Approximate methods for large molecular systems

Approximate methods for large molecular systems

Marcus Elstner

Physical and Theoretical Chemistry, Technical University of Braunschweig

C60-trimer Si1600 MoS2

4H-SiC-surfacesdefects, doping GaN-devices

Structure-formation, atomic-scale related properties and processes

MotivationMotivation

Si21 a-SiCN-ceramics

Alcohol DeHydrogenase

Photosynthetic Reaction Center

Reactions in biological SystemsReactions in biological Systems

Aquaporin

Photochemistry

bRCatalysis

Proton Transfer

Photochemistry

Electron/Energy Transfer

Need QM description

Computational challangeComputational challange

~ 1.000-10.000 atoms

~ ns molecular dynamics simulation

(MD, umbrella sampling)

- weak bonding forces

- chemical reactions

- treatment of excited states

‚multiscale business‘‚multiscale business‘

CI, MPCASPT2

CI, MPCASPT2

Length scale

predictivity

Continuum electrostatics Continuum electrostatics

Molecular MechanicsMolecular Mechanics

f

s

ps

ns

t i

me

SE-QMapprox-DFT

SE-QMapprox-DFT

HF, DFTHF, DFT

nm

Size problem: Size problem:

number of structures

MD, MC, GA

time scale of processMD, MC -- RP, TST

size of system: number of atoms

ab initio, SE MM

Size problem: QM-Methods Size problem: QM-Methods

Hybride methods: QM/MM, QM/QM

Linear scaling: O(N)

SE/approx. Methods

Semi-empirical /approximate methodsSemi-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-DFTB

Self consistent - charge density functional tight-binding

Approximate density-functional theory:SCC-DFTB

Self 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: Extensions and Combinations:

O(N)-QM/MMdivide+conquer

H. Liu W. YangDuke Univ

QM/MMAMBER: Han, Suhai DKFZ CHARMM: Cui, Karplus; HarvardTINKER: Liu, Yang; Duke CEDAR: Hu, Hermans; NC Univ

DISPERSIONP. Hobza, Prague

TD-DFTB-LR

TD-DFTBR. Allen Texas A&M

SCC-DFTB

SolventCosmo: W. Yang

GB: H. Liu

Electron TransportA. Di Carlo

SCC-DFTB: 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: 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

Transport calculations

SCC-DFTB: Reviews SCC-DFTB: Reviews

1) 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) 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-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, 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: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-bonded systems: water 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)

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

Secondary-structure elements for Glycine und Alanine-based polypeptides

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

Secondary-structure elements for Glycine und Alanine-based polypeptides

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

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 quite bad

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

Strength of SCC-DFTBStrength of SCC-DFTB

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.

Structure of large molecules

- dynamics

- relative energies

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!)

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

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

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

DFT Problems: (very) selective publications DFT Problems: (very) selective publications

(1) Ex: PRB 23 (1981) 5048, JCP 109 (1998) 2604

(2) Ex: JCP 113 (2000) 8918, Mol. Phys. 97 (1999) 859.

(3) Ex: JPCA 104 (2000) 4755, JCP 119 (2003) 2943.

(4) Ec: JCP 114 (2001) 5149

(5) Ec: Angew. Chem. Int. Ed. 2006, 45, 4460 –4464

(6) Koch, Wolfram / Holthausen, Max C.A Chemist's Guide to Density Functional Theory, Wiley

Problems of DFT-GGAProblems of DFT-GGA

- overbinding of small molecules: CO... B3LYP, rev-PBE 10 kcal

- transition metals: B3LYP, PB86 ..., spin states, energetics 10-20 kcal

- vib. Freqencies:

- conjugate systems: GGAs overpolarize PA‘s of respective proton donors 10 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