Quantum Monte-Carlo Studies of B, Al, and C clusters

Ching-Ming Wei

Institute of Atomic & Molecular Sciences,Academia Sinica, TAIWAN

In collaboration with: Cheng-Rong Hsing, Hsin-Yi Chen Neil Drummond, Richard Needs

International WorkshopQuantum Monte Carlo in the Apuan Alps III

Saturday 21st - Saturday 28th July 2007The Towler Institute, Vallico Sotto, Tuscany

Outline1. Motivation2. Results

• B18 and B20 (July ~ Oct. 2006)

• Al13 and Al55 (May ~ June 2007)

• C20 (June ~ July 2007)• graphene ribbon (Jan. ~ May 2007)

3. Summary and Conclusion

Quantum Size Effects in Metallic Nanoparticles

C. M. Wei1, C. M. Chang2 and C. Cheng3

1 Institute of Atomic and Molecular Sciences,

Academia Sinica, Taiwan 2 National Dong-Hwa University, Taiwan

3 National Cheng-Kung University, Taiwan

Motivation?

Possible shell structures of nano particles

Icosahedral: 20 (111) faces

Decahedral:10 (111) faces +

5 (100) faces

Cubotohedral: 8 (111) faces +6 (100) faces

Quantum Size Effects in Metallic Nanoparticles:

No. of particles for icosahedral, decahedral & cubotohedral

N= 10/3 n3+ 5 n2 +11/3 n+1N= 13 (n=1) ; 55 (n

=2) 147 (n=3) ; 309 (n

=4)561 (n=5) ; 923 (n

=6) …………

V & S of 3 structures is basically the same !

Stability & structural transition ?

Icosahedron

Cubotohedron

Etot = a V + b S

= a v0 N + b cV2/3

= a v0 N + b’ (v0 N )2/3

= a v0 N + b’ v0 2/3

N2/3

= a’ N + b”

N2/3

Ecoh(N) = Etot / N = a’ + b” N -1/3

Cohesive energy of metallic nanoparticles

0.0 0.1 0.2 0.3 0.4 0.5-3.4

-3.2

-3.0

-2.8

-2.6

-2.4

-2.2

-2.0

Au20

stable

Au13

Au309

Au147

Au55

Au32

stable

eV

Cohesive Energy v.s. N-1/3

N -1/3

CUBO DECA ICOSA

The cohesive energy of Au13 deviate from N-1/3 curve is a sign of QSE!

Hollow Au20 & Au32 is stable because lower than N-1/3 curve!

Johansson et al. Angew.

Chem. Int. Ed. 2004

Li et al., Science 2003

For Lennard-Jones Cluster: Eico < Edeca < Ecubo J. Chem. Soc. Faraday Trans. 87, p215 (1991)

-6

-4

-2

0

2

4

6

FCC

FCCFCC

ICOICO

ICO

Pb923

Pb561Pb

147Pb

309Pb

55Pb

13

eV

Relative stability of ico and fcc Pb clusters

0.0 0.1 0.2 0.3 0.4

-3.6

-3.5

-3.4

-3.3

-3.2

-3.1

1. oscillating stability2. ico has a strong QSE

Pb13

Pb147

Pb561

Pb55

Pb309

Pb923

Cohesive energy for Pb nanoclusters

eV

N-1/3

cubotohedron icosahedron

Structure phase transition of Icosahedral CubotohedralMackay transition Acta Cryst. 15, p916 (1962)

ico if fcc if s= 0

1 2 3 4-50

0

50

100

150

200

barrier high divided by no. of cluster atoms

meV

shell index

Barrier heights (~10 meV) of ICO FCC transition of Pb clusters oscillate with the shell index (or radius of cluster) indicates th

e possible Quantum Size Effect of the melting points ?

Which Au38 is a more stable structure?

Efcc= - 100.40 eV (PBE) EO_h= - 101.86 eV Efcc= - 97.50 eV (GGA) EO_h= - 99.02 eV Efcc= - 131.98 eV (LDA) EO_h= - 130.81 eV

QMC needed?

