From owner-chemistry@ccl.net Thu Jan 28 21:48:01 2021 From: "Cory Pye cpye%ap.smu.ca" To: CCL Subject: CCL: why is there no 2d subshell in atoms Message-Id: <-54263-210128213841-13024-rqcJMfMwtDZzFoprJLxzbg++server.ccl.net> X-Original-From: Cory Pye Content-Type: multipart/mixed; BOUNDARY="-1198781419-1365227688-1611887906=:21108" Date: Thu, 28 Jan 2021 22:38:21 -0400 (AST) MIME-Version: 1.0 Sent to CCL by: Cory Pye [cpye:_:ap.smu.ca] ---1198781419-1365227688-1611887906=:21108 Content-Type: text/plain; charset=UTF-8; format=flowed Content-Transfer-Encoding: 8BIT On Tue, 22 Dec 2020, 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? > It comes from the solution of the Schrodinger equation for a hydrogen atom. Basically, for a hydrogen atom, the system can be solved by writing the Hamiltonian in terms of spherical polar coordinates. In this case, it can be shown that the three-dimensional partial differential equation can be separated into three one-dimensional ordinary differential equations. 1) The phi equation, which is very easy to solve. 2) The theta equation, which is fairly hard. 3) The r (radial) equation, which is also fairly hard. #2 and #3 require the use of infinite series solutions. When this is done, one derives a recursion formula for the coefficients. If the series are infinite, one can show that they can diverge unless one of the coefficients happens to be zero. Thus the infinite series becomes a polynomial. Some extra math shows then that the solutions to equations 2 and 3 are the associated Legendre functions or associated Laguerre polynomials, respectively. The associated Legendre functions involve taking the |m|-th derivative of a polynomial of order l. If |m| was greater than l, then the derivative would be zero everywhere, and thus the associated Legendre function is also zero. This explains why m is between -l and +l, inclusive. I have a J. Chem Ed. paper from 2006 on this (p460) if you are interested. Because l appears in the radial equation, it also appears in its solution. In order to get rid of the singularity at r = 0, you essentially need to multiply by r^l. When solving the series solution, it naturally commes out that n = k + l, where k is a positive integer. Therefore l < n. -Cory Pye - ************* ! Dr. Cory C. Pye ***************** ! Associate Professor *** ** ** ** ! Theoretical and Computational Chemistry ** * **** ! Department of Chemistry, Saint Mary's University ** * * ! 923 Robie Street, Halifax, NS B3H 3C3 ** * * ! cpye(0)crux.stmarys.ca http://apwww.stmarys.ca/~cpye *** * * ** ! Ph: (902)-420-5654 FAX:(902)-496-8104 ***************** ! ************* ! Les Hartree-Focks (Apologies to Montreal Canadien Fans) ---1198781419-1365227688-1611887906=:21108--