DFT and VdW interactions

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DFT and VdW interactions
Marcus Elstner Physical and Theoretical
Chemistry, Technical University of Braunschweig

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DFT and VdW interactions

• 2 Problems
• Pauli repulsion exchange effect
• exp(R??) or 1/R12??
• - attraction due to correlation
• -1/R6??

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DFT Problem

• B88 exchange too repulsive ?
• PBEx/PW91x too attractive
• already at Ex only level
• LDA finds often binding!

Ex ??

• fix Ex
• correlation Ec?

Ec ??
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Ar2 with Ex only

• B too repulsive,
• PW91x too attractive
• Complete mess with
• DFT

Wu et al. JCP 115 (2001) 8748
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Popular Functionals role of Ex
BPW91 BLYP B3LYP PW91
B3LYP contains only 20 HF exchange!

Xu Yang JCP 116 (2002) 515
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Popular Functionals role of Ec
Xu Yang JCP 116 (2002) 515
BPW91 BLYP B3LYP PW91

• BPW91 vs PW91 attraction only due to
  exchange!!!!!
• Correlation not significant for PW91 and LYP

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Popular Functionals role of Ec
Perez-Jorda et al. JCP 110 (1999) 1916 DFT
HFx Ec some Ec lead to (over-) binding,
some dont!

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Does overlap matter?
GGA
DFTB

Xu Yang JCP 116 (2002) 515
Elstner et al. JCP 114 (2001) 5149
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DFT and VdW interactions the problem
Exc ??
Ec 0
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DFT and VdW interactions solutions
Adding empirical dispersion Elstner et al. JCP
114 (2001) 5149 Xu Yang JCP 116 (2002)
515 Zimmerli et al. JCP 120 (2004) 2693 Grimme
JCC 25 (2004) 1463 DFT model for empircal
dispersion on top of HF Becke Johnson JCP 124
(2006) 014104 Put it into the pseudopotential v.
Lilienfeld et al. PRB 71 (2005) 195119 Find a
new dispersion functional Dion, et al. Phys.
Rev. Lett. 92 (2004) 246401 JCP 124 (2006)
164106 Kamiya et al. JCP 117 (2002) 6010.
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Adding empirical dispersion
Following the idea of HFdis Add ???f
(R??) C6 /R6?? to DFT total energy
C6 empirical values Elstner, Hobza et al.
JCP 114 (2001) 5149 To be successfull Ex
should be well-behaved (i.e. like HF) Ec
double counting

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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)
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How to get Dispersion coefficients?Halgren JACS
114 (1992) 7827
London, Phys. Chem. (Leipzig) B 11(1930) 222
Slater Kirkwood. Phys. Rev. 37 (1931)
682. Kramer Herschbach J. Chem. Phys. 53
(1970) 2792 effective electron number
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DFTB input
Etot ESCC-DFTB - ???f (R??) C6 /R6??
f(R??) 1-exp(-3(R??/R0)7)3

• R0 e.g. 3.8 for ONC
• Atomic polarizabilities
• hybridisation dependent
• Effective electron number (from Halgren)

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DFTB dispersion
Sponer et al. J.Phys.Chem. 100 (1996) 5590 Hobza
et al. J.Comp.Chem. 18 (1997) 1136stacking
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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DFT empirical dispersion 1st generation

• 1) Problem of unbalanced Ex
• 2) Problem of Ec?? Which one to choose?
• ? Large variation in results when adding
  dispersion
• Wu and Wang 2002
• Zimmerli et al 2004

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DFT and empirical dispersion
Does not work for all Exc functionals properly Wu
and Wang 2002 Zimmerli et al.2004

From Wu and Yang 2002
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DFT empirical dispersion 2nd generation

• 1) Problem of unbalanced Ex
• 2) Problem of Ec?? Which one to choose?
• ? Large variation when adding dispersion
• Grimme 2004 scale BLYP dispersion with 1.4
• scale PW91 dispersion with 0.7

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???f (R??) ? C6 /R6??

• choice of C6 coefficients
• Choice of damping function

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Choice of C6 coefficients

• hybridisation dependence vs. atomic values
• empirical values
• ? Very similar in various approaches

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Choice of damping function

• various functional forms
• - Fermi-function
• - f(R??) 1-exp(-3(R??/R0)7)3
• choice of cutoff radius

from Grimme 2004
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Choice of fdamp

• fdamp balances several effects
• - contribution from Ex/Ec in overlap region
• - double counting of Ec
• BSSE and BSIE
• missing higher order terms 1/R8
• Determination completely empirical
• Choose, to reproduce interaction energies for
  large set of stacked compounds

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Choice of fdamp
However, form of fdamp may be crucial
Location of minimum For A-A stack

From Wu and Yang 2002
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Grimme JCC 25 (2004) 1463

• s6
• PW91 0.7
• BLYP 1.4
• Scaling
• -Basis set dependent
• functional dependent

