Simulations Moléculaires
Wei DONG
Laboratoire de Chimie,UMR 5182 CNRS - Ecole Normale
Supérieure de Lyon,46,Allée d’Italie,69364 Lyon Cedex 07,
France
Tél,0472728844
Email,Wei.Dong@ens-lyon.fr
Bureau,LR6 A008
Plan
Introduction
Brief recall of statistical mechanics and thermodynamics
Molecular dynamics method (MD)
Monte Carlo method (MC)
Illustration of a few applications of MD and MC in the research
works carried out in our laboratory
Tutorials on MD and MC
What can one do with molecular simulations?
Study complex systems,i.e.,those which cannot be described
by simple theories,
Calculate thermodynamic properties of complex systems (e.g.,
liquids),internal energy,pressure,chemical potential,Henry’s
constant etc.
Calculate transport properties,e.g.,diffusion coefficient,
viscosity,coefficient of thermal conduction etc.
Calculate the rate of different types of chemical reactions
Determine different types of distribution functions
Application domains
Simple liquids,molecular liquids;
liquid mixtures,electrolyte solutions;
inhomogeneous fluids (interfaces),fluids confined in porous solids;
polymers,proteins;
self-assembling systems (amphiphile,micelle,micro emulsion) etc.
What is the relation between simulation,theory
and experiment?
Simulation
Theory Experiment
Difference and relation between Molecular
Simulation and Quantum Chemistry
Different scopes (lenght and time scales):
- Quantum chemistry
Electronic level,one or a few atoms or molecules,
Microscopic length and time scales.
Bond energy,Transition state etc..
- Molecular simulation
Atomic and molecular level,large assembly of atoms and molecules,
Larger length and time scales.
Effects of inter-molecular interactions,dynamics (non reacting
and reacting systems).
What is a molecular simulation or the aim of it?
generating microscopic configurations or trajectories of model
systems.
sampling phase space to determine structural,thermodynamic and
transport properties through calculating averages and probabilities.
What are the main methods:
Molecular dynamics
Monte Carlo
Molecular mechanics
Basic principles of MD and MC:
Molecular dynamics
From a given initial condition (positions and momenta),the
trajectories of all the molecules are generated by solving the
equations of motion (Newton equation for classical dynamics
or time-dependent Schr?dinger equation for quantum
dynamics).
Monte Carlo
Different configurations are generated by using a stochastic
methods.
A short history:
Monte Carlo:
N,Metropolis,A.W,Rosenbluth,M.N,Rosenbluth,A.H,Teller
and E,Teller,J,Chem,Phys,21,1087,(1953).
Molecular dynamics:
B.J,Alder and T.E,Wainwright,J,Chem,Phys,27,1208,
(1957); ibid,31,459,(1959).
State of the art:
Molecular simulation spreads more and more widely in
various scientific disciplines,e.g.,physics,chemistry,biology,
chemical engineering etc..
Systems tractable nowadays,fluids with a million of
particles,bio-polymers etc..
Force fields
...),,(),()( 321
rrrurruru kijk jiijijij iii i
U
Atomic systems:
where u1,u2 and u3 are respectively one-body,two-body,
three-body interaction potentials.
Pair potential models are widely used in simulations.
Model interaction potentials
Hard-sphere potential:
uHS(r) =
r < s
0 r > s
r
u(r)
s
Square-well potential:
r < s1
uSW(r) = -e s1 < r < s2
0 r > s2
Lennard-Jones potential:
uLJ(r) = 4e ( (s/r)12 - (s/r)6)
-e
s1 s2
u(r)
r
-e
u(r)
r
Molecular systems:
U(r1,r2,…,rN) = uintra + uinter
)4/())/,()/,(( 0
,
6
6
12
,12i n t rqqrcrcu ijrji jiijijjier
jiji ee
d i h e d r a l sa ng l e sb on d s bra
nKKbbKu )c o s (100 )()( 22i n t
This type of force-field is implemented in numerous MD
codes,e.g.,AMBER,CHARMM,GROSMOS etc..
