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c
°CCLRC
Section 2.2
7. OPLS torsion potential: (opls)
U (φ) = A0 +
1
{A1 (1 + cos(φ)) + A2 (1 − cos(2φ)) + A3 (1 + cos(4φ))}
2
(2.37)
In these formulae φijkn is the dihedral angle defined by
φijkn = cos−1 {B(rij , rjk , rkn )} ,
with
(
(rij × rjk ) · (rjk × rkn )
|rij × rjk ||rjk × rkn |
B(rij , rjk , rkn ) =
(2.38)
)
.
(2.39)
With this definition, the sign of the dihedral angle is positive if the vector product
(rij × rjk ) × (rjk × rkn ) is in the same direction as the bond vector rjk and negative if in the
opposite direction.
The force on an atom arising from the dihedral potential is given by
f`α = −
∂
U (φijkn ) ,
∂r`α
(2.40)
with ` being one of i, j, k, n and α one of x, y, z. This may be expanded into
∂
− α U (φijkn ) =
∂r`
(
1
sin(φijkn )
)
∂
∂
U (φijkn ) α B(rij , rjk , rkn ) .
∂φijkn
∂r`
(2.41)
The derivative of the function B(rij , rjk , rkn ) is
1
∂
∂
B(rij , rjk , rkn ) =
{(rij × rjk ) · (rjk × rkn )} −
α
∂r`
|rij × rjk ||rjk × rkn | ∂r`α
cos(φijkn )
2
(
1
1
∂
∂
|r × rjk |2 +
|r × rkn |2
|rij × rjk |2 ∂r`α ij
|rjk × rkn |2 ∂r`α jk
)
,
(2.42)
with
∂
α
{(rij × rjk ) · (rjk × rkn )} = rij
([rjk rjk ]α (δ`k − δ`n ) + [rjk rkn ]α (δ`k − δ`j )) +
∂r`α
α
rjk
([rij rjk ]α (δ`n − δ`k ) + [rjk rkn ]α (δ`j − δ`i )) +
α
rkn
([rij rjk ]α (δ`k − δ`j ) + [rjk rjk ]α (δ`i − δ`j )) +
α
2rjk
[rij rkn ]α (δ`j − δ`k ) ,
(2.43)
∂
α
|r × rjk |2 = 2rij
([rjk rjk ]α (δ`j − δ`i ) + [rij rjk ]α (δ`j − δ`k )) +
∂r`α ij
α
2rjk
([rij rij ]α (δ`k − δ`j ) + [rij rjk ]α (δ`i − δ`j )) ,
(2.44)
∂
α
|r × rkn |2 = 2rkn
([rjk rjk ]α (δ`n − δ`k ) + [rjk rkn ]α (δ`j − δ`k )) +
∂r`α jk
α
2rjk
([rkn rkn ]α (δ`k − δ`j ) + [rjk rkn ]α (δ`k − δ`n )) .
(2.45)
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