Conductance Length Dependence in Carbon-Conjugated Nanoscale Systems …
591
conductivity. Recall that each cumulene molecule with even number of C atoms
contains two, short and long, orthogonal π -subsystems, and for the conductivity
problem one can take into consideration only the long one which will be symbolized
by (-C = C- ) n . In the cumulenes a weak interaction between the subsystems can be
safely neglected. In [13] the focus is on alternation of resonance integrals, and two
types of their alternation were investigated, the normal bond alternation (NBA) and
reverse bond alternation (RBA). In the NBA case ‘double bond’ resonance integral
is β c=c = β 0 , and that of ‘simple’ bond is β c−c = 0.935β 0 with β 0 = −3.0 eV; for
RBA we must take β c=c = 0.935β 0 and β c−c = β 0 . In this section we make using this
TB parametric scheme of [13], and to the TB results of the cited work we will add
appropriate data from the more advanced π -models (RHF, EQC, HPHF, and FCI).
For linear n-polyene-like chains with n = 1 ÷ 7, the obtained results are displayed
numerically in Tables 1 and 2, and graphically in Fig. 1.
It can be seen from the tables that, if excluding the TB scheme, the approximate
(relative to FCI) GF results, particularly the HPHF ones, are very close to those of
the FCI gold standard. The principal outcome from these calculations is that with
increasing the polyene-like chain (n in the tables) no increase of GF (and thereby
Table 1 Comparison of π -electron GF element (G X
0 ) 1,2n for various models (X = FCI, HPHF,
EQC, RHF, and TB) in n-polyene-like chains with NBA
n
FCI
HPHF
EQC
RHF
TB
2
−0.557
−0.575
−0.564
−0.606
−0.935
3
0.369
0.383
0.382
0.395
0.874
4
−0.256
−0.264
−0.270
−0.264
−0.817
5
0.182
0.186
0.196
0.180
0.764
6
−0.131
−0.132
−0.145
−0.123
−0.715
7
0.096
0.094
0.109
0.085
0.668
Table 2 Comparison of π -electron GF element (G X
0 ) 1,2n for various models (X = FCI, HPHF,
EQC, RHF, and TB) in n-polyene-like chains with RBA
n
FCI
HPHF
EQC
RHF
TB
2
−0.632
−0.647
−0.645
−0.699
−1.144
3
0.470
0.481
0.489
0.516
1.223
4
−0.370
−0.374
−0.389
−0.396
−1.308
5
0.300
0.300
0.317
0.312
1.399
6
−0.249
−0.245
−0.262
−0.250
−1.497
7
0.209
0.202
0.216
0.202
1.601
591
conductivity. Recall that each cumulene molecule with even number of C atoms
contains two, short and long, orthogonal π -subsystems, and for the conductivity
problem one can take into consideration only the long one which will be symbolized
by (-C = C- ) n . In the cumulenes a weak interaction between the subsystems can be
safely neglected. In [13] the focus is on alternation of resonance integrals, and two
types of their alternation were investigated, the normal bond alternation (NBA) and
reverse bond alternation (RBA). In the NBA case ‘double bond’ resonance integral
is β c=c = β 0 , and that of ‘simple’ bond is β c−c = 0.935β 0 with β 0 = −3.0 eV; for
RBA we must take β c=c = 0.935β 0 and β c−c = β 0 . In this section we make using this
TB parametric scheme of [13], and to the TB results of the cited work we will add
appropriate data from the more advanced π -models (RHF, EQC, HPHF, and FCI).
For linear n-polyene-like chains with n = 1 ÷ 7, the obtained results are displayed
numerically in Tables 1 and 2, and graphically in Fig. 1.
It can be seen from the tables that, if excluding the TB scheme, the approximate
(relative to FCI) GF results, particularly the HPHF ones, are very close to those of
the FCI gold standard. The principal outcome from these calculations is that with
increasing the polyene-like chain (n in the tables) no increase of GF (and thereby
Table 1 Comparison of π -electron GF element (G X
0 ) 1,2n for various models (X = FCI, HPHF,
EQC, RHF, and TB) in n-polyene-like chains with NBA
n
FCI
HPHF
EQC
RHF
TB
2
−0.557
−0.575
−0.564
−0.606
−0.935
3
0.369
0.383
0.382
0.395
0.874
4
−0.256
−0.264
−0.270
−0.264
−0.817
5
0.182
0.186
0.196
0.180
0.764
6
−0.131
−0.132
−0.145
−0.123
−0.715
7
0.096
0.094
0.109
0.085
0.668
Table 2 Comparison of π -electron GF element (G X
0 ) 1,2n for various models (X = FCI, HPHF,
EQC, RHF, and TB) in n-polyene-like chains with RBA
n
FCI
HPHF
EQC
RHF
TB
2
−0.632
−0.647
−0.645
−0.699
−1.144
3
0.470
0.481
0.489
0.516
1.223
4
−0.370
−0.374
−0.389
−0.396
−1.308
5
0.300
0.300
0.317
0.312
1.399
6
−0.249
−0.245
−0.262
−0.250
−1.497
7
0.209
0.202
0.216
0.202
1.601