Atomic structures of 13-atommetallic clusters by DFT

Hsin-Yi,Tiffany, Chen

Ching-Ming Wei

Institute of Atomic & Molecular Sciences,Academia Sinica, Taiwan

Motivation?

MotivationMotivation To determine the ground-state structures of 44 metallic (Tab.1) 13-atom clusters

Find out the possible regularity existed and then try to understand the reasons accounting for the regularity.

Tab.1 Selected 13-atom clusters of the Group1A~3A, 3d~5d series and Pb13 in a periodic table (44 elements)

Group 1A

Group 2A

Group 3A

K19 bcc

2D+ico

Ba56 bcc

ico

Sc21 hcp

icoTi

22 hcp

icoV

23 bcc

bcc+2D Cr

24 bcc

dec(?)Mn

25 cubiccomplex

tbp(?)

Ru44 hcp

2D-cag

Co27 hcp

2D-tbpNi

28 fcc

icoCu

29 fcc

2D-gclZn

30 hcp

2DGa

31complex

dec+hcp

Pb82 hcp

ico

Rb37 bcc

2D+ico

Na11 bcc

2D+icoMg

12 hcp

2D

Li3 bcc

icoBe

4 hcp

2D+ico

Sr38 fcc

ico

Ca20 fcc

ico

Y39 hcp

ico

La57 hex

ico

Zr40 hcp

ico

Hf72 hcp

ico

Nb41 bcc

bcc

Ta73 bcc

bcc+ico

Mo42 bcc

dec(?)

W74 bcc

dec

Tc43 hcp

2D-tbp

Re75 hcp

2D-tbp

Fe26 bcc

ico

Os76 hcp

2D-cagIr

77 fcc

2D-cag

Rh45 fcc

2D-cagPd

46 fcc

2D-tbp

Pt78 fcc

2D

Ag47 fcc

2D-gclCd

48 hcp

hcp(?)In

49 tetr

dec+hcp

Au79 fcc

2D-gclTl

81 hcp

dec+hcp

B5 hcp

2D-bbp

Al13 hcp

ico

C6

Si14

Ge32

Sn50

Cs55 bcc

Hg80

Group 3B

Group 4B

Group 5B

Group 6B

Group 7B

Group 8B

Group 8B

Group 8B

Group 1B

Group 2B

Two questions we are asking: (1) If the highest symmetry icosahedral structure would always be the most stable in each element?

(2) Are there any relations between clusters and their bulk crystal structures?

Method & Calculated Materials

Calculated Materials

Chosen elements : Group 1 ~ Group 13, and Pb in the periodic table 9 available and familiar atomic structures of ground-state from literature

searches were calculated to find out the lowest energy in each element.

Method

Software : Vienna Ab Initio Simulation Program (VASP) Pseudopotential method : PAW Compare 3 exchange-correlation functional : LDA, PW91, PBE K points sampling : gamma point Supercell Dimensions : 20 Å × 20 Å × 20 Å

1 2 13

1211109876543

Materials – 9 available 13-atom atomic structures fccico dec 5 High Symmetry → 3D

icosahedral (ico), Ih

cuboctahedral (fcc), Oh

decahedral (dec), D5v

body-center cubic (bcc) D4h

hexagonal-close packed (hcp), D3v

buckled biplanar (bbp) <ref1> garrison-cap(gcl)

4 Low Symmetry → 2D buckled biplanar (bbp), C2v

triangular biplanar (tbp), C3v

garrison-cap layer (gcl) C2v

cage (cag), C1h

hcpbcc

triangular biplanar(tbp)

hexagonal array (7) + central square (4)+(2)side

atoms

triangle (3)+ (7) atoms

+ triangle (3)<Ref 1> C. M. Chang, M. Y. Chou, Phys. Rev. Lett. 93, 133401 (2004) , <Ref 2> Y. C. Bae, et al, Phys. Rev. B 72, 125427 (2005)

hexagonal array (7)

+ triangle (6)

cage(cag) <ref2>

(1) atom+ 2 square (12)

top view

side view top view side view top view side view

How do we compare 3 exchange correlation functionals?

