Introducción a la asignatura de Computación Avanzada Grado en

Introducción a la asignatura de
Computación Avanzada
Grado en Física
Javier Junquera
Datos identificativos de la asignatura
Vicerrectorado de Ordenación Académica
Facultad de Ciencias
1. DATOS IDENTIFICATIVOS DE LA ASIGNATURA
Título/s
Grado en Física ( Optativa )
Centro
Facultad de Ciencias
Módulo / materia
MATERIA COMPUTACIÓN AVANZADA
MENCION FISICA APLICADA TODAS LAS OPTATIVAS
MENCION FISICA FUNDAMENTAL TODAS LAS OPTATIVAS
OPTATIVAS TRANSVERSALES G-FISICA
Código y
denominación
Créditos ECTS
G80 - Computación Avanzada
Curso /
Cuatrimestre
CUATRIMESTRAL (1)
Web
http://www.ctr.unican.es/asignaturas/Computacion_Avanzada_4_F/
Idioma de
impartición
Inglés
Forma de
impartición
Presencial
Departamento
DPTO. ELECTRONICA Y COMPUTADORES
Profesor
responsable
JULIO LUIS MEDINA PASAJE
E-mail
[email protected]
Número despacho
Facultad de Ciencias. Planta: + 3. DESPACHO - COMPUTADORES TIEMPO REAL (3051)
Otros profesores
FRANCISCO JAVIER JUNQUERA QUINTANA
6
Bibliography:
M. P. Allen and D. J. Tildesley
Computer Simulation of Liquids
Oxford Science Publications
ISBN 0 19 855645 4
How to reach me
Javier Junquera
Ciencias de la Tierra y Física de la Materia Condensada
Facultad de Ciencias, Despacho 3-12
E-mail: [email protected]
URL: http://personales.unican.es/junqueraj
In the web page, you can find:
- The program of the course
- Slides of the different lecture
- The code implementing the simulation of a
liquid interacting via a Lennard-Jones potential
Office hours:
- At the end of each lecture
- At any moment, under request by e-mail
Physical problem to be solved during this course
Given a set of
classical particles (atoms or molecules)
whose microscopic state may be specified in terms of:
- positions
Note that the classical description has to be
- momenta
adequate. If not we can not specify at the same time
the coordinates and momenta of a given molecule
and whose Hamiltonian may be written as the sum of kinetic and potential
energy functions of the set of coordinates and momenta of each molecule
Solve numerically in the computer the equations of motion which governs
the time evolution of the system and all its mechanical properties
Physical problem to be solved during this course
In particular, we will simulate numerically the evolution with time of
classical particles interacting via a two-body potential
(the Lennard-Jones potential)
Why molecular dynamics?
Why Lennard-Jones potential?
Advantages
It allows to solve time dependent problems:
- Reactions
- Collisions
Standard two-body effective pair potential
- Diffusion
- Growth
Well known parameters for many elements
- Vibrations
- Fractures
Provides reasonable description of closed
- Radiation damage
shells atoms (such as inert gases)
-…
It is more parallelizable than other
methods, such as Monte Carlo techniques
Disadvantages
More complex (differential equations)
Less adaptable
Ergodicity problems
It require forces
Note about the generalized coordinates
May be simply the set of cartesian coordinates
system
of each atom or nucleus in the
Sometimes it is more useful to treat the molecule as a rigid body.
In this case,
will consist of:
- the Cartesian coordinates of the center of mass of each molecule
- together with a set of variables that specify the molecular orientation
In any case,
stands for the appropriate set of conjugate momenta
Kinetic and potential energy functions
Usually the kinetic energy
takes the form
molecular mass
runs over the different
components of the momentum
of the molecule
The potential energy contains the interesting information
regarding intermolecular interactions
Potential energy function of an atomic system
Consider a system containing
atoms.
The potential energy may be divided into terms depending on the coordinates
of individual, pairs, triplets, etc.
Expected to
be small
One body
potential
One body potential
Represents the effect
of an external field
(including, for
example, the
contained walls)
Particle interactions
Pair potential
Depends only on the
magnitude of the pair
separation
The notation
Three particle potential
Significant at liquid densities.
Rarely included in computer
simulations (very time
consuming on a computer)
indicates a summation over all
distinct pairs
without containing any pair twice.
