Hybridization of atomic orbitals: Valence Bond Theory
We generate the three sp2 hybrid orbitals and represent them graphically.
i := 0 .. 30
θ i :=
π⋅i
30
j := 0 .. 40
φ j :=
2 ⋅π ⋅ j
40
The two p orbitals are represented by the corresponding spherical harmonics.
The s orbital can be represented simply by a constant:
( )
( )
pxi , j := sin θ i ⋅ cos φ j
( )
( )
pyi , j := sin θ i ⋅ sin φ j
Define the three hybrid orbitals, using the s orbital radius to be 1/3:
sp21i , j :=
0.333
The second hybrid orbital, sp2(2): sp22i , j :=
0.333
sp23i , j :=
0.333
The first hybrid orbital, sp2(1):
The third hybrid orbital, sp2(3):
3
3
3
+
2
⋅ pxi , j
3
−
1
−
1
6
6
⋅ pxi , j −
1
⋅ pxi , j +
1
2
2
⋅ pyi , j
⋅ pyi , j
Define the x, y, z coordinates for the parametric plots:
( ) ( )
y1i , j := sp21i , j⋅ sin ( θ i) ⋅ sin ( φ j)
z1i , j := sp21i , j⋅ cos ( θ i)
x1i , j := sp21i , j⋅ sin θ i ⋅ cos φ j
( ) ( )
y3i , j := sp23i , j⋅ sin ( θ i) ⋅ sin ( φ j)
z3i , j := sp23i , j⋅ cos ( θ i)
x3i , j := sp23i , j⋅ sin θ i ⋅ cos φ j
( ) ( )
y2i , j := sp22i , j⋅ sin ( θ i) ⋅ sin ( φ j)
z2i , j := sp22i , j⋅ cos ( θ i)
x2i , j := sp22i , j⋅ sin θ i ⋅ cos φ j
Now plot (x1, y1, z1) for the first hybrid orbital, (x2, y2, z2) for the second, and
(x3, y3, z3) for the third. By keeping the "tilt" and "rotations" the same for all
three, their relative orientations are clearly observable.
sp2(1)
( x1 , y1 , z1)
sp2(2)
( x2 , y2 , z2)
sp2(3)
( x3 , y3 , z3)