intermolecular forces and monte-carlo integration
TRANSCRIPT
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Intermolecular Forces and Monte-Carlo Integration
열역학 특수 연구
2003.3.28
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Source of the lecture note.
J.M.Prausnitz and others, “Molecular Thermodynamics of Fluid Phase Equiliria”
Atkins, “Physical Chemistry” Lecture Note, Prof. D.A.Kofke, University at Buffalo Lecture Note, R.J.Sadus, Swineburn University
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Tasks of Molecular Simulation
−
−>=<
rr
rrr
dkTU
dkTUAA
)/)((
)/)()((
Model for Intermolecular Forces
Method of Integration forMultiple vector space
Part 1 of this lecture
Part 2 of this lecture
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Intermolecular Forces
Intermolecular forces Force acting between the molecules of given mixture or
pure species
It is essential to understand the nature of intermolecular forces for the study of molecular simulation
Only simple and idealized models are available (approximation)
Our understanding of intermolecular forces are far from complete.
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Types of intermolecular forces
Electrostatic forces Charged particles and permanent dipoles
Induced forces Permanent dipole and induced dipole
Force of attraction between nonpolar molecules
Specific forces Hydrogen bonding, association and complex
formation
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Potential Energy Function and Intermolecular Forces
Potential Energy : Energy due to relative position to one another
If additional variables are required for potential energy function …
drdF Γ−=
,...),,(,...),,( φθφθ rrF Γ−∇=
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1. Electrostatic Force
Due to permanent charges (ions,…) Coulomb’s relation (inverse square law)
Two point charges separated from distance r
For two charged molecules (ions) ,
2ree
F ji= .constree ji
ij +=ΓdrdF Γ−=
Drzz ji
ij
2ε=Γ
Dielectric constant of given medium
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Nature of Electrostatic forces
Dominant contribution of energy …. Long range nature Force is inversely proportional to
square of the distance Major difficulties for concentrated
electrolyte solutions
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Electrostatic forces between dipoles Dipole
Particles do not have net electric charge Particles have two electric charges of same
magnitude e but opposite sign. Dipole moment
Potential Energy between two dipolesel=μ
[ ])cos(sinsincoscos23 jijijiji
ij rφφθθθθ
μμ−−=Γ
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Energies of permanent dipole, quadrupoles Orientations of molecules are governed by two
competing factors Electric field by the presence of polor molecules Kinetic energy random orientation
Dipole-Dipole
Dipole-Quadrupole
Quadrupole-Quadrupole
...8
22
+−=ΓkTrQji
ij
μ
...32
6
22
+−=ΓkTr
jiij
μμ
...40 10
22
+−=ΓkTr
QQ jiij
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2. Induced Forces
Nonpolar molecules can be induced when those molecules are subjected to an electric field.
Ei αμ =
Polarizability
Electric Field Strength
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Mean Potential Energies of induced dipoles Permanent Dipole + Induced Dipole
Permanent Dipole + Permanent Dipole
Permanent Quadrupole + Permanent Quadrupole
6
2
rji
ij
μα−=Γ
( )8
22
23
rQQ ijji
ij
αα +−=Γ
( )6
22
rijji
ij
μαμα +−=Γ
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3. Intermolecular Forces between Nonpolar Molecules
1930, London There was no adequate explanation for the forces
between nonpolar molecules Instant oscillation of electrons Distortion of electron
arrangement was sufficient to caus temporary dipole moment
On the average, the magnitude and direction averages zero, but quickly varying dipoles produce an electric field. induces dipoles in the surrounding molecules
Induced dipole-induced dipole interaction
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London dispersion force
( ))(2
36
ji
jijiij IIr
II+
−=Γαα
jjiiij ΓΓ=Γ
Potential energy between two nonpolar molecules are :
independent of temperature and
varies inversely as sixth power of the distance between them .
