1 Physical Chemistry III 01403343 Molecular Simulations Piti Treesukol Chemistry Department Faculty...

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Physical Chemistry III01403343

Molecular Simulations

Piti TreesukolChemistry Department

Faculty of Liberal Arts and Science

Kasetsart University : Kamphaeng Saen Campus

Chem:KU-KPSPiti Treesukol

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Statistical Mechanics Individual Molecular Properties

Modes of motions; Energy levelsState Variables

T, V, P etc.

Partition function

Thermodynamics Properties

No interaction between molecules!


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Molecular InteractionsElectron distribution

Permanent dipole Induced dipole

Coulombic interactionVan der Waals interaction

Dipole-Dipole Dipole-Induced dipole Dispersion Hydrogen bonding


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Molecular Mechanics SimulationSimulate the interaction between

moleculesChanges of system configuration:

A collection of configurations are concerned Molecular dynamics

Time space

Monte Carlo methodEnsemble space


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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.


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+

+

+

++

+

V=V(r,t) F=dV(r,t)/dr


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+

+

+

+

+

+

+

+

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


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| | | | | | | |


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10 -

9 -

8 -

7 -

6 -

5 -

4 -

3 -

2 -

1 -

0 -

X=

V=

F=


10


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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.


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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?


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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’


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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


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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.

http://www.matdl.org/matdlwiki/index.php/Image:Rdf_schematic.jpg


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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


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Gas

Liquid

Solid


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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


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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.

http://upload.wikimedia.org/wikipedia/commons/8/87/Focal_stability.png


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position

velo

city

A system at time t is represented by 1 point only!

Dt

Dt


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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.


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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)


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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!


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g(r)

4.0 –

3.5 –

3.0 –

2.5 –

2.0 –

1.5 –

1.0 –

0.5 –

0.0 –

| | | | | | |

0.0 0.2 0.4 0.6 0.8 1.0 1.2

Distance (nm)

Radial Distribution Function of water at 298 K

gHH

gOH

gOO

1 Physical Chemistry III 01403343 Molecular Simulations Piti Treesukol Chemistry Department Faculty...

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