# Derivation of the Chargedensity Formula

The charge density operator for an electron system is given by a sum of delta functions on the location of the individual electrons in a system carrying an elementary charge :

 (161)

Using spherical coordinates the delta function in this expression can be rewritten as

 (162)

The delta function of spherical harmonics can be expressed by spherical and tesseral harmonic functions (see appendix F):

 (163)

Using the above equations and assuming the same radial part of the wave function for all electrons in an unfilled shell the radial integrals in the expectation value of the charge density operator can be substituted by the radial wavefunction and we obtain the expression

 (164)

If higher multiplets are neglected, the expectation values of tesseral harmonics at the right side of equation (175) can be rewritten using the operator equivalent method by Stevens[22]:

 (165)

For the notation see [22]: the pre factors of harmonic tesseral functions as given in table IV in [22] and in appendix F, and corresponds to the number of electrons () in the configuration for and to the Stevens factors , and for  2,4,6, respectively.

This results in the following expression for the chargedensity:

 (166)

Martin Rotter 2017-01-10