1. Quantum Monte Carlo in the Apuan Alps V
Vallico Sotto, Tuscany, July 25~August 1, 2009
QMC studies of (i) transition metal clusters and (ii) surface adsorption
Ching-Ming Wei
Institute of Atomic & Molecular Sciences,
Academia Sinica, TAIWAN
In collaboration with: Cheng-Rong Hsing, Jyh-Pin Chou, Hsin-Yi Chen
Chun-Ming Chang (NDHU), Cheng Ching (NCKU)
Neil Drummond, Richard Needs (Cambridge)

2. Institute of Atomic & Molecular Sciences, Academia Sinica, Taiwan
Outline
Systems Studied by plane wave based DFT
Surface states of Be(1010), Mg(1010) and Al(111)
QSE of metallic thin films Pb(111), Al(111), Pb(100)
New structural model of Nan / Si(111) 7x7 surface
QSE and Mackay Transition of Pb clusters (up to Pb923)
Failure of DFT  Quantum Monte Carlo needed ……
Quantum Monte Carlo in real material systems
(i) B, C, Al clusters; (ii) Li, Be, Mg, Na clusters (iii) Cu and Au clusters  DFT is a good tool for the study of metallic clusters
Single atom adsorbed on graphite
CO/Pt(111) adsorption puzzle?
3. Summary and Conclusion
Ab-Initio Simulation Lab.
Institute of Atomic & Molecular Sciences, Academia Sinica, Taiwan

3. Surface states of Be(1010 ) Phys Rev. B 2008, S.-J. Tang
What can we
learn from DFT?

9. Quantum Size Effect (QSE) in metal thin films
Pb/Si(111):
Flat-top islands of selective heights
Heights of abundant islands differ by bilayer increments
The in-plane lattice constant of Pb(111) is 9% smaller than that of the Si(111) substrate.
W. B. Su et al. PRL 86, 5116 (2001)
STM image (300 nm  300 nm), taken with the sample bias of 2 V, of Pb
islands grown on Si(111) substrate at 200 K and coverage of 3.2 ML.

10. Quantum Size Effect (QSE) in metal thin films
STM results by Hanno H. Weitering’s group (2004)
Why such flat films can grow?
Why a bi-layer growth mode?
What is the role of substrate?
Si(111)(√3×√3)R30°-Pb(α) (a ~ f)

11. Quantum Size Effects in Metal Thin Films
(b)
Fermi level cuts the p band, determine kF with reference to L, kF ~ 0.45 p/d, DD = lF/2 = 2.2 d (d is the interlayer distance).
A clear bi-layer oscillation feature of the system properties will be seen due to the Fermi wave vector along (111) direction; e.g. surface energy (a) and work function (b) of Pb(111) films as a function of the film thickness.

13. Surface Relaxations of Pb(111) Metal Thin Films
First three interlayer spacings (d12, d23, d34) shifted from the bulk positions of Pb (111) thin films as a function of film thickness (N). A clear bi-layer feature (N < 12) is seen due to Fermi wave vector along the (111) direction.

14. S. Hasagewa et al. (2008)

15. It is possible to study how the dangling bond
orbitals changes after Na atoms adsorption
in a very short time (a few days)!

16. Mackay Structural Transition of Metallic Nanoparticles:
possible shell structures of nano particles
Decahedral:
10 (111) faces +
5 (100) faces
Icosahedral:
20 (111) faces
Cubotohedral:
8 (111) faces +
6 (100) faces
No. of particles for icosahedral, decahedral & cubotohedral
N= 10/3 n3+ 5 n2 +11/3 n+1
N= (n=1) ; (n=2)
147 (n=3) ; (n=4)
561 (n=5) ; 923 (n=6)
…………
V & S of 3 structures
is basically the same !
Stability &
structural transition ?

