# When calculating a dihedral angle for a chemical bond, how is the direction defined? What are the most common convention?

I understand the geometry and calculation of a dihedral angle, discussed in this question. It is not clear how the direction is defined for a chemical bond. On the wiki page for dihedral angles, it states that, "the dihedral angle $\phi$ is the counterclockwise angle..." between the planes $P_{123}$ and $P_{234}$ (where $P_{123}$ is defined as the plane containing atoms 1, 2, and 3 in the diagram below, and likewise for $P_{234}$).

$\hspace{45 mm}$ This counterclockwise definition means that the direction of the dihedral angle is defined using the left-hand rule about the line connecting atom 2 to atom 3. The left-hand rule is unusual in mathematics and geometry. Alternatively, this example may be defining the directionality of $\phi$ simply so the result is a positive angle less than $180^\circ$.

I have not been able to find any chemistry standards on this. Is there a common convention?

## 1 Answer

The IUPAC Gold Book states that,

In a Newman projection the torsion angle is the angle (having an absolute value between 0° and 180°) between bonds to two specified (fiducial) groups, one from the atom nearer (proximal) to the observer and the other from the further (distal) atom. The torsion angle between groups A and D is then considered to be positive if the bond A-B is rotated in a clockwise direction through less than 180° in order that it may eclipse the bond C-D: a negative torsion angle requires rotation in the opposite sense.

This implies a range of -180 to 180, which is by far the most typical range used, but I certainly have seen reports ranging from 0..360, e.g., 225 degree dihedrals.

• When they report a negative dihedral angle, or an angle larger than 180, there has to be some way they are determining directionality. Otherwise the results are ambiguous. – Steven C. Howell Nov 4 '15 at 20:15
• The link you provided does describe the orientation, "In a Newman projection the torsion angle is the angle (having an absolute value between 0° and 180°) between bonds to two specified (fiducial) groups, one from the atom nearer (proximal) to the observer and the other from the further (distal) atom. The torsion angle between groups A and D is then considered to be positive if the bond A-B is rotated in a clockwise direction through less than 180° in order that it may eclipse the bond C-D: a negative torsion angle requires rotation in the opposite sense." – Steven C. Howell Nov 4 '15 at 20:17
• I am not familiar with the IUPAC Gold Book but it seem valid. According to this definition, the wiki page is incorrect and the torsion angle is defined using the right-hand rule along the line connecting atoms 2 and 3. – Steven C. Howell Nov 4 '15 at 20:27
• My point is that despite the Gold Book, different software (or people) report different references. – Geoff Hutchison Nov 4 '15 at 21:00
• This is good to note. I think the most important point is that they use the same directionality (but maybe this is not the case). If a positive angle is always defined using the right-hand rule about the B-C line (simplified version of the IUPAC Gold Book), converting from -60 to +300 is straightforward. Without a common orientation, anything but 0-180 is ambiguous. – Steven C. Howell Nov 4 '15 at 21:05