1
Dipole angle restraints
We are going to restraint the dipole orientation of any molecule, to do this we will apply an harmonic
restraint to θ among the chosen axis and the molecule’s dipole, so the harmonic restraint is given
by
V (~r, q) =
1
k(θr − θref )2
2
(1)
and the force F
~ (~r, q)
F~ (~r, q) = −∇V
= −k(θr − θref )
∂θr
∂θr
∂θr
x̂ +
ŷ +
ẑ
∂x
∂y
∂z
Now, we replace θr as a function of dipole vector,
µ
~ (~r, q)com · ~r
θr = arccos
= arccos (µ̂com · r̂)
|~
µ(~r, q)com ||~r|
(2)
(3)
with cos θ = µ̂com · r̂ and where
r̂ = p
(x, y, z)
x2 + y 2 + z 2
and dipole vector for a polar molecule is defined by
µ
~ (~r, q)com =
n
X
qi (~ri − ~rcom )
(4)
i=1
where qi is the charge of atom i, ri position of atom i and ~rcom center of mass defined by
~rcom =
n
1 X
mi~ri = (xcom , ycom , zcom )
M i=1
(5)
where M is total mass, mi mass of atom i and n number of atoms of the molecule. Now we derive
equation (3)
∂θr
=
∂x
∂ arccos
µ
~ (~
r ,q)com ·r̂
|~
µ(~
r ,q)com |
∂x
∂θ ∂ cos θ
=
∂ cos θ ∂x
−1
∂ cos θ
=√
2
1 − cos θ ∂x
so, from the second term of equation (8) we have
1
(6)
(7)
(8)
∂ cos θr
1
∂
∂
1
=
µ
~ com · r̂ + (~
µcom · r̂)
∂x
|~
µcom | ∂x
∂x |~
µcom |
where
∂
x
µ
~ com · r̂ =
∂x
|~r|
mj X
qj −
qi
M i
(9)
!
(10)
and
1
µxcom
∂
=−
∂x |~
µcom |
|~
µcom |3
n
mj X
qj −
qi
M i=1
!
(11)
Now we replace (11) and (10) in (9) and obtain
!
!
n
mj X
mj X
µxcom
qj −
qj −
qi − (~
qi
µcom · r̂)
M i
|~
µcom |3
M i=1
!
n
mj X
x
µxcom
qj −
qi
− (µ̂com · r̂)
M i=1
|~r|
|~
µcom |
1
x
∂ cos θr
=
∂x
|~
µcom | |~r|
=
1
|~
µcom |
(12)
(13)
analogous for y and z. Finally the force given by (2) results
∂ arccos(µ̂com · r̂)
∂ arccos(µ̂com · r̂)
∂ arccos(µ̂com · r̂)
x̂ +
ŷ +
ẑ
∂x
∂y
∂z
!
n
~r
1
mj X
µ
~ com
qi
= k(θr − θref ) √
qj −
− (µ̂com · r̂)
M i=1
|~r|
|~
µcom |
1 − cos2 θ|~
µcom |
!
n
1
mj X
= k(θr − θref ) p
qj −
qi [r̂ − (µ̂com · r̂)µ̂com ]
M i=1
1 − (µ̂com · r̂)2 |~
µcom |
F~j (~r, q) = −k(θr − θref )
for each j atom.
Alejandro Bernardin S
2
(14)