Transcript Kein Folientitel

The SCC-DFTB method applied to organic and
biological systems: successes, extensions
and problems.
Marcus Elstner
Physical and Theoretical Chemistry
Technical University of Braunschweig
DFTB: non-self-consistent scheme
Consider a case, where you know the DFT ground state
density G already (exactly or in good approximation in ):
 1 2

  2   eff  in   i   i  i 
occ
out    i
i
Then the energy can given by (Foulkes& Haydock PRB 1989):
  0   E  out   O( )
2
2
DFTB: non-self-consistent scheme
DFTB: consider input density 0 as superposition of neutral atomic
densities
 1 2

  2   eff  0   i   i  i 
LCAO basis:
 i    c
i
occ
   i
2
i
0
i
i
H
[

]
c


S
c
  0  i   


TB energy:
1   r    r
       i  
d rd r    K   xc           r  d r
xc
2
r  r
i
occ
occ
       i   rep   
i
DFTB: non-self-consistent scheme
 1 2






eff  0   i   i  i
 2

0
i
i
H
[

]
c


S
c
  0  i   


• No charge transfer between atoms  very good results for
homonuclear systems (Si, C), hydrocarbons etc.
• Complete transfer of one charge between atoms Also does not
fail for ionic systems (e.g. NaCl):
- Harrison
- Slater, (Theory of atoms and molecules)
• Problematic case: everything in between
DFTB: non-self-consistent scheme
 1 2

  2   eff  0   i   i  i
0
i
i
H
[

]
c


S
c
  0  i   


Problems:
• HCOOH: C=O and C-O bond lengths equalized
• H2N-CH=O and peptides: N-C and C=O bond lengths equalized
non-CT systems:
• CO2 vibrational frequencies
• C=C=C=C=C.. chains, dimerization, end effects
DFTB: non-self-consistent scheme
 1 2








eff
0  i   i i
 2

H  [  ]c   S  c



0
i
0
i
i
Problem: charge transfer between atoms overestimated due to
electronegativity differences between atoms need balancing
force: onsite e-e interaction of excess charge is missing!
0
   H 
[ 0 ]
C
2pC
O
C: 0
O: 0
C: +1
O: -1
2p scf scheme ok:
Non
C: +0.5
O: -0.5
- no charge transfer
C
- 2p
transfer
of one electron 2pO
C
2pO
2pO
DFTB: non-self-consistent scheme
 1 2






eff  0   i   i  i
 2

0
H  [  ]c   S  c



0
i
0
i
i

 1 2








eff
 2
  i   i  i
Try to keep H0 since it works well for many systems
 eff     eff  0   
 veff
 dr

0
H  [  ]  H 
[ 0 ]  ??
DFT total energy
  r  Z
1   r    r 
ˆ
       i T  i   
dr  
d rd r    xc      K
2
r  r
R  r
i

occ
1   r    r
ˆ
      i T  eff     i  
d rd r    K   xc           r  d r
xc
2
r  r
i
occ
occ
       i   rep   
i
Second order expansion of DFT total energy


r

r



1
      i Tˆ  eff     i  
d rd r    K   xc           r  d r
xc
2
r  r
i
occ
Expand E[ ] at 0, which is the reference density used to calculate the H0
 0  r  0  r  
1
       i Tˆ  eff  0   i   
d rd r    K   xc  0     0  0  r  d r
xc
2
r  r
i
occ
2



 xc     
1
1
  

   r    r   d rd r 
2  r  r     r    r    
0

Second order expansion of DFT total energy
Write density fluctuations as a sum of
atomic contributions
  r      r 
occ
       i Tˆ  eff  0   i 
(I)
i
1  0  r  0  r  
 
d rd r    K   xc  0     0  0  r  d r
xc
2
r  r
2


  xc     
1
1
   

    r     r   d rd r 
2   r  r     r    r    
0

(II)
(III)
(I) Hamiton matrix elements
Introduce LCAO basis:
 i    c
i
occ
occ
i
i
i i
0
ˆ
   i T  eff  0   i   ni c c H 

