quantum monte-carlo studies of b, al, and c clusters
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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!
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