CCL: C2v pointgroup & coordinate system convention

 Sent to CCL by: "Jimmy J. Stewart" [MrMOPAC!^!]
 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.
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