Introduction to DFTB+

Martin Persson

Accelrys, Cambridge

• DFTB– Why DFTB?– Basic theory DFTB– Performance

• DFTB+ in Materials Studio– Energy, Geometry, Dynamics, Parameterization– Parameterization• Basic theory• Setting up a parameterization

Outline

Why DFTB+

• DFT codes are good for small systems • Nano structures and bio molecules are often too large for

DFT but their electronic properties are still of interest– hence quantum mechanical description is needed.

• Classical force field based codes can handle large systems but are missing the QM part

• Empirical TB has been applied to systems up to a few million atoms– No charge self consistency– Limited transferability– Using simplified energetic expressions

QM vs. CM

• DFTB merges the reliability of DFT with the computational efficiency of TB – Parameters are based on an atomic basis– The parameters can be made transferable– Charge self consistent– Describes both electronic as well as energetic

properties– Can handle thousands of atoms

This is where DFTB+ comes in

Examples of what can be done with DFTB+

Diamond nucleation Novel SiCN ceramics Si cluster growth

Magnetic Fe clusters WS2 nanotubes

Basic DFTB Theory

• DFTB– Pseudo atomic orbital basis– Non SCC Hamiltonian elements are parameterized– 2nd order charge self consistent theory– Charges are treated as Mulliken charges– Short range potential is used to correct the

energetics – Hamiltonian matrix is sparse and can partly be

treated with O(N) methods

DFTB theory in short

• Minimal basis set • Pseudo atomic orbitals– Slater orbitals– Spherical harmonics

DFTB basis set

vv

vv

v

mlnml

rnlnv r

rerar,,,

Pseudo atomic orbitals

S P1 P2 P3

D5D4D3D2D1

Silicon sp3d5 orbitals

For Silicon the d-orbitals are un-occupied but needed to properly model the conduction band.

Hamiltonian elements

otherwise

if if

0

atom free

0 BAVVTH BBA

A

• Diagonal elements use free atom energies

• Two centre integrals• Tabulated values

1. Expand the Kohn-Sham total energy expression of DFT to 2nd order in terms of electron and magnetization density fluctuations

2. Represent the Hamiltonian elements in a minimal basis of pseudo-atomic orbitals

3. Express the charge density in terms of Mulliken charges4. Expand the magnetization density in terms of non-overlapping

spherically symmetric functions5. Replace the remaining terms with a short range repulsive energy

DFT DFTB

N

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1. Expand the Kohn-Sham total energy expression of DFT to 2nd order in terms of electron and magnetization density fluctuations

2. Represent the Hamiltonian elements in a minimal basis of pseudo-atomic orbitals

3. Express the charge density in terms of Mulliken charges4. Expand the magnetization density in terms of non-overlapping

spherically symmetric functions5. Replace the remaining terms with a short range repulsive energy

DFT DFTB

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1. Expand the Kohn-Sham total energy expression of DFT to 2nd order in terms of electron and magnetization density fluctuations

2. Represent the Hamiltonian elements in a minimal basis of pseudo-atomic orbitals

3. Express the charge density in terms of Mulliken charges4. Expand the magnetization density in terms of non-overlapping

spherically symmetric functions5. Replace the remaining terms with a short range repulsive energy

DFT DFTB

rrddrrnnrdnnV

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nEWpp

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xc

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1. Expand the Kohn-Sham total energy expression of DFT to 2nd order in terms of electron and magnetization density fluctuations

2. Represent the Hamiltonian elements in a minimal basis of pseudo-atomic orbitals

3. Express the charge density in terms of Mulliken charges4. Expand the magnetization density in terms of non-overlapping

spherically symmetric functions5. Replace the remaining terms with a short range repulsive energy

DFT DFTB

rep

N

l lllll

occ

iiiitot

EWpp

qq

HccnE

21

21

,

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DFTB+ Performance

Performance figures

N2.9

N1.5

•10x10 CNT• 32 atoms/unitcell• Run on single core• Intel(R) Xeon(TM) CPU 3.00GHz

•Small systems (<300 atoms) O(N) processes dominate•Large systems (>300) O(n) eigenvalue solver dominates•Around 100 times faster then normal DFT

DFTB+ in Materials Studio 6.0

• First official release that includes the DFTB+ module

• Supported tasks– Energy– Geometry optimization– Dynamics– Parameterization

• Also support– Dispersion correction– Spin unrestricted calculations

DFTB+ in Materials Studio 6.0

• Slater-Koster libraries instead of DFT Functionals– CH, CHNO and SiGeH

• What if I don’t have the needed library?– Download academic

libraries at www.dftb.org• mio, C-H-N-O-S-P• pbc, Si-F-O-N-H|Fe• matsci, various parameters

– Make your own

Starting a DFTB+ job

• Need to register to get access.• The downloaded parameters will

contain many different Slater Koster files

Downloading parameters

•To be used in MS-DFTB+ the parameters need to be packed up in a .skflib format.• The .skflib file is just a tagged concatenation of the different files

• [Begin section] [End section], surrounds list of all files• [Begin file <filename>] [End file <filename>], surrounds content of file.

