Biomedical Engineering Reference
In-Depth Information
Tabl e 7. 3
Degeneracy, energy, and symmetry of the ground states
of the isolated benzene molecule for different particle numbers. We
choose the on-site and inter-site Coulomb interactions to be
U
=
10
e
V,
V
5
e
V. Notice, however, that
screening effects from the leads and the dielectric are expected to
renormalize the energy of the benzene many-body states. From [
20
]
=
6
e
V, and the hopping to be
b
=
−
2
.
Energy[
e
V]
(at
Symmetry behavior
under
N
Degeneracy
ξ
=
0)
Symmetry
σ
v
01
0
A
1
g
sym
−
12
22
A
2
u
sym
−
.
21
42
25
A
1
g
sym
34
−
57
.
42
E
1
g
2 sym, [2 antisym]
4[ ]
[
−
68
.
87
]
[
A
2
g
]
[antisym]
2
−
68
.
37
E
2
g
1 sym, [1 antisym]
54
−
76
.
675
E
1
g
2 sym, [2 antisym]
61
−
81
.
725
A
1
g
sym
74
−
76
.
675
E
2
u
2 sym, [2 antisym]
8[ ]
[
−
68
.
87
]
[
A
2
g
]
[antisym]
2
−
68
.
37
E
2
g
1 sym, [1 antisym]
94
−
57
.
42
E
2
u
2 sym, [2 antisym]
10
1
−
42
.
25
A
1
g
sym
11
2
−
22
B
2
g
sym
12
1
0
A
1
g
sym
certain irreducible representation of the point group. Table
7.2
shows an overview of
the states of the neutral molecule (the 6-particle states) sorted by
S
z
and symmetries.
The eigenstates of the interacting benzene molecule have either
A
-,
B
-or
E
-type
symmetries. While orbitals having
A
or
B
symmetries can only be spin degenerate,
states with an
E
symmetry show an additional twofold orbital degeneracy, essential
for the explanation of the transport features occurring in the meta configuration.
Transport at low bias is described in terms of transitions between ground states
with different particle number. Table
7.3
shows the symmetries of the ground states
(and of some first excited states) of interacting benzene for all possible particle
numbers. Ground state transitions occur both between orbitally non-degenerate
states (with
A
and
B
symmetry), as well as between orbitally degenerate and non-
degenerate states (
E
-to
A
-type states).
The interacting benzene Hamiltonian commutes with all the symmetry opera-
tions of the
D
6
h
point group, thus it has a set of common eigenvectors with each
operation. The element of
D
6
h
of special interest for the
para
configuration is
σ
v
,
i.e., the reflection about the plane through the contact atoms and perpendicular to the
molecular plane. The molecular orbitals with
A
and
B
symmetry are eigenstates of
σ
v
with eigenvalue
±
1, i.e., they are either symmetric or antisymmetric with respect
to the
σ
v
is basis dependent,
yet one can always choose a basis in which one orbital is symmetric and the other
one antisymmetric.
σ
v
operation. The behavior of the
E
-type orbitals under