The parameters are determined by fitting against either
experimental results or those of quantum mechanical
calculations.
Where to find the force-field parameters for a
given system?
There exists a very extensive literature on force-field.
Large data bases can be found in various simulation codes.
AMBER,CHARMM,widely used organic compounds
GROMOS,biomolecules
OPLS,liquid phase,solutions
Procedure for fitting force-filed parameters
Preliminary fitting:
Experimental results of second virial coefficient,B(T).
Main fitting:
Heat of evaporation => energy parameter
Pressure => size parameter
Refinement:
Tune partial charges
0
2 1))/(ex p (2)( kTrudrTB r?
Exercise
Calculate the second virial coefficient,B(T),for a HS potential
and a SW potential,Define a reduced temperature as T*=kT/e and
plot B(T*) as a function of T* (0.5 < T* < 5) in both cases.
Periodic boundary condition and minimum image convention
Why one needs the periodic boundary condition?
To avoid undesired boundary effect and to mimic a macroscopic
system.
Minimum image convention
1 1
2
2
3
3
4
4
5
5
6
7
8
Exercise
Write a piece of program for dealing with periodic boundary
condition and the minimum image.
To simplify,consider a 2D system,Let N be the number of
particles in the system,Consider a square simulation box with
box length equal to L,Use two arrays for the coordinates of
particles,e.g.,RX (N),RY(N).
References
Statistical physics:
D,Chandler,Introduction to modern statistical Mechanics,
(Oxford University Press,1987)
W,Brenig,Théorie statistique de la chaleur,(Dunon,1998)
J.P,Hansen and I.R,McDonald,Theory of simple liquids,2nd
Edition (Academic Press,1986)
Simulation methods:
M.P,Allen and D.J,Tildesley,Computer Simulation of Liquids,
(Clarendon Press,Oxford,1987)
B,Smit and D,Frankel,Understanding Molecular Simulation,
(Academic Press,1996).
D,Landau and K,Binder,Aguide to Monte Carlo Simulations in
Statistical Physics,(Cambridge University Press,2000),
Wei DONG
Laboratoire de Chimie,UMR 5182 CNRS - Ecole Normale
Supérieure de Lyon,46,Allée d’Italie,69364 Lyon Cedex 07,
France
Tél,0472728844
Email,Wei.Dong@ens-lyon.fr
Bureau,LR6 A008
Plan
Introduction
Brief recall of statistical mechanics and thermodynamics
Molecular dynamics method (MD)
Monte Carlo method (MC)
Illustration of a few applications of MD and MC in the research
works carried out in our laboratory
Tutorials on MD and MC
What can one do with molecular simulations?
Study complex systems,i.e.,those which cannot be described
by simple theories,
Calculate thermodynamic properties of complex systems (e.g.,
liquids),internal energy,pressure,chemical potential,Henry’s
constant etc.
Calculate transport properties,e.g.,diffusion coefficient,
viscosity,coefficient of thermal conduction etc.
Calculate the rate of different types of chemical reactions
Determine different types of distribution functions
Application domains
Simple liquids,molecular liquids;
liquid mixtures,electrolyte solutions;
inhomogeneous fluids (interfaces),fluids confined in porous solids;
polymers,proteins;
self-assembling systems (amphiphile,micelle,micro emulsion) etc.
What is the relation between simulation,theory
and experiment?
Simulation
Theory Experiment
Difference and relation between Molecular
Simulation and Quantum Chemistry
Different scopes (lenght and time scales):
- Quantum chemistry
Electronic level,one or a few atoms or molecules,
Microscopic length and time scales.
Bond energy,Transition state etc..
- Molecular simulation
Atomic and molecular level,large assembly of atoms and molecules,
Larger length and time scales.
Effects of inter-molecular interactions,dynamics (non reacting
and reacting systems).
What is a molecular simulation or the aim of it?
generating microscopic configurations or trajectories of model
systems.
sampling phase space to determine structural,thermodynamic and
transport properties through calculating averages and probabilities.
