Electronic StructureTheory, Part 1.
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VOLUME 12
Energetic Materials, Part 1. Decomposition, Crystal
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Energetic Materials, Part 2. Detonation,
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Molecular and Nano Electronics:
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VOLUME 18
Nanomaterials: Design and
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P.B. Balbuena and J.M. Seminario (Editors)
⑦
18
THEORETICAL AND COMPUTATIONAL CHEMISTRY
Nanomaterials:Design and Simulation
Edited by
Perla B. Balbuena
Jorge M. Seminario
Department of Chemical Engineering
and Department of Electrical and Computer Engineering

Texas A&M University
College Station, Texas,USA.
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Contents
Preface vii
1 Electrical Characteristics of Bulk-Molecule Interfaces 1

Perla B. Balbuena, Lina R. Saenz,
Carolina Herrera, and Jorge M. Seminario
2 Structural Properties of Pure and Binary Nanoclusters Investigated by
Computer Simulations 35
Giulia Rossi and Riccardo Ferrando
3 Computer Simulation of the Solid–Liquid Phase Transition in Alkali Metal
Nanoparticles 59
Andrés Aguado and José M. López
4 Multiscale Modeling of the Synthesis of Quantum Nanodots and their
Arrays 85
Narayan Adhikari, Xihong Peng, Azar
Alizadeh, Saroj Nayak, and Sanat K. Kumar
5 Structural Characterization of Nano- and Mesoporous Materials by
Molecular Simulations 101
Lourdes F. Vega
6 Hydrogen Adsorption in Corannulene-based Materials 127
Yingchun Zhang, Lawrence G. Scanlon, and Perla B. Balbuena
7 Toward Nanomaterials: Structural, Energetic and Reactivity Aspects
of Single-walled Carbon Nanotubes 167
T. C. Dinadayalane and Jerzy Leszczynski
8 Thermal Stability of Carbon Nanosystems: Molecular-Dynamics
Simulations 201
¸Sakir Erkoç, Osman BarI¸s MalcIo
˘
glu and Emre Ta¸scI
9 Modeling and Simulations of Carbon Nanotubes 227
Alper Buldum
10 Nano-Confined Water 245
Alberto Striolo
v

vi Contents
11 Ab Initio Simulations of Photoinduced Molecule–Semiconductor Electron
Transfer 275
Walter R. Duncan, William Stier and Oleg V. Prezhdo
12 Nano-Particulate Photocatalysts for Overall Water Splitting under
Visible Light 301
Kazuhiko Maeda and Kazunari Domen
Index 317
Preface
Since the Richard Feynman dictum “there is plenty of room at the bottom” in 1959,
several approaches have been developed to design materials using a bottom-up approach,
i.e., designing nanostructured or molecularly structured materials. This novel avenue has
revolutionized practically all fields of science and engineering, providing an additional
design variable, the feature size of the nanostructures, which can be tailored to provide
new materials with very special characteristics. A particularly important role towards
a rational design of nanostructures is played by atomistic modeling, including a variety
of methods from ab initio electronic structure techniques, ab initio molecular dynam-
ics, to classical molecular dynamics, also being complemented by coarse-graining and
continuum methods. Such rainbow of computational tools offers the great possibility of
exploring nanoscopic details and using such information for the prediction of physical
and chemical properties in some cases impossible to be obtained experimentally, and in
others providing an invaluable new instrument for guiding and interpreting experiments.
This volume covers several aspects of the simulation and design of nanomaterials
analyzed by a selected group of active researchers in the field. The editors thank all
the contributors for their kind collaboration, effort, and patience to make this book a
reality. The editors would like to recognize the effort and dedication of Ms Mery Diaz,
who helped putting together this camera-ready volume. The editors also acknowledge
the US Army Research Office and the US Department of Energy for their sustained
interest in several aspects of the nanomaterials field.
Perla B. Balbuena and Jorge M. Seminario

Texas A&M University
vii
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Nanomaterials: Design and Simulation
P. B. Balbuena & J. M. Seminario (Editors)
© 2007 Elsevier B.V. All rights reserved.
Chapter 1
Electrical Characteristics of Bulk-Molecule
Interfaces
a
Perla B. Balbuena,
a
Lina R. Saenz,
a
Carolina Herrera,
and
ab
Jorge M. Seminario
a
Department of Chemical Engineering and
b
Department of Electrical and Computer Engineering, Texas A&M University,
3122 TAMU, College Station, TX 77842 USA
1. Introduction
Electron transfer reactions between molecules and surfaces of nanosized clusters and of
bulk materials are of paramount importance for the development of new materials for
catalysis, sensors, and power source devices. For example, the oxygen reduction reaction
is one of the surface electrode reactions of low temperature fuel cells, characterized by
its slow kinetics even when taking place on Pt, the best catalyst currently known for
this reaction [1]. Improving the performance of such catalytic process may significantly

