Structure prediction
issue: due to greediness and isotropy, large variety of possible
morphologies (structural motifs) which determine properties
Icosahedron Decahedron Truncated Leary TD Poly-Icosahedron
Octahedron (hybrid)
possible interconversion
how to predict the cluster structure(s) ?
International School on NanoAlloys (ISNA) Tirrenia, May 21th – 25th 2012
Structure prediction
interfacial metallic/oxidic nanosystems:
structure of supported metal nanoparticles
not only “small” particles (N ≤ 100)
(PtCo)64/MgO(100) Au32/MgO(100)
but also “large” particles (100 ≤ N ≤ 10000)
how to predict the cluster structure(s) ?
International School on NanoAlloys (ISNA) Tirrenia, May 21th – 25th 2012
Structure prediction
hierarchical algorithms for structure prediction
• “small” systems (N ≤ 40):
density-functional basin-hopping (DF-BH) algorithm
• “intermediate” systems (40 ≤ N ≤ 200):
density-functional empirical-potential (DF-EP) algorithm
• “large” systems (200 ≤ N ≤ 1000):
empirical-potential global optimization (EP-GO) algorithm
• “very large” systems (1000 ≤ N ≤ 10000):
extrapolation based on structural motifs
underlying theory: global optimization
International School on NanoAlloys (ISNA) Tirrenia, May 21th – 25th 2012
Length rescaling approach
1-20 Å 10-50 Å 5-20 nm
Quantum Mechanics Classical Mechanics Effective Models
Empirical Potentials
0.1-1 µ 1-10 µ 0.1-1 mm
Coarse Graining Constitutive Equations Finite Elements
International School on NanoAlloys (ISNA) Tirrenia, May 21th – 25th 2012
Length rescaling approach: renormalization
renormalization renormalization
5Å 50 Å 20 nm
Quantum Mechanics Empirical Potentials Effective Models
overlap regions between two length scales in which a self-consistent cross-check is
possible of the link between the various length scales thus allowing a (partial) validation of
the renormalization process
International School on NanoAlloys (ISNA) Tirrenia, May 21th – 25th 2012
Density-Functional Basin-Hopping (DF-BH)
Let’s consider a metal cluster composed by N atoms.
How can I single out the lowest-energy structure?
The algorithm we are going to use is the DF-BH:
Density-Functional Basin Hopping
i.e. a basin hopping in which energies and forces are evaluated
using DFT
DF-BH
STARTING BEST !
RANDOM
Tirrenia, May 21th – 25th 2012
International School on NanoAlloys (ISNA)
Density-Functional Basin-Hopping (DF-BH)
The algorithm:
1) transform the potential energy surface (PES) into a multidimensional
staircase function;
2) make a Metropolis Monte Carlo walk on the staircase PES – in the
field of free Empirical Potential (EP) clusters BH pioneered by Doye
and Wales
energy
Accept the move if Metropolis
criterion is satisfied:
space exp(-E/kBT) > random number
International School on NanoAlloys (ISNA) random number [0,1]
kBT is the MC parameter
Tirrenia, May 21th – 25th 2012
Density-Functional Basin-Hopping (DF-BH)
Application to a pure system:
AgN (N=1 – 8)
1.in the gas-phase
2.supported on Fs-defected MgO(100)
Chem. Eur. J. (2007) : 1st DF-BH application to supported metal clusters
International School on NanoAlloys (ISNA) Tirrenia, May 21th – 25th 2012
Density-Functional Basin-Hopping (DF-BH)
regular MgO(100) terrace Fs-defected MgO(100) terrace
Fs-defected MgO(100) step Fs-defected MgO(100) corner
International School on NanoAlloys (ISNA) Tirrenia, May 21th – 25th 2012
Density-Functional Basin-Hopping (DF-BH)
Ag2 & Ag3
+0.00 +0.03 +0.05
Ag4
+0.00 +0.01 +0.03
International School on NanoAlloys (ISNA) Tirrenia, May 21th – 25th 2012
Density-Functional Basin-Hopping (DF-BH)
Ag6
+0.00 +0.20 +0.64
+0.09 +0.22 +0.08 +0.03 +0.05 +0.00
International School on NanoAlloys (ISNA) Tirrenia, May 21th – 25th 2012
Density-Functional Basin-Hopping (DF-BH)
Ag8
+0.00 +0.14 +0.16
+0.00 +0.03 +0.04 +0.29 +0.03
International School on NanoAlloys (ISNA) Tirrenia, May 21th – 25th 2012
