6
1 Operations
Fig. 1.4 The dashed orbital
is obtained by rotating the
2p x orbital, counterclockwise
through an angle α
which depends on the azimuthal angle, and Φ x (φ) indicates how the function depends on the angle φ in the xy plane, measured from the positive x-direction. One
has:
Φ x (φ) =
1
√
π
cos φ
Φ y (φ) =
1
√
π
sin φ
(1.10)
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 transformed 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:
ˆ
R(cos φ 1 ) = cos φ 0 = cos(φ 1 − α)
= cos φ 1 cos α + sin φ 1 sin α
ˆ
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 2p-orbitals:
ˆ
R2p x = 2p x cos α + 2p y sin α
ˆ
R2p y =−2p x sin α + 2p y cos α
(1.12)
Again we should get accustomed to read these expressions almost visually. For instance, when the angle is 90 ◦ , one has 2p ′
x = 2p y and 2p ′
y =−2p x .T h i ss i m -
ply means: take the 2p x orbital, rotate it over 90 ◦ counterclockwise around the zdirection, and it will become 2p y . If the same is done with 2p y , it will go over into
−2p x since the plus and minus lobes of the dumbell become congruent with the
oppositely signed lobes of the 2p x orbital.