Define the average energy of 9 atomic structures as “reference point”

dE atomic structure

= E atomic structure – [(Ebbp+Egbp+Ebcc+Edec+Efcc+Ehcp+Eico+Etbp+Ecag)/ 9]

reference point = Ebbp+Egbp+Ebcc+Edec+Efcc+Ehcp+Eico+Etbp+Ecag)/ 9

Define “relative energy”, dE(eV) = Total energy of the atomic structure– reference point

Equ.

Equ.

remark We use “relative energy” to compare 3 exchange correlation functionals

1

2

dE bbp = Ebbp– [(Ebbp+Egbp+Ebcc+Edec+Efcc+Ehcp+Eico+Etbp+Ecag)/ 9]

bbp bcc cbp cag dec fcc gcl hcp ico

-1.0

-0.8

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

In LDA PW91 PBE

dE (e

V)

9 Atomic Structurestbp

bbp bcc cbp cag dec fcc gcl hcp ico-4

-3

-2

-1

0

1

2

3

4

Re LDA PW91 PBE

dE (e

V)

9 Atomic Structurestbp

bbp bcc cbp cag dec fcc gcl hcp ico-2.5

-2.0

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

2.0Ba LDA

PW91 PBE

dE (e

V)

9 Atomic Structures

Consistency in 3 exchange correlation functionals

For Ba13

the lowest energyall occur in Icosahedral

Remark

For Re13

the lowest energyall occur in garrison-cap layer (2D-gcl, low symmetry)

For In13, ∵ the energies of dec and hcp

→ too close and competitive

∴Atomic structure of ground-state could be dec or hcp

Rem

ark

tbp

Remark

relative stability

bbp bcc cbp cag dec fcc gcl hcp ico --

-4

-3

-2

-1

0

1

2

3

4

LDA PW91 PBE

dE

(e

V)

9 Atomic Structures

Cr

tbp-- bbp bcc cbp cag dec fcc gcl hcp ico --

-2.0

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

2.0

2.5

LDA PW91 PBE

dE

(eV

)

9 Atomic Structures

Mo

tbp

Consistency and Inconsistency of LDA, PW91, & PBE occurred in Group 6

bbp bcc cbp cag dec fcc gcl hcp ico ---2.5

-2.0

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

2.0

2.5

3.0 LDA PW91 PBE

dE

(eV

)

9 Atomic Structures

WCr13

For LDA, PW91, & PBE → the lowest energies all occur in “dec”

Bulk

Cr24 bcc

dec(?)

Mo42 bcc

dec(?)

W74 bcc

dec

Group 6B

(III) dec

Cluster

consistencyLDA

3

Mo13

For PW91 & PBE → the lowest energy only occur in “dec”

bcc

gcl

tbp

BUT

For LDA → lower energies occur in

“dec” & ico”

Inconsistent “relative stabilities” occur in “bcc” & “gcl”

BUTBUT

Ga

ico

complex31

49

Intetr

dec+hcp

Group 1

Group 2

Group 13

K19 bcc

2D+ico

Ba56 bcc

ico

Sc21 hcp

icoTi

22 hcp

icoV

23 bcc

bcc+2D Cr

24 bcc

dec(?)Mn

25 cubiccomplex

2D-tbp(?)

Ru44 hcp

2D-cag

Co27 hcp

2D-tbpNi

28 fcc

icoCu

29 fcc

2D-gclZn

30 hcp

2D dec+hcp

Pb82 hcp

ico

Rb37 bcc

2D+ico

Na11 bcc

2D+icoMg

12 hcp

2D

Li3 bcc

icoBe

4 hcp

2D+ico

Sr38 fcc

ico

Ca20 fcc

ico

Y39 hcp

ico

La57 hex

ico

Zr40 hcp

ico

Hf72 hcp

ico

Nb41 bcc

bcc

Ta73 bcc

bcc+ico

Mo42 bcc

dec(?)

W74 bcc

dec

Tc43 hcp

2D-tbp

Re75 hcp

2D-tbp

Fe26 bcc

ico

Os76 hcp

2D-cagIr

77 fcc

2D-cag

Rh45 fcc

2D-cagPd

46 fcc

2D-tbp

Pt78 fcc

2D

Ag47 fcc

2D-gclCd

48 hcp

hcp(?)

