A.2 Levi-Civita Antisymmetric Tensor
195
Fig. 7.6 (Left) Schematic
picture of the quadrupole
(Q zz ) distribution in the
interface system, where the
lower bound of the integral
z = z b is shown by red line.
(Right) Divided quadrupole
moments (Q yz , Q zz ) by the
threshold z = z b bring net
dipole moments (μ y , μ z ) in
the integral region z > z b
z
0
Q zz
Q yz
z
y > 0
z > 0
y
z b
into play at the lower bound z = z b . The molecules located across the threshold
z = z b partially contribute to the integral, since the divided quadrupole moments
Q yz and Q zz bring net dipole moments μ y and μ z , respectively, in the integral of
P I . The contribution necessarily arises from the uniformly distributed quadrupole,
irrespective of the location of the threshold z = z b . We note that this mechanism is
essentially common to the role of quadrupole on the surface potential [22].
The integral of quadrupole contributions could be understood without resorting
to the arbitrary threshold of z b . This mechanism of χ IQB is related to the infinite
summation of oscillating terms. A quadrupole is regarded as a pair of antiparallel
dipoles, as illustrated in Fig. 7.6. Thus the sum of all quadrupole contributions
becomes equivalent to the infinite summation of a pair of antiparallel dipoles,
(μ − μ) + (μ − μ) + (μ − μ) + · · · , where each pair (μ − μ) corresponds to
a quadrupole moment. This infinite summation could be defined on the basis of
Abel summability [9],
(μ − μ) + (μ − μ) + · · · =
∞
n=0
μ (−1)
n
= lim
x→−1+0
μ
1 + x
=
μ
2
,
which yields a net dipole contribution. The above definition of this infinite sum can
be obtained by using a proper convergence factor. Such oscillating sum elucidates
the χ IQB contribution to the net dipole P I . We will encounter the analogous
mechanism in the NaOH aqueous solution surface in Sect. 9.3.
A.2 Levi-Civita Antisymmetric Tensor
The Levi-Civita antisymmetric tensor ε ij k is defined as follows.
ε ij k =
⎧
⎨
⎩
1 (ij k = xyz, yzx, zxy) ,
−1 (ij k = yxz, zyx, xzy) ,
0 (otherwise) .
(7.132)
This symbol is quite convenient when manipulating various formulas in the vector
analysis.
195
Fig. 7.6 (Left) Schematic
picture of the quadrupole
(Q zz ) distribution in the
interface system, where the
lower bound of the integral
z = z b is shown by red line.
(Right) Divided quadrupole
moments (Q yz , Q zz ) by the
threshold z = z b bring net
dipole moments (μ y , μ z ) in
the integral region z > z b
z
0
Q zz
Q yz
z
y > 0
z > 0
y
z b
into play at the lower bound z = z b . The molecules located across the threshold
z = z b partially contribute to the integral, since the divided quadrupole moments
Q yz and Q zz bring net dipole moments μ y and μ z , respectively, in the integral of
P I . The contribution necessarily arises from the uniformly distributed quadrupole,
irrespective of the location of the threshold z = z b . We note that this mechanism is
essentially common to the role of quadrupole on the surface potential [22].
The integral of quadrupole contributions could be understood without resorting
to the arbitrary threshold of z b . This mechanism of χ IQB is related to the infinite
summation of oscillating terms. A quadrupole is regarded as a pair of antiparallel
dipoles, as illustrated in Fig. 7.6. Thus the sum of all quadrupole contributions
becomes equivalent to the infinite summation of a pair of antiparallel dipoles,
(μ − μ) + (μ − μ) + (μ − μ) + · · · , where each pair (μ − μ) corresponds to
a quadrupole moment. This infinite summation could be defined on the basis of
Abel summability [9],
(μ − μ) + (μ − μ) + · · · =
∞
n=0
μ (−1)
n
= lim
x→−1+0
μ
1 + x
=
μ
2
,
which yields a net dipole contribution. The above definition of this infinite sum can
be obtained by using a proper convergence factor. Such oscillating sum elucidates
the χ IQB contribution to the net dipole P I . We will encounter the analogous
mechanism in the NaOH aqueous solution surface in Sect. 9.3.
A.2 Levi-Civita Antisymmetric Tensor
The Levi-Civita antisymmetric tensor ε ij k is defined as follows.
ε ij k =
⎧
⎨
⎩
1 (ij k = xyz, yzx, zxy) ,
−1 (ij k = yxz, zyx, xzy) ,
0 (otherwise) .
(7.132)
This symbol is quite convenient when manipulating various formulas in the vector
analysis.