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.

The SCC-DFTB method applied to organic and biological systems: successes, extensions

and problems.

Marcus Elstner

Physical and Theoretical ChemistryTechnical University of Braunschweig

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

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

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

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

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

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

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

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

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)

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

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)

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

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)

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

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

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

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

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

Klopman-Ohno vs DFTB GammaKlopman-Ohno vs DFTB Gamma

2 21 1

1

14

R R U U

r r

1/r

DFTB-

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

Hamilton-MatrixelementsHamilton-Matrixelements

•non-scc: neglect of red contributions

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

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

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

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

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

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

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

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

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

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!

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

Gamma: Rcov ~ 1/U ? Gamma: Rcov ~ 1/U ?

U-Hubbard

N

B-F

Si-Cl

H

R covalent

Gamma requires : 3.2*Rcov= 1/U?Gamma requires : 3.2*Rcov= 1/U?

H

U vs Rcov: Hydrogen atomU vs Rcov: Hydrogen atom

H

Si

C

O

R covalent

U-Hubbard

N

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

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!

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

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

H-bonds: water clusterMP2 from KS Kim et al 2000

H-bonds: water clusterMP2 from KS Kim et al 2000

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

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

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.

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

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)

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

SCC-DFTB: TestsSCC-DFTB: Tests

4) Transition metal complexes

5) Properties

IR, Raman, NMR

excited states with TD-DFT

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.

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

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

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

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.

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

Performance of DFTBPerformance of DFTB

Small molecules don’t tell the whole story

Test for large ones:

- peptides

- DNA, sugar

- other extended structures

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)

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

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

TD-DFTB and excited statesTD-DFTB and excited states

Problems of TD-DFT:

Combination of DFTB and OM2!

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!)

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

DFT and VdW interactionsDFT and VdW interactions

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

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

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

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