contribute to an enhanced fuel cell performance, which is strongly needed in order to
make fuel cells a commercial reality [2]. Thus, achieving such goal involves first to
develop a thorough understanding of the Oxygen Reduction Reaction mechanism, which
is still debated in spite of vast research efforts [3].
According to the generally accepted knowledge, the oxygen reduction reaction on Pt
surfaces in acid medium may proceed via a direct four-electron pathway that reduces O
2
to H
2
O via a sequence of four electron and proton transfers, or via a series pathway where
H
2
O
2
is produced as an intermediate, which is then reduced to water [3]. Recently Bal-
buena et al. have reported ab initio molecular dynamics [4] and density functional theory
[5] results which suggest that a parallel (direct and series) mechanism may be in place
at the fuel cell operational conditions, with the direct as the dominant pathway. The elec-
trolyte used in proton-exchange membrane fuel cells is a polyelectrolyte membrane, which
conducts protons from the anode, where a fuel such as H
2
is oxidized, to the cathode
where O
2
is reduced [2, 6]. In such environment, the production of H
2
O
2
is highly
undesired because of its potential for generating radicals that degrade the polymer mem-

brane [7], one of the most expensive materials of the fuel cell, along with the catalyst itself.
1
2 P. B. Balbuena et al.
Several reaction intermediates (OOH, OH, O, H
2
O
2
) and H
2
O (the reaction product)
of the oxygen reduction reaction are produced by combined electron and proton
transfer which take place sequentially or simultaneously with the O
2
dissociation on
the catalytic surface [4, 8]. In previous work [5, 9–11] we investigated the adsorption
of intermediate species of the oxygen reduction reaction on bimetallic clusters using
density functional theory in small clusters, and we identified the variation of the
adsorption strength produced by the presence of one or more foreign atoms (such as Cr,
Co, Ni) located in a neighbor position to a Pt active site. Moreover, we analyzed [12]
the effect of the metal on the discrete molecular orbitals of O
2
. One important feature
is given by the broadening of the peaks corresponding to the energy states of the
molecule. The energy of the HOMO orbital of the O
2
molecule is −654 eV, lying just
below the Fermi level of Pt (−593 eV), and that of the LUMO orbital is −511 eV. In
the molecule–metal system, we found that those states are shifted revealing the transfer
of electrons from the metal to the antibonding 
∗

states of O
2
[12].
On the other hand, in a totally different application, which uses the same techniques
shown here, electron transfer reactions at molecule–bulk interfaces are examined because
of their vital importance for the development of molecular electronics circuits able to
continue the scaling-down of integrated circuits after present CMOS technology reaches
its physical limits [13–22].
In this chapter, we illustrate with a few examples details of electron transfer at
the metal–molecule and semiconductor–molecule interfaces as calculated from first
principles. First, we analyze the effect on the local density of states of Pt and Pt-alloys
of the main intermediate species that adsorb on the catalytic surface (H
2
O
2
, OOH,
O, OH, and H
2
O) during the oxygen reduction reaction and for comparative purposes
similar examinations are done for the adsorption of H atoms on Pt clusters. Second, a
semiconductor–hydrogen interface is analyzed with the same methods. In all cases, the
molecule-cluster system is embedded in a continuum bulk, thus adding carefully the
non-local effects while keeping high accuracy for the local effects.
2. Computational Methods
Three different types of first principles methods were used in our approach to understand
electron dynamics on metal surfaces. The single molecule or cluster calculations, which
given the advance on hardware and software are able to contain all atoms needed to
consider local effects, i.e., most of the chemistry. The second group of methods are the
ab initio methods for extended systems, which are very similar to the single molecule
calculations except that they allow us to calculate a single cell with periodic boundary

