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Spherically Averaged Coulomb Potential

The next step is to superimpose the spherically-symmetric potentials on
neighboring atoms to find the spherically symmetric muffin-tin Coulomb
potential at any particular atom. Consider a potential, V_{1}(__r___{1}),
for atom 1 positioned in the lattice at center 1. Due to the spherical
symmetry about center 1, the vector from center 1 to a point P in space
can be taken as just a distance, __r___{1}=r_{1}, and
the potential can be written as V_{1}(r_{1}).

Now, we want to define the effect of the potential as determined from
a different perspective. Look at the potential at point P due to atom 1
but referenced about center 2. This is no longer a definition of the potential
from a center that correlates to the origin of the potential, so symmetry
is lost. However, the potential about center 2 can be defined in terms
of a spherically symmetric potential about center 2 using an expansion
in spherical harmonics;

.

Since, physically, V_{1}(r_{1}) = V_{2}(__r___{2}),
we get

and

using the orthogonality property of the spherical harmonics. This can
be simplified using

and noting that the integral over f_{2}
is a delta function about m = 0 to give

.

This is the contribution of atom 1 (whose position is center 1) to
the potential at center 2. At this point, we make a simplification and
look only at the spherically symmetric contribution of the potential of
atom 1 about center 2. This is the zeroth order contribution, l=0, m=0,
P_{0}=1;

Note that the integral must be performed over a spherical shell with
its origin at center 2 and of radius r_{2}. This is the process
of spherically averaging about center 2. To accomplish this, we need to
redefine the integral to look at values of r_{1} lying on the spherical
shell about center 2. From the geometry shown in the figure, we see that

so

.

This gives the zeroth order spherically symmetric component of the
potential of atom 1 (at center 1) as referenced at center 2;

.

Repeating this process for all near neighbors and adding together with
the potential of atom 2, one obtains the final spherically averaged Coulomb
contribution to the potential V_{c}(r) at center 2; i.e. V_{c}(r)
at center 2 = [V_{2,00 }of atom 1] + [V_{2}] + [V_{2,00
}of
atom 3] + ...
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*Created: April 9, 1999 ---- Last Updated: April 12, 1999*

*By Mark D. Pauli*