From owner-chemistry@ccl.net Wed Dec 23 03:48:00 2020 From: "Jan Halborg Jensen jhjensen**chem.ku.dk" To: CCL Subject: CCL: why is there no 2d subshell in atoms Message-Id: <-54245-201223034647-27713-HeHgdGz0wQpta+b9wjWuIQ++server.ccl.net> X-Original-From: Jan Halborg Jensen Content-ID: <6531995E47538941BF7B44EAC4E40159++alumni.ku.dk> Content-Language: en-US Content-Transfer-Encoding: 8bit Content-Type: text/plain; charset="utf-8" Date: Wed, 23 Dec 2020 08:46:35 +0000 MIME-Version: 1.0 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 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 From owner-chemistry@ccl.net Wed Dec 23 05:24:00 2020 From: "Raghu R raghu.rama.chem,,gmail.com" To: CCL Subject: CCL: why is there no 2d subshell in atoms Message-Id: <-54246-201223052037-19548-JlIneMCaW8AOmX7tfEcuhw-.-server.ccl.net> X-Original-From: Raghu R Content-Type: multipart/alternative; boundary="0000000000006393d605b71f0a39" Date: Wed, 23 Dec 2020 15:50:18 +0530 MIME-Version: 1.0 Sent to CCL by: Raghu R [raghu.rama.chem^-^gmail.com] --0000000000006393d605b71f0a39 Content-Type: text/plain; charset="UTF-8" It's the 1/r Coulomb potential for the electronic problem. For nucleons, the potential is different allowing 1f functions. On Wed, Dec 23, 2020 at 10:32 AM 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 > --0000000000006393d605b71f0a39 Content-Type: text/html; charset="UTF-8" Content-Transfer-Encoding: quoted-printable
It's the 1/r Coulomb potential for the electronic p= roblem. For nucleons, the potential is different allowing 1f functions.

On Wed, Dec 23, 2020 at 10:32 AM Thomas Manz thomasamanz++gmail.com <owner-chemistry]*[ccl.net> wrote:
Dear colleagues,

I am looking for a reference to cite that=C2=A0provides mathematical = details as to why a 2d subshell does not exist for an atom. I understand th= e traditional=C2=A0pat answer that n >=3D L+1 where L is angular quantum= number ( L =3D 0 for s, 1 for p, 2 for d, etc.) and n is the principal qua= ntum number. I would like to understand the mathematical and physical reaso= n for this, preferably with some kind of mathematical derivation. Does anyo= ne know a good reference for this?

Although the ab= ove question seems "simple", I believe=C2=A0there more to it than= first meets the eye. Specifically, such a rule does not apply to the nucle= ons inside an atomic nucleus. In nuclear=C2=A0models (e.g., nuclear shell m= odel), 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 sp= in 1/2 fermions? The physical interaction (coupling regime) must have somet= hing to do with whether or not the 1f orbital exists for a particular fermi= on. In the case of nucleons, there is a very strong pairing so that two nuc= leons practically pair to make an effective boson; however, it is my unders= tanding that for nucleons with odd-numbered nucleons, the odd nucleon can s= till exist in orbitals such as 1f. The spin-orbit coupling is substantial f= or nucleons, but also substantial for electrons in heavy elements.

I would appreciate any mathematical or physical insights a= s well references to understand what is going on here.

