Introduction To Molecular Simulation

고려대 화공생명공학과

강정원

Computer Simulation …

Computer experiments (simulation ) become a general research tool.

Motivation of computer …– Development of Nuclear Weapons– Code breaking

MANIAC, 1952 – Metreopolis was interested in solving broad spectrum of

problems on this machine.

ScientificCalculation

Metropolis Monte-CarloSimulation Method

Method before computer simulation

Approximate TheoriesMechanical Simulation – Plastic foam balls– Metal bearings

Tedious, laborious Quite realistic

Molecular SimulationA study of state of matter using computer. – Gas state– Liquid State– Solid State– Other specialized state : nano-space, structured polymers ,…

Why computer ? – We cannot solve many-body problems even using simple

Newtonian mechanics. (What about quantum mechanics ?)– There is no hope to get answer to many-body problem using pencil

and paper….Before computer simulation …– Approximate theories

Van der Waals equation for non-polar fluidsDebye-Huckel for electrolytes

Use of Molecular Simulation

RealProblem

Model

Experimental Result

Exact Resultfor Model

TheoreticalPredictions

Test ofModel

Test of Theories

Perform Experiments

Carry outComputer Simulation

ConstructApproximate

Theories

Compare Compare

Liquid StateGas StatePolymerNano space

Lennard-Jones potentialHard Sphere potentialSquare-well potential

Van der WaalsDebye-Huckel

Use of Molecular SimulationTest of model– Test of model potential, model structure– Comparison with experimental data

Test of approximate theories– Comparison with theoretical prediction– Computer-generated exact result

Prediction of properties– Replacement of experimental data– Computer does not care about the condition….

Simulation at 10,000 K (?) Discovery of new fact – Alder and Wainwright (1950s) : predicted 1st order freezing

transition for harsh short range repulsive molecules.

Procedure to perform molecular simulation

ModelStatisticalAveraging

Method

ComputerImplementation

SimulationResult

Interaction Energy model

StructuralModel

StatisticalMechanics

Ensemble Average

Random NumberGeneration

RandomWalk

Test of Model

Test of Theory

Property Prediction

New Discovery

Statistical Treatment

Method of Integration

Need to study molecular simulation…

Computer simulations (computer experiments) become general research tool. Understanding the “Black box” greatly improve the efficiency of using it. The techniques can be applied to various field of science and engineering. – Polymer science– Nano technology– Biological materials– Special structures : Zeolites, Supercritical fluid,

Aerogels,….

Recent Research Topics2001-2003

Molecular Simulation of Diblock copolymer filmsAdsorption of materials in a single-wall carbon nano-tube ZeolitesDrug delivery devicesViscosity in nano spacingNanoscale heat transfer Supercritical behavior Aerogels

Prerequisite for the course

Programming skill (FORTRAN or C/C++) Statistical Mechanics – Will be covered shortly in 2 week lecture.

Basic Thermodynamics

Statistical Thermodynamics

Link between microscopic properties and bulk properties

Microscopic Properties

T,PU,H,A,G,S

μ,Cp,…

Microscopic Properties

Statistical Thermodynamics

Molecular Properties Thermodynamic Properties

uij

r

)1( += JhcBJEJvhcvE ~)

21( +=ν

2

22

8mXhnEn =

Potential Energy

Kinetic Energy

Crash course in statistical mechanics

Mechanics

ClassicalMechanics

Quantum Mechanics

Statistical Thermodynamics

MolecularPartition Functions

EnsembleAveraging

Method

Phase Space Integration

−

−>=<

NN

NNN

dU

dUAA

rr

rrr

))(exp(

))(exp()(

β

β

Classical Mechanics …

Hamiltonian : Total Energy of System– r : position vectors (N)– p: momentum vectors (N)

Using Legendre Transformation Technique,

),...,,(2

),(

energy) alPE(potenti energy) KE(kinetic),(

21 Ni i

iNN

NN

Um

H

H

rrrppr

pr

+=

+=

ii

ii

H

H

rp

pr

=

∂∂

−=

∂∂

CanonicalRelationship

Ex) if N= 1.E24 then

6.E24 initial values6.E24 set of 1st order differential equations

Can you solve it ?

Quantum Mechanics …

Failure of classic mechanics– Blackbody – radiation– The Planck distribution– Heat capacities at low T– Atomic and molecular spectra

Wave-particle duality – Waves have characteristics of particles– Particles have characteristics of waves

Conclusion of Quantum Mechanics

Particles can only have discrete values of energiesThe energy values can be calculated using Schrodinger equation

Ψ=Ψ+Ψ∇−i

ii

EUm

h 22

2

8π

Second order differential equationEigen value problem : series of allowed solutions

Available energy values

Examples of solution to Schrodinger Equation

Translational motion of a free particle

Vibrational Motion (harmonic motion)

Rotational motion of a linear rotor...3,2,1,0),1( =+= JJhcBJEr

...3,2,1,0,,)21(

2/1

=

=+= v

mkvEv ωω

...3,2,1,0,2

cossin)(22

==

+=

km

kE

kxDkxCx

k

k

ψ