Conductance Length Dependence in Carbon-Conjugated Nanoscale Systems …
589
Now for the retarded GF at the Fermi energy let us consider the customary approximations along with the schemes elaborated recently (for more detail see Appendices).
They all can be encompassed by a general one-electron resolvent of the form:
G
X
0 = (i0
+
− h
X
)
−1
,
(2)
where X specifies the used model and the respective effective Hamiltonian h
X . The
occurrence of the imaginary infinitesimal i0
+ is not essential for our purposes here.
More specifically, take the simplest known model, i.e. the standard TB approximation
for π -electrons. As usual, the TB Hamiltonian matrix, h
TB , is identified (up to a factor)
with an adjacency matrix of the corresponding graph (it is in fact the Hückel energy
matrix). Then
G
TB
0 = (i0
+
− h
TB
)
−1
(3)
is a suitable working formula having a huge body of applications. The merits and
demerits of the Hückel π -electron approximation are well discussed in literature
([20] and many others).
The characteristic drawback of TB is neglecting direct long-distance electronic
effects (see, e.g., [19, 21]). One can partially avoid this difficulty by using the
restricted Hartree-Fock (RHF) scheme:
h
RHF
= f,
(4)
with f being the usual Fock matrix which is computed easily within the conventional
Pariser-Parr-Pople approximation. Additionally, it is suitable to shift all orbital energy
in (4) in such a way that the Fermi energy becomes a zero reference point. The
shifted and renormalized h
RHF matrix is further used for evaluating G
RHF
0
and for
other models. For more detail see Appendices; in particular, the renormalization of
h
RHF is made by factor 1/β eff where β eff is written in (23), and the energy shift value
is identified with the Fermi level given by (30).
However, do not forget that RHF can suffer from serious disadvantages (RHF
instability and ignoring electron correlation) which are particularly significant for
large-scale problems. At the same time, a very simple way to improve TB model has
been was proposed [22, 23] in the form of a quasi-correlated TB (QCTB) scheme.
As shown in [18], in QCTB we in fact replace h
TB by the formal Hamiltonian
h
QCTB
⇒ [(h
TB
)
2
+ δ
2
]/ h
TB
,
(5)
589
Now for the retarded GF at the Fermi energy let us consider the customary approximations along with the schemes elaborated recently (for more detail see Appendices).
They all can be encompassed by a general one-electron resolvent of the form:
G
X
0 = (i0
+
− h
X
)
−1
,
(2)
where X specifies the used model and the respective effective Hamiltonian h
X . The
occurrence of the imaginary infinitesimal i0
+ is not essential for our purposes here.
More specifically, take the simplest known model, i.e. the standard TB approximation
for π -electrons. As usual, the TB Hamiltonian matrix, h
TB , is identified (up to a factor)
with an adjacency matrix of the corresponding graph (it is in fact the Hückel energy
matrix). Then
G
TB
0 = (i0
+
− h
TB
)
−1
(3)
is a suitable working formula having a huge body of applications. The merits and
demerits of the Hückel π -electron approximation are well discussed in literature
([20] and many others).
The characteristic drawback of TB is neglecting direct long-distance electronic
effects (see, e.g., [19, 21]). One can partially avoid this difficulty by using the
restricted Hartree-Fock (RHF) scheme:
h
RHF
= f,
(4)
with f being the usual Fock matrix which is computed easily within the conventional
Pariser-Parr-Pople approximation. Additionally, it is suitable to shift all orbital energy
in (4) in such a way that the Fermi energy becomes a zero reference point. The
shifted and renormalized h
RHF matrix is further used for evaluating G
RHF
0
and for
other models. For more detail see Appendices; in particular, the renormalization of
h
RHF is made by factor 1/β eff where β eff is written in (23), and the energy shift value
is identified with the Fermi level given by (30).
However, do not forget that RHF can suffer from serious disadvantages (RHF
instability and ignoring electron correlation) which are particularly significant for
large-scale problems. At the same time, a very simple way to improve TB model has
been was proposed [22, 23] in the form of a quasi-correlated TB (QCTB) scheme.
As shown in [18], in QCTB we in fact replace h
TB by the formal Hamiltonian
h
QCTB
⇒ [(h
TB
)
2
+ δ
2
]/ h
TB
,
(5)