Problem 4.18

Apply the group theoretical treatment to obtain the bonding description in NO₂^-.

Answer:

First, find the point group (C_2v) and draw the axis systems (except for z on N, this is arbitrary).

 

The character table (from Appendix C) is:
 

C2v	E	C2	v(xz)	'v(yz)
a1	1	1	1	1	z	x2, y2, z2
a2	1	1	-1	-1	Rz	xy
b1	1	-1	1	-1	x, Ry	xz
b2	1	-1	-1	1	y, Rz	yz

The irreducible representations of the N basis orbitals are found directly from the character table as 2s (a₁), 2p_x (b₁), 2p_y (b₂), and 2p_z (a₁).
The irreducible representations of the O basis orbitals, also 2s and 2p, must be found by using the symmetry:
 

Total representations for O orbitals:
 

basis	E	C2	v(xz)	'v(yz)
2s	2	0	2	0
2px	2	0	2	0
2py	2	0	2	0
2pz	2	0	-2	0

So the O 2s orbitals transform as a₁ + b₁, the O 2p_x orbitals transform as a₁ + b₁, the O 2p_y orbitals transform as a₁ + b₁, and the O 2p_z orbitals transform as a₂ + b₂.
 

Next, generate the group orbitals using projection operators:
 

basis	E	C2	v(xz)	'v(yz)
Rs1	s1	s2	s1	s2
Rpx1	px1	px2	px1	px2
Rpy1	py1	py2	py1	py2
Rpz1	pz1	-pz2	-pz1	pz2

After normalization:

Combining orbitals of like symmetry and energy: