An introduction to DFTB+

Report
Introduction to DFTB+
Martin Persson
Accelrys, Cambridge
Outline
• DFTB
– Why DFTB?
– Basic theory DFTB
– Performance
• DFTB+ in Materials Studio
– Energy, Geometry, Dynamics, Parameterization
– Parameterization
• Basic theory
• Setting up a parameterization
Why DFTB+
QM vs. CM
• 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
This is where DFTB+ comes in
• 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
Examples of what can be done with DFTB+
Diamond nucleation
Novel SiCN ceramics
Magnetic Fe clusters
Si cluster growth
WS2 nanotubes
Basic DFTB Theory
DFTB theory in short
• 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 basis set
• Minimal basis set
• Pseudo atomic orbitals
– Slater orbitals
– Spherical harmonics
 v r  

n , , l v , m v
a n r
lv  n
e
 r
r 
 lv m v  
r
Pseudo atomic orbitals
Silicon sp3d5 orbitals
S
D1
P1
D2
P2
D3
P3
D4
For Silicon the d-orbitals are un-occupied but needed to
properly model the conduction band.
D5
Hamiltonian elements
• Diagonal elements use free
atom energies
• Two centre integrals
• Tabulated values
0
H 
  free atom
if   
 A
B
    T  V A  V B  if A  B
0
otherwise

DFT  DFTB
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 pseudoatomic 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
occ
E tot 
 
  , 
i
 E xc n ,   
2


1
n i   i 
v
2
2

1
2
N


Z Z 
R  R 
n ( r ) d r 
3

r  r
 i



DFT  DFTB
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 pseudoatomic 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
occ
E tot 
 
  , 

1
2

1
2

i
2


1
n i   i 
v
2
2


 V xc n 0 , 0   i  
r  r

n d r 
3

n  n0  n
2
 1

 E xc
3
3


| n 0 , 0  n  n d rd r 
 r  r  nn



 E xc
2
   
| n 0 , 0   d rd r   E xc n 0 , 0  
3
  V xc n 0 , 0 n 0 d r 
3
1

2
3
n 0 n 0
r  r
d rd r 
3
3
  0  
1
2
N


Z Z 
R  R 
DFT  DFTB
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 pseudoatomic 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
occ
E tot 
 
  , 

1
2

1
2
i

n i  
 



0
c  i  c i  H  

     q   q 


N


p  l p  l W  l l   E xc n 0 , 0  
l  l 
  V xc n 0 , 0 n 0 d r 
3
1

2
n 0 n 0
r  r
1
2
d rd r 
3
3
N


Z Z 
R  R 
DFT  DFTB
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 pseudoatomic 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
occ
E tot 
 
  , 

1
2

1
2
i

n i  
 



0
c  i  c i  H  

     q   q 


N


l  l 
p  l p  l W  l l   E rep
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
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
Starting a DFTB+ job
•
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
Downloading parameters
• Need to register to get access.
• The downloaded parameters will
contain many different Slater
Koster files
•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
DFTB+ Analysis
•
•
•
•
•
•
•
Band structure
Density of states
Electron density
Fermi surface
Orbitals
Slater-Koster parameters
Dynamics analysis is done
using the Forcite analysis
tools
Materials Studio 6.0 Parameterization tool
The DFTB+ 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 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
occ
E tot 
 
  , 

1
2

1
i

n i  
 



0
c  i  c i  H  

     q   q 

The remaining terms, Erep, will be
described using fitted repulsive pair
potentials.

