and the overlap matrix is

S¼

1 :0000 0: 5017

0 :5017 1: 0000



ð 5 : 112 Þ

Step 3– Calculating the orthogonalizing matrix
Calculating the orthogonalizing matrixS#1/2(see Eqs.5.67–5.69and the dis-
cussion referred to inChapter 4):
DiagonalizingS

S¼

0 :7071 0: 7071

0 : 7071 # 0 : 7071



1 :5017 0: 0000

0 :0000 0: 4983



0 :7071 0: 7071

0 : 7071 # 0 : 7071



PDP#^1

ð 5 : 113 Þ

CalculatingD#1/2

D#^1 =^2 ¼^1 :^5017

1 = (^20) : 0000

0 : 0000 0 : 4983 #^1 =^2



¼

0 :8160 0: 000

0 :0000 1: 4166



ð 5 : 114 Þ

CalculatingS#1/2

S#^1 =^2 ¼PD#^1 =^2 P#^1 ¼

1 : 1163 # 0 : 3003

# 0 :3003 1: 1163



ð 5 : 115 Þ

• 
  
  
  
  • 
    
  
  
  
  

––

• 
  


• 
  

––

(22|11)

f 1 superposed right on f 1

(11|11)

f 2 f 1

(21|11)

f 2 f 1

(21|21)

f 2 f 1

Fig. 5.10 Schematic depictions of the physical meaning of some two-electron repulsion integrals
(Section 5.2.3.6.5). Each basis functionfis normally centered on an atomic nucleus. The integrals
shown here are one-center and two-center two-electron repulsion integrals – they are centered on
one and on two atomic nuclei, respectively. For molecules with three nuclei three-center integrals
arise, and for molecules with four or more nuclei, four-center integrals arise

220 5 Ab initio Calculations