Electrostatic properties and the multipole expansion

While the calculation of the electrostatic functions from the multipole parameters parallels that of the calculation of the atomic electrostatic moments, there is an 

The electron density centered at M is the only central contributor at the nuclear position M, as in this case the nucleus coincides with the field point P, which is excluded from the integrals. For transition metal atoms, the central contributions are the largest contributors to the properties at the nuclear position, which can be compared directly with results from other experimental methods. The electric field gradient at the nucleus, for instance, can be measured very accurately for certain nuclei with nuclear quadrupole resonance and/or Mdssbauer spectroscopic methods, while the electrostatic potential at the nucleus is related to the inner-shell ionization energies of atoms, which are accessible by photoelectron and X-ray spectroscopic methods. 

The operators for the potential, the electric field, and the electric field gradient have the same symmetry, respectively, as those for the atomic charge, the dipole moment, and the quadrupole moment discussed in chapter 7. In analogy with the moments, only the spherical components on the density give a central contribution to the electrostatic potential, while the dipolar components are the sole central contributors to the electric field, and only quadrupolar components contribute to the electric field gradient in its traceless definition. 

We start with the spherical terms. Substitution of the density expressions (8.39) and (8.40) into the second term of Eq. (8.35), and integration over the coordinates, gives for the central electrostatic potential  

The last term in Eq. (8.41) originates from the Slater-type monopole in the scattering formalism, with n0 as power of r, and Co as exponent. 

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