6rB
ii −=Γ
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Repulsive force and total interaction When molecules are squeezed, electronic replusion and
rising of eletronic kinetic energy began to dominate the attractive force
The repulsive potential can be modeled by inverse-power law
The total potential is the sum of two separate potential
mrA=Γ
nmij rB
rA −=Γ
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General form of intermolecular potential curve
Mie’s Potential
Lennard-Jones Potential
−
=Γ
612
4rrσσε
−
−=−=Γ
− mnmnmn
nm rrmnmn
rB
rA σσε )/(1)/(
The parameters for potential models can be estimatedfrom variety of physical properties (spectroscopic and molecular-beam experiments)
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Specific (Chemical) Forces
Association : The tendency to from polymer Solvation : The tendency to form complexes from
different species
Hydrogen Bond and Electron Donor-Acceptor complexes
The models for specific forces are not well established.
The most important contribution in bio-molecules (proteins, DNA, RNA,…)
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Simplified Potential Models for Molecular Simulations
)( 0)()( )(
σσ
≥=Γ<∞=Γ
rrrr
HS
HS
vSS rar =Γ )(
Soft-Sphere Potential withRepulsion parameter = 1 Soft-Sphere Potential with
Repulsion parameter = 12
)( 0)()( )(
)( )(
2
21
σσσε
σ
≥=Γ≤≤−=Γ
<∞=Γ
rrrr
rr
SW
SW
SW
vSS rar =Γ )(
Square Well Potential Hard Sphere Potential
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Calculation of Potential in Molecular Monte Carlo Simulation
There are no contribution of kinetic energy in MMC simulation Only “configurational” terms are calculated
...),,(),()( 321 +Γ+Γ+Γ= kjijii rrrrrrU
Effect of external field
Potential between pairs of particles
Potential between particles of triplets
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Using reduced units…
Dimensionless units are used for computer simulation purposes
εσεε
ρσρ
///
3*
*
*
3*
PPEEkTT
====
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Contribution to Potential energy
Two-body interactions are most important term in the calculation
For some cases, three body interactions may be important.
Including three body interactions imposes a very large increase in computation time.
m: number of interactions
mNt ∝
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Short range and long range forces
Short range force Dispersion and Replusion
Long range force Ion-Ion and Dipole-Dipole interaction
Interaction Type Dependence Typical E(kJ/mol)
Comment
Ion-Ion 1/r 250Ion-Dipole 1/r2 15Dipole-Dipole 1/r3 2 Stationary Dipole-Dipole 1/r6 0.6 RotatingLondon 1/r6 2
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Short range and long range interactions
Computation time-saving devices for short range interactions Periodic boundary condition Neighbor list
Special methods are required for long range interactions. (The interaction extends past the length of the simulation box)
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Naïve energy calculation
−
= +=
Γ=1
1 12 )(
N
i
N
ijijrU
Summation are chosen to avoid “self” interaction
Loop i = 1, N-1Loop j = i+1,N
Evaluate rijEvaulate Uij
Accumulate EnergyEnd j Loop
End j Loop
Pseudo Code
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Problems
Simulations are performed typically with a few hundred molecules arranged in a cubic lattice. Large fraction of molecules can be expected at the
surface rather than in the bulk.
Periodic Boundary Conditions (PBC) are used to avoid this problem
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Periodic Boundary Condition
Infinite Replica of the lattice of the cubic simulation box
Molecules on any lattice have a mirror image counter part in all the other boxes
Changes in one box are matched exactly in the other boxes surface effects are eliminated.
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Another difficulty…
Summation over infinite array of periodic images This problem can be overcame using
Minimum Image Convention (MIC)
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Minimum Image Convention (MIC)
Nearest images of colored sphere
For a given molecule, we positionthe molecule at the center of a box with dimension identical to the simulation box.
All the coordinates lie within therange of ½ L and – ½ L
Assume that the central molecule only interacts with all molecules whose center fall within this region.
