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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.

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  • $\begingroup$ Without more information about why/to what level of precision you need to calculate this (is it homework?), it is difficult to be sure, but it sounds like you could benefit by reading up about Density Functional Theory. There are existing libraries that you could use to do this (rolling your own is likely to be non-trivial). Typically you'd setup some sort of a variational problem, where you need to find a density matrix for the system (in this case, a gold atom's) that minimises its energy. $\endgroup$
    – Martin C.
    Apr 15 '20 at 9:15
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    $\begingroup$ @MartinC. It is not homework it is work. Thanks for the comment. $\endgroup$
    – mattiav27
    Apr 15 '20 at 9:17
  • $\begingroup$ @MartinC. see my edit I added some details of my final goal. $\endgroup$
    – mattiav27
    Apr 15 '20 at 9:35
  • $\begingroup$ I see. In that case I am not sure that DFT is the appropriate tool... $\endgroup$
    – Martin C.
    Apr 15 '20 at 9:57
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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.

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

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