conditions and therefore providing the effects of the continuum or bulk material. And
finally, a Green’s function approach that allows us to consider the effects of the bulk
on the single molecule, providing a precise interpretation of any physics and chemistry
taking place at the reaction sites.
2.1. Calculation of Single Molecules and Clusters
Except indicated otherwise, density functional theory (DFT) as implemented in Gaussian
2003 [23] is used to study the optimized structures, binding energies, Mulliken charge
Electrical Characteristics of Bulk-Molecule Interfaces 3
distribution, and vibrational frequencies of molecule and clusters. All calculations were
performed using the B3PW91 hybrid functional, which uses a combination of the Becke3
(B3) [24] exchange functionals and the Perdew–Wang (PW91) [25, 26] correlation
functionals. The combined functional B3PW91 is used with the quasi-relativistic pseu-
dopotential LANL2DZ (Los Alamos National Laboratory, doble-) to describe the 1s to
4f core electrons for Pt, Ni, Co, and Cr using effective core pseudopotentials [27–29].
For hydrogen, first and second row atoms we use the standard basis set 6-31G(d) also
named 6-31G*; this combination of functional, basis set, and pseudopotential used to
describe the cluster-molecule complexes in this study has been found to provide excel-
lent results in several related applications [9, 11, 12]. In particular, when tested against
results from precise experiments of molecules containing only first and second row
atoms, results are of chemical accuracy [30, 31]. Validations on systems containing
higher atoms from third row or higher are difficult to make because the scarcity of
experimental data with chemical accuracy. Following the structural optimization by
calculating the derivatives of the energy with respect to the Cartesian coordinates of all
atoms, which assures that the atomistic system is in equilibrium, i.e., forces are approx-
imately zero in all atoms, a second derivative calculation is performed to determine the
existence of a true local minimum and to find zero-point energies. The second derivative
is needed because the fact that the forces are zero does not guarantee a local minimum
for such geometrical structure. If negative eigenvalues are found in the Hessian matrix,
the geometry is modified in order to get away from those negative eigenvalues whose
eigenvectors determine dissociative states. The self-consistency of the non-interactive

wave function was performed with a convergence threshold on the density matrix of
10
−6
and 10
−8
for the root-mean-square and maximum density matrix error between
iterations, respectively. These settings provide correct energies of at least five decimal
figures, three for the atom lengths and one for the bond angles within the level of theory.
2.2. Electron Transfer at Interfaces Bulk-Molecule
Using a combination of the Green’s function theory and density functional theory
we study the electron transport characteristics of different Metal–molecule interfaces
[32–34]. In its present version, our approach is able to handle two different bulk materials
interfaced to the molecule, although for most practical applications in catalysis only
one bulk material would be enough. The case with two interfaces is very important to
determine the electrical characteristics of single molecules approached by two different
electrodes [21]. In general, an interface, [Bulk
1
-Cluster
1
]-[M]-[Cluster
2
-Bulk
2
], consists
of an extended cluster that is composed of a molecule or central cluster M, a cluster
of Bulk
1
atoms from the bulk interfacing the molecule and a cluster of Bulk
2
atoms

interfacing the molecule on another site 2; both clusters are followed by a semi-infinite
bulk (Bulk
2
) on the left, and a semi-infinite bulk material (Bulk
2
), as shown in Figure 1.
All the chemistry takes place at site M and the two bulk materials do not react with
each other.
Partial DOS of the bulk materials represent the s p, and d band contributions to the
bulk or catalyst materials, which are attached to discrete clusters of the bulk materials
atoms, representing the interface to the system M. The system M is a molecule or
cluster containing a finite number of atoms where the electron transfer reactions are
4 P. B. Balbuena et al.
Figure 1 A molecule embedded in a Bulk
2
material (usually vacuum, air, or solvent) is inter-
acting with a Bulk
1
material (usually a catalyst). An extended molecule, M, is defined as the
reactive species (H
2
O
2
in this case) augmented with the nearest atoms from the bulk materials
(Pt
3
in this example)
calculated. To do this and to keep the chemistry of reactant and products as close to
reality, calculations of M are actually performed on an extended molecule or cluster
Cluster

1
-[M]-Cluster
2
thus M has the local effect of the contacts. A Green’s function
approach takes the discrete characteristics of the extended molecule and includes the
nonlocal effect of the contacts. Preparing a calculation of electron transport through a
discrete chemical system interconnected to two large contacts requires extreme care as
performing the experiment on a single molecule. This is why these calculations need to be
performed with rigorous ab initio methods, avoiding any empirical or phenomenological
theory.
Basically, three different types of calculations are performed in our approach: the
ab initio DFT calculations for the extended cluster or molecule, the ab initio DFT
calculations with periodic boundary conditions for the bulk materials, and the Green’s
function calculation for the electron transport through the junction.
Crystal-2003 [35] is used to study the bulk materials using the linear combination of
atomic orbital approximation with periodic boundary conditions. Crystal-2003 is a suite
of programs that can calculate electronic structure, total energy, and wave functions,
including its band structure, density of states (DOS), electron charge distribution, elec-
tron momentum distribution, Compton profile, Mulliken charges, electrostatic potentials,
and X-ray structure factors [36] using a DFT with a linear combination of atomic orbitals
with periodic boundary conditions [37]. The basis set expands the Bloch functions built
using, s p, and d Gaussian functions.
In order to include the chemistry of the junction in the calculations, full geometry
optimizations and second derivative evaluation are performed for the extended molecule.
The extended cluster or molecule [Bulk
1
]-[M]-[Bulk
2
], calculation optimizes the geom-
etry to make sure that all forces in all atoms are zero, then the second derivative of the

