Constructing wave function for Beryllium (Z=4)

[adwodon]

Homework Statement

Using an independent particle approximation to construct the wave functions of the ground and the first singlet and triplet excited states of the Beryllium atom in terms of the hydrogen-like atomic orbitals, which satisfy the Pauli Exclusion Principle.

Homework Equations

n/a

The Attempt at a Solution

Ok so it's been a while since I've done any molecular physics so I'm getting a bit stumped at the basics and need a bit of pointing in the right direction.

So the ground state will be

1s2 2s2 right?
With spins like this:

2s $\uparrow$ $\downarrow$
1s $\uparrow$ $\downarrow$

1s2 2s 2p would be the first excited state?

2p $\uparrow$ or $\downarrow$
2s $\uparrow$ or $\downarrow$
1s $\uparrow$ $\downarrow$

Is this the triplet state? ie you can have 3 different orientations of the spins, both up, both down, or a mix as there are 2 unpaired electrons?

Would the singlet state be 1s2 2p2? As the electrons are paired?

Ok so from there I have to construct the wave functions, is this a case of summing the products of spatial and spin functions or is it simpler than that?

I get that for a two electron system the spin functions are:

$\frac{1}{\sqrt{2}}$[$\alpha$(1)$\beta$(2)-$\alpha$(1)$\beta$(2)] - singlet

$\alpha$(1)$\alpha$(2) - triplet

$\frac{1}{\sqrt{2}}$[$\alpha$(1)$\beta$(2)+$\beta$(1)$\alpha$(2)] -triplet

$\beta$(1)$\beta$(2) - triplet

How do you extend this to a 4 electron system, and will all spin functions be relevant?

Obviously you'd have

$\alpha$(1)$\alpha$(2)$\alpha$(3)$\alpha$(4)
$\beta$(1)$\beta$(2)$\beta$(3)$\beta$(4)

Im guessing the rest would be combinations of α & β, are there any specifics I should watch out for?

Then spatial wavefunctions would just be of the form:

u$_{nlm}$(r) = R$_{nl}$(r)Y$_{lm}$($\vartheta$,$\varphi$)

And I just pair the relevant spin functions with their spatial functions to get the wavefunctions I want.

Any help would be appreciated.

 

[mark726]

Hello,

It seems like you have a good understanding of the basic concepts involved in constructing the wave functions for the Beryllium atom. To answer your question about the singlet and triplet states, you are correct in thinking that the singlet state would have paired spins, while the triplet state would have unpaired spins.

To extend this to a 4 electron system, you would need to consider all possible combinations of α and β electrons, just as you did for the 2 electron system. However, in this case, you would have to take into account the Pauli exclusion principle, which states that no two identical fermions (such as electrons) can occupy the same quantum state simultaneously. This means that certain combinations of α and β electrons will not be allowed, so you will have to be careful in choosing the relevant spin functions for each spatial wavefunction.

Overall, you are on the right track with your approach. Just remember to consider the Pauli exclusion principle and make sure that your spin functions are consistent with the number of electrons in the system. Good luck with your calculations!