The same care must be taken for triplets, etc.
The effective pair potential
The potential energy may be divided into terms depending on the coordinates
of individual, pairs, triplets, etc.
The pairwise approximation gives a remarkably good description because the
average three body effects can be partially included by defining an “effective”
pair potential
The pair potentials appearing in computer simulations are generally to be regarded as
effective pair potentials of this kind, representing many-body effects
A consequence of this approximation is that the effective pair potential needed to
reproduce experimental data may turn out to depend on the density, temperature, etc.
while the true two-body potential does not.
Types of bonds
Types of materials.
Cations
Anions
Ions Cations
Ionic
Valence electrons
Ions
Metal
Ions
Bonds
Covalent
Polarized atoms
Molecular
Types of simulations
What do we want regarding…?
The model:
-Realistic or more approximate
The energy:
-Prevalence of repulsion?
-Bonds are broken?
The scale:
-Atoms?
-Molecules?
-Continuum?
speed equipment. The possibility that a similar phenomenon for hard spheres in two dimensions may have
been missed in the original Monte Carlo calculationsl
will also be investigated.
an hour 2000, 1000, and 500 collisions, respectively, can
256 and
particles
are
be calculated. The results for For
the500other
compounds
measured the spin-orbit planation of this behavior will be attempted in
not now presented due to inadequate
couplingstatistics.
constants thus obtained are either unreason- present paper.
the acThe equation of state shown
Fig. or
1 of
ablyinlarge
small.
ACKNOWLEDGMENTS
particles
is
for, the
companying paper6 for 32 and 108
(CH3NH3hIrCI
6 (C6H6NhIrCI6 and (<p4AshIrCI 6
intermediate region of density, where disagreement was
wish Dto thank461
Research Corpora
S T 1] DIEdisplay
SIN a111 rather
0 LEe peculiar
U L A R magnetic
D Y N A 1\1 behavior
I C S. I. with
G ENE R A L 1\1 E THO
found with the previous Monte Carlo results. The
moments considerably below the expected values over Umverslty Research Council, and American V
volume, v, is given relative to the volume of close
themore
whole
range
No and
ex- then
Corporation
for financial
support
time interval
by recalculating
a new
force of this research
since
size temperature
a crystallite
haveinvestigated.
Vo. Plotted
alsothe
aremaximum
the
extended
Monte can
packing,distorted
particles.these
Equally
serious is the to apply for the next short-time interval, and so on.
would the
be, agreement
say, 500 between
Carlo results;
three syswithin thethat
present
of thecrystal
pressure
tems isnecessity
this accuracy
SOO-particle
solidifies not This method could also handle particles with anisodetermination.
agreement
provides
only in aThis
given
lattice type
but an
alsointeresting
in a special orienta- tropic potentials and with rotational and other degrees
of the
postulates
of boundary,
statistical mechanics
confirmation
tion with
respect
to the
affecting its proba- of freedom, provided that classical description is adefor thisbility
system.
of appearance. Thus the only way solids formed quate. The accuracy of such calculations would depend
THE JOURNAL
VOLUME 31, NUMBER 2
AUGUST,
paper shows OF
two CHEMICAL PHYSICS
Figure 1 of the accompanying
from a melt can differ is by a translational displacement. on the length of the time interval. Since it was desired
separate and overlapping branches. In the overlapping
Evidence
that inexist
order
at presented
a given density,
in to
twoexamine the in the present work to make no approximations in the
region the
system will
can, be
Studies in Molecular Dynamics. I. General Method*
properties
of
a
heterogeneous
system
it
is
states with considerably different pressures. As thenecessary to calculations, a simple potential was chosen for which
use proceeds
many more
particlesis than
in jump
the case
calculation
the pressure
seen to
sud- of a single the force is truly constant (zero) for short-time interJ. ALDER
AND T. E. WAINWRIGHT
phase
homogeneous
solid
which
the particles are allowed to move.
posi-region itself vals B.during
denly from
onesystem.
level to In
the the
other.
