How do I calculate the electron density of a gold atom? I have to calculate the electron density of a gold atom. As far as I know, it is given by $\rho=e|\psi|^2$ if $\psi$ is the wave function of the electrons. The only way I know for calculate the wave function $\psi$ is the Hartree-Fock method, but with gold I would have 79 kinetic terms and $\dfrac{79*(79-1)}{2}=3081$ exchange terms. I am sure someone came up a more clever idea. 
Can anyone point out a way to solve this problem?
It looks like some more details are needed. My goal is to calculate the scattering amplitude of electrons impinging on a test mass of gold. In order to do this, I have to solve the differential equation:
$$-\dfrac{\hbar}{2m} \dfrac{1}{r^2} \dfrac{d}{dr} \left( r^2\dfrac{d}{dr}R_l(r) \right) + \dfrac{\hbar}{2m} \dfrac{l(l+1)}{r^2} R_l(r) - \left( -\dfrac{Ze^2}{r} + V_{ext}(r) \right)R_l(r)=E \,R_l(r)$$
where the potential $V_{ex}(r)\propto \rho(r)$ where $\rho(r)$ is the electron density of the gold atom.
 A: The Hartree-Fock method is a numerical method for getting
approximate solutions for the Schrödinger equation in a multi-electron atom.
It tries to model the mutual electrical repulsion and
Pauli's exclusion principle between the electrons.
As results it predicts the electron shells (for the ground state
and for excited states) quite well.
It was famous for the big computer-power needed for the calculations.
A simpler approach is the Thomas-Fermi model.
It models the electrons in a semi-classical way as moving in the
same common potential, which in turn is created by the charge
density of all electrons.
This approximation is bad for atoms with only a few electrons,
but it gives a reasonable result for atoms with many electrons
(like the gold atom with 79 electrons).
Of course this model doesn't predict the electron shells,
but instead gives a broader view glossing over these "details".
The image below compares the electron density of the Neon
atom (i.e. 10 electrons) as calculated by Hartree-Fock and Thomas-Fermi
model. 

(image from Analytic model of a multi-electron atom)