(I)
Second order expansion of DFT total energy
Write density fluctuations as a sum of
atomic contributions
  r      r 
occ
       i Tˆ  eff  0   i 
(I)
i
1  0  r  0  r  
 
d rd r    K   xc  0     0  0  r  d r
xc
2
r  r
2


  xc     
1
1
   

    r     r   d rd r 
2   r  r     r    r    
0

(II)
(III)
(II) Repulsive energy contribution
1  0  r  0  r  
Erep [ 0 ]   
d rd r    K   xc  0     0  0  r  d r
xc
2
r  r
0   

1
 
2 

0
0  r   0  r 
1 Z Z 
 r  r d rd r  2 
 R
1
Erep [ 0 ]   U
2 
  C
•pair potentials
•exponentially
decaying

Second order expansion of DFT total energy
Write density fluctuations as a sum of
atomic contributions
  r      r 
occ
       i Tˆ  eff  0   i 
(I)
i
1  0  r  0  r  
 
d rd r    K   xc  0     0  0  r  d r
xc
2
r  r
2


  xc     
1
1
   

    r     r   d rd r 
2   r  r     r    r    
0

(II)
(III)
(III) Second order term
 1   2    
1
xc



    r    r  d rd r


2   r  r    r   r   
0






q
F
Monopolapproximation:

 00
Two limits:
1) | r  r ' | 
2)| r  r ' | 0
q q
1
 [  ,  0 ] 
 
2 
R
2
2 at

E
1
1
2
2
2
 [  , 0 ] 


q

U

q



2 q2
2
New parameter U: calculated for every
element from DFT
(III) Second order term
 1   2    
1
xc



    r    r  d rd r


2   r  r    r   r   
0

2pC
2 at

E
1
1
2
2
2
C
 [  , 0 ] 


q

U

q


 2p
2
2 q
2
2pC
2pO
2pO
2pO
Combine the two limits
 1   2    
1
xc



    r    r  d rd r


2   r  r    r   r   
0

1) | r  r ' | 
2)| r  r ' | 0
q q
1
 [  ,  0 ] 
 
2 
R
2
2 at

E
1
1
2
2
2
 [  , 0 ] 


q

U

q



2 q2
2
(III) Second order term: Klopman-Ohno
approximation
 1
  2  xc       
  rr  rr     rr   rr   F00 F00    

 0 


1
r r 2 1 1 1 2
R  R  U  U  
4


  

1
r r 2 1
2
R  R  U1  U 1 
4

R
Determination of Gamma in DFTB
- Consider atomic charge densities
~ Rcov
 ~ exp((r  R ) /  )
-Calculate coulomb integrals ( ) for 2 spherical charge densities:
 1
  2  xc       
  rr  rr     rr   rr   F00 F00   

 0 

-deviation from 1/R for small R
R=0:
1/ = 3.2 UHubbard
Klopman-Ohno vs DFTB Gamma
1/r
DFTB-
  

1
r r 2 1 1 1 2
R  R  U  U  
4

Approximate density-functional theory
Elstner et al. Phys. Rev. B 58 (1998) 7260
occ
        i Tˆ  eff  0   i 
i
r
r
r r
1 0  r  0  r   r r
 
d rd r    K   xc  0      0  0  r  d r
r r
xc
2
r  r
 1
  2  xc     
r
r r r
1


   r r  
r
r    r     r   d rd r 

2   r  r     r    r     
0 

occ
tot   ni c c H 
i
i

i
0
1
  rep  0    q q  
2 
Hamilton-Matrixelements
•non-scc: neglect of red contributions
Comparison to SE models: Matrix elements
H 
1 0
 H   S   q (     ),    ,  
2

0
0

q

q

q
H    



0
0
H 
  | T  V (    ) |  
•Extended Hueckel (can be derived from DFT)
0
H 
 Ip
0
H 
1 0
0
0
 S  K ( H 
 H
)
2
H   I p  QU '
Q  q  Z
Comparison to SE models: Matrix elements
H 
1 0
 H   S   q (     ),
2

0
H 
 
0
   ,  
0
H 
  | T  V (    ) |  
•Fenske Hall
H   I p   Q  AB

H 
1 0
 S  ( I   I )  S   Q (     )
2

Comparison to SE models: Matrix elements
•formal similarity in Hamiltonmatrixelements
•Very different in determination of
matrixelements
•DFTB: incorporate strengths, but also
fundamental weaknesses of DFT
Differences w.r. to SE models: e.g. J
e.g. MNDO
AA
H 
 U    P (      / 2)