energy is calculated to guarantee that the Hessian matrix has no negative eigenvalues
confirming that the optimized structure of the extended cluster is a local minimum [38].
If the optimized structure is not a local minimum, the structure of the extended cluster
is adjusted and re-optimized until a local minimum is reached. Chemically speaking all
these are very important steps in order to consider the important local effects at the
Electrical Characteristics of Bulk-Molecule Interfaces 5
interface. Then, a series of single point calculations of the extended cluster or molecule
under the effect of different applied electric fields are performed; and the Hamiltonian
and overlap matrices obtained for each external electrical field are used in the Green’s
function approach. All the calculations of the extended clusters and molecules were
performed with the Gaussian 2003 program [23]. The partial DOS, s p d
eg
, and d
t2g
bands for the bulk catalysts are obtained and used to construct the self-energy matri-
ces for Green’s function approach. For the sake of consistency, both the discrete and
continuous systems are calculated using the same method and basis sets.
The extended molecule and the bulk calculations are both first principles quantum
mechanics calculations at the B3PW91/LANL2DZ level of theory, which corresponds
to a Kohn–Sham (KS) Hamiltonian [39, 40] with the Becke-3 hybrid exchange func-
tional [41] and the generalized-gradient approximation (GGA) Perdew–Wang 91 cor-
relation functional [42, 26]. The basis set used is the LANL2DZ, which also includes
effective core potentials for heavy atoms [27, 28, 43].
The Green’s function approach requires as input: (a) the Hamiltonian and overlap
matrices of the extended cluster or molecule under different electric field biases and
(b) the partial DOS of the bulk materials. From the Green’s function matrix, the DOS
at the interfaces are calculated [32].
In the electron transfer calculation, the matrix representation of Green’s function is
used. The retarded Green’s function matrix, G
R

, satisfies the following matrix equa-
tion [32, 44]:
EI −H
interface
G
R
=I (1)
where H
interface
is the interface Hamiltonian matrix, I is the unit matrix, and E is the
energy of the injecting electron. Since the interface is an open system, the interface
Hamiltonian matrix should count the whole interface system including the cluster or
molecule, the contacts, and the semi-infinite bulk. The contributions from the quasi-
infinite bulk are modeled by self-energy 
1
and 
2
terms for the Bulk
1
and the Bulk
2
materials, respectively [45, 46]. In this calculation, the construction of self-energy
follows the procedure introduced in [32]. The contribution to the interface Hamiltonian
from the cluster and bulk is evaluated from the Kohn–Sham Hamiltonian H
KS
Extended
and overlap matrix S
Extended
of the extended cluster (cluster + bulk) as follows: A
transformation,

H =S
−1
Extended
H
KS
Extended
(2)
is performed because the atomic basis sets are not orthogonal [47]. And then, H is
rearranged into sub-matrices [48]:
H =
⎡
⎣
H
11
H
1C
H
12
H
C1
H
CC
H
C2
H
21
H
2C
H
22

⎤
⎦
(3)
The sub-matrices H
11
, H
22
, H
CC
, H
12
(H
21
), H
1C
(H
C1
), and H
2C
(H
C2
) correspond
to matrix elements of the left bulk and right bulk materials, the cluster, the coupling
between the two bulk materials, the coupling between the cluster and left bulk, and
6 P. B. Balbuena et al.
the coupling between the cluster and right bulk, respectively. Thus, the corresponding
Green’s function is expressed as,
GE V  =ESV −HV
−1
=

⎡
⎣
g
−1
1
−
1
0
−
+
1
ES
M
V  −H
M
V  −
+
2
0 −
2
g
−1
2
⎤
⎦
−1
and the interface Hamiltonian can then be written as
H
interface
=H

CC
+
1
+
2
(4)
where 
i
E = 
+
i
g
i

i
for j = 1 2. As opposed to an isolated cluster or molecule,
which has real eigenvalues, an interface has complex eigenvalues since the interface
Hamiltonian matrix is not Hermitian as a result of the imaginary nature of the self-
energy [49]:


=
0
−

+i

/2 (5)
With complex eigenvalues, the originally discrete electronic states of the isolated cluster
become broadened peaks with width 


