Computational Chemistry

(Steven Felgate) #1

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

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