Density-Functional Tight-Binding (DFTB) as fast approximate DFT method - An introduction

All lectures online at: http://qc.chem.nagoya-u.ac.jp/presentations.html

2

Density-Functional Tight-Binding (DFTB) as fast approximate DFT method

Helmut Eschrig Gotthard Seifert Thomas Frauenheim Marcus Elstner

Lecture II:Introduction to the Density-Functional Tight-Binding (DFTB) Method

3

Density-Functional Tight-Binding

1. Tight-Binding

2. Density-Functional Tight-Binding (DFTB)

3. Bond Breaking in DFTB

4. Extensions

5. Performance and Applications

4


1. Tight-Binding



4. Extensions


55

Lecture II 1. Tight-Binding

Resources1. http://www.dftb.org

2. 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.

http://www.dftb.org/

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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) and is the successor of the old Paderborn DFTB and Dylax codes. 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

DFTB in the Gaussian code

Amber is a package of molecular simulation programs distributed by UCSF, developed mainly for biomolecular simulations. The current version of Amber includes QM/MM support, whereby part of the system can be treated quantum mechanically, and DFTB is among the quantum mechanical methods available. Amber also has a stand-alone (pure QM) implementation.

CHARMm (Chemistry at HARvard Macromolecular Mechanics)

DFTB integrated in the Amsterdam Density Functional (ADF) program suite.

DFTB+

DFTB+/Accelrys

deMon

GAUSSIAN G09

AMBER

CHARMm

ADF

Implementations


7

• 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


8

• ei 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


9

• Based on approximation by W. Wolfsberg and L. J. Helmholz (1952) H Ci = i S Ci

• H – Hamiltonian matrix constructed using nearest neighbor relationships

• Ci – column vector of the i-th molecular orbital coefficients

• i – orbital energy

• S – overlap matrix

• H - choose as a constant – valence shell ionization potentials

• H = K S (H + H)/2

• K – Wolfsberg Helmholz constant, typically 1.75

Extended Huckel (EHT) Method


Categories of TB approaches

10

Source: http://beam.acclab.helsinki.fi/~akrashen/esctmp.html


Slater-Koster (SK) Approximation (I)

11



SK Approximation (II)

12



SK Approximation (III)

13



SK Approximation (IV)

14



15

SK Tables



16


1. Tight-Binding



4. Extensions


17

DFTB MethodQuick review I

Taken from Oliviera, Seifert, Heine, Duarte, J. Braz. Chem. Soc.

20, 1193-1205 (2009)

...open access

Thomas Heine

Helio Duarte

18

DFTB MethodQuick review II

Density Functional Theory (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

at convergence:

Various criteria for convergence possible: • Electron density• Potential• Orbitals• Energy• Combinations of above quantities

Walter Kohn/John A. Pople 1998

19

DFTB MethodQuick review III

Phys. Rev. B, 39, 12520 (1989)

Foulkes + Haydock Ansatz

20

Self-consistent-charge density-functional tight-binding (SCC-DFTB)

M. Elstner et al., Phys. Rev. B 58 7260 (1998)

Approximate density functional theory (DFT) method!

Second order-expansion of DFT energy in terms of reference density r0 and charge fluctuation r1 ( ≅ r r0 + r1) 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)

Lecture II DFTB

21

DFTB and SCC-DFTB methods

where ni and i — occupation and orbital energy ot the ith Kohn-Sham

eigenstate Erep — distance-dependent diatomic repulsive potentials

qA — induced charge on atom A

AB — distance-dependent charge-charge interaction functional; obtained from chemical hardness (IP – EA)

Lecture II DFTB

22

DFTB method Repulsive diatomic potentials replace usual nuclear repulsion

energy

Reference density 0 is constructed from atomic densities

Kohn-Sham eigenstates i are expanded in Slater basis of valence

pseudoatomic orbitals i

The DFTB energy is obtained by solving a generalized DFTB eigenvalue problem with H0 computed by atomic and diatomic DFT

Lecture II DFTB

23

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

A B D

C

A

B

DC

Situation I Situation II

Both approximations are justified by the screening argument: Far away, neutral atoms have no Coulomb contribution.

