1
Physical Chemistry III01403343
Molecular Simulations
Piti TreesukolChemistry Department
Faculty of Liberal Arts and Science
Kasetsart University : Kamphaeng Saen Campus
Chem:KU-KPSPiti Treesukol
2
Statistical Mechanics Individual Molecular Properties
Modes of motions; Energy levelsState Variables
T, V, P etc.
Partition function
Thermodynamics Properties
No interaction between molecules!
Chem:KU-KPSPiti Treesukol
3
Molecular InteractionsElectron distribution
Permanent dipole Induced dipole
Coulombic interactionVan der Waals interaction
Dipole-Dipole Dipole-Induced dipole Dispersion Hydrogen bonding
Chem:KU-KPSPiti Treesukol
4
Molecular Mechanics SimulationSimulate the interaction between
moleculesChanges of system configuration:
A collection of configurations are concerned Molecular dynamics
Time space
Monte Carlo methodEnsemble space
Chem:KU-KPSPiti Treesukol
5
Molecular Dynamics From the molecular positions, the forces acting
on each molecule are calculated; these are used to advance the positions and velocities through a small time-step, and then the procedure is repeated. Principal features:
Solution of Newton's equations of motion by a step-by-step algorithm.
Simulation times from picoseconds to nanoseconds.
The method provides thermodynamic, structural and dynamic properties.
Chem:KU-KPSPiti Treesukol
7
+
+
+
+
+
+
+
+
V=V(r,t) F=dV(r,t)/dr
F(tn)=m·a(tn)
r(tn+1)= r(tn) + ½ a(tn) dt2
F(tn+1)=m·a(tn+1)
r(tn+2)= r(tn+1) + ½ a(tn+1) dt2
Chem:KU-KPSPiti Treesukol
11
Monte Carlo At each stage, a random move of a molecule is
attempted; random numbers are used to decide whether or not to accept the move, and the decision depends on how favorable the energy change would be. Then the procedure is repeated. Principal features:
Sampling configurations from a statistical ensemble by a random walk algorithm.
No true analogue of time. Possible to devise special sampling methods. Provides thermodynamic and structural
properties.
Chem:KU-KPSPiti Treesukol
12
Random Walk
# up down left right
1 0.3 0.7 0.5 0.4
2 0.5 0.2 0.7 0.9
3 0.8 0.3 0.7 0.5
4 0.9 0.5 0.1 0.3
5 0.1 0.2 0.4 0.3
6 0.7 0.5 0.6 0.2
7 0.1 0.2 0.5 0.3
8 0.3 0.6 0.2 0.5
x
Random Number
Gravity
Increase the possibility to move down, how?
Chem:KU-KPSPiti Treesukol
13
Ising Model
2D-Ising Model1D-Ising Model
ji
jiij SSJE
If E’ < E then E’
If E’ > E then if random # > 0.5 then E’
Chem:KU-KPSPiti Treesukol
14
Molecular Simulation Molecular Dynamics Monte Carlo
Initial x,v
Calculate F(x)
Calculatenew a
Calculate new v
Calculate new x
dt
Initial x
Possible new x’s
CalculateE(possible x)
Calculateq, p
Move to new x
random
Chem:KU-KPSPiti Treesukol
15
Radial Distribution Function Radial distribution function, g(r)
key quantity in statistical mechanics quantifies correlation between atom pairs
The radial distribution function, also known as RDF, g(r), or the pair correlation function, is a measure to determine the correlation between particles within a system.
Specifically, it is a measure of, on average, the probability of finding a particle at a distance of r away from a given reference particle.
Chem:KU-KPSPiti Treesukol
16
The RDF is usually determined by calculating the distance between all particle pairs and binning them into a histogram. The histogram is then normalized with respect to an ideal gas, where particle histograms are completely uncorrelated. For three dimensions, this normalization is the number density of the system multiplied by the volume of the spherical shell, which mathematically can be expressed as Ni.g(r) = 4πr2ρdr, where ρ is the number density.
( )( )
( )id
r dg r
r d
r
r
Number of atoms at r for ideal gas
Number of atoms at r in actual system
( )id Nr d d
V r r
dr
4
3
2
1
0
543210
Hard-sphere g(r) Low density High density
Chem:KU-KPSPiti Treesukol
19
The radial distribution function is an important measure because several key thermodynamic properties, such as potential energy and pressure can be calculated from it. For a 3-D system where particles interact via pair-wise
potentials, the potential energy of the system can be calculated as follows:
where N is the number of particles in the system, ρ is the number density, u(r) is the pair potential.
The pressure of the system can also be calculated by relating the 2nd virial coefficient to g(r). The pressure can be calculated as follows:
0
242
drrgrVrN
V
0
32
3
2drrgr
dr
rdVdrTkp B
Chem:KU-KPSPiti Treesukol
20
Ergodic Theorem Phase space, introduced by Willard Gibbs in 1901, is a space in
which all possible states of a system are represented, with each possible state of the system corresponding to one unique point in the phase space. Usually the phase space usually consists
of all possible values of position and momentum variables.
A plot of position and momentum variables as a function of time is sometimes called a phase diagram.
A central aspect of Ergodic theorem is the behavior of a dynamical system when it is allowed to run for a long period of time. Under certain conditions, the time average of a function along
the trajectories exists almost everywhere and is related to the space average.
For the special class of ergodic systems, the time average is the same for almost all initial points: statistically speaking, the system that evolves for a long time "forgets" its initial state.
Chem:KU-KPSPiti Treesukol
21
position
velo
city
A system at time t is represented by 1 point only!
Dt
Dt
Chem:KU-KPSPiti Treesukol
22
An ensemble (also statistical ensemble or thermodynamic ensemble) is an idealization consisting of a large number of mental copies (sometimes infinitely many) of a system, considered all at once, each of which represents a possible state that the real system might be in.
Chem:KU-KPSPiti Treesukol
23
Macroscopic properties of a system define its macrostate, but they actually arise from the configuration of its microscopic components (microstate).
Phase SpaceProbability of each microstates depends on its energy
(long) time average = ensemble average
Microstate
Pi = P(Ei)Ei qi,pi
Ei ni,x, ni,y, ni,z
iE
ii eQZ
Q
1q
(pos
ition
)
p (momentum)
Chem:KU-KPSPiti Treesukol
24
Final ExaminationExam date: 28 February 2009
Definitions Partition Functions Ensembles Thermodynamic Properties Polarization Molecular Interactions
Presentation date: The exam-problems would be online 1-2
days before the exam date!