Computational Chemistry

and thus theP’s, change from SCF cycle to cycle. The change in the electron
repulsion matrixGcorresponds to that in the molecular wavefunction as thec’s
change (recall the LCAO expansion); it is the wavefunction (squared) which
represents the time-averaged electron distribution and thus the electron/charge
cloud repulsion (Sections 5.2.3.2,5.2.3.5and 5.2.3.6.2).
Step 5– TransformingFtoF^0 , the Fock matrix that satisfiesF^0 ¼C^0 eC^0 #^1
As in Section 4.4.1.2, we use the orthogonalizing matrixS#1/2(ofStep 3) to
transformFto a matrixF^0 which when diagonalized gives the energy levelseand a
coefficient matrixC^0 which is subsequently transformed to the matrixCof the
desiredc’s (see Section 5.2.3.6.2):

F^00 ¼

1 : 1163 # 0 : 3003

# 0 :3003 1: 1163



# 0 : 7511 # 0 : 9508

# 0 : 9508 # 1 : 3092



1 : 1163 # 0 : 3003

# 0 :3003 1: 1163



S#^1 =^2 F 0 S#^1 =^2

¼

# 0 : 4166 # 0 : 5799

# 0 : 5799 # 1 : 0617



F^00

ð 5 : 129 Þ

Step 6– DiagonalizingF^0 to obtain the energy level matrixeand a coefficient
matrixC^0

F^00 ¼

0 :5069 0: 8620

0 : 8620 # 0 : 5069



# 1 :4027 0: 0000

# 0 : 0000 # 0 : 0756



0 :5069 0: 8620

0 : 8620 # 0 : 5069



C^01 e 1 C^01 #^1

ð 5 : 130 Þ

The energy levels (the eigenvalues ofF^00 ) from this first SCF cycle are#1.4027 h
and#0.0756 h (h¼hartrees, the unit of energy in atomic units), corresponding to
the occupied MOc 1 and the unoccupied MO c 2. The MO coefficients (the
eigenvectors ofF^00 ) ofc 1 andc 2 ,for the transformed, orthonormal basis functions,
are, fromC^01 (actually hereC^01 and its inverse,C^01 #^1 are the same):

v^01 ¼

0 : 5069

0 : 8620



and v^02 ¼

0 : 8620

# 0 : 5069



ð 5 : 131 Þ

v^01 is the first column ofC^01 andv^02 is the second column ofC^01. These coefficients
are the weighting factors that with the transformed, orthonormal basis functions
give the MO’s:

c 1 ¼ 0 : 5069 f^01 þ 0 : 8620 f^02 and c 2 ¼ 0 : 8620 f^01 # 0 : 5069 f^02 ð 5 : 132 Þ

wheref^01 andf^02 are not our original basis functions, but ratherlinear combinations
of our original basis functionsf 1 andf 2. The original basis functionsfwere

224 5 Ab initio Calculations