1
Density-Functional Tight-Binding (DFTB) as fast approximate DFT method
Helmut Eschrig Gotthard Seifert Thomas Frauenheim Marcus Elstner
Introduc+ontotheDensity-Func+onalTight-Binding(DFTB)Method
PartI
Workshop at IACS April 8 & 11, 2016
2
Density-Func,onalTight-Binding
PartI1. Tight-Binding2. Density-Func,onalTight-Binding(DFTB) PartII
3. BondBreakinginDFTB4. Extensions5. PerformanceandApplica,ons
3
Density-Func,onalTight-Binding
PartI1. Tight-Binding2. Density-Func,onalTight-Binding(DFTB) PartII
3. BondBreakinginDFTB4. Extensions5. PerformanceandApplica,ons
44
1. Tight-Binding
Resources1. hBp://www.dGb.org2. DFTB Porezag, D., T. Frauenheim, T. Köhler, G. Seifert, and R. Kaschner, Construction
of tight-binding-like potentials on the basis of density-functional theory: application to carbon. Phys. Rev. B, 1995. 51: p. 12947-12957.
3. DFTB Seifert, G., D. Porezag, and T. Frauenheim, Calculations of molecules, clusters, and solids with a simplified LCAO-DFT-LDA scheme. Int. J. Quantum Chem., 1996. 58: p. 185-192.
4. SCC-DFTB Elstner, M., D. Porezag, G. Jungnickel, J. Elsner, M. Haugk, T. Frauenheim, S. Suhai, and G. Seifert, Self-consistent-charge density-functional tight-binding method for simulations of complex materials properties. Phys. Rev. B, 1998. 58: p. 7260-7268.
5. SCC-DFTB-D Elstner, M., P. Hobza, T. Frauenheim, S. Suhai, and E. Kaxiras, Hydrogen bonding and stacking interactions of nucleic acid base pairs: A density-functional-theory based treatment. J. Chem. Phys,, 2001. 114: p. 5149-5155.
6. SDFTB Kohler, C., G. Seifert, U. Gerstmann, M. Elstner, H. Overhof, and T. Frauenheim, Approximate density-functional calculations of spin densities in large molecular systems and complex solids. Phys. Chem. Chem. Phys., 2001. 3: p. 5109-5114.
7. DFTB3 Gaus, M.; Cui, C.; Elstner, M. DFTB3: Extension of the Self-Consistent-Charge Density-Functional Tight-Binding Method (SCC-DFTB). J. Chem. Theory Comput., 2011. 7: p. 931-948.
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Standalone fast and efficient DFTB implementation with several useful extensions of the original DFTB method. It is developed at the Bremen Center for Computational Materials Science (Prof. Frauenheim, Balint Aradi). ased on previous DYLAX code. Free for non-commercial use. DFTB+ as part of Accelrys' Materials Studio package, providing a user friendly graphical interface and the possibility to combine DFTB with other higher or lower level methods. DFTB integrated in the ab initio DFT code deMon (Thomas Heine) DFTB in the Gaussian code (Keiji Morokuma) Amber is a package of molecular simulation programs distributed by UCSF, developed mainly for biomolecular simulations. The current version of Amber includes QM/MM. (Marcus Elstner et al.) CHARMm (Chemistry at HARvard Macromolecular Mechanics) (QianCui.) DFTB integrated in the Amsterdam Density Functional (ADF) program suite. (Thomas Heine) DFT /2/3 and FMO2-DFTB1/2/3 (Yoshio Nishimoto Dmitri edorov, Stephan Irle
DFTB+ DFTB+/Accelrys deMon GAUSSIAN G09 AMBER CHARMm ADF GAMESS-US
Implementations 1. Tight-Binding
6
• Tight binding (TB) approaches work on the principle of treating electronic wavefunction of a system as a superposition of atom-like wavefunction (known to chemists as LCAO approach)
