Monte Carlo, Molecular Dynamics
http://allos.up.univ-mrs.fr/simulations.html
Bogdan KuchtaLaboratoire Chimie Provence (LCP)
Université de Provence, Marseille
1990
2000
Evolution of Computational MethodsEvolution of Computational Methods
Combinationwith DFT
Molecular
Gradient corrections
Car-Parinello
Multiscale methodslink to macroscopic models
Quantum Monte Carlo
2010 Mesoscale modeling
Ab initio molecular dynamics
1960
1980
1970
Solid State Physics Quantum ChemistryStatistical Mechanics
Molecular dynamics andMonte Carlo
Molecular Mechanics
Force fieldsSemi-empirical
parameters
Total energies
Analytical forcesgeometry optimization
Analytical frequencies
SCF
Hartree-Fock
Total energies
Forces
Car-Parinello
Density functional
Tight-binding
Energy band structures
example: 2003 Chemistry Nobel Prize“for discoveries concerning channels in cell membranes”
Simulating water transport throughAquaporin channel
“But the depth of science has increased dramatically, and
Alfred Nobel would be astonished by the changes. Now in
the 21st century, the boundaries separating chemistry,
physics, and medicine have become blurred, and as
happened during the Renaissance, scientists are following
From Peter From Peter Agre’sAgre’s Nobel Banquet SpeechNobel Banquet Speech
http://www.nobel.se
happened during the Renaissance, scientists are following
their curiosities even when they run beyond the formal
limits of their training. This year a former physics student
shares the Economics Prize, a philosophy student shares
the Physics Prize, chemistry and mathematics students
share the Medicine Prize, and medical students share the
Chemistry Prize.
Modeling Modeling –– interdisciplinary scienceinterdisciplinary science
Numerical Modeling
Numerical simulations
[atomistic]
Quantum Chemistry
(electrons)
[atomistic]
(Optimization,Monte Carlo,
Molecular Dynamics)
Finite elements
(objects > µµµµm)
10-6
10-3
100
(µµµµs)
(ms)
TIME/s
Mesoscale methods
Semi-empirical
Continuum
Lattice Monte CarloBrownian dynamicsDissipative particle dyn
Methods
Theory and Simulation Scales
Based on SDSC Blue Horizon (SP3)512-1024 processors1.728 Tflops peak performanceCPU time = 1 week / processor
Atomistic Simulation Methods
10-15
10-12
10-9
10-10 10-9 10-8 10-7 10-6 10-5 10-4
(nm) (µµµµm)
(fs)
(ps)
(ns)
LENGTH/meters
Semi-empiricalmethods
Ab initiomethods
Monte Carlomolecular dynamics
tight-bindingMNDO, INDO/S
1. Discrete character of variables, ( ∆x, ∆t ) – defined by the problem
2. Constant number of independent variables
Limitations due to the speed and the memory capacity of computers
N – number of independent variables
(electrons, atoms or finite elements)
nmµµµµ m –nm≥µ≥µ≥µ≥µ m
N finite
elementsN atoms N electrons
(electrons, atoms or finite elements)
Simulations
numériquesExpérience
Real systems
(nature)
Theories(analytical solution)
Model (parameters, interaction, ...)
Numerical
experiments
experiment
Why simulations ?Why simulations ?
1. Fundamental
studies, e.g. to
determine range of
validity of Kelvin’s
equation, Fick’s law of
diffusion, Newton’s
law of viscosity, etc.
2. Test theories by
comparing theory and
Exact solution
(trajectories of all particles)
Tests of model Tests of theories
data (spectra)
experimental
Approximate
solutions
comparing theory and
simulation
3. Test model by comparing simulated and experimental properties. Then
use model in further simulations to carry out “experiments” not possible
in the laboratory, e.g. critical points for molecules that decompose below Tc,
properties of molten salts, long-chain hydrocarbon properties at very high
pressures, properties of confined nano-phases, etc.
Construction of
modelMethod
Model of
interaction
Numerical simulations: how ?Numerical simulations: how ?
Real systems
(molecules in pore)
Pores structure and
adsorbed atoms
Force Field
Numerical Simulations in
statistical ensemble
Generation of
states
Trajectory
analysis
interaction
Interaction with
environment
Equation of
motion
T = const.
