Atomic Radii for Uniform Valence Density

It is clear from previous section that in order to derive the atomic radii it is necessary to solve the equations regarding the Mulliken electronegativity in the different scales presented. However, from the AUred-Rochow electronegativity scale is quite simple to get the atomic radii with the form (Putz, 2012b,e) 

The problem appears when we like to solve above electronegativity equations respecting the atomic radii R, because the appearance of the 

Returning to the classical electronegativity estimations, Boyd-Markus above formulation can be further developed with the radial electronic density having the above homogeneous form. Arranging the terms as an equation respecting to atomic radii we arrive at the equality  

Quantum Nanochemistry— Volume II Quantum Atoms and Periodicity 

At this point we like to simplify the fourth order equation, because the small numerical values of atomic radii, but in the present case we are not allowed. This, because canceling the fourth order radius term we loose the connection with the information comprised in the Mulliken electronegativity, and more, the remaining equation will give an imaginary result that is false. Thus we must to solve the entirely fourth-order Boyd-Maikus radii equation that will provide the acceptable result as being 

---