Relativistic
parameterization of the
SCC-DFTB method
Henryk Witek
Institute of Molecular Science & Department of Applied Chemistry
National Chiao Tung University
Hsinchu, Taiwan
232nd ACS meeting in SF, 12.09.2006
Aims
Provide the DFTB community with a
general and easy-to-use tool for
developing Slater-Koster files

Develop
a reliable set of SCC-DFTB
parameters suitable for modeling
chemical reactions
232nd ACS meeting in SF, 13.09.2006
Requirements

Important issues of the project
general character
 relativistic framework
 well-defined procedure
 high automaticity
 error control – test suite

232nd ACS meeting in SF, 13.09.2006
Theoretical framework

4-component Dirac-Kohn-Sham equation

Modification of relativistic Dirac-Slater code
of J.P. Desclaux
 Comp.
Phys. Comm. 1, 216 (1969)
 Comp. Phys. Comm. 9, 31 (1975)
Density confinement
 Spinor confinement

232nd ACS meeting in SF, 13.09.2006
Slater-Koster files

One-center quantities
orbital energies
 orbital hardness
 orbital spin-densities interaction parameters


Two-center quantities
Hamiltonian integrals
 overlap integrals
 repulsive potentials

232nd ACS meeting in SF, 13.09.2006
Input description

Atomic information




nuclear charge
number of electrons
shell occupations
Method information



exchange-correlation functional type
confinement radius
way to construct molecular XC potential
density superposition
 potential superposition

232nd ACS meeting in SF, 13.09.2006
Output: spinors of carbon
* atom electronic structure and final shell energies:
shell type
occupation
final energy
======== ======== ==========
1 S1/2
2.00
-11.29598
2 S1/2
2.00
-0.44465
2 P1/2
1.00
-0.12665
2 P3/2
1.00
-0.12623
* radial overlap integrals for spinors
spinor 1
======
1 S1/2
spinor 2
======
2 S1/2
overlap integral
===========
-0.000000000022
232nd ACS meeting in SF, 13.09.2006
Output: spinors of lead
* atom electronic structure and final shell energies:
shell type
occupation
final energy
======= ======== =========
1 S1/2
2.00
-3256.80560
2 S1/2
2.00
-585.97772
2 P1/2
2.00
-564.09214
2 P3/2
4.00
-482.19388
3 S1/2
2.00
-141.89459
…
5
5
6
6
6
…
D3/2
D5/2
S1/2
P1/2
P3/2
4.00
6.00
2.00
2.00
0.00
…
-0.79336
-0.68107
-0.33752
-0.09002
-0.04704
232nd ACS meeting in SF, 13.09.2006
Output: spinors of lead
* radial overlap integrals for spinors
spinor 1
======
1 S1/2
1 S1/2
2 S1/2
2 P1/2
2 P3/2
spinor 2
======
2 S1/2
3 S1/2
3 S1/2
3 P1/2
3 P3/2
…
2
3
4
5
P3/2
P3/2
P3/2
P3/2
overlap integral
===========
0.000000000068
0.000000000016
0.000000000186
0.000000000099
0.000000000094
…
6
6
6
6
P3/2
P3/2
P3/2
P3/2
…
0.000000000048
-0.000000000358
-0.000000001312
0.000000000096
232nd ACS meeting in SF, 13.09.2006
Output: atomic density
* error for the fitted atomic density at grid points
density
======
dn
norm1
=======
0.000010
norm2
======
0.000019
norm∞
======
0.000104
C
* renormalization of fitted density
=> density renormalized from 5.999981 to 6.000000 electrons
* error for the fitted atomic density at grid points
density
======
dn
norm1
=======
0.030532
norm2
======
0.049705
norm∞
======
0.147628
Pb
* renormalization of fitted density
=> density renormalized from 82.000529 to 82.000000 electrons
232nd ACS meeting in SF, 13.09.2006
radial density of lead
232nd ACS meeting in SF, 13.09.2006
Semi-relativistic orbitals

Scalar relativistic valence orbitals are
obtained by:
neglecting small component
 averaging spin-orbit components of every
scalar orbital

1
l g l  l  1g l 1 
g l r  
2l  1
V.Heera, G. Seifert, P. Ziesche, J. Phys. B 17, 519 (1984)
232nd ACS meeting in SF, 13.09.2006
Large vs. small component
232nd ACS meeting in SF, 13.09.2006
Averaging spin-orbit split
components of a spinor
232nd ACS meeting in SF, 13.09.2006
Output: orbitals of carbon
* info about scalar atomic orbitals
num
orbital
occupation
final energy
==== ===== ======== =========
1
1s
2.00
-11.29598
2
2s
2.00
-0.44465
3
2p
2.00
-0.12637
type
=====
core
valence
valence
* error for the fitted curve at grid points
orbital
=====
2s
2p
norm1
======
0.000231
0.000013
norm2
======
0.000721
0.000025
norm∞
======
0.005025
0.000108
* renormalization after fit and neglecting small component
=> orbital 2s renormalized from
0.999957 to
1.0d0
=> orbital 2p renormalized from
0.999957 to
1.0d0
232nd ACS meeting in SF, 13.09.2006
Output: orbitals for lead
* info about scalar atomic orbitals
num
orbital
occupation
final energy
==== ====== ======== ==========
1
1s
2.00
-3256.80560
2
2s
2.00
-585.97772
3
2p
6.00
-509.49330
4
3s
2.00
-141.89459
5
3p
6.00
-119.52024
6
3d
10.00
-94.16394
7
4s
2.00
-32.79553
8
4p
6.00
-25.30912
9
4d
10.00
-15.92391
10
4f
14.00
-5.84011
11
5s
2.00
-5.53058
12
5p
6.00
-3.33518
13
5d
10.00
-0.72598
14
6s
2.00
-0.33752
15
6p
2.00
-0.06137
232nd ACS meeting in SF, 13.09.2006
type
=====
core
core
core
core
core
core
core
core
core
core
valence
valence
valence
valence
valence
Output: orbitals for lead
* fitting valence orbitals with gaussians
* error for the fitted curve at grid points
orbital
=====
5s
5p
5d
6s
6p
norm1
======
0.000048
0.000047
0.000143
0.000108
0.000026
norm2
======
0.000138
0.000094
0.000245
0.000257
0.000045
norm∞
=======
0.002025
0.000988
0.000807
0.003610
0.000371
* renormalization after fit and neglecting small component
=> orbital 5s renormalized from
0.999235 to
1.0d0
=> orbital 5p renormalized from
0.990674 to
1.0d0
=> orbital 5d renormalized from
0.998799 to
1.0d0
=> orbital 6s renormalized from
0.999913 to
1.0d0
=> orbital 6p renormalized from
0.991615 to
1.0d0
232nd ACS meeting in SF, 13.09.2006
Relativistic vs. non-relativistic atomic
orbitals: carbon atom
232nd ACS meeting in SF, 13.09.2006
Relativistic vs. non-relativistic atomic
orbitals: carbon atom
232nd ACS meeting in SF, 13.09.2006
Relativistic vs. non-relativistic atomic
orbitals: lead atom
232nd ACS meeting in SF, 13.09.2006
Relativistic vs. non-relativistic atomic
orbitals: lead atom
232nd ACS meeting in SF, 13.09.2006
Confinement potential

