# CCL: C2v pointgroup & coordinate system convention

*From*: "Jimmy J. Stewart" <MrMOPAC^-^att.net>
*Subject*: CCL: C2v pointgroup & coordinate system
convention
*Date*: Wed, 24 May 2006 19:24:19 -0400

Sent to CCL by: "Jimmy J. Stewart" [MrMOPAC!^!att.net]
Peter Burger asked:
>the following has puzzled me for a while...
>In the C2v point group different choices have been made in the literature
with regard to the x and y axis.
>Several books on symmetry and point groups provide different answers... and
sometimes even choose use them either way in different chapters.
>
>For water for instance, most of them opt for the xz plane as the molecular
plane with y pependicular to it while others refer to the yz as molecular plane
with x perpendicular to it. This obviously leads to a change of the b1 and b2
symbols and makes comparisons somehow complicated...
>
>So I guess my question is - use Al Cottons or Tinkhams book as authorities?
Is there somewhere the _real_ truth?
This is a very sophisticated question. The problem only arises with the groups
C2v and D2, of the Abelian groups. For C2v there are two options for assigning
b1 and b2 irreducible representations, and for D2 there are six ways to assign
b1, b2, and b3.
The convention used in the symmetry package (written by David Danovich) within
MOPAC conforms with Cotton's definition, and is, I believe, the most general.
In this, the b1 IR is unsymmetric in the plane that has the most atoms. In
Cotton's C2v group table the plane that has the most atoms is the yz plane.This
is a definition.
This is in accord with Dr Johnson's comment that "For planar C2v molecules
the convention is to have the x-axis out of the plane." I would, however,
extend this to "For C2v molecules, the convention is to have the x-axis
perpendicular to the plane of that has the most atoms." Similar rules
hold for D2. Although these rules cover most cases, there is still some
ambiguity in C2v systems of the type AB2C2: is there a convention for those
cases where there are the same number of atoms in the various planes? If anyone
knows the rule here, please inform me.
Jimmy
.
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