What are the main methods:
Molecular dynamics
Monte Carlo
Molecular mechanics
Basic principles of MD and MC:
Molecular dynamics
From a given initial condition (positions and momenta),the
trajectories of all the molecules are generated by solving the
equations of motion (Newton equation for classical dynamics
or time-dependent Schr?dinger equation for quantum
dynamics).
Monte Carlo
Different configurations are generated by using a stochastic
methods.
A short history:
Monte Carlo:
N,Metropolis,A.W,Rosenbluth,M.N,Rosenbluth,A.H,Teller
and E,Teller,J,Chem,Phys,21,1087,(1953).
Molecular dynamics:
B.J,Alder and T.E,Wainwright,J,Chem,Phys,27,1208,
(1957); ibid,31,459,(1959).
State of the art:
Molecular simulation spreads more and more widely in
various scientific disciplines,e.g.,physics,chemistry,biology,
chemical engineering etc..
Systems tractable nowadays,fluids with a million of
particles,bio-polymers etc..
Force fields
...),,(),()( 321
rrrurruru kijk jiijijij iii i
U
Atomic systems:
where u1,u2 and u3 are respectively one-body,two-body,
three-body interaction potentials.
Pair potential models are widely used in simulations.
Model interaction potentials
Hard-sphere potential:
uHS(r) =
r < s
0 r > s
r
u(r)
s
Square-well potential:
r < s1
uSW(r) = -e s1 < r < s2
0 r > s2
Lennard-Jones potential:
uLJ(r) = 4e ( (s/r)12 - (s/r)6)
-e
s1 s2
u(r)
r
-e
u(r)
r
Molecular systems:
U(r1,r2,…,rN) = uintra + uinter
)4/())/,()/,(( 0
,
6
6
12
,12i n t rqqrcrcu ijrji jiijijjier
jiji ee
d i h e d r a l sa ng l e sb on d s bra
nKKbbKu )c o s (100 )()( 22i n t
This type of force-field is implemented in numerous MD
codes,e.g.,AMBER,CHARMM,GROSMOS etc..
The parameters are determined by fitting against either
experimental results or those of quantum mechanical
calculations.
Where to find the force-field parameters for a
given system?
There exists a very extensive literature on force-field.
Large data bases can be found in various simulation codes.
AMBER,CHARMM,widely used organic compounds
GROMOS,biomolecules
OPLS,liquid phase,solutions
Procedure for fitting force-filed parameters
Preliminary fitting:
Experimental results of second virial coefficient,B(T).
Main fitting:
Heat of evaporation => energy parameter
Pressure => size parameter
Refinement:
Tune partial charges
0
2 1))/(ex p (2)( kTrudrTB r?
Exercise
Calculate the second virial coefficient,B(T),for a HS potential
and a SW potential,Define a reduced temperature as T*=kT/e and
plot B(T*) as a function of T* (0.5 < T* < 5) in both cases.
Periodic boundary condition and minimum image convention
Why one needs the periodic boundary condition?
To avoid undesired boundary effect and to mimic a macroscopic
system.
Minimum image convention
1 1
2
2
3
3
4
4
5
5
6
7
8
Exercise
Write a piece of program for dealing with periodic boundary
condition and the minimum image.
To simplify,consider a 2D system,Let N be the number of
particles in the system,Consider a square simulation box with
box length equal to L,Use two arrays for the coordinates of
particles,e.g.,RX (N),RY(N).
References
Statistical physics:
D,Chandler,Introduction to modern statistical Mechanics,
(Oxford University Press,1987)
W,Brenig,Théorie statistique de la chaleur,(Dunon,1998)
J.P,Hansen and I.R,McDonald,Theory of simple liquids,2nd
Edition (Academic Press,1986)
Simulation methods:
M.P,Allen and D.J,Tildesley,Computer Simulation of Liquids,
(Clarendon Press,Oxford,1987)
B,Smit and D,Frankel,Understanding Molecular Simulation,
(Academic Press,1996).
D,Landau and K,Binder,Aguide to Monte Carlo Simulations in
Statistical Physics,(Cambridge University Press,2000),
