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