, and their positions shift from 
0
to 
0
−

.
These broadened peaks are described by a continuous function, the DOS:
DOSE =


1
2


E −
0
+


2
+

/2
2
(6)
3. Adsorption of OOH, OH, O, H
2
O

2
and H
2
O
on a Platinum Surface
A number of oxygenated species result upon oxygen reduction in acid medium catalyzed
by platinum. Among them, the radical OOH results from the interaction of a physisorbed
or chemisorbed O
2
– the strength of the interaction depending on pH, solvent, degree
of coverage of the surface – with a hydrated proton; this radical has a short life on the
surface and rapidly becomes dissociated into adsorbed O and OH, which in turn become
involved in new electron and proton transfer reactions producing water molecules.
As an alternative pathway, the adsorbed OOH radical before dissociation may combine
with a proton yielding weakly adsorbed H
2
O
2
. This molecule may easily desorb from
the surface and decompose, providing highly reactive radical species that may cause the
degradation of the electrolyte membrane.
Thus, it is important to characterize the differences among the metal–molecule inter-
face for each of these species. We have chosen four typical cases to analyze the effect of
the adsorbates on the local density of states of bulk Pt. In all cases, the local interaction
involves one or more O atoms interacting with Pt top sites. The optimized geometries are
shown in Figure 2, and Figure 3 depicts possible schemes of the interface molecule–bulk
metal.
The adsorption geometries shown in Figure 2 were obtained from DFT optimizations
using the hybrid B3PW91 functional together with the LANL2DZ pseudopotential and
basis set for Pt atoms and the 6-31G* basis set for H and O atoms.

Electrical Characteristics of Bulk-Molecule Interfaces 7
Figure 2 DFT optimized structures for adsorption of intermediate species of the oxygen reduc-
tion reaction on Pt
3
. Top left: H
2
O
2
adsorbed through one of its oxygen atoms on top of one of
the Pt atoms; Top right: two water molecules each adsorbed via O–Pt interaction; Bottom left:
O + OH; Bottom right: adsorption of radical OOH
Figure 3 Most likely metal–molecule interfaces for the geometries of Figure 2. Eventually, our
procedure can provide a specific bulk DOS for any shape of the material
The calculated local density of states is shown in Figure 4 for the cases displayed in
Figure 3. The DOS corresponding to Pt (blue curve) is characterized by a broad band of
electronic states located approximately at the Fermi level of the metal (∼593eV). Note
that the presence of the adsorbates strongly modifies the local electronic characteristics.
8 P. B. Balbuena et al.
*

Pt
3
3
2.5
2
1.5
DOS
1
0.5
0–1–3 –2–4–5

Energy (ev)
–6–7–8–9–10
*

O

+

OH
*

2H
2
O
*

H
2
O
2
*

OOH
*

H
2
O
2
Figure 4 Local density of states (arbitrary units) of the systems shown in Figure 3. The exper-

imental Fermi level of Pt is located at −593 eV. The various curves describe the local effect
of the adsorbate on the DOS of the bulk metal. Such local DOS is found very sensitive to the
position of the adsorbate site
Platinum is an electron donor for the O atom that becomes negatively charged,
especially in the cases of adsorbed OOH, O, and OH. Thus, in these cases (yellow and
purple cases), even though there is a depletion of states (compared to pure Pt) in the
range −5to−7 eV, the DOS in that range shows a relatively high population which is
attributed to the electronic states of the adsorbates. In contrast, adsorption of H
2
O and
H
2
O
2
are much weaker, and those adsorbates do not contribute new states at energies
between −7 and −5eV.
This example clearly illustrates the analysis that can be performed to investigate
details of the electronic states at the interface of the reactants, intermediates, and products
with the catalytic sites. In the next sections we describe details of the DFT calculations
carried out to analyze the metal–cluster and metal–semiconductor systems.
4. Toward a Platinum Testbed
A challenging problem is to find a testbed to perform reactions simulating events on a
platinum surface. The initial step for this is to study systematically the characteristics
of small clusters resembling one of the typical surfaces where reactions are going to
take place. For this preliminary work we have chosen the (111) surface of platinum.
The goal is to obtain a cluster of platinum atoms showing a geometry compatible with
a (111) surface that is in static equilibrium and that corresponds to a local minimum.
Then molecules of interest are attached in order to study the effects on the electron
characteristics of the interface. We focus mainly in these studies on the DOS at the
interfaces. The structure and energies of platinum, hydrogenated platinum clusters, and

oxygenated platinum clusters have been performed using the B3PW91 functional.
Electrical Characteristics of Bulk-Molecule Interfaces 9
4.1. Platinum Clusters
The LANL2DZ and SDD basis sets were employed to study the platinum clusters
[50–53]. The geometries and energies of the Pt
n
n = 1 2 3 clusters are shown in
Table 1 and Figure 5.
The lowest energy corresponds to the triplet state using the SDD basis set for platinum
monomer and dimer. As shown in Table 1, the bond lengths obtained with the two basis
sets are different. We have found recently that the SDD basis set does not provide better
results than the LANLDZ. For the Pt trimer there is not a marked difference between
the singlet and triplet states. Using the LANDZ basis set the singlet and the triplets
correspond to D
3h
and C
2v
point groups.
The bond lengths, angles, and energies of Pt
4
are shown in Table 2. The geometry is
displayed in Figure 6. The SDD basis set provides lower energy values than LANL2DZ
basis set. The minimum energy corresponds to triplet state. The bond distances of Pt
4
with symmetry D
2h
are similar for both basis sets.
The bond distances and angles of Pt
5
for different multiplicities are shown in Table 3