• Valence electrons are tightly bound to the cores (not allowed to delocalize beyond the confines of a minimal LCAO basis)
• Semi-empirical tight-binding (SETB): Hamiltonian Matrix elements are approximated by analytical functions (no need to compute integrals)
• TB energy for N electrons, M atoms system:
• This separation of one-electron energies and interatomic distance-dependent potential vj,k constitutes the TB method
Tight-Binding
1. Tight-Binding
7
• ει are eigenvalues of a Schrodinger-like equation
• solved variationally using atom-like (minimum, single-zeta) AO basis set,
leading to a secular equation:
where H and S are Hamiltonian and overlap matrices in the basis of the AO functions. In orthogonal TB, S = 1 (overlap between atoms is neglected)
• H and S are constructed using nearest-neighbor relationships; typically only nearest-neighbor interactions are considered: Similarity to extended Hückel method
Tight-Binding
1. Tight-Binding
8
• Basedonapproxima+onbyM.WolfsbergandL.J.Helmholz(1952)H Ci = εi S Ci
• H – Hamiltonian matrix constructed using nearest neighbor relationships
• Ci – column vector of the i-th molecular orbital coefficients • εi – orbital energy • S – overlap matrix • Hµµ - choose as a constant – valence shell ionization
potentials
• Hµν = K Sµν (Hµµ + Hνν)/2 • K – Wolfsberg Helmholz constant, typically 1.75
Extended Huckel (EHT) Method
1. Tight-Binding
Categories of TB approaches
9
Source: http://beam.acclab.helsinki.fi/~akrashen/esctmp.html
1. Tight-Binding
Slater-Koster (SK) Approximation (I)
10
Source: http://beam.acclab.helsinki.fi/~akrashen/esctmp.html
1. Tight-Binding
SK Approximation (II)
11
Source: http://beam.acclab.helsinki.fi/~akrashen/esctmp.html
1. Tight-Binding
SK Approximation (III)
12
Source: http://beam.acclab.helsinki.fi/~akrashen/esctmp.html
1. Tight-Binding
SK Approximation (IV)
13
Source: http://beam.acclab.helsinki.fi/~akrashen/esctmp.html
1. Tight-Binding
14
SK Tables
Source: http://beam.acclab.helsinki.fi/~akrashen/esctmp.html
1. Tight-Binding
15
Density-Func,onalTight-Binding
PartI1. Tight-Binding2. Density-Func,onalTight-Binding(DFTB) PartII
3. BondBreakinginDFTB4. Extensions5. PerformanceandApplica,ons
16
Taken from Oliviera, Seifert, Heine, Duarte, J. Braz. Chem. Soc. 20, 1193-1205 (2009)
...open access
Thomas Heine
Helio Duarte
DFTB
17
DensityFunc+onalTheory(DFT)
[ ] ( ) ( )
[ ] ( ) ( )
2 3
1
3 3
, 1
1
'1 '2 '
'1 1 '2 ' 2
M
i i ext ii
N
xc
M
i i repi
rE n v r d r
r r
Z Zr rE d rd r
r r R R
n E
α β
α β α βα β
ρρ ψ ψ
ρ ρρ
ε
=
=≠
=
= − ∇ + +−
+ − +− −
= +
∑ ∫
∑∫∫
∑
rr r rr r
r rr r
at convergence:
Various criteria for convergence possible: • Electron density • Potential • Orbitals • Energy • Combinations of above quantities
Walter Kohn/John A. Pople 1998
DFTB
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Phys. Rev. B, 39, 12520 (1989) Foulkes + Haydock Ansatz
DFTB
19
Self-consistent-charge density-functional tight-binding (SCC-DFTB)
M. Elstner et al., Phys. Rev. B 58 7260 (1998)
€
E ρ[ ] = ni φiˆ H ρ0[ ] φi
i
valenceorbitals
∑1
+ ni φi
ˆ H ρ0[ ] φii
coreorbitals
∑2
+ Exc ρ0[ ]
3
−12
ρ0VH ρ0[ ]R3∫
4
−
− ρ0Vxc ρ0[ ]R3∫
5
+ Enucl6 +
12
ρ1VH ρ1[ ]R3∫
7
+12
δ 2Exc
δρ12
ρ0
ρ12
R3∫∫
8
+ο 2( )
Approximate density functional theory (DFT) method!