N

2

p  l p  l W  l l   E rep
l  l 
pairs
E rep  E
tot
DFT
E
tot
DFTB , bare

U
type ( ij )
( rij )
i j
Pair potentials
The pair potentials are fitted against a
basis of cutoff polynomials
 ( r  rcutoff ) n if r  rcutoff
f n (r )  
otherwise
0

Systems
•
•
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...
Bond order fitting
Use weight distributions to combine several
bond orders into a single potential fit
Parameterization job results
• 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
Evaluating the result
benzene
------DMol3 C3-C2 = 1.39838
DFTB+ C3-C2 = 1.41171
Diff C3-C2 = 0.01333
C3-H9 = 1.09097
C3-H9 = 1.10386
C3-H9 = 0.01289
DMol3 C2-C7-C6 = 120.00000
DFTB+ C2-C7-C6 = 119.99783
Diff C2-C7-C6 = -0.00217
H12-C7-C6 = 120.00000
H12-C7-C6 = 120.00930
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
DFTB+ C1-C2-H6 = 121.55765
Diff C1-C2-H6 = -0.09384
H4-C1-H3 = 116.69702
H4-C1-H3 = 116.88453
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.
SiGeH
• sp3d5 basis
• LDA(PWC)
• Fitted against
–
–
–
–
–
Si, Ge and SiGe solids
Si2H6, Si2H4
Ge2H6, Ge2H4
SiGeH6, SiGeH4
SiH4, GeH4 and H2
• Tested against:
Si vacancy Formation energy
E f  E N 1 
N 1
N
EN
Ef(eV)
DFTB+
2.6
DMol3
2.7
–
–
–
–
Solids
Nanowires
Nanoclusters
Si vacancy
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.
• 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: Larger molecules
CNT-6x6
Bond
Diff (Å)
C-C
0.005
Caffeine
N-AA
Bond
Diff (Å)
C-C
0.0095
C-N
0.0075
C-O
0.0078
C-H
0.0028
Bond
Diff (Å)
C-C
0.0148
C-N
0.0118
C-O
0.0100
C-H
0.0114
N-H
0.0127
O-H
0.0019
• Successfully tested
for:
– CNT
– C60
– Caffeine
– Glucose
– Porphine
– N-Acetylneuraminic
acid
Thanks for your attention
Other contributors:
Paddy Bennett (Cambridge, Accelrys)
Bálint Aradi (Bremen, CCMS)
Zoltan Bodrog (Bremen, CCMS)
Generating the orbitals
2




r
at
ˆ
T  V eff ( r )       ( r )      ( r )
r 

 0  
• The Kohn-Sham equation is solved for a single
atom.
• Using an added extra confining potential to
better model molecules and solids
DFT  DFTB
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 pseudoatomic 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
occ
E tot 
 
  , 
i
 E xc n ,   
2


1
n i   i 
v
2
2

1
2
N


Z Z 
R  R 
n ( r ) d r 
3

r  r
 i



DFT  DFTB
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 pseudoatomic 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
occ
E tot 
 
  , 

1
2

1
2

i
2


1
n i   i 
v
2
2


 V xc n 0 , 0   i  
r  r

n d r 
3

n  n0  n
2
 1

 E xc
3
3


| n 0 , 0  n  n d rd r 
 r  r  nn



 E xc
2
   
| n 0 , 0   d rd r   E xc n 0 , 0  
3
  V xc n 0 , 0 n 0 d r 
3
1

2
3
n 0 n 0
r  r
d rd r 
3
3
  0  
1
2
N


Z Z 
R  R 
DFT  DFTB
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 pseudoatomic 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
occ
E tot 
 
  , 

1
2

1
2

i

n i  
 



*
0
c  i  c i  H  

2
 1

 E xc
3
3


| n 0 , 0  n  n d rd r 
 r  r  nn



 E xc
2
    |
  d rd r   E xc n 0 , 0  
3
n0 ,0
  V xc n 0 , 0 n 0 d r 
3
1

2
3
n 0 n 0
r  r
d rd r 
3
3
 i 
 c 
i


1
2
N


Z Z 
R  R 
0
0
H     i  Hˆ   i 
DFT  DFTB
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 pseudoatomic 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
occ
E tot 
 
  , 




0
c  i  c i  H  

n 
     q   q 
2
1
2
 q  q  q
0

 E xc
2
   
| n 0 , 0    r  r   E xc n 0 , 0  
3
  V xc n 0 , 0 n 0  r 
3
1
2
  n
a
1


i

n i  
 

3
n 0 n 0
r  r
 r r 
3
3
1
2
N


Z Z 
R  R 

   
Hubbard U 
 Coulomb
interacion
αβ
αβ
DFT  DFTB
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 pseudoatomic 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
occ
E tot 
 
  , 

1
2

1
2
i

n i  
 



0
c  i  c i  H  

p al  p l   p  l 
     q   q 


N


p  l p  l W  l l   E xc n 0 , 0  
l  l 
  V xc n 0 , 0 n 0 d r 
3
1

2
n 0 n 0
r  r
1
2
d rd r 
3
3
N


Z Z 
R  R 
W  ll   spin interactio
n
DFT  DFTB
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 pseudoatomic 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
occ
E tot 
 
  , 

1
2

1
2
i

n i  
 



0
c  i  c i  H  

     q   q 


N


l  l 
p  l p  l W  l l   E rep
Calculation time vs. structure size
• 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
Performance figures
N2.9
N1.5
•10x10 CNT
•32 atoms/unitcell
•Run on single core
•Intel(R) Xeon(TM) CPU 3.00GHz
#cpu
OpenMP
Speedup Efficiency
1
1.0
2
0.87
•Small systems (<300 atoms) O(N) processes dominate
3
•Large systems (>300) eigenvalue solver dominates
4
0.80
0.72
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
• 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
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.
Starting a DFTB+ job: Electronic
•
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: Properties
• 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: Job Control
• 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
During a DFTB+ job
• 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
DFTB+ Result files
Visible files
• <>.xsd
– Final structure
• <>.xtd (dynamics)
– Dynamics trajectory
• <>.txt
– Compilation of the results
• <>.dftb
– The last output from DFTB+
• <>.skflib (parameterization)
– Slater-Koster library
Hidden files
• *.tag
– Final output data
• *.cube
– Density and orbital
data
• *.bands
– Band structure data
Zn compounds using DFTB+
Working with Zn containing compounds
• 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
N. H. Moreira, J. Chem. Theory Comput. 2009, 5 , 605
Zn Solids
W-ZnO DFTB+
w-ZnO
zb-ZnS
Method
Ecoh
a(Å)
b(Å)
B0(GPa)
DFTB+
9.77
3.28
5.25
161
PBE
8.08
3.30
5.34
124
EXP
7.52
3.25
5.20
208
DFTB+
7.93
5.43
-
44.2
LDA
7.22
5.35
-
82
EXP
6.33
5.40
-
76.9
• Reasonable solid state properties
N. H. Moreira, J. Chem. Theory Comput. 2009, 5 , 605
W-ZnO PBE
ZnO Surface stability
DFTB+
DFT
•Predicts correct order and magnitude for the
cleavage energy
•Bond and angle deviation ~1-2%
F. Claeyssens J. Mat. Chem.
2005, 15 139
N. H. Moreira, J. Chem. Theory Comput. 2009, 5 , 605
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
Small molecule surface interaction
ZnO (1010)-CO2
ZnO (1010)-NH3
CO2
• Bond difference 1-2%
• Binding too strong
~0.5 eV/CO2
• Turn over point for
monolayer well
described
NH3
E abs  ( E T  E ZnO 10 1 0  n  ) / 2
N. H. Moreira, J. Chem. Theory Comput. 2009, 5 , 605
• Overall good
agreement with
experiments and
DFT calculations
Electronic settings
• 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)
Polynomial fitting setup
• Each fitting is done
using different
polynomial orders
• Fittings are done
for a set of cutoff
radius scale factors
 ( r  rcutoff ) n if r  rcutoff
f n (r )  
otherwise
0

Possible future extensions to DFTB+
DFTB+ features outside of Material Studio
• Optical Properties
– LR-TD-DFTB
• Electronic transport
– NEG-DFTB
• QM/MM
• Vibrational modes
Please let us know what extensions and enhancements you
would like to see for DFTB+ in the future.

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