Computational Chemistry

Step 10– TransformingF 1 toF^01 (cf.Step 5)

F^01 ¼

1 : 1163 # 0 : 3003

# 0 :3003 1: 1163



# 0 : 7982 # 0 : 9791

# 0 : 9791 # 1 : 3448



1 : 1163 # 0 : 3003

# 0 :3003 1: 1163



S#^1 =^2 F 1 S#^1 =^2

¼

# 0 : 4595 # 0 : 5900

# 0 : 5900 # 1 : 0913



F^01

ð 5 : 142 Þ

Step 11– DiagonalizingF^01 to obtain the energy levelseand a coefficient matrix
C^0 (cf. Step 6)

F^01 ¼

0 :5138 0: 8579

0 : 8579 # 0 : 5138



# 1 :4447 0: 0000

0 : 0000 # 0 : 1062



0 :5138 0: 8579

0 : 8579 # 0 : 5138



C^02 e 2 C^02 #^1

ð 5 : 143 Þ

The energy levels from this second SCF cycle are#1.4447 h and#0.1062 h. To
get the MO coefficients corresponding to these MO energy levels in terms of the
original basis functionsf 1 andf 2 we now transformC^02 toC 2.
Step 12– TransformingC^02 toC 2 (cf. Step 7)

C 2 ¼

1 : 1163 # 0 : 3003

# 0 :3003 1: 1163



0 :5138 0: 8579

0 : 8579 # 0 : 5138



¼

0 :3159 1: 1120

0 : 8034 # 0 : 8319



S#^1 =^2 C^02 C 2

ð 5 : 144 Þ

This completes the second SCF cycle. We now have the MO energy levels and
basis function coefficients:
From Eq.5.143:

e 1 ¼# 1 :4447 and e 2 ¼# 0 : 1062 ð 5 : 145 Þ

From Eq.5.144:

c 1 ¼ 0 : 3159 f 1 þ 0 : 8034 f 2 and c 2 ¼ 1 : 1120 f 1 # 0 : 8319 f 2 ð 5 : 146 Þ

Step 13– Comparing the density matrix from the latestc’s with the previous
density matrix to see if the SCF procedure has converged
The density matrix elements based on thec’s ofC 2 are

P 11 ¼ 2 c 11 c 11 ¼ 2 ð 0 : 3159 Þ 0 : 3159 ¼ 0 : 1996

P 12 ¼ 2 c 11 c 21 ¼ 2 ð 0 : 3159 Þ 0 : 8034 ¼ 0 : 5076

P 22 ¼ 2 c 21 c 21 ¼ 2 ð 0 : 8034 Þ 0 : 8034 ¼ 1 : 2909

5.2 The Basic Principles of the ab initio Method 227