Page 188 - PRINCIPLES OF QUANTUM MECHANICS as Applied to Chemistry and Chemical Physics

6.4 Atomic orbitals 179
5=2
1 Z
j2p x i 2 ÿ1=2 (j2p 1 i‡j2p ÿ1 i) ˆ re ÿ Zr=2a 0 sin è cos j (6:61a)
4(2ð) 1=2 a 0
5=2
1 Z
j2p y i ÿi2 ÿ1=2 (j2p 1 iÿj2p ÿ1 i) ˆ re ÿ Zr=2a 0 sin è sin j
4(2ð) 1=2 a 0
(6:61b)

where equations (A.32) and (A.33) have been used. These new orbitals j2p x i
and j2p y i are orthogonal to each other and to all the other eigenfunctions
jnlmi. The factor 2 ÿ1=2 ensures that they are normalized as well. Although
these new orbitals are simultaneous eigenfunctions of the Hamiltonian operator
^
^
^ 2
H and of the operator L , they are not eigenfunctions of the operator L z .
If we now substitute equations (5.29a), (5.29b), and (5.29c) into (6.61a),
(6.61b), and (6.60a), respectively, we obtain for the set of three real 2p orbitals
5=2
1 Z
j2p x iˆ xe ÿ Zr=2a 0 (6:62a)
4(2ð) 1=2 a 0
5=2
1 Z
j2p y iˆ ye ÿ Zr=2a 0 (6:62b)
4(2ð) 1=2 a 0
5=2
1 Z
j2p z iˆ ze ÿ Zr=2a 0 (6:62c)
ð 1=2 2a 0
The subscript x, y,or z on a 2p orbital indicates that the angular part of the
orbital has its maximum value along that axis. Graphs of the square of the
angular part of these three functions are presented in Figure 6.2. The mathema-
tical expressions for the real 2p and 3p atomic orbitals are given in Table 6.2.

d orbitals
The ®ve wave functions for n ˆ 3, l ˆ 2 are
7=2
1 Z 2 ÿ( Zr=3a 0 ) 2
j3d 0 iˆj320iˆ p  r e (3 cos è ÿ 1) (6:63a)
81 6ð a 0
7=2
1 Z 2 ÿ( Zr=3a 0 ) ij
j3d 1 iˆj32 1iˆ p  r e sin è cos è e (6:63b)
81 ð a 0
7=2
1 Z 2 ÿ( Zr=3a 0 ) 2 i2j
j3d 2 iˆj32 2iˆ p  r e sin è e (6:63c)
162 ð a 0
The orbital j3d 0 i is real. Substitution of equation (5.29c) into (6.63a) and a
change in notation for the subscript give