1. Molecular dynamics (4) Treatment of long-range interactions Computing properties from simulation results

2. Treatment of long-range interactions

3. Division of „through space” interactions
Short range: n>3
(converges fast; cutoff suffices)
Long range: n<=3
(converges slowly; advanced techniques needed to accelerate convergence).

4. Computing electrostatic-interaction energy with a periodic box

5. Ewald summation
point charges screened by Gaussian charge densities wit opposite net charge. The energy converges fast in real space.
Original charge density.
Energy converges slowly
Compensating charge
density.
The energy converges fast in reciprocal space after Fourier transformation.

6. Ewald summation
P. Ewald, Ann. Phys., 1921, 369 (3): 253–287

7. Particle mesh Ewald summation
Lagrange interpolation of the reciprocal contribution to the potential (Frec) from each „mesh” cell.
Darden, York, Pedersen, J. Chern. Phys., 1993, 98,

8. Computing properties

9. Kinds of properties Equilibrium properties
Thermodynamic properties (average energy, heat capacity),
Mechanical properties (pressure, density, volume, radius of gyration),
Structural properties (correlation function),
Non-equilibrium properties (transport)
Diffusion coefficients,
Shear viscosity,
Heat conductivity.

10. Equilibrium properties

11. Averages and errors For independent measurements.
For MD simulations; correlated „measurements”.

12. Block average method

13. Block average method
Variation of the estimated s with block size (Leach, Principles of Molecular Modeling)

14. Pressure
d: dimension of the system (usually d=3)

15. Heat capacity However for microcanonical systems
Numerical differentiation of the E(T) curve also is an option.

16. Structural properties

17. Radial distribution functions

18. Radial distribution functions are related to structure factors and can be determined from X-ray/neutronographic data.

19. Comparison of the radial distribution function (left) and the structure factor (right) in liquid water at t=25oC. Data from J.K. Soper, J. Chem. Phys., 101, , (1994).

20. Calculation of radial distribution functions from MD simulations

21. Radial distribution function of a Lennard-Jones fluid at various reduced temperatures T*=kBT/e; the reduced density is r=s3r.

22. Radial distribution functions can be used to evaluate pressure

23. Non-equilibrium (transport) properties

24. Time-correlation functions
Autocorrelation function
Normalized autocorrelation function

25. Monotonic behavior

26. Non-monotonic behavior (velocity autocorrelation function)

27. Calculation from MD time series

28. Diffusion z uz uy
D: diffusion coefficient y ux x

29. Computing for MD simulations
Direct evaluation
From velocity autocorrelation (Green-Kubo equation)

30. Shear viscosity
y dimension
Navier-Stokes equation

31. Thermal conductivity
Fourier equation
excess energy of atom i