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## Approximate methods for large molecular systems

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Approximate methods for large molecular systems

Marcus Elstner

Physical and TheoreticalChemistry, Technical Universityof Braunschweig

Motivation

Structure-formation, atomic-scale related properties and processes

Si1600

MoS2

a-SiCN-ceramics

Si21

C60-trimer

defects, doping

GaN-devices

4H-SiC-surfaces

Reactions in biological Systems

Alcohol DeHydrogenase

Aquaporin

Photosynthetic Reaction Center

Catalysis

Proton Transfer

Photochemistry

Electron/Energy Transfer

bR

Need QM

description

Photochemistry

Computational challange

- ~ 1.000-10.000 atoms
- ~ ns molecular dynamics simulation
- (MD, umbrella sampling)
- weak bonding forces
- chemical reactions
- treatment of excited states

Molecular Mechanics

SE-QM

approx-DFT

HF, DFT

‚multiscale business‘fs ps ns time

CI, MP

CASPT2

Length scale

nm

predictivity

Size problem:

number of

structures

MD, MC, GA

time scale of process

MD, MC -- RP, TST

ab initio, SE MM

size of system: number of atoms

Semi-empirical /approximate methods

- approximation, neglect and parametrization of interaction integrals from ab-initio and DFT methods
- HF-based:
- CNDO, INDO, MNDO, AM1, PM3, MNDO/d, OM1,OM2
- DFT-based:
- SCC-DFTB, DFT- 3center- tight binding (Sankey)
- Fireballs --- > Siesta DFT code
- ~ 1000 atoms, ~ 100 ps MD

Approximate density-functional theory:SCC-DFTBSelf consistent - charge density functional tight-binding

- Seifert (1980-86): Int. J. Quant Chem., 58, 185 (1996).
- O-LCAO; 2-center approximation: approximate DFT
- http://theory.chm.tu-dresden.de
- Frauenheim et al. (1995): Phys. Rev. B 51, 12947 (1995).
- efficient parametrization scheme: DFTB
- www.bccms.uni-bremen.de
- Elstner et al. (1998): Phys. Rev. B 58, 7260 (1998).
- charge self-consistency: SCC-DFTB
- www.tu-bs.de/pci

approximate DFT

Extensions and Combinations:

TD-DFTB-LR

O(N)-QM/MM

divide+conquer

H. Liu W. Yang

Duke Univ

QM/MM

AMBER: Han, Suhai DKFZ

CHARMM: Cui, Karplus; Harvard

TINKER: Liu, Yang; Duke

CEDAR: Hu, Hermans; NC Univ

SCC-DFTB

Solvent

Cosmo: W. Yang

GB: H. Liu

DISPERSION

P. Hobza, Prague

Electron

Transport

A. Di Carlo

TD-DFTB

R. Allen Texas A&M

SCC-DFTB:

- available for H C N O S P Zn

(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: 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: Tests

4) Transition metal complexes

5) Properties

- IR, Raman, NMR
- excited states with TD-DFT
- Transport calculations

SCC-DFTB: Reviews

- Application to biological molecules
- M. Elstner, et al. ,A self-consistent carge density-functional based tight-binding scheme for large biomolecules, phys. stat. sol. (b) 217 (2000) 357.
- Elstner, et al. An approximate DFT method for QM/MM simulations of biological structures and processes. J. Mol. Struc. (THEOCHEM), 632 (2003) 29.
- M. Elstner, The SCC-DFTB method and its application to biological systems, Theoretical Chemistry Accounts, in print 2006.

2) Focus on solids and nanostructures

- T. Frauenheim, et al., Atomistic Simulations of complex materials: ground and excited state properties, J. Phys. : Condens. Matter 14 (2002) 3015.
- Th. Frauenheim et al. A self-consistent carge density-functional based tight-binding method for predictive materials simulations in physics, chemistry and biology, phys. stat. sol. (b) 217 (2000) 41.
- G. Seifert, in: Encyclopedia of Computational Chemistry (Wiley&Sons 2004)

SCC-DFTB Tests 1: Elstner et al., PRB 58 (1998) 7260

- Performance for small organic molecules
- (mean absolut deviations)
- Reaction energiesa): ~ 5 kcal/mole
- Bond-lenghtsa) : ~ 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, 269
- 1981.

