Atomic Orbitals - Methods of Molecular Quantum Mechanics - page 47

As we have already said, the only difference with STOs from the
hydrogenic AOs (3.51)-(3.53) is in the radial part R n (r), which is now
independent of ' . This implies a wrong behaviour of the ns functions

with n > 1, which are now zero at the origin, lacking the characteristic

cusp present in HAOs and which must be eventually corrected by
Schmidt orthogonalization of the ns STO against all

its inner s

orbitals.
Bearing in mind the characteristic integrals (Abramowitz and Stegun,
1965)

¥

n !
c n þ 1

drr n exp ð cr Þ¼

ð 3 :

56 Þ

0

< <

2

ð

p

cos 2 m w

ð 1 þd

0m Þ

d w ¼ p ð 3 : 57 Þ

: :

sin 2 m w

ð 1 d

0m Þ

0

ð

1

' þ 1 ð' þ m Þ!

2

2

dx ½ P m

' ð x Þ

¼ m ¼j m j 0 ð 3 : 58 Þ

2 ð' m Þ!

1

the overall normalization factor N of the general STO is obtained as

" # 1 = 2

1 = 2

2n þ 1

ð 2c Þ 2 ' þ 1 ð' m Þ!

ð' þ m Þ!

N ¼ N r N W ¼ ð 3 : 59 Þ

ð 2n Þ! 2 p ð 1 þd 0m Þ

where N r andN W are separate radial and angular normalization factors. It
must be further recalled that the real spherical harmonics (tesseral
harmonics) are orthogonal and normalized to 1:

ð d

W Y c ; s

' 0 m 0 ðWÞ Y c ; s ' m ðWÞ¼d '' 0 d mm 0

ð 3 : 60 Þ

where W stands for the angular variables ðu; wÞ and d W ¼ sin u d u d w .