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Implementing PBC & MIC
Two Approaches Decision based : if statement Function based : rounding, truncation, modulus
BOXL2 = BOXL/2.0 IF(RX(I).GT.BOXL2) RX(I)=RX(I)-BOXLIF(RX(I).LT.-BOXL2) RX(I) = RX(I) + BOXL
RX(I) = RX(I) – BOXL * AINT(RX(I)/BOXL)
Decision Function
Nearest integer
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Implementing PBC & MIC
Loop i = 1, N -1Loop j = I + 1, N
Evaluate rijConvert rij to its periodic image (rij’) if (rij’ < cutOffDistance)
Evaluate U(rij)Accumulate Energy
End ifEnd j Loop
End i Loop
Pseudo CODE
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Improvement due to PBC & MIC(Compared with naïve calculation)
Accumulated energies are calculated for the periodic separation distance.
Only molecules within cut-off distance contribute to the calculated energy.
Caution : cut-off distance should be smaller than the size of the simulation box Violation to MIC
Calculated potential Truncated potential
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Long range correction to PCB
Adding long range correction…
drrurNE
XXX
crlrc
lrccfull
∞
=
+=
)(2 2ρπ
For NVT ensemble, density and no. of particles are const.LRC and be added after simulationFor other ensembles, LRC terms must be added during simulation
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Technique to reduce computation time Neighbor List
1967, Verlet proposed a new algorithm. Instead of searching for neighboring
molecules, the neighbor of the molecules are stored and used for the calculation.
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Neighbor List
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Neighbor List
Variable d is used to encompass molecules slightly outside the cut-off distance (buffer).
Update of the list Update of the list per 10-20 steps Largest displacement exceed d value.
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Algorithm for Integration
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Method of Integration
Methodological Approach Rectangular Rule, Triangular Rule,
Simpson’s Rule
( )b
a
I f x d x= ( )f x
x
Quadrature Formula
1 1( ) ( )
n n
i ii i
b aI x f x f xn= =
−≈ Δ = Uniformly separated points
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Monte Carlo Integration
Stochastic Approach Same quadrature formula, different
selection of points
1( )
n
ii
b aI f xn =
−≈
( )xπ
x
Points are selected from uniform distribution ( )xπ
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Example …(from Univ. at Buffalo, School of Eng. And Appl. Science, Prof. David Kofke)
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Example …(from Univ. at Buffalo, School of Eng. And Appl. Science, Prof. David Kofke)
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Why Monte Carlo Integration ?
Comparison of errors Methodological Integration Monte Carlo Integration
MC error vanishes much slowly for increasing n For one-dimensional integration, MC offers no
advantage The conclusion changes when dimension of
integral increases Methodological Integration Monte Carlo Integration
22 / nxE Δ∝2/1/1 nE ∝
dnxE /22 /Δ∝2/1/1 nE ∝
MC “wins” about d = 4
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Shape of High Dimensional Region
Two (and Higher) dimensional shape can be complex
How to construct weighted points in a grid that covers the region R ?
Problem : mean-square distance from the origin
+
>=<dxdy
dxdyyxr
)( 222
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Shape of High Dimensional Integral It is hard to formulate
methodological algorithm in complex boundary
Usually we do not have analytical expression for position of boundary
Complexity of shape can increase unimaginably as dimension of integral grows
We want 100 + dimensional integrals
?iw
−
−>=<
rr
rrr
dkTU
dkTUAA
)/)((
)/)()((
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Nature of the problem …
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Integration over simple shape ?
0.5 0.5 2 22 0.5 0.5
0.5 0.5
0.5 0.5
( ) ( , )
( , )
dx dy x y s x yr
dx dys x y
+ +
− −+ +
− −
+=
1 inside R
0 outside Rs =
Grid must be fine enough !
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Sample Integration
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Sample Integration
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Integration over simple shape ?
Statistical mechanics integrals typically have significant contribution from miniscule regions of the integration space.
Ex ) 100 spheres at freezing fraction = 10-260
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Importance Sampling
Put more quadrature points in the region where integral recieves its greatest contribution
Choose quadrature points according to some distribution function.
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A sample integration.
2( ) 3f x x=
( ) 2x xπ =
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Importance Sampling Integral
Using Rectangular-Rule : Use unevenly spaced intervals
1xΔ 2xΔ 3xΔ nxΔ…
1( )
n
i ii
I f x x=
≈ Δ
1( )i
i
b axn xπ−Δ =