A study of the
that can
as long
the system
Radiation
Laboratory,
Universitytoofdeal
California,
Livermore,potentials,
California it
tions ofvarious
the particles
reveals
lattice
types
beasstudied
to Lawrence
overcome
Although
it is feasible
with realistic
stays onthe
theabove
lower branch
of thedifficulty
curve the and
particles
mentioned
the are
one of lowest entails (Received
a considerable
slowing
down
of
the
calculation
February 19, 1959)
all confined
the narrow
region in space
determined
by have been and involves the problem of having to cope with refree toenergy
determined.
Examples
of this
their neighbors,
while on the upper branch of the curveis
by which
it is possible
to calculate
the behavior
of several exhundred interworked out.
pulsive
collisions
where exactly
the forces
the particles
claSSical
The
of this many-body problem is carried out by an electronic computer
the particles
have acquired
freedom
torepresent
exchange
Although
smallenough
systems
can which
an
infinite
perience
change
very
rapidly.
solves
are numencally the simultaneous equations of motion. The limitations of this numerical scheme
with the surrounding particles. Since the spheres
system
remarkably
the statistical
of
The
most
interaction
potential
which
has so
are enum.erat:?
important
steps
in generalthe
program efficient
on the
computers
are indicated.
originally
in ordered
positions,well,
the system
starts
outfluctuations
on and
The.
al?phcablhty
?f
to
the
solutIOn
of
many
problems
in
both
equilibrium
and
nonequilibrium
any
property
(the
pressure,
for
example)
must
be
far
been
used
is
the
square-well
potential,
V,
the lower branch; the first jump to the upper branch can
statistical
IS discussed.
From the
of mechamcs
making
most
is
require examined.
very many collisions.
Thepoint
trend,ofas view
expected,
V= 00
r<<TI
ofnon-trivial
computing
time,
these
efficient
useofmore
that at
higher
densities
collisions arephase
necessary
for statistical
Discovery
transitions,
V=Vo
O"I<r<<T2
the firstfluctuations
transition, however,
therereduced
are largeby
deviations.
are best
averaging a large
not
just
looking
the
equations
V=O
r><T2,
At vlvo=
1.60,evident
5000
collisions
were
at 1.55,
number
of calculations
withrequired;
a small system
rather than
INTRODUCTION
The most common scheme is to let a represen
25000; making
while at 1.54
only
400;
at
1.535,
7000;
at
1.53,
a smaller number of calculations in a large where r is the magnitude of the separation of the
particle experience the potential of the rest of the
75 000; system.
and at 1.525, 95 000. Runs NE
in excess
of 200
000 difficulties in centers
of the
great
the present
dayof molecules and <TI, <T2, and Vo are
of a pair
cles held fixed in an average position. This av
collisions at
vivo
of
1.55
and
1.53
have
not
shown
any
to describe
physical
Both the Monte Carlo theoretical
method andattempts
the dynamical
constants.
Theand
hard sphere
potential
a special case.
potential
can beis obtained
from a definite ph
return to the lower branch, while
at
1.525
the
system
has
chemical
systems
the inadequate
method can have difficulty
due to
the slowis convergence
This mathematical
interaction potential
of events
returned several times, however only for relatively few
modelallows
or inthea sequence
self-consistent
way. The next b
apparatus
which
has
been available
the many-system to be described by a series of
of The
the lowest
systemdensity
to theat
equilibrium
configuration.
In the to
in solve
a many-body
collisions.
which the system
did
approximation in such a scheme would be to le
body
problem.
Thus,
propertiescollisions.
of an That is, since a particle does not
dynamical
method,
presently
machines,the two-body
at 1.50,
howeveravailable
the
run although
not jump
to the upper
curve iswith
molecules move in the potential of the rest o
isolated
molecule
well established
and the
is practical
follow
a small
system
ofare
molecules
for experience
any elechangesystem.
in velocity
at theand
instant
extends it
only
to 50 000 to
collisions
and
at that
density
it
Thisexcept
procedure
several variations
mentary
processes
occur when
twoitsuch
mole- from another
onlyvery
about
millimicrosecond
at lowwhich
temperature
might take
manyone
collisions
before the
appropriate
when
is separated
particle
by
<TI or <T2,
have
indeed
been
worked
out
fluctuation
escape
from
its
culestointeract
are described
by awell-known
thein a finite system, be more than twofor various phy
few hundred
thousand
collisions
for
(theoccurs
order for
of a molecule
there willlaws
never,
models. However, the calculations are so compli
108
neighborhood.
