Introduction to DFTB+

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Dec 13, 2013 (3 years and 8 months ago)

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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


2
nd

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














v
v
v
v
v
m
l
n
m
l
r
n
l
n
v
r
r
e
r
a
r
,
,
,




Pseudo atomic
orbitals

S

P
1

P
2

P
3

D
5

D
4

D
3

D
2

D
1

Silicon sp
3
d
5

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
B
A
V
V
T
H
B
B
A
A










Diagonal elements use

free
atom energies


Two centre integrals


Tabulated values

1.
Expand the Kohn
-
Sham total energy expression of DFT to 2
nd

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

charges

4.
Expand the magnetization density in terms of non
-
overlapping
spherically symmetric functions

5.
Replace the remaining terms with a short range repulsive energy

DFT


DFTB































N
xc
occ
i
i
i
i
tot
R
R
Z
Z
n
E
r
r
r
d
r
n
v
n
E












2
1
,
)
(
2
1
2
,
3
2
1.
Expand the Kohn
-
Sham total energy expression of DFT to 2
nd

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

charges

4.
Expand the magnetization density in terms of non
-
overlapping
spherically symmetric functions

5.
Replace the remaining terms with a short range repulsive energy

DFT


DFTB







r
rd
d
r
r
n
n
r
d
n
n
V
R
R
Z
Z
n
E
r
rd
d
E
r
rd
d
n
n
n
n
E
r
r
n
V
r
r
r
d
n
v
n
E
xc
N
xc
n
xc
n
xc
occ
i
i
xc
i
i
tot


























































3
3
0
0
3
0
0
0
3
3
0
,
2
3
3
0
,
2
,
0
3
2
2
1
0
,
2
1
0
,
|
2
1
|
1
2
1
0
,
2
1
2
0
0





















n
n
n



0




0
1.
Expand the Kohn
-
Sham total energy expression of DFT to 2
nd

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

charges

4.
Expand the magnetization density in terms of non
-
overlapping
spherically symmetric functions

5.
Replace the remaining terms with a short range repulsive energy

DFT


DFTB





r
rd
d
r
r
n
n
r
d
n
n
V
R
R
Z
Z
n
E
W
p
p
q
q
H
c
c
n
E
xc
N
xc
N
l
l
l
l
l
l
occ
i
i
i
i
tot







































3
3
0
0
3
0
0
0
,
0
2
1
0
,
2
1
0
,
2
1
2
1


























1.
Expand the Kohn
-
Sham total energy expression of DFT to 2
nd

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

charges

4.
Expand the magnetization density in terms of non
-
overlapping
spherically symmetric functions

5.
Replace the remaining terms with a short range repulsive energy

DFT


DFTB

rep
N
l
l
l
l
l
l
occ
i
i
i
i
tot
E
W
p
p
q
q
H
c
c
n
E

















































2
1
2
1
,
0
DFTB+ Performance

Performance figures

N
2.9

N
1.5


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

rep
N
l
l
l
l
l
l
occ
i
i
i
i
tot
E
W
p
p
q
q
H
c
c
n
E

















































2
1
2
1
,
0





pairs
j
i
ij
ij
type
tot
bare
DFTB
tot
DFT
rep
r
U
E
E
E
)
(
)
(
,
The remaining terms,
E
rep
, will be
described using fitted repulsive pair
potentials.

The pair potentials are fitted against a
basis of
cutoff

polynomials







otherwise
r
r

if
0
)
(
)
(
cutoff
n
cutoff
n
r
r
r
f
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
-
03

C
-
H = 1.00915e
-
02

=================

Total Average = 9.45112e
-
03


Angle Error Statistics:

HCH = 1.87511e
-
01

CCC = 2.16738e
-
03

HCC = 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


Si
2
H
6
, Si
2
H
4


Ge
2
H
6
, Ge
2
H
4


SiGeH
6
, SiGeH
4


SiH
4
, GeH
4

and H
2


Tested against:


Solids


Nanowires


Nanoclusters


Si vacancy

SiGeH


N
N
f
E
N
N
E
E
1
1




Si vacancy Formation energy

E
f
(
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


C
60


Caffeine


Glucose


Porphine


N
-
Acetylneuraminic

acid


CHNO: Larger molecules

Bond

Diff (
Å
)

C
-
C

0.0095

C
-
N

0.0075

C
-
O

0.0078

C
-
H

0.0028

Bond

Diff (
Å
)

C
-
C

0.005

Bond

Diff (
Å
)

C
-
C

0.0148

C
-
N

0.0118

C
-
O

0.0100

C
-
H

0.0114

N
-
H

0.0127

O
-
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

)
(
)
(
)
(
ˆ
2
0
r
r
r
r
r
V
T
at
eff

























1.
Expand the Kohn
-
Sham total energy expression of DFT to 2
nd

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

charges

4.
Expand the magnetization density in terms of non
-
overlapping
spherically symmetric functions

5.
Replace the remaining terms with a short range repulsive energy

DFT


DFTB































N
xc
occ
i
i
i
i
tot
R
R
Z
Z
n
E
r
r
r
d
r
n
v
n
E












2
1
,
)
(
2
1
2
,
3
2
1.
Expand the Kohn
-
Sham total energy expression of DFT to 2
nd