N variable
(Statistical ensemble)
Stochastic
(Monte Carlo)
Molecular Dynamics
Monte Carlo algorithm
Mean values and
fluctuations of:
Energies
Number of atoms
Classical Force FieldsClassical Force Fields
Consist of: 1. Analytical form of the interatomic potential energy
U=Ee(R) as a function of the atomic coordinates of
the molecule
2. Parameters which enter U
------------------------------
• Fundamental to everything is the Schrödinger equation
–
– wave function
– H = Hamiltonian operator
– time independent form
H it
∂ΨΨ =∂
h
( , , )R r tΨNuclear coordinates
Electronic coordinates
H EΨ = Ψ
22 im
H K U U= + = − ∇ +∑h
Solution of Schrödinger equation :
Born-Oppenheimer approx. Ψ(r,R)≈ψ (r|R)Θ(R)
– electrons relax very quickly compared to nuclear motions
– nuclei move in presence of potential energy obtained by solving electron distribution for
fixed nuclear configuration
Equation of motion for electrons: ( Ke + V ) ψ(r, R) = Ee(R) ψ(r, R)
Potential Energy Surface (PES)
PES
e e
Equation of motion for nuclei:
(motion on the PES)
quantum classical (MD, MC)
(R)Edt
Rdm e−∇=
2
2
( Kn + Ee(R) ) Θ(R) = ET(R) Θ(R)
Intermolecular forces.Intermolecular forces.Intermolecular forces.Intermolecular forces.
long range:interaction energy is proportional
to some inverse power
of molecular separation
electrostatic: from static charge
distribution (attractive or repulsive)
induction: from distortion caused by
short range:interaction energy exponentially decays
with molecular separation
at small intermolecular distance,
an overlap of the molecular wave
functions causes electronic exchange
or repulsion
Contributions to PES
induction: from distortion caused by
molecular field of neighbors (always
attractive)
dispersion: from instantaneous
fluctuations caused by electron
movement (always attractive)
or repulsion
In theory, it is possible to
calculate the intermolecular
interactions from first principles (ab initio).(in practice, only for small systems)
non pair - additive
No simple theory exists.No simple theory exists.
Repulsion (overlap) Repulsion (overlap)
forces.forces.
Repulsion (overlap) Repulsion (overlap)
forces.forces.
( ) ( ) Broverlap erAru −≈
For simple molecules, can be approximated byFor simple molecules, can be approximated by
inconvenient to use.
People often approximate this by
Contributions to PES
( ) noverlap Arru −≈ with n = 8 to ∞
People often approximate this by
( )
−
=612
4r
σ
r
σεru
Example: Lennard-Jones potential
repulsione
r
u
s
rmin
• Total pair energy breaks into a sum of terms
Intramolecular only
• U - stretch • UvdW - van der WaalsRepulsion
polelvdWcrosstorsbendstrN
e UUUUUUUrURE ++++++=⇒ )()(
Intermolecular only
Contributions to PES
• Ustr - stretch
• Ubend - bend
• Utors - torsion
• Ucross - cross
• UvdW - van der Waals
• Uel - electrostatic
• Upol - polarization
taken from Dr. D. A. Kofke’s lectures on Molecular Simulation, SUNY Buffalohttp://www.eng.buffalo.edu/~kofke/ce530/index.html
Mixed terms
Repulsion- +-+
-+ -+
The potential energy of N interacting particles can be evaluated as:
Calculation of interaction energyCalculation of interaction energyCalculation of interaction energyCalculation of interaction energy
( ) ( ) ( ) ....,,, 321 +++= ∑∑ ∑∑∑∑> >>>
N
i
N
ijkji
N
ijk
N
iji
N
iji
N
ipot rrrurruruE
effect of an
external field
interactions
between pairs
of particles
interactions
between
particle triplets
many body
interactions
interactions between particles
particle triplets
Typically, it is assumed that only two-body term is important
( Epot is truncated after the second term)
How can we use this PES (interaction energy) in simulations ?
In some particular cases three-body term should be considered
Classical Force FieldsClassical Force Fields
J Simple, fixed algebraic form for every type of interaction.
J Variable parameters depend on types of atoms involved.
CHARMM* force field:
*Chemistry at HARvard Macromolecular Mechanics
Force fieldsForce fields(potential energy, interaction model, …)(potential energy, interaction model, …)
Electrons: H(r,R)ΨR(r) = ERΨR(r) - ab initio
Atoms: Ecovalent = ∑kij(rij-roij)2 + ∑kijk(Θijk-Θoijk)
2 + .......
Edisp(rij) = -Cdisp/rij6
E (r ) = (σ/r )n
k, C, σσσσ, .... Semi-empirical
AtomisticErep(rij) = (σ/rij)n
................
