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

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 

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 ??

Ar 2 with E x only B too repulsive, PW91x too “attractive” Complete mess with DFT Wu et al. JCP 115 (2001) 8748

Popular Functionals: role of E x Xu & Yang JCP 116 (2002) 515 BPW91 BLYP B3LYP PW91 B3LYP contains only 20% HF exchange!

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

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

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

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

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) Put it into the pseudopotential v. Lilienfeld et al. PRB 71 (2005) Find a new dispersion functional Dion, et al. Phys. Rev. Lett. 92 (2004) ; [JCP 124 (2006) ] Kamiya et al. JCP 117 (2002) 6010.

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

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) damping f(R  ) = [1-exp(-3(R  /R 0 )7 )] 3 R 0 = 3.8Å (für O, N, C) E ~ 1/R 6

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

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)

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

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

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

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

  f (R  )  C 6 /R 6  -choice of C6 coefficients -Choice of damping function

Choice of C6 coefficients - hybridisation dependence vs. atomic values - empirical values  Very similar in various approaches

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

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

Choice of fdamp However, form of fdamp may be crucial From Wu and Yang 2002 Location of minimum For A-A stack

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

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

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

Applications of DFTB-D

Benzene (from Irle/Morokuma, Emory)

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

Hybride materials

O(N)-QM/MM-molecular-dynamics for DNA-dodecamer in H 2 O Elstner et al. in preparation DNA-Dodecamer 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

Intercalation: Ethidium – AT Reha et al JACS 2003

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) 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

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

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?

Secondary structure formation Elstner et al., Chem. Phys. 256 (2000) 15 DFT/DFTB ? helix  R -helix peptide size DFT: crossover only for N~20 !!  solvation?? E N

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

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

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

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

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

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

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

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

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

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

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

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...

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!