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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 E ~ 1/R 6 2 Problems: - Pauli repulsion: exchange effect ~ exp( R ) or 1 /R 12 - attraction due to correlation ~ -1 /R 6

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DFT Problem - B88 exchange: too repulsive ? - PBEx/PW91x: too attractive already at Ex only level - LDA finds often binding! E ~ 1/R 6 - fix E x - correlation E c ? E c ?? E x ??

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Ar 2 with E x 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 E x Xu & Yang JCP 116 (2002) 515 BPW91 BLYP B3LYP PW91 B3LYP contains only 20% HF exchange!

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BPW91 vs PW91: attraction only due to exchange!!!!! Correlation not significant for PW91 and LYP BPW91 BLYP B3LYP PW91 Popular Functionals: role of E c Xu & Yang JCP 116 (2002) 515

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Perez-Jorda et al. JCP 110 (1999) 1916 DFT HF x + E c : some Ec lead to (over-) binding, some don’t! Popular Functionals: role of E c

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Does overlap matter? Xu & Yang JCP 116 (2002) 515 Elstner et al. JCP 114 (2001) 5149 GGA DFTB

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DFT and VdW interactions: the problem E ~ 1/R 6 E c = 0 E xc = ??

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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 HF+dis: Add f (R ) C 6 /R 6 to DFT total energy C 6 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 interactions Elstner et al. JCP 114 (2001) 5149 E tot = E SCC-DFTB - f (R ) C 6 /R 6 C 6 via Slater-Kirckwood combination rules of atomic polarizibilities after Halgreen, JACS 114 (1992) 7827. damping f(R ) = [1-exp(-3(R /R 0 )7 )] 3 R 0 = 3.8Å (für O, N, C) E ~ 1/R 6

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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 f(R ) = [1-exp(-3(R /R 0 )7 )] 3 E tot = E SCC-DFTB - f (R ) C 6 /R 6 R 0 : 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) 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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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 From Wu and Yang 2002 Does not work for all E xc functionals properly Wu and Wang 2002 Zimmerli et al.2004

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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 ) C 6 /R 6 -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 /R 0 )7 )] 3 - choice of “cutoff” radius from Grimme 2004

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Choice of f damp f damp balances several effects - contribution from E x /E c in overlap region - double counting of E c - BSSE and BSIE - missing higher order terms 1/R**8 … 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 From Wu and Yang 2002 Location of minimum For A-A stack

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Grimme JCC 25 (2004) 1463 - hybridisation dependence - empirical vs. new fits Very similar in various approaches s6: PW91: 0.7 BLYP: 1.4 Scaling: -Basis set dependent -functional dependent

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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 DFT + empirical dispersion: 2nd generation Grimme JCC 25 (2004) 1463: scale BLYP + disp with 1.4 scale PW91 + disp with 0.7 3rd generation: revPBE, XLYP and s6=1

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Applications of DFTB-D

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Benzene (from Irle/Morokuma, Emory)

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RHF, MP2 (both CP corrected) and DFTB E on benzene dimers: Benzene (from Irle/Morokuma, Emory)

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Hybride materials

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O(N)-QM/MM-molecular-dynamics for DNA-dodecamer in H 2 O Elstner et al. in preparation DNA-Dodecamer 758 + 2722 H 2 O + 22 Na periodic BC-Ewald-summation dispersion in QM-region MD-simulation at 300 K parallel-16 processors SP2 energy/forces: 1 – 2 sec. 10 ps/day 1-st stable QM/MM ns-scale dynamic simulation

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Intercalation: Ethidium – AT Reha et al JACS 2003

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Secondary-structure elements for Glycine und Alanine- based polypeptides: ß-sheets, helices and turn Elstner, et a.. Chem. Phys. 256 (2000) 15 N = 1 (6 stable conformers) 3 10 - helix stabilization by internal H-bonds N-fold periodicity between i and i+3 N R -helix between i and i+4 For increasing N: energetics of different conformers, geometries, vibrations

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Glycine and Alanine based polypeptides in vacuo Elstner et al., Chem. Phys. 256 (2000) 15 N = 1 (6 stable conformers) N Relative energies, structures and vibrational properties: N=1-8 (6-31G*) C 7 eq C 5 ext C 7 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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Polypeptides in vacuo Effect of dispersion: favors more compact structures N = 2 (6-31G*) Ace-Ala 2 -Nme BLYP B3LYP MP2 HF SCC-DFTB C 7 eq C 5 ext 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 DFT/DFTB ? 3 10 - helix R -helix peptide size DFT: crossover only for N~20 !! solvation?? E N

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Secondary structure : Influence of aqueous solution Cui et al, JPCB 105 (2001) 569 3 10 - helix R -helix 3 10 – helix: occurence for N<8 in database QM/MM MD of octa-Alanine: 3 10 - helix converts into R -helix within 10 ps Situation in Protein?

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Molecular-dynamics for Crambin in H 2 O-solution O(N)-QM/MM simulation Liu et al. PROTEINS 44 (2001) 484 Crambin (639) + 2400 H 2 O MD simulation for 0.35 ns energy and interatomic forces parallel (16-node SP2): 2 sec.

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Influence of Dispersion Liu et al. PROTEINS 44 (2001) 484 QM/MM MD-Simulation Crambin in Solution HF DFT/DFTB ? SCC-DFTB + DIS MP2

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Enkephalin: ~30 local minima 3 cluster Jalkanen et al. to be published Enkephalin: ~30 local minima 3 cluster Jalkanen et al. to be published C5 compact extended double bend single bend

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Enkephalin: MP2/6-31G* vs DFTB-dis//DFTB-dis compact extended conformer kcal Rel. energy (kcal) vs. conformer b a c

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Enkephalin: MP2/6-31G* vs DFTB//DFTB-dis compact extended conformer kcal

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Enkephalin: MP2 vs B3LYP//DFTB-dis compact extended conformer kcal

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Enkephalin: MP2 vs B3LYP-dis//DFTB-dis compact extended conformer kcal

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Enkephalin: MP2 vs PBE+dis//DFTB-dis compact extended conformer kcal

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Enkephalin: MP2 vs PBE//DFTB-dis compact extended conformer kcal

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Enkephalin: MP2 vs PBE+dis//DFTB-dis compact extended conformer kcal

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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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DFT+large soft matter structures: don‘t do without dispersion! - large impact on relative energies - stabilizes more compact structures: relevant secondary structures may not be stable without!

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