Wafers, platelets, rods and spheres: Using DFTB to determine the structural

minima of atomic clusters

Koblar Jackson

Physics Department

Central Michigan University

The structure problem

Bulk Si: diamond structure

Bulk fragment

Cluster

- Bulk fragments not favorable for clusters due to surface dangling bonds

- What happens when chemical intuition doesn’t work???

The Jackson Living Room

Piano

Sofa

Chair

TableB

ooks

Pian

o

Sofa

Chair

Table

Boo

ks

The Jackson Living Room

Piano

Sofa Chair

Table Boo

ks

The Jackson Living Room

“NP hard problem”: the number of local minima grows exponentially with cluster size

Ene

rgy

R

Energy vs structure (R)

Probing the energy surface

Gradient optimization – following forces

Global minimum Local

minimum

Random starting structure

Local minimum

Random starting structure

Local minimum

-Optimal volume compression ~ (1/5)3

-Ground states found up to N = 105-As efficient as genetic algorithm up to N = 40

Efficiency vs Box size: Lennard-Jones clusters

Role of DFTB in Search Process

• Need quantum description: good vs bad bonds; electron kinetic energy

• DFT (PBE-GGA) accurate, but computationally demanding

• DFTB mimics DFT, but is 102 – 103 times faster• Use DFTB (Frauenheim et al.) to probe energy

landscape

H[]iii

Ordering minima: Si24

0.00 (1)0.00 (1)

0.06 (2) 0.42 (3) 0.46 (4) 0.46 (5)

0.47 (6) 0.50 (7) 0.57 (8) 0.58 (9) 0.59 (10)

0.31 (3) 1.00 (5) 2.21 (9) 2.31 (10)

1.09 (6) 1.78 (8) 0.64 (4) 1.25 (7) 0.30 (2)

E in eV (DFTB rank)E in eV (DFT rank)

DFT vs DFTB energy surfaces

A

DFT

DFTB

DFT1

-DFT1 energy ordering improves DFTB

E

Q

Si25 100 local minima

DFT Relaxed vs

DFTB

DFT Relaxed vs

DFT1

Big Bang Search Methodology: Parallel method for finding global minima

- Done in parallel~1 x 10^6 local minima~2 x 10^3 stored

Compressed Geometries & DFTB

relaxation

-Exact DFT ordering~30 lowest structures

Full DFT optimization

-Approximate DFT ordering-lowest ~300 structures

Reorder using DFT1

Jackson et al., Comp. Mat. Sci. 35, 232 (2006)

SiN Shape Transition: ExperimentHudgins et al, Journal of Chemical Physics 111, 7864

(1999)

# of

clu

ster

s

Drift Time (ms)

# of

clu

ster

s

Drift Time (ms)

Abrupt change in cluster shape across 24-28

Sample

LaserDrift Tube

Si21+

+0.08 eV

+0.26 eV

+0.37 eV

+0.45 eV

0.00 eV

+0.39 eV

Rich structural variety: unbiased search

Cs 3.666

20

21

22

23

24

25

26

27

C1 3.557

Cs 3.551

C2v 3.583C2v 3.565

Cs 3.616

C1 3.600

C1 3.635

C1 3.627

Cs 3.652

C1 3.649

Cs 3.666

C2v 3.691

Cs 3.687

C1 3.697

C1 3.691

Best prolate vs best compact structure: shape evolution of SiN

+ global minima

Stability crossover at n=25: shape transition driven by thermodynamics!

Jackson et al., Phys. Rev. Lett. 93, 013401 (2004)

Lowest-energy isomers reproduce data across transition region

Number of Atoms, n

20 21 22 23 24 25 26 27

Inve

rse

Mob

ilit

y, V

s/m

2

2200

2300

2400

2500

2600

2700

2800

Expt minor

Expt major

Th local min

Th ground state

Compact

Prolate

Stretched

Predicted vs observed ion mobilities

Hudgins et al., J. Chem. Phys. 111, 7865 (1999)

Jackson et al., Phys. Rev. Lett. 93, 013401 (2004)

( E(M+) + E(N-M) ) – E(N+)

Theory

Expt: Jarrold and Honea, J. Phys. Chem. 95, 9181 (1991)

Fragments

Local

Global

ED

N+

M+

N-M

Expt

Dissociation Energy

Minimum-energy structures reproduce dissociation E data

Stretched:

3 subunits

Prolate:

2 subunits

Compact:

1 subunit

n=22

Structural Families

Recent DFTB-based work (X. C. Zeng) extending Si structure searches to larger

sizes:

1. Bai J, Cui LF, Wang JL, et al., Structural evolution of anionic silicon clusters Sin (20 <= n <= 45) J. Phys. Chem. A 110 (3): 908-912 JAN 26 2006

2. Yoo S, et al., Structures and relative stability of medium-sized silicon clusters. V. Low-lying endohedral fullerene-like clusters, Si31 – Si40 and Si45. J. Chem. Phys. 124 124 (16): 164311 APR 28 2006

Cu, Ag clusters Empirical/semi-empirical predictions:

icosahedral growth pattern

Tight-Binding Molecular Dynamics Search Kabir et al. Phy. Rev. A 69:43203(2004)

Cu Clusters (N = 10 – 15): DFT Predictions (limited sampling)

Guvelioglu et al. Phys. Rev. Lett. 94:26103(2005)

10 11 11 12 12 13

Fernandez et al. Phys. Rev. B 70:165403(2004)

DFT: no icosahedral ordering; but no agreement on minima

9A(0.00) 9B(0.02) 9C(0.02) 10A(0.00) 10B(0.09) 10C(0.20)