• Will prevent accidental mixing of files between libraries and makes handling easier

• Band structure• Density of states• Electron density• Fermi surface• Orbitals• Slater-Koster parameters• Dynamics analysis is done

using the Forcite analysis tools

DFTB+ Analysis

Materials Studio 6.0 Parameterization tool

• DFTB+ depends on parameters– Hamiltonian and overlap integrals– Hubbard terms (orbital resolved)– Spin constants– Wave function coefficients– Short range repulsive potential

The DFTB+ Parameterization Tool

The DFTB+ parameterization tool enables you to make your own parameterizations. It calculates all of the needed parameters. The result is packed up in a single file (.skflib)

Repulsive fitting

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21

21

,

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pairs

jiijijtype

totbareDFTB

totDFTrep rUEEE )()(,

The remaining terms, Erep, will be described using fitted repulsive pair potentials.

The pair potentials are fitted against a basis of cutoff polynomials

otherwiserr if

0)(

)( cutoffn

cutoffn

rrrf

Pair potentials

• Short range pair potentials are fitted against small molecules or solids• Path generators

– Stretch, Perturb, Scale, Trajectory• Fitting against Energy and optionally forces• Use of spin unrestricted calculations• Steps, weights and width are set under Details...

Systems

Bond order fitting

Use weight distributions to combine several bond orders into a single potential fit

• C-H.txt- Job summary• Best fit (C-H.skflib)

returned in the base folder• Fits for alternative cutoff

factors are returned in the Alternatives folder

Parameterization job results

Evaluating the result

benzene-------DMol3 C3-C2 = 1.39838 C3-H9 = 1.09097 DFTB+ C3-C2 = 1.41171 C3-H9 = 1.10386 Diff C3-C2 = 0.01333 C3-H9 = 0.01289

DMol3 C2-C7-C6 = 120.00000 H12-C7-C6 = 120.00000 DFTB+ C2-C7-C6 = 119.99783 H12-C7-C6 = 120.00930 Diff C2-C7-C6 = -0.00217 H12-C7-C6 = 0.00930

Atomization Diff = -111.42032==============================================

ethene------DMol3 C2-C1 = 1.33543 C2-H5 = 1.09169 DFTB+ C2-C1 = 1.33114 C2-H5 = 1.09898 Diff C2-C1 = -0.00429 C2-H5 = 0.00729

DMol3 C1-C2-H6 = 121.65149 H4-C1-H3 = 116.69702 DFTB+ C1-C2-H6 = 121.55765 H4-C1-H3 = 116.88453 Diff C1-C2-H6 = -0.09384 H4-C1-H3 = 0.18751

Atomization Diff = -48.44673==============================================

Bond Error Statistics:C-C = 8.81072e-03C-H = 1.00915e-02=================Total Average = 9.45112e-03

Angle Error Statistics:HCH = 1.87511e-01CCC = 2.16738e-03HCC = 5.15662e-02=================Total Average = 7.32028e-02

1. Initial evaluation against small set of structures

2. Final evaluation against larger set of structures

3. Validation against larger structures

Materials Studio supplies a MS Perl script which compares geometry and atomization energy for structures.

• sp3d5 basis• LDA(PWC)• Fitted against

– Si, Ge and SiGe solids– Si2H6, Si2H4

– Ge2H6, Ge2H4

– SiGeH6, SiGeH4

– SiH4, GeH4 and H2

• Tested against:– Solids– Nanowires– Nanoclusters– Si vacancy

SiGeH

NNf ENNEE 1

1

Si vacancy Formation energy

Ef(eV)

DFTB+ 2.6

DMol3 2.7

• sp3 basis• GGA(PBE)• Tested against a large set

(~60) of organic molecules• Also, validated against a

smaller set of larger molecules

• Good diamond cell parameter, 3.590 (3.544) Å

CHNO

Bond type Average difference (Å)C-C 0.0108

C-N 0.0131

C-O 0.0105

C-H 0.0081

N-N 0.0070

N-O 0.0123

N-H 0.0087

Average bond difference: 0.0096 Å

Average angle difference: 1.16 degrees

Accuracy is comparative to that of the Mio library.