=
Sincerest thanks,

Tom Manz
--0000000000006393d605b71f0a39-- From owner-chemistry@ccl.net Wed Dec 23 06:18:01 2020 From: "Mariusz Radon mariusz.radon[]gmail.com" To: CCL Subject: CCL: why is there no 2d subshell in atoms Message-Id: <-54247-201223061426-14504-kV+a7wlTQwQj6ExW2YdOMw##server.ccl.net> X-Original-From: Mariusz Radon Content-Transfer-Encoding: 8bit Content-Type: text/plain; charset=utf-8 Date: Wed, 23 Dec 2020 12:14:15 +0100 Mime-Version: 1.0 (Mac OS X Mail 13.4 \(3608.120.23.2.4\)) Sent to CCL by: Mariusz Radon [mariusz.radon^gmail.com] > On 23 Dec 2020, at 04:53, Thomas Manz thomasamanz++gmail.com 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 Dear Tom: I believe the answer is quite simple. If you perform SCF-LCAO calculations for single atoms, the first (in terms of orbital energy) d-type orbital obtained has no radial nodes, and therefore it is (normally) called 3d for analogy with 3d orbital of hydrogen atom (which also has no radial nodes). The same holds true about first (radial nodeless) f-type orbital, which is then called 4f per analogiam with hydrogen atom. But in principle you could label these orbitals as “1d” and “1f” meaning "the first d orbital”, “the first f orbital”. ( By the way, although I am inexpert in nuclear physics, this could be the reason why you encounter things like "1f orbitals”? ) Whereas the angular quantum number (L) is well defined in any atom (due to spherical symmetry), the principal quantum number (N) is no longer strictly defined in multi-electron atoms. Although N remains a very valuable concept (to keep analogy with hydrogen atom) you can live without it. Therefore, labeling of real atomic orbitals as 3d (to stress analogy with hydrogen atom) or 1d (to say it is the first d orbital of this particular atom) is, in principle, equally well founded, although the first way is much more common, at least in chemistry. It is even more tricky when it comes to atomic basis sets. In some basis sets all d functions have the form of 3d orbital of an H atom, i.e. they are all radial nodeless. Then, if you perform SCF-LCAO calculation for an atom, you obtain combinations of these function having zero, one, two, … radial nodal planes. The resulting orbitals are conventionally labelled as 3d, 4d, 5d, … in analogy to the number of nodal planes in these hydrogen atom orbitals. But you do not need to know or use the N quantum number in any way to perform these calculations and (probably in most cases) even to construct the underlying atomic basis sets. Best wishes, Mariusz -- Mariusz Radon, Ph.D., D.Sc. Assistant Professor Faculty of Chemistry, Jagiellonian University Address: Gronostajowa 2, 30-387 Krakow, Poland Room C1-06, Phone: 48-12-686-24-89 E-mail: mradon()chemia.uj.edu.pl (mariusz.radon()uj.edu.pl) Web: http://www.chemia.uj.edu.pl/~mradon ORCID: https://orcid.org/0000-0002-1901-8521 From owner-chemistry@ccl.net Wed Dec 23 12:15:00 2020 From: "Dr.N Sukumar n.sukumar::snu.edu.in" To: CCL Subject: CCL: why is there no 2d subshell in atoms Message-Id: <-54248-201223024215-5878-abj6zFEF+vxut26RVtuBog,server.ccl.net> X-Original-From: "Dr.N Sukumar" Content-Type: multipart/alternative; boundary="000000000000f966fb05b71cd33f" Date: Wed, 23 Dec 2020 13:11:55 +0530 MIME-Version: 1.0 Sent to CCL by: "Dr.N Sukumar" [n.sukumar ~ snu.edu.in] --000000000000f966fb05b71cd33f Content-Type: text/plain; charset="UTF-8" Content-Transfer-Encoding: quoted-printable It has to do with the nature of the potential. Atomic orbitals are solutions of the Hamiltnian in a spherically symmetric potential; the radial part of the solutions are the spherical harmonics Ylm. In a disk jellium model, for instance (often used in 2d clusters), you do find 1f and 2d orbitals. *N. SukumarProfessor of ChemistryDirector, Center for Informatics**Shiv Nadar University, India* https://chemistry.snu.edu.in/people/faculty/n-sukumar "The belief that there is only one truth is the deepest root of evil in the world." =E2=80=94 Max Born On Wed, Dec 23, 2020 at 11:14 AM Thomas Manz thomasamanz++gmail.com < owner-chemistry,,ccl.net> wrote: > Dear colleagues, > > I am looking for a reference to cite that provides mathematical details a= s > to why a 2d subshell does not exist for an atom. I understand the > traditional pat answer that n >=3D L+1 where L is angular quantum number = ( L > =3D 0 for s, 1 for p, 2 for d, etc.) and n is the principal quantum numbe= r. 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 ha= ve > something to do with whether or not the 1f orbital exists for a particula= r > 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 nucle= on > 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 > --000000000000f966fb05b71cd33f Content-Type: text/html; charset="UTF-8" Content-Transfer-Encoding: quoted-printable
It has to do with the nature of the potential. Atomic orbi= tals are solutions of the Hamiltnian in a spherically symmetric potential; = the radial part of the solutions are the spherical harmonics Ylm. In a disk= jellium model, for instance (often used in 2d clusters), you do find 1f an= d 2d orbitals.
=
N. S= ukumar
Professor of Chemistry
Director, Center for Informatics
Shiv Nadar University, India
https://chemistry.snu.edu.in/people/faculty/n-suk= umar

"The belief that there is only one truth is = the deepest root of evil in the world."

