Chemistry Reference
In-Depth Information
Fig. 1.4
The dashed orbital
is obtained by rotating the
2
p
x
orbital, counterclockwise
through an angle
α
which depends on the azimuthal angle, and
Φ
x
(φ)
indicates how the function de-
pends on the angle
φ
in the
xy
plane, measured from the positive
x
-direction. One
has:
1
√
π
cos
φ
Φ
x
(φ)
=
(1.10)
1
√
π
sin
φ
Φ
y
(φ)
=
Both
r
and
θ
are invariant under a rotation around the
z
-direction,
θ
1
=
θ
0
, and
r
1
=
r
0
; hence, only the
φ
part will matter when we rotate in the plane. The trans-
formed functions are easily determined starting from the general equation and using
the matrix expression for the coordinate rotation, where we replace
α
by
−
α
, since
we need the inverse operation here:
R(
cos
φ
1
)
=
cos
φ
0
=
cos
(φ
1
−
α)
=
cos
φ
1
cos
α
+
sin
φ
1
sin
α
R(
sin
φ
1
)
=
sin
φ
0
=
sin
(φ
1
−
α)
=
sin
φ
1
cos
α
−
cos
φ
1
sin
α
(1.11)
Multiplying with the radial and azimuthal parts, we obtain the desired functional
transformation of the in-plane 2
p
-orbitals:
R
2
p
x
=
2
p
x
cos
α
+
2
p
y
sin
α
(1.12)
R
2
p
y
=−
2
p
x
sin
α
+
2
p
y
cos
α
Again we should get accustomed to read these expressions almost visually. For in-
stance, when the angle is 90
◦
, one has 2
p
x
=
2
p
y
and 2
p
y
=−
2
p
x
.Thissim-
ply means: take the 2
p
x
orbital, rotate it over 90
◦
counterclockwise around the
z
-
direction, and it will become 2
p
y
. If the same is done with 2
p
y
, it will go over into
−
2
p
x
since the plus and minus lobes of the dumbell become congruent with the
oppositely signed lobes of the 2
p
x
orbital.
