From: toms@fcs260c2.ncifcrf.gov (Tom Schneider)
Newsgroups: bionet.info-theory
Subject: Re: dspace question and relatedly, D
Date: 14 Apr 92 23:28:44 GMT
Organization: NCI Supercomputer Facility, Frederick, MD

In article <gapdrr.703287864@gsusgi1.gsu.edu> gapdrr@gsusgi2.gsu.edu
(Daniel R T Rothman) writes:

>	[Toms] paper states that the orientations of the components are not
>    germaine, and thus may be ignored, yielding, by my calculations (6-3)n
>    degrees of freedom. tom uses the formula 3n-6 for dspace.

The formula is used in molecular spectroscopy.  Look up under vibrations of
molecules also.  Suppose there are n atoms.  Then to define the location of the
n atoms in a coordinate system, we need 3n numbers.  How a molecular machine
works (usually) is not affected by its orientation (unless it is relative to
some other object, a case we will ignore for simplicity).  There are 3 numbers
that define the center of gravity of the molecule, and another three numbers
for its rotational orientation in space.  So 6 of the 3n numbers are not
important in a description of what the machine is doing, so we get 3n-6.  This
is important in molecular spectroscopy because it determines the number of
modes of vibration of the molecule.  The published version of the theory
doesn't deal in molecular vibrations (directly) so the connection is not so
clear.  (I mentioned my reason for not doing this in an earlier posting.)

Note that the description of the velocities takes another 3n-6 numbers, so
the complete description of the positions and velocities of the molecular
machine takes 2(3n-6) numbers.

>	this formula appears to break down for small n ie 1 or 2. 3n-6 degrees
>    of freedom in a 1 component system suggests that -3 coordinates are
>    nece[s]sary to describe the system. i will yield that this result may be
>    interpreted as saying that there is no information requisite to describe
>    a system with one component, when a natural coordinate system is used.

Good observation!  It is true it breaks down.  It is unfortunate for
understanding since one often tries simple molecules first to understand the
formula!  A water molecule (n=3) has 3n-6=3 modes of vibration.  If my head
were the oxygen and my hands the hydrogens, then the modes are:

1.  pulling my head forward while bringing my arms to my head
2.  pulling my head forward while opening my arms
3.  assymetric mode: head goes right while left arm comes in and right arm goes
out.

Repeat.  Whew, this exercise is tough!  Source:

@inproceedings{Mulliken1983,
author = "R. S. Mulliken",
editor = "S. P. Parker",
title = "Molecular structure and spectra",
booktitle = "McGraw-Hill Encyclopedia of Physics",
publisher = "McGraw-Hill Book Company, Inc.",
address = "New York",
year = "1983",
pages = "662",
isbn = "0-07-045253-9",
callnumber = "QC5.M425"}

The formula 3n-6 breaks down if the molecule is linear, like carbon dioxide.
Then the modes collapse into each other.  I don't know a simpler way to explain
it.  Perhaps one of our chemist friends could help.

>second: it occu[r]red to me that the vectorization that is done to the
>    informational degrees of freedom was very similar to the vectorization done
>    in non-linear diff-eq's. i haven't done enough work with large non-linear
>    systems to be authoritative about this, but was wondering if someone could
>    offer more insights into this.

This is potentially an important clue of what mathematics may be useful here!

  Tom Schneider
  National Cancer Institute
  Laboratory of Mathematical Biology
  Frederick, Maryland  21702-1201
  toms@ncifcrf.gov