Physical Chemistry IV 01403343 Molecular Simulations 01403343 Chem:KU-KPS Physical Chemistry IV 01403343 Molecular Simulations Piti Treesukol Chemistry Department Faculty of Liberal Arts and Science Kasetsart University : Kamphaeng Saen Campus

นัดสอบปลายภาค 15 /16 / 22 พฤษภาคม 2559 เวลา 13:00-16:00 น.

คำถาม 16 การจำลองแบบคืออะไร MD ใช้อะไรเป็นตัวกำหนดการเปลี่ยนแปลง MC ใช้อะไรเป็นตัวกำหนดการเปลี่ยนแปลง

Statistical Mechanics Individual Molecular Properties Modes of motions; Energy levels State Variables T, V, P etc. Partition function No interaction between molecules! Thermodynamics Properties

อนาคตจะเป็นอย่างไร ขึ้นกับอะไร What’s the next move ? อนาคตจะเป็นอย่างไร ขึ้นกับอะไร กฏแห่งกรรม โชค ดวง ทำไมทำดีถึงไม่ได้ดี ? ตั้งใจอ่านหนังสือทำไมได้คะแนนน้อย ?

Molecular Interactions Electron distribution Permanent dipole Induced dipole Coulombic interaction Van der Waals interaction Dipole-Dipole Dipole-Induced dipole Dispersion Hydrogen bonding

Molecular Mechanics Simulation Simulate the interaction between molecules Changes of system configuration: A collection of configurations are concerned Molecular dynamics Time space Monte Carlo method Ensemble space

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.

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

+ + + + + + V=V(r,t) F=dV(r,t)/dr + + F(tn)=m·a(tn) r(tn+1)= r(tn) + ½ a(tn) dt2 + + V=V(r,t) F=dV(r,t)/dr + + F(tn+1)=m·a(tn+1) r(tn+2)= r(tn+1) + ½ a(tn+1) dt2 + +

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5 – 4 – 3 – 2 – 1 – 0 – -1 – -2 – -3 – -4 – -5 – X= V= F=

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.

Random Walk # U D L R P 1 2 3 4 x Gravity 10% 50% 20% 1 2 3 4 x Gravity Increase the possibility to move down, how?

Ising Model 1D-Ising Model 2D-Ising Model If E’ < E then E’ If E’ > E then if random # > 0.5 then E’

Molecular Simulation Molecular Dynamics Monte Carlo Initial x,v Calculate F(x) Possible new x’s Calculate new a Calculate E(possible x) Calculate new v Calculate q, p dt random Calculate new x Move to new x

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.

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. Number of atoms at r for ideal gas Number of atoms at r in actual system

Gas Liquid Solid

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