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Truncating the potential and the minimum image convention

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The stupid bag of tricks

Truncating the potential and the minimum image conventionTruncating the potential and the minimum image convention, MM 283-294. p. 1

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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