I Weakly interacting electrons in an atom

[Isaac0427]

TL;DR Summary

What is the structure of the wavefunction in multi-electron atoms? Are they slater determinants? If so, how do we determine its angular momentum and spin? And how do we know what term symbols violate Pauli exclusion?

Consider electrons in atom, and let's mostly ignore interactions between the electrons for now. What I mean by that is that the lowest energy level is the doubly degenerate 1s, then the doubly degenerate 2s, then the 6-fold degenerate 2p, etc.

Textbooks like Griffiths use term symbols $^{2S+1}L_J$ to describe the wavefunction. Then, to determine what is disallowed by the Pauli exclusion principle, it says that the wavefunction is a spatial part times a spin part; if the spatial part is symmetric under exchange, the spin part is antisymmetric under exchange, and vice versa. Also, even L is symmetric and odd L is antisymmetric, and for two particles even S is antisymmetric and odd S is symmetric.

I see how this works for two electrons. However, this breaks down quickly when you consider more than two particles. For instance, you can't have a three-particle purely spin state with a spin-1/2 electron. It seems like the full shells are often ignored, but I have yet to see rigorous justification for this; after all, the total wavefunction must be antisymmetric upon exchange of any two electrons in the atom.

So, I was thinking that the states could be, in general, slater determinants. That's what those electron configuration diagrams seem to show. But, if that is the case, is there a general formula for the L, S, and J quantum numbers for a slater determinant? Are those slater determinants even eigenstates of those operators? And, how do we know what term symbols violate Pauli exclusion?

The video below gives a very hand-wavy argument for what term symbols are forbidden by the Pauli exclusion principle, but seems to use slater determinants. This is why I am confused; different sources seem to contradict each other on the precise form of the wavefunction and how we determine which term symbols violate Pauli exclusion.

Thank you in advance!

 

[gentzen]

What is the structure of the wavefunction in multi-electron atoms? Are they slater determinants?

No, they are superpositions (i.e. linear combinations) of slater determinants.

If so, how do we determine its angular momentum and spin? And how do we know what term symbols violate Pauli exclusion?

Each slater derminant in the superposition individually satisfies Pauli's exclusion principle. So you may restrict your attention to slater determinants. Griffiths and the video you linked probably tell you more about the details than I would be willing to write here.

So, I was thinking that the states could be, in general, slater determinants. That's what those electron configuration diagrams seem to show. But, if that is the case, is there a general formula for the L, S, and J quantum numbers for a slater determinant? Are those slater determinants even eigenstates of those operators?

Those configuration diagrams focus on single slater determinants. Those are typically not eigenstates of the Hamiltonian, but can still be eigenstates of suitable quantum number operators. (I am too lazy at the moment to lookup some of the more subtle details, maybe L, S, and J are all just fine, I am just not sure.)

There is also a

Atomic and Condensed Matter​

forum here, where people might be less lazy when it comes to details.

 

[Isaac0427]

Each slater derminant in the superposition individually satisfies Pauli's exclusion principle. So you may restrict your attention to slater determinants. Griffiths and the video you linked probably tell you more about the details than I would be willing to write here.

Right, I am talking about the term symbols -- i.e., for carbon $^3D$ is forbidden, since #L# is even and #S# is odd. I don't know how that works.

There is also a

Atomic and Condensed Matter​

forum here, where people might be less lazy when it comes to details.

Perhaps a mentor can move this post there, if it is appropriate.