Atomic Orbitals - Methods of Molecular Quantum Mechanics - page 45

written as

Y c Y s Y ' 0 ;

' m / Q ' m cos m w; ' m / Q ' m sin m w

ð m > 0 Þ ð 3 : 50 Þ

The first few (un-normalized) HAOs in real form are hence simply
given by

<

1s / exp ð cr Þ ð c ¼ Z Þ
2s / exp ð cr Þð 1 cr Þ 2c ¼ Z Þ
2p z / exp ð cr Þ z
2p x / exp ð cr Þ x
2p y / exp ð cr Þ y

ð 3 : 51 Þ

:

!

<

2
3 c 2 r 2

3s / exp ð cr Þ 1 2cr þ ð 3c ¼ Z Þ

!

1
2 cr

3p z / exp ð cr Þ 1 z

ð 3 : 52 Þ

!

:

1
2 cr

3p x / exp ð cr Þ 1 x

!

1
2 cr

3p y / exp ð cr Þ 1 y

<

3z 2 r 2

2

3d z 2 / exp ð cr Þ

3d xy / exp ð cr Þ xy
3d zx / exp ð cr Þ zx
3d yz / exp ð cr Þ yz
3d x 2 y 2 / exp ð cr Þð x 2

ð 3 : 53 Þ

:

y 2 Þ

where x, y, and z are given by Equation 1.45. Hence, our real AOs have
the same transformation properties of (x,y,z)-coordinates or of their