Finite elements: (example: diffusion)
∂∂=
∂∂
2
2 ),(),(
x
txcD
t
txc
D, .... Mesoscale
Atomistic
Continuous
Components of a Force FieldsComponents of a Force Fields
Any force field contains the necessary building blocks
for the calculation of energy and force:
- a list of atom types
- a list of atomic charges
- rules for atom-types
- functional forms of the components of the energy expression
- parameters for the function terms
- rules for generating parameters that have not been explicitly defined
(for some force fields)
- a defined way of assigning functional forms and parameters (for some
force fields)
Why atomistic simulations depend on
Statistical Thermodynamics (Physics) ?
The language of computer simulations and real experiments is different; The language of computer simulations and real experiments is different;
simulations are using the notions ofsimulations are using the notions of::
Phase spacePhase space
Ensemble, probability, distribution of statesEnsemble, probability, distribution of states
Instantaneous (microscopic) configurations Instantaneous (microscopic) configurations
(positions and velocities)(positions and velocities)(positions and velocities)(positions and velocities)
Experiments measures average propertiesExperiments measures average properties: :
structures, energies, density, etc… structures, energies, density, etc… -- macroscopic propertiesmacroscopic properties
In order to interpret microscopic measurements to compare with experimental In order to interpret microscopic measurements to compare with experimental
average values we need average values we need
the statistical physics (thermodynamics)the statistical physics (thermodynamics)
Ergodic hypothesis
Ensemble of atoms defined by their positions in a cell – many body
problem
Numerical SimulationsNumerical Simulations
Why atomistic simulations depend on
Statistical Thermodynamics (Physics) ?
Ergodic hypothesis
<A(r(t), p(t))>time = <A(r, p)>ensemble
Molecular Dynamics Monte Carlo
Le « Volume » de l’espace des phases : dΩΩΩΩ = dr1dp1 dr2dp2 .... drNdpN
L’espace des phases (N~1023):
(rN(t), pN(t)) = (r1(t),r2(t),r3(t),…,rN(t), p1(t), p2(t), p3(t), …pN(t)) – chaque
point définit un état microscopique du système
The Semi-Classical Approximation
1 1 2 2 N N
L’évolution de ΩΩΩΩ(t) au cours du temps définit
une trajectoire de phase qui est donc parfaitement
déterminée par la donnée du point initial ΩΩΩΩo.
Alors, un système qui est en équilibre au niveau
macroscopique est dynamique à l'échelle
microscopique.
Exemple: la pression d’un gaz dans un container
dans les intervalles du temps très courts Atom i: ri, pi
La mécanique quantique permet de préciser le nombre d’états
microscopiques distincts.
Cela grâce au principe d’incertitude de Heisenberg :
The Semi-Classical Approximation
Interprétation de l’extension en phase (r + p)
Cela grâce au principe d’incertitude de Heisenberg :
dpx drx ∼ h (constante de Planck)
Une analyse plus précise conduit à d’une évaluation du nombre des états :
dn = dΩΩΩΩ/h3N
Monte Carlo (MC)
• If we want to know some (mechanical) property ( )NN rpA ,
we can get it from an ensemble average :
( ) ( )∫ ∫= NNNNNN rpPrpArdpdA , ,
In the canonical ensemble,
∑−
− EkT
Ei
i
e
Why atomistic simulations depend on
Statistical Physics : brief account of MC, MD
In the canonical ensemble,
∑−
==i
kT
EkT
i
i
eZZ
eP
• In MC a random number generator is used to move the molecules. The
moves are accepted or rejected according to a recipe that ensures
configurations have the Boltzmann probability proportional to : kT
Ei
e−
•First MC in 1953: N. Metropolis, A.W. Rosenbluth, M.N. Rosenbluth, A.H. Teller
and E.Teller, J. Chem. Phys. 21, 1087 (1953)
Distribution Law for Canonical EnsembleDistribution Law for Canonical Ensemble
A system at fixed N,V,T can exist in A system at fixed N,V,T can exist in
many possible quantum states, many possible quantum states,
1,2,….n,… with energies 1,2,….n,… with energies
EE11,E,E22,…E,…Enn,… The probability of ,… The probability of
observing the system in a observing the system in a
particular state n is given by the particular state n is given by the
Boltzmann distribution law,Boltzmann distribution law,
(7)n n
n
E E
n E
n
e eP
Q e
β β
β
− −
−= =∑
(8)nE
nQ e β−=∑
where where β β = 1/= 1/kTkT
and k = R/Nand k = R/NA A = 1.38066x10= 1.38066x10--2323 JKJK--11 is the is the
Boltzmann constant. Boltzmann constant.