Density-Functional Basin-Hopping (DF-BH)
Ag10
+0.00 +0.35 +0.12 +0.12
+0.26 +0.11 +0.98 +0.50 +0.29 +0.00
International School on NanoAlloys (ISNA) Tirrenia, May 21th – 25th 2012
Density-Functional Basin-Hopping (DF-BH)
Structural transitions: Ag6 Ag8 Ag10
Ag6
AgN gas-phase:
• planar N 6 Ag10
• five-fold compact N 8
AgN absorbed: Ag4
• planar N 4
• five-fold compact N = 6,8
• (distorted) fcc N = 10
Ag8
International School on NanoAlloys (ISNA) Tirrenia, May 21th – 25th 2012
Density-Functional Basin-Hopping (DF-BH) Ag10
0.1-0.3
fluxional character
E Ag4 Ag6 Ag8
gas-phase - 0.2-0.6 0.1-0.2
absorbed 0.01-0.03 0.03-0.20 0.03-0.30 0.11-0.98
International School on NanoAlloys (ISNA) Tirrenia, May 21th – 25th 2012
Density-Functional Basin-Hopping (DF-BH)
magic vs non-magic character
gas-phase
absorbed
International School on NanoAlloys (ISNA) Tirrenia, May 21th – 25th 2012
Density-Functional Basin-Hopping (DF-BH)
AgN supported on the defected surface:
1. enhanced fluxionality (whence low melting point ?)
2. transition from compact 5-fold symmetric to compact fcc
(distorted) structures at N = 10 because of bond strain
Excess energy (eV)
gas-phase
absorbed
N=6 N=8 N = 10 adsorbed Ag8 NON-magic
Tirrenia, May 21th – 25th 2012
compact 5-fold compact 5-fold compact fcc
structure structure structure
International School on NanoAlloys (ISNA)
Density-Functional Basin-Hopping (DF-BH)
adhesion of metal clusters on Fs-defect
fluxional character, no magic clusters
how to obviate to this problem?
binary clusters
International School on NanoAlloys (ISNA) Tirrenia, May 21th – 25th 2012
Density-Functional Basin-Hopping (DF-BH)
Application to a nanoalloy system:
PdAgN (N=1 – 8)
1.in the gas-phase
2.supported on Fs-defected MgO(100)
J. Chem. Phys. C (2007) : 1st DF-BH application to metal nanoalloy [free &
supported]
International School on NanoAlloys (ISNA) Tirrenia, May 21th – 25th 2012
Density-Functional Basin-Hopping (DF-BH) PdAgN gas-phase
Gap = 0.87 eV
+0.15 eV
Gap = 1.1 eV
Gap = 1.51 eV Gap = 0.97 eV
2
if N even, Pd retains ~ 4d10 5s0 S = 0
Tirrenia, May 21th – 25th 2012
Pd – Ag distances elongated
International School on NanoAlloys (ISNA)
Density-Functional Basin-Hopping (DF-BH)
PdAgN absorbed
International School on NanoAlloys (ISNA) Tirrenia, May 21th – 25th 2012
Density-Functional Basin-Hopping (DF-BH) PdAgN absorbed
E(N)-E(N-1) 2
highest formation energy lowest formation energy
for PdAg6 cluster for the PdAg7 cluster
International School on NanoAlloys (ISNA) Tirrenia, May 21th – 25th 2012
Density-Functional Basin-Hopping (DF-BH)
M8 = 8 electrons = electronic shell closure (jellium model)
1. Pure Ag clusters Ag8 not! magic
2. binary Pd-Ag clusters Pd1Ag6 magic! (2e-+Pdd10)
highest incremental
formation energy
for the N cluster
low incremental
formation energy
for the N +1 cluster
the first example of a binary magic
metal cluster absorbed on an oxide
surface
International School on NanoAlloys (ISNA) Tirrenia, May 21th – 25th 2012
Density-Functional Basin-Hopping (DF-BH)
Electronic shell closure
gas-phase
6 Ag atoms + 1 Pd atom
Each Ag is d10s1
The Pd is d10s0
Total number of valence electrons: 6
6 is a magic number of planar jellium model
Fs-supported
6 Ag atoms + 1 Pd atom + Fs-vacancy
Each Ag is d10s1
The Pd is d10s0
Fs vacancy has 2 electrons
Total number of valence electrons: 8
8 is a magic number of spherical jellium model
International School on NanoAlloys (ISNA) Tirrenia, May 21th – 25th 2012
Quantum Mechanics H^ = E
building of analytic expressions
global optimization Empirical Potentials E = Σi E atom self-consistent
with structural refinement of
recognition i
empirical
medium-size systems: potentials
DF-EP algorithm
EE
pIh Ih Dh
fcc fcc pIh
Ih Dh
PCCP (2008) cross-check of energy ordering via quantum
mechanics
Multi-scale approach: renormalization
renormalization renormalization
