6.2 Diatomic Molecules
289
Fig. 6.10 Cartesian and
spherical polar coordinate
systems
θ
φ
X
Y
Z
(ρ,θ,φ)
or
(x, y, z )
z =ρcosθ
ρ sin θ
O
y =ρsinθsinφ
x =ρsinθcosφ
In addition it is easy to see that ρ is given in terms of x, y, z by ρ = (x 2 + y 2 +
z 2 )
1
2 . The operation of inversion, ı, can be represented in Cartesian coordinates by
(x, y, z)
ı
→ (x
, y
, z
) = (−x, −y, −z) ,
(6.2.82)
and in spherical polar coordinates by
(ρ, θ, φ)
ı
→ (ρ
, θ
, φ
) ,
(6.2.83)
in which ρ , θ , φ are as yet unknown. From the definition of ρ in terms of x , y , z ,
viz. ρ = [x 2 + y 2 + z 2 ]
1
2 , we see immediately that
ρ
= [(−x)
2
+ (−y)
2
+ (−z)
2
]
1
2 = ρ .
(6.2.84)
The equations relating Cartesian and spherical polar coordinates allow us to make
the following observations regarding the effect of inversion of the coordinate
system:
z → −z ⇒ cos θ
= − cos θ ⇒ θ
= π − θ ,
(6.2.85)
x → −x ⇒ cos φ
= − cos φ ⇒ φ
= φ ± π ,
(6.2.86)
y → −y ⇒ sin φ
= − sin φ ⇒ φ
= φ + π .
(6.2.87)