It is generally accepted that a closed shell structure is a very good picture for inert atoms. But how good is it quantitatively? Suppose we do both the Hartree-Fock approximation and the (in-principle exact) configuration-interaction for an inert atom, how close to each other are the results? How close is the exact ground state to a Slater determinant?
1
-
3$\begingroup$ Many people knowledgeable in these topics also come to mattermodeling.stackexchange.com $\endgroup$– Vladimir F Героям славаMar 26 at 20:08
Add a comment
|
2 Answers
$\begingroup$
$\endgroup$
6
This paper gives a bunch of Hartree-Fock results that can be compared to experimental ground state energies. Page 1587 has energies of all the ground states. These ground state energies can be compared to the energy it takes to fully ionize an atom, given in this NIST query. For example, for Beryllium, we get from the NIST database 0.3426+0.6692+5.6556+8.0010=14.668 Hartree (Hartree is the atomic unit of energy = 27.2114eV) to fully ionize vs the Hartree-Fock value 14.573. We should always expect the Hartree-Fock value to be lower, since it's technically a variational estimate of the groundstate energy.
Another important metric for Hartree-Fock accuracy is whether or not it correctly predicts the total angular momentum and multiplicity of the ground state. I haven't yet found a good table that gives this information. These notes tell us how to take the "configuration" reported by a Hartree-Fock result (also reported in the paper from before), and find the "groundstate multiplet."
-
1$\begingroup$ "Another important metric for Hartree-Fock accuracy is whether or not it correctly predicts the total spin of the ground state. I haven't yet found a good table that gives this information." Even for a diatomic molecule like Fe2, it is unknown experimentally or theoretically whether the ground state spin multiplicity is 7 or 9: mattermodeling.stackexchange.com/a/1247/5. For that reason, in many cases I do not expect Hartree-Fock to be able to correctly predict the total spin of the ground state. $\endgroup$ Mar 27 at 13:47
-
$\begingroup$ I meant the technique described here bohr.physics.berkeley.edu/classes/221/notes/atomstruc.pdf which takes the "orbital" result of Hartree-Fock and finding the "ground state multiplet" - which tells you the multiplicity of the ground state. I'm only personally aware of how this works in single atoms, and that's what I meant. I'm not surprised to find that there are examples where it's hard, especially in molecules. $\endgroup$– AXensenMar 27 at 13:57
-
1$\begingroup$ Since the question was about inter atoms, perhaps you're correct that I'm over-complicating things by mentioning molecules. Also, since the paper that you mentioned in 30 pages long and has quite a large number of tables, perhaps if you mentioned where the calculated energies are compared to experimental energies, it would be helpful to future readers. I believe that Table 2 on page 1587 does not have experimental energies. $\endgroup$ Mar 27 at 14:03
-
$\begingroup$ I should have been more clear, I meant for the OP to do the comparison themselves, using, for example this query on the NIST database. The groundtate energy can be compared to the energy it takes to fully ionize an atom (for Be: 0.3426+0.6692+5.6556+8.0010=14.668 vs the Hartree-Fock value 14.573) physics.nist.gov/cgi-bin/ASD/… $\endgroup$– AXensenMar 27 at 14:18
-
$\begingroup$
$\endgroup$
It is generally accepted that a closed shell structure is a very good picture for inert atoms. But how good is it quantitatively?
It depends on the atom and it depends on the observable of interest and it depends on what you think "good" is.
Table 7-4 in Bethe and Jackiw's textbook "Intermediate Quantum Mechanics" compares the energy separation ratio $\frac{{}^1S - {}^1D}{{}^1D - {}^3P}$ from the Hartree-Fock theory with Experimental Results.
The theoretical result is 1.50.
The experimental result for carbon is 1.13.
The experimental results for silicon is 1.48.
The experimental result for germanium is 1.50.
The experimental result for tin is 1.39.
Other comparisons can be found in Condon and Shortley's textbook "The Theory of Atomic Spectra."
