Wiki Helium Article mistake?

[conway]

The Wikipedia article on the Helium Atom goes through the Thomas-Fermi approximation and shows that when you use a screening parameter Z you can optimize the energy of the atom according to the following formula:

$$\langle H \rangle = [-2Z^2 + \frac{27}{4}Z]E_1$$

($$E_1$$ is elsewhere defined as the Rydberg unit -13.6 eV)

The minimum value for energy occurs for a choice of Z=1.69, and you can plug in the numbers (I did) to confirm that this gives a value of -77.45 eV for the helium atom.

QUESTION: The article then concludes by saying that using this value "we obtain the most accurate result yet:"

$$\frac{1}{2} \Bigg(\frac{3}{2}\Bigg)^6 = -77.5 eV$$

What is this formula? Of course it is a mistake because it is missing $$E_1$$; but that's not the problem. We can correct the mistake and just multiply 729/128 by 13.6 eV to get the quoted result. But how does this come from the original calculation? Where is the value Z=1.69 which came from the optimization?

 

[Frame Dragger]

The Wikipedia article on the Helium Atom goes through the Thomas-Fermi approximation and shows that when you use a screening parameter Z you can optimize the energy of the atom according to the following formula:

$$\langle H \rangle = [-2Z^2 + \frac{27}{4}Z]E_1$$

($$E_1$$ is elsewhere defined as the Hartree unit -13.6 eV]

The minimum value for energy occurs for a choice of Z=1.69, and you can plug in the numbers (I did) to confirm that this gives a value of -77.45 eV for the helium atom.

QUESTION: The article then concludes by saying that using this value "we obtain the most accurate result yet:"

$$\frac{1}{2} \Bigg(\frac{3}{2}\Bigg)^6 = -77.5 eV$$

What is this formula? Of course it is a mistake because it is missing $$E_1$$; but that's not the problem. We can correct the mistake and just multiply 729/128 by 13.6 eV to get the quoted result. But how does this come from the original calculation? Where is the value Z=1.69 which came from the optimization?

It's a Wiki article... GOK.

 

[conway]

Oops. Found the answer. The Z value that comes from the minimization is actually an exact fraction, 27/16.