First, assume we have a collection of charged particles in a cubic box
with side length , with periodic boundary conditions. The
collection is assumed neutral; there is an equal number of positive
and negative charges. The total Coulombic energy in this system is
given by:
![]() |
(226) |
is the electrostatic potential at position
:
![]() |
(227) |
To evaluate
efficiently, we break it into two parts:
How can we do this? The idea of Ewald is to do two things: first,
screen each point charge using a diffuse cloud of opposite charge
around each point charge, and then compensate for these screening
charges using a smoothly varying, periodic charge density. The
screening charge is constructed to make the electrostatic potential
due to a charge at position decay rapidly to near zero at a
prescribed distance. These interactions are treated in real space.
The compensating charge density, which is the sum of all screening
densities except with opposite charges, is treated using a Fourier
series.
The standard choice for a screening potential is Gaussian:
![]() |
(228) |
So for each charge, we add such a screening potential. Now, to
evaluate
, we have to evaluate the potential
of a charge density that compensates for the screening charge
densities at each particle. This is done in Fourier space.
The potential of a given charge distribution is given by Poisson's equation:
Now, the compensating charge distribution, denoted , can be
written:
![]() |
(230) |
Now, consider the Fourier transform of Poisson's equation:
![]() |
(231) |
![]() |
![]() |
![]() |
(232) |
![]() |
![]() |
(233) |
![]() |
(234) |
We can use Eq. 229 to solve for
:
Fourier inverting
gives
![]() |
(236) |
![]() |
(237) |
So, the total Coulombic energy due to the compensating charge distribution
is
![]() |
(241) |
Notice that this does indeed include a spurious self-self interaction, because
the point charge at interacts with the compensating charge
cloud also at
. This self-interaction is the potential
at the center of a Gaussian charge distribution. First, we solve
Poisson's equation for the potential due to a Gaussian charge
distribution (details in F&S):
![]() |
(242) |
![]() |
(243) |
![]() |
(244) |
![]() |
(245) |
So the total self-interaction energy becomes
![]() |
(246) |
Finally, the real-space contribution of the point charge at is the
screened potential:
![]() |
(247) |
![]() |
(248) |
Putting it all together:
![]() |
(252) |
Now, the arbitrariness left to us at this point is in a choice for the
parameter . Clearly, very small alphas make the Gaussians tighter
and therefore the compensating charge distribution less smoothly varying.
This means a Fourier series representation of
with
a given number of terms is more accurate for larger
. We'll
evaluate choice of
in Sec. 6.3.
cfa22@drexel.edu