17. Ab-Initio Simulation Lab.
Institute of Atomic & Molecular Sciences, Academia Sinica, Taiwan

18. Structure Phase Transition of Icosahedral  Cubotohedral
Mackay transition Acta Cryst. 15, p916 (1962)
ico if
fcc if s= 0
Ab-Initio Simulation Lab.
Institute of Atomic & Molecular Sciences, Academia Sinica, Taiwan

19. Barrier heights (~10 meV) of ICO  FCC transition of Pb clusters oscillate with the shell index (or radius of cluster) indicates the possible Quantum Size Effect of the melting points ?
Ab-Initio Simulation Lab.
Institute of Atomic & Molecular Sciences, Academia Sinica, Taiwan

20. Density Functional Theory is great, and why do we need to bother by
Quantum Monte Carlo ?
For Al55 cluster :
QMC need 64CPU run 2days,
but DFT only takes ~ 1 cpu hour !
A factor of 3000 or even more!

21. Which Au38 is a more stable structure?
Efcc= eV (LDA) EO_h= eV
Efcc= eV (PBE) EO_h= eV
Efcc= eV (PW91) EO_h= eV
DErel > 2.5 eV… need more accurate methods? QMC
Ab-Initio Simulation Lab.
Institute of Atomic & Molecular Sciences, Academia Sinica, Taiwan

22. Motivation?
Metal atom adsorbed on Graphite
Will DFT with different ExCs give a consistent and correct adsorption energy?
Ab-Initio Simulation Lab.
Institute of Atomic & Molecular Sciences, Academia Sinica, Taiwan

23. DFT has little predictive power! Need QMC!!!
Ab-Initio Simulation Lab.
Institute of Atomic & Molecular Sciences, Academia Sinica, Taiwan

24. trans-stilbene/Ag-Ge(111) cis-stilbene/Ag-Ge(111)
Eads= eV (LDA)
Eads= eV (LDA)
LDA agrees expt., but… Eads ~ 0.40 & 0.20 eV (PW91)
and again DFT with very little predicting power!!!
Ab-Initio Simulation Lab.
Institute of Atomic & Molecular Sciences, Academia Sinica, Taiwan

25. QMC studies of metallic clusters
B, C, Al clusters
Li, Be, Mg, Na clusters
Cu and Au clusters
Let us start with DFT……

26. Methodology (44 metallic elements) n : normal (without semi-core)
s : s-type semi-core state
p : p-type semi-core state
d : d-type semi-core state
Three kinds of PPs
Two kinds of PPs
Group 1
Group 2
Group 13 3
bcc 4
hcp
Only one kind of PP 5
hcp 6 Li Be B C
(n, ,s)
(n, ,s) 11
bcc 12
hcp 13
hcp 14 Na Mg Al Si
Group 3
Group 4
Group 5
Group 6
Group 7
Group 8
Group 9
Group 10
Group 11
Group 12
(n,p,s)
(n,p, )
(n, , ) 19
bcc 20
fcc 21
hcp 22
hcp 23
bcc 24
bcc 25
cubic complex 26
bcc 27
hcp 28
fcc 29
fcc 30
hcp 31
complex 32
diamond K Ca Sc Ti V Cr Mn Fe Co Ni Cu Zn Ga Ge
( ,p,s)
( , ,s)
( , ,s)
(n,p,s)
(n,p,s)
(n,p, )
(n,p, )
(n,p, )
(n, , )
(n,p, )
(n,p, )
(n, , ,d)
(n, , ,d) 37
bcc 38
fcc 39
hcp 40
hcp 41
bcc 42
bcc 43
hcp 44
hcp 45
fcc 46
fcc 47
fcc 48
hcp 49
tetr 50
diamond Rb Sr Y Zr Nb Mo Tc Ru Rh Pd Ag Cd In Sn
( ,p,s)
( , ,s)
( , ,s)
( , ,s)
( ,p,s)
(n,p, )
(n,p, )
(n,p, )
(n,p, )
(n,p, )
(n, , )
(n, , ,d)
(n, , ,d) 55
bcc 56
bcc 57
hex 72
hcp 73
bcc 74
bcc 75
hcp 76
hcp 77
fcc 78
fcc 79
fcc 80 81
hcp 82
hcp Cs Ba La Hf Ta W Re Os Ir Pt Au Hg Tl Pb
( , ,s)
( , ,s)
(n, , )
(n,p, )
(n,p, )
(n,p, )
(n,p, )
(n,p, )
(n, , )
(n, , )
(n, , )
(n, , )
(n, , ,d)