and Figure 7. The bond length values yielded by the two basis sets are very similar.
Table 4 displays the total energies for the lowest states of each multiplicity. The lowest
energy corresponds unambiguously to the quintet state.
Table 1 Structure and Energies for the Pt
n
n =1 2 3 cluster by using the B3PW91 functional
with the LANL2DZ and SDD basis sets
Molecule m Bond lengths (Å) Energy (Ha)
LANL SDD LANL SDD
Pt 3 – – −11911851 −11934386
1 2.662 2.672 −238256 87 −23871101
3 2.356 2.528 −238335 96 −23876748
Pt
2
5 2.395 2.533 −23827572 −23873340
7 2.394 2.388 −238114 65 −23857253
1 2.499 2.499 −357560 49 −35823354
1–2 2.558
Pt
3
3 1–3 2.478 2.527 −357560 23 −35823270
2–3 2.558
7 2.545 2.537 −357484 07 −35815741
Figure 5 Optimized structures of Pt
2
and Pt
3
. The bond length values for different multiplicities
are shown in Table 1
10 P. B. Balbuena et al.

Table 2 Structure and energies of Pt
4
by using LANL2DZ and SDD basis set using the B3PW91
functional
m Basis set Bond length and angle optimized Energy (Ha)
(Å) (

)
R1 R2 
LANL 2.557 2.495 121.6 −47675690
1 SDD 2.559 2.494 121.7 −477657 01
LANL 2.577 2.471 122.7 −477657 01
3 SDD 2.551 2.702 116.0 −477660 73
5 LANL 2.556 2.542 120.4 −476763 96
SDD 2.552 2.616 118.3 −477658 51
LANL 2.529 2.731 114.6 −47673542
7 SDD 2.526 2.720 114.8 −47763384
R1
R
2
θ
Figure 6 Structure optimized of Pt
4
Table 3 Bond lengths (Å) of Pt
5
for different multiplicities using the B3PW91 functional with
the LANL2DZ and SDD basis sets
m = 1 m =3 m = 5 m = 7
LANL SDD LANL LANL SDD LANL SDD
1–2 2.475 2.504 2.531 2.493 2.504 2.510 2.501

1–3 2.797 2.660 2.925 3.143 2.636 2.776 2.648
1–4 2.797 2.661 2.638 2.612 2.636 2.776 2.648
1–5 2.475 2.504 2.535 2.549 2.504 2.510 2.501
2–3 2.520 2.484 2.487 2.513 2.568 2.491 2.584
3–4 2.520 2.768 2.695 2.540 2.652 2.673 2.514
4–5 2.520 2.484 2.488 2.489 2.568 2.491 2.584
Electrical Characteristics of Bulk-Molecule Interfaces 11
Figure 7 Optimized Structure of Pt
5
Table 4 Energies (Ha) of Pt
5
using the B3PW91 func-
tional with the LANL2DZ and SDD basis sets for several
multiplicity states. The triplet state of Pt
5
using SDD did
not optimize
Multiplicity LANL SDD
1 −59595631 −59709459
3 −595980 17 –
5 −59599458 −59711328
7 −59598635 −59710210
4.2. Hydrogenated Platinum Clusters
The LANL2DZ and the SDD basis sets were used to study Pt
n
H
m
clusters (n =3 7 13)
for their singlet, triplet, and quintet states.
Also several other theoretical and experimental studies have been already performed

[52, 54–56]. The geometrical parameters for PtH
m
(m =1, 2, 3, 4) are shown in Table 5.
We can observe in Table 5 that the difference in predicted geometries between the two
basis sets is larger for the smaller systems and it improves for the larger systems.
Tables 6–12 display the geometries and energies of PtH
m
(m =5 611) clusters.
All distances are reported in Angstroms (Å). Table 6 shows that the lowest state
corresponds to a quartet (three unpaired electrons), which is roughly at 68 kcal/mol
below the lowest doublet state (one unpaired electron).
Table 7 shows the results for PtH
6
. The ground state is definitely a triplet, which is
several eV below the nearest lowest quintet state. In the case of PtH
7
the doublet state is
the ground state. The sextet state is far above. Table 9 shows the results for the singlet
and triplet of PtH
8
. The singlet is the ground state.
For PtH
9
the doublet is the ground state followed by the sextet (with five unpaired
electrons) and then the quartet (three unpaired electrons).
The singlet of PtH
10
becomes the ground state followed by the triplet and then the
quintet (Table 11).
PtH

