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Or The stupid bag of tricks. Truncating the potential and the minimum image convention. 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

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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

• 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

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

• 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

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

• 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

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

• 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

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

• 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

Truncating the potential and the minimum image convention, MM 283-294. p. 12

Energy, kcal/mol

• Energy is continuous

• Total energy is conserved

• Problems with MD stability

• Thermodynamic properties are changed by the cut-off

Truncating the potential and the minimum image convention, MM 283-294. p. 13

Energy, kcal/mol

• Both energy and gradients are continuous

• Total energy is conserved

• Stable MD

• Thermodynamic properties are changed

Truncating the potential and the minimum image convention, MM 283-294. p. 14

• Both energy and gradients are continuous

• Total energy is conserved

• Stable MD

• Thermodynamic properties are not affected

Energy, kcal/mol