11A(0.00) 11B(0.07) 11C(0.08) 12A(0.00) 12B(0.08) 12C(0.13)

13A(0.00) 13B(0.01) 13C(0.07) 14A(0.00) 14B(0.13) 14C(0.14)

AgN N = 9 - 14

M. Yang, K. Jackson, J. Jellinek, J. Chem. Phys. (to appear)

Ground-state structures of Cu clusters N = 10 – 16: “platelets”

Top view

Side view10 11 12 13 14 15 16

M. Yang, K. Jackson, C. Koehler, Th. Frauenheim, and J. Jellinek, J. Chem. Phys. 124, 024308 (2006)

Ground-state structures of Cu clusters N = 17 – 20 : “spheres”

17 18 19 20

Icosahedral core

M. Yang, K. Jackson, C. Koehler, Th. Frauenheim, and J. Jellinek, J. Chem. Phys. 124, 024308 (2006)

-IP can distinguish isomers

-Lowest-energy structures generally in best agreement

5.0

5.4

5.8

6.2

6.6

7.0

7 9 11 13 15 17 19 21

Cluster size

IP (

eV)

exptIsomer 1Isomer 2Isomer 3

IP (

eV

)

Cluster size

CuN: Calculated and measured vertical ionization potentials

M. Knickelbein, CPL 192,129(1992)

Shape evolution and shell filling: AgN vs “ultimate jellium”

0.4

0.6

0.8

1

1.2

1.4

8 10 12 14 16 18 20N

<I>

AgN

JN

<Ii> = 3*Ii/(I1+ I2+ I3)

Sphere: I1= I2=I3 = 1

Prolate: I1,I2 > 1

Oblate: I1,I2 < 1

JN : M. Koskinen et al.,

Z. Phys. D:At., Mol. Clusters

35, 285 (1995).

M. Yang, K. Jackson, J. Jellinek, J. Chem. Phys. (to appear)

Summary

• DFTB plays essential role in structural search algorithm: scan energy surface for likely structures

• Search methodology yields structures consistent with known expt data

• Clusters display an array of shapes at small sizes: wafers, platelets, rods, and spheres

Thanks to:

• J. Barra, J. Boike, J. Juen, I. Rata, A. Balakrishnan (students)

• M. Yang, M. Horoi (CMU)

• Frauenheim, Seifert, Koehler, Hajnal (DFTB friends)

• A. Shvartsburg (PNNL)

• J. Jellinek (ANL)

Support

• This work is supported by the Office of Basic Energy Sciences, Division of Chemical Sciences, Geosciences, and Biosciences, U. S. Department of Energy, under contract DE-FG02-03ER15489

0.5

1.0

1.5

2.0

2.5

3.0

0 2 4 6 8 10 12 14 16 18 20

Cluster size

Co

he

siv

e E

ne

rgy

(e

V/a

tom

) Wafers Platelets Spheres

N = 5

N = 12N = 19

N = 16

Shape fluctuations in Cu clusters

Cohesive energy of layered and compact Cu clusters

2.0

2.1

2.2

2.3

2.4

10 12 14 16 18 20

Coh

esiv

e en

ergy

(e

V)

Cluster size

layered

compact

M. Yang, K. Jackson, C. Koehler, Th. Frauenheim, and J. Jellinek, J. Chem. Phys. 124, 024308 (2006)

Calculated and measured vertical detachment energies of Cu anions

Cha et al. J. Chem. Phys. 99:6308(1993)

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

0 2 4 6 8 10 12 14 16 18 20

Cluster size

VD

E /e

V

layeredcompactmeasuredI

VD

E (

eV)

Cluster size

AgN vs CuN

2.01

2.03

2.05

2.07

2.09

1.351.361.371.381.391.4

Ecoh of Ag10 (eV)

Eco

h

of

Cu

10

(eV

)

Thirty lowest-energy isomers of Cu10 vs. corresponding isomers of Ag10

Excellent correlation: structures found in CuN search can be used as candidate structures for AgN

M. Yang, K. Jackson, J. Jellinek, J. Chem. Phys. (to appear)

17A(0.00) 17B(0.24) 17C(0.26) 18A(0.00) 18B(0.06) 18C(0.13)

19A(0.00) 19B(0.11) 19C(0.18) 20A(0.00) 20B(0.05) 20C(0.24)

AgN N = 9 – 20 (cont’d)

15A(0.00) 15B(0.03) 15C(0.06) 16A(0.00) 16B(0.09) 16C(0.14)

M. Yang, K. Jackson, J. Jellinek, J. Chem. Phys. (to appear)

AgN HOMO-LUMO Gaps

0.0

0.4

0.8

1.2

1.6

8 10 12 14 16 18 20

N

En

erg

y g

aps

(eV

)

PES

TH (neutral)

TH (anion)

+

M. Yang, K. Jackson, J. Jellinek, J. Chem. Phys. (to appear)

AgN: Impact of shape on properties

1.2

1.3

1.4

1.5

1.6

1.7

-1.0

-0.5

0.0

0.5

8 10 12 14 16 18 20

E(2) (eV)

ECoh (eV)

E(2

) = 2

E(N

) –

E(N

+1)

– E

(N)

Ecoh = [NE(1) – E(N)]/N

M. Yang, K. Jackson, J. Jellinek, J. Chem. Phys. (to appear)

Shape vs dipole polarizability

6.0

7.0

8.0

9.0

0 2 4 6 8 10 12 14 16 18 20

N

Po

lari

zab

ilit

ies

(A3 /

N)

Planar to layered

Layered to compact

M. Yang, K. Jackson, J. Jellinek, J. Chem. Phys. (to appear)

12A11 13

14 1716A

CuN- PES: expt vs theory