1-12-2005 1

Force Fields and Numerical Solutions

Christian Hedegaard Jensen

- within Molecular Dynamics

1-12-2005 2

Outline

• General Introduction• Force Fields• Numerical Solutions• Test

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Outline

• General Introduction• Force Fields• Numerical Solutions• Test

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General Introduction

• From last time we have that the problem is

• Force Fields = What is V ?• Numerical Solutions = How to solve the

equation numerically ?

ii Fxm

ii x

xVF

,...)( 1

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Outline

• General Introduction• Force Fields• Numerical Solutions• Test

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Force Fields

• A force field may look like this (taken from [1])

ij

ji

coulombic ij

LJ ij

ij

ij

ijij

improperseq

dihedrals

angleseq

bondseqr

r

qq

r

rr

KnK

KrrKV

0

612

2

22

)(4

1

2

cos1

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Force Fields

• Dihedral

22cos1 cos1

nK cos1

2cos1

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Force Fields

• Lennard-Jones

or

612

2ij

ij

ij

ijij rr

“Stolen” from [2]

612ij

ij

ij

ij

r

B

r

A

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Force Fields

• Coulomb

→ (r) model the effect of a solvent.→This can also be modelled explicitly in which case

(r) = 1.

ij

ji

ij r

qq

r 0)(4

1

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Outline

• General Introduction• Force Fields• Numerical Solutions• Test

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Numerical Solutions

• Predictor-corrector algorithm

)()(

)()()(

)()()()(

)()()()()(2

21

3612

21

tbttb

ttbtatta

tbtttatvttv

tbttatttvtrttr

ipi

iipi

iiipi

iiiipi

)()()( ttattatta pi

cii

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Numerical Solutions

)()()(

)()()(

)()()(

)()()(

3

2

1

0

ttacttbttb

ttacttatta

ttacttvttv

ttacttrttr

ici

ci

ici

ci

ici

ci

ici

ci

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Numerical Solutions

• Verlet algorithm

)()()()( 221 tatttvtrttr iiii

)()()()( 221 tatttvtrttr iiii

)()(2)()( 2 tattrttrttr iiii

)()()(2)( 2 tatttrtrttr iiii

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Numerical Solutions

• Errors→If you start with slightly perturbed initial

conditions the trajectories will diverge from each other eventually.

→Fluctuations in energy. For longer time steps the Verlet algorithm is better.

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Numerical Solutions

• Thermostats (If you want to sample at constant temperature)→Example Andersen Thermostat

• Pick random atom/molecule at intervals• Set the velocity so that it is chosen randomly from

the Maxwell-Boltzmann distribution• This corresponds to introducing collisions with

“virtual” heat bath particles.

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Outline

• General Introduction• Force Fields• Numerical Solutions• Test

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Test

• 1. What is the following ?

→Coublumb interaction→Lennard-Jones interaction→Verlet interaction

612

2ij

ij

ij

ijij rr

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Test

• The Andersen … is used for what ?→ To solve N2 at constant Energy→ To calculate forces in the system→ To ”solve” N2 at constant Tempreture

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Test

• How is the force on a particle (in one direction) found from the potential ?→

→

→

i

nii x

xxxVF

),...,,...,( 1

i

nii x

xxxVF

),...,,...,( 1

t

xxxVF nii

),...,,...,( 1

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Answers

• Question 1. Lennard-Jones interaction• Question 2. To ”solve” N2 at constant

Temperature • Question 3.

i

nii x

xxxVF

),...,,...,( 1

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References

• [1] N. Rathore et al.; Density of states simulations of proteins; J. Chem. Phys.; v.118 n. 9 4285; 2002

• [2] http://employees.csbsju.edu/hjakubowski/classes/ch331/protstructure/olunderstandconfo.html

• M.P. Allen and D. J. Tildesley; Computer Simulations of Liquids; Oxford; 1987