Q Q is the is the canonical partition functioncanonical partition function
T = const
N, V
E1,E2,….
dq
Canonical Distribution Law: ProofCanonical Distribution Law: Proof
Basic Postulate:Basic Postulate:
For a closed system (constant For a closed system (constant NN) at fixed volume ) at fixed volume VV and temperature and temperature TT, the , the
only dynamical variable that only dynamical variable that PPnn depends on is the energy of the state depends on is the energy of the state nn, , EEnn, ,
i.e. i.e. PPnn=f(E=f(Enn))
Remember !
En depend on N and V only ! (Schroedinger equation depends on N and the En depend on N and V only ! (Schroedinger equation depends on N and the
boundary conditions – En are the quantum solutions)
They do not depend on T !!! - Pn depends on T !!!
Higher temperature, probability of the
higher energy is larger
E1 E2 E3
E5E4
E6 ……..
(7)n n
n
E E
n E
n
e eP
Q e
β β
β
− −
−= =∑
Derivation of Boltzmann Distribution LawDerivation of Boltzmann Distribution Law
Consider a body (the system) at (N,V) immersed in a Consider a body (the system) at (N,V) immersed in a
large heat bath at temperature T. Quantum states large heat bath at temperature T. Quantum states
1,2,…n,… are available to the body and1,2,…n,… are available to the body and
Suppose a second body, with (N’,V’) is immersed in Suppose a second body, with (N’,V’) is immersed in
the same heat bath, and has possible quantum states the same heat bath, and has possible quantum states
1’,2’, ….k, …, so that1’,2’, ….k, …, so that
( ) (9)n nP f E=
'( ) (10)k kP f E= ( ) (10)k kP f E=
T = const
N, V
E1,E2,….
dq N’, V’
E’1,E’2,….
dq’( ) (9)n nP f E= '( ) (10)k kP f E=
'( ) (11)nk n kP f E E= +
Derivation of Boltzmann Distribution LawDerivation of Boltzmann Distribution Law
If If PPnknk is the probability that body 1 is in state n and body 2 is in state k, thenis the probability that body 1 is in state n and body 2 is in state k, then
If the heat bath is large enough the 2 bodies will behave independently. If the heat bath is large enough the 2 bodies will behave independently.
The time body 1 spends in a particular quantum state will be unaffected The time body 1 spends in a particular quantum state will be unaffected
by the state of body 2, soby the state of body 2, so
(12)P P P= (12)nk n kP P P=So, we haveSo, we have
The The only only function for which this relation is obeyed is an exponential one, function for which this relation is obeyed is an exponential one,
i.e. i.e.
' '( ) ( ) ( ) (13)n k n kf E E f E f E+ =
'
1 2; (14)n kE En kP C e P C eβ β− −= =
1)(
)()(
)(
)()(
yx
yxf
y
yx
yx
yxf
y
yxf
+∂+∂=
∂+∂
+∂+∂=
∂+∂
y
yfxf
y
yxf
∂∂=
∂+∂ )(
)()(
Derivation of Boltzmann Distribution LawDerivation of Boltzmann Distribution Law
Proof: f(x+y) = f(x)f(y)
Similar derivation with respect to x:
x
xfyf
yx
yxf
∂∂=
+∂+∂ )(
)()(
)(y
yfxf
yx
yxf
∂∂=
+∂+∂ )(
)()(
)(
xyx ∂+∂ )(
x
xf
y
yf
x
xf
xfy
yf
yf ∂∂=
∂∂
⇒∂
∂=∂
∂ ))(ln())(ln()(
)(
1)(
)(
1
Left hand side is independent of y, right hand side is independent of x
It means that both sides are independent of x and y, that is, is equal to constant :
-βxCexfcxxf ββ −=⇒+−= )())(ln(
X = En, f = pn ∑pn = 1
1nnP =∑
1 1/
(15)
n
n n
n
E
n
E E
n E
C e
e eP
e Q
β
β β
β
−
− −
−
=
= =
∑
∑
Since the probabilities are normalized, , we haveSince the probabilities are normalized, , we have from (14)from (14)
Derivation of Boltzmann Distribution LawDerivation of Boltzmann Distribution Law
Note that CNote that C11 and Cand C22 are different for the two bodies since in general their volumes and are different for the two bodies since in general their volumes and
number of molecules will be different. However, number of molecules will be different. However, ββ must be the same for the two must be the same for the two
bodies for eqn. (13) to hold. Since they are in the same heat bath this suggests that bodies for eqn. (13) to hold. Since they are in the same heat bath this suggests that ββ is is
related to temperature; related to temperature; ββ must be positive, otherwise must be positive, otherwise PPnn would become infinite when would become infinite when
EEnn approaches infinity. approaches infinity.