• hybridisation dependence
• empirical vs. new fits
• ? Very similar in various approaches

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DFT empirical dispersion 3rd generation

• 1) Problem of unbalanced Ex
• 2) Problem of Ec?? Which one to choose?
• ? Large variation in results when adding
  dispersion
• mix PW91x and Bx
• revPBE
• meta GGA??
• balanced damping function, no scaling

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DFT empirical dispersion 1st generation

• 1) Problem of unbalanced Ex
• 2) Problem of Ec?? Which one to choose?
• ? Large variation in results when adding
  dispersion
• Wu and Wang JCP 116 (2002) 515
• Zimmerli et al. JCP 120 (2004) 2693

3rd generation revPBE, XLYP and s61
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Applications of DFTB-D

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Benzene (from Irle/Morokuma, Emory)
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Benzene (from Irle/Morokuma, Emory)
RHF, MP2 (both CP corrected) and DFTB DE on
benzene dimers
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Hybride materials
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O(N)-QM/MM-molecular-dynamics for DNA-dodecamer
in H2OElstner et al. in preparation

• DNA-Dodecamer 758 2722 H2O 22 Na
• periodic BC-Ewald-summation
• dispersion in QM-region
• MD-simulation at 300 K
• parallel-16 processors SP2energy/forces 1 2
  sec. ? 10 ps/day

1-st stable QM/MM ns-scale dynamic simulation
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Intercalation Ethidium ATReha et al JACS 2003

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Secondary-structure elements for Glycine und
Alanine-based polypeptides ß-sheets, helices and
turnElstner, et a.. Chem. Phys. 256 (2000) 15
For increasing N energetics of different
conformers, geometries, vibrations
N 1 (6 stable conformers)
aR-helix
310 - helix
N-fold periodicity
stabilization by internal H-bonds
between i and i3
between i and i4
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Glycine and Alanine based polypeptides in
vacuoElstner et al., Chem. Phys. 256 (2000) 15
Relative energies, structures and vibrational
properties N1-8
N 1 (6 stable conformers)
E relative energies (kcal/mole)
B3LYP
(6-31G)
MP2
MP4-BSSE
SCC-DFTB
Ace-Ala-Nme
C7eq C5ext C7ax

MP4-BSSE Beachy et al, BSSE corrected at MP2
level
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Polypeptides in vacuoEffect of dispersion
favors more compact structures
(6-31G)
N 2
BLYP B3LYP
HF
MP2
SCC-DFTB
Ace-Ala2-Nme

C7eq C5ext BI
BII BI BII
DFT relative stability of compact vs. extended
structures?
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Secondary structure formation Elstner et al.,
Chem. Phys. 256 (2000) 15
E
DFT/DFTB ?
310 - helix
aR-helix
N
peptide size
DFT crossover only for N20 !! ? solvation??
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Secondary structureInfluence of aqueous
solutionCui et al, JPCB 105 (2001) 569
310 helix occurence for Nlt8 in database
QM/MM MD of octa-Alanine 310 - helix
converts into aR-helix within 10 ps
310 - helix
aR-helix
Situation in Protein?
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Molecular-dynamics for Crambin in H2O-solution
O(N)-QM/MM simulationLiu et al. PROTEINS 44
(2001) 484
Crambin (639) 2400 H2O
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Influence of Dispersion Liu et al. PROTEINS 44
(2001) 484
QM/MM MD-Simulation Crambin in Solution
HF?
DFT/DFTB ?
MP2
SCC-DFTB DIS ?
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Enkephalin 30 local minima? 3 cluster Jalkanen
et al. to be published
single bend
double bend
compact
extended
C5
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Enkephalin MP2/6-31G vs DFTB-dis//DFTB-dis
compact ? extended
kcal
c
a
b
conformer
Rel. energy (kcal) vs. conformer
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Enkephalin MP2/6-31G vs DFTB//DFTB-dis
compact ? extended
kcal
conformer
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Enkephalin MP2 vs B3LYP//DFTB-dis
compact ? extended
kcal
conformer
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Enkephalin MP2 vs B3LYP-dis//DFTB-dis
compact ? extended
kcal
conformer
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Enkephalin MP2 vs PBEdis//DFTB-dis
compact ? extended
kcal
conformer
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Enkephalin MP2 vs PBE//DFTB-dis
compact ? extended
kcal
conformer
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Enkephalin MP2 vs PBEdis//DFTB-dis
compact ? extended
kcal
conformer
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CONCLUSIONS

• Dispersion favors compact structures 15
  kcal/mole
• MP2/6-31G
• - internal BSSE
• - higher level correlation contribution
• -PBE and B3LYP differ in stability of extended
  (C5) confs
• -B3LYP overestimates Pauli repulsion N-H...

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DFTlarge soft matter structures dont do
without dispersion!

• large impact on relative energies
• stabilizes more compact structures
• relevant secondary structures may
• not be stable without!