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pair_dipole_sf.tex
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rLAMMPS lammps
pair_dipole_sf.tex
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\documentclass
[12pt]
{
article
}
\begin
{
document
}
\begin
{
eqnarray*
}
E
_{
LJ
}
&
=
&
4
\epsilon
\left\{
\left
[
\left
(
\frac
{
\sigma
}{
r
}
\right
)
^{
\!
12
}
-
\left
(
\frac
{
\sigma
}{
r
}
\right
)
^{
\!
6
}
\right
] +
\left
[ 6
\left
(
\frac
{
\sigma
}{
r
_
c
}
\right
)
^{
\!
12
}
-
3
\left
(
\frac
{
\sigma
}{
r
_
c
}
\right
)
^{
\!
6
}
\right
]
\left
(
\frac
{
r
}{
r
_
c
}
\right
)
^{
\!
2
}
- 7
\left
(
\frac
{
\sigma
}{
r
_
c
}
\right
)
^{
\!
12
}
+
4
\left
(
\frac
{
\sigma
}{
r
_
c
}
\right
)
^{
\!
6
}
\right\}
\\
E
_{
qq
}
&
=
&
\frac
{
q
_
i q
_
j
}{
r
}
\left
(1-
\frac
{
r
}{
r
_
c
}
\right
)
^{
\!
2
}
\\
E
_{
pq
}
&
=
&
E
_{
ji
}
= -
\frac
{
q
}{
r
^
3
}
\left
[ 1 -
3
\left
(
\frac
{
r
}{
r
_
c
}
\right
)
^{
\!
2
}
+
2
\left
(
\frac
{
r
}{
r
_
c
}
\right
)
^{
\!
3
}
\right
] (
\vec
{
p
}
\bullet\vec
{
r
}
)
\\
E
_{
qp
}
&
=
&
E
_{
ij
}
=
\frac
{
q
}{
r
^
3
}
\left
[ 1 -
3
\left
(
\frac
{
r
}{
r
_
c
}
\right
)
^{
\!
2
}
+
2
\left
(
\frac
{
r
}{
r
_
c
}
\right
)
^{
\!
3
}
\right
] (
\vec
{
p
}
\bullet\vec
{
r
}
)
\\
E
_{
pp
}
&
=
&
\left
[1-4
\left
(
\frac
{
r
}{
r
_
c
}
\right
)
^{
\!
3
}
+
3
\left
(
\frac
{
r
}{
r
_
c
}
\right
)
^{
\!
4
}
\right
]
\left
[
\frac
{
1
}{
r
^
3
}
(
\vec
{
p
_
i
}
\bullet
\vec
{
p
_
j
}
) -
\frac
{
3
}{
r
^
5
}
(
\vec
{
p
_
i
}
\bullet
\vec
{
r
}
) (
\vec
{
p
_
j
}
\bullet
\vec
{
r
}
)
\right
]
\\
\end
{
eqnarray*
}
\begin
{
eqnarray*
}
F
_{
LJ
}
&
=
&
\left\{\left
[48
\epsilon
\left
(
\frac
{
\sigma
}{
r
}
\right
)
^{
\!
12
}
-
24
\epsilon
\left
(
\frac
{
\sigma
}{
r
}
\right
)
^{
\!
6
}
\right
]
\frac
{
1
}{
r
^
2
}
-
\left
[48
\epsilon
\left
(
\frac
{
\sigma
}{
r
_
c
}
\right
)
^{
\!
12
}
- 24
\epsilon
\left
(
\frac
{
\sigma
}{
r
_
c
}
\right
)
^{
\!
6
}
\right
]
\frac
{
1
}{
r
_
c
^
2
}
\right\}\vec
{
r
}
\\
F
_{
qq
}
&
=
&
\frac
{
q
_
i q
_
j
}{
r
}
\left
(
\frac
{
1
}{
r
^
2
}
-
\frac
{
1
}{
r
_
c
^
2
}
\right
)
\vec
{
r
}
\\
F
_{
pq
}
&
=
&
F
_{
ij
}
= -
\frac
{
3q
}{
r
^
5
}
\left
[ 1 -
\left
(
\frac
{
r
}{
r
_
c
}
\right
)
^{
\!
2
}
\right
](
\vec
{
p
}
\bullet\vec
{
r
}
)
\vec
{
r
}
+
\frac
{
q
}{
r
^
3
}
\left
[1-3
\left
(
\frac
{
r
}{
r
_
c
}
\right
)
^{
\!
2
}
+
2
\left
(
\frac
{
r
}{
r
_
c
}
\right
)
^{
\!
3
}
\right
]
\vec
{
p
}
\\
F
_{
qp
}
&
=
&
F
_{
ij
}
=
\frac
{
3q
}{
r
^
5
}
\left
[ 1 -
\left
(
\frac
{
r
}{
r
_
c
}
\right
)
^{
\!
2
}
\right
] (
\vec
{
p
}
\bullet\vec
{
r
}
)
\vec
{
r
}
-
\frac
{
q
}{
r
^
3
}
\left
[1-3
\left
(
\frac
{
r
}{
r
_
c
}
\right
)
^{
\!
2
}
+
2
\left
(
\frac
{
r
}{
r
_
c
}
\right
)
^{
\!
3
}
\right
]
\vec
{
p
}
\\
F
_{
pp
}
&
=
&
\frac
{
3
}{
r
^
5
}
\Bigg\{\left
[1-\left(\frac{r}{r_c}\right)^{\!4}\right]
\left
[(
\vec
{
p
_
i
}
\bullet\vec
{
p
_
j
}
) -
\frac
{
3
}{
r
^
2
}
(
\vec
{
p
_
i
}
\bullet\vec
{
r
}
)
(
\vec
{
p
_
j
}
\bullet
\vec
{
r
}
)
\right
]
\vec
{
r
}
+
\\
&
&
\left
[1 -
4
\left
(
\frac
{
r
}{
r
_
c
}
\right
)
^{
\!
3
}
+3
\left
(
\frac
{
r
}{
r
_
c
}
\right
)
^{
\!
4
}
\right
]
\left
[ (
\vec
{
p
_
j
}
\bullet
\vec
{
r
}
)
\vec
{
p
_
i
}
+ (
\vec
{
p
_
i
}
\bullet
\vec
{
r
}
)
\vec
{
p
_
j
}
-
\frac
{
2
}{
r
^
2
}
(
\vec
{
p
_
i
}
\bullet
\vec
{
r
}
)
(
\vec
{
p
_
j
}
\bullet
\vec
{
r
}
)
\vec
{
r
}
\right
]
\Bigg\}
\\
\end
{
eqnarray*
}
\end
{
document
}
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