nE
ne Q∑
EnsemblesEnsembles
We must first choose a set of We must first choose a set of independent variables (ensemble),independent variables (ensemble), and then derive the and then derive the distribution law for the probability the system is in a certain (quantum or classical) distribution law for the probability the system is in a certain (quantum or classical) state. The most commonly used ensembles are:state. The most commonly used ensembles are:
N,V,E N,V,E MicrocanonicalMicrocanonical ensembleensemble
N,V,T Canonical ensembleN,V,T Canonical ensemble
µµµµµµµµ,V,T Grand canonical ensemble,V,T Grand canonical ensemble
N,P,T IsothermalN,P,T Isothermal--isobaric ensembleisobaric ensemble
Here :Here :
N=number of particles, N=number of particles,
V=volume, V=volume,
T=temperature, T=temperature,
E=total energy, E=total energy,
P=pressure, P=pressure,
µµµµµµµµ=chemical potential.=chemical potential.
Initially we worked in the Initially we worked in the canonical ensemblecanonical ensemble
L’univers – ensemble microcanonique absolu (NVE)
Il est constitué de systèmes identiques de point de vue macroscopique et de même
énergie.
En pratique, expérimentalement il est impossible de fixer exactement l’énergie pour un
système donné.
Probabilité d’un état i: Pi = 1/W
EnsemblesEnsembles
L’ensemble canonique – fluctuation d'énergie à cause de contact du système avec
l'environnement (NVT)
Un système est plongé dans un thermostat avec lequel il peut échanger de l’énergie
sous forme de chaleur. L’état d’équilibre correspond du point de vue macroscopique à
l'égalité des températures du système et du thermostat. Cet équilibre a une caractère
dynamique, avec d’incessantes fluctuations de l’énergie moyenne.
Probabilité d’un état i: Pi = e-Ui/kT/Z
L’ensemble grand canonique – on autorise les fluctuation du nombre de particules
(µVT)
Le système est en relation avec un « réservoir de particules » avec lequel il peut
échanger non seulement de l’énergie, mais aussi des particules.
Probabilité d’un état i: Pi = e-(Ui-µµµµN
i)/kT/Z ;
EnsemblesEnsembles
Rappel : <A> = ∑PiAi ( = ∫p(x)A(x) dx )
Probabilité d’un état i: Pi = e i i /Z ;
µµµµ - potentiel chimique ;
Z = ∑ e-(Ui-µµµµN
i)/kT
• With the distribution law known, it is straightforward to derive relations for the
thermo properties in terms of Z.
• Internal Energy, U
U is simply the average energy of the system
nE
nn
VN
eEZ β
β−∑−=
∂∂
,
Expressions for Thermodynamic Properties
∂∂−=== ∑∑ −
ββ Z
eEZ
EPN
U
n
Enn
nn
nln1
ββ
β
β
β
ddZ
Zeg
egU
ZegN
eZNgU
N
UN
NU
i
i
i
ii
i
i
i
ii
i
i
i
ii
iU
iU
iU
iU
1−====∑∑
∑
∑∑∑
−
−
−
−
Proof:
Using β=1/kT, this can be rewritten as
Ideal gas: Ar, Kr,Xe
Z = qN q = (2πmkT/h2)3/2
lnQ = 3/2NlnT + C
U = 3/2NkTT
ZNkTU
∂∂= ln2
Expressions for Thermodynamic Properties
C'est-à-dire pour une mole (N=L) T
ZRTU
∂∂= ln2
Free energy A
A = U - TS
• Helmholtz Free Energy,A
• A=U-TS is related to U via the Helmholtz eqn.,
( )2
/ (18)
A T U
T T
∂ = − ∂
dA = -SdT – Vdp
A = U –TS
A/T = U/T – S
∂(A/T)/∂T = 1/T(∂A/∂T) – A/T2 =
= S/T –A/T2 = -U/T2
Expressions for Thermodynamic Properties
so that from we have
2
,
(18)N V
T T= − ∂
lnln (19)
A Qk dT k Q C
T T
∂= − = − +∂∫