• Successfully tested for:– CNT– C60

– Caffeine– Glucose– Porphine– N-Acetylneuraminic

acid

CHNO: Larger molecules

Bond Diff (Å)C-C 0.0095C-N 0.0075C-O 0.0078C-H 0.0028

Bond Diff (Å)C-C 0.005

Bond Diff (Å)C-C 0.0148C-N 0.0118C-O 0.0100C-H 0.0114N-H 0.0127O-H 0.0019

CNT-6x6

Caffeine

N-AA

Thanks for your attention

Other contributors:Paddy Bennett (Cambridge, Accelrys)

Bálint Aradi (Bremen, CCMS) Zoltan Bodrog (Bremen, CCMS)

• The Kohn-Sham equation is solved for a single atom.

• Using an added extra confining potential to better model molecules and solids

Generating the orbitals

)()()(ˆ2

0

rrrrrVT at

eff

1. Expand the Kohn-Sham total energy expression of DFT to 2nd order in terms of electron and magnetization density fluctuations

2. Represent the Hamiltonian elements in a minimal basis of pseudo-atomic orbitals

3. Express the charge density in terms of Mulliken charges4. Expand the magnetization density in terms of non-overlapping

spherically symmetric functions5. Replace the remaining terms with a short range repulsive energy

DFT DFTB

N

xc

occ

iiiitot

RRZZ

nE

rrrdrnvnE

21,

)(21

2,

32

1. Expand the Kohn-Sham total energy expression of DFT to 2nd order in terms of electron and magnetization density fluctuations

2. Represent the Hamiltonian elements in a minimal basis of pseudo-atomic orbitals

3. Express the charge density in terms of Mulliken charges4. Expand the magnetization density in terms of non-overlapping

spherically symmetric functions5. Replace the remaining terms with a short range repulsive energy

DFT DFTB

rrddrrnnrdnnV

RRZZ

nErrddE

rrddnnnnE

rr

nVrrrdnvnE

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1. Expand the Kohn-Sham total energy expression of DFT to 2nd order in terms of electron and magnetization density fluctuations

2. Represent the Hamiltonian elements in a minimal basis of pseudo-atomic orbitals

3. Express the charge density in terms of Mulliken charges4. Expand the magnetization density in terms of non-overlapping

spherically symmetric functions5. Replace the remaining terms with a short range repulsive energy

DFT DFTB

ii HH 00 ˆ

ii c

1. Expand the Kohn-Sham total energy expression of DFT to 2nd order in terms of electron and magnetization density fluctuations

2. Represent the Hamiltonian elements in a minimal basis of pseudo-atomic orbitals

3. Express the charge density in terms of Mulliken charges4. Expand the magnetization density in terms of non-overlapping

spherically symmetric functions5. Replace the remaining terms with a short range repulsive energy

DFT DFTB

rrrrnnrnnV

RRZZ

nErrE

qq

HccnE

xc

N

xcnxc

occ

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033

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2

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210,|

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a

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1. Expand the Kohn-Sham total energy expression of DFT to 2nd order in terms of electron and magnetization density fluctuations

2. Represent the Hamiltonian elements in a minimal basis of pseudo-atomic orbitals

3. Express the charge density in terms of Mulliken charges4. Expand the magnetization density in terms of non-overlapping

spherically symmetric functions5. Replace the remaining terms with a short range repulsive energy

DFT DFTB

rrddrrnnrdnnV

RRZZ

nEWpp

qq

HccnE

xc

N

xc

N

l lllll

occ

iiiitot

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0

,

0

210,

210,

21

21

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1. Expand the Kohn-Sham total energy expression of DFT to 2nd order in terms of electron and magnetization density fluctuations

2. Represent the Hamiltonian elements in a minimal basis of pseudo-atomic orbitals

3. Express the charge density in terms of Mulliken charges4. Expand the magnetization density in terms of non-overlapping

spherically symmetric functions5. Replace the remaining terms with a short range repulsive energy

DFT DFTB

rep

N

l lllll

occ

iiiitot

EWpp

qq

HccnE

21

21

,

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• Most of DFTB+ is running with O(N) routines • Two exceptions – DFTB+ SCC

• Ewald-summation, O(N2)– DFTB+ eigenvalue solvers

• LAPACK solvers, O(N3)

• Small systems (<300 atoms), the O(N) processes dominate

• Large systems (>300), the eigenvalue solver dominates

Calculation time vs. structure size

Performance figures

N2.9

N1.5

•10x10 CNT• 32 atoms/unitcell• Run on single core• Intel(R) Xeon(TM) CPU 3.00GHz

•Small systems (<300 atoms) O(N) processes dominate•Large systems (>300) eigenvalue solver dominates

#cpu Speedup Efficiency

1 1.0

2 0.87

3 0.80

4 0.72

OpenMP

• DFTB+ is significantly faster than a normal DFT code• Depending on what DFT code we compare to its a factor 102-103 faster• DFTB+ compared to DMol3 is a factor of 30-80 faster

DMol3 vs. DFTB+

Atoms TimeDFTB+(s) TimeDMol3(s) TimeDMol3/TimeDFTB+

32 4 233 58

64 8 632 79

96 17 872 51

128 26 1092 42

160 46 1501 33

Starting a DFTB+ job: Setup

• Available tasks• Energy• Geometry optimization• Dynamics• Parameterization

• Dispersion correction• Spin unrestricted

The parameterization dialogs are accessed through the More... Button.