27. Methodology ICO FCC DEC BCC HCP BBP TBP GCL CAG
(nine isomer structure)
Five high-symmetry three-dimensional structures and
four low-symmetry layer-type structures.
ICO
FCC
DEC
BCC
HCP
BBP
TBP
GCL
CAG
Top view
Top view
Top view
Side view
Side view
Side view
hexagonal array (7) + central square (4) + side atoms (2)
hexagonal array (7) + triangle (6)
triangle (3) + atoms (7) + triangle (3)
one atom (1) + double square (6) + double square (6)

28. Methodology A n-dimensional displacement vector Here, n=9
XC functional
A n-dimensional displacement vector
M : metal element
(a) : semi-core state
Here, n=9
The relative energy definition :
: the total energy of a certain isomer
: the average energy of all the isomers

29. Results and Discussion
Group A (XC dependence)
(XC dependence)
Hereafter, (pv) means PP with p-type semi-core state
(sv) means PP with s-type semi-core state
(d) means PP with d-type semi-core state

30. Results and Discussion
Group A (PP dependence)
(PP dependence)
The consistency between GGAs is better than that between LDA and GGA.
Despite a small deviation in the relative energy for one or two isomers in some elements, the calculated results from LDA and GGA are consistent and reliable.
PPs without semi-core lead to reliable results.

31. Results and Discussion
Group B (XC dependence)
(XC dependence)
The consistency between GGAs is better than that between LDA and GGA

32. Results and Discussion
Group B (XC dependence)
(XC dependence)

33. Results and Discussion
Group B (XC dependence)
(XC dependence)
The consistency of the three XC functionals in 4d and 5d are better than that in 3d.
For Cr, Mn, Fe, Co, Ni, and Rh clusters, the usages of LDA and GGA produce different results.

34. Results and Discussion
Group B (PP dependence)
(PP dependence)
The usage of deeper semi-core states in the PPs is inevitable for V, Cr, and Mo.

35. CASINO code : QMC Methods
R.J. Needs, M.D. Towler, N.D. Drummond and P. López Ríos, CASINO version 2.1 User Manual, University of Cambridge, Cambridge (2007).

36. DFT & QMC calculation of – John and Needs pseudopotentials
C20, B18, B20, Al13, and Al55 clusters
– John and Needs pseudopotentials

37. Clusters
C20
B18
B20
DFT is much better at predicting realtive energy ordering of the metallic clusters than the covalent ones.
Different XC functionals predicts relative stabilities of the metallic clusters consistently, but this is not the case for the covalent clusters.
Al13
Al55
Cluseter θ L
LDA
PBE
C20
58.49
22.59
1.189
0.312
B18
72.73
21.11
0.592
0.596
B20
57.64
16.01
0.639
0.691
Al13
3.21
0.44
0.844
0.845
Al55
1.51
0.722
0.788

38. CASTEP & CASINO calculation of
Li, Na, K, Be, and Mg clusters – comparison of optimal and TM pseudopotentials (opium code)

39. OPT Pseudopotentials angle θ L Li13 0.67 0.99 Na13 3.32 1.11 K13 4.28
1.06

40. angle θ L
Be13
4.29
1.06
Mg13
3.21
1.02

41. Li13
3E-TM-1.00A
1E-opt-1.75A and 3E(VASP)

42. Li13 3E-TM-1.00A 1E-opt-1.75A DLDA-1E DVMC DVASP degree 4.40 27.76
4.25 L
1.01
1.51
1E-opt-1.75A
DLDA-1E
DVMC
DVASP
degree
5.33
5.83
7.59 L
1.04
1.43
0.83

