The question How Good is the Quantum Mechanical Explanation of the Periodic System? is answered by Eric Scerri by: Not Very Good:
- I would like to issue a caution regarding the extent to which the periodic table, for example, is truly explained by quantum mechanics so that chemical educators might refrain from overstating the success of this approach.
Quantum mechanics offers a standard explanation in terms of the Hydrogen orbitals combined with Pauli's exclusion principle identifying electrons by 4 quantum numbers (n, l, m, s) with n = 1, 2, 3,... the principal shell number, l = 0,..., n -1 the azimuthal quantum number, m ranging from - l to + l the magnetic quantum number and s = +- 1/2 the spin quantum number.
The basic idea is that shells are sequentially filled with electrons in a certain order (s,p,d,e,..,) within each shell. The trouble is that shells are not sequentially filled with an inner shell being full before an outer shell starts to get filled. The observed order is instead (usually) by the number n + l referred to as Madelung's rule complemented by Hund's rule, which both appear to be introduced ad hoc without quantum mechanical rationale.
The quantum mechanical explanation of the periodic table thus requires three additional ad hoc rules (Pauli, Madelung and Hund) and the advice of Scerri to not overstate its success seems to be well founded.
It is in fact unclear what the standard quantum mechanical explanation based on Hydrogen orbitals, does offer as explanation of the periodic table. Even the basic numbers 2, 8, 8, 18, 18, 32,..., of electrons in filled shells require doubling from the sequence of Hydrogen orbitals 1, 4, 9, 16,..., by ad hoc introduction of spin combined with Pauli's exclusion principle.
And so it remains to give a more convincing explanation of the periodic table using a form of quantum mechanics without ad hoc attributes. There are indications that there is such a version, which will be the subject of upcoming posts. As preparation, browse The Periodic Table: Its Story and Significance, by Eric Scerri.
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Are you ignoring my question about your interpretation and a free charged particle?
SvaraRaderaHave you thought about it earlier and have an answer? Or do you lack any answer at this moment?
I see no reason to question the standard view of a one-electron system.
SvaraRaderaYou don't seem to understand the question. Let's repeat it again.
SvaraRaderaI don't really get what the physical reality would be when you use your interpretation for a free electron that scatters against a potential.
For one electron your equation is the same as the Schrödinger equation. But the solution then scatters in all directions.
What physical reality does this corresponds to?
You write in your text that the square of the modulus of the wavefunction shall be interpreted as the actual density of the electron. So when a free electron scatter against a potential it gets smeared spherically (with varying intensity in the different directions) out in the surrounding space according to your interpretation. This since that is what the resulting wavefunction does when solving for it.
Do you really mean that this is the standard view? An elementary charge that gets smeared all over the place, is there really observations that support this?
An electron as a pointwise particle does not make sense, only as a wave does.
SvaraRaderaSo you really mean that an electron bouncing of a much heavier potential is being smeared all over space in all directions (some directions more pronounced than others)? What happens to the elementary charge when the electron is so smeared that it no longer, in good faith, can be called an electron any more?
SvaraRaderaWhat observations supports this?
It may be that Schrödinger's wave equation does not describe this event. In any case there are no particles only waves at least according to Schrödinger (and myself).
SvaraRaderaI'm in the process of reading a whole set of books on electronic structures calculations.
SvaraRaderaCan you help me sort one thing out.
What differs your equation (3.2) in your Many minds QM draft from an ordinary set of one electron Hartree equations? I see no difference.
One more question.
SvaraRaderaWhat properties constitute a particle, that is what definition of a particle are you renouncing?
Further, you seem to have sign errors on page 15, bottom equation for the energy.
SvaraRaderaYou should have a minus sign in front of the double sum.
Still further, continuing on page 16.
SvaraRaderaThe quantity E in your resulting equation must be Lagrange multiplier connected with your normalization and not the total energy. You need to shift this value to get the energy of the system.
If you arrived at E=-2.90 that's an unfortunate coincidence, because you still need to shift your solution to get the energy consistent with your problem.
Yes I suggest to explore the Hartree model which was abandoned too soon and replaced by Hartree-Fock. The fact that the true ground state of Helium is not 1s^2 has brought atomic phyiscs into a wrong direction, as far as I can understand.
SvaraRaderaParticle properties are contradictory: Something occupying space without extension. Meaningless.
SvaraRaderaI don't find the Hartree equation energy for Helium in the literature, which is strange. So what is it? It could well be - 2.90.
SvaraRaderaI don't find the Hartree equation energy for Helium in the literature, which is strange. So what is it? It could well be - 2.90.
SvaraRaderaThat is not strange at all. Solving the Hartree self consistent problem for Helium is a task suitable for an undergraduate course in computational physics.
A related use can be found here:
A Hartree self-consistent field calculaiton on the helium atom
The article lists a result of -2.8549 for the Helium ground state energy for a Hartree self consistent calculation.
This value is close to the full Hartree-Fock treatment which isn't surprising. The Hartree method treats the electrons as distinguishable particles, hence neglecting exchange correlation. But this energy is really small compared to the Hartree potential contribution. So for a system of only two particles, this effect isn't that prominent, and the results should be expected to be similar.
This is not true Hartree because of the Schiff orbital approx. and so true Hartree is better than 2.85 but reference is still missing.
SvaraRaderaThe orbital approximation is not Schiff's, Schiff is a textbook on quantum mechanics. If you look that one up he gives the Hartree theory and also cites the original paper from 1928.
SvaraRaderaAlso looking at the equation in A Hartree self-consistent field calculaiton on the helium atom , it is the equation Hartree proposed and same equation that I just looked up in four different textbooks dealing with electronic structures calculatiosn. It is also the same equation that you give in your text on page 16 (up to a sign error).
The orbital approximation mentioned must be the Hartree ansatz of writing the many-body wave function as a product of single particle orbitals. Exactly as you do yourself.
What is true Hartree? Reference?
What I call true Hartree is the Hartree equation without simplification as in your reference with only radial dependence. I think the secret lies in angular dependence and I don't find any reference to true Hartree so if you can supply one I would be interested.
SvaraRadera