602
A. V. Luzanov
Appendix B: Rough Estimations and Correlations Between
Long-Range GF Elements and Bond Orders
Before discussing the estimations and statistical correlations let us consider the
behavior of the full G = G (E) near E = E F . As usual, essential contributions
arise due to frontier orbitals (HOMO and LUMO). But we first look at the problem
being approached from a general GF theory. Let us take a spectral representation of
GF (see (5.76) in [9] or (22.A.2) in [19]) and write down the contribution, G, arising
from lowest electron-detached and electron-attached states of the initial N-electron
system with its singlet ground-state energy E N :
G (E) = (E + I 0 )
−1
|II| + (E + A 0 )
−1
|AA| ..
(28)
Here I 0 = E N −1 − E N and A 0 = E N − E N +1 are the lowest ionization potential
and electron affinity; |I, |A are the associated spin-free Dyson orbitals for ionization
and attachment processes, respectively. In the case of TB these orbitals are exactly
the frontier MOs. By analogy we name orbitals |I and |A as the frontier Dyson
orbitals.
Now recall the Hush-Pople-McLachlan theorem [52, 53] for the chemical potential
(in fact the Fermi energy) E F . The theorem is valid for bipartite π -system within PPP
theory [54]. It states that E F which can be defined as
E F = −(I 0 + A 0 )/2,
(29)
is a constant for all bipartite:
E F = W C + γ C /2.
(30)
Here W C is the standard valence state ionization potential for the C atom, and
γ C is the corresponding Coulomb one-center integral (the both quantities are taken
as semiempirical parameters). Then, combining (29) with (17) gives the following:
I 0 = c /2 − E F and A 0 = − c /2 − E F , so that (28) becomes
G (E F ) = 2 (|II| − |AA| )// c ,
(31)
where c is defined by (17). From (31) we can obtain
G p, p ≡ G p, p (E F ) = p| G (E F )
p
,
(32)
that is the contribution to G p, p from the frontier Dyson orbitals. Here | p and
p
are the atomic π -orbitals of sites p and p
, respectively. When simplifying (32) one
must keep in mind that scalar products p | I,
p
I
, etc. are expansion coefficients
of the Dyson orbitals. For them the same sign rule as for the usual TB orbitals can
be proven [25] in the case of bipartite π -structures. It comes to the equality of the I
A. V. Luzanov
Appendix B: Rough Estimations and Correlations Between
Long-Range GF Elements and Bond Orders
Before discussing the estimations and statistical correlations let us consider the
behavior of the full G = G (E) near E = E F . As usual, essential contributions
arise due to frontier orbitals (HOMO and LUMO). But we first look at the problem
being approached from a general GF theory. Let us take a spectral representation of
GF (see (5.76) in [9] or (22.A.2) in [19]) and write down the contribution, G, arising
from lowest electron-detached and electron-attached states of the initial N-electron
system with its singlet ground-state energy E N :
G (E) = (E + I 0 )
−1
|II| + (E + A 0 )
−1
|AA| ..
(28)
Here I 0 = E N −1 − E N and A 0 = E N − E N +1 are the lowest ionization potential
and electron affinity; |I, |A are the associated spin-free Dyson orbitals for ionization
and attachment processes, respectively. In the case of TB these orbitals are exactly
the frontier MOs. By analogy we name orbitals |I and |A as the frontier Dyson
orbitals.
Now recall the Hush-Pople-McLachlan theorem [52, 53] for the chemical potential
(in fact the Fermi energy) E F . The theorem is valid for bipartite π -system within PPP
theory [54]. It states that E F which can be defined as
E F = −(I 0 + A 0 )/2,
(29)
is a constant for all bipartite:
E F = W C + γ C /2.
(30)
Here W C is the standard valence state ionization potential for the C atom, and
γ C is the corresponding Coulomb one-center integral (the both quantities are taken
as semiempirical parameters). Then, combining (29) with (17) gives the following:
I 0 = c /2 − E F and A 0 = − c /2 − E F , so that (28) becomes
G (E F ) = 2 (|II| − |AA| )// c ,
(31)
where c is defined by (17). From (31) we can obtain
G p, p ≡ G p, p (E F ) = p| G (E F )
p
,
(32)
that is the contribution to G p, p from the frontier Dyson orbitals. Here | p and
p
are the atomic π -orbitals of sites p and p
, respectively. When simplifying (32) one
must keep in mind that scalar products p | I,
p
I
, etc. are expansion coefficients
of the Dyson orbitals. For them the same sign rule as for the usual TB orbitals can
be proven [25] in the case of bipartite π -structures. It comes to the equality of the I