Subject: determining point groups

From: "R.G.A.Bone" <[email-masked]>
Date: Wed, 16 Feb 94 14:38:40 -0800

Concerning the various perspectives on deducing the geometric
symmetry of a molecule from its nuclear coordinates:

i)  The Gaussian package readily does this, (one of its more irritating
    features I should add), though you can turn symmetry off with a whole
    manner of flags at the various levels of calculation, you know, dep-
    ending on how you want to "fix" your result (cf. 'symmetry-broken'
    solutions in UHF).

ii) Philosophically (regrettably) given that assemblies of nuclei (i.e.,
    molecules) are not static - there are continuous vibrations, etc., the
    only molecules which will have any geometric symmetry at all, at any
    instant, are triatomics (a plane of symmetry) and diatomics.  The latter
    will have the infinite-fold rotation axis (and a whole "bunch" of others
    if homonuclear).  Thus, only if two nuclei have symmetry-related coordinates
    (to some arbitrary level of precision) is geometric symmetry present.
    Of course, one could specify this level of precision to be the dimension of
    a nucleus (typically femtometres).  But, being chemists, with an under-
    standing of spectroscopy, we know a little better and assume that, for
    all intents and purposes point-symmetry operations commute with the
    vibronic Hamiltonian so this issue does not arise and the 'time-averaged'
    position of the nuclei, or the geometry at the well in the potential is
    what counts.

iii) But, if you are a computer, to examine a number of nuclear coordinates
    and determine symmetric-relationships between them requires some intell
    igent decision concerning what is "near-symmetry" and what is "exact-
    symmetry".  Suppose there are small rounding errors in the data:
    e.g., 2 nuclear positions:
    1.00000000   0.50000000    2.34567890
    1.00000000   0.50000000   -2.34567889
    Are these 2 nuclei symmetrically-related?  Well, surely yes, although
    their z-coordinates differ by a trivial amount.  The algorithm must
    contain a threshold-cutoff which copes with cases like this.

iv) In response to the comment that you need to know the nuclear identity
    as well as the coordinates in order to deduce symmetry.  Well, except
    for the trivial case of diatomics, I challenge anybody to find 2 nuclei
    in a molecule which are NOT identical but which are in exactly-symmetry-
    related coordinates, to say 10^-6 Angstrom precision.  Of course, there may
    be circumstances under which it is desirable to label (for example iso-
    topic)  substituents as symmetrically-equivalent, and cases where it is
    not.  But that's moving the goal-posts.  It also depends on the source
    of the data: is it an experimental or theoretical geometry for the
    isotopomer?  (i.e., have zero-point effects been included or not ?)

Basically, this matter of deducing the point group from a set of coordinates
does have a slight algorithmic difficulty, which is perhaps why it has not
been widely implemented.   Arguably also, let's face it: almost all molecules
on this planet don't have any symmetry at all; (theoretical) chemists' obsession
with symmetry comes from 1) the small size of molecules they are accustomed
to dealing with (symmetry is more preponderent in small molecules),  2) the
fact that, even for small molecules, use of symmetry can make a big calculation
more practical.  Furthermore, there are hardly any molecules for which the
point group can't be deduced "by inspection", unless there happens to be a
potential confusion between "near-symmetry" and actual-symmetry.  One might
argue that that difficulty increases as molecules increase in size, but then
the amount of symmetry typically decreases in the same way (crystal-lattice
unit-cells, excepted).  

Richard Bone

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

R. G. A. Bone.
Molecular Research Institute,
845 Page Mill Road,
Palo Alto,
CA 94304-1011,
U.S.A.

Tel. +1 (415) 424 9924 x110
FAX  +1 (415) 424 9501

E-mail  [email-masked]

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