14.2.2 The Singly Occupied Molecular Orbital and Spin
Density Distribution in Free Radicals of Conjugated
Systems
The other important theorem [25] is relative to the systems with an odd (2n + 1)
number of sites, i.e. concerns the free radicals. Let us consider that the graph
implies n + 1 red atoms and n blue atoms. Then the theorem tells that the (n + 1)th
eigenenergy is zero,
e n þ 1 ¼ 0
ð14:7Þ
and that the corresponding coefficients are zero on the minor (blue) color sites:
c n þ 1;q
0 ¼ 0:
ð14:8Þ
The SOMO exhibits nodes on minor (blue) atoms. The proof follows the same
logics. The non-zero coefficients on the major (red) sites satisfy the following
equations (one for each atom q′ of minor color):
X
p
t pq 0 c n þ 1;p ¼ 0;
ð14:9Þ
where p and q are bonded atoms and n + 1 is the number of the non-bonding MO. If
the molecule is neutral, the MO u n þ 1 is singly occupied (SOMO). The distribution
of the spin density is given by the squared coefficients of this MO on the various
atoms
q p = (c n þ 1;p Þ
2
ð14:10Þ
q q 0 ¼ 0:
ð14:11Þ
From the eigenequation H u n þ 1
¼ 0 or Eq. (14.9) one can easily determine the
coefficients of the SOMO and the spin densities. The relative amplitudes of the
SOMO on the different atoms may be obtained directly, on the back of an envelope.
The so-obtained SOMO is not normalized. The norm is the sum of the squares of
the amplitudes, and the atomic spin density is the square of the coefficient divided
by the norm. Giving for instance a coefficient 1 to the most isolated (less connected)
atom of major color, one satisfies Eq. (14.9) for each minor color atom q′ successively. Let us start with an even regular 1D chain (with equal hopping integrals
t) to which an external atom is attached through a weak bond of hopping integral τ.
This atom receives an arbitrary coefficient 1. Then atom 2 of the chain has a
coefficient c 2 = −τ/t. The amplitudes on the even numbered atom are the same all
along the chain c 2 = −c 4 = c 6 …. The external spin introduces a non-damped spin
wave in the chain, whatever the weakness of the hopping integral τ, as illustrated
below when τ/t = ½.
14 Magnetic Properties of Conjugated Hydrocarbons …
365
Density Distribution in Free Radicals of Conjugated
Systems
The other important theorem [25] is relative to the systems with an odd (2n + 1)
number of sites, i.e. concerns the free radicals. Let us consider that the graph
implies n + 1 red atoms and n blue atoms. Then the theorem tells that the (n + 1)th
eigenenergy is zero,
e n þ 1 ¼ 0
ð14:7Þ
and that the corresponding coefficients are zero on the minor (blue) color sites:
c n þ 1;q
0 ¼ 0:
ð14:8Þ
The SOMO exhibits nodes on minor (blue) atoms. The proof follows the same
logics. The non-zero coefficients on the major (red) sites satisfy the following
equations (one for each atom q′ of minor color):
X
p
t pq 0 c n þ 1;p ¼ 0;
ð14:9Þ
where p and q are bonded atoms and n + 1 is the number of the non-bonding MO. If
the molecule is neutral, the MO u n þ 1 is singly occupied (SOMO). The distribution
of the spin density is given by the squared coefficients of this MO on the various
atoms
q p = (c n þ 1;p Þ
2
ð14:10Þ
q q 0 ¼ 0:
ð14:11Þ
From the eigenequation H u n þ 1
¼ 0 or Eq. (14.9) one can easily determine the
coefficients of the SOMO and the spin densities. The relative amplitudes of the
SOMO on the different atoms may be obtained directly, on the back of an envelope.
The so-obtained SOMO is not normalized. The norm is the sum of the squares of
the amplitudes, and the atomic spin density is the square of the coefficient divided
by the norm. Giving for instance a coefficient 1 to the most isolated (less connected)
atom of major color, one satisfies Eq. (14.9) for each minor color atom q′ successively. Let us start with an even regular 1D chain (with equal hopping integrals
t) to which an external atom is attached through a weak bond of hopping integral τ.
This atom receives an arbitrary coefficient 1. Then atom 2 of the chain has a
coefficient c 2 = −τ/t. The amplitudes on the even numbered atom are the same all
along the chain c 2 = −c 4 = c 6 …. The external spin introduces a non-damped spin
wave in the chain, whatever the weakness of the hopping integral τ, as illustrated
below when τ/t = ½.
14 Magnetic Properties of Conjugated Hydrocarbons …
365