Entropie S
Nous avons vu que S = Nk ln Z + ββββkU = Nk ln Z + U/T
Since A=U-TS we have S=(U-A)/T2
,
ln (17)
N V
QU kT
T
∂ = ∂
Z
ZkT
Aln−=
Expressions for Thermodynamic Properties
ln Q∂
• Entropy and the Probability Distribution
• From the distribution law, (15), ,we have/ /nE kTnP e Q−=
ln / lnn nP E kT Q= − −
ZkT
ln−=
,
lnln (20)
N V
QS kT k Q C
T
∂ = + − ∂
Comparison with eqn. (20) shows that1
∑
Multiplying throughout by Pn and summing over all n gives
∑∑∑ −−=n nnn nnn n PQPE
kTPP ln
1ln
= = = −∑ ∑
Expressions for Thermodynamic Properties
Thus, entropy is directly related to the probability distribution
1ln lnn nn
P P U QkT
= − −∑ n n nn nU P E E e= = = −∑ ∑
ln (21)n nnS k P P C= − −∑
• Pressure, P
TNTN V
QkTP
V
AP
,,
ln
∂
∂=⇒
∂∂−= (22)
• Chemical potential, m
lnN
QkTµ
N
Aµ
∂∂−=⇒
∂∂= (23)
Expressions for Thermodynamic Properties
'' ,,,, NVTiNVTi NkTµ
Nµ
∂
−=⇒
∂
= (23)
• Heat capacity, Cv
VNVNv
VNv
T
QkT
T
QkTC
T
UC
,2
22
,,
lnln2
∂∂+
∂
∂=⇒
∂∂= (24)
• Enthalpy, H
TNVN V
QkTV
T
QkTHPVUH
,,
2 lnln
∂
∂+
∂
∂=⇒+= (25)
La définition statistique de la capacité calorifique –fluctuations d'énergie
( ) ( ) EEEEE 2222δ =><−>=<><−=
( ) VVN
CkTT
EkTE
2
,
22 =
∂∂=δ
( ) ( )
( ) VN
VNVN
ii
iii
i
EZ
ZZZZ
EPEP
,
22
,22
,221
2
2
//ln
//
ββ
ββ
∂><∂−=
∂∂=
=∂∂−∂∂=
=
−=
−−
∑∑
In the semi-classical approximation, we assume that the translational and
(extended) motions can be treated classically.
• Exceptions would be light molecules (H2, He, HF, etc.) at low temperatures
• In this approximation, the Hamiltonian operator can be written as:
H = H cl + H qu
H
Semi-classical approximation
where H cl corresponds to coordinates that can be treated classically
(translation, rotation),
and H qu to those that must be treated quantally (electronic, vibration)
• We further assume that there are two independent set of quantum states,
corresponding to H cl and H qu, respectively.
→ this implies neglect of interaction between vibrations, translations and
rotations
qun
clnn PPP = and • We can now write quclQQQ = , where
∑∑ −−
−−
==
==
n
Eβqu
n
Eβcl
qu
Eβqu
ncl
Eβcl
n
qun
cln
qun
cln
eQeQ
Q
eP
Q
eP
,
,
Semi-classical approximation
∑∑nn
Because the intermolecular forces are assumed to have no effect on the quantum states,
Enqu is just a sum of single-molecule quantum energies, which are themselves mutually
independent:
nNnnnqun εεεεE ++++= ...321
and Nququ qQ = , where ==∑
−
j
βεNqu
qujeq molecular partition function
• Therefore, Qqu is independent of density: it is the same for a liquid, solid
or ideal gas.
• In classical statistical mechanics, the probability distribution
clEβcl
n QePcln−=
is replaced by a continuous probability density
( )Nω
NNN pωprP ,,,for the classical states in phase space. Here:
Semi-classical approximation
for the classical states in phase space. Here:
== NN rrrrr ,...,,, 321 locations of centers of molecules 1, 2, … N
==N
N ppppp ,...,,,321 translational momenta conjugate to Nrr ,,1 K
== NN ωωωωω ,...,,, 321 orientations of molecules 1, 2, … N
== NωωωωNω ppppp ,...,,, 321 orientational momenta conjugate to Nωω ,...,1
Classical Partition Function
( )∫ ∫
−==
kT
,p,ωp,rdpωdpdrd
hNhN
ZQ
Nω
NNNNω
NNNNfNfcl
Hexp
!
1
!'
