Computational Chemistry

centered on atomic nuclei and were normalized but not orthogonal (Section 4.3.3),
while the transformed basis functionsf^0 are delocalized over the molecule and are
orthonormal (Section 4.4.1.1). Note that the sum of the squares of the coefficients of
f^01 andf^02 is unity, as must be the case if the basis functions are orthonormal
(Section 4.3.6). In the next stepC^01 is transformed to obtain the coefficients of the
original basis functionsfin the MO’s. We want the MOs in terms of the original,
atom-centered basis functions (roughly, atomic orbitals –Section 5.3) because such
MOs are easier to interpret.
Step 7– TransformingC^0 toC, the coefficient matrix of the original, nonortho-
gonal basis functions
As in Section 4.4.1.2, we use the orthogonalizing matrixS#1/2to transformC^0 toC:

C 1 ¼

1 : 1163 # 0 : 3003

# 0 :3003 1: 1163



0 :5069 0: 8620

0 : 8620 # 0 : 5069



¼

0 :3070 1: 1145

0 : 8100 # 0 : 8247



S#^1 =^2 C^01 C 1

ð 5 : 133 Þ

This completes the first SCF cycle. We now have the first set of MO energy
levels and basis function coefficients:
From Eq.5.130

e 1 ¼# 1 :4027 and e 2 ¼# 0 : 0756 ð 5 : 134 Þ

From Eq.5.133(cf. Eq.5.132):

c 1 ¼ 0 : 3070 f 1 þ 0 : 8100 f 2 and c 2 ¼ 1 : 1145 f 1 # 0 : 8247 f 2 ð 5 : 135 Þ

Note that the sum of the squares of the coefficients off 1 andf 2 is not unity,
since these atom-centered functions are not orthogonal (contrast the simple H€uckel
method, Section 4.3.4).
Step 8– 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 1 (Eq.5.133) can be compared
with those (Eq.5.123) based on the initial guess:

P 11 ¼ 2 c 11 c 11 ¼ 2 ð 0 : 3070 Þ 0 : 3070 ¼ 0 : 1885

P 12 ¼ 2 c 11 c 21 ¼ 2 ð 0 : 3070 Þ 0 : 8100 ¼ 0 : 4973

P 22 ¼ 2 c 21 c 21 ¼ 2 ð 0 : 8100 Þ 0 : 8100 ¼ 1 : 3122

ð 5 : 136 Þ

Suppose our convergence criterion was that the elements ofPmust agree with
those of the previousPmatrix to within one part in 1,000. Comparing Eqs.5.136
with Eqs.5.123we see that this has not been achieved: even the smallest change is
|(1.312#1.503)/1.503|¼0.127, far above the required 0.001. Therefore another
SCF cycle is needed.

5.2 The Basic Principles of the ab initio Method 225