60
M. Senami and A. Fukushima
α
ij
V (t) =
1
V
V
ˆ
α
ij (x)dr.
(3.14)
The quantity of operator sandwiched by , , which are state vectors, bra and ket,
means the expectation value. This definition of the average is reasonable. If the
region V is capacitor and the region M is parallel plates, this average corresponds to
the polarization induced by the parallel plates over D. The average of local dielectric
constant is defined as
ij
V (t) =
1
V
V
ˆ
ij (x)dr.
(3.15)
The local dielectric constant and local polarizability are interesting physical
quantities by themselves, and moreover, these quantities are important for the
representation of the relation between other quantities. In this article, we introduce
one example, the relation between local conductivities mentioned in the previous
section, and we refer Ref. [25] as another relation, where the variation of Lorentz
force density by the change of electric field is given with local dielectric constant.
Ordinary electrical conductivity is the linear response of the electric current to
electric field between electrodes. A local counterpart of conductivity is the matrix
relation of the electric current density of region A to electric displacement of region
M. This definition of local conductivity is given as [15, 17, 18]
ˆ
j
i (x) = ˆ
σ
ij
ext (x) ˆ
D
j (x).
(3.16)
This definition of conductivity corresponds to ordinary conductivity as an average
value discussed above. In another definition, we can consider the relation between
the local electric current and the electric field E,
ˆ
j
i (x) = ˆ
σ
ij
int (x) ˆ
E
j (x).
(3.17)
These two definition are related through the local dielectric constant as,
ˆ
j
i (x) = ˆ
σ
ij
ext (x) ˆ
D
j (x)
= ˆ
σ
ij
ext (x)ˆ
jk (x) ˆ
E
k (x)
= ˆ
σ
ij
int (x) ˆ
E
j (x).
(3.18)
Before we introduce works using local dielectric constant, some limitation in
these works are mentioned. First, the time variation of vector potential in estimate
of E, D, and P is neglected. Only steady states are considered in the works,
and it has been reported that effects of vector potential can be negligible even
for conductive states where wave packets are prepared by computations based on
quantum mechanics [26]. In following sections, system region A is chosen to be
a whole cluster model, and electric displacement D(x) is simply assumed to be
M. Senami and A. Fukushima
α
ij
V (t) =
1
V
V
ˆ
α
ij (x)dr.
(3.14)
The quantity of operator sandwiched by , , which are state vectors, bra and ket,
means the expectation value. This definition of the average is reasonable. If the
region V is capacitor and the region M is parallel plates, this average corresponds to
the polarization induced by the parallel plates over D. The average of local dielectric
constant is defined as
ij
V (t) =
1
V
V
ˆ
ij (x)dr.
(3.15)
The local dielectric constant and local polarizability are interesting physical
quantities by themselves, and moreover, these quantities are important for the
representation of the relation between other quantities. In this article, we introduce
one example, the relation between local conductivities mentioned in the previous
section, and we refer Ref. [25] as another relation, where the variation of Lorentz
force density by the change of electric field is given with local dielectric constant.
Ordinary electrical conductivity is the linear response of the electric current to
electric field between electrodes. A local counterpart of conductivity is the matrix
relation of the electric current density of region A to electric displacement of region
M. This definition of local conductivity is given as [15, 17, 18]
ˆ
j
i (x) = ˆ
σ
ij
ext (x) ˆ
D
j (x).
(3.16)
This definition of conductivity corresponds to ordinary conductivity as an average
value discussed above. In another definition, we can consider the relation between
the local electric current and the electric field E,
ˆ
j
i (x) = ˆ
σ
ij
int (x) ˆ
E
j (x).
(3.17)
These two definition are related through the local dielectric constant as,
ˆ
j
i (x) = ˆ
σ
ij
ext (x) ˆ
D
j (x)
= ˆ
σ
ij
ext (x)ˆ
jk (x) ˆ
E
k (x)
= ˆ
σ
ij
int (x) ˆ
E
j (x).
(3.18)
Before we introduce works using local dielectric constant, some limitation in
these works are mentioned. First, the time variation of vector potential in estimate
of E, D, and P is neglected. Only steady states are considered in the works,
and it has been reported that effects of vector potential can be negligible even
for conductive states where wave packets are prepared by computations based on
quantum mechanics [26]. In following sections, system region A is chosen to be
a whole cluster model, and electric displacement D(x) is simply assumed to be