Au79 fcc

2D-gclTl

81 hcp

dec+hcp

B5 hcp

2D-bbp

Al13 hcp

C6

Si14

Ge32

Sn50

Cs55 bcc

Hg80

Group 3

Group 4

Group 5

Group 6

Group 7

Group 8

Group 9

Group 10

Group 11

Group 12

(I) ico or 2D+ico except Mg13

(IV) 2D (tbp, cag, gcl)(II) bcc (III) dec (V) dec+hcpcompetitive

Overall Results of Regularity

Bulk structure

Cluster structure

cluster’s structures are the same as bulks’ only in Group 5

2D+ico: cluster structure could be“2D Low symmetry” or “ico”

dec(?): undetermined structure

2D-tbp: cluster structure is “2D low symmetry--tbp”

dec+hcp: cluster structure could be“dec” or “hcp”

If all these DFT results are reliable?

Ag adsorbed on Graphite

ExC Ag+Graphite Ag/Graphite Ead

LDA -323.063 eV -323.614 eV 0.551 eV PW91 -295.830 eV -295.876 eV 0.046 eV PBE -295.447 eV -295.481 eV 0.034 eV

Motivation? DFT is no predict power!!!

trans-stilbene cis-stilbene

Etrans= 0.0 eV Ecis= 0.204 eV

Ag-Ge(111)-IET

trans-stilbene/Ag-Ge(111) cis-stilbene/Ag-Ge(111)

Eads= 1.059 eV (LDA) Eads= 0.887 eV (LDA)

LDA agrees expt., but… Eads = 0.40 & 0.20 eV (PW91)and again DFT without any predicting power!!!

QMC results of B18 and B20

• To check if tube structure will become favor in B20?

• To check if the hollow B18 (Oh) will become the most stable cluster?

Boron18 cluster

(1) (2) (3) (4)VASP(PBE) 0.717 eV 0.739 eV 0.000 eVCASTEP(PBE) 0.740 eV 0.752 eV 0.000 eV 0.970 eV -1366.606 -1366.594 -1367.346 -1366.376 CASTEP(LDA) 0.601 eV 0.836 eV 0.000 eV 0.192 eV -1362.212 -1361.977 -1362.813 -1362.620-----------------------------------------------------------------------------------------------------------------------------------------QMC(dt=0.005) -1361.680(39) (1.55) -1362.316(42) (0.91) -1363.227(35) eV -1361.836(42) (1.39)QMC(dt=0.010) -1361.598(25) (1.51) -1362.148(28) (0.96) -1363.107(27) eV -1361.716(27) (1.39)

B18-1 B18-2 B18-3 B18-4-0.2

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8 castep-LDA castep-PBE Q-M-C

eV

DFT & QMC results of B18

cluster

Boron20 clusters

5

Above 4 structures are described by J. Chem. Phys. 124, 154310 (2006),but 5th structure is found by me recently with a comparable low energy with structures 2, 3, and 4.

(1) (2) (3) (4) (5)VASP(PBE) 0.00 0.65 0.46 0.59 0.48CASTEP(PBE) 0.00 0.44 0.57 0.49 -1520.216 -1519.774 -1519.646 -1519.727 CASTEP(LDA) 0.00 0.28 0.39 0.08 -1514.946 -1514.667 -1514.554 -1514.863------------------------------------------------------------------------------------------------------------------------------------------------

QMC(dt=0.005) (~100000 steps) -1516.171(36) -1515.417(35) -1515.426(51) -1515.358(45)

0.75 0.74 0.81QMC(dt=0.010) (~100000 steps) -1515.953(33) -1515.292(31) -1515.267(57) -1515.115(35) 0.66 0.69 0.84QMC(dt=0.010) (=300000 steps) -1515.961(19) -1515.278(19) -1515.239(23) -1515.125(20) 0.68 0.72 0.84

5(C1h)

