Sent to CCL by: Martin Kaupp [martin.kaupp-$-tu-berlin.de] Dear Tom,For a mathematical explanation, look at the solution of the radial differential equation for the hydrogen atom, which provides products of associated Laguerre polynomials and an exponential as solutions. In the derivation you have to solve some recursion equations, which lead to the condition n-l-1 >= 0 (or l <= n-1). These mathematical conditions are indeed related to the radial and angular nodes in the solutions (and to the required orthogonality of the eigenfunctions of a Hermitian operator), mentioned by Jan. This translates then to many-electron atoms, which still have the same angular solutions and modified radial ones, which still have the same nodal relationships. You can find the derivations for the hydrogen atom in many text books on quantum mechanics, e.g. in the appendix of Atkins/Freedman.
Many regards, Martin Am 23.12.2020 um 09:46 schrieb Jan Halborg Jensen jhjensen**chem.ku.dk:
Sent to CCL by: Jan Halborg Jensen [jhjensen~!~chem.ku.dk] Dear Tom One way to think about it is that the principal quantum number n is related to the number of nodes of the AO: number of nodes = n-1. The minimum number of nodes in a d-orbital is 2 (you can’t have a d-orbital shape without 2 nodes). So the minimum value of n for d-orbitals is 3. So why is the number of nodes related to n? One way to think about n, at least for the H atom, is in terms of the orbital energies: orbitals with the same energy have the same n. In other words the energy of the electron is a function of the number of nodes. The nodes increase the energy because they cause the electron to be further away from the nucleus (on average). The nodes are there to keep the orbitals orthogonal so that the Pauli exclusion (i.e. anti-symmetry) principle is satisfied, among other things. Hope this helps. Best regards, JanOn 23 Dec 2020, at 04.53, Thomas Manz thomasamanz++gmail.com <owner-chemistry::ccl.net> wrote: Dear colleagues, I am looking for a reference to cite that provides mathematical details as to why a 2d subshell does not exist for an atom. I understand the traditional pat answer that n >= L+1 where L is angular quantum number ( L = 0 for s, 1 for p, 2 for d, etc.) and n is the principal quantum number. I would like to understand the mathematical and physical reason for this, preferably with some kind of mathematical derivation. Does anyone know a good reference for this? Although the above question seems "simple", I believe there more to it than first meets the eye. Specifically, such a rule does not apply to the nucleons inside an atomic nucleus. In nuclear models (e.g., nuclear shell model), for example, they encounter things such as the 1f orbitals. Why does such an orbital exist for nucleons but not for electrons, when both are spin 1/2 fermions? The physical interaction (coupling regime) must have something to do with whether or not the 1f orbital exists for a particular fermion. In the case of nucleons, there is a very strong pairing so that two nucleons practically pair to make an effective boson; however, it is my understanding that for nucleons with odd-numbered nucleons, the odd nucleon can still exist in orbitals such as 1f. The spin-orbit coupling is substantial for nucleons, but also substantial for electrons in heavy elements. I would appreciate any mathematical or physical insights as well references to understand what is going on here. Sincerest thanks, Tom Manz>
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