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

charges

4.
Expand the magnetization density in terms of non
-
overlapping
spherically symmetric functions

5.
Replace the remaining terms with a short range repulsive energy

DFT


DFTB







r
rd
d
r
r
n
n
r
d
n
n
V
R
R
Z
Z
n
E
r
rd
d
E
r
rd
d
n
n
n
n
E
r
r
n
V
r
r
r
d
n
v
n
E
xc
N
xc
n
xc
n
xc
occ
i
i
xc
i
i
tot


























































3
3
0
0
3
0
0
0
3
3
0
,
2
3
3
0
,
2
,
0
3
2
2
1
0
,
2
1
0
,
|
2
1
|
1
2
1
0
,
2
1
2
0
0





















n
n
n



0




0




r
rd
d
r
r
n
n
r
d
n
n
V
R
R
Z
Z
n
E
r
rd
d
E
r
rd
d
n
n
n
n
E
r
r
H
c
c
n
E
xc
N
xc
n
xc
n
xc
occ
i
i
i
i
tot














































3
3
0
0
3
0
0
0
3
3
0
,
2
3
3
0
,
2
,
0
*
2
1
0
,
2
1
0
,
|
2
1
|
1
2
1
0
0
























1.
Expand the Kohn
-
Sham total energy expression of DFT to 2
nd

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

charges

4.
Expand the magnetization density in terms of non
-
overlapping
spherically symmetric functions

5.
Replace the remaining terms with a short range repulsive energy

DFT


DFTB









i
i
H
H
0
0
ˆ









i
i
c
1.
Expand the Kohn
-
Sham total energy expression of DFT to 2
nd

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

charges

4.
Expand the magnetization density in terms of non
-
overlapping
spherically symmetric functions

5.
Replace the remaining terms with a short range repulsive energy

DFT


DFTB





r
r
r
r
n
n
r
n
n
V
R
R
Z
Z
n
E
r
r
E
q
q
H
c
c
n
E
xc
N
xc
n
xc
occ
i
i
i
i
tot



































3
3
0
0
3
0
0
0
3
3
0
,
2
,
0
2
1
0
,
2
1
0
,
|
2
1
2
1
0






























β
α

β
α

interacion

Coulomb
U
Hubbard











a
n
n



0



q
q
q



1.
Expand the Kohn
-
Sham total energy expression of DFT to 2
nd

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

charges

4.
Expand the magnetization density in terms of non
-
overlapping
spherically symmetric functions

5.
Replace the remaining terms with a short range repulsive energy

DFT


DFTB





r
rd
d
r
r
n
n
r
d
n
n
V
R
R
Z
Z
n
E
W
p
p
q
q
H
c
c
n
E
xc
N
xc
N
l
l
l
l
l
l
occ
i
i
i
i
tot







































3
3
0
0
3
0
0
0
,
0
2
1
0
,
2
1
0
,
2
1
2
1






























l
l
al
p
p
p


n
interactio
spin



l
l
W

1.
Expand the Kohn
-
Sham total energy expression of DFT to 2
nd

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

charges

4.
Expand the magnetization density in terms of non
-
overlapping
spherically symmetric functions

5.
Replace the remaining terms with a short range repulsive energy

DFT


DFTB

rep
N
l
l
l
l
l
l
occ
i
i
i
i
tot
E
W
p
p
q
q
H
c
c
n
E

















































2
1
2
1
,
0

Most of DFTB+ is running with O(N) routines


Two exceptions


DFTB+ SCC


Ewald
-
summation, O(N
2
)


DFTB+
eigenvalue

solvers


LAPACK solvers, O(N
3
)



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


Large systems (>300), the
eigenvalue

solver
dominates


Calculation time vs. structure size

Performance figures

N
2.9

N
1.5


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 10
2
-
10
3
faster


DFTB+ compared to DMol3 is a factor of 30
-
80 faster

DMol3 vs. DFTB+

Atoms

Time
DFTB
+
(s)

Time
DMol3
(s)

Time
DMol3
/
Time
DFTB
+

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 t
asks


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


ZnH
2
, Zn(CH
3
)
2
, Zn(NH
3
)
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, CO
2

and NH
3
)


Zn in biological systems

Working with Zn containing compounds

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

Zn Solids

Method

E
coh

a(
Å
)

b(
Å
)

B
0
(
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

CO
2


Bond difference 1
-
2%


Binding too strong
~0.5
eV
/CO
2


Turn over point for
monolayer well
described

NH
3


Overall good
agreement with
experiments and
DFT calculations

Small molecule surface interaction

ZnO

(10
1
0)
-
CO
2

ZnO

(10
1
0)
-
NH
3

2
/
)
(
0
1
10

n
E
E
E
ZnO
T
abs




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







otherwise
r
r

if
0
)
(
)
(
cutoff
n
cutoff
n
r
r
r
f
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.