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