Calculating Sodium Atom Ground State Degeneracy

[hanson1011]

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

 

[Amok]

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).

 

[hanson1011]

Cheers for the reply Amok,

So does the ground state correspond to the highest occupied n level state?

 

[Amok]

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?

 

[hanson1011]

Thanks very much, that makes a lot more sense! Its a stand alone question with no prior sections.

Many thanks