Oops, typo! I meant Laguerre polynomial, not Legendre function.
I
should have written:
Mathematically, the 1-electron Coulombic (eg, H-atom) wavefunction
has:
* l angular nodes that arise from the associated Legendre function
Y^m_l (a polynomial of order l) in the angular part of the
wavefunction
* n-l-1 radial nodes that arise from the associated Laguerre
polynomial L^(2l+1)_(n+l) (a polynomial of order n-l-1) in the
radial part of the wavefunction
If l > n-1 (eg, 2d for which n=2 and l=2) , then the associated
Laguerre polynomial L^(2l+1)_(n+l) is undefined (qualitatively it
would have a negative number of nodes, which is unphysical).
So it is the Laguerre polynomial in the wavefunction arising from
the 1-electron Coulombic potential that enforces l < n.
Will
On 12/23/20 11:26 AM, William F. Polik
wrote:
Tom,
The node argument very appealing. Here is a mathematical basis
for it.
Mathematically, the 1-electron Coulombic (eg, H-atom) wavefunction
has:
* l angular nodes that arise from the associated Legendre
function Y^m_l (a polynomial of order l) in the angular part of
the wavefunction
* n-l-1 radial nodes that arise from the associated Legendre
function L^(2l+1)_(n+l) (a polynomial of order n-l-1) in the
radial part of the wavefunction
If l > n-1 (eg, 2d for which n=2 and l=2) , then the associated
Legendre function L^(2l+1)_(n+l) is undefined (qualitatively it
would have a negative number of nodes).
So it is the Legendre function arising from the 1-electron
Coulombic potential wavefunction that enforces l < n.
One could probably reference almost any quantum mechanical or
quantum chemistry textbook that provides sufficient detail into
the derivation of the H-atom wavefunction for this.
Will
On 12/23/20 3:46 AM, Jan Halborg
Jensen jhjensen**
chem.ku.dk wrote:
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, Jan
On 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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Dr. William F. Polik
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Department of Chemistry
Schaap Science Center 2122
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tel: (616) 395-7639
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--
Dr. William F. Polik
Hofma Professor of Chemistry
Department of Chemistry
Schaap Science Center 2122
Hope College
35 East 12th Street
Holland, MI 49422-9000
USA
polik|-|hope.edu
http://www.chem.hope.edu/~polik
tel: (616) 395-7639
fax: (616) 395-7118