DFT and VdW interactionsDFT and VdW interactions

Marcus Elstner

Physical and Theoretical Chemistry, Technical University of Braunschweig

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

DFT ProblemDFT Problem

- B88 exchange: too repulsive ?

- PBEx/PW91x: too attractive

already at Ex only level

- LDA finds often binding!

E ~ 1/R6

- fix Ex

- correlation Ec?Ec ??

Ex ??

Ar2 with Ex onlyAr2 with Ex only

• B too repulsive,

• PW91x too “attractive”

Complete mess with

DFTWu et al. JCP 115 (2001) 8748

Popular Functionals: role of ExPopular Functionals: role of Ex

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 EcPopular Functionals: role of Ec

Xu & Yang JCP 116 (2002) 515

Perez-Jorda et al. JCP 110 (1999) 1916

DFT HFx + Ec:

some Ec lead to (over-) binding, some don’t!

Popular Functionals: role of EcPopular Functionals: role of Ec

Does overlap matter?Does overlap matter?

Xu & Yang JCP 116 (2002) 515

Elstner et al. JCP 114 (2001) 5149

GGA

DFTB

DFT and VdW interactions: the problemDFT and VdW interactions: the problem

E ~ 1/R6

Ec = 0

Exc = ??

DFT and VdW interactions: solutionsDFT and VdW interactions: solutions

Adding empirical dispersionElstner 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 HFBecke & Johnson JCP 124 (2006) 014104

Put it into the pseudopotentialv. Lilienfeld et al. PRB 71 (2005) 195119

Find a new dispersion functionalDion, et al. Phys. Rev. Lett. 92 (2004) 246401; [JCP 124 (2006) 164106]

Kamiya et al. JCP 117 (2002) 6010.

Adding empirical dispersionAdding empirical dispersion

Following the idea of HF+dis:

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

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

How to get Dispersion coefficients?Halgren JACS 114 (1992) 7827

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 inputDFTB input

f(R) = [1-exp(-3(R/R0)7)]3

Etot = ESCC-DFTB - f (R) C6 /R6

• R0: e.g. 3.8 for ONC

• Atomic polarizabilities:

hybridisation dependent

• Effective electron number (from Halgren)

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

DFT + empirical dispersion: 1st generationDFT + 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 dispersionDFT and empirical dispersion

From Wu and Yang 2002

Does not work for all Exc functionals properly

Wu and Wang 2002

Zimmerli et al.2004

DFT + empirical dispersion: 2nd generationDFT + 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) C6 /R6

f (R) C6 /R6

-choice of C6 coefficients

-Choice of damping function

Choice of C6 coefficientsChoice of C6 coefficients

- hybridisation dependence vs. atomic values

- empirical values

Very similar in various approaches

Choice of damping functionChoice of damping function

- various functional forms

- Fermi-function

- f(R) = [1-exp(-3(R/R0)7)]3

- choice of “cutoff” radius

from Grimme 2004

Choice of fdampChoice 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/R**8 …

Determination completely empirical

Choose, to reproduce interaction energies for large set of stacked compounds

Choice of fdampChoice 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) 1463Grimme 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

DFT + empirical dispersion: 3rd generationDFT + 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 generationDFT + 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 generationDFT + 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=13rd generation: revPBE, XLYP and s6=1

Applications of DFTB-DApplications of DFTB-D

C

C

C

C

C

C

1.396

1.099

CC C

CC C

CCC

C CC

3.670

CCC

CC

C

CC

CC

C

C

5.295

CCC

CCC

CCC

CCC

6.403

3.325

CC

CC

CC

CC

CCCC

3.454

5.193

M S T

PD 1.108

3.556

DFTBDFTB-Dmonomer geometriesremain unchangedthroughout (but optimized)

(MP2/aug-cc-pVDZ (monomer frozen)){MP2/aug-cc-pVTZ (monomer frozen)}

(3.8){3.7}

(5.0){4.9}

(3.4){3.4}

(1.6){1.6}

Benzene (from Irle/Morokuma, Emory) Benzene (from Irle/Morokuma, Emory)

Monomer S-Dimer T-Dimer PD-DimerE [kcal/mol] E [kcal/mol] E [kcal/mol] E [kcal/mol]

RHF/cc-pVDZ//MP2/aug-cc-pVTZ 0.00 4.36 0.82 4.00RHF/aug-cc-pVDZ//MP2/aug-cc-pVDZ 0.00 3.60 -0.11 4.04RHF/aug-cc-pVTZ//MP2/aug-cc-pVTZ 0.00 5.10 1.42 5.02 MP2/cc-pVDZ//MP2/aug-cc-pVTZ 0.00 -2.71 -3.76 -4.23MP2/aug-cc-pVDZ//MP2/aug-cc-pVDZ 0.00 -2.90 -3.16 -4.28MP2/aug-cc-pVTZ//MP2/aug-cc-pVTZ 0.00 -3.26 -3.46 -4.67MP2/aug-cc-pVQZ//MP2/aug-cc-pVTZ 0.00 -3.37 -3.54 -4.79CCSD(T)/CBS (based on MP2-R12) 0.00 -1.81 -2.74 -2.78