  Z B  1/ RB 
B
Coulomb part J:
e.g. CNDO
 
Approx. by multipolemultipole interaction
    
AA
H 
 U    qB AB  Vne  ..
B
Differences to SE models: e.g. J
CNDO
AA
H 
 U    qB AB  Vne  ..
B
r
r
1   r    r r r
1
d rd r    q AqB AB
r r

2
r  r
2 AB
Coulomb part accounts for e-e interaction due to interaction of
atomic charges: looks similar to 2nd order term in DFTB.
MNDO: simple charge-charge higher multipoles
Differences to SE models: e.g. J
DFTB: how is e-e interaction treated? consider J
r
r
     0
1   r    r r r
d rd r  
r r

2
r  r
r
r
1   r     r   r r
1
d rd r 

qAqB AB
r r

2
r  r
2 AB
r
r
1 0  r  0  r   r r
 Erep
 
d rd r 
r r
2
r  r
r
r
r
0
0  r 
H 
  r  0  r   r r
   r r dr 
 
d rd r 
r r
r  r
r  r

Extensions of DFTB
FAQS:
- better basis sets (e.g. double zeta)
- higher order expansion
- monopole  multipole
- other reference density
- why Mulliken charges?
- better fitting of Erep
Approximate density-functional theory
occ
        i Tˆ  eff  0   i 
i
r
r
r r
1 0  r  0  r   r r
 
d rd r    K   xc  0      0  0  r  d r
r r
xc
2
r  r
 1
  2  xc     
r
r r r
1


   r r  
r
r    r     r   d rd r 

2   r  r     r    r     
0 

occ
tot   ni c c H 
i
i

i
0
1
  rep  0    q q  
2 
Extensions
occ
        i Tˆ  eff  0   i 
i
r
r


r

r
r r
1
0  0  r r

 
d rd r   K   xc  0     0  0  r  d r
r r
xc
2
r  r
occ
0
tot   ni ci ci H 
  rep  0 
i

FAQS:
- better basis sets  much higher cost
Extensions
 1
  2  xc     
r
r r r
1


r r  
r
r    r     r   d rd r 


2   r  r     r    r     
0 

FAQS:
- higher order expansion
1
q q  

2 
- monopole  multipole
- inspection of gamma?
No additional cost!
Determination of Gamma
deviation from 1/R for small R
R=0:
1/ = 3.2 Uhubbard
 ~ exp((r  R ) /  )
Is this valid throughout the periodic table?
What is the relation between
‚atomic size‘ and
chemical hardness?
Gamma: Rcov ~ 1/U ?
Si-Cl
R covalent
B-F
H
U-Hubbard
N
Gamma requires : 3.2*Rcov= 1/U?
H
U vs Rcov: Hydrogen atom
U-Hubbard
O
H
N
C
Si
R covalent
U vs Rcov: H not in line!
U-Hubbard
H
N
In DFTB, H is 0.73A
instead of 0.33A!
 Gamma requires: 3.2*Rcov= 1/U
 size of H overestimated based on hardness
value: H has same size like N!
On-site interaction and coulomb scaling: H
• UH for the on-site interaction of H should not be changed!
• However, UH is a bad measure for the size of H!
Leads to too ‚large‘ H-atoms! I.e. coulomb interaction is
damped too fast due to ‚artificial‘ overlap effect!
 modify coulomb-scaling for H!
Modified Gamma for H-bonding
change only X-H interaction!
   1 / R  S     1/ R  S * fdamp
  
1
1 1 1 2
R  U  U   exp( R 2 )
4
2
Modified Gamma for H-bonding
-Water dimer: 3.3 kcal
4.6 kcal
standard DFTB: H-bonds ~ 1-2 kcal too low
mod Gamma: ~0.3-0.5 kcal too low
H-bonds: water cluster
MP2 from KS Kim et al 2000
Expansion to higher order?
occ
        i Tˆ  eff  0   i 
i
r
r
r r
1 0  r  0  r   r r
 
d rd r    K   xc  0      0  0  r  d r
r r
xc
2
r  r
 1
  2  xc     
r
r r r
1


   r r  
r
r    r     r   d rd r 

2   r  r     r    r     
0 

occ
tot   ni c c H 
i
i

i
0
1
  rep  0    q q  
2 
Charged systems with localized charge
H2O  OH- + H+
E.g.:
1
q q  