All calculations were performed using Gaussian 03, Revision C.02 package.24 For neutral clusters, full geometry optimizations were performed using the second-order Møller-Plesset perturbation theory25,26 MP2 method as well as DFT methods in generalized gradient approximations GGAs with two hybrid exchange-correlation functionals, namely, B3LYP Ref. 27 and PBE1PBE,28 and a recently developed hybrid metafunctional TPSS1KCIS.29 A modest cc-pVDZ basis set30 Dunning’s correlation consistent polarized valence double zeta, contracted 3s2p plus polarization set 1d was chosen with the MP2 method and a large ccpVTZ basis set30 Dunning’s correlation consistent polarized valence triple zeta, contracted 4s3p plus polarization set 2d1f with DFTs. Next, the harmonic vibrationalfrequency analyses were carried out to assure that the optimized structures give no imaginary frequencies. To determine the energy ordering, several high-level ab initio molecular-orbital methods were employed to calculate single-point energies of the four neutral isomers with the optimized structures at the MP2/cc-pVDZ level of theory: 1 the fourth-order Møller-Plesset perturbation theory31 MP4 with cc-pVTZ basis set for neutral isomers; 2 a coupled-cluster32 method at the CCSDT1Diag/6-311Gd level of theory to examine possible multireference quality for the top-two lowest-energy isomers; and 3 the coupledcluster method including single, double, and noniteratively perturbative triple excitations at the CCSDT/6-311Gd level of theory.

B20-1 B20-3 B20-5 B20-4-0.1

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

castep-LDA castep-PBE Q-M-C

eV

DFT & QMC results of B20

cluster

QMC study of Al13 and Al55

bbp bcc cag cbp dec fcc hcp ico t13-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

2.0

2.5

gga lda pbe

eV

DFT results of Al13

cluster

bbp bcc cag cbp dec fcc hcp ico t13-3

-2

-1

0

1

2

3

4

5

gga lda pbe

eV

DFT results of Mn13

cluster

• To answer if DFT can be used to the study of metallic clusters?• Which ExC approximation might be better if LDA, PW91, and PBE do not give consistent results?

DEC FCC ICO-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

castep-LDA q-m-c vasp-LDA vasp-PW91 vasp-PBE

eV

DFT & QMC results of Al13

cluster

MD simulation at 500 K starting from Al55 ICO structure========PW91===LDA===PBE========================= 1 ps -3.009 -2.618 -2.772 eV 2 ps -3.357 -2.099 -2.432 eV 3 ps -2.821 -3.116 -3.053 eV 4 ps -2.726 -2.854 -3.044 eV 5 ps -3.131 -3.595 -3.049 eV 6 ps -2.993 -3.678 -2.828 eV 7 ps -3.531 -3.687 -3.354 eV 8 ps -3.402 -3.541 -3.238 eV 9 ps -3.479 -3.670 -3.223 eV 10 ps -3.501 -3.672 -3.211 eV 11 ps -3.257 -3.313 -3.411 eV 12 ps -2.673 -3.297 -3.382 eV 13 ps -3.039 -3.183 -3.383 eV 14 ps -3.408 -3.443 -3.502 eV 15 ps -3.544 -3.454 -3.712 eV 16 ps -3.201 -3.447 -3.396 eV 17 ps -3.128 -3.453 -3.384 eV 18 ps -3.186 -3.450 -3.306 eV 19 ps -2.946 -3.104 -2.970 eV 20 ps -3.481 -3.500 -3.107 eV

Question: if we really find the local minimum?

1. Relax the structure using the relaxed structure obtained by LDA at 7 ps with PBE potential, then using this relaxed structure but with LDA potential again, it happens the relaxed structure go back to original structure !

2. Relax the structure using the relaxed structure obtained by PBE at 15 ps with LDA potential, then using this relaxed structure but with PBE potential again, it happens the relaxed structure go back to original structure !

Answer : YES

DECICO

FCCAMOR

DEC FCC ICO AMOR-3.0

-2.5

-2.0

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

2.0

2.5

vasp-PBE castep-LDA Q-M-C

eV

DFT & QMC results of Al55

cluster

QMC study of C20

• To see if DFT with new developed ExC (like PBE) can describe well the energy difference of local minimum structures?

