Most material in this seminar has been produced by Bert de Groot at the MPI in G ö ttingen.

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Most material in this seminar has been produced by Bert de Groot at the MPI in G ö ttingen.

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Most material in this seminar has been produced by Bert de Groot at the MPI in G ö ttingen.

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Molecular dynamicsSome random notes on molecular dynamics simulationsSeminar based on work by Bert de Groot and many anonymous Googelable colleagues

Most material in this seminar has been produced by Bert de Groot at the MPI in Göttingen.

Schrödinger equation

Born-Oppenheimer approximation

Nucleic motion described classically

Empirical force field

Inter-atomic interactions

=

=

R

Motions of nuclei are described classically:

Non-bonded interactions

Covalent bonds

Eibond

approximated

exact

KBT {

0

|R|

Potential function Eel describes the electronic influence on motions of the nuclei and is approximated empirically „classical MD“:

„Force-Field“

Coulomb potential

Lennard-Jones potential

Now we need to give all atoms some initial speed, and then, evolve that speed over time using the forces we now know. The average speed of nitrogen in air of 300K is about 520 m/s. The ensemble of speeds is best described by a Maxwell distribution.

Back of the enveloppe calculation:

500 m/s = 5.10 Å/s

Let’s assume that we can have things fly 0.1 A in a straight line before we calculate forces again, then we need to recalculate forces every 20 femtosecond; one femtosecond is 10 sec.

In practice 1 fsec integration steps are being used.

12

-15

http://en.wikipedia.org/wiki/Verlet_integration

http://en.wikipedia.org/wiki/Maxwell_speed_distribution

Knowing the forces (and some randomized Maxwell distributed initial velocities) we can evolve the forces over time and get a trajectory. Simple Euler integration won’t work as this figure explains.

You can imagine that if you know where you came from, you can over-compensate a bit. These overcompensation algorithms are called Verlet-algorithm, or Leapfrog algorithm.

If you take bigger time steps you overshoot your goal. The Shake algorithm can fix that. Shake allows you larger time steps at the cost of little imperfection so that longer simulations can be made in the same (CPU) time.

http://en.wikipedia.org/wiki/Verlet_integration

Molecule: (classical) N-particle system

Newtonian equations of motion:

Integrate numerically via the „leapfrog“ scheme:

with

Δt 1fs!

(equivalent to the Verlet algorithm)

BPTI: Molecular Dynamics (300K)

Solve the Newtonian equations of motion:

Molecular dynamics is very expensive ...

Example: A one nanosecond Molecular Dynamics simulation of F1-ATPase in water (total 183 674 atoms) needs 106 integration steps, which boils down to 8.4 * 1017 floating point operations.

on a 100 Mflop/s workstation:ca 250 years

...but performance has been improved by use of:

+ multiple time steppingca. 25 years

+ structure adapted multipole methods*ca. 6 years

+ FAMUSAMM*ca. 2 years

+ parallel computers ca. 55 days

* Whatever that is

MD-Experiments with Argon Gas

Explicit or implicit?

Box or droplet?

periodic boundary conditions

H. Frauenfelder et al., Science229 (1985) 337

Limits of MD-Simulations

classical description:

chemical reactions not describedpoor description of H-atoms (proton-transfer)poor description of low-T (quantum) effectssimplified electrostatic modelsimplified force fieldincomplete force field

only small systems accessible (104 ... 106 atoms)only short time spans accessible (ps ... μs)