Second order-expansion of DFT energy in terms of reference density ρ0 and charge fluctuation ρ1 (ρ ≅ ρ0 + ρ1) yields:
Density-functional tight-binding (DFTB) method is derived from terms 1-6
Self-consistent-charge density-functional tight-binding (SCC-DFTB) method is derived from terms 1-8
o(3)
DFTB
20
DFTB and SCC-DFTB methods
v where Ø niand εi—occupation and orbital energy ot the ith Kohn-Sham
eigenstate Ø Erep—distance-dependent diatomic repulsive potentials Ø ΔqA—induced charge on atom A Ø γAB—distance-dependent charge-charge interaction functional;
obtained from chemical hardness (IP – EA)
EDFTB = niεii
valenceorbitals
∑term1
+12
ErepAB
A≠B
atoms
∑terms 2−6
ESCC−DFTB = niεii
valenceorbitals
∑term1
+12
γABΔqAΔqBA,B
atoms
∑terms 7−8
+12
ErepAB
A≠B
atoms
∑terms 2−6
DFTB
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DFTB method v Repulsive diatomic potentials replace usual nuclear repulsion
energy
v Reference density ρ0is constructed from atomic densities
v Kohn-Sham eigenstatesφiare expanded in Slater basis of valence pseudoatomic orbitals χi
v The DFTB energy is obtained by solving a generalized DFTB eigenvalue problem with H0 computed by atomic and diatomic DFT
€
ρ0 = ρ0A
A
atoms
∑
€
φi = cµiχµµ
AO
∑
€
H0C = SCε with Sµν = χµ χν
Hµν0 = χµ
ˆ H ρ0M ,ρ0
N[ ] χν
DFTB
22
Traditional DFTB concept: Hamiltonian matrix elements are approximated to two-center terms. The same types of approximations are done to Erep.
From Elstner et al., PRB 1998
[ ] [ ][ ] [ ] [ ]
0
0
(Density superposition)
(Potential superposition)eff eff A B
eff eff A eff B
V V
V V V
ρ ρ ρ
ρ ρ ρ
≈ +
≈ +
A B D
C
A
B
D C
Situation I Situation II
Both approximations are justified by the screening argument: Far away, neutral atoms have no Coulomb contribution.
Approximations in the DFTB HamiltonianDFTB
SCC-DFTB matrix elements LCAO ansatz of wave function
( )∑ −=Ψν
αννφ Rrii c
secular equations ( ) 0=−∑
νµνµνν ε SHc i
ivariational
principlepseudoatomic orbital
Atom 1 – 4 are the same atom & have only s shell�
1
4
2
3
r12
r23
r14
r34
r13
r24
How to construct?�
ü two-center approximation ü nearest neighbor off-diagonal elements only (choice of cutoff values)
Hamiltonian Overlap
pre-computed parameter • Reference Hamiltonian H0
• Overlap integral Sµν
SCC-DFTB matrix elements LCAO ansatz of wave function
( )∑ −=Ψν
αννφ Rrii c
secular equations ( ) 0=−∑
νµνµνν ε SHc i
ivariational
principlepseudoatomic orbital
H11
H22
H33
H44
Atom 1 – 4 are the same atom & have only s shell�Diagonal term
Orbital energy of neutral free atom (DFT calculation)
1
4
2
3
r12
r23
r14
r34
r13
r24
Hamiltonian Overlap
( )∑ Δ++=ξ
ξβξαξµµµ γγε qH21
Charge-charge interaction function
Induced charge
SCC-DFTB matrix elements LCAO ansatz of wave function
( )∑ −=Ψν
αννφ Rrii c
secular equations ( ) 0=−∑
νµνµνν ε SHc i
ivariational
principlepseudoatomic orbital
H11
H22
H33
H41 H44
Atom 1 – 4 are the same atom & have only s shell�
1
4
2
3
r12
r23
r14
r34
r13
r24
r14
Two-center integral
( )∑ Δ++=ξ
ξβξαξµνµνµν γγ qSHH210
Charge-charge interaction function
Induced charge
Hamiltonian Overlap
Lookup tabulated H0
and S at distance r
r14
SCC-DFTB matrix elements LCAO ansatz of wave function
( )∑ −=Ψν
αννφ Rrii c
secular equations ( ) 0=−∑
νµνµνν ε SHc i
ivariational
principlepseudoatomic orbital
H11
H22
H33
H41 H43 H44
Atom 1 – 4 are the same atom & have only s shell�
1
4
2
3
r12
r23
r34
r13
r24
r34
Two-center integral
( )∑ Δ++=ξ
ξβξαξµνµνµν γγ qSHH210
Charge-charge interaction function
Induced charge
Hamiltonian Overlap
Ø Repeat until building off-diagonal term
Lookup tabulated H0
and S at distance r
27
DFTB parameters
DFTB
• •+
• •
1s
σ1s
H H
H2
Δρ = ρ – Σa ρa H2 difference density 1s
DFTB repulsive potential Erep
Which molecular systems to include?
