1. 1 Physical Chemistry III 01403343 Molecular Simulations Piti Treesukol Chemistry Department Faculty of Liberal Arts and Science Kasetsart University : Kamphaeng Saen Campus

2. Chem:KU-KPS Piti Treesukol 2 Statistical Mechanics Individual Molecular Properties Individual Molecular Properties  Modes of motions; Energy levels State Variables State Variables  T, V, P etc. Partition function Thermodynamics Properties No interaction between molecules!

3. Chem:KU-KPS Piti Treesukol 3 Molecular Interactions Electron distribution Electron distribution  Permanent dipole  Induced dipole Coulombic interaction Coulombic interaction Van der Waals interaction Van der Waals interaction  Dipole-Dipole  Dipole-Induced dipole  Dispersion  Hydrogen bonding

4. Chem:KU-KPS Piti Treesukol 4 Molecular Mechanics Simulation Simulate the interaction between molecules Simulate the interaction between molecules Changes of system configuration: A collection of configurations are concerned Changes of system configuration: A collection of configurations are concerned  Molecular dynamics Time space Time space  Monte Carlo method Ensemble space Ensemble space

5. Chem:KU-KPS Piti 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: 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. Solution of Newton's equations of motion by a step-by-step algorithm. Simulation times from picoseconds to nanoseconds. Simulation times from picoseconds to nanoseconds. The method provides thermodynamic, structural and dynamic properties. The method provides thermodynamic, structural and dynamic properties.

6. Chem:KU-KPS Piti Treesukol 6 + + + + + + V=V(r,t) F=dV(r,t)/dr

7. Chem:KU-KPS Piti Treesukol 7 + + + + + + + + V=V(r,t)F=dV(r,t)/dr F(t n )=m·a(t n ) r(t n+1 )= r(t n ) + ½ a(t n ) dt 2 F(t n+1 )=m·a(t n+1 ) r(t n+2 )= r(t n+1 ) + ½ a(t n+1 ) dt 2

8. Chem:KU-KPS Piti Treesukol |||||||||||||||| 8

9. Chem:KU-KPS Piti Treesukol 9 10 - 9 - 8 - 7 - 6 - 5 - 4 - 3 - 2 - 1 - 0 - X= V= F=

10. Chem:KU-KPS Piti Treesukol 10

11. Chem:KU-KPS Piti 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: 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. Sampling configurations from a statistical ensemble by a random walk algorithm. No true analogue of time. No true analogue of time. Possible to devise special sampling methods. Possible to devise special sampling methods. Provides thermodynamic and structural properties. Provides thermodynamic and structural properties.

12. Chem:KU-KPS Piti Treesukol 12 Random Walk # updownleftright 1 0.30.70.50.4 2 0.50.20.70.9 3 0.80.30.70.5 4 0.90.50.10.3 5 0.10.20.40.3 6 0.70.50.60.2 7 0.10.20.50.3 8 0.60.20.5  x Random Number Gravity Increase the possibility to move down, how?

13. Chem:KU-KPS Piti Treesukol 13 Ising Model 2D-Ising Model 1D-Ising Model If E’ < E then E’ If E’ > E then if random # > 0.5 then E’

14. Chem:KU-KPS Piti Treesukol Molecular Simulation Molecular Dynamics Molecular Dynamics Monte Carlo Monte Carlo 14 Initial x,v Calculate F(x) Calculate new a Calculate new a Calculate new v Calculate new x dt Initial x Possible new x’s Possible new x’s Calculate E(possible x) Calculate E(possible x) Calculate q, p Calculate q, p Move to new x random

15. Chem:KU-KPS Piti Treesukol 15 Radial Distribution Function Radial distribution function, g(r) 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. 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. 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.

16. Chem:KU-KPS Piti 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 N i.g (r) = 4πr2ρdr, where ρ is the number density. 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 N i.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

17. Chem:KU-KPS Piti Treesukol 17 Gas Liquid Solid

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19. Chem:KU-KPS Piti 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. 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:

20. Chem:KU-KPS Piti 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.

21. Chem:KU-KPS Piti Treesukol 21 position velocity A system at time t is represented by 1 point only! tt tt

22. Chem:KU-KPS Piti 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. 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.

23. Chem:KU-KPS Piti Treesukol 23 Macroscopic properties of a system define its macrostate, but they actually arise from the configuration of its microscopic components (microstate). Macroscopic properties of a system define its macrostate, but they actually arise from the configuration of its microscopic components (microstate). Phase Space Probability of each microstates depends on its energy (long) time average = ensemble average Microstate P i = P(E i ) E i  q i,p i E i  n i,x, n i,y, n i,z q (position) p (momentum)

24. Chem:KU-KPS Piti Treesukol 24 Final Examination Exam date: 28 February 2009 Exam date: 28 February 2009  Definitions  Partition Functions  Ensembles  Thermodynamic Properties  Polarization  Molecular Interactions Presentation date: Presentation date: The exam-problems would be online 1-2 days before the exam date!

25. Chem:KU-KPS Piti Treesukol 25 g(r) 4.0 – 3.5 – 3.0 – 2.5 – 2.0 – 1.5 – 1.0 – 0.5 – 0.0 – ||||||| 0.00.20.40.60.81.01.2 Distance (nm) Radial Distribution Function of water at 298 K g HH g OH g OO