2 
Description of OH-:
O is very ‚negative‘,
is the approximation of a constant Hubbard value
(chemical hardness) appropriate?
Deprotonation
energy
B3LYP/6-311++G(2d2p):
397 kcal/mole
SCC-DFTB:
424 kcal/mole
1
2
qU

2 
U  U (q)
Problems with charged systems: inclusion of third
order correction into DFTB
•charge dependent Hubbard
U(q) = U(q0) + dU/dq *(q-q0)
•Calculate dU/dq through U(q)
consider atoms for different charge states.
Deprotonation energies
B3LYP vs SCC-DFTB and 3rd order correction Uq:
- basis set dependence
- large charges on anions
-U(q): changes “size” of atom: Rcov~ 1/U
SCC-DFTB:
 ‚organic set‘: available for
H C N O S P Zn
solids: Ga,Si, ...
 all parameters calculated from DFT
 computational efficiency as NDO-type methods
(solution of gen. eigenvalue problem for valence electrons in minimal basis)
SCC-DFTB: Tests
1) Small molecules: covalent bond
 reaction energies for organic molecules
 geometries of large set of molecules
 vibrational frequencies,
2) non-covalent interactions
 H bonding
 VdW
3) Large molecules (this makes a difference!)
 Peptides
 DNA bases
SCC-DFTB: Tests
4) Transition metal complexes
5) Properties
 IR, Raman, NMR
 excited states with TD-DFT
SCC-DFTB Tests 1: Elstner et al., PRB 58 (1998) 7260
Performance for small organic molecules
(mean absolut deviations)
• Reaction energiesa): ~ 5 kcal/mole
• Bond-lenghtsb) : ~ 0.014 A°
• Bond anglesb): ~ 2°
•Vib. Frequenciesc): ~6-7 %
a) J. Andzelm and E. Wimmer, J. Chem. Phys. 96, 1280 1992.
b) J. S. Dewar, E. Zoebisch, E. F. Healy, and J. J. P. Stewart, J. Am.
Chem. Soc. 107, 3902 1985.
c) J. A. Pople, et al., Int. J. Quantum Chem., Quantum Chem. Symp. 15, 269
1981.
SCC-DFTB Tests 2: T. Krueger, et al., J.
Chem. Phys. 122 (2005) 114110.
With respect to G2:
mean ave. dev.: 4.3 kcal/mole
mean dev.:
1.5 kcal/mole
SCC-DFTB Tests 3: Sattelmeyer & Jorgensen, (to be
published)
Mean Absolute Errors in Calculated Heats of Formation for
Neutral Molecules Containing the
Elements C, H, N and O (kcal/mol).
N
AM1 PM3 PDDG/PM3 SCC-DFTB
Hydrocarbons 254
5.6 3.6
2.6
4.8
All Molecules
622
6.7 4.4
3.2
5.9
Training Set
134
6.1 4.3
2.7
7.0
Test Set
488
6.8 4.4
3.3
5.6
SCC-DFTB Tests 3: Sattelmeyer & Jorgensen, (to be
published)
Absolute Errors for Additional Molecular Properties of CHNO-containing
Species.
N
AM1
PM3
PDDG/PM3 SCC-DFTB
Bond lengths (Å)
218 0.017 0.012
0.013
0.012
Bond angles (deg.)
126 1.5
1.7
1.9
1.0
Dihedral angles (deg.) 30
2.8
3.2
3.7
2.9
Dipole moments (D)
0.23
0.25
0.23
0.39
47
• ok: H-bonds, ions
• quite bad: S
SCC-DFTB Tests:
Accuracy for vib. freq., problematic case e.g.:
Special fit for vib. Frequencies:
Mean av. Err.: 59 cm-1  33 cm-1
Malolepsza, Witek & Morokuma: CPL 412 (2005) 237.
Witek & Morokuma, J Comp Chem. 25 (2004) 1858.
for CH
H-bonds
Han et al. Int. J. Quant. Chem.,78 (2000) 459.
Elstner et al. phys. stat. sol. (b) 217 (2000) 357.
Elstner et al. J. Chem. Phys. 114 (2001) 5149.
Yang et al., to be published.
-~1-2kcal/mole too weak
- relative energies reasonable
Coulomb
interaction
- structures well reproduced
H2O-dimer complexes Cs, C2v
NH3-NH3- and NH3-H2O-dimer
Model peptides: N-Acetyl-(L-Ala)n
N‘-Methylamide (AAMA) + 4 H2O
Performance of DFTB
Small molecules don’t tell the whole story
Test for large ones:
- peptides
- DNA, sugar
- other extended structures
Secondary-structure elements for Glycine und
Alanine-based polypeptides
N = 1 (6 stable conformers)
310 - helix
R-helix
stabilization by internal H-bonds
between i and i+3
between i and i+4
N
DFTB very good for:
main problem for DFT(B): dispersion!
- relative energies
 AM1, PM3, MNDO not convincing
- geometries
 OM2 much improved (JCC 22 (2001) 509)
- vib. freq. o.k.!
Glycine and Alanine based polypeptides in vacuo
Elstner et al., Chem. Phys. 256 (2000) 15
Elstner et al. Chem. Phys. 263 (2001) 203
Bohr et al., Chem. Phys. 246 (1999) 13
Relative energies, structures and vibrational properties: N=1-8
N=1
(6 stable conformers)
E relative energies (kcal/mole)
B3LYP
(6-31G*)
MP2
MP4-BSSE
SCC-DFTB

