Download - The SCC-DFTB method applied to organic and biological systems: successes, extensions and problems
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The SCC-DFTB method applied to organic and biological systems: successes, extensions
and problems.
The SCC-DFTB method applied to organic and biological systems: successes, extensions
and problems.
Marcus Elstner
Physical and Theoretical ChemistryTechnical University of Braunschweig
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DFTB: non-self-consistent schemeDFTB: non-self-consistent scheme
221
2
occ
eff in i i i out ii
Consider a case, where you know the DFT ground state
density G already (exactly or in good approximation in ):
Then the energy can given by (Foulkes& Haydock PRB 1989):
20 ( )outE O
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DFTB: non-self-consistent schemeDFTB: non-self-consistent scheme
220
1
2
occ
eff i i i ii
occ
i repi
1
2
occ
i K xc xci
r rd rd r r d r
r r
TB energy:
DFTB: consider input density 0 as superposition of neutral atomic densities
LCAO basis:
ii c
00[ ] i i
iH c S c
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DFTB: non-self-consistent schemeDFTB: non-self-consistent scheme
20
1
2
eff i i i
00[ ] i i
iH c S c
• 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
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DFTB: non-self-consistent schemeDFTB: non-self-consistent scheme
20
1
2
eff i i i 00[ ] i i
iH c S c
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
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DFTB: non-self-consistent schemeDFTB: non-self-consistent scheme
20
1
2
eff i i i 00[ ] i i
iH c S c
Problem: charge transfer between atoms overestimated due to electronegativity differences between atoms need balancing force: onsite e-e interaction of excess charge is missing!
00[ ]H
2pO
2pO2pC
2pO2pC
2pC
C: 0 O: 0
C: +1 O: -1
C: +0.5 O: -0.5
C O
Non scf scheme ok:
- no charge transfer- transfer of one electron
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DFTB: non-self-consistent schemeDFTB: non-self-consistent scheme
20
1
2
eff i i i 0
0[ ] i iiH c S c
21
2
eff i i i
0
Try to keep H0 since it works well for many systems
0
eff
eff eff
vdr
00[ ] [ ] ??H H
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DFT total energyDFT total energy
1ˆ2
occ
i eff i K xc xci
r rT d rd r r d r
r r
1ˆ2
occ
i i xc Ki
r Z r rT d r d rd r
r rR r
��������������
occ
i repi
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Expand E[ ] at 0, which is the reference density used to calculate the H0
0
000 0 0 0
2
1ˆ2
1 1
2
occ
i eff i K xc xci
xc
r rT d rd r r d r
r r
r r d rd rr r r r
Second order expansion of DFT total energySecond order expansion of DFT total energy
1ˆ2
occ
i eff i K xc xci
r rT d rd r r d r
r r
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Write density fluctuations as a sum of
atomic contributions
0
0
000 0 0
2
ˆ
1
2
1 1
2
occ
i eff ii
K xc xc
xc
T
r rd rd r r d r
r r
r r d rd rr r r r
Second order expansion of DFT total energySecond order expansion of DFT total energy
r r
(I)
(II)
(III)
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Introduce LCAO basis:
(I) Hamiton matrix elements(I) Hamiton matrix elements
(I)
ii c
00
ˆocc occ
i ii eff i i
i i
T n c c H
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Write density fluctuations as a sum of
atomic contributions
0
0
000 0 0
2
ˆ
1
2
1 1
2
occ
i eff ii
K xc xc
xc
T
r rd rd r r d r
r r
r r d rd rr r r r
Second order expansion of DFT total energySecond order expansion of DFT total energy
r r
(I)
(II)
(III)
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000 0 0 0
1[ ]
2rep K xc xc
r rE d rd r r d r
r r
(II) Repulsive energy contribution(II) Repulsive energy contribution
0 0
0 0
1 1
2 2
r r Z Zd rd r
r r R
0
1[ ]
2repE U
•pair potentials
•exponentially decayingC
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Write density fluctuations as a sum of
atomic contributions
0
0
000 0 0
2
ˆ
1
2
1 1
2
occ
i eff ii
K xc xc
xc
T
r rd rd r r d r
r r
r r d rd rr r r r
Second order expansion of DFT total energySecond order expansion of DFT total energy
r r
(I)
(II)
(III)
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0
21 1
2xc r r d rd r
r r r r
(III) Second order term(III) Second order term
Monopolapproximation: 00q F
Two limits:
1) | ' |r r
2)| ' | 0r r
20
1
2[ , ]
q q
R
220
22
2[ , ]
1 1
2 2
atEq U q
q
New parameter U: calculated for every element from DFT
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0
21 1
2xc r r d rd r
r r r r
(III) Second order term(III) Second order term
220
22
2[ , ]
1 1
2 2
atEq U q
q
2pO
2pO2pC
2pO2pC
2pC
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0
21 1
2xc r r d rd r
r r r r
Combine the two limitsCombine the two limits
1) | ' |r r
2)| ' | 0r r
20
1
2[ , ]
q q
R
220
22
2[ , ]
1 1
2 2
atEq U q
q
