The Central Field Approximation for Many-Electron Atoms

[rtareen]

TL;DR Summary

My book (Young and Freedman 14th) doesn't go into detail about the central field approximation other than saying the potential energy only has a radial component.

Attached is my book's section on many-electron atoms. It says that in the central field approximation, an electron's potential energy is a function of its distance from the nucleus. Later on it says there is an effective atomic number. Does this mean that in this approximation, all charges (protons and electrons) are taken to be in the nucleus? That's very simple to understand. But if that's not the case, how does it actually work? Are the other electrons given certain distances from the electron of interest? How would it work for electrons that closer to the nucleus or farther out?

 

[PeterDonis]

Does this mean that in this approximation, all charges (protons and electrons) are taken to be in the nucleus?

No, for two reasons.

First, as noted in the discussion around equation 41.45, if all of the charges except the single electron being considered were treated as being in the nucleus (or at least closer to it than that single electron), then $Z_{eff}$ would be exactly $1$. In fact it is larger than $1$, so only a portion of the other electrons' charges are being treated as screening the nuclear charge.

Second, if all of the charge except for the single electron being considered was treated as being in the nucleus, then the potential would just be proportional to $1 / r$. But, as noted, the potential function $U(r)$ is not that simple. That effectively means that the charge is spread out, not all at the center.

Are the other electrons given certain distances from the electron of interest?

Not as far as I know; my understanding is that the potential $U(r)$ is derived empirically, not from any specific theoretical assumption about distances of the other charges from the center.

How would it work for electrons that closer to the nucleus or farther out?

As far as I know, the central field approximation works best for electrons in the outermost shell, i.e., the ones farthest from the nucleus.

 

[rtareen]

No, for two reasons.

First, as noted in the discussion around equation 41.45, if all of the charges except the single electron being considered were treated as being in the nucleus (or at least closer to it than that single electron), then $Z_{eff}$ would be exactly $1$. In fact it is larger than $1$, so only a portion of the other electrons' charges are being treated as screening the nuclear charge.

Second, if all of the charge except for the single electron being considered was treated as being in the nucleus, then the potential would just be proportional to $1 / r$. But, as noted, the potential function $U(r)$ is not that simple. That effectively means that the charge is spread out, not all at the center.

Not as far as I know; my understanding is that the potential $U(r)$ is derived empirically, not from any specific theoretical assumption about distances of the other charges from the center.

As far as I know, the central field approximation works best for electrons in the outermost shell, i.e., the ones farthest from the nucleus.

Every time you explain something it is always so clear-cut and easy to understand. I always feel satisfied with your answers. Unfortunately the next chapter is about molecules and condensed matter, which is a subforum I can see you have never posted in. Would you be willing to look out for me over there?

 

[PeterDonis]

Every time you explain something it is always so clear-cut and easy to understand. I always feel satisfied with your answers.

Thanks! Glad I could help.

Would you be willing to look out for me over there?

If the question is a general question about how QM models more complex systems like molecules, it could also be asked in this forum.

 

[hilbert2]

My inorganic chemistry textbook (Housecroft & Sharpe) says that for the helium atom, an effective nuclear charge of 1.69 gives the closest approximation to the ground state energy in the central field approximation. If the helium atom is put in an excited state where one electron occupies the 1s orbital and another is somewhere much higher, like 50s orbital, the electron in the high excited state sees an effective nuclear charge of approximately 1.00 because of its greater distance from the nucleus. The same applies to a muonic helium atom where one electron and one muon orbit the nucleus.