SPC/E Water Reference Calculations  Ewald Summation

n  Lattice vector of periodic cell images 
k  Fourierspace vector of periodic cell images 
k  modulus of k ; k^{2} = k^{2} 
q_{j}  Value of charge at site j 
α 
Ewald damping parameter 
N  Total number of charged sites 
M  Total number of molecules 
N_{j}  Total number of charged sites in molecule j 
κ, λ 
Indices of sites in a single molecule 
V  Volume of the simulation cell, L_{x}L_{y}L_{z} 
ε_{0}  Permittivity of vacuum (see below) 
i  Imaginary unit, (1)^{1/2} 
r_{j}  Cartesian vector coordinate of site j 
r_{jl}  r_{j } r_{l} 
erf(x)  Error Function computed for abscissa x 
erfc(x)  Complimentary Error Function computed for abscissa x 
In this form, the superscript "†" (dagger) in E_{real} indicates that the sum skips all pairs i=j inside the original simulation cell (n = 0). The superscript "†^{1}" in E_{intra} 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 = 2e_{x}+e_{y}+4e_{z} where e_{i} 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 [3]:
We note that the realspace term now includes multiplication by the Heaviside Step Function, Θ(r_{ij}  r_{cut}), which functionally truncates that term at r_{ij} = r_{cut}.
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(L_{x},L_{y},L_{z}) 
k_{max}  5 ; also only include k for which k^{2} < k_{max}^{2} +2, i.e. k^{2} < 27. 
r_{cut}  10 Å 
Truncation  
Dispersion 
Truncate at r_{cut}, apply analytic longrange corrections 
Coulomb 
Truncate realspace term at r_{cut} 
Boundary Conditions 
Periodic and tinfoil (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:
Name  Symbol  Value  Units 
Boltzmann Constant 
k_{B}  1.3806488E23 
J/K 
Avogadro Constant 
Na  6.02214129E+23 
mol^{1} 
Planck constant 
h  6.62606957E34 
J s 
Elementary charge 
e  1.602176565E19  C 
Permittivity of Vacuum 
ε_{0}  8.854187817E12  C^{2}/(J m) 
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.
Configuration > 
1 
2  3  4 
M (number of SPC/E molecules) 
100  200  300  750 
L_{x}=L_{y}=L_{z} (Å) 
2.00000E+01  2.00000E+01  2.00000E+01  3.00000E+01 
E_{disp}/k_{B} (K)  9.95387E+04  1.93712E+05  3.54344E+05  4.48593E+05 
E_{LRC}/k_{B} (K) 
8.23715E+02  3.29486E+03  7.41343E+03  1.37286E+04 
E_{real}/k_{B} (K) 
5.58889E+06  1.19295E+06  1.96297E+06  3.57226E+06 
E_{fourier}/k_{B} (K) 
6.27009E+03  6.03495E+03  5.24461E+03  7.58785E+03 
E_{self}/k_{B} (K) 
2.84469E+06  5.68938E+06  8.53407E+06  1.42235E+07 
E_{intra}/k_{B} (K) 
2.80999E+06  5.61998E+06  8.42998E+06  1.41483E+07 
E_{total}/k_{B} (K)  4.88604E+05  1.06590E+06  1.71488E+06  3.20501E+06 
References
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)