The computer algorithms for matrix diagonalization use some version of the
Jacobi rotation method [ 37 ], which proceeds by successive numerical approxima-
tions (textbooks describe a diagonalization method based on expanding the deter-
minant corresponding to the matrix; this is not used in computational chemistry).
Therefore in order to diagonalize our Fock matrices we need numbers in place ofa
andb. In methods more advanced than the SHM, like the extended H€uckel method
(EHM), other semiempirical methods, and ab initio methods, theHijintegrals are
calculated to give numerical (in energy units) values. In the SHM we simply use
energy values in |b| units relative toa(recall thatbis a negative quantity: Fig.4.13).
The matrix of Fig.4.14athen becomes
H¼
ab
ba¼
0 " 1
" 10
(4.63)
An electron in an MO represented by a 1,2-type interaction is lower in energy
than one in a p orbital (1,1-type interaction) by one |b| energy unit. Similarly, theH
matrix of Fig.4.14bbecomes
H¼
0 " 10
" 10 " 1
0 " 10
0
B
@
1
C
A (4.64)
and theHmatrix of Fig.4.14cbecomes
H¼
0 " 10 " 1
" 10 " 10
0 " 10 " 1
" 10 " 10
0
B
B
@
1
C
C
A (4.65)
TheHmatrices can be written down simply by setting alli, i-type interactions
equal to 0, and alli, j-type interactions equal to"1 whereiandjrefer to atoms that
are bonded together, and equal to 0 wheniandjrefer to atoms that are not bonded
together.
Diagonalization of the two-basis-function matrix of Eq.4.63gives
H¼
0 " 1
" 10
¼
0 :707 0: 707
0 : 707 " 0 : 707
" 10
01
0 :707 0: 707
0 : 707 " 0 : 707
C e C"^1
(4.66)Comparing Eq.4.66with Eq.4.60, we see that we have obtained the matrices we
want: the coefficients matrixCand the MO energy levels matrix«. The columns of
4.3 The Application of the Schr€odinger Equation to Chemistry by H€uckel 129