c CCLRC
Section 2.2
Following through, the (extremely tedious!) differentiation gives the result:
f
=
1
sin(
ijkn
)
ijkn
U (
ijkn
) ×
(2.59)
-(
j
-
i
)
cos(
ijkn
)
r
2
ij
r
ij
+
1
r
ij
w
kn
(
j
-
i
){(r
ij
· ^
u
kn
)^
u
kn
+ (r
ij
· ^
v
kn
)^
v
kn
}
+(
k
-
i
)
r
ij
· ^
u
kn
u
kn
r
ik
r
ij
- (r
ij
· ^
u
kn
)^
u
kn
- (r
ij
· r
ik
- (r
ij
· ^
u
kn
)(r
ik
· ^
u
kn
))
r
ik
r
2
ik
+(
k
-
i
)
r
ij
· ^
v
kn
v
kn
r
ik
r
ij
- (r
ij
· ^
v
kn
)^
v
kn
- (r
ij
· r
ik
- (r
ij
· ^
v
kn
)(r
ik
· ^
v
kn
))
r
ik
r
2
ik
+(
n
-
i
)
r
ij
· ^
u
kn
u
kn
r
in
r
ij
- (r
ij
· ^
u
kn
)^
u
kn
- (r
ij
· r
in
- (r
ij
· ^
u
kn
)(r
in
· ^
u
kn
))
r
in
r
2
in
+(
n
-
i
)
r
ij
· ^
v
kn
v
kn
r
in
r
ij
- (r
ij
· ^
v
kn
)^
v
kn
- (r
ij
· r
in
- (r
ij
· ^
v
kn
)(r
in
· ^
v
kn
))
r
in
r
2
in
This general formula applies to all atoms
= i, j, k, n. It must be remembered however, that
these formulae apply to just one of the three contributing terms (i.e. one angle ) of the full
inversion potential: specifically the inversion angle pertaining to the out-of-plane vector r
ij
. The
contributions arising from the other vectors r
ik
and r
in
are obtained by the cyclic permutation of
the indices in the manner described above. All these force contributions must be added to the final
atomic forces.
Formally, the contribution to be added to the atomic virial is given by
W = -
4
i=1
r
i
· f
i
.
(2.60)
However, it is possible to show by thermodynamic arguments (cf [
],) or simply from the fact that
the sum of forces on atoms j,k and n is equal and opposite to the force on atom i, that the inversion
potential makes no contribution to the atomic virial.
If the force components f
for atoms
= i, j, k, n are calculated using the above formulae, it is
easily seen that the contribution to be added to the atomic stress tensor is given by
= r
ij
f
j
+ r
ik
f
k
+ r
in
f
n
.
(2.61)
The sum of the diagonal elements of the stress tensor is zero (since the virial is zero) and the matrix
is symmetric.
In DL POLY 3 inversion forces are handled by the routine inversions forces.
2.2.8
Tethering Forces
DL POLY 3 also allows atomic sites to be tethered to a fixed point in space, r
0
, taken as their
position at the beginning of the simulation (t = 0). This is also known as position restraining. The
specification, which comes as part of the molecular description, requires a tether potential type and
the associated interaction parameters.
Note, firstly, that application of tethering potentials means that the momentum will no longer be
a conserved quantity of the simulation. Secondly, in constant pressure simulations, where the MD
cell changes size or shape, the tethers' reference positions are scaled with the cell vectors.
23