Chemistry Reference
In-Depth Information
coordinates are rotated in the opposite direction (hence
α
is replaced by
−
α
), one
obtains
x
=
x
cos
α
+
y
sin
α
(1.22)
y
=−
x
sin
α
+
y
cos
α
Invert these equations to express
x
and
y
as a function of
x
and
y
:
x
cos
α
y
sin
α
x
=
−
(1.23)
x
sin
α
y
cos
α
y
=
+
The partial derivatives needed in the chain rule can now be obtained by direct deriva-
tion:
∂x
∂x
=
cos
α
∂y
∂x
=
sin
α
(1.24)
∂x
∂y
=−
sin
α
∂y
∂y
=
cos
α
Hence, the transformation of the derivatives is entirely similar to the transformation
of the
x,y
functions themselves:
R
∂x
∂y
cos
α
∂y
∂x
−
sin
α
∂
∂
=
(1.25)
sin
α
cos
α
In an operator formalism, we should denote this as
R,
∂
∂x
cos
α
∂
sin
α
∂
∂y
=
∂x
+
(1.26)
R,
∂
∂y
sin
α
∂
cos
α
∂
∂y
=−
∂x
+
As a further example, consider a symmetry transformation of a symmetry operator
itself. Take, for instance, a rotation,
C
2
, corresponding to a rotation about the
x
-axis
of 180
◦
and rotate this 90
◦
counterclockwise about the
z
-direction by
C
4
. Applying
the general expression for an operator transformation yields
C
2
=
C
4
C
2
C
4
−
1
(1.27)
The result of this transformation corresponds to an equivalent twofold rotation
around the
y
-direction,
C
2
. The rotation around
x
is thus mapped onto a rotation
