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Chapter 4. Reduced Cell Multipole Method

Extending the CMM to handle systems with periodic boundary conditions
substantially expands the range of problems that can be attacked. PBC is
especially important for the simulation of bulk materials.
The Reduced Cell Multipole Method (RCMM) [1] is a relatively simple way of
extending the CMM to periodic systems that maintains the overall scaling and
memory usage advantages of the CMM. It builds on the idea of representing
distant atoms with multipole expansions.

First, an enlarged set of multipoles (through at least the hexadecapole
moments) is computed from the atoms in the unit cell. This is an extended
version of the standard CMM multipole computation and uses the same octree
structure to ensure that it is *O*(*N*) in the number of atoms.
Next, a random set of points within the unit cell is chosen. Charges are
placed on these points so as to reproduce the enlarged multipole set determined
above. The number of random points is equal to the number of multipole
coefficients, so this amounts to merely solving a set of simultaneous linear
equations. This "reduced set" of random points and charges then can be used to
substitute for the unit cell at large enough distances without substantially
decreasing accuracy.

In order to handle the infinite periodic lattice, a standard Ewald summation
method is used [2,3]. In this method, we divide the effective field of the
infinite system at a given point into two pieces: a sum over atoms close to the
unit cell in real space and a sum over terms in reciprocal space.
Computing an Ewald sum over all of the atoms in the unit cell would be
computationally infeasible, so we only compute the sum over the reduced set,
which is much smaller. This still reproduces all of the effects of the
infinite system.

The real space part of the effective field is composed of the fields due to
each of the nearby (reduced set) atoms, modified by a screening function.

(1)

where
and
is a parameter that determines the cutoff range for the real space sum.

If there is an atom that is very close to the point at which the field is being
evaluated, a different expression is used for the term due to that atom to
account for the singularity in the field.

(2)

The sum of these screened terms converges very rapidly as the distance from the
evaluation point increases.

To correct for the screening functions introduced in real space, we must add
terms that represent their complement. This sum now converges slowly in real
space, but its Fourier transform converges rapidly in reciprocal space.

(3)

where
is the reciprocal lattice vector and
is the volume of the unit cell.

Finally, a correction is added to handle the case of charged unit cells.

(4)

(5)

At shorter ranges, the reduced set may not be an adequate representation of the
atoms in the unit cell. Therefore, we will use a different technique (an
extended CMM) to compute interactions with the nearest-neighbor unit cells.
The Ewald sum over the reduced set already attempts to include these
interactions, however, so we subtract out the field contribution from the
nearest-neighbor unit cell images. Since this is within the range of the
real-space portion of the Ewald sum, this subtraction is quite easy; the
screening function in Equation 1 above is simply modified by subtracting 1.
The Ewald sum is evaluated at a specific set of points within the unit cell.
Taylor series coefficients representing the resulting field values are then
computed by interpolation. These coefficients compose the farfield due to the
infinite array of atoms, except for those in the nearest-neighbor unit cells.
This is exactly the correct farfield to use for the (level 0) unit cell at the
beginning of the induction step of the standard CMM.
To compute the interactions with the neighboring unit cells, we can extend the
CMM into those cells. We use the multipole expansions and fields calculated
for the unit cell, but translated to the neighboring images. This adds extra
calculations for CMM cells that used to be at the edge of the bounding box but
now have image cells as neighbors, as well as additional calculations at level
1 in the octree, since the cells at that level now have PNCs in the image unit
cells.

If the unit cell is not cubic, adjustments have to be made to the CMM. The
method chosen is to compute a transformation matrix that converts the unit cell
to a unit cube. The octree decomposition is then performed on the cube, with
the cell coordinates being mapped back to real-space Cartesian coordinates by
the inverse of the transformation matrix. The points at which the Ewald sum is
computed are expressed in terms of transformed (unit cube) coordinates; the
resulting interpolated Taylor series coefficients are then transformed back to
real-space coordinates.

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Kian-Tat Lim