43. Be13
TM-1.8A
opt-1.35A

44. Be13 TM-1.8A opt-1.35A DLDA DVMC DVASP degree 13.70 10.80 11.57 L 1.02
1.26
0.97
opt-1.35A
DLDA
DVMC
DVASP
degree
13.44
9.47
10.78 L
1.02
1.23
0.97

45. Mg13
TM-1.8A
opt-1.65A

46. Mg13 TM-1.8A opt-1.65A DLDA DVMC DVASP degree 5.66 11.15 6.76 L 0.95
0.87
0.98
opt-1.65A
DLDA
DVMC
DVASP
degree
5.09
10.35
6.14 L
0.95
0.88
0.98

47. Na13
DLDA
DVMC
DVASP
degree
17.59
53.50
15.23 L
1.12
0.75
1.01

48. DLDA
DVASP
degree
17.59
15.23 L
1.12
1.01

49. – opium pseudopotentials
QMC calculation of
Cu13 and Au13 clusters
– opium pseudopotentials
Cluseter θ L
LDA
PBE
C20
58.49
22.59
1.189
0.312
B18
72.73
21.11
0.592
0.596
B20
57.64
16.01
0.639
0.691
Al13
3.21
0.44
0.844
0.845
Al55
1.51
0.722
0.788
Cu13
16.79
8.32
0.996
0.908
Au13
22.75
6.14
0.663
0.755

51. DFT (PBE) is reliable in predicting
the relative energy of metallic clusters

52. Which Au38 is a more stable structure?
Efcc= eV (LDA) EO_h= eV (X)
Efcc= eV (PBE) EO_h= eV
Efcc= eV (PW91) EO_h= eV
Hollow Au38 is more stable!
Ab-Initio Simulation Lab.
Institute of Atomic & Molecular Sciences, Academia Sinica, Taiwan

53. C10 : 2D PES D5h D10h D(E) = 0.079 eV CASTEP – LDA
= (20) eV DMC (DFT geometry)
6-311+g = eV B3LYP
cc-pVDZ = eV B3LYP
cc-pVTZ = eV B3LYP
cc-pVQZ = eV B3LYP
How about Quantum Chemistry Methods: HF, MP2, MP4, CCSD(T)?

54. High level methods are more important than increasing local basis sets!

55. DFT with XCs are better than HF, MP2 and MP4

56. Single graphene
Metal atom adsorbed on Graphene
Will DFT with different ExCs give a consistent and correct adsorption energy?

57. LDA & PBE predict an over-binding effect
Single graphene B C Si Al
LDA & PBE predict an over-binding effect
(The results using periodic 4x4 unit cell is the same…)

58. Pt(111) surface
Times Cited: 222 (July 1, 2009)

59. All different kinds of DFT results : But Experimental result :
Pt(111) surface
All different kinds of DFT results :
FCC site
But Experimental result :
Atop site
[ref] Blackman, G. S. et al. Phys. Rev. Lett. 1988, 61, 2352

60. Pt(111) surface

61. Diffusion Monte Carlo predicts a correct adsorption site
Pt(111) surface
DMC result : atop site
Ead (fcc) = eV
Ead (atop) = eV
Diffusion Monte Carlo predicts
a correct adsorption site
and
adsorption energy

62. Summary and Conclusion
DFT is a very powerful tool
to study the nanostructure systems
with a few hundred atoms
and/or thousand electrons.
However, it is perhaps about time to do more Quantum Monte-Carlo in real material systems
Ab-Initio Simulation Lab.
Institute of Atomic & Molecular Sciences, Academia Sinica, Taiwan