N

Ace-Ala-Nme
C7eq
C5ext
C7ax

2

MP4-BSSE: Beachy et al, BSSE corrected at MP2 level
R

P
SCC-DFTB vs. NDDO (MNDO, AM1, PM3)
DFTB:
 energetics of ONCH ok, S, P problematic
 very good for structures of larger Molecules
 vibrational frequencies mostly sufficient (less accurate than DFT)
NDDO:
 very good for energetics of ONCH (and others, even better than DFT)
 structures of larger Molecules often problematic !!!
 do NOT suffer from DFT problems e.g. excited states
 Mix of DFTB and NDDO to combine strengths of both worlds
TD-DFTB and excited states
Problems of TD-DFT:
Combination of DFTB and
OM2!
Problems:
 same Problems as DFT
 additional Problems:
- except for geometries, in general lower accuracy than
DFT
- slight overbinding (probably too low reaction barriers?!)
- too weak Pauli repulsion
- H-bonding (will be improved)
- hypervalent species, e.g. HPO4 or sulfur compounds
- transition metals: probably good geometries, ... ?
- molecular polarizability (minimal basis method!)
DFT Problems:

E : wrong asymptotic form; -
(1) Ex: Self interaction error.
(2)
x
J- Ex = 0 !: Band gaps, barriers
HOMO
<< Ip: virtual KS orbitals
(3) Ex: ‚somehow too local‘; overpolarizability, CT excitations
(4) Ec: ‚too local‘: Dispersion forces missing
(5) Ec: even much more ‚too local‘: isomerization reactions
(6) Multi-reference problem
DFT and VdW interactions
DFT and VdW interactions
2 Problems:
- Pauli repulsion: exchange effect
~ exp(R) or 1/R12
- attraction due to correlation
E ~ 1/R6
~ -1/R6
Dispersion forces - Van der Waals interactions
Elstner et al. JCP 114 (2001) 5149

Etot = ESCC-DFTB - f (R) C6 /R6
C6 via Slater-Kirckwood combination rules of atomic
polarizibilities after Halgreen, JACS 114 (1992) 7827.
damping f(R) = [1-exp(-3(R/R0)7)]3
E~
1/R6
R0 = 3.8Å (für O, N, C)
DFTB + dispersion
Sponer et al. J.Phys.Chem. 100 (1996) 5590; Hobza et al. J.Comp.Chem. 18 (1997) 1136
stacking energies in MP2/6-31G* (0.25), BSSE-corrected ( + MP2-values)
 Hartree-Fock, no stacking
 AM1, PM3, repulsive interaction (2-10) kcal/mole
 MM-force fields strongly scatter in results
vertical dependence twist-dependence
With help from
QM/MM: DFTB
Q. Cui, Madison
H. Hu, J. Herrmans
UNC
Morokuma, Witek
Zheng, Irle
IR, RAMAN, metals
D. York, Minnesota
A. Roitberg, Florida
DFTB:
Frauenheim, Seifert
& Suhai groups
Dispersion, DNA
P. Hobza,
Nat. Academie,
Prague
H. Liu, W. Yang, Duke
O(N), COSMO, GB
DFG,
Univ. Paderborn