• Select Slater-Koster library– CH, CHNO and SiGeH– Use Browse... to access local

library• What if I don’t have the

needed library?– Download academic libraries at

www.dftb.org• mio, C-H-N-O-S-P• pbc, Si-F-O-N-H|Fe• matsci, various parameters

– Make your own

Starting a DFTB+ job: Electronic

• Select any properties that should be calculated– Band structure– DOS– Electron density– Orbitals– Population analysis

• Properties will be calculated at the end of the job

Starting a DFTB+ job: Properties

• Select server or run on local machine

• DFTB+ support OpenMP but not MPI

• On a cluster it will run on the cores available to it on the first node

• Parameterization is always run as a serial job

Starting a DFTB+ job: Job Control

• The DFTB+ calculations are run by Materials Studio as an energy server

• Geometry optimization and Dynamics jobs are controlled by the same code that is used during a Forcite job

During a DFTB+ job

• <>.xsd– Final structure

• <>.xtd (dynamics)– Dynamics trajectory

• <>.txt – Compilation of the results

• <>.dftb– The last output from DFTB+

• <>.skflib (parameterization)– Slater-Koster library

DFTB+ Result files

• *.tag– Final output data

• *.cube– Density and orbital

data• *.bands– Band structure data

Visible files Hidden files

Zn compounds using DFTB+

• Zn-X (X = H, C, N, O, S, Zn)• Can be downloaded at www.DFTB.org (znorg-0-1)• Reference systems during fitting

– ZnH2, Zn(CH3)2, Zn(NH3)2, Zn(SH)2

– fcc-Zn, zb-ZnO• Applied to:

– Zinc solids, Zn, ZnO, ZnS– Surfaces, ZnO– Nanowires and Nanoribbons, ZnO– Small species interaction with ZnO surface (H, CO2 and

NH3)– Zn in biological systems

Working with Zn containing compounds

N. H. Moreira, J. Chem. Theory Comput. 2009, 5 , 605

Zn Solids

Method Ecoh a(Å) b(Å) B0(GPa)

w-ZnO DFTB+ 9.77 3.28 5.25 161

PBE 8.08 3.30 5.34 124

EXP 7.52 3.25 5.20 208

zb-ZnS DFTB+ 7.93 5.43 - 44.2

LDA 7.22 5.35 - 82

EXP 6.33 5.40 - 76.9

W-ZnO DFTB+

W-ZnO PBE

• Reasonable solid state properties

N. H. Moreira, J. Chem. Theory Comput. 2009, 5 , 605

ZnO Surface stability

F. Claeyssens J. Mat. Chem. 2005, 15 139

N. H. Moreira, J. Chem. Theory Comput. 2009, 5 , 605

•Predicts correct order and magnitude for the cleavage energy•Bond and angle deviation ~1-2%

DFTB+

DFT

ZnO nanowires

•Good geometries and electronic structure•Excellent agreement with DFT results• Surface Zn atoms move inwards

N. H. Moreira, J. Chem. Theory Comput. 2009, 5 , 605

CO2• Bond difference 1-2%• Binding too strong

~0.5 eV/CO2

• Turn over point for monolayer well described

NH3

• Overall good agreement with experiments and DFT calculations

Small molecule surface interaction

ZnO (1010)-CO2 ZnO (1010)-NH3

2/)( 0110 nEEE ZnOTabs

N. H. Moreira, J. Chem. Theory Comput. 2009, 5 , 605

• Choose functional (LDA(PWC) or GGA(PBE))• The electronic fitting can be done in two modes

– Potential mode, confinement potential for wave function– Density mode, confinement potentials for wave function and

electron density• Each element will have its own settings

– What basis to use– Electron configuration– Confinement potential(s)

Electronic settings

• Each fitting is done using different polynomial orders

• Fittings are done for a set of cutoff radius scale factors

Polynomial fitting setup

otherwiserr if

0)(

)( cutoffn

cutoffn

rrrf

Possible future extensions to DFTB+

• Optical Properties– LR-TD-DFTB

• Electronic transport – NEG-DFTB

• QM/MM• Vibrational modes

DFTB+ features outside of Material Studio

Please let us know what extensions and enhancements you would like to see for DFTB+ in the future.