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Determination of V(R)

•  Pointwise QM determination of the full 3N dim PES ⇒ only practicable for very small molecules

•  PES determined on the fly where: Car-Parrinello MD ⇒ <1000 atoms

First-Principles Surfaces:

Empirical Interaction Potentials: •  Choice of functional form (2-body? Many-body? Nonpolarizable/Polarizable ? All atom/united atom?) •  Parameterized with experimental or QC data on small gas phase molecules (plus adaption to condensed phase environment)

Exp. Dipole moment H2O 1.85D (gas phase) ~3 D (water)

Force Fields •  molecules modeled as classical mechanical objects with electrostatic charge interactions •  no explicit electrons only set of classical particles or interaction sites •  no quantum effects

Picture from wikipedia

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chemical bonds (2 adjacent atoms): -> described by mechanical springs: bond potential (harmonic, anharmonic, Morse etc..) force constants e.g. from stretching modes bond angles (3 adjacent atoms): ditto (harmonic, anharmonic etc..), → force constants e.g. from bending modes Torsional Potentials (4 adjacent atoms)

•  electrostatic interactions: Coulomb interaction between effective (atom centered or off-site) point charges •  van der Waals interactions (Pauli repulsion & dispersion): Lennard-Jones 12-6, n-m, Williams exponential

C C

H

H H

H

Standard (Bio)molecular Force Field ∑ ∑ +−+−=b

ijkkbijrbkMMHθ

θθθ2

0212

021 )()(

∑∑⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

⎟⎟⎠

⎞⎜⎜⎝

⎛ σ−⎟

⎟⎠

⎞⎜⎜⎝

⎛ σε+

πε+

oplm

6

op

12

oplm0

mlrr

4r4qq

•  GROMOS, AMBER, CHARMM, OPLS-AA, MM3, SYBIL, UFF, SPC, SPC/E, TIP3P, TIP4P, TIP5P etc..

[ ]∑ −+∑+φ

φφ )cos( 01 ijkhnnkn

electrostatics Lennard-Jones 12-6

Bond term angle term

Torsional term

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Treatment of Long-Range Electrostatic Interactions

è many different approaches:

- non-bonded forces O(N2): computationally most expensive part - periodic boundary conditions require infinite sum of interactions with all periodic replicas

•  neighbor lists •  spherical cutoff (shifting fcts, ε(r)) •  reaction field •  Ewald summation

⇒ particle-mesh Ewald (PME), smooth particle-mesh Ewald (SPME), particle-particle particle mesh Ewald (P3M)

•  (Hierarchical) Fast Multipole Expansions

(Hockney&Eastwood, Darden)

(Greengard&Rohklin)

MOLECULAR DYNAMICS PACKAGES AMBER Peter Kollman, UCSF www.amber.ucsf.edu/amber/amber.html, 400$ CHARMM Martin Karplus, Harvard www.yuri.harvard.edu, 600$ DL-POLY CCP5, UK ww.dl.ac.uk/TCSC/Sofware/DL_POLY/main.html EGO VIII Paul Tavan, Univ. Munich www.imo.physik.uni-muenchen.de GROMOS Wilfred vanGunsteren, ETH Zurich igc.ethz.ch/gromos NAMD Univ. Illinois www.ks.uiuc.edu/Research/namd TINKER Robert Paine, Univ. New Mexico, Jay Ponder dasher.wustl.edu/tinker, free (incl. source) X-PLOR Axel Brünger, Stanford Univ. atb.slac.standford.edu

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Limitations of Empirical Force Fields ⇒ Transferability Problem empirical force fields are only parameterized for a given electronic environment, cannot adjust to large changes in the electron distribution (e.g. different types of chemical bonding) ⇒ cannot treat breaking and forming of chemical bonds ⇒ no chemical reactions! ⇒ many-body effects (polarization)! ⇒ transition metals difficult to treat!

⇒ parameter-free first-principles MD

Car - Parrinello Molecular Dynamics

Roberto Car

Michele Parrinello

{ }{ }[ ]( ) ( ){ }[ ]ijji

ijij

Iii iiI IIdrrr

RERMLδφφφφφµ

−Λ+−+=

∫∑∑∑ ,2/1 2

Equations of Motion

III R

ERMδδ

−=

jj ijii H φφφµ ∑ Λ+−=