From owner-chemistry@ccl.net Thu Dec 24 09:54:00 2020 From: "William F. Polik polik#%#hope.edu" To: CCL Subject: CCL: why is there no 2d subshell in atoms Message-Id: <-54252-201224092853-18140-7jq+J5JjBhj9qQY+IcucEw###server.ccl.net> X-Original-From: "William F. Polik" Content-Language: en-US Content-Type: multipart/alternative; boundary="------------49943C588A8E8E3A69F047A5" Date: Thu, 24 Dec 2020 09:28:43 -0500 MIME-Version: 1.0 Sent to CCL by: "William F. Polik" [polik]![hope.edu] This is a multi-part message in MIME format. --------------49943C588A8E8E3A69F047A5 Content-Type: text/plain; charset=utf-8; format=flowed Content-Transfer-Encoding: 8bit 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 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>> >> E-mail to subscribers:CHEMISTRY#,#ccl.net or use:>> >> E-mail to administrators:CHEMISTRY-REQUEST#,#ccl.net or use>> >> Before posting, check wait time at:http://www.ccl.net >> >> Job:http://www.ccl.net/jobs >> Conferences:http://server.ccl.net/chemistry/announcements/conferences/ >> >> Search Messages:http://www.ccl.net/chemistry/searchccl/index.shtml>> >> RTFI:http://www.ccl.net/chemistry/aboutccl/instructions/ >> >> >> . > > -- > ------------------------------------------------------------------------ > 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 > ------------------------------------------------------------------------ -- ------------------------------------------------------------------------ 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 ------------------------------------------------------------------------ --------------49943C588A8E8E3A69F047A5 Content-Type: text/html; charset=utf-8 Content-Transfer-Encoding: 8bit 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
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

--
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
--------------49943C588A8E8E3A69F047A5-- From owner-chemistry@ccl.net Thu Dec 24 17:26:00 2020 From: "Thomas Manz thomasamanz : gmail.com" To: CCL Subject: CCL: why is there no 2d subshell in atoms Message-Id: <-54253-201224172440-6127-gNq1fizwfRRfdjpqm+MBpw^server.ccl.net> X-Original-From: Thomas Manz Content-Type: multipart/alternative; boundary="000000000000a074ef05b73d452b" Date: Thu, 24 Dec 2020 15:24:21 -0700 MIME-Version: 1.0 Sent to CCL by: Thomas Manz [thomasamanz/a\gmail.com] --000000000000a074ef05b73d452b Content-Type: text/plain; charset="UTF-8" Content-Transfer-Encoding: quoted-printable Hi William, Thanks for your response. I think that answers my question. Tom On Thu, Dec 24, 2020 at 9:49 AM William F. Polik polik#%#hope.edu < owner-chemistry###ccl.net> wrote: > Oops, typo! I meant Laguerre polynomial, not Legendre function. I shoul= d > 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=3D2 and l=3D2) , then the associated Lague= rre > 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=3D2 and l=3D2) , then the associated Legen= dre > 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 relat= ed to the number of nodes of the AO: number of nodes =3D n-1. The minimum n= umber of nodes in a d-orbital is 2 (you can=E2=80=99t have a d-orbital shap= e 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 th= e 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 t= hey cause the electron to be further away from the nucleus (on average). Th= e nodes are there to keep the orbitals orthogonal so that the Pauli exclusi= on (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 a= s to why a 2d subshell does not exist for an atom. I understand the traditi= onal 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 number. I wo= uld like to understand the mathematical and physical reason for this, prefe= rably with some kind of mathematical derivation. Does anyone know a good re= ference for this? > > Although the above question seems "simple", I believe there more to it th= an first meets the eye. Specifically, such a rule does not apply to the nuc= leons inside an atomic nucleus. In nuclear models (e.g., nuclear shell mode= l), for example, they encounter things such as the 1f orbitals. Why does su= ch an orbital exist for nucleons but not for electrons, when both are spin = 1/2 fermions? The physical interaction (coupling regime) must have somethin= g 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 nucleo= ns practically pair to make an effective boson; however, it is my understan= ding that for nucleons with odd-numbered nucleons, the odd nucleon can stil= l 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 referenc= es to understand what is going on here. > > Sincerest thanks, > > Tom Manz > > CHEMISTRY|-|ccl.net or use:> > E-mail to administrators: CHEMISTRY-REQUEST|-|ccl.net or usehttp://www.ccl.net/c= hemistry/sub_unsub.shtml > > . > > > -- > ------------------------------ > 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 > ------------------------------ > > > -- > ------------------------------ > 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 > ------------------------------ > --000000000000a074ef05b73d452b Content-Type: text/html; charset="UTF-8" Content-Transfer-Encoding: quoted-printable
Hi William,

Thanks for= your response. I think that answers my question.

= Tom


On Thu, Dec 24, 2020 at 9:49 AM William F. Polik po= lik#%#hope.edu <owner-chemistry###ccl.net> wrote:
=20 =20 =20
Oops, typo!=C2=A0 I meant Laguerre polynomial, not Legendre function.= =C2=A0 I should have written:

Mathematically, the 1-electron Coulombic (eg, H-atom) wavefunction has:
=C2=A0 * l angular nodes that arise from the associated Legendre functi= on Y^m_l (a polynomial of order l) in the angular part of the wavefunction
=C2=A0 * 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
=C2=A0
If l > n-1 (eg, 2d for which n=3D2 and l=3D2) , 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:
=20 Tom,

The node argument very appealing.=C2=A0 Here is a mathematical basis for it.

Mathematically, the 1-electron Coulombic (eg, H-atom) wavefunction has:
=C2=A0 * 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
=C2=A0 * 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
=C2=A0
If l > n-1 (eg, 2d for which n=3D2 and l=3D2) , then the associate= d 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**ch= em.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 =3D n-1. The minimum num=
ber of nodes in a d-orbital is 2 (you can=E2=80=99t 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 le=
ast 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 the=
y 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 tradition=
al 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 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 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 sh=
ell 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 a=
re 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 tw=
o nucleons practically pair to make an effective boson; however, it is my u=
nderstanding that for nucleons with odd-numbered nucleons, the odd nucleon =
can still exist in orbitals such as 1f. The spin-orbit coupling is substant=
ial 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

CHEMISTRY|-|ccl.n=
et or use:
      http://www.ccl.net/cgi-bin/ccl/send_ccl_message

E-mail to administrators: CHEMISTRY-REQUEST|-|ccl.net or use
      http://www.ccl.net/cgi-bin/ccl/send_ccl_messagehttp://www.c=
cl.net/chemistry/sub_unsub.shtml

Before posting, check wait time at: http://www.ccl.net

Job: http://www.ccl.n=
et/jobs=20
Conferences: http://server.ccl.net/chemistry/announcements/co=
nferences/

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.

--
=20
Dr. William F. Polik
Hofma Professor of Chemistry

Department of Chemistry
Schaap Science Center 2122
Hope College
35 East 12th Street
Holland, MI=C2=A0 49422-9000
USA

polik|-|h= ope.edu
htt= p://www.chem.hope.edu/~polik
tel: (616) 395-7639
fax: (616) 395-7118

--
=20
Dr. William F. Polik
Hofma Professor of Chemistry

Department of Chemistry
Schaap Science Center 2122
Hope College
35 East 12th Street
Holland, MI=C2=A0 49422-9000
USA

polik|-|hop= e.edu
http:= //www.chem.hope.edu/~polik
tel: (616) 395-7639
fax: (616) 395-7118
--000000000000a074ef05b73d452b--