For
comparison,
the
first
jump
for
of systems
of whether
many interacting
32-particle system). behavior
It is sometimes
a worry
are changing.
particles at a time whose
that velocities
it is necessary
to seekThis
numerical solution
particlesaoccurred
vlvo=
1.55cannot,
and 1.60
at general,
about 2000
with theoretically
system for
is in
equilibrium
orinwhether
it be
is indealt
a metapotential hasintheanqualitative
features
of
a
real
molecular
means of automatic computers.
It is interesting to
collisions. This is fewer collisions
per
particle
than
for
exact way.
Even a be
three-particle system presents great
stable
state.
question
can
usually
of simplicity
is indicative
of
larger
possible resolved by potential and still some
the smaller
system
andThis
that elements
to calculate
the actualwhich
dynamics of the m
not theories relatively easy to apply.
analytical
difficulty.
starting theinsystem
variousApparently,
initial
configurations
and difficulties
make the are
analytical
density fluctuations
larger in
systems.
the Since these
particle system is, in some cases, not a greater pro
Example of ideal effective pair potentials
* Work performed under the auspices of the U. S. Atomic
Energy Commission.
1 M. N. Rosenbluth and A. W. Rosenbluth, J. Chern. Phys. 22,
881 (1954).
2 Metropolis, Rosenbluth, Rosenbluth, Teller, and Teller, J.
Chern. Phys. 21, 1087 (1953).
3 B. J. Alder and T. Wainwright, J. Chern. Phys. 27,1208 (1957).
4 W. W. Wood and F. R. Parker, J. Chern. Phys. 27, 720 (1957).
This paper discusses the Monte Carlo method in some detail, as
well as giving computational results for Lennard-Jones molecules.
6 Kirkwood, Maun, and Alder, J. Chern. Phys. 18, 1040 (1950).
Hard-sphere potential
Phase Transition for a Hard
Sphere System
B. J. ALDER AND T. E. WAINWRIGHT
University of California Radiation LalJoratory, Livermore, California
(Received August 12, 1957)
CALCULATION of molecular dynamic motion
has been designed principally to study the reA
laxations accompanying various nonequilibrium phenomena. The method consists of solving exactly (to the
number of significant figures carried) the simultaneous
classical equations of motion of several hundred particles by means of fast electronic computors. Some of the
details as they relate to hard spheres and to particles
having square well potentials of attraction have been
described. l •2 The method has been used also to calculate
equilibrium properties, particularly the equation of
state of hard spheres where differences with previous
Monte Carl03 results appeared.
The calculation treats a system of particles in a
rectangular box with periodic boundary conditions. 4
Initially, the particles are in an ordered lattice with
velocities of equal magnitude but with random orientations. After a very short initial runl •2 the system reached
the Maxwell-Boltzmann velocity distribution so that
the pressure could thereafter be evaluated directly by
means of the virial theorem, that is by the rate of change
of the momentum of the colliding particles. l •2 The
pressure has also been evaluated from the radial distribution function. 6 Agreement between the two methods
is within the accuracy of the calculation.
Soft sphere potential
(ν=1)
No attractive part
Square-well potential
O
Soft sphere potential
(ν=12)
The soft-sphere potential
becomes progressively
harder as ν increases
conceptual
computers
observing whether the
same finalbut
statemathematical,
is reached. high-speed
Furthermore,
it is possible to make theoretical extenthan the calculations required for the models.
are well
to dealwhich
with is
them.
Still another limitation
on suited
the method
con- sions of the results to more realistic potentials by
One of the aims of the exact numerical solution
To of
take
account
interaction techniques.
of a
nected with the smallness
theexplicitly
system, isinto
that
mole- the
perturbation
It is important to develop
compare
results with these analytical the
large number
of adequately
particles involves
either multi- techniques inthe
cules with long-rangefairly
potentials
cannot be
such perturbation
order to overcome one
are more clean-cut than compar
d.imensional
integralswould
or high-order
differential
studied since the field
of one molecule
extend of
the most equasevere limitations
of numerical
schemes,
WIth
expenments
on natural
systems because
Downloaded 20 Sep 2013 to 193.144.179.175. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jcp.aip.org/about/rights_and_permissions
expressions
can bethat
reduced
outside the periodictIOns.