Approximations in the DFTB Hamiltonian

Lecture II DFTB

SCC-DFTB matrix elements

24

LCAO ansatz of wave function

Rri

i c

secular equations

0

SHc ii

variational

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?

two-center approximationnearest neighbor off-diagonal

elements only

Hamiltonian Overlap

pre-computed parameter• Reference Hamiltonian

H0

• Overlap integral Sμν


25


Rri

i c

secular equations

0

SHc ii

variational


H11

H22

H33

H44


Diagonal term

Orbital energy of neutral free atom(DFT calculation)

1

4

2

3

r12

r23

r14

r34

r13

r24

Hamiltonian Overlap

qH2

1

Charge-charge interaction function

Induced charge


26


Rri

i c

secular equations

0

SHc ii

variational


H11

H22

H33

H41 H44


1

4

2

3

r12

r23

r14

r34

r13

r24

r14

Two-center integral

qSHH2

10


Induced charge

Hamiltonian Overlap

Lookup tabulated H0

and S at distance r

r14


27


Rri

i c

secular equations

0

SHc ii

variational


H11

H22

H33

H41 H43 H44


1

4

2

3

r12

r23

r34

r13

r24

r34

Two-center integral

qSHH2

10


Induced charge

Hamiltonian Overlap

Repeat until building off-diagonal term

Lookup tabulated H0

and S at distance r

28

DFTB parameters

Lecture II DFTB

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)

29

Lecture II DFTB

30

Additional induced-charges term allows for a proper description of charge-transfer phenomena

Induced charge qA on atom A is determined from Mulliken

population analysis

Kohn-Sham eigenenergies are obtained from a generalized, self-consistent SCC-DFTB eigenvalue problem

SCC-DFTB method (I)

Lecture II DFTB

31

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):

Lecture II DFTB

32

SCC-DFTB method (III)

Lecture II DFTB

33

SCC-DFTB method (IV)

Several possible formulations for gab: Mataga-Nishimoto < Klopmann-Ohno < DFTB

Klopmann-Ohno:

Lecture II DFTB

34

Gradient for the DFTB methods

The DFTB force formula

The SCC-DFTB force formula

computational effort: energy calculation 90%

gradient calculation 10%

Lecture II DFTB

35

Spin-polarized DFTB (SDFTB)

Lecture II DFTB

for systems with different and spin densities, we have total density = + magnetization density S = -

2nd-order expansion of DFT energy at (0,0) yields

The Spin-Polarized SCC-DFTB (SDFTB) method is derived from terms 1-9

o(3)

36

where pA l — spin population of shell l on atom A

WA ll’ — spin-population interaction functional

Spin populations pA l and induced charges qA are obtained from Mulliken population analysis


Lecture II DFTB

37


Lecture II DFTB

Kohn-Sham energies are obtained by solving generalized, self-consistent SDFTB eigenvalue problems

where

M,N,K: indexing specific atoms

38

SCC-DFTB w/fractional orbital occupation numbers

12

2tot i i repi

E f E q q

0vi iv

c H S Fractional occupation numbers fi of Kohn-Sham eigenstates replace integer ni

TB-eigenvalue equation

Lecture II DFTB

E

2fi0 1 2

m

Finite temperature approach (Mermin free energy EMermin)

1

exp / 1ii B e

fk T

2 ln 1 ln 1e B i i i ii

S k f f f f

Te: electronic temperatureSe: electronic entropy

0 1

2N

repi i i i

i

EH H SF f c c q q

SR R R R

0 1if

Atomic force

M. Weinert, J. W. Davenport, Phys. Rev. B 45, 13709 (1992)

EMermin = Etot - TeSe

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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

f

x

x

x

electHF pulay charge

i ii i i i

i i i

F F F F

ff f

x x x

Correction term arising from Fermi distribution function cancels out

Lecture II DFTB

0 1 2 3 4 50

20

40

60

80

40

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 problemSCC 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 4

0

20

40

60

80

0 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

DFTBLecture II

41


1. Tight-Binding



4. Extensions


Bond breakingLecture II

SCC-DFTB and SDFTB Dissociation of H2+





(correct)

