where x, y, and z denote the Cartesian coordinates. Let S be an orthogonal matrix that
transforms the xyz-coordinate system to x’y’z’-coordinate system. We suppose that
S is expressed as
S ¼
s 11 s 12 s 13
s 21 s 22 s 23
s 31 s 32 s 33
0
B
@
1
C
A and
x
0
y
0
z
0
0
B
@
1
C
A ¼
s 11 s 12 s 13
s 21 s 22 s 23
s 31 s 32 s 33
0
B
@
1
C
A
x
y
z
0
B
@
1
C
A: ð19:30Þ
Since S is an orthonormal matrix, we have
x
y
z
0
B
@
1
C
A ¼
s 11 s 21 s 31
s 12 s 22 s 32
s 13 s 23 s 33
0
B
@
1
C
A
x
0
y
0
z
0
0
B
@
1
C
A:
ð19:31Þ
The equation is due to
SS
T
¼ S
T S ¼ E,
ð19:32Þ
where E is a unit matrix. Then we have
∂
∂x 0 ¼
∂x
∂x 0
∂
∂x
þ
∂y
∂x 0
∂
∂y
þ
∂z
∂x 0
∂
∂z
¼ s 11
∂
∂x
þ s 12
∂
∂y
þ s 13
∂
∂z
:
ð19:33Þ
Partially differentiating (19.33), we have
∂
∂x 0 2 ¼ s 11
∂
∂x 0
∂
∂x
þ s 12
∂
∂x 0
∂
∂y
þ s 13
∂
∂x 0
∂
∂z
¼ s 11 s 11
∂
∂x
þ s 12
∂
∂y
þ s 13
∂
∂z
∂
∂x
þ s 12 s 11
∂
∂x
þ s 12
∂
∂y
þ s 13
∂
∂z
∂
∂y
þs 13 s 11
∂
∂x
þ s 12
∂
∂y
þ s 13
∂
∂z
∂
∂z
¼ s 11
2 ∂
2
∂x 2 þ s 11 s 12
∂
2
∂y∂x
þ s 11 s 13
∂
2
∂z∂x
þs 12 s 11
∂
2
∂x∂y
þ s 12
2 ∂
2
∂y 2 þ s 12 s 13
∂
2
∂z∂y
þs 13 s 11
∂
2
∂x∂z
þ s 13 s 12
∂
2
∂y∂z
þ s 13
2 ∂
2
∂z 2 :
ð19:34Þ
736
19 Applications of Group Theory to Physical Chemistry