5Å 50 Å 20 nm
Quantum Mechanics Empirical Potentials Effective Models
overlap regions between two length scales in which a self-consistent cross-check is
possible of the link between the various length scales thus allowing a (partial) validation of
the renormalization process
International School on NanoAlloys (ISNA) Tirrenia, May 21th – 25th 2012
EP-GO and extrapolation
“large” systems: EP-GO algorithm
crucial choice of moves,
structure recognition and
other (“smart”) tricks
“very large” systems: extrapolations based
on structural motifs
cross-over among
structural families
(or motifs)
International School on NanoAlloys (ISNA) Tirrenia, May 21th – 25th 2012
Multi-scale approach 10-50 Å 5-20 nm
Classical Mechanics Effective Models
1-20 Å Empirical Potentials
Quantum Mechanics
0.1-1 µ 1-10 µ 0.1-1 mm
Coarse Graining Constitutive Equations Finite Elements
International School on NanoAlloys (ISNA) Tirrenia, May 21th – 25th 2012
Effective Hamiltonians
For large clusters (N 1000-2000 atoms) one can use Effective
Hamiltonians, e.g. Ising-type models, to evaluate the energy of the
system:
E H ms m 1 Vmns ms n
2
m m
where the energy is a sum over all sites m (s is the occupation, 0 or 1)
and Vmn are the bond contributions between first-neighbors [NB: you
do NOT relax the structure]
In the simplest example, the total energy of the cluster is a sum of
atomic site energies (depending only on coordination number)
especially if the structural motif is known
The parameters are usually obtained by interpolating surface energies
of extended systems:
International School on NanoAlloys (ISNA) Tirrenia, May 21th – 25th 2012
Effective Hamiltonians and surface energies
Ag is linear
Pt is highly NON-linear
(111) (100)
International School on NanoAlloys (ISNA) Tirrenia, May 21th – 25th 2012
Specific issue of nanoalloys: chemical ordering
Let’s consider a bimetallic cluster composed by (NA , NB) atoms
NA
Ntot
NB
Two structures characterized by the same structure but a different
chemical ordering are called HOMOTOPS. It can be easily demonstrated
that the number of homotops is given by the following relation:
number of = Ntot = Ntot! ≈ 2Ntot It’s a huge number!
homotops
NA NA! NB!
International School on NanoAlloys (ISNA) Tirrenia, May 21th – 25th 2012
Effective Hamiltonians and chemical ordering
Let us suppose that the structural problem is solved and the
structural motifs are known.
If the lattice mismatch between the two species is not large, so that
chemical rearrangement does not produce structural deformations,
one can use Effective Hamiltonians, e.g. Ising-type models, to predict
the chemical ordering
E Hms m 1 Vmns ms n
2
m m
where the energy is a sum over all sites m (s is the occupation, 0 or 1)
and Vmn are the bond contributions between first-neighbors, which can
be distinguished according to each binary pair
International School on NanoAlloys (ISNA) Tirrenia, May 21th – 25th 2012
The Orbit Approach to Chemical Ordering
There exist specific chemical compositions which realize magic clusters,
i.e. clusters with structural shell-closure
Icosahedron (Ih) Decahedron (Dh) Truncated octahedron (Oh)
(from Ferrando’s presentation)
In these structures, the atoms can be distinguished in a smaller number
of symmetry-equivalent groups (called shells or orbits). Some examples:
International School on NanoAlloys (ISNA) Tirrenia, May 21th – 25th 2012
The Orbit Approach to Chemical Ordering
6
3 4
5
2
1
79-atom TO: 6 orbits 116-atom TO: 7 orbits 201-atom TO: 12 orbits
309-atom Ih: 98-atom Leary-Td:
11 orbits 9 orbits
International School on NanoAlloys (ISNA) Tirrenia, May 21th – 25th 2012
The Orbit Approach to Chemical Ordering
In the “hands-on” session
We will work on this
system considering
Pd and Pt
and using Empirical
Potential (Gupta type)
0 1 1 0 0 1 001 110
201-atom TO orbits 12 11 10 9 8 7 ...
