Truncating the potential and the minimum image convention

1. Or The stupid bag of tricks Truncating the potential and the minimum image convention Truncating the potential and the minimum image convention, MM 283-294. p. 1

2. Truncation of the potential • Calculation of the non-bonded forces/energies is the most time-consuming part of a simulation • Simple ways of reducing the expense • Minimal image convention • Non-bonded cutoffs • Caveat: all techniques we will discuss today are formally justified only for rapidly decaying (short-range) interactions Truncating the potential and the minimum image convention, MM 283-294. p. 2

3. Short- and long-range interactions For extended 3D systems: • 1/r interactions formally diverge; results depend on the boundary conditions • 1/r2 and 1/r3 interactions are conditionally convergent • 1/rn (n>3) interactions are absolutely convergent Lennard-Jones – r-6 Dipole-dipole – r-3 Point charge-point charge – r-1 Truncating the potential and the minimum image convention, MM 283-294. p. 3

4. D D D D Minimum image convention • Periodic boundary conditions • Each atom interacts with at most one image of every atom • Approximates isolated-molecule or cluster calculations • Inter-cell dipole-dipole interactions increase acetone’s dipole moment • Minimum image convention removes this interaction Truncating the potential and the minimum image convention, MM 283-294. p. 4

5. Cutoffs: Achieving the savings • Even with the cutoffs, we still need to decide whether a non-bonded interaction has to be included. • In a naïve implementation, this means that all N(N-1)/2 possible pair interactions have to be examined. • Still a win, if pair interaction are expensive to compute • Greater payoff is possible with more sophisticated techniques • Non-bonded neightbour list • Cell index method Truncating the potential and the minimum image convention, MM 283-294. p. 5

6. Atoms outside the cut-off, but still “close” Atom 1 4 6 2 8 5 9 10 3 5 6 7 Atoms within the cut-off Atom 2 … … Non-bonded neighbour list • Only atoms within the neighbour list need to be considered • Including “close” atoms avoids recalculation of the neigbour list on each iteraction • Displacement-based criteria for recalculation of the neighbour list Truncating the potential and the minimum image convention, MM 283-294. p. 6

7. Cell index method • Assign each atom to a “cell”, with cell size  cutoff radius • When recomputing the neighbour list, only the 27 nearest cells (3D) need to be considered Truncating the potential and the minimum image convention, MM 283-294. p. 7

8. Scaling of the cut-off techniques N – number of particles R – cost of a distance calculation E – cost of an energy evaluation m – number of steps between neighbour list updates k – average number of particles per “cell” Truncating the potential and the minimum image convention, MM 283-294. p. 8

9. Cut-off artifacts, and patching them up • A fraction of the potential energy is excluded • Partial inclusion of (long-range) group interactions • Discontinuities in potential energy surfaces and gradients Truncating the potential and the minimum image convention, MM 283-294. p. 9

10. Energy corrections For monoatomic liquid, the total energy is (5.16): Replaced with rc in a cut-off calculation Assuming ideal gas radial distribution g(r), the correction is (5.29): Truncating the potential and the minimum image convention, MM 283-294. p. 10

11. TIP3P water, 8Å cutoff TIP3P water, no cutoff 1x(O-O) = +29 4x(O-H) = -59 4x(H-H) = +29 Partial inclusion of long-range group interactions • Group-based cutoffs – all contributions are included (or omitted) together • Groups should be neutral or almost so • Groups should be much smaller than the cut-off radius Truncating the potential and the minimum image convention, MM 283-294. p. 11

12. Discontinuities • Both energy and gradients are discontinuous • Total energy is not conserved • Problems with MD stability • Thermodynamic properties are not affected by the cut-off Energy, kcal/mol Gradient, kcal/mol/Å Truncating the potential and the minimum image convention, MM 283-294. p. 12

13. Discontinuities: Shifted potential Energy, kcal/mol • Energy is continuous • Gradients are discontinuous • Total energy is conserved • Problems with MD stability • Thermodynamic properties are changed by the cut-off Gradient, kcal/mol/Å Truncating the potential and the minimum image convention, MM 283-294. p. 13

14. Discontinuities: Switching (A) Energy, kcal/mol • Both energy and gradients are continuous • Total energy is conserved • Stable MD • Thermodynamic properties are changed Gradient, kcal/mol/Å Truncating the potential and the minimum image convention, MM 283-294. p. 14

15. Discontinuities: Switching (B) • Both energy and gradients are continuous • Total energy is conserved • Stable MD • Thermodynamic properties are not affected Energy, kcal/mol Gradient, kcal/mol/Å Truncating the potential and the minimum image convention, MM 283-294. p. 15

16. Summary • Cutoff are useful in MC and MD simulations • Simplify simulations of molecules and clusters, when the periodic boundary conditions must be used (minimum image convention) • Can make simulation less expensive • Cutoffs must be used with care, or nasty artifacts may arise Truncating the potential and the minimum image convention, MM 283-294. p. 16