4.9 Application: Hückel Theory
85
to transform the atomic basis into SALCs according to the irreps of the point group.
Molecular orbitals have a definite irrep and component symmetry and thus contain
only SALCs with these same symmetry characteristics. Transforming the atomic
basis into SALCs will reduce the Hamiltonian matrix to a set of smaller symmetry
blocks. From the eigenfunctions one can determine π -contributions to properties
such as the on-site atomic population, q r , and the inter-site π -bond order, p rs .The
population of atom r and the bond order of the bond between atoms r and s are
given by
q r =
k
n k |c
k
r |
2
p rs =
k
n k c k
r c
k
s
(4.112)
Here, the index k runs over the eigenfunctions: n k is the occupation number (0,1,
or 2) of the kth eigenlevel, and c k
r is the coefficient of the |φ r atomic orbital for
the normalized eigenfunction. The atomic populations are simply the densities or
weights at the atomic sites and may vary between 0 and 2. The neutral atom has a
population of one p z -electron, and sites with q r < 1 are cationic and with q r > 1
anionic. The bond order adopts the form of a correlation coefficient between two
sites. The π -bond order for a full π -bond in ethylene is equal to 1, and for benzene,
it is 2/3. Below we shall examine in detail some special cases where symmetry
plays an important role.
Cyclic Polyenes
Cyclic polyenes, also known as annulenes, are hydrocarbon rings, C n H n . Each carbon atom contributes one p z -orbital, perpendicular to the plane of the ring, which
gives rise to conjugated π -bonding. The prototype is the aromatic molecule benzene. The adjacency matrix has the form of a circulant matrix. This is a matrix
where each row is rotated one element to the right relative to the preceding row.
Because each atom is linked to only two neighbors, each row contains only two
elements. These are arranged left and right of the diagonal, which is characteristic
for a chain, but with additional nonzero elements in the upper right and lower left
corners, where both ends of the chain meet. For benzene, it is given by
A =
⎛
⎜
⎜
⎜
⎜
⎜
⎜
⎝
010001
101000
010100
001010
000101
100010
⎞
⎟
⎟
⎟
⎟
⎟
⎟
⎠
(4.113)
85
to transform the atomic basis into SALCs according to the irreps of the point group.
Molecular orbitals have a definite irrep and component symmetry and thus contain
only SALCs with these same symmetry characteristics. Transforming the atomic
basis into SALCs will reduce the Hamiltonian matrix to a set of smaller symmetry
blocks. From the eigenfunctions one can determine π -contributions to properties
such as the on-site atomic population, q r , and the inter-site π -bond order, p rs .The
population of atom r and the bond order of the bond between atoms r and s are
given by
q r =
k
n k |c
k
r |
2
p rs =
k
n k c k
r c
k
s
(4.112)
Here, the index k runs over the eigenfunctions: n k is the occupation number (0,1,
or 2) of the kth eigenlevel, and c k
r is the coefficient of the |φ r atomic orbital for
the normalized eigenfunction. The atomic populations are simply the densities or
weights at the atomic sites and may vary between 0 and 2. The neutral atom has a
population of one p z -electron, and sites with q r < 1 are cationic and with q r > 1
anionic. The bond order adopts the form of a correlation coefficient between two
sites. The π -bond order for a full π -bond in ethylene is equal to 1, and for benzene,
it is 2/3. Below we shall examine in detail some special cases where symmetry
plays an important role.
Cyclic Polyenes
Cyclic polyenes, also known as annulenes, are hydrocarbon rings, C n H n . Each carbon atom contributes one p z -orbital, perpendicular to the plane of the ring, which
gives rise to conjugated π -bonding. The prototype is the aromatic molecule benzene. The adjacency matrix has the form of a circulant matrix. This is a matrix
where each row is rotated one element to the right relative to the preceding row.
Because each atom is linked to only two neighbors, each row contains only two
elements. These are arranged left and right of the diagonal, which is characteristic
for a chain, but with additional nonzero elements in the upper right and lower left
corners, where both ends of the chain meet. For benzene, it is given by
A =
⎛
⎜
⎜
⎜
⎜
⎜
⎜
⎝
010001
101000
010100
001010
000101
100010
⎞
⎟
⎟
⎟
⎟
⎟
⎟
⎠
(4.113)