Development of (semi-)automatic fitting: • Knaup, J. et al., JPCA, 111, 5637, (2007) • Gaus, M. et al., JPCA, 113, 11866, (2009) • Bodrog Z. et al., JCTC, 7, 2654, (2011)
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DFTB
29
v Additional induced-charges term allows for a proper description of charge-transfer phenomena
v Induced charge ΔqAon atom A is determined from Mulliken population analysis
v Kohn-Sham eigenenergies are obtained from a generalized, self-consistent SCC-DFTB eigenvalue problem
�€
ΔqA = ni cµicνiSµνν
AO
∑µ∈A∑
i
MO
∑ − qA0
€
HC = SCε with Sµν = χµ χν and
Hµν = χµˆ H ρ0
M ,ρ0N[ ] χν +
12
Sµν γMK + γNK( )ΔqKK
atoms
∑
SCC-DFTB method (I)
DFTB
30
SCC-DFTB method (II)
Basic assumptions: • Only transfer of net charge between atoms • Size and shape of atom (in molecule) unchanged
Only second-order terms (terms 7-8 on slide 16):
DFTB
31
SCC-DFTB method (III)
DFTB
32
SCC-DFTB method (IV)
Several possible formulations for γαβ: Mataga-Nishimoto < Klopmann-Ohno < DFTB
Klopmann-Ohno:
DFTB
33
Gradient for the DFTB methods The DFTB force formula
The SCC-DFTB force formula
�
computational effort: energy calculation 90% gradient calculation 10%
€
Fa = − ni cµicνi∂Hµν
0
∂a−εi
∂Sµν
∂a
&
' (
)
* +
µν
AO
∑i
MO
∑ −∂E rep
∂a
€
Fa = − ni cµicνi∂Hµν
0
∂a− εi −
12
γMK + γNK( )ΔqKK
atoms
∑)
* +
,
- . ∂Sµν
∂a
/
0 1
2
3 4
µν
AO
∑i
MO
∑ −
−ΔqA∂γAK∂a
ΔqKK
atoms
∑ −∂E rep
∂a
DFTB
34
Spin-polarized DFTB (SDFTB)
DFTB
v forsystemswithdifferent↑and↓spindensi+es,wehaveØ totaldensityρ=ρ↑+ρ↓Ø magne+za+ondensityρS=ρ↑-ρ↓
v 2nd-orderexpansionofDFTenergyat(ρ0,0)yields
€
E ρ,ρS[ ] = ni φiˆ H ρ0[ ] φi
i
valenceorbitals
∑1
+ ni φi
ˆ H ρ0[ ] φii
coreorbitals
∑2
+ Exc ρ0[ ]
3
−12
ρ0VH ρ0[ ]R3∫
4
−
− ρ0Vxc ρ0[ ]R3∫
5
+ Enucl6 +
12
ρ1VH ρ1[ ]R3∫
7
+12
δ 2Exc
δρ12
ρ0 ,0( )
ρ12
R3∫∫
8
+12
δ 2Exc
δρS( )2
ρ0 ,0( )
ρS( )2
R3∫∫
9
+ο 2( )
The Spin-Polarized SCC-DFTB (SDFTB) method is derived from terms 1-9
�
o(3)
wherepAl—spinpopula+onofshelllonatomA
WAll’—spin-popula+oninterac+onfunc+onal
v Spinpopula+onspAlandinducedchargesΔqAareobtainedfromMullikenpopula+onanalysis
€
E SDFTB = ni↑εi
↑
i
valenceorbitals
∑ + ni↓εi
↓
i
valenceorbitals
∑term1
+12
γABΔqAΔqBA≠B
atoms
∑terms 7−8
+12
E repAB
A≠B