K
• f = number of classical degrees of freedom per molecule
= 6 for non-linear molecules
= 5 for linear molecules
• The factor (N !)-1 arises because molecules are indistinguishable• The factor (N !)-1 arises because molecules are indistinguishable
• The factor h-Nf corrects for the fact that the phase coordinates cannot be
precisely defined (uncertainty principle, see Appendix 3D GG)
where
( )∫ ∫
−=
kT
,p,ωp,rdpωdpdrdZ
Nω
NNNNω
NNN Hexp'
K
Is called phase integral
• The classical partition can be factorized:
NfclhN
ZQ
!= crtcl QQQQ =
with232 2
exp1 N
iN mkTπppdQ
=
−= ∫ ∑
Factoring the partition function Q
23
22
exp1
i
iNNt
h
mkTπ
mkT
ppd
hQ
=
−= ∫ ∑
orN
ttQ 3Λ
−= , where
=
=
212
2Λ
mkTπ
ht
thermal de Broglie wavelength of the molecules
where
rΛkTIπ
h
kTIπ
h
kTIπ
h
π zyx
21
2
221
2
221
2
2
888
1
= (nonlinear)
Factoring the partition function Q
( )N
rαi α
αiNNf
N
r kTI
JJd
hQ −
− =
−= ∫ ∑ Λ
2exp
Ω2
3
IkTπ
h
kTIπkTIπkTIπ zyx
2
2
8
888
=
(linear)
( )N
ckTωrNNNc
N
ZueωdrdN
QNN
Ω!Ω!
1 , == ∫−
and
In these equations,
∫= ωdΩ
πφdθθd
πχdφdθθd
π
o
π
o
π
o
π
o
π
o
4sin
8sin
2
222
==
==
∫∫
∫∫∫ (nonlinear)
(linear)
Factoring the partition function Q
oo
• Qc is the only part of Q that depends on ( ),N NωrU , and the only part
that depends on the volume (density).
• For an ideal gas, 0=U and!N
VQ
N
c = , so
kTρV
NkT
V
QkTP
TN
==
∂
∂=,
ln
This equation of state is known to be true independently from kinetic theory.
σ2= <f2> - <f>2 where ∑∑==
==n
ii
n
ii xf
nfxf
nf
1
2
1
2 )(1
)(1
The accuracy depends on the number of trials in the MC method. One possible measure of the error is the variance σσσσ2:
σ cannot be a direct measure of the error !!!!
One way to obtain an estimate for the error is to make additional runs of n trials each. Each run of n trials yields a mean value (or measurement) which we denote as Mα. The
Simulation error analysis
Each run of n trials yields a mean value (or measurement) which we denote as Mα. The magnitude of the differences between the measurements is a measure of the error
associated with a single measurement:
σmm2= <M2> - <M>2 where
∑∑==
==mm
Mm
MMm
M1
2
1
2 11
αα
αα
Error estimation (standard mean deviation) : σm= σ/m1/2
Simulation error analysis
G
What is the minimal length n ?
n steps
Trajectory mn
m bins
2
2
limσσσσσσσσ m
nnc
nn
∞→=
n
Error estimation (standard mean deviation) : σm= σ/n1/2
Proof:
One may show that this relation is exact in the limit of a very large number of measurements
∑∑∑
∑
= ==
=
==
=
m n
ii
m
n
ii
xnm
Mm
M
xn
M
1 1,
1
1,
11
1
ααα
αα
The difference between measurement α and the mean : −= MMe
Simulation error analysis
The difference between measurement α and the mean :
∑=
=
−=m
m em
MMe
1
22 1
αα
αα
σ
We now relate σm to the variance of the individual trials : Mxd ii −= ,, αα
We write: ∑∑∑===
=−=−=−=n
ii
n
ii
n
ii d
nMx
nMx
nMMe
1,
1,
1,
1)(
11ααααα
== ∑∑ ∑∑== ==
n
jj
m n
ii
m
m dn
dnm
em 1
,1 1
,1
22 1111α
αα
αασ
We may calculate :
We expect that dα,I are independent and equally positive or negative on average. Hence in the limit of a very large number of measurements, we
Simulation error analysis
∑∑∑= ==
==m n
i
i
m
m dmn
em 1 1
,2
21
22 11
αα
αασ
nm
22 σσ =
average. Hence in the limit of a very large number of measurements, we expect that only the terms with i=j will survive:
Because the definition of the variance, we have:
= ∑∑= =
m n
iim d
mn 1 1
2,
2 1
αασ
Ab Initio Methods
Calculate properties from first principles, solving the
Schrödinger (or Dirac) equation numerically.
