Biomedical Engineering Reference
In-Depth Information
2
d
0
y
+
id
0
x
(2
n
+
1)
1
G
mn
(
r
0
,
d
0
)
=
n
(
n
+
1
)
b
m
−
1,
n
(
r
0
)
2
d
0
y
−
id
0
x
(2
n
+
1)
1
−
n
(
n
+
1
)
(
n
−
m
)(
n
+
m
+
1)
b
m
+
1,
n
(
r
0
)
+
id
0
z
(2
n
+
1)
n
(
n
+
1)
mb
mn
(
r
0
),
(4.B.5)
where
(
n
−
m
)!
(
n
+
m
)!
ζ
n
(
k
0
r
0
)
P
n
(cos
θ
0
)
e
−
im
ϕ
0
.
b
mn
(
r
0
)
=
(4.B.6)
k
0
r
0
ϕ
0
arethesphericalcoordinates
of the position of a chiral molecule, and the function
ζ
n
(
k
0
r
0
) is
defined above [see (4.B.2)]. In the case of
m
0
=
0, the expressions
(4.B.3) coincide with the known results [14]. Explicit expressions
forthefunctions
F
mn
(
r
0
,
−
i
m
0
)and
G
mn
(
r
0
,
−
i
m
0
)canbeobtained
from (4.B.4) and (4.B.5) by changing all the Cartesian components
of the vector
d
0
to corresponding components of the vector
−
i
m
0
.
In a special case of a chiral molecule located on the
z
−
axis, one can
obtain (
θ
0
=
0or
θ
0
=
π
)
θ
0
,
In(4.B.4),(4.B.5),and(4.B.6),
r
0
,
2
δ
m
1
d
0
x
−
id
0
y
−
δ
m
,
−
1
d
0
x
+
id
0
y
n
(
n
+
1)
1
F
mn
(
r
0
,
d
0
)
=−
n
(
n
+
1)
ζ
n
(
k
0
r
0
)
(2
n
+
1)
×
P
n
+
1
(cos
θ
0
)
k
0
r
0
1)
ζ
n
(
k
0
r
0
)
+
δ
m
0
d
0
z
(2
n
+
(
k
0
r
0
)
2
P
n
+
1
(cos
θ
0
),
(4.B.7)
2
δ
m
1
d
0
y
+
id
0
x
−
δ
m
,
−
1
d
0
y
−
id
0
x
n
(
n
+
1)
×
1
G
mn
(
r
0
,
d
0
)
=
(2
n
+
1)
n
(
n
1)
ζ
n
(
k
0
r
0
)
P
n
(cos
θ
0
),
(4.B.8)
+
k
0
r
0
where the prime near the function means its derivative, and
δ
mp
is
the Kronecker's delta.
The vector spherical harmonics
N
ζ
mn
and
M
ζ
mn
are defined
in (4.B.2), and the coe
cients
C
(0)
mn
and
D
(0)
mn
can be obtained
from the expressions for coe
cients
A
(0)
mn
and
B
(0)
mn
[see (4.B.3)],
correspondingly, by changing
b
mn
to
(
n
−
m
)!
(
n
+
m
)!
ψ
n
(
k
0
r
0
)
P
n
(cos
θ
0
)
e
−
im
ϕ
0
,
c
mn
(
r
0
)
=
(4.B.9)
k
0
r
0
where the function
ψ
n
(
k
0
r
0
) is defined above [(see (4.B.1)].
