SPC/E Water Reference Calculations - Ewald Summation
|n||Lattice vector of periodic cell images
|k||Fourier-space vector of periodic cell images
|k||modulus of k ; k2 = |k|2|
|qj||Value of charge at site j|
||Ewald damping parameter
|N||Total number of charged sites
|M||Total number of molecules
|Nj||Total number of charged sites in molecule j|
||Indices of sites in a single molecule
|V||Volume of the simulation cell, LxLyLz
|ε0||Permittivity of vacuum (see below)
|i||Imaginary unit, (-1)1/2|
|rj||Cartesian vector coordinate of site j
|rjl||rj - rl|
|erf(x)||Error Function computed for abscissa x
|erfc(x)||Complimentary Error Function computed for abscissa x
In this form, the superscript "†" (dagger) in Ereal indicates that the sum skips all pairs i=j inside the original simulation cell (n = 0). The superscript "†-1" in Eintra indicates that the sum is over site pairs within molecules in the original simulation cell. Additionally, the Fourier vectors (k) in this equation are composed of integer elements, e.g. k = 2ex+ey+4ez where ei is the unit vector for Cartesian direction i. The Fourier space term can alternatively be written using k vectors with elements proportional to 2∏. In practice, the above equation is not how the Ewald Summation is actually implemented. Typically, one makes the following assumptions/reductions to simplify the summation:
Thus, the practical implementation of the Ewald Summation is :
We note that the real-space term now includes multiplication by the Heaviside Step Function, Θ(rij - rcut), which functionally truncates that term at rij = rcut.
For the reference calculations given below, we use the following parameters and apply certain conditions to the calculation of both the dispersion interactions and the Ewald Summation:
|α||5.6 / min(Lx,Ly,Lz)
|kmax||5 ; also only include k for which k2 < kmax2 +2, i.e. k2 < 27.
||Truncate at rcut, apply analytic long-range corrections
||Truncate real-space term at rcut|
||Periodic and tin-foil (conducting) boundary conditions for all Cartesian Directions|
|erfc(x)||Implementation of Numerical Recipes ERFCC function; Ref. 4, page 164.
The reference calculations given below were done using fundamental constants of physics and chemistry recommended by CODATA in 2010 [5,6]. We report these constants because the calculation of each contribution to the intermolecular energy will depend, ever so slightly, on the choice of fundamental physical constants and, in particular, the number of digits in those constants that are carried in the simulation. We use the full constants (untruncated) given in the CODATA 2010 recommendation:
|Permittivity of Vacuum
Four sample configurations of SPC/E molecules are available for download as a gzipped tarball archive. This archive contains five files: the four sample configuration files and one metadata file that explains the format of the sample configurations. These configurations should be converted to the configuration file format native to a user's simulation software.
|M (number of SPC/E molecules)
1. H. J. C. Berendsen, J. R. Griger, and T. P. Straatsma, J. Phys. Chem., 91, 6269 (1987)
2. P. Ewald, Ann. Phys., 369, 253 (1921)
3. M. P. Allen and D. J. Tildesley, Computer Simulation of Liquids (Oxford University Press, New York, 1989).
4. W. H. Press, et al., Numerical Recipes: The Art of Scientific Computing (Cambridge University Press, New York, 1986).
5. P. J. Mohr, B. N. Taylor, and D. B. Newell, Rev. Mod. Phys., 84, 1527 (2012)
6. P. J. Mohr, B. N. Taylor, and D. B. Newell, J. Phys. Chem. Ref. Data, 41, 043109 (2012)