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0
2
00 002 21 1
1 1
14
xc F Fr r r r
R R U U
r r r r r r
(III) Second order term: Klopman-Ohno approximation
(III) Second order term: Klopman-Ohno approximation
R
2 21 1
1
14
R R U U
r r
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Determination of Gamma in DFTB Determination of Gamma in DFTB
- Consider atomic charge densities
~ Rcov
-Calculate coulomb integrals ( ) for 2 spherical charge densities:
-deviation from 1/R for small R
R=0: 1/ = 3.2 UHubbard
~ exp( ( ) / )r R
0
2
00 00
1 xc F Fr r r r
r r r r
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Klopman-Ohno vs DFTB GammaKlopman-Ohno vs DFTB Gamma
2 21 1
1
14
R R U U
r r
1/r
DFTB-
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0
0 00 0 0
2
0
1 1
2
ˆ
2
1
occ
i eff i
K x x
c
i
x
c c
r rd rd r r d r
r
T
r
r r d rd rr r r r
r rr r r
r r rr r
rr
r
r
rr
Approximate density-functional theory Elstner et al. Phys. Rev. B 58 (1998) 7260
Approximate density-functional theory Elstner et al. Phys. Rev. B 58 (1998) 7260
00 1
2tot re
occi i
i pi
n c c H q q
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Hamilton-MatrixelementsHamilton-Matrixelements
•non-scc: neglect of red contributions
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Comparison to SE models: Matrix elementsComparison to SE models: Matrix elements
0 01(
2,),H qH S
•Extended Hueckel (can be derived from DFT)
0pH I
0 0 00 1( )
2S K HH H
0H
0 | ( ) |H T V
'p QUH I
Q q Z
0q q q
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Comparison to SE models: Matrix elementsComparison to SE models: Matrix elements
0 01(
2,),H qH S
•Fenske Hall
0H
0 | ( ) |H T V
p ABH QI
0(
1( )
2) S QH S I I
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Comparison to SE models: Matrix elementsComparison to SE models: Matrix elements
•formal similarity in Hamiltonmatrixelements
•Very different in determination of matrixelements
•DFTB: incorporate strengths, but also fundamental weaknesses of DFT
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Differences w.r. to SE models: e.g. JDifferences w.r. to SE models: e.g. J
( / 2)
1/
AA
B BB
H U P
Z R
e.g. MNDO
Coulomb part J: Approx. by multipole-multipole interaction
e.g. CNDO
..AAB AB ne
B
H U q V
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Differences to SE models: e.g. JDifferences to SE models: e.g. J
Coulomb part accounts for e-e interaction due to interaction of atomic charges: looks similar to 2nd order term in DFTB.
..AAB AB ne
B
H U q V
1 1
2 2 A B ABAB
r rd rd r q q
r r
r r
r rr r
CNDO
MNDO: simple charge-charge higher multipoles
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Differences to SE models: e.g. JDifferences to SE models: e.g. J
DFTB: how is e-e interaction treated? consider J
0 0
0
1
2
1
2
1
2
r rd rd r
r r
r rd rd r
r r
r rd rd r
r r
r rd rd r
r r
r rr r
r r
r rr r
r r
r rr r
r r
r rr r
r r
1
2 A B ABAB
q q
repE
0 0H rdr
r r
r
r r
0
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Extensions of DFTBExtensions 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
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0
0 00 0 0
2
0
1 1
2
ˆ
2
1
occ
i eff i
K x x
c
i
x
c c
r rd rd r r d r
r
T
r
r r d rd rr r r r
r rr r r
r r rr r
rr
r
r
rr
Approximate density-functional theoryApproximate density-functional theory
00 1
2re
occi i
t t i pi
o n c c H q q
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0
0 00 0 0
1
ˆ
2
occ
i eff i
K xc xc
i
r rd rd r r d r
r
T
r
r r
r r r rr r
ExtensionsExtensions
00
occi i
ii
ro pt t en c c H
FAQS:
- better basis sets much higher cost
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0
21 1
2xc r r d rd r
r r r r
r r r r
r r r r
1
2q q
ExtensionsExtensions
FAQS:
- higher order expansion
- monopole multipole
- inspection of gamma?
No additional cost!
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Determination of Gamma Determination of Gamma
deviation from 1/R for small R
R=0: 1/ = 3.2 Uhubbard
Is this valid throughout the periodic table?
What is the relation between
‚atomic size‘ and
chemical hardness?
~ exp( ( ) / )r R
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Gamma: Rcov ~ 1/U ? Gamma: Rcov ~ 1/U ?
U-Hubbard
N
B-F
Si-Cl
H
R covalent
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Gamma requires : 3.2*Rcov= 1/U?Gamma requires : 3.2*Rcov= 1/U?
H
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U vs Rcov: Hydrogen atomU vs Rcov: Hydrogen atom
H
Si
C
O
R covalent
U-Hubbard
N
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U vs Rcov: H not in line!U vs Rcov: H not in line!
H
U-Hubbard
Gamma requires: 3.2*Rcov= 1/U
size of H overestimated based on hardness value: H has same size like N!
In DFTB, H is 0.73A instead of 0.33A!N
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On-site interaction and coulomb scaling: HOn-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!
