CHEM699.08 Lecture #9 Calculating Time-dependent Properties June 28, 2001 MM 1 st Ed. Chapter 6 -- 333-342 MM

1. CHEM699.08 Lecture #9 Calculating Time-dependent Properties June 28, 2001 MM1stEd. Chapter 6 -- 333-342 MM2ndEd. Chapter 7.6 -- 374-382 [ 1 ]

2. Calculating Time-dependent Properties ¤ An advantage of a molecular dynamics (MD) simulation over a Monte Carlo simulation is that each successive iteration of the system is connected to the previous state(s) of the system in time. ¤ The evolution of a MD simulation over time allows the data, or some property, at one time (t) to be related to the same or different properties at some other time (t+dt). ¤ A time correlation coefficient is a calculated measurement of the degree of correlation for an observed time-dependent property. [ 2 ]

3. Calculating Time-dependent Properties ¤ Our “simple” 2D MD simulation of a single hard sphere moving through an arbitrarily chosen plane. [ 3 ]

4. Calculating Time-dependent Properties y ¤ Our “simple” 2D MD simulation of a single hard sphere moving through an arbitrarily chosen plane. x [ 3 ]

5. Calculating Time-dependent Properties y ¤ Our “simple” 2D MD simulation of a single hard sphere moving through an arbitrarily chosen plane. ¤ Is the movement of the sphere in the x direction related to the motion in the y direction? x [ 3 ]

6. Calculating Time-dependent Properties y ¤ Our “simple” 2D MD simulation of a single hard sphere moving through an arbitrarily chosen plane. ¤ Is the movement of the sphere in the x direction related to the motion in the y direction? x [ 3 ] t = 0

7. Calculating Time-dependent Properties y ¤ Our “simple” 2D MD simulation of a single hard sphere moving through an arbitrarily chosen plane. ¤ Is the movement of the sphere in the x direction related to the motion in the y direction? x [ 3 ] t = 1 t = 0

8. Calculating Time-dependent Properties y ¤ Our “simple” 2D MD simulation of a single hard sphere moving through an arbitrarily chosen plane. ¤ Is the movement of the sphere in the x direction related to the motion in the y direction? x [ 3 ] t = 1 t = 2 t = 0

9. Calculating Time-dependent Properties y ¤ Our “simple” 2D MD simulation of a single hard sphere moving through an arbitrarily chosen plane. ¤ Is the movement of the sphere in the x direction related to the motion in the y direction? x [ 3 ] t = 1 t = 2 t = 3 t = 0

10. Calculating Time-dependent Properties y ¤ Our “simple” 2D MD simulation of a single hard sphere moving through an arbitrarily chosen plane. ¤ Is the movement of the sphere in the x direction related to the motion in the y direction? x [ 3 ] t = 4 t = 1 t = 2 t = 3 t = 0

11. Calculating Time-dependent Properties y ¤ Our “simple” 2D MD simulation of a single hard sphere moving through an arbitrarily chosen plane. ¤ Is the movement of the sphere in the x direction related to the motion in the y direction? x [ 3 ] t = 4 t = 5 t = 1 t = 2 t = 3 t = 0

12. Calculating Time-dependent Properties ¤ If there are two sets of data, x and y, the correlation between them (Cxy) can be defined as: (1) [ 4 ]

13. Calculating Time-dependent Properties ¤ If there are two sets of data, x and y, the correlation between them (Cxy) can be defined as: (1) ¤ This can also be normalized to a value between -1 and +1 by dividing by the rms of x and y: (2) [ 4 ]

14. Calculating Time-dependent Properties ¤ A value of cxy = 0 would indicate no correlation between the values of x and y, while a value of 1 indicates a high degree of correlation. [ 5 ]

15. Calculating Time-dependent Properties ¤ A value of cxy = 0 would indicate no correlation between the values of x and y, while a value of 1 indicates a high degree of correlation. ¤ If x and y are found to only fluctuate around some average value as would be the case for bond lengths, for example, Equation 2 is commonly expressed only as the fluctuating part of x and y. (3) [ 5 ]

16. Calculating Time-dependent Properties ¤ One drawback to Equation 3 is that the mean values of x and y can’t accurately be known until the MD simulation has completed all M steps. [ 6 ]

17. Calculating Time-dependent Properties ¤ One drawback to Equation 3 is that the mean values of x and y can’t accurately be known until the MD simulation has completed all M steps. ¤ Tired of waiting for those pesky MD simulations to finish before generating your time-correlation coefficients? ¤ Well there’s a way around this. [ 6 ]

18. Calculating Time-dependent Properties ¤ Equation 3 can be re-written without the mean values of x and y: (4) ¤ This expression allows for the calculation of cxy on the fly, as the MD simulation progresses! [ 7 ]