(wrong)


SCC-DFTB and SDFTB Dissociation of H2

M. Lundberg, Y. Nishimoto, SI, Int. J. Quant. Chem. 112, 1701 (2012)

46


1. Tight-Binding



4. Extensions


47

Extensions

M=Sc, Ti, Fe, Co, NiX=M,CHON

d-elements with X partners:Geometries very good -“ballpark” energies only

Recommendation:Use ONIOM(QM:QM),esp. ONIOM(DFT:DFTB)

Lecture II

48/25

New Confining Potentials

Wa

Conventional potential

r0

Woods-Saxon potential

k

R

rrV

0

)(

R0 = 2.7, k=2

)}(exp{1)(

0rra

WrV

r0 = 3.0, a = 3.0, W = 3.0

Typically, electron density contracts under covalent bond formation.

In standard ab initio methods, this problem can be remedied by including more basis functions.

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 density1s

Henryk Witek

Electronic Parameters DFTB Parameterization

48

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

49

Band structure for Se (FCC)

Brillouin zone50


Particle swarm optimization (PSO)


51

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.

•

••

••

•

• •••

•

•••

••

• •••

• ••••••••

•••••••••••

particle

1)

4)

2)

3)

Particle Swarm Optimization DFTB Parameterization

52

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

orbitala [2.5, 3.5]W [0.1, 0.5]r0 [3.5, 6.5]

densitya [2.5, 3.5]W [0.5, 2.0]r0 [6.0, 10.0]

Particle Swarm Optimization

53

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

54

Electronic Parameters

55

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

No POTCAR file for Z ≥ 84 in VASP

a

55

56

Band structure fitting for FCC crystal structures



• space group No. 225


a


56

57

Band structure fitting for SCL crystal structures





a


57

58

Band structure fitting for HCP crystal structures




• 2 lattice constants (a, c)

c

a


58

59

Band structure fitting for Diamond crystal structures





a


59

DFTB ParameterizationTransferability of optimum parameter sets for different structures

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.

60

Influence of virtual orbital energy (3d) to Al (fcc) band structure

OPT

The bands of upper part are shifted up constantly as orb (3d) becomes larger

61

Influence of W(orb) to Al (fcc) band structure

OPT

The bands of upper part go lower as W(orb) becomes larger 62

Influence of a(orb) to Al (fcc) band structure

OPT

Too small a(orb) gives the worse band structure 63

Influence of r(orb) to Al (fcc) band structure

OPT

r(orb) strongly influences DFTB band structure 64

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)

In particular for main group elements, there seems to be a correlation between r(orb) and atomic diameter.

65

DFTB ParameterizationElectronic 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• CuC

l• ZnS• GaA

s…

Others

• Wurtzite (BeO, AlO, ZnO, GaN, …)

• Hexagonal (BN, WC)• Rhombohedral (ABCABC

stacking sequence, BN)

more than 100 pairs tested

66

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)

Further improvement can be performed for specific purpose but this preliminary sets will work as good starting points 67

68

Extensions

Analytical Hessian for DFTB and SCC-DFTBH. A. Witek, S. Irle, K. Morokuma, J. Chem. Phys. 121, 5163 (2005)

Lecture II

69

Extensions

To determine response of molecular orbitals to nuclear perturbation, one has to solve a set of iterative coupled-perturbed SCC-DFTB equations

Coupled Perturbed SCC-DFTB Equations

Lecture II

70

Frequency calculationsperformance test

all-transpolyenesCnHn+2

Performance of DFTBLecture II

71

How to treat interlayer interactions in graphite cheaply?

Addition of empirical London dispersion term!

• Ahlrichs et al., Chem. Phys. 19, 119 (1977): HFD (Hartree-Fock Dispersion), Ci parameters from expt.

• Mooij et al., J. Phys. Chem. A 103, 9872 (1999): PES with London dispersion, C6 parameters from MP2 calculations

• Elstner et al., J. Chem. Phys. 114, 5149 (2001): SCC-DFTB-D (dispersion-augmented SCC-DFTB), C6 parameters via Halgren,J. Am. Chem. Soc. 114, 7872 (1992) from expt.

