Approximate methods for large molecular
systems
Marcus Elstner
Physical and Theoretical Chemistry, Technical
University of Braunschweig
Motivation
Structure-formation, atomic-scale related properties and
processes
Si21
C60-trimer
defects, doping
Si1600
MoS2
4H-SiC-surfaces
a-SiCN-ceramics
GaN-devices
Reactions in biological Systems
Alcohol DeHydrogenase
Aquaporin
Photosynthetic Reaction Center
Catalysis
bR
Photochemistry
Proton Transfer
Photochemistry
Electron/Energy Transfer
Need QM
description
Computational challange
~ 1.000-10.000 atoms
~ ns molecular dynamics simulation
(MD, umbrella sampling)
- weak bonding forces
- chemical reactions
- treatment of excited states
‚multiscale business‘
Continuum electrostatics
Molecular Mechanics
SE-QM
approx-DFT
HF, DFT
CI, MP
CASPT2
predictivity
nm
Length scale
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
Size problem: QM-Methods
Hybride methods: QM/MM, QM/QM
Linear scaling: O(N)
SE/approx. Methods
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-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:
TD-DFTB-LR
O(N)-QM/MM
QM/MM
divide+conquer
AMBER: Han, Suhai DKFZ
CHARMM: Cui, Karplus; Harvard
TINKER: Liu, Yang; Duke
CEDAR: Hu, Hermans; NC Univ
H. Liu W. Yang
Duke 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
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 tightbinding 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
Malolepsza, Witek & Morokuma: CPL 412 (2005) 237.
Witek & Morokuma, J Comp Chem. 25 (2004) 1858.
for CH
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-bonds
Han 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
Coulomb
interaction
- structures well reproduced
H2O-dimer complexes Cs, C2v
NH3-NH3- and NH3-H2O-dimer
Model peptides: N-Acetyl-(L-Ala)n
N‘-Methylamide (AAMA) + 4 H2O
Secondary-structure elements for Glycine und
Alanine-based polypeptides
Elstner, et al.. Chem. Phys. 256 (2000) 15
N = 1 (6 stable conformers)
310 - helix
aR-helix
stabilization by internal H-bonds
between i and i+3
between i and i+4
N
DFTB very good for:
main problem for DFT(B): dispersion!
- relative energies
 AM1, PM3, MNDO quite bad
- geometries
 OM2 much improved (JCC 22 (2001) 509)
- vib. freq. o.k.!
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
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

N

Ace-Ala-Nme
C7eq
C5ext
C7ax

2
a
MP4-BSSE: Beachy et al, BSSE corrected at MP2 level
R
a
P
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:
(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
(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-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