Spherically Averaged Coulomb Potential

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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, V1(r1), 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, r1=r1, and the potential can be written as V1(r1).

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, V1(r1) = V2(r2), we get

                        and

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

and noting that the integral over f2 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, P0=1;

Note that the integral must be performed over a spherical shell with its origin at center 2 and of radius r2. This is the process of spherically averaging about center 2. To accomplish this, we need to redefine the integral to look at values of r1 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 Vc(r) at center 2; i.e. Vc(r) at center 2 = [V2,00 of atom 1] + [V2] + [V2,00 of atom 3] + ...

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Created: April 9, 1999 ---- Last Updated: April 12, 1999
By Mark D. Pauli