Eq. (12) that the laser intensity dependence in the electron angular momentum is
involved in the imaginary part of the off-diagonal density matrices, Imq ba ðtÞ.
An electric current passing through a surface S at time t is generally defined as
JðtÞ
h
i
Z
S
d
2 r ? ~ n ? Á WðtÞj ^ Jð~ rÞjWðtÞ
;
ð13Þ
where ~ n ? is a unit vector perpendicular to a surface S,
^ Jð~ rÞ ¼
e h
2m e i
~
r À r
;
ð14Þ
is the current density operator. Here, ~
rðr Þ denotes the nabra operating the atomic
orbital on the right-hand (left-hand) side.
This can be expressed as
JðtÞ
h
i ¼
2ne h
m e
X
a\b
Imq ba ðtÞ
X
ij
d ab c
Ã
a 0 i c b 0 j þ d a 0 b 0 c
Ã
ai c bj
À
Á
Z
S
d
2 r ? v
Ã
i ~ n ? Á ~
rv j
X
ij
JðtÞ
h
i ij
ð15Þ
Here, surface integration in Eq. (13) is carried out over a half-plane S.
Since the L and R rings are not round but consist of nonequivalent C–C bonds,
we introduce the bond current JðtÞ
h
i ij from the nearest neighbor atoms at sites j to i,
which is given in terms of inter-atomic bond current (IABC) J
IABC
ij
as
JðtÞ
h
i ij ¼
2ne h
m e
X
a\b
Imq ba ðtÞ d ab c
Ã
a 0 i c b 0 j À c
Ã
b 0 i c a 0 j
À
Á þ d a 0 b 0 c
Ã
ai c bj À c
Ã
bi c aj
À
Á
À
Á
J
IABC
ij
;
ð16Þ
where
J
IABC
ij
¼
Z
S
d
2 r ? v
Ã
i ~ n ? Á ~
rv j J
IABCðS at centerÞ
ij
:
ð17Þ
Here, surface S is set to be perpendicular to the C i and C j bond at the center, and
~ n ? ¼
~ r i À~ r j
~ r i À~ r j
j j
.
164
H. Mineo and Y. Fujimura