11
yield the quartet as ground state 50 kcal/mol below the doublet. Looking
carefully at Figure 8, it can be determined that PtH
11
has dissociated into the triplet
PtH
10
leaving one free hydrogen atom.
12 P. B. Balbuena et al.
Table 5 Geometries and Energies to PtH
m
(m =1, 2, 3, 4) by using LANL2DZ and SDD basis
sets for different multiplicities. PtH
3
with multiplicity 2 did not optimize by using SDD basis set
Bond length (Å) Angle (

) Energy (Ha)
m LANL SDD LANL SDD LANL SDD
2 1.536 1.722 −119742 63 11995957
PtH 4 1.722 1.701 −11964312 −11987478
6 3.182 3.549 −11942870 −119671 10
1 1.656 1.663 1800 1800 −120112 64 −120490 78
PtH
2
3 1.683 1.672 1800 1800 −120267 20 −120497 05
5 2.920 1.673 1800 1800 −120062 12 −120327 43
2 1.874 – 1200 1200 −120612 28 –
PtH
3

4 1.666 1.654 1200 1200 −120845 02 −121081 44
6 2.827 4.154 1200 1200 −120592 35 −120849 39
PtH
4
3 1.667 1.661 900900 −12141851 −121658 62
5 1.676 1.664 900900 −121242 55 −121482 77
Table 6 Geometries and energies of PtH
5
for doublet and quartet states
Bond lengths and angles 2 4
R11560 1653
R22792 1665
1 

 900900
2 

 1200 1200
Energy (Ha) −121886 31 −12199474
Table 7 Geometries and energies of PtH
6
for triplet and quintet states. All
bond lengths are the same and the angle between hydrogen–platinum–hydrogen
is 90

Bond lengths 3 5
Pt–H 1664 1506
Energy (Ha) −122574 39 −122112 52
Table 8 Geometries and energies of PtH
7

for doublet and sextet states
Bond lengths 2 6
1–2, 1–3, 1–5, 1–6, 1–7 164 201
1–4 164 166
1–8 167 166
Energy (Ha) −123230 13 −122902 90
Electrical Characteristics of Bulk-Molecule Interfaces 13
Table 9 Geometries and energies of PtH
8
for singlet and triplet
states. The value of platinum–hydrogen bond length is the same
Bond lengths 1 3
Pt–H 165 228
Energy (Ha) −123852 47 −12334608
Table 10 Geometries and energies of PtH
9
for doublet, quartet, and sextet states
Bond lengths and angles 2 4 6
1–2, 1–3, 1–4 184 207 495
1–5, 1–6, 1–7, 1–8, 1–9, 1–10 164 228 164
Energy (Ha) −12429834 −123993 43 −124127 34
Table 11 Geometries and energies of PtH
10
for singlet, triplet, and quintet
Bond lengths and angles 1 3 5
1–2, 1–3, 1–4, 1–5, 1–6, 1–7, 1–9, 1–10 165 173 188
1–8, 1–11 258 179 174
Energy (Ha) −124768 39 −12474565 −12459776
Table 12 Geometries and energies of PtH
11

for doublet and quartet
states
Bond lengths and angles 2 4
1–2 186 162
1–3, 1–5 192 208
1–4 345 517
1–6, 1–7, 1–8 186 162
1–9, 1–12 170 205
1–10, 1–11 192 208
Energy (Ha) −125284 32 −12536680
The lowest energy of PtH
5
 PtH
6
, and PtH
7
correspond to multiplicities 4, 1, and 6,
respectively. The lowest energy singlet states of PtH
8
 PtH
9
and PtH
10
correspond to
D
4h
 C
3v
, and D
2h

, point groups, respectively. Notice that the hydrogen atoms tend to
join or to move away from the molecule due their short distance between themselves
in the PtH
m
clusters for m =6 711. The PtH
12
molecule with symmetry O
h
was
run with different bond distances but none of them optimized. This was very important
to simulate a single atom connected to the 12 nearest neighbors as it takes place in the
face-centered cubic structure of platinum. Table 13 shows the bond lengths that were
tested with their corresponding energies.
14 P. B. Balbuena et al.
(a) (b) (c)
(d) (e) (f)
(g) (h) (i)
(
j) (k) (l)
6
6
3
3
1
1
7
1
1
1
1

10
11
9
7
8
3
3
3
3
7
7
2
2
2
2
2
2
4
4
4
7
4
4
4
5
5
5
5
5
5

10
6
6
6
6
8
8
8
9
9
Figure 8 Optimized structures of a. PtH, b. PtH
2
, c. PtH
3
, d. PtH
4
, e. PtH
5
, f. PtH
6
, g. PtH
7
,
h. PtH
8
, i. PtH
9
, j. PtH
10
, k. PtH