atoms
∑terms 2−6
+12
pA l pA l 'WAll 'l '∈A∑
l∈A∑
A
atoms
∑term 9
€
ΔqA = ni↑cµi
↑ cνi↑ + ni
↓cµi↓ cνi
↓( )Sµνν
AO
∑µ∈A∑
i
MO
∑ − qA0
pA l = ni↑cµi
↑ cνi↑ − ni
↓cµi↓ cνi
↓( )Sµνν
AO
∑µ∈A ,l∑
i
MO
∑35
Spin-polarized DFTB (SDFTB)
DFTB
36
Spin-polarized DFTB (SDFTB)
DFTB
v Kohn-Shamenergiesareobtainedbysolvinggeneralized,self-consistentSDFTBeigenvalueproblems
where
€
H↑C↑ = SC↑ε↑
H↓C↓ = SC↓ε↓
€
Sµν = χµ χν
Hµν↑ = χµ
ˆ H ρ0M ,ρ0
N[ ] χµ +12
Sµν γMK + γNK( )ΔqKK
atoms
∑ + δMN12
Sµν WAll ' + WAll"( )pM l"l"∈M∑
Hµν↓ = χµ
ˆ H ρ0M ,ρ0
N[ ] χµ +12
Sµν γMK + γNK( )ΔqKK
atoms
∑ −δMN12
Sµν WAll ' + WAll"( )pM l"l"∈M∑
M,N,K: indexing specific atoms
37
SCC-DFTB w/fractional orbital occupation numbers
122tot i i rep
i
E f E q qαβ α βαβ
ε γ= + + Δ Δ∑ ∑
( ) 0vi iv
c H Sµν µνε− =∑
Fractional occupation numbers fi of Kohn-Sham eigenstates replace integer ni
TB-eigenvalue equation
DFTB
E
2fi 0 1 2
µ
Finite temperature approach (Mermin free energy EMermin)
( )1
exp / 1ii B e
fk Tε µ
=− +⎡ ⎤⎣ ⎦
( ) ( )2 ln 1 ln 1e B i i i ii
S k f f f f∞
= − + − −∑
Te: electronic temperature Se: electronic entropy
0 1
2N
repi i i i
i
EH H SF f c c q q
SR R R Rµν µν µν αξ
α µ ν α ξµν ξµνα α α α
γε
⎡ ⎤⎛ ⎞ ∂∂ ∂ ∂= − − − −Δ Δ −⎢ ⎥⎜ ⎟⎜ ⎟∂ ∂ ∂ ∂⎢ ⎥⎝ ⎠⎣ ⎦
∑ ∑ ∑r r r r r
0 1if≤ ≤
Atomic force
M. Weinert, J. W. Davenport, Phys. Rev. B 45, 13709 (1992)
EMermin=Etot-TeSe
38
Fermi-Dirac distribution function: Energy derivative for Mermin Free Energy
M. Weinert, J. W. Davenport, Phys. Rev. B 45, 13709 (1992)
( ) ( )
elect TSHF pulay charge TS
i ii
i ii i
i i
ii
e
ii
i
i
F F F F F
fx
f
T
fx x
f
S
fx
x
x
α
ε
ε
εε
ε
−
∞
∞ ∞
∞
∞
∂ −
≡ + + +
∂= +
∂
∂ ∂= +
∂ ∂
∂=
−∂
∂
∂
∂
∑
∑ ∑
∑
∑
r r r r( )
electHF pulay charge
i ii i i i
i i i
F F F F
ff fx x x
α
εε ε
∞ ∞ ∞
≡ + +
∂ ∂∂= = +
∂ ∂ ∂∑ ∑ ∑
r r r
Correction term arising from Fermi distribution function cancels out
DFTB
0 1 2 3 4 50
20
40
60
80
39
0 1 2 3 4 50
20
40
60
80
0 1 2 3 4 50
20
40
60
80
0 1 2 3 4 50
20
40
60
80
0 1 2 3 4 50
20
40
60
80
Time[ps] Time[ps] Time[ps]
Te= 0K
Te= 1500K
Te= 10kK
Te=0 K always yields SCC convergence problem SCC iterations(time) Maximum iteration number is 70
0 1 2 3 4 50
20
40
60
80
0 1 2 3 4 50
20
40
60
800 2 40
20
40
60
800 1 2 3 4 50
20
