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Sometime back, I encountered the Self Consistent Field Method in Quantum Mechanics, which is used to compute wave functions in complex atoms. The book I read this from is "Practical Inorganic Chemistry" by Clyde and Day. The method is explained through an argument about the potential energy function (V) of the Helium atom (a sum of three terms due to the three particle system).

According to it, in the case of Helium which has two electrons (and one nucleus with a +2e charge), the wave equation is

[tex]

\frac{1}{m_1}\nabla^2_{1} \psi_{T} +

\frac{1}{m_2}\nabla^2_{2} \psi_{T} +

\frac{8\pi^{2}}{h^{2}} (E-V)\psi_{T} = 0

[/tex]

where [tex]m_{1}[/tex] and [tex]m_{2}[/tex] are electron masses. Now, I think this is incomplete and should include a third term for the mass of the nucleus with charge +2e. So in my opinion, the wave equation should be,

[tex]

\frac{1}{m_1}\nabla^2_{1} \psi_{T} +

\frac{1}{m_2}\nabla^2_{2} \psi_{T} +

\frac{1}{m_3}\nabla^2_{3} \psi_{T} +

\frac{8\pi^{2}}{h^{2}} (E-V)\psi_{T} = 0

[/tex]

where [tex]m_{3}[/tex] is the mass of the nucleus. The book adopts a convention I have not come across elsewhere (so far): there is a laplacian operator for each particle (hence the subscripts). This of course, leads to an intitally tedious looking set of expressions while separation of variables.

Please tell me which of the abovementioned equations is correct, as I am wondering why such an advanced book should fail to explain it.

Cheers

Vivek

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# Wave Equation for the Helium Atom/No of terms = 2 or 3?

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