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 $e$ :

$\displaystyle \hat\rho(\mathbf r) = \sum_i -\vert e\vert \delta(\mathbf r_i - \mathbf r)$ (201)

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

$\displaystyle \delta(\mathbf r_i - \mathbf r) = \frac{1}{r^2}\delta(r-r_i) \delta(\Omega- \Omega_i)$ (202)

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

$\displaystyle \delta(\Omega- \Omega_i)= \sum_{l,m} Y_l^{m\star}(\Omega_i) Y_l^{m}(\Omega)= \sum_{l,m} Z_l^{m}(\Omega_i) Z_l^{m}(\Omega)$ (203)

Using the above equations and assuming the same radial part $R(r)$ 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 $R(r)$ and we obtain the expression

$\displaystyle \langle \hat\rho(\mathbf r)\rangle= -\vert e\vert \vert R(r)\vert^2 \sum_{l,m}Z_l^{m}(\Omega) \langle\sum_i Z_l^{m}(\Omega_i)\rangle$ (204)

If higher multiplets are neglected, the expectation values of tesseral harmonics at the right side of equation (220) can be rewritten using the operator equivalent method by Stevens[26]:

$\displaystyle \langle\sum_i Z_l^{m}(\Omega_i)\rangle = \vert p_{lm}\vert \theta_{l} \langle O_l^m(\mathbf J)\rangle_T$ (205)

For the notation see [26]: $p_{nm}$ the pre factors of harmonic tesseral functions $Z_{lm}(\Omega)$ as given in table IV in [26] and in appendix F, and $\theta_l$ corresponds to the number of $4f$ electrons ( $\nu$ ) in the $4f^{\nu}$ configuration for $l=0$ and to the Stevens factors $\alpha$ , $\beta$ and $\gamma$ for $l=$ 2,4,6, respectively.

This results in the following expression for the chargedensity:

$\displaystyle \langle \hat\rho(\mathbf r)\rangle=-\vert e\vert\vert R_{4f}(r)\... ...\vert p_{lm}\vert \theta_{l} \langle O_l^m(\mathbf J)\rangle_T Z_{lm}(\Omega)$ (206)