40
60
80(A) H10C60 Fe38 (B) Fe13C10 (C) Fe6C2
kbTe(10kK) ~0.87 eV ~half-width of 3d band in Fe38
DFTB
40/25
New Confining Potentials
W a
Conventional potential
r0
Woods-Saxon potential
k
RrrV ⎟⎟⎠
⎞⎜⎜⎝
⎛=
0
)(
R0 = 2.7, k=2
)}(exp{1)(
0rraWrV
−−+=
r0 = 3.0, a = 3.0, W = 3.0
Ø Typically, electron density contracts under covalent bond formation. Ø In standard ab initio methods, this problem can be remedied by including more basis functions.
Ø DFTB uses minimal valence basis set: the confining potential is adopted to mimic contraction
• •+
• •
1s
σ1s
H H
H2 Δρ = ρ – Σa ρa
H2 difference density 1s
Henryk Witek
Electronic Parameters DFTB Parameterization
40
2). DFTB band structure fitting • Optimization of parameter sets for Woods-Saxon confining potential (orbital and density) and unoccupied orbital energies • Fixed orbital energies for electron occupied orbitals • Valence orbitals : [1s] for 1st row [2s, 2p] for 2nd row [ns, np, md] for 3rd – 6th row (n ≥ 3, m = n-1 for group 1-12, m = n for group 13-18) • Fitting points : valence bands + conduction bands (depending on the system, at least including up to ~+5 eV with respect to Fermi level)
Electronic Parameters DFTB Parameterization
1). DFT band structure calculations • VASP 4.6 • One atom per unit cell • PAW (projector augmented wave) method • 32 x 32 x 32 Monkhorst-Pack k-point sampling • cutoff = 400 eV • Fermi level is shifted to 0 eV
41
Band structure for Se (FCC)
Brillouin zone 42
Electronic Parameters DFTB Parameterization
Particle swarm optimization (PSO)
Electronic Parameters DFTB Parameterization
43
1) Particles (=candidate of a solution) are randomly placed initially in a target space. 2) – 3) Position and velocity of particles are updated based on the exchange of information between particles and particles try to find the best solution. 4) Particles converges to the place which gives the best solution after a number of iterations.
•
••
••
•
• •••
•
•••••
• •••
• ••••••••
•••••••••••
par+cle
1)
4)
2)
3)
Particle Swarm Optimization DFTB Parameterization
44
Each particle has randomly generated
parameter sets (r0, a, W) within some region
Generating one-center quantities (atomic
orbitals, densities, etc.)