• Yang et al., J. Chem. Phys. 116, 515 (2002): DFT+vdW (DFT including van der Waals interactions), C6 parameters from expt.

• Grimme, J. Comput. Chem. 25, 1463 (2004): DFT-D, C6 parameters from expt.

• … several more plus many reviews and benchmarks

Etot = EQM - f (R) C6

/R6 (+1/R8

etc. terms)

f (R): damping function Fritz London1/R6: 1922

ExtensionsLecture II

72

Extensions

London Dispersion

Lecture II

73

DFTB+DispersionElstner et al., JCP 114, 5149 (2001)

Lecture IX Extensions

74

E ~ 1/R6

damping f(R) = [1-exp(-3(rij/Rij,vdW)7)]3

Rij,vdW = 3.8 Å (for 1st row), 4.8 A (for 2nd row)

DFTB+DispersionElstner et al., JCP 114, 5149 (2001)

DFTB-D choice of C6 parameters:

- generally hybridization dependent (i.e. not simply atomic values)

- use “empirical” values for parameters to match BSSE-corrected MP2 interaction energies

R [Å]

E [ha]


75

Application to GraphiteExtensions

DFTB-D

Lecture II

Principles of GRRM

ADDAnharmonic downward distortion

S. Maeda and K. Ohno, J. Phys. Chem. A, 2005, 109, 5742.

76

UDC

DDC

One of the most powerful and reliable methods to find reaction pathways.

Extensions

GRRM with DFTB

Lecture II

Example: Formaldehyde (RB3LYP/6-31G(d))

4EQs

9TSs

3DDCs

9UDCs

DDC at DFTB

DFTB B3LYP Match

EQ 4 4 4

TS 9 9 6 ~ 8

DDC 4 3 3

UDC 7 9 7: Reproduced by DFTB

: Found

TS at DFTB

77


78


1. Tight-Binding



4. Extensions


79

Performance for small organic molecules (mean absolut deviations)• Reaction energies: ~ 5 kcal/mole

• Bond-lenghts: ~ 0.014 A°

• Bond angles: ~ 2°

• Vib. Frequencies: ~6-7 %

Performance

SCC-DFTB: general comparison with experiment

Lecture II

80

Performance

Accuracy of DFTB Geometries and Energies for Fullerene Isomers

Fullerene Isomers Geometries and Energies vs B3LYP/6-31G(d)G. Zheng, SI, M. Elstner, K. Morokuma, Chem. Phys. Lett. 412, 210 (2005)

Geometries

Energetics

• 102 Fullerene Isomers• small cage non-IPR C20-C36 (35), large cage IPR C70-C86 (67)

R2: Energy linear regression between E(Method) and E(B3LYP)

Lecture II

What about non-cage carbon cluster structures? Some C28 isomers as example (from Scuseria, CPL 301, 98 (1999))

PerformanceLecture II

Performance

What about non-cage carbon cluster structures?Some C28 isomers as example TACC’04 Symp. Proceedings

AM1 and PM3 are performing very bad!DFTB includes effects of polarization functions through parameterization

Wrong minimum structures!

Lecture II

83

Performance

Accuracy of DFTB GeometriesTable 3. The relative energies (in eV) of 11 different isomeric singlet structures of C28 calculated with B3LYP/6-31G(d), and

DFTB.

Structure a B3LYP/6-31G(d) DFTBB3LYP/6-31G(d)

// DFTBbuckyD2 0.00 0.00 0.00

ring 3.32 8.10 3.43

c24-6 3.17 3.56 3.66

2+2r14 5.08 9.66 5.22

2+2r16 6.01 10.25 6.13

c20-6o 5.41 5.52 5.96

c20-6 m 5.57 5.62 6.09

2 + 4 7.97 10.28 8.52

central7 5.86 6.07 6.47

8 + 8 7.43 9.43 7.41

4 + 4 9.91 14.27 10.20

R2 b 0.75 0.99

a Structures illustrated below, with the labels taken from “Portmann, S.; Galbraith, J. M.; Schaefer, H. F.; Scuseria, G. E.; Lüthi, H. P. Chem. Phys. Lett. 1999, 301, 98-104.”b Squared correlation coefficients R2 in the linear regression analysis with B3LYP/6-31G(d) energies.