• To see if DFT can describe well the energy difference due to Jahn-Teller distortion?

cage

IhC31.44A 1.4~1.5A

C20 structures

ring

20h10h1.28A 1.24A 1.32A

bowl

C5v1.24A

1.40~1.43A

bowl_C_5v cage_C_3 ring_10h-2.0

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

2.0

castep-LDA vasp-LDA vasp-PW91 vasp-PBE Q-M-C

eV

DFT & QMC results of C20

cluster

PBE does give the correct

energy order!

bowl_C_5v cage_I_h cage_C_3 ring_20h ring_10h-3.0-2.5-2.0-1.5-1.0-0.50.00.51.01.52.02.53.03.5

SS LL

castep-LDA vasp-LDA vasp-PW91 vasp-PBE Q-M-C

eV

DFT & QMC results of C20

cluster

L

DFT fail to give a correct E due to Jahn-Teller effect!

QMC results of graphene ribbon

For N=5 Ribbon, the energy difference of nanoribbon states obtained by DFT are:-------------------------------------------------------------Code & ExC E (NM-AF) E (FM-AF)------------------------------------------------------------CASTEP LDA 32.5 meV 1.5 meV VASP LDA 36.3 meV 2.0 meVVASP PW91 55.6 meV 3.3 meVVASP PBE 79.1 meV 5.7 meV------------------------------------------------------------------------------------------------------------------------------

Crystal B3LYP 290 meV 49 meV ref: Harrison et al. in PRB 75, 2007

-------------------------------------------------------

FIG. 1. Color online A monohydrogenated ribbon of width N=5 along y. The system is periodic only along x and the dashedlines delimit the periodic unit cell of length a.

FIG. 7. Color online Isovalue surfaces of the spin density forthe antiferromagnetic case (a) and ferromagnetic case (b) of ribbon of width N=10 .

FIG. 4. Color online Electron density of a nonmagnetic, monohydrogenated,N=10 ribbon contributed by a the states near the Fermi level and b the rest of the occupied states.

For N=5 Graphene Ribbon, the QMC energies obtained are : AF (VMC) : Final energy = -1564.659 (38) eVNM (VMC) : Final energy = -1563.803 (41) eVAF (DMC) : Final energy = -1577.092 (31) eV (~9000 CPU hour)NM (DMC) : Final energy = -1576.658 (35) eV (~9000 CPU hour)And the energy difference obtained by QMC are E (NM-AF) = 856 meV VMCE (NM-AF) = 434 meV DMC (T=0.005)(Here E_cut = 600 eV, BLIP = 2.0 and 1x1x6 unit cell)

It seems that DMC favors B3LYP!

N=5 RibbonK-point (PBE) E (NM-AF) E (FM-AF)---------------------------------------------------------- 1x1x 6 59.0 meV 3.6 meV 1x1x 9 76.5 meV 5.0 meV 1x1x12 79.1 meV 5.7 meV 1x1x15 70.8 meV 9.9 meV 1x1x30 67.4 meV 10.9 meV 1x1x60 67.5 meV 13.0 meV----------------------------------------------------------

For N=5 Ribbon, the energy difference of nanoribbon states obtained by DFT are:------------------------------------------------------------------------Code & ExC E (NM-AF) E (FM-AF)--------------------------------------------------------------CASTEP LDA 32.5 meV 1.5 meV VASP LDA 36.3 meV 2.0 meVVASP PW91 55.6 meV 3.3 meVVASP PBE 79.1 meV 5.7 meV------------------------------------------------------------------------

Crystal B3LYP 290 meV 49 meV ref: Harrison et al. in PRB 75, 2007

-----------------------------------------------------------------------

-

LDA, PW91, PBE are at least afactor of 4~5 less than B3LYP!!!

Summary and Conclusion• In general, DFT should be able to use in the study of t

he metallic clusters judging from the QMC results of Al13 and Al55 clusters, and PBE is perhaps better!

• DFT fails to describe B18 and B20 clusters, and QMC is needed!

• DFT fails to describe well C20 clusters, however, PBE can perhaps describe the energy difference of local minimums!

• DFT with LDA, PW91, PBE ExC fails to describe the energy difference of AF and NM states of graphene ribbon!