“onecent”
Computing two-center overlap and Hamiltonian integrals for wide range of interatomic distances
“twocent”
“DFTB+”
Calculating DFTB band structure
Update the parameter sets of each particle
Memorizing the best fitness value and parameter sets
Evaluating “fitness value” (Difference DFTB – DFT band
structure using specified fitness points) “VASP”
DFTB Parameterization
orbital a [2.5, 3.5] W [0.1, 0.5] r0 [3.5, 6.5]
density a [2.5, 3.5] W [0.5, 2.0] r0 [6.0, 10.0]
Particle Swarm Optimization
45
Example: Be, HCP crystal structure
DFTB Parameterization
Total density of states (left) and band structure (right) of Be (hcp) crystral structure
2.286
3.584
• Experimental lattice constants • Fermi energy is shifted to 0 eV
46
Electronic Parameters
Band structure fitting for BCC crystal structures • space group No. 229
• 1 lattice constant (a)
Transferability checked (single point calculation) Reference system in PSO Experimental lattice constants available
Ø No POTCAR file for Z ≥ 84 in VASP a
47
Band structure fitting for FCC crystal structures
Reference system in PSO Experimental lattice constants available
• space group No. 225
• 1 lattice constant (a) a
Transferability checked (single point calculation)
48
Band structure fitting for SCL crystal structures
Reference system in PSO Experimental lattice constants available
• space group No. 221
• 1 lattice constant (a) a
Transferability checked (single point calculation)
49
Band structure fitting for HCP crystal structures
Reference system in PSO Experimental lattice constants available
• space group No. 194
• 2 lattice constants (a, c) c
a
Transferability checked (single point calculation)
50
Band structure fitting for Diamond crystal structures
Reference system in PSO Experimental lattice constants available
• space group No. 227
• 1 lattice constant (a) a
Transferability checked (single point calculation)
51
DFTB Parameterization Transferability of optimum parameter sets for different structures
Ø Artificial crystal structures can be reproduced well
e.g. : Si, parameters were optimized with bcc only
W (orb) 3.33938
a (orb) 4.52314
r (orb) 4.22512
W (dens) 1.68162
a (dens) 2.55174
r (dens) 9.96376
εs -0.39735
εp -0.14998
εd 0.21210
3s23p23d0
bcc 3.081
fcc 3.868
scl 2.532
diamond 5.431
Parameter sets:
Lattice constants: bcc fcc
scl diamond
Expt.
52
Influence of virtual orbital energy (3d) to Al (fcc) band structure
OPT
Ø The bands of upper part are shifted up constantly as orbε(3d) becomes larger 53
Influence of W(orb) to Al (fcc) band structure
OPT
Ø The bands of upper part go lower as W(orb) becomes larger 54
Influence of a(orb) to Al (fcc) band structure
OPT
Ø Too small a(orb) gives the worse band structure 55
Influence of r(orb) to Al (fcc) band structure
OPT
Ø r(orb) strongly influences DFTB band structure 56
Correlation of r(orb) vs. atomic diameter
Atomic Number Z
Ato
mic
dia
met
er [a
.u.]
Empirically measured radii (Slater, J. C., J. Chem. Phys., 41, 3199-3204, (1964).)
Calculated radii with minimal-basis set SCF functions (Clementi, E. et al., J. Chem. Phys., 47, 1300-1307, (1967).)
Expected value using relativistic Dirac-Fock calculations (Desclaux, J. P., Atomic Data and Nuclear Data Tables, 12, 311-406, (1973).) This work r(orb)
Ø In particular for main group elements, there seems to be a correlation between r(orb) and atomic diameter.
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DFTB Parameterization Electronic Parameters
Straightforward application to binary crystal structuresRocksalt (space group No. 225)
• NaCl • MgO • MoC • AgCl …
• CsCl • FeAl …
B2 (space group No. 221)
Zincblende (space group No. 216)
• SiC • CuCl • ZnS • GaAs …
Others
• Wurtzite (BeO, AlO, ZnO, GaN, …) • Hexagonal (BN, WC) • Rhombohedral (ABCABC stacking sequence, BN)
Ø more than 100 pairs tested
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Selected examples for binary crystal structures
element name
Ga, As hyb-0-2
B, N matsci-0-2
Reference of previous work :
• d7s1 is used in POTCAR (DFT)
Ø Further improvement can be performed for specific purpose but this preliminary sets will work as good starting points
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Acknowledgements
• MarcusElstner• JanKnaup• AlexeyKrashenninikov• YasuhitoOhta• ThomasHeine• KeijiMorokuma• MarcusLundberg• YoshioNishimoto• Someothers
Acknowledgements