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312 9MOLECULAR STRUCTURE
As explained in the text, in order to �nd the energies it is su�cient to diagonal-
ize the matrix on the right, and this is convenient because most mathematical
so�ware packages are only able to diagonalize numerical matrices.�e matrix
which achieves the diagonalization is
( 0.707 0.707
0.707 −0.707 )
and the diagonal elements are ±1.00, implying that the energies are α ± 1.00 β.
�e secular determinant and hamiltonian matrix for butadiene are
RRRRRRRRRRRRRRRRRR
α − E β 0 0
β α − E β 0
0 β α − E β
0 0 β α − E
RRRRRRRRRRRRRRRRRR
H = α1 + β
⎛
⎜⎜⎜
⎝
0 1 0 0
1 0 1 0
0 1 0 1
0 0 1 0
⎞
⎟⎟⎟
⎠
�e matrix which achieves the diagonalization is
⎛
⎜⎜⎜
⎝
0.372 0.602 0.602 0.372
0.602 0.372 −0.372 −0.602
0.602 −0.372 −0.372 0.602
0.372 −0.602 0.602 −0.372
⎞
⎟⎟⎟
⎠
�e diagonal elements are ±1.62, and ±0.618, implying that the energies are
α ± 1.62 β and α ± 0.618 β.
�e secular determinant and hamiltonian matrix for hexatriene are
RRRRRRRRRRRRRRRRRRRRRRRRRRR
α − E β 0 0 0 0
β α − E β 0 0 0
0 β α − E β 0 0
0 0 β α − E β 0
0 0 0 β α − E β
0 0 0 0 β α − E
RRRRRRRRRRRRRRRRRRRRRRRRRRR
H = α1 + β
⎛
⎜⎜⎜⎜⎜⎜⎜⎜
⎝
0 1 0 0 0 0
1 0 1 0 0 0
0 1 0 1 0 0
0 0 1 0 1 0
0 0 0 1 0 1
0 0 0 0 1 0
⎞
⎟⎟⎟⎟⎟⎟⎟⎟
⎠
�is is diagonalized by the matrix
⎛
⎜⎜⎜⎜⎜⎜⎜⎜
⎝
0.232 0.418 0.521 0.521 0.418 0.232
0.418 0.521 0.232 −0.232 −0.521 −0.418
0.521 0.232 −0.418 −0.418 0.232 0.521
0.521 −0.232 −0.418 0.418 0.232 −0.521
0.418 −0.521 0.232 0.232 −0.521 0.418
0.232 −0.418 0.521 −0.521 0.418 −0.232
⎞
⎟⎟⎟⎟⎟⎟⎟⎟
⎠
�e diagonal elements are ±1.80, ±1.25, and ±0.445, implying that the energies
are α ± 1.80 β, α ± 1.25 β, and α ± 0.445 β