A Toy Model of Chemical Reaction Networks - TBI - Universität Wien

3.6. FRONTIER MOLECULAR ORBITAL THEORY 33
3.6 Frontier Molecular Orbital Theory
It is natural to continue with the qualitative theory of Frontier Molecular
Orbital (FMO) theory after having used EHT for molecular property calculation.
From perturbation theory, the energy increment incurred by the
reactants A and B by interacting at the start of a reaction is the Klopman-
Salem formula [71, 99]
∆E = ∑
G ab + ∑ q(a)q(b)
ɛr ab
a∈A,b∈B a∈A,b∈B
( occ
)
∑
unocc
∑ ∑occ
unocc
∑
− F α,ζ + F α,ζ
α∈A ζ∈B α∈B ζ∈A
G ab = − ∑ ∑
(q i + q j )H ij S ij ,
i@a j@b
( ) 2
F α,ζ 2 ∑ ∑∑
∑
=
C α,i C ζ,j H ij ,
E ζ − E α
a∈A
i@a
b∈B
j@b
(3.16)
where r ab is the bond length, ɛ is the dielectric constant of the reaction
medium and α ∈ A and ζ ∈ B is an occupied and an unoccupied MO, respectively.
This increment is extrapolated in FMO theory from the initial
stage of the reaction to the transition state and may thus serve to approximate
the reaction rate. It can be derived to predict relative reactivities and
regioselectivity as described in [38]. The reactivity is then inversely proportional
to the difference of the HOMO and LUMO energies of the reactants.
The regioselectivity is determined by the MO coefficients at the reactive sites
i, such that ∑ i C HOMO,i C LUMO,i is maximal.
With the abbreviation
we obtain a four-point term
occ
F ab;a ′ b ′ = 2 ∑
W αζ
ab = ∑ i@a
α∈A
unocc
∑
ζ∈B
∑
C α,i C ζ,j H ij (3.17)
j@b
W αζ
ab W αζ
a ′ b ′
E ζ − E α
+
∑occ
α∈B
unocc
∑
ζ∈A
W αζ
ba W αζ
b ′ a ′
E ζ − E α
(3.18)
that allows us to write ∆E as an expansion of atom pairs and quadruples.
Within the approximation of the Toy Model all contributions (with the exception
of the Coulomb term) that do not belong to new bonds (or bonds