11
, l. PtH
12
Table 13 Trial and final bond lengths and their corresponding energies. None opti-
mized correctly
Bond length (Å) Energy (Ha) Final bond length (Å)
1.7 −125330 1.913
1.8 −125545 1.789
1.9 −125545 1.959
2.0 −125530 1.827
2.5 −125489 2.045
3.0 −125502 2.051
Table 14 summarizes the bond lengths, multiplicities, and energies for the lowest states
of PtH
n
clusters. Their corresponding optimized structures are shown in the Figure 8.
Once the chemistry of a single platinum atom interacting with a group of hydrogen
atoms is understood, we decided to increase the number of platinum atoms in our
Electrical Characteristics of Bulk-Molecule Interfaces 15
Table 14 Symmetries, multiplicities, and energies of the lowest states of the PtH
n
clusters using
the B3PW91 with the LANL2DZ and SDD basis sets
Molecule Symmetry Multiplicity Less energy
(Ha/particle)
Basis set
PtH D
h
2 −119742 6 LANL
−1199596 SDD

PtH
2
D
3h
3 −120267 2 LANL
−120497 1 SDD
PtH
3
D
3h
4 −120845 0 LANL
−1210814 SDD
PtH
4
D
4h
3 −121418 5 LANL
−1216586 SDD
PtH
5
D
3h
4 −121994 7 LANL
PtH
6
O
h
3 −122574 4 LANL
PtH
7

D
5h
2 −123230 1 LANL
PtH
8
D
4h
1 −123852 5 LANL
PtH
9
C
3h
2 −124298 3 LANL
PtH
10
D
2h
1 −124768 4 LANL
PtH
11
C
2v
4 −125366 8 LANL
PtH
12
O
h
did not optimize LANL
clusters toward forming a stable testbed representing a surface of platinum. The process
was not straightforward because finding a small cluster of platinum atoms resembling a

(111) surface is not trivial because these structures are very unstable.
In addition, the optimization of these structures is not straightforward and some-
times impossible to converge as is the case of a planar cluster of platinum atoms. This
compelled us to insert hydrogen atoms to the cluster in order to improve their static
stabilization. The energies and the number of imaginary frequencies of Pt
3
H
n
clusters
with n = 12 22, and 30 are shown in the Table 15. For all these cases, the number
of imaginary frequencies is too large to be of practical interest. Imaginary frequen-
cies correspond to negative eigenvalues in the Hessian matrix and relate to unstable
geometrical conformations.
The Pt
7
H
m
clusters were developed in order to find an appropriated geometry that can
yield zero imaginary frequencies. The number of hydrogen atoms m was varied in order
to reach the goal of zero imaginary frequencies. The results are shown in Table 16, and
Table 15 Energies and number of imaginary frequencies for Pt
3
H
m
clusters
for different multiplicities
Molecule Multiplicity Energy (Ha) # of imaginary frequencies
Pt
3
H

12
1 −36471327 4
3 −36462596 13
Pt
3
H
22
1 −37049622 3
3 −370324 38 11
5 −370324 37 10
Pt
3
H
30
3 −37500981 9
16 P. B. Balbuena et al.
Table 16 Energies and number of imaginary frequencies for Pt
7
H
m
clusters
Multiplicity Energy (Ha) # Negative frequencies
Pt
7
H
18
3 −84504861 28
Pt
7
H

18
1 −84509302 15
5 −84502561 20
Pt
7
H
12
1 −84155051 11
Pt
7
H
12
1 −84166017 7
3 −84150887 17
Pt
7
H
10
1 −84045510 9
Pt
7
H
9
3 −83981335 10
Pt
7
H
6
1 −83796763 4
3 −83799021 6

5 −83804509 5
Pt
7
H
6
1 −83810145 0
5 −83809669 0
Figure 9 displays the optimized geometries. It was fortunate to find out that the three
hydrogen atoms strategically located on the cluster make the seven platinum clusters
stable and still with a structure that resembles the (111) surface of platinum.
This structure corresponds to a singlet state (or spin zero structure) and have a quintet
excited state at only 3 kcal/mol above. Figures show that steric effects are the main
reasons for unstable structures.
The lowest energy and stable structure corresponds to Figure 9h. The advantage
of having actually two totally different electronic structures for practically the same
geometric conformation it is a great advantage for the testbed. We can say that this
is the smallest testbed that we can use. The corresponding geometries are shown in
Table 17. It can be observed that the singlet state has larger Pt–Pt distances, however
but shorter Pt–H distances are.
(a) (b) (c) (d)
(e) (f) (g) (h)
Figure 9 Structures of Pt
7
H
m
clusters: a. and b. Pt
7
H
18
, c. and d. Pt

7
H
12
, e. Pt
7
H
10
, f. Pt
7
H
9
,
and g. and h. Pt
7
H
6