Lecture II

84

Performance

Accuracy of DFTB frequencies

84

calculated frequencies scaled frequenciesTesting set of 66 molecules, 1304 distinct vibrational modes

Lecture II

85

Performance

Accuracy of DFTB frequenciesTesting set of 66 molecules, 1304 distinct vibrational modes

mean absolute deviation

standard deviation

maximal absolute deviation

scalingfactor

SCC-DFTB 56 cm-1 82 cm-1 529 cm-1 0.9933

DFTB 60 cm-1 87 cm-1 536 cm-1 0.9917

AM1 69 cm-1 95 cm-1 670 cm-1 0.9566

PM3 74 cm-1 102 cm-1 918 cm-1 0.9762

HF/cc-pVDZ 30 cm-1 49 cm-1 348 cm-1 0.9102

BLYP/cc-pVDZ 34 cm-1 47 cm-1 235 cm-1 1.0043

B3LYP/cc-pVDZ 29 cm-1 42 cm-1 246 cm-1 0.9704

Lecture II

triphenylene

Energy, frequencies, andRaman and IR intensitiescalculations

DFT BLYP/cc-pVDZLinux 2.4GHz machine

32 hours

SCC-DFTBLinux 333MHz machine

24 seconds

Performance

Efficiency of DFTB frequencies

Lecture II

Performance

Improvement of DFTB ERep

H. A. Witek, et al, J. Theor. Comp. Chem. 4, 639 (2005) and others

Lecture II

88

H. A. Witek, SI, G. Zheng, W. A. de Jong, K. Morokuma, J. Chem. Phys. 125, 214706 (2005)

Performance

C28 D2

Harmonic IR and Raman Spectra of C28

Lecture II

89


Performance


C60 D2h~Ih

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90


Performance


C70 D5h

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91

Performance

SCC-DFTB: Systematic comparison with other methods by Walter Thiel et al.

Thiel and coworkers, J. Phys. Chem. A 111, 5751 (2007)

(reference: exptl. DH0)

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92

Performance

Thiel and coworkers, J. Phys. Chem. A 111, 5751 (2007)

SCC-DFTB: Systematic comparison with other methods by Walter Thiel et al.

Lecture II

93

Performance

But: NCC- and SCC-DFTB for radicals: own comparison with B3LYP/6-311G**

Lecture II

94

Performance

NCC- and SCC-DFTB for radicals: own comparison

NCC-DFTB

B3LYP/6-311G(d,p)

SCC-DFTB

B3LYP/6-311G(d,p)

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95

A B C

DFT:PW91[1] -6.24 -5.63 -1.82

SCC-DFTB[2] -5.17 -4.68 -1.86

Adhesion energies (eV/atom)

A B C

PW91: An ultrasoft pseudopotential with a plane-wave cutoff of 290 eV for the single metal and the projector augmented wave method with a plane-wave cutoff of 400 eV for the metal cluster

Fe-Fe and Fe-C DFTB parameters from: G. Zheng et al., J. Chem. Theor. Comput. 3, 1349 (2007)

[1] Phys. Rev. B 75, 115419 (2007) [2] Fermi broadening=0.13 eV

H10C60Fe H10C60Fe H10C60Fe55

Fe55 icosahedron

H-terminated (5,5) armchair SWNT + Fe atom/cluster

PerformanceLecture II

96

Acknowledgements

• Marcus Elstner• Jan Knaup• Alexey Krashenninikov• Yasuhito Ohta• Thomas Heine• Keiji Morokuma• Marcus Lundberg• Yoshio Nishimoto• Some others

Acknowledgements

Lecture II

Density-Functional Tight-Binding (DFTB) as fast approximate DFT method - An introduction

Technology

tightbinding resources

tight binding tb approaches

dftb seifert

empirical tightbinding

construction of tight

dftb porezag

original dftb method

sccdftb elstner