19. Calculating Time-dependent Properties ¤ As the MD simulation proceeds the values of one property can be compared to the same, or another property at a later time: (5) [ 8 ]

20. Calculating Time-dependent Properties ¤ As the MD simulation proceeds the values of one property can be compared to the same, or another property at a later time: (5) ¤ If x and y are different properties, then Cxy is referred to as a cross-correlation function. If x and y are the same property, then this is referred to as an autocorrelation function. ¤ The autocorrelation function can be though of as an indication of how long the system retains a “memory” of its previous state. [ 8 ]

21. Calculating Time-dependent Properties ¤ An example is the velocity autocorrelation coefficient which gives an indication of how the velocity at time (t) correlates with the velocity at another time. (6) ¤ We can normalize the velocity autocorrelation coefficient thusly: (7) [ 9 ]

22. Calculating Time-dependent Properties ¤ For properties like velocities, the value of cvv at time t = 0 would be 1, while at loner times cvv would be expected to go to 0. ¤ The time required for the correlation to go to 0 is referred to as the correlation time, or the relaxation time. The MD simulation must be at least long enough to meet the relaxation time, obviously. [ 10 ]

23. Calculating Time-dependent Properties ¤ For properties like velocities, the value of cvv at time t = 0 would be 1, while at loner times cvv would be expected to go to 0. ¤ The time required for the correlation to go to 0 is referred to as the correlation time, or the relaxation time. The MD simulation must be at least long enough to meet the relaxation time, obviously. ¤ For long MD simulations the relaxation times can be calculated relative to several starting points in order to reduce the uncertainty. Fig.1 [ 10 ]

24. Calculating Time-dependent Properties ¤ Shown here are the velocity autocorrelation functions for the MD simulations of argon at two different densities. ¤ At time t = 0 the velocity autocorrelation function is highly correlated as expected, and begins to decrease toward 0. cvv(t) Fig.2 [ 11 ] Time (ps)

25. Calculating Time-dependent Properties ¤ The long time tail of cvv(t) has been ascribed to “hydrodynamic vortices” which form around the moving particles, giving a small additive contribution to their velocity. Fig.3 [ 12 ]

26. Calculating Time-dependent Properties ¤ This slow decay of the time correlation toward 0 can be problematic when trying to establish a time frame for the MD simulation, and also in the derivation of some properties. ¤ Transport coefficients require the correlation function to be integrated between time t = 0 and t = ¥. ¤ In cases where the time correlation has a long time-tail there will be fewer blocks of data over a sufficiently wide time span to reduce the uncertainty in the correlation coefficients. [ 13 ]

27. Calculating Time-dependent Properties ¤ Another example is the net dipole moment of the system. This requires the summation of the individual dipoles (vector quantities) of each molecule in the system -- which will change over time. (8) [ 14 ]

28. Calculating Time-dependent Properties ¤ Another example is the net dipole moment of the system. This requires the summation of the individual dipoles (vector quantities) of each molecule in the system -- which will change over time. (8) ¤ The total dipole correlation function is expressed as: (9) [ 14 ]

29. Calculating Time-dependent Properties ¤ Transport Properties ¤ A mass or concentration gradient will give rise to a flow of material from one region to another until the concentration is even throughout. ¤ The word “transport” suggests the system is at non-equilibrium. ¤ Here we will deal with calculating non-equilibrium properties by considering local fluctuations in a system already at equilibrium. ¤ Examples: temperature gradient, mass gradient, velocity gradient, etc. [ 15 ]

30. Calculating Time-dependent Properties ¤ The flux (transport of some quantity) can be expressed by Fick’s first law of diffusion thusly: Jz = -D (dN / dz) (10) [ 16 ]

31. Calculating Time-dependent Properties ¤ The flux (transport of some quantity) can be expressed by Fick’s first law of diffusion thusly: Jz = -D (dN / dz) (10) ¤ The time dependence (time-evolution of some distribution) is expressed by Fick’s second law: ¶N (z,t) ¶2N (z,t) (11) = D ¶ t ¶ z2 [ 16 ]

32. Calculating Time-dependent Properties ¤ Einstein showed that the diffusion coefficient (D) is related to the mean square of the distance, and in 3-dimensions this is given by: (12) 3D = [ 17 ]

33. Calculating Time-dependent Properties ¤ Einstein showed that the diffusion coefficient (D) is related to the mean square of the distance, and in 3-dimensions this is given by: (12) 3D = ¤ It is important to point out that Fick’s law only applies at long time durations, such as the case above. To a good approximation some duration where “t” effectively approaches infinity as far as the simulation is concerned will be sufficient. [ 17 ]

34. Calculating Time-dependent Properties ~ fin ~ [ 18 ]