15 Molecular Theory of Graphene
253
Criterion 2
N D = 0.
Here, N D is the total number of effectively unpaired electrons. The number is determined as
N D = tr D
r|r
= 0 and N D =
A
D A .
(15.2)
Here, D(r|r ) [12] and D A [13] present the total and atom-fractioned spin density
caused by the spin asymmetry due to the location of electrons with different spins
in different spaces.
Criterion 3
ˆ
S
2
≥ 0.
(15.3)
Here,
ˆ
S
2
= ˆ
S
2
U − S(S + 1)
presents the misalignment of squared spin. ˆ
S 2
U is the squared spin calculated within
the applied unrestricted technique while S(S + 1) presents the exact value of ˆ
S 2 .
Criterion 1 follows from the well known fact that the electron correlation, if available, lowers the total energy [14]. Criterion 2 highlights the fact that the electron correlation is accompanied with the appearance of effectively unpaired electrons that
provide the molecule radicalization [12, 13, 15]. Those electrons total number N D
depends on interatomic distance: when the latter is under a critical value R crit
cov , two
adjacent electrons are covalently bound and N D = 0. However, when the distance
exceeds R crit
cov , the two electrons become unpaired; N D ≥ 0, the more, the larger is
the interatomic spacing. In the case of the sp 2 C–C bonds, R crit
cov = 1.395 Å [16].
Criterion 3 is the manifestation of the spin contamination of unrestricted singledeterminant solutions [13, 15]; the stronger electron correlation, the bigger spin
contamination of the studied spin state.
Table 15.1 presents sets of the three parameters evaluated for a number of
graphene molecules presented by rectangular (n a , n z ) fragments of graphene (n a
and n z count the benzenoid units along armchair and zigzag edges of the fragment,
respectively [19]), (n a , n z ) nanographenes (NGrs) below, by using the AM1 version
of the semiempirical UHF approach implemented in the CLUSTER-Z1 codes [18].
To our knowledge, only this software allows for getting all the above three parameters within one computing session. As seen in the table, the parameters are certainly
not zero, obviously greatly depending on the fragment size while their relative values are practically non size-dependent. The attention should be called to rather large
N D values, both absolute and relative. The finding evidences that the length of C–
C bonds in the considered molecules exceed the critical value R crit
cov = 1.395 Å. It
should be added as well that the relation N D = 2 ˆ
S 2
U , which is characteristic for
253
Criterion 2
N D = 0.
Here, N D is the total number of effectively unpaired electrons. The number is determined as
N D = tr D
r|r
= 0 and N D =
A
D A .
(15.2)
Here, D(r|r ) [12] and D A [13] present the total and atom-fractioned spin density
caused by the spin asymmetry due to the location of electrons with different spins
in different spaces.
Criterion 3
ˆ
S
2
≥ 0.
(15.3)
Here,
ˆ
S
2
= ˆ
S
2
U − S(S + 1)
presents the misalignment of squared spin. ˆ
S 2
U is the squared spin calculated within
the applied unrestricted technique while S(S + 1) presents the exact value of ˆ
S 2 .
Criterion 1 follows from the well known fact that the electron correlation, if available, lowers the total energy [14]. Criterion 2 highlights the fact that the electron correlation is accompanied with the appearance of effectively unpaired electrons that
provide the molecule radicalization [12, 13, 15]. Those electrons total number N D
depends on interatomic distance: when the latter is under a critical value R crit
cov , two
adjacent electrons are covalently bound and N D = 0. However, when the distance
exceeds R crit
cov , the two electrons become unpaired; N D ≥ 0, the more, the larger is
the interatomic spacing. In the case of the sp 2 C–C bonds, R crit
cov = 1.395 Å [16].
Criterion 3 is the manifestation of the spin contamination of unrestricted singledeterminant solutions [13, 15]; the stronger electron correlation, the bigger spin
contamination of the studied spin state.
Table 15.1 presents sets of the three parameters evaluated for a number of
graphene molecules presented by rectangular (n a , n z ) fragments of graphene (n a
and n z count the benzenoid units along armchair and zigzag edges of the fragment,
respectively [19]), (n a , n z ) nanographenes (NGrs) below, by using the AM1 version
of the semiempirical UHF approach implemented in the CLUSTER-Z1 codes [18].
To our knowledge, only this software allows for getting all the above three parameters within one computing session. As seen in the table, the parameters are certainly
not zero, obviously greatly depending on the fragment size while their relative values are practically non size-dependent. The attention should be called to rather large
N D values, both absolute and relative. The finding evidences that the length of C–
C bonds in the considered molecules exceed the critical value R crit
cov = 1.395 Å. It
should be added as well that the relation N D = 2 ˆ
S 2
U , which is characteristic for