SCC-DFTB Tests 2: T. Krueger, et al., J.Chem. Phys. 122 (2005) 114110.

With respect to G2:

mean ave. dev.: 4.3 kcal/mole

mean dev.: 1.5 kcal/mole

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 CH

Malolepsza, Witek & Morokuma: CPL 412 (2005) 237.

Witek & Morokuma, J Comp Chem. 25 (2004) 1858.

H-bonded systems: water

CCSD(T): 5.0 kcal/mole (Klopper et al PCCP 2000 2, 2227)

BLYP: 4.2 kcal/mole

PBE: 5.1 kcal/mole

B3LYP: 4.6 kcal/mole

HF: 3.7 kcal/mole

(from Xu&Goddard, JCPA 2004)

For larger systems:

DFTB: 3.3 kcal/mole

HF: 5.7 kcal/mole @ 6-31G*

B3LYP: 6.8 kcal/mole @ 6-31G* ~2 kcal/mole BSSE (BSIE)

NH3-NH3- and NH3-H2O-dimer

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.

Coulomb

interaction

- ~1-2kcal/mole too weak
- relative energies reasonable
- structures well reproduced

Model peptides: N-Acetyl-(L-Ala)nN‘-Methylamide (AAMA) + 4 H2O

Secondary-structure elements for Glycine und Alanine-based polypeptidesElstner, et al.. Chem. Phys. 256 (2000) 15

aR-helix

N = 1 (6 stable conformers)

310 - helix

stabilization by internal H-bonds

between i and i+4

between i and i+3

- main problem for DFT(B): dispersion!
- AM1, PM3, MNDO quite bad
- OM2 much improved (JCC 22 (2001) 509)

- DFTB very good for:
- relative energies
- geometries
- vib. freq. o.k.!

Glycine and Alanine based polypeptides in vacuoElstner et al., Chem. Phys. 256 (2000) 15 Elstner et al. Chem. Phys. 263 (2001) 203 Bohr et al., Chem. Phys. 246 (1999) 13

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

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

Strength of SCC-DFTB

Structure of large molecules

- dynamics

- relative energies

DNA:

A. V. Shiskin, et al., Int. J. Mol. Sci. 4 (2003) 537.

O. V. Shishkin, et al., J. Mol. Struc. (THEOCHEM) 625 (2003) 295.

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!)

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

DFT Problems:

Ex: Self interaction error. J- Ex = 0 !: Band gaps, barriers

Ex: wrong asymptotic form; - HOMO << Ip: virtual KS orbitals

Ex: ‚somehow too local‘; overpolarizability, CT excitations

Ec: ‚too local‘: Dispersion forces missing

Ec: even much more ‚too local‘: isomerization reactions

Multi-reference problem

(1) –(3) of course related, cure: exact exchange!

DFT Problems: (very) selective publications

- Ex: PRB 23 (1981) 5048, JCP 109 (1998) 2604
- Ex: JCP 113 (2000) 8918, Mol. Phys. 97 (1999) 859.
- Ex: JPCA 104 (2000) 4755, JCP 119 (2003) 2943.
- Ec: JCP 114 (2001) 5149
- Ec: Angew. Chem. Int. Ed. 2006, 45, 4460 –4464
- Koch, Wolfram / Holthausen, Max C.A Chemist's Guide to Density Functional Theory, Wiley

Problems of DFT-GGA

- - overbinding of small molecules: CO... B3LYP, rev-PBE10 kcal
- transition metals: B3LYP, PB86 ..., spin states, energetics10-20 kcal
- - vib. Freqencies:
- conjugate systems: GGAs overpolarize PA‘s of respective proton donors10 kcal
- - H-bonds: ok with DFT, HF (cancellation of errors), water structure?
- proton transfer (PT) barriers: GGA<B3LYP < MP2< CCSD 2-4 kcal with B3LYP!
- Solution1: don‘t worry or don‘t care different functionals VERY different accuracy
- Solution2: use something else
- VdW- problem (dispersion)complete failure
- ‚Solution‘: empirical dispersion for GGAs
- excited states within TD-DFT: ionic, CT states, double excitations, Rydberg states
- Solution: exact exchange or CI-based methods

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