DFTB//MP2/aug-cc-pVTZ 0.00 0.56 -0.31 0.38DFTB//MP2/aug-cc-pVTZ w/DISP 0.00 -4.02 -2.68 -4.36DFTB//DFTB 0.00 0.54 -0.34 -0.16DFTB//DFTB w/DISP 0.00 -4.54 -2.74 -4.60

RHF, MP2 (both CP corrected) and DFTB E on benzene dimers:

Benzene (from Irle/Morokuma, Emory) Benzene (from Irle/Morokuma, Emory)

Hybride materials Hybride materials

O(N)-QM/MM-molecular-dynamics for DNA-dodecamer in H2O

Elstner et al. in preparation

O(N)-QM/MM-molecular-dynamics for DNA-dodecamer in H2O

Elstner 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

Intercalation: Ethidium – ATReha et al JACS 2003

Intercalation: Ethidium – ATReha 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

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) 310 - helix

stabilization by internal H-bonds

N-fold periodicity

between i and i+3N

R-helix

between i and i+4

For increasing N: energetics of different conformers, geometries, vibrations

Glycine and Alanine based polypeptides in vacuoElstner et al., Chem. Phys. 256 (2000) 15

Glycine and Alanine based polypeptides in vacuoElstner et al., Chem. Phys. 256 (2000) 15

N = 1 (6 stable conformers)

N

Relative energies, structures and vibrational properties: N=1-8

2 L 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

Polypeptides in vacuoEffect of dispersion: favors more compact structures

Polypeptides in vacuoEffect of dispersion: favors more compact structures

N = 2

(6-31G*)

Ace-Ala2-Nme

BLYPB3LYP

MP2

HF

SCC-DFTB

C7eq C5

ext BI BII BI` BII`

DFT: relative stability of compact vs. extended structures?

Secondary structure formation Elstner et al., Chem. Phys. 256 (2000) 15

Secondary structure formation Elstner et al., Chem. Phys. 256 (2000) 15

DFT/DFTB ? 310 - helix R-helix

peptide size

DFT: crossover only for N~20 !! solvation??

E

N

Secondary structure:Influence of aqueous solutionCui et al, JPCB 105 (2001) 569

Secondary structure:Influence of aqueous solutionCui et al, JPCB 105 (2001) 569

310 - helix R-helix

310 – helix: occurence for N<8 in database

QM/MM MD of octa-Alanine:

310 - helix converts into R-helix within 10 ps

Situation in Protein?

Molecular-dynamics for Crambin in H2O-solution O(N)-QM/MM simulation

Liu et al. PROTEINS 44 (2001) 484

Molecular-dynamics for Crambin in H2O-solution O(N)-QM/MM simulation

Liu et al. PROTEINS 44 (2001) 484

Crambin (639) + 2400 H2O

MD simulation for 0.35 ns energy and interatomic forcesparallel (16-node SP2): 2 sec.

Influence of Dispersion Liu et al. PROTEINS 44 (2001) 484

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 clusterJalkanen et al. to be published

Enkephalin: ~30 local minima 3 clusterJalkanen et al. to be published

C5

compact

extended

double bendsingle bend

Enkephalin: MP2/6-31G* vs DFTB-dis//DFTB-disEnkephalin: 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-disEnkephalin: MP2/6-31G* vs DFTB//DFTB-dis

compact extended

conformer

kcal

Enkephalin: MP2 vs B3LYP//DFTB-disEnkephalin: MP2 vs B3LYP//DFTB-dis

compact extended

conformer

kcal

Enkephalin: MP2 vs B3LYP-dis//DFTB-disEnkephalin: MP2 vs B3LYP-dis//DFTB-dis

compact extended

conformer

kcal

Enkephalin: MP2 vs PBE+dis//DFTB-disEnkephalin: MP2 vs PBE+dis//DFTB-dis

compact extended

conformer

kcal

Enkephalin: MP2 vs PBE//DFTB-disEnkephalin: MP2 vs PBE//DFTB-dis

compact extended

conformer

kcal

Enkephalin: MP2 vs PBE+dis//DFTB-disEnkephalin: MP2 vs PBE+dis//DFTB-dis

compact extended

conformer

kcal

CONCLUSIONS 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!

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!