Ed,
OK, I need to think about this more when I have time, but at this point
> I think it is a semantic difference- For me the first and last rotation are
> about the same Z axis because as you say they are both around the screen Z
> axis
> and both operators look like cos,sin, 0, -sin, cos, 0, 0, 0, 1; i.e.
> rotation
> about "THE" z axis; and it is not helpful to consider it a different z
> axis just
> because the atoms moved.
>
Indeed since, by definition, if the z axis is fixed (i.e. not rotated),
then it must be the same z axis! Conversely if it is rotated then it must
be a different (i.e. 'new z') axis. As I mentioned in an earlier post, the
only co-ordinate basis axes that are relevant to defining both rotation
axis directions and atomic co-ordinates are the 'screen' axes, regardless
of whether you are describing rotations as around fixed (screen) or rotated
(molecule) axes. Switching between different bases only adds to the
confusion!
As you say the Rz matrices all contain terms of the form cos,sin, 0, -sin,
cos, 0, 0, 0, 1 regardless of whether you are talking about the first or
last rotation, or whether you are describing it in terms of fixed or
rotated axes, indeed the equations are identical in both cases. As you say
it is only a semantic difference, i.e. the way we translate the equation
into words. The equation in both cases is:
x' = Rz(a) Ry(b) Rz(g) x
The conventional textbook way of interpreting this is:
x' = Rz(a) (Ry(b) (Rz(g) x))
i.e. the Rz(g) matrix is first applied to the vector x, then Ry(b) is
applied to the result, then Rz(a) is applied to the result of that to get
the final vector x'. What is perhaps less known is that if you instead
refer to rotated ('new') axes then this conventional order must be reversed
(I bet that's not in the textbooks!). This is in large part a source of
the confusion surrounding the 'new axis' interpretation of the equation.
We come up with the same conclusions with our different ways of thinking
> about it:
> for one, deriving the concatenated simple operators to represent a general
> rotation,
> and the commutativity: I would say the operators do not commute as long as
> the axes
> they rotate about are kept fixed, but if the axes rotate the same as the
> molecule
> then the z axis will always be passing through the atoms the same way.
> Then rotations would commute, because the z axis would always represent
> the same molecular axis. Which I am sure is NOT what you meant by saying
> "new z axis".
>
Simple rule: in the general case rotation matrices _never_ commute (only in
the special cases I described earlier). The fallacy in your argument is
that you are referring to molecular basis axes: in the screen basis
convention that we are using the rotation axis directions must always be
with respect to the screen basis axes, even if we are talking about the
rotated axis interpretation of the above equation; in that case they are
therefore by definition in general different axes, both in name ('z' & 'new
z') and in direction, and therefore cannot possibly commute.
The z axis is 'new' because it's a different axis in our screen basis. The
fact that it's the same axis in another basis is totally irrelevant for the
simple reason that we are not using that basis! As I said before, if you
don't stick to one basis this is going to get very confusing!
I anticipate that you will argue that this means that in the fixed axis
case because the first and last z axes are in the same direction it follows
that they must commute. Not so! Because of the intervening y axis
rotation, although the two Rz matrices are identical the molecule in each
case is in a different starting orientation, so the effect of the z
rotation on the molecule is different. Hence no commutation in this case
either.
Cheers
-- Ian
