# How to calculate dihedral (not torsion) angles?

islam islam at icrf.icnet.uk
Wed Mar 17 07:56:03 EST 1999

```The definition of a torsion angle between 4 points,
does not require them to be bonded ! The dihedral angle
is simply the angle between the two planes ABC and BCD
(in fact the torsion angle is the complement of the dihedral angle
i.e. torsion = 180.0 - dihedral ). You can simply calculate
the plane for ABC (or 1-3-5 below) & BCD (or 1-2-3 below) and then
the angle between them.
____________________________________________
Suhail A Islam
Biomolecular Modelling Laboratory
Imperial Cancer Research Fund, P.O. Box 123
44 Lincoln's Inn Fields, London WC2A 3PX
Tel: (0171) 269 3380
Fax: (0171) 269 3258
email: islam at icrf.icnet.uk
http://www.icnet.uk/bmm/
____________________________________________

Antonio wrote:
>
> We know how to calculate a torsion angle about an axis, for example
> if we have 4 points in 3-D (for example, a butane molecule)
>              A                D
>           r1  \              /  r3
>                 B- - - -C
>                      r2
> in which the points A, B, C and D are connected by the vectors
> r1, r2 and r3, we can obtain the torsion angle phi about r2 by applying:
>
>                      p1= r1 x r2
>                      p2= r2 x r3
>                      p1 . p2 = |p1| |p2| cos (phi)
>                      r2 . (p2 x p1) = |p1| |p2| |r2| sin(phi)
> and finally phi=atan[sin(phi)/cos(phi)] through a function like ATAN2
> in Fortran. Phi is obviously also the angle between the A-B-C and B-C-D
> planes.
> The problem is, how to calculate a dihedral angle when the four centers
> defining the two planes are
> not directly connected? For example, if we have a six ring
> molecule:
>                                           1 ---2
>                                          /        \
>                                         6        3
>                                           \       /
>                                           5 -- 4
>
> and we need the angle between the 1-3-5 and 1-2-3 planes, now there isn't
> a common bond, as r2 in the previous case (indeed now we can't
> speak of 'torsion' angle, but more generally of 'dihedral', even if both are
> angles between planes).  Therefore, the previous expressions
> aren't directly applicable. I tried to define a 'dummy' common bond, e.g
> 1--3, in order to apply the above expressions to a system like:
>                   5                 2
>                     \               /
>                       1--------3
>
> actually, with this trick we should obtain the angle between planes
> 5-1-3 and 1-3-2, which should be the same as that between 1-3-5
> and 1-2-3 planes. However, this doesn't seem to work well. Maybe
> the last assumption (dihedral 513/132= dihedral 135/123) is not
> correct,  or some adjustment are needed in the above expressions, to make
> them valid in this case? Or do you know of a
> different method to evaluate such dihedral (not torsion) angles?