box. These
This mathematical
limitation could
be namely,
they are
only
valid
for
the
specific
case
possible
to
set
up
artificial
many-particle
systems
for dilute solved.
systems since, in
probably overcome tobymanageable
introducingequations
approximations
interactions
which
are
both
simple
and
exactly
k
case, thetheories.
behaviorThat
of the
can
similar to those used that
in analytical
is, system
the
In be
theconceived
dynamical and
calculation
the analytical
molecules theories
are all are relatively
for which
?f properly
as
succession
of essentially
unrelated
particles could interact
at short range
but the given
initialbinary
velocities
From then much
on, more detaile
to and
workpositions.
out. Furthermore,
mteractions.
In the by
caseanofaverage
nondiluteofsystems
is
long-range interactions
could be replaced
course, that
the future
behavior
of
the
system
is
deterforma.tio.n is
from calculations of this
when thesolution
range of
small
potential. A direct numerical
of intermolecular
the quantum- forces
mined.is Various
initial
conditions
have
been to
used
than
It IS ever
possIble
getbut
from real experim
compared
to has
the not
average
distancethe molecules have been given equal
at- most frequently
mechanical many-body
problem
been intermolecular
No attractive part
It is useful to divide realistic potentials in separate
attractive and repulsive components
Attractive interaction
Van der Waals-London or fluctuating dipole interaction
Classical argument
C. Kittel
Introduction to Solid State Physics (3rd Edition)
John Wiley and sons
Electric field produced by dipole 1 on position 2
Instantaneous dipole induced by this field on 2
Potential energy of the dipole moment
Always attractive
Is the unit vector directed from 1 to 2
J. D. Jackson
Classical Electrodynamics (Chapter 4)
John Wiley and Sons
It is useful to divide realistic potentials in separate
attractive and repulsive components
Attractive interaction
Van der Waals-London or fluctuating dipole interaction
Quantum argument
Hamiltonian for a system of two interacting oscillators
Where the perturbative term is the dipole-dipole interaction
From first-order perturbation theory, we can compute the change in energy
C. Kittel
Introduction to Solid State Physics (3rd Edition)
John Wiley and sons
It is useful to divide realistic potentials in separate
attractive and repulsive components
Repulsive interaction
As the two atoms are brought together, their
charge distribution gradually overlaps,
changing the energy of the system.
The overlap energy is repulsive due to the
Pauli exclusion principle:
No two electrons can have all their
quantum numbers equal
When the charge of the two atoms overlap there is a tendency for electrons from atom B to
occupy in part states of atom A already occupied by electrons of atom A and viceversa.
Electron distribution of atoms with closed shells can overlap only if accompanied by a
partial promotion of electrons to higher unoccpied levels
Electron overlap increases the total energy of the system and gives a repulsive
contribution to the interaction
The repulsive interaction is exponential
Born-Mayer potential
It is useful to divide realistic potentials in separate
attractive and repulsive components
Buckingham potential
37
2.2 THE FORCE FIELD ENERGY
2500
Energy (kJ/mol)
2000
"Exact"
Lennard-Jones
Buckingham
Morse
1500
F. Jensen
Introduction to Computational Chemistry
John Wiley and Sons
1000
500
0
–500
0.0
1.0
2.0
3.0
4.0
5.0
Distance (Å)
Figure 2.10
Comparison of Evdw functionals for the H2—He potential
kJ/mol)
Because the exponential term converges to a constant as
,
while the term diverges,
the Buckingham potential “turns over” as
1.0
becomes small.
0.8
This may be problematic
when dealing "Exact"
with a structure with very short
Lennard-Jones
0.6
Buckingham
interatomic distances
Morse
0.4
It is useful to divide realistic potentials in separate
attractive and repulsive components
Lennard-Jones potential
37
2.2 THE FORCE FIELD ENERGY
2500
Energy (kJ/mol)
2000
"Exact"
Lennard-Jones
Buckingham
Morse
1500
1000
F. Jensen
Introduction to Computational Chemistry
John Wiley and Sons
500
0
–500
0.0
1.0
2.0
3.0
4.0
5.0
Distance (Å)
Figure 2.10
Comparison of Evdw functionals for the H2—He potential
The repulsive term has no theoretical justification.