Pros:
• Can handle processes that involve bond
breaking/formation, or electronic rearrangement (e.g.
chemical reactions).
• Methods offer ways to systematically improve on the
Electron localization function for
(a) an isolated ammonium ion
and (b) an ammonium ion with
its first solvation shell, from ab
initio molecular dynamics. From
Y. Liu, M.E. Tuckerman, J. Phys.
Chem. B 105, 6598 (2001)
• Methods offer ways to systematically improve on the
results, making it easy to assess their quality.
• Can (in principle) obtain essentially exact properties
without any input but the atoms conforming the system.
Cons:
• Can handle only small systems, about O(102) atoms.
• Can only study fast processes, usually O(10) ps.
• Approximations are usually necessary to solve the eqns.
Semi-empirical Methods
Use simplified versions of equations from ab initio methods,
e.g. only treat valence electrons explicitly; include
parameters fitted to experimental data.
Pros:
• Can also handle processes that involve bond
breaking/formation, or electronic rearrangement.
• Can handle larger and more complex systems than ab
Structure of an oligomer of
polyphenylene sulfide
phenyleneamine obtained with
the semiempirical method. From
R. Giro, D.S. Galvão, Int. J. Quant.
Chem. 95, 252 (2003)
• Can handle larger and more complex systems than ab
initio methods, often of O(103) atoms.
• Can be used to study processes on longer timescales
than can be studied with ab initio methods, of about
O(10) ns.
Cons:
• Difficult to assess the quality of the results.
• Need experimental input and large parameter sets.
Atomistic Simulation Methods
Use empirical or ab initio derived force fields, together
with semi-classical statistical mechanics (SM), to
determine thermodynamic (MC, MD) and transport (MD)
properties of systems. SM solved ‘exactly’.
Pros:
• Can be used to determine the microscopic structure of
more complex systems, O(105-6) atoms.
Structure of solid Lennard-Jones
CCl4 molecules confined in a
model MCM-41 silica pore. From
F.R. Hung, F.R. Siperstein, K.E.
Gubbins, in progress.
more complex systems, O(10 ) atoms.
• Can study dynamical processes on longer timescales, up
to O(1) ms
Cons:
• Results depend on the quality of the force field used to
represent the system.
• Many physical processes happen on length- and
timescales inaccessible by these methods, e.g. diffusion
in solids, many chemical reactions, protein folding,
micellization.
Mesoscale Methods
Introduce simplifications to atomistic methods to remove
the faster degrees of freedom, and/or treat groups of
atoms (‘blobs of matter’) as individual entities interacting
through effective potentials.
Pros:
• Can be used to study structural features of complex
systems with O(108-9) atoms.
• Can study dynamical processes on timescales inaccessible
Phase equilibrium between a
lamellar surfactant-rich phase
and a continuous surfactant-
poor phase in supercritical CO2,
from a lattice MC simulation.
From N. Chennamsetty, K.E.
Gubbins, in progress.
• Can study dynamical processes on timescales inaccessible
to classical methods, even up to O(1) s.
Cons:
• Can often describe only qualitative tendencies, the
quality of quantitative results may be difficult to
ascertain.
• In many cases, the approximations introduced limit the
ability to physically interpret the results.
Applications: Applications: Mesoporous MaterialsMesoporous Materials
Surfactant Silica 9600 surfactant chains17400 silica unitsReduced temperature 6.5
-130
-132
Synthesis of MCM-41
-132
-134
-136
-138
-140
-142
-144
-146
-148
0 100000 200000 300000 400000 500000
cycles
Ene
rgy
per
mol
ecul
e
Equilibrium
Ref: 1) F. R. Siperstein, K. E. Gubbins, Langmuir, 19, 2049(2003)
Continuum Methods
Assume that matter is continuous and treat the properties
of the system as field quantities. Numerically solve
balance equations coupled with phenomenological
equations to predict the properties of the systems.
Pros:
• Can in principle handle systems of any (macroscopic)
size and dynamic processes on longer timescales.
Temperature profile on a laser-
heated surface obtained with
the finite-element method.
From S.M. Rajadhyaksha, P.
Michaleris, Int. J. Numer. Meth.
Eng. 47, 1807 (2000)
size and dynamic processes on longer timescales.
Cons:
• Require input (viscosities, diffusion coeffs., eqn of state,
etc.) from experiment or from a lower-scale method that
can be difficult to obtain.
• Cannot explain results that depend on the electronic or
molecular level of detail.