Calculating Sodium Atom Ground State Degeneracy

hanson1011
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Hi guys,
Just got a question I'm a little stuck on and would love a push in the right direction

Q) Using Hartree's theory calculate the degeneracy of the ground state of the Sodium atom.

Its a previous exam question and I'm struggling to find much descriptive information about the topic so any help is great

Many thanks

H
 
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I'm not sure about this, but using the fact that the electronic configuration for the sodium atom is 1s2 2s2 2p6 3s1, I'd say the ground state is twice degenerate (two states can be the ground state).

(electronic configurations in chemistry are based on the hartree approximation).
 
Cheers for the reply Amok,

So does the ground state correspond to the highest occupied n level state?
 
No, the ground state is the ground electronic configuration. 1s2 2s2 2p6 3s1 is the ground state. I said it is twice degenrate (actually that was a mistake, it is once-degenerate), because the electron in the 3s orbital can have spin up or spin down.

Hartree suggested that the SE for a many-electron atom could be approximately solved by using a "mean-field" approximation (like a Hartree-Fock approximation without exchange). This results in a wave function that is a product of wave functions each depending on a single-electron coordinate (the positions of each electron are uncorrelated):

\psi(x_1, x_2, x_3, ... x_n)= \phi(x_1) \phi(x_2) \phi(x_3) ... \phi(x_n)

Where the phis are spin-orbitals (single electron wavefunctions) and x is a position and spin coordinate. They are the product of a "position wavefunction" with a "spin wavefunction".

In the case of the sodium atom this would give:

\psi(x_1, x_2, x_3, ... x_{11})= \phi(x_1) \phi(x_2) \phi(x_3) ... \phi(x_{11})

The last orbital can be written as:

\phi(x_{11}) = \varphi(r_{11}) \alpha

or as:

\phi(x_{11}) = \varphi(r_{11}) \beta

Where alpha and beta are spin wavefunctions. The ground state is therefore once-degenerate.

Now knowing what you have learned, I'm not sure this is the answer they want. The question seems kinda weird to me. In which context does this question appear?
 
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Thanks very much, that makes a lot more sense! Its a stand alone question with no prior sections.

Many thanks
 
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