1.0
It is used because
it approximates the Pauli repulsion well, and
is more convenient
due to the relative computational efficiency
0.8
"Exact"
of calculating r12 as the
square of r6.
Lennard-Jones
kJ/mol)
0.6
0.4
Buckingham
Morse
0
Comparison of effective two body potentials
–500
Buckingham potential
0.0
1.0
2.0
3.0
4.0
5.0
Distance (Å)
Lennard-Jones
Figure 2.10
Comparison of Evdw functionals for the H2—He potential
Morse
1.0
0.8
"Exact"
Lennard-Jones
Buckingham
Morse
F. Jensen
Introduction to Computational Chemistry
John Wiley and Sons
Energy (kJ/mol)
0.6
0.4
0.2
0.0
–0.2
–0.4
2.0
2.5
3.0
3.5
4.0
4.5
5.0
5.5
6.0
Distance (Å)
Figure 2.11 Comparison of Evdw functionals for the attractive part of the H2—He potential
The Lennard-Jones potential
The well depth is often quoted as in units of temperature,
where
is the Boltzmann’s constant
For instance, to simulate liquid Argon, reasonable values are:
We must emphasize that these are not the values which
would apply to an isolated pair of argon atoms
The Lennard-Jones potential
The well depth is often quoted as in units of temperature,
where
is the Boltzmann’s constant
Suitable energy and length parameters for interactions
between pairs of identical atoms in different molecules
WARNING:
The parameters are not
designed to be transferable:
the C atom parameters in CS2
are quite different from the
values appropriate to a C in
graphite
Interactions between unlike
atoms in different molecules
can be approximated by the
Lorentz-Berthelot mixing rules
(for instance, in CS2)
Is realistic the Lennard-Jones potential?
Dashed line: 12-6 effective Lennard-Jones potential for liquid Ar
Solid line: Bobetic-Barker-Maitland-Smith pair potential for liquid Ar
(derived after considering a large quantity of experimental data)
Lennard-Jones
Steeply rising
repulsive wall at
short distances,
due to non-bonded
overlap between
the electron clouds
Optimal
Attractive tail at large
separations, due to
correlation between
electron clouds
surrounding the
atoms.
Responsible for
cohesion in
condensed phases
Separation of the Lennard-Jones potential into
attractive and repulsive components
Steeply rising
repulsive wall at
short distances,
due to non-bonded
overlap between
the electron clouds
Attractive tail at large
separations, due to
correlation between
electron clouds
surrounding the
atoms.
Responsible for
cohesion in
condensed phases
Separation of the Lennard-Jones potential into
attractive and repulsive components: energy scales
repulsive
attractive
(u DDQ + uDQQ + u QQQ + u DDD4) are the higher multipole contributions.
The triple -dipole potential can be evaluated from the formula proposed by Axilrod
Beyond
the two body potential:
and Teller [Axi43]:
the
Axilrod-Teller potential
u DDD =
(
v DDD 1 + 3 cos θ i cos θ j cos θ k
)
(rij rik r jk )
3
(2.17)
where the angles and intermolecular separations refer to a triangular configuration of
atoms (see Figure 2.1) and where vDDD is the non-additive coefficient which can be
estimated from observed oscillator strengthsAxilrod-Teller
[Leo75].
potential:
Three body potential that results from a third-order perturbation correction to
the attractive Van der Waals-London dipersion interactions
Figure 2.1 Triplet configuration of atoms i, j and k .
The contribution of the AT potential can be either negative or positive depending on
For ions or charged particles, the long range
Coulomb interaction has to be added
Where
and
are the charges of ions and ,
is the permittivity of free space
How to deal with molecular systems
Solution
Treat the molecule as a rigid or semi-rigid unit with fixed bond-lengths
and, sometimes fixed bond and torsion angles
Justification
Bond vibrations are of very high frequency (difficult to handle in classical
simulations), but of low amplitude (unimportant for many liquid properties)
MODEL SYSTEMS A N D INTERACTION POTENTIALS
13
Fig. 1.6 An atom-atom model of a diatomic molecule.
pairwise contributions from distinct sites a in molecule i, at position ria,and b
in molecule j, at position rjb
A diatomic molecule with a
strongly binding interatomic
potential energy surface can be
simulated by a dum-bell with a
rigid interatomic bond
Interaction between nuclei and electronic
charge clouds of a pair of molecules
Complicated function if relative positions
and
and orientations
MODEL SYSTEMS A N D INTERACTION POTENTIALS
and
13
Interaction sites:
usually centered
more or less on the
position of the nuclei
in the real molecule
Fig. 1.6 An atom-atom model of a diatomic molecule.
