[Phys-l] atomic wavefunctions, spherical harmonics, multipole moments

[John Denker <jsd@av8n.com>]

In case anybody was wondering why I was fussing with
color wheels:

I just now put up an improved version of
  http://www.av8n.com/physics/wavefunctions.htm

In particular, this section is hugely upgraded:
  http://www.av8n.com/physics/wavefunctions.htm#sec-ylm

All in all, the document gives several ways of visualizing
wavefunctions, including hands-on demos as well as three
different kinds of online animations.

The task is a difficult one:  We have a two-dimensional
ordinate (a complex number) as a function of a four-
dimensional abscissa (x,y,z,t).  There is no way AFAIK
to visualize the whole thing at once, but I have amassed
a collection of ways of visualizing various parts of
the situation.  If you understand each of the parts and
put them together in your mind, you can get a reasonably
complete picture of what's going on.

=====================

Also related:

I wrote a little proggy to calculate the multipole expansion 
coefficients for some geometries representative of simple 
molecules.

Some output is at:
  http://www.av8n.com/physics/multipole-expansion.txt

The code is at
  http://www.av8n.com/physics/multipole-expansion.c

The usage message says:

> Calculate a few terms in the multipole expansion
> aka multipole series
> including monopole, dipole, quadrupole, octupole, and hexadecapole.
>
> We use spherical harmonics (Ylm) to account for the angular dependence.
> Then we account for the radial dependence.
>
> We apply this to a few configurations with more-or-less
> the same symmetry as certain simple molecules:
>   SO4 CO CO2 XeF4 BCl3 PF5 CF4 SF6
>
> The output format is:
> For each molecule there is a headline giving the name of
> the molecule, followed by a stanza of results.
> In each stanza there are (1+Lmax) lines, one for each L value.
> On each line, there are 2L+1 entries, one for each M value.
> Each entry is a complex number.
> In the margin of each line there is a label giving the
> multipole order, i.e. 2^L
> with an asterisk if it is the /leading/ order,
> i.e. if that line contains the lowest-order non-zero term.
>
> For the examples considered here, all the entries have
> zero imaginary part, but this is not true in general.
> Example:  Try re-orienting the CO dipole along the Y axis.
>
> The inputs model the general shape of the molecule but /not/ 
> its size, so the results are not normalized.  This could be 
> fixed easily.

I looked at the results, and they seem plausible ... but this
has not undergone any review or any testing worthy of the name.