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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




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


1. Tight-Binding

SK Approximation (II)

11


1. Tight-Binding

SK Approximation (III)

12


1. Tight-Binding

SK Approximation (IV)

13


1. Tight-Binding

14

SK Tables


1. Tight-Binding

15




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

18

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

21

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


( )∑ −=Ψν

αννφ Rrii c



ivariational


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


( )∑ −=Ψν

αννφ Rrii c



ivariational


H11

H22

H33

H41 H44


1

4

2

3

r12

r23

r14

r34

r13

r24

r14

Two-center integral

( )∑ Δ++=ξ

ξβξαξµνµνµν γγ qSHH210


Induced charge

Hamiltonian Overlap

Lookup tabulated H0

and S at distance r

r14


( )∑ −=Ψν

αννφ Rrii c



ivariational


H11

H22

H33

H41 H43 H44


1

4

2

3

r12

r23

r34

r13

r24

r34

Two-center integral

( )∑ Δ++=ξ

ξβξαξµνµνµν γγ qSHH210


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)

28

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


DFTB

36


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


atoms

∑ + δMN12

Sµν WAll ' + WAll"( )pM l"l"∈M∑

Hµν↓ = χµ

ˆ H ρ0M ,ρ0

N[ ] χµ +12


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)


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


Particle swarm optimization (PSO)


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





49

Band structure fitting for HCP crystal structures



• 2 lattice constants (a, c) c

a


50

Band structure fitting for Diamond crystal structures





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.

57

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

58

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

density-functional tight-binding (dftb) as fast ... · pdf file1 density-functional...

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