force fields and numerical solutions
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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
1-12-2005 3
Outline
• General Introduction• Force Fields• Numerical Solutions• Test
1-12-2005 4
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
1-12-2005 5
Outline
• General Introduction• Force Fields• Numerical Solutions• Test
1-12-2005 6
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
r
rr
KnK
KrrKV
0
612
2
22
)(4
1
2
cos1
1-12-2005 7
Force Fields
• Dihedral
22cos1 cos1
nK cos1
2cos1
1-12-2005 8
Force Fields
• Lennard-Jones
or
612
2ij
ij
ij
ijij rr
“Stolen” from [2]
612ij
ij
ij
ij
r
B
r
A
1-12-2005 9
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
r 0)(4
1
1-12-2005 10
Outline
• General Introduction• Force Fields• Numerical Solutions• Test
1-12-2005 11
Numerical Solutions
• Predictor-corrector algorithm
)()(
)()()(
)()()()(
)()()()()(2
21
3612
21
tbttb
ttbtatta
tbtttatvttv
tbttatttvtrttr
ipi
iipi
iiipi
iiiipi
)()()( ttattatta pi
cii
1-12-2005 12
Numerical Solutions
)()()(
)()()(
)()()(
)()()(
3
2
1
0
ttacttbttb
ttacttatta
ttacttvttv
ttacttrttr
ici
ci
ici
ci
ici
ci
ici
ci
1-12-2005 13
Numerical Solutions
• Verlet algorithm
)()()()( 221 tatttvtrttr iiii
)()()()( 221 tatttvtrttr iiii
)()(2)()( 2 tattrttrttr iiii
)()()(2)( 2 tatttrtrttr iiii
1-12-2005 14
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.
1-12-2005 15
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.
1-12-2005 16
Outline
• General Introduction• Force Fields• Numerical Solutions• Test
1-12-2005 17
Test
• 1. What is the following ?
→Coublumb interaction→Lennard-Jones interaction→Verlet interaction
612
2ij
ij
ij
ijij rr
1-12-2005 18
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
1-12-2005 19
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
1-12-2005 20
Answers
• Question 1. Lennard-Jones interaction• Question 2. To ”solve” N2 at constant
Temperature • Question 3.
i
nii x
xxxVF
),...,,...,( 1
1-12-2005 21
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
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