=E2=80=94 Max = Born


=
On Wed, De= c 23, 2020 at 11:14 AM Thomas Manz thomasamanz++gmail.com <owner-chemis= try,,ccl.net> wrote:
Dear colleagues,

I am lookin= g for a reference to cite that=C2=A0provides mathematical details as to why= a 2d subshell does not exist for an atom. I understand the traditional=C2= =A0pat answer that n >=3D L+1 where L is angular quantum number ( L =3D = 0 for s, 1 for p, 2 for d, etc.) and n is the principal quantum number. I w= ould like to understand the mathematical and physical reason for this, pref= erably with some kind of mathematical derivation. Does anyone know a good r= eference for this?

Although the above question see= ms "simple", I believe=C2=A0there more to it than first meets the= eye. Specifically, such a rule does not apply to the nucleons inside an at= omic nucleus. In nuclear=C2=A0models (e.g., nuclear shell model), for examp= le, 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 practicall= y 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 or= bitals 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 reference= s to understand what is going on here.

Sincerest t= hanks,

Tom Manz
--000000000000f966fb05b71cd33f-- From owner-chemistry@ccl.net Wed Dec 23 12:50:00 2020 From: "Martin Kaupp martin.kaupp!A!tu-berlin.de" To: CCL Subject: CCL: why is there no 2d subshell in atoms Message-Id: <-54249-201223052314-20205-R8thgn7r/D7X7Db17EOgJA[A]server.ccl.net> X-Original-From: Martin Kaupp Content-Transfer-Encoding: 8bit Content-Type: text/plain; charset="utf-8"; format=flowed Date: Wed, 23 Dec 2020 11:23:01 +0100 MIME-Version: 1.0 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, Jan > >> On 23 Dec 2020, at 04.53, Thomas Manz thomasamanz++gmail.com 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> > -- Prof. Dr. Martin Kaupp Technische Universität Berlin Institut für Chemie Theoretische Chemie Sekr. C 7 Strasse des 17. Juni 135 D-10623 Berlin Gebäude C, Ostflügel, EG, Raum C 78 Telefon +49 30 314 79682 Telefax +49 30 314 21075 email: martin.kaupp-x-tu-berlin.de www: http://www.quantenchemie.tu-berlin.de/ From owner-chemistry@ccl.net Wed Dec 23 19:17:01 2020 From: "Jo o Brand o jbrandao^-^ualg.pt" To: CCL Subject: CCL: why is there no 2d subshell in atoms Message-Id: <-54250-201223191217-31429-TgP+mlqmobB3Synu7uIEhQ]|[server.ccl.net> X-Original-From: "Jo o Brand o" Date: Wed, 23 Dec 2020 19:12:12 -0500 Sent to CCL by: "Jo o Brand o" [jbrandao{}ualg.pt] Dear Tom Manz It comes from the resolution of the Schrdinger equation for the hydrogen atom, that you can find in any quantum mechanics book. First we solve the angular part and get the l- quantum number. Then, when solving the radial part, the radial function and the n values depend on the angular momentum quantum number. There, this restriction appears naturally. Best regards Joo Brando Dep. de Qumica e Farmcia Universidade do Algarve 8005-139 Faro, Portugal e-mail: jbrandao()ualg.pt From owner-chemistry@ccl.net Wed Dec 23 21:56:00 2020 From: "William F. Polik polik[]hope.edu" To: CCL Subject: CCL: why is there no 2d subshell in atoms Message-Id: <-54251-201223112628-20250-vPl+R4nzmpiZZ97CD6T/2A[]server.ccl.net> X-Original-From: "William F. Polik" Content-Language: en-US Content-Type: multipart/alternative; boundary="------------EDA0720F1AF67F05E6D7FB3F" Date: Wed, 23 Dec 2020 11:26:19 -0500 MIME-Version: 1.0 Sent to CCL by: "William F. Polik" [polik,hope.edu] This is a multi-part message in MIME format. --------------EDA0720F1AF67F05E6D7FB3F Content-Type: text/plain; charset=utf-8; format=flowed Content-Transfer-Encoding: 8bit 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 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> > > . -- ------------------------------------------------------------------------ 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-x-hope.edu http://www.chem.hope.edu/~polik tel: (616) 395-7639 fax: (616) 395-7118 ------------------------------------------------------------------------ --------------EDA0720F1AF67F05E6D7FB3F Content-Type: text/html; charset=utf-8 Content-Transfer-Encoding: 8bit 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
Hofma Professor of Chemistry

Department of Chemistry
Schaap Science Center 2122
Hope College
35 East 12th Street
Holland, MI  49422-9000
USA

polik-x-hope.edu
http://www.chem.hope.edu/~polik
tel: (616) 395-7639
fax: (616) 395-7118
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