Simplified
“atom-atom”
pairwise contributions
from distinct sitesor
a in “site-site”
molecule i, at positionapproach
ria,and b
in molecule j, at position rjb
Pairwise contributions from distinct sites
in molecule
at position
and site in molecule at position
Nitrogen, Fluorine,…
typically considered as
two Lennard-Jones
atoms separated by
fixed bond-lengths
Here a, b take the values 1,2, vab is the pair potential acting between sites a and
b, and rabis shorthand for the inter-site separation rab= Iris I,.-'
The interaction sites are usually centred, more or less, on the positions of the
nuclei in the real molecule, so as to represent the basic effects of molecular
'shape'. A very simple extension of the hard-sphere model is to consider a
diatomic composed of two hard spheres fused together [Streett and Tildesley
19761, but more realistic models involve continuous potentials. Thus,
nitrogen, fluorine, chlorine etc. have been depicted as two 'Lennard-Jones
atoms' separated by a fixed bond length [Barojas et al. 1973; Cheung and
Powles 1975; Singer, Taylor, and Singer 19771. Similar approaches apply to
polyatomic molecules.
The description of the molecular charge distribution may be improved
somewhat by incorporating point multipole moments at the centre of charge
Pair potential acting
between
and
,
Incorporate pole multipole moments at the center of
charge to improve molecular charge distribution
Might be equal to the known (isolated molecule) variable
14
INTRODUCTION
or
distribute fictitious multipoles in a similar way [Price, Stone, and Alderton
May be “effective” values chosen to give better description of the
19841. For example, the electrostatic part of the interaction between nitrogen
molecules may be modelled using five partial charges placed along the axis,
thermodynamic properties
while, for methane, a tetrahedral arrangement of partial charges is appropriate. These are illustrated in Fig. 1.7. For the case of N,, the quadrupole
moment Q is given by [Gray and Gubbins 19841
Alternative
Use “partial charges” distributed in a “physically
reasonable way” around the molecule to reproduce
the known multipole moments
Electrostatic part of the interaction
between N2 molecule might be modelled
using five partial charges placed along
the axis
(first non-vanishing moment: quadrupole)
For methane, a tetrahedral arrangement of
partial charges is appropriate
(first non-vanishing moment: octupole)
Fig. 1.7 Partial charge models. (a) A five-chargemodel for N2.There is one charge at the bond
centre, two at the positions of the nuclei, and two more displaced beyond the nuclei. Typical values
are (in units of the magnitude of the electronic charge) z = +5.2366, z' = -4.0469, giving
For large molecules, the complexity can be reduced
by fixing some internal degrees of freedom
Model for butane:
CH3-CH
MODEL SYSTEMS
A N D INTERACTION
POTENTIALS
2-CH2-CH
3
17
Represent the molecule as a four-center
molecule with fixed bond-lengths and
bond-bending angles derived from known
experimental data
Whole group of atoms (CH3 and CH2) are
condensed into spherically symmetric
effective “united atoms”
Interaction between such groups may be
represented by Lennard-Jones potential
with empirically chosen parameters
C1-C2, C2-C3, and C3-C4 bond lengths fixed
Trans-conformer
butane
Gauge
conformations
Fig. 1.8 (a) A model of butane [Ryckaert and Bellemans 19751.
[Marechal and Ryckaert 19831.
and angles fixed (can be done by
constraining the distances C1-C3 and C2-C4
Just one internal degree of freedom is left
unconstrained: the rotation about C2-C3 (the angle)
For each molecule, an extra term in the
potential energy appears in the Hamiltonian
(b) The torsional potential
Reduced units
Lennard-Jones parameters:
-  He:
20.0
0.0
-  Ar:
Energy (K)
-20.0
-40.0
He
Ar
The functional form is the
same in both cases.
Only the parameters
change
-60.0
-80.0
-100.0
-120.0
0.0
If the simulation for He
predicts a phase
transition at
1.0
2.0
Distance (nm)
To know these critical points, should we
perform two different simulations for
essentially the same interatomic potential?
