Ewald Summation

1
Ewald Summation

• What is the potential due to a point charge? 1/r
• This potential should be screened at large
  distances, and periodic with respect to box
  repeats. -gt add in some terms to make this happen
• Solve and write for efficient summation

Equations from The Potential Distribution Theorem
and Models of Molecular Solutions.
2
A Numerical Example

• The total electrostatic potential is
  pairwise-additive with one caveat
• Calculating the caveat
• erfc term disappears for large enough h
• K-space sum approx. zero for large enough k

3
Python Numpy Source
gtgtgt calc_eps(ones(3)10.0, .54, 10) Error
4.04980576401e-16 mad -2.83729748 -2.83729748
-2.83729748
4
What about the usual MD Codes?

• System
• 1 sodium ion in a (32 A)3 box
• PME grid size 643
• x/2L -61.594 kJ/mole
• NAMD 2.6 E 0.0 kJ/mole
• Gromacs 3.3.1 E -61.595734 kJ/mole
• Despite citation of Darden's (traditional)
  method, the total charge correction is
  implemented in Gromacs!

U. Essman, L. Perela, M. L. Berkowitz, T. Darden,
H. Lee and L. G. Pedersen (1995). J. Chem. Phys.
103 8577-92.