12 orbits
International School on NanoAlloys (ISNA) Tirrenia, May 21th – 25th 2012
The Orbit Approach to Chemical Ordering
Pd-rich side: At 25 % of Pt, we observe a strong tendency towards
size 201 segregation on (111) facets only
+0.00 eV +3.82 eV +4.13 eV +5.30 eV
crown (111) facets subsurface + subsurface + subsurface +
internal (1) internal (2) internal (3)
Pt-rich side: At 75 % of Pt, we observe a tendency towards segregation of
size 201 Pd on the surface in correspondence of low-coordinated sites
International School on NanoAlloys (ISNA) Tirrenia, May 21th – 25th 2012
The Orbit Approach to Chemical Ordering
mid- At 50 % composition, a joint tendency towards maximization of hetero-bonds
compositions: and the tendency of Pt towards (111) surface segregation determines the
size 201 formation of a “patchy multishell” chemical ordering
+0.00 eV +0.73 eV +1.13 eV +1.34 eV +1.65 eV +2.11eV
patchy almost patchy
All the low-lying homotops are characterized by surface segregation of Pt in the “crown” sites
International School on NanoAlloys (ISNA) Tirrenia, May 21th – 25th 2012
The Orbit Approach to Chemical Ordering
mid- In the patchy structure the chemical ordering is dictated by the
compositions: ordering of the external shell, where Pt segregates at the (111)
size 201 faces and Pd on the other sites.
In the shell beneath the ordering is opposite! and so on...
Pd102Pt99
International School on NanoAlloys (ISNA) NanoLetters 11, 1766 (2011)
Tirrenia, May 21th – 25th 2012
Dynamic phenomena
the role of theory and computation
dynamic phenomena: diffusion and growth
STEP 1: DF determination of the low-energy structures for the
clusters for various MN through GO
STEP 2: at each N, determination of the transitions between local
minima giving rise to diffusion over the surface
STEP 3: calculation of the diffusion barriers by applying the NEB
(Nudged Elastic Band) method within DF
STEP 4: Monte Carlo simulation of the cluster growth
International School on NanoAlloys (ISNA) Tirrenia, May 21th – 25th 2012
Dynamic phenomena: diffusion Ag
&
Pd Au
Trimer (tetramer) walking
Tetramer rocking/rolling along [110]
IntTerrnimateiornawl Saclhkoinogl oannNdantoeAtlrloayms (eISrNrAo) lling Tirrenia, May 21th – 25th 2012
Dynamic phenomena: diffusion
results of DF-NEB diffusion energy barriers
(values in eV) Pd Au Ag
Monomer 0.39 0.22 0.10
Dimer 0.39 0.62 0.22
Trimer 0.30 0.19 0.12
Tetramer 0.38 0.42 0.21
DFT ~ 0.38 ~ 0.20 ~ 0.10
interesting to note in passing: M2 and M4 for Ag and Au
diffuse more slowly than M1 and M3
International School on NanoAlloys (ISNA) Tirrenia, May 21th – 25th 2012
Dynamic phenomena: diffusion and growth
comparison with growth experiments
• study of nucleation on flat terraces island density as f(T)
extraction of effective activation energies Ag 0.08 eV
Au 0.12 eV
(C. Henry) Pd 0.16-22 eV
NB: in reasonable agreement with theoretical estimates of diffusion coefficients
International School on NanoAlloys (ISNA) Tirrenia, May 21th – 25th 2012
Dynamic phenomena: diffusion and growth
• deposition of Pd atoms on a highly defected
MgO(100) surface Haas et al. PRB (2000)
experimental evidence: constant density of Pd
metal islands over a broad range of
temperatures (200 – 600 K)
interpretation: Pd small clusters diffuse on the
surface, reach the defects, are trapped there
where islands nucleate and grow
results of Monte Carlo simulation of island
nucleation where:
only monomers diffuse
small clusters up to the tetramer diffuse
with 3 min postdeposition
International School on NanoAlloys (ISNA) Tirrenia, May 21th – 25th 2012
Dynamic phenomena: diffusion and growth
conclusions
small coinage metal clusters are highly mobile on
the regular MgO (100) surface: they can move
fast on terraces till they reach defect sites
where they get trapped
on defect sites the growth of larger metal
aggregates takes place
PRL (2005) + review: R. Ferrando and A. Fortunelli, J.
Phys.: Condens Matt., 21, 264001 (2009) + review chapter:
Chemical Sensors – Simulation and Modeling (request)
International School on NanoAlloys (ISNA) Tirrenia, May 21th – 25th 2012