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Modified Gamma for H-bondingchange only X-H interaction!
Modified Gamma for H-bondingchange only X-H interaction!
22 1 1 2
1
1exp( )
4R U U R
/ 1/ *1 dampS fR R S
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Modified Gamma for H-bondingModified 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
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H-bonds: water clusterMP2 from KS Kim et al 2000
H-bonds: water clusterMP2 from KS Kim et al 2000
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0
0 00 0 0
2
0
1 1
2
ˆ
2
1
occ
i eff i
K x x
c
i
x
c c
r rd rd r r d r
r
T
r
r r d rd rr r r r
r rr r r
r r rr r
rr
r
r
rr
Expansion to higher order?Expansion to higher order?
00 1
2re
occi i
t t i pi
o n c c H q q
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Charged systems with localized chargeCharged systems with localized charge
E.g.: H2O OH- + H+
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
2q q
21
2q U
( )U U q
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Problems with charged systems: inclusion of third order correction into DFTB
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.
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Deprotonation energiesDeprotonation 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
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SCC-DFTB: 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)
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SCC-DFTB: TestsSCC-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
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SCC-DFTB: TestsSCC-DFTB: Tests
4) Transition metal complexes
5) Properties
IR, Raman, NMR
excited states with TD-DFT
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SCC-DFTB Tests 1: Elstner et al., PRB 58 (1998) 7260SCC-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, 2691981.
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SCC-DFTB Tests 2: T. Krueger, et al., J.Chem. Phys. 122 (2005) 114110.
SCC-DFTB Tests 2: T. Krueger, et al., J.Chem. Phys. 122 (2005) 114110.
With respect to G2:mean ave. dev.: 4.3 kcal/molemean dev.: 1.5 kcal/mole
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SCC-DFTB Tests 3: Sattelmeyer & Jorgensen, (to be published)
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
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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) 47 0.23 0.25 0.23 0.39
SCC-DFTB Tests 3: Sattelmeyer & Jorgensen, (to be published)
SCC-DFTB Tests 3: Sattelmeyer & Jorgensen, (to be published)
• ok: H-bonds, ions• quite bad: S
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SCC-DFTB Tests: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 for CHMalolepsza, Witek & Morokuma: CPL 412 (2005) 237.
Witek & Morokuma, J Comp Chem. 25 (2004) 1858.
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H-bondsHan 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.
H-bondsHan 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
- structures well reproduced
Model peptides: N-Acetyl-(L-Ala)n N‘-Methylamide (AAMA) + 4 H2O
H2O-dimer complexes Cs, C2v
NH3-NH3- and NH3-H2O-dimer
Coulomb interaction
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Performance of DFTBPerformance of DFTB
Small molecules don’t tell the whole story
Test for large ones:
- peptides
- DNA, sugar
- other extended structures
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Secondary-structure elements for Glycine und Alanine-based polypeptides
Secondary-structure elements for Glycine und Alanine-based polypeptides
N = 1 (6 stable conformers) 310 - helix
stabilization by internal H-bonds
between i and i+3
N
R-helix
between i and i+4
DFTB very good for:
- relative energies
- geometries
- vib. freq. o.k.!
main problem for DFT(B): dispersion!
AM1, PM3, MNDO not convincing
OM2 much improved (JCC 22 (2001) 509)
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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
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
N = 1 (6 stable conformers)
N
Relative energies, structures and vibrational properties: N=1-8
2 R P
(6-31G*)
C7
eq C5ext C7
ax
MP4-BSSE
MP2
B3LYP
SCC-DFTB
E relative energies (kcal/mole)
MP4-BSSE: Beachy et al, BSSE corrected at MP2 level
Ace-Ala-Nme
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SCC-DFTB vs. NDDO (MNDO, AM1, PM3)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
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TD-DFTB and excited statesTD-DFTB and excited states
Problems of TD-DFT:
Combination of DFTB and OM2!
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Problems: 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!)
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DFT Problems: DFT Problems:
(1) Ex: Self interaction error. J- Ex = 0 !: Band gaps, barriers
(2) Ex: wrong asymptotic form; - 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
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DFT and VdW interactionsDFT and VdW interactions
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DFT and VdW interactionsDFT and VdW interactions
E ~ 1/R6
2 Problems:
- Pauli repulsion: exchange effect
~ exp(R) or 1/R12
- attraction due to correlation
~ -1/R6
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Dispersion forces - Van der Waals interactionsElstner et al. JCP 114 (2001) 5149
Dispersion forces - Van der Waals interactionsElstner 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 R0 = 3.8Å (für O, N, C)
E ~ 1/R6
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DFTB + dispersionDFTB + 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
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With help fromWith help from
QM/MM: DFTB
Q. Cui, Madison
H. Hu, J. Herrmans UNC
D. York, Minnesota
A. Roitberg, Florida
Morokuma, Witek Zheng, Irle
IR, RAMAN, metals
Dispersion, DNA P. Hobza, Nat. Academie, Prague
H. Liu, W. Yang, Duke
O(N), COSMO, GB
DFTB:Frauenheim, Seifert
& Suhai groups
DFG, Univ. Paderborn