3.0
The same phase
transition will occur for
Ar, although at a different
temperature
The use of reduced units avoids the possible embarrasement
of conducting essentially duplicate simulations
The interatomic potential is completely specified by two parameters:
Take them as fundamental units for energy and length.
Units for other quantities (pressure, time, momentum,…) follow directly
1.0
Energy (K)
0.5
Reduced units
0.0
-0.5
-1.0
0.0
1.0
2.0
Distance (nm)
3.0
The molecular dynamic simulation
is carried out only once.
The transformation from reduced
to other units, will be done
afterwards, taking into account the
real values of
2
Calculating the potential
I.
ENERGY FUNCTIONAL FOR A DIELECTRIC INSIDE AN ELECTRIC FIELD.
C The potential energy will be stored in a variable V,
C which is zeroed initially
V
= 0.0
C
C
C
C
C
C
Outer loop begins
DO 100 I = 1, N - 1
We assume that the coordinate vectors of our atoms
are stored in three FORTRAN arrays
RX(I), RY(I), and RZ(I),
with the particle index I
running from 1 to N (the number of particles)
RXI = RX(I)
RYI = RY(I)
RZI = RZ(I)
C
C
Inner loop begins.
We take care to count each pair only once
DO 99 J = I + 1, N
C
C
Temporary variables RXI, RYI, RZI are used
not to make a large number of array references in the inner loop
RXIJ = RXI - RX(J)
RYIJ = RYI - RY(J)
RZIJ = RZI - RZ(J)
RIJSQ = RXIJ ** 2 + RYIJ ** 2 + RZIJ ** 2
C
C
C
99
For the Lennard-Jones potential, it is useful to
have precomputed the value of sigma^2,
which is stored in the variable SIGSQ
SR2
= SIGSQ / RIJSQ
SR6
= SR2 * SR2 * SR2
SR12 = SR6 ** 2
V
= V + SR12 - SR6
ENDDO
100
ENDDO
C
C
C
The factor 4 x epsilon_0, which appears in every pair potential term,
is multiplied only once, at the very end, rather than many times
within the crucial inner loop.
V = 4.0 * EPSLON * V
Ionic Systems
Ionic Systems
Ionic systems: the Born-Mayer
potential
Repulsive interaction Coulomb
between electronic attraction
clouds
Ionic radii
1st+2nd neighbors
V
Repulsive, VR
“universal” constants
Equilibrium
distance
Two universal constants
Ro
Bonding energy
V(Ro)
Total, V
Coulomb, VC
R
Ionic systems: the Born-Mayer potential.
Validity of the model
Model adequate only for very ionic molecules.
differs less than 10% from the experimental value for NaCl
•  In this model, ions are
considered spherical
•  Improvement: consider
possible deformations of their
charge distributions
(polarizabilities)
•  With these extra polarizability
terms, the errors in
are
smaller than 3%.
+
-
R→∞
+
R
-
Shell model
Shell Model.
Shell model: linear chain
Courtesy of Ph. Ghosez,
Troisime Cycle de la Physique en Suisse
Romande
Each unit cell of lattice parameter
contains two atoms:
One Cation of mass
and static charge
One Anion of mass
and static charge
The cation is connected to the
anion through a string of force
constant
The anion core and shell are
connected through a spring
of force constant
Anion:
Spherical atomic shell of negligible mass and
charge
coupled to an ione core of charge
and mass
Charge neutrality
For the
cell, the
relative displacements of
the cation, ion core and
ion-shell are respectively
Spring constant
Shell model
Basic idea: Vibrations
aroundallows
an equilibrium
poin
The first force constant
to
describe vibrations around an
equilibrium position
!(k)
These parameters
Parameters
can
be fitted tocan be fitte
to experiment
experiment
The second force constant, that
accounts for the polarizability of the
electronic cloud, acknowledges the
internal structure of the atom
ABO3 perovskite
Covalent model without bond breaking
Morse
potential
Covalent Bond. Bonding Potentials.
Bond stretching
Harmonic
potential
Morse Potential.
Bond bending
Bond Stretching
Bond torsion
Bond Bending
Van der Waals
Bond Torsion
Hydrogen bridge
Electrostatic