Additional term Vconf in Dirac-Kohn-Sham
effective potential
V  VDKS  Vconf
k
Vconf



r
   , k  2 or 4
 r0 
contraction of orbital’s exponential tail
relaxation of basis set
additional variational parameter in the formalism
232nd ACS meeting in SF, 13.09.2006
Effect of the confinement potential
radial density of Pb
232nd ACS meeting in SF, 13.09.2006
Repulsive potentials

Effective two-center, distance-dependent
potentials accounting for
repulsion between atomic chemical cores
 double counting terms in electronic part


Total DFTB energy is
EDFTB  E
elec
E
rep
E
elec
1
rep
  E AB rAB 
2 A B
232nd ACS meeting in SF, 13.09.2006
Constructing C-C repulsive potential
M. Sternberg, Ph.D. Thesis
232nd ACS meeting in SF, 13.09.2006
repulsive C-C potential
Malolepsza, Witek, and Morokuma, ChPL 412, 237 (2005)
232nd ACS meeting in SF, 13.09.2006
performance of new C-C potential
Malolepsza, Witek, and Morokuma, ChPL 412, 237 (2005)
232nd ACS meeting in SF, 13.09.2006
Resultant repulsive potentials
232nd ACS meeting in SF, 13.09.2006
Derivatives of repulsive
potentials
232nd ACS meeting in SF, 13.09.2006
Analytical form of potentials
EDFTB  E
elec
E
rep
E
elec
1
rep
  E AB rAB 
2 A B
rep
rAB     iAB exp   iAB rAB 
E AB
K
i 1
EDFTB
K
1
elec
AB
AB
 E    i exp   i rAB 
2 A B i 1
232nd ACS meeting in SF, 13.09.2006
Analytical form of potentials

Atomization energies
K
1

  elec
AB
AB
A
elec 
 i exp  i rAB   EDFTB   EDFTB    E   E A 

2 A B i 1
A
A

 



K
1

AB
AB
A   elec
elec 
 i exp  i rAB   EDFT   EDFT    E   E A 

2 A B i 1
A
A

 



K
1

AB
AB
A   elec
elec 
 i exp  i rAB   Eexp   Eexp    E   E A 

2 A B i 1
A
A

 



232nd ACS meeting in SF, 13.09.2006
Analytical form of potentials

Equilibrium structures
EDFTB
K
K
1
elec
AB
AB
 E    i exp   i rAB 
2 A B i 1
 
B  A i 1
where

AB AB
i
i

exp   i rAB
AB

rAB Eelec

v A
v A
v  x, y, z over all atoms A
232nd ACS meeting in SF, 13.09.2006
First derivatives of repulsive potential
1st derivative of HH repulsive potential
1st derivative of OO repulsive potential
0.1
0.1
NO2
O3
NO2-
0
-0.1
H2 O 2
H2 O 2
H2 O
0
H3 O +
NH3
O3
-0.2
-0.3
H2
O2
-0.1
-0.4
0
0.5
1
1.5
2
2.5
3
0
0.5
1
1.5
dist (Å)
dist (Å)
232nd ACS meeting in SF, 13.09.2006
2
2.5
3
First derivatives of repulsive potential
1st derivative of NO repulsive potential
1st derivative of NH repulsive potential
0.1
0
0
-0.1
NO2-
-0.2
-0.3
-0.5
HNO
-0.1
NO2, HNO
-0.4
NO
-0.6
NH3
-0.7
-0.2
0
0.5
1
1.5
2
2.5
3
0
0.5
1
1.5
dist (Å)
dist (Å)
1st derivative of OH repulsive potential
HNO
0
H2 O 2
H2 O 2
-0.1
H3 O + H O
2
-0.2
0
0.5
1
1.5
2
2.5
232nd ACS meetingdist(Å)
in SF, 13.09.2006
3
2
2.5
3
Conclusions
Convenient relativistic tool for automatic
DFTB parameterization is suggested
 New form of potential parameterization is
proposed

232nd ACS meeting in SF, 13.09.2006
Acknowledgements
Christof Köhler
 Keiji Morokuma